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<journal-meta>
<journal-id journal-id-type="publisher-id">Front. Soft Matter</journal-id>
<journal-title>Frontiers in Soft Matter</journal-title>
<abbrev-journal-title abbrev-type="pubmed">Front. Soft Matter</abbrev-journal-title>
<issn pub-type="epub">2813-0499</issn>
<publisher>
<publisher-name>Frontiers Media S.A.</publisher-name>
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<article-meta>
<article-id pub-id-type="publisher-id">1359128</article-id>
<article-id pub-id-type="doi">10.3389/frsfm.2024.1359128</article-id>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Soft Matter</subject>
<subj-group>
<subject>Original Research</subject>
</subj-group>
</subj-group>
</article-categories>
<title-group>
<article-title>Geometric modeling of phase ordering for the isotropic&#x2013;smectic A phase transition</article-title>
<alt-title alt-title-type="left-running-head">Zamora Cisneros et al.</alt-title>
<alt-title alt-title-type="right-running-head">
<ext-link ext-link-type="uri" xlink:href="https://doi.org/10.3389/frsfm.2024.1359128">10.3389/frsfm.2024.1359128</ext-link>
</alt-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname>Zamora Cisneros</surname>
<given-names>David Uriel</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
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<contrib contrib-type="author">
<name>
<surname>Wang</surname>
<given-names>Ziheng</given-names>
</name>
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<sup>1</sup>
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<contrib contrib-type="author">
<name>
<surname>Dorval Courchesne</surname>
<given-names>No&#xe9;mie-Manuelle</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
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<contrib contrib-type="author">
<name>
<surname>Harrington</surname>
<given-names>Matthew J.</given-names>
</name>
<xref ref-type="aff" rid="aff2">
<sup>2</sup>
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<contrib contrib-type="author" corresp="yes">
<name>
<surname>Rey</surname>
<given-names>Alejandro D.</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
<xref ref-type="corresp" rid="c001">&#x2a;</xref>
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<aff id="aff1">
<sup>1</sup>
<institution>Department of Chemical Engineering</institution>, <institution>McGill University</institution>, <addr-line>Montreal</addr-line>, <addr-line>QC</addr-line>, <country>Canada</country>
</aff>
<aff id="aff2">
<sup>2</sup>
<institution>Department of Chemistry</institution>, <institution>McGill University</institution>, <addr-line>Montreal</addr-line>, <addr-line>QC</addr-line>, <country>Canada</country>
</aff>
<author-notes>
<fn fn-type="edited-by">
<p>
<bold>Edited by:</bold> <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/970330/overview">Frank Alexis</ext-link>, Universidad San Francisco de Quito, Ecuador</p>
</fn>
<fn fn-type="edited-by">
<p>
<bold>Reviewed by:</bold> <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/1544902/overview">Anne-laure Fameau</ext-link>, Institut National de recherche pour l&#x2019;agriculture, l&#x2019;alimentation et l&#x2019;environnement (INRAE), France</p>
<p>
<ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/2564557/overview">Konstantin Kornev</ext-link>, Clemson University, United States</p>
</fn>
<corresp id="c001">&#x2a;Correspondence: Alejandro D. Rey, <email>alejandro.rey@mcgill.ca</email>
</corresp>
</author-notes>
<pub-date pub-type="epub">
<day>21</day>
<month>05</month>
<year>2024</year>
</pub-date>
<pub-date pub-type="collection">
<year>2024</year>
</pub-date>
<volume>4</volume>
<elocation-id>1359128</elocation-id>
<history>
<date date-type="received">
<day>20</day>
<month>12</month>
<year>2023</year>
</date>
<date date-type="accepted">
<day>16</day>
<month>04</month>
<year>2024</year>
</date>
</history>
<permissions>
<copyright-statement>Copyright &#xa9; 2024 Zamora Cisneros, Wang, Dorval Courchesne, Harrington and Rey.</copyright-statement>
<copyright-year>2024</copyright-year>
<copyright-holder>Zamora Cisneros, Wang, Dorval Courchesne, Harrington and Rey</copyright-holder>
<license xlink:href="http://creativecommons.org/licenses/by/4.0/">
<p>This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.</p>
</license>
</permissions>
<abstract>
<sec>
<title>Background</title>
<p>Liquid crystal (LC) mesophases have an orientational and positional order that can be found in both synthetic and biological materials. These orders are maintained until some parameter, mainly the temperature or concentration, is changed, inducing a phase transition. Among these transitions, a special sequence of mesophases has been observed, in which priority is given to the direct smectic liquid crystal transition. The description of these transitions is carried out using the Landau&#x2013;de Gennes (LdG) model, which correlates the free energy of the system with the orientational and positional order.</p>
</sec>
<sec>
<title>Methodology</title>
<p>This work explored the direct isotropic-to-smectic A transition studying the free energy landscape constructed with the LdG model and its relation to three curve families: (I) level-set curves, steepest descent, and critical points; (II) lines of curvature (LOC) and geodesics, which are directly connected to the principal curvatures; and (III) the Casorati curvature and shape coefficient that describe the local surface geometries resemblance (sphere, cylinder, and saddle).</p>
</sec>
<sec>
<title>Results</title>
<p>The experimental data on 12-cyanobiphenyl were used to study the three curve families. The presence of unstable nematic and metastable plastic crystal information was found to add information to the already developed smectic A phase diagram. The lines of curvature and geodesics were calculated and laid out on the energy landscape, which highlighted the energetic pathways connecting critical points. The Casorati curvature and shape coefficient were computed, and in addition to the previous family, they framed a geometric region that describes the phase transition zone.</p>
</sec>
<sec>
<title>Conclusion and significance</title>
<p>A direct link between the energy landscape&#x2019;s topological geometry, phase transitions, and relevant critical points was established. The shape coefficient delineates a stability zone in which the phase transition develops. The methodology significantly reduces the impact of unknown parametric data. Symmetry breaking with two order parameters (OPs) may lead to novel phase transformation kinetics and droplets with partially ordered surface structures.</p>
</sec>
</abstract>
<kwd-group>
<kwd>liquid crystals</kwd>
<kwd>smectic A</kwd>
<kwd>phase transition</kwd>
<kwd>energy landscape</kwd>
<kwd>shape coefficient</kwd>
<kwd>free energy</kwd>
<kwd>Landau&#x2013;de Gennes</kwd>
</kwd-group>
<contract-sponsor id="cn001">Fonds de recherche du Qu&#xe9;bec &#x2013; Nature et technologies<named-content content-type="fundref-id">10.13039/501100003151</named-content>
</contract-sponsor>
<contract-sponsor id="cn002">Natural Sciences and Engineering Research Council of Canada<named-content content-type="fundref-id">10.13039/501100000038</named-content>
</contract-sponsor>
<contract-sponsor id="cn003">Consejo Nacional de Ciencia y Tecnolog&#xed;a<named-content content-type="fundref-id">10.13039/501100003141</named-content>
</contract-sponsor>
<custom-meta-wrap>
<custom-meta>
<meta-name>section-at-acceptance</meta-name>
<meta-value>Liquid Crystals</meta-value>
</custom-meta>
</custom-meta-wrap>
</article-meta>
</front>
<body>
<sec id="s1">
<title>1 Introduction</title>
<p>Synthetic and biological liquid crystals (LCs) are anisotropic soft materials with partial degrees of orientational and positional order, conveying fluidity as viscous liquids and anisotropy as in the crystalline order (<xref ref-type="bibr" rid="B108">Rey and Denn, 2002</xref>; <xref ref-type="bibr" rid="B38">Donald et al., 2006</xref>; <xref ref-type="bibr" rid="B107">Rey, 2010</xref>; <xref ref-type="bibr" rid="B116">Sonnet and Virga, 2012</xref>; <xref ref-type="bibr" rid="B90">Petrov, 2013</xref>; <xref ref-type="bibr" rid="B65">Lagerwall, 2016</xref>; <xref ref-type="bibr" rid="B115">Selinger, 2016</xref>; <xref ref-type="bibr" rid="B20">Collings and Goodby, 2019</xref>; <xref ref-type="bibr" rid="B122">Stewart, 2019</xref>; <xref ref-type="bibr" rid="B141">Zannoni, 2022</xref>). Importantly, possible LC mesogens include rod-, board-, disk-, and screw-like molecules with flexibilities ranging from semi-flexible to rigid, involving monomers or main/side-chain polymers and colloids (<xref ref-type="bibr" rid="B38">Donald et al., 2006</xref>; <xref ref-type="bibr" rid="B31">Demus et al., 2008b</xref>; <xref ref-type="bibr" rid="B32">Demus et al., 2011</xref>). The synthesis and formation of these mesophase materials follow equilibrium self-assembly processes driven by temperature (thermotropic), concentration (lyotropic), or both (<xref ref-type="bibr" rid="B12">Bowick et al., 2017</xref>; <xref ref-type="bibr" rid="B136">Wang et al., 2023b</xref>). The presence of multiple components, as in nanoparticle-loaded mesophases, gives rise to couplings between self-assembly and phase separation with states that can combine the crystallinity (positional order) of one component with the liquid crystallinity (various degrees of positional and orientational order) of the other (<xref ref-type="bibr" rid="B117">Soul&#xe9; et al., 2012a</xref>; <xref ref-type="bibr" rid="B118">Soul&#xe9; et al., 2012b</xref>; <xref ref-type="bibr" rid="B119">Soul&#xe9; et al., 2012c</xref>; <xref ref-type="bibr" rid="B121">Soule and Rey, 2012</xref>; <xref ref-type="bibr" rid="B75">Milette et al., 2013</xref>; <xref ref-type="bibr" rid="B47">Gurevich et al., 2014</xref>). Various external fields (flow, electromagnetic, and thermal) can be used to generate self-organized structures not seen in purely equilibrium self-assembly. The natural setting to describe self-assembly starts with the energy landscape and its geometrical properties (curvatures, cusps, domes, <italic>etc.</italic>), while for non-equilibrium organization, the natural setting will include the entropy production landscape and its defining geometric measures. In this paper, we present a widely applicable geometry-based methodology for describing self-assembly in anisotropic soft matter and target one specific transition that exhibits complex ordering processes. Without ambiguity, for convenience and brevity, below we refer to self-assembly as phase ordering and/or phase transition, as we do not include phase separation and conserved quantities. In addition, we refer to the degree of quench as the equivalent of the thermodynamic driving force for phase ordering.</p>
<p>The spectrum of self-assembly processes is significantly enriched when considering the sequence of phase transformations. This progression follows an increase in order as the magnitude of the thermodynamic driving force is increased, corresponding to a decrease in temperature for thermotropes or an increase in concentration for lyotropes. The sequence then goes from a high-symmetry isotropic state to an orientationally ordered (nematic) state. This state is then followed by the addition of the positional order (smectic). Then, the crystalline organization is reached with a further increase in the driving force. The usual sequence of isotropic&#x2013;orientational&#x2013;positional phase ordering (<xref ref-type="bibr" rid="B86">Oswald and Pieranski, 2005a</xref>; <xref ref-type="bibr" rid="B57">J&#xe1;kli and Saupe, 2006</xref>; <xref ref-type="bibr" rid="B11">Blinov, 2011</xref>) is sometimes reordered to a direct isotropic&#x2013;positional/orientational ordering, as observed in monomeric thermotropes (certain cyano-biphenyls (CB) and oxy-cyanobiphenyls (OCB)) (<xref ref-type="bibr" rid="B85">Oh, 1977</xref>; <xref ref-type="bibr" rid="B54">Idziak et al., 1996</xref>; <xref ref-type="bibr" rid="B87">Oswald and Pieranski, 2005b</xref>; <xref ref-type="bibr" rid="B38">Donald et al., 2006</xref>; <xref ref-type="bibr" rid="B3">Abukhdeir and Rey, 2008</xref>; <xref ref-type="bibr" rid="B17">Chahine et al., 2010</xref>; <xref ref-type="bibr" rid="B46">Gudimalla et al., 2021</xref>; <xref ref-type="bibr" rid="B84">Nesrullajev, 2022</xref>) and biological liquid crystals (BLCs) such as collagen mesophase precursors in the mussel byssus (<xref ref-type="bibr" rid="B61">Knight and Vollrath, 1999</xref>; <xref ref-type="bibr" rid="B127">Viney, 2004</xref>; <xref ref-type="bibr" rid="B38">Donald et al., 2006</xref>; <xref ref-type="bibr" rid="B107">Rey, 2010</xref>; <xref ref-type="bibr" rid="B109">Rey and Herrera-Valencia, 2012</xref>; <xref ref-type="bibr" rid="B110">Rey et al., 2013</xref>; <xref ref-type="bibr" rid="B101">Renner-Rao et al., 2019</xref>; <xref ref-type="bibr" rid="B72">Manolakis and Azhar, 2020</xref>; <xref ref-type="bibr" rid="B51">Harrington and Fratzl, 2021</xref>; <xref ref-type="bibr" rid="B9">Berent et al., 2022</xref>). This important and non-classical behavior is the focus of this paper: the direct isotropic-to-smectic A (I-SmA) LC phase transition, where SmA denotes the simplest smectic phase. A collection of compounds exhibiting this scheme can be found (<xref ref-type="bibr" rid="B52">Hawkins and April 1983</xref>; <xref ref-type="bibr" rid="B54">Idziak et al., 1996</xref>; <xref ref-type="bibr" rid="B67">Lenoble et al., 2007</xref>; <xref ref-type="bibr" rid="B3">Abukhdeir and Rey, 2008</xref>; <xref ref-type="bibr" rid="B77">Mohieddin Abukhdeir and Rey, 2008a</xref>; <xref ref-type="bibr" rid="B68">Li et al., 2009</xref>; <xref ref-type="bibr" rid="B139">Wojcik et al., 2009</xref>; <xref ref-type="bibr" rid="B17">Chahine et al., 2010</xref>; <xref ref-type="bibr" rid="B97">Pouget et al., 2011</xref>; <xref ref-type="bibr" rid="B111">Salamonczyk et al., 2016</xref>; <xref ref-type="bibr" rid="B13">Bradley, 2019</xref>; <xref ref-type="bibr" rid="B101">Renner-Rao et al., 2019</xref>; <xref ref-type="bibr" rid="B46">Gudimalla et al., 2021</xref>; <xref ref-type="bibr" rid="B56">Jackson et al., 2021</xref>; <xref ref-type="bibr" rid="B58">Jehle et al., 2021</xref>; <xref ref-type="bibr" rid="B59">Khadem and Rey, 2021</xref>; <xref ref-type="bibr" rid="B84">Nesrullajev, 2022</xref>), which is largely driven by attractive forces in thermotropic LCs [e.g., cyanobiphenyl family (n-CB, n &#x3e; 10) (<xref ref-type="bibr" rid="B8">Bellini et al., 2002</xref>)] and excluded volume interactions in BLCs (e.g., collagen in mussel byssus and Ff phages). In both cases, a common geometric feature is the presence of rigid or semi-rigid rod-like cores and sufficiently long semi-flexible ends. Currently, the best-characterized materials that follow the direct I-SmA are low-molar mass thermotropic LCs, such as 10CB and 12CB (<xref ref-type="bibr" rid="B19">Collings, 1997</xref>; <xref ref-type="bibr" rid="B126">Urban et al., 2005</xref>; <xref ref-type="bibr" rid="B30">Demus et al., 2008a</xref>; <xref ref-type="bibr" rid="B68">Li et al., 2009</xref>; <xref ref-type="bibr" rid="B17">Chahine et al., 2010</xref>; <xref ref-type="bibr" rid="B46">Gudimalla et al., 2021</xref>; <xref ref-type="bibr" rid="B140">Zaluzhnyy et al., 2022</xref>). To avoid introducing a plethora of unknown parameters and material data, we focus on them as model systems. In the future, our ultimate goal is to expand the findings to lyotropic polymeric and colloidal smectics that are found in collagen precursors of the mussel byssus (<xref ref-type="bibr" rid="B101">Renner-Rao et al., 2019</xref>; <xref ref-type="bibr" rid="B58">Jehle et al., 2021</xref>; <xref ref-type="bibr" rid="B130">Waite and Harrington, 2022</xref>). It is noted that even though we only consider thermotropic LCs in the current manuscript, using correspondence principles such as those provided in the following references (<xref ref-type="bibr" rid="B92">Picken, 1990</xref>; <xref ref-type="bibr" rid="B120">Soule and Rey, 2011</xref>; <xref ref-type="bibr" rid="B43">Golmohammadi and Rey, 2009</xref>; <xref ref-type="bibr" rid="B79">Doi, 1981</xref>), we can, in the future, use the current findings for collagen-based LCs.</p>
<p>A key feature of the I-SmA transition is the strong coupling of the positional and orientational order parameters (OPs) (<xref ref-type="bibr" rid="B93">Pikin, 1991</xref>; <xref ref-type="bibr" rid="B45">Gorkunov et al., 2007</xref>; <xref ref-type="bibr" rid="B11">Blinov, 2011</xref>; <xref ref-type="bibr" rid="B125">Turek et al., 2020</xref>; <xref ref-type="bibr" rid="B48">Gurin et al., 2021</xref>), which results in the enhancement of the orientational OP over and above what a nematic phase could show at these conditions of temperature or concentration. A practical consequence of this, as we know from synthetic liquid crystal polymers (LCPs), is that in the solid state, the enhanced orientational order parameter has a strong impact on the mechanical properties (e.g., Young modulus) that can be seen in LCP fibers (<xref ref-type="bibr" rid="B138">Ward, 1993</xref>; <xref ref-type="bibr" rid="B143">Ziabicki, 1993</xref>; <xref ref-type="bibr" rid="B38">Donald et al., 2006</xref>; <xref ref-type="bibr" rid="B125">Turek et al., 2020</xref>; <xref ref-type="bibr" rid="B15">Bunsell et al., 2021</xref>). This order parameter coupling decreases the free energy of the system to result in a smectic A phase at minimal quenches from the stable disordered state. Polymeric and biological liquid-crystalline materials, being part of the lyo/thermotropic spectrum, also share the preferred native mode of fiber-formation, as seen in their <italic>in vivo</italic> and <italic>in vitro</italic> states (<xref ref-type="bibr" rid="B74">Matthews et al., 2002</xref>; <xref ref-type="bibr" rid="B127">Viney, 2004</xref>; <xref ref-type="bibr" rid="B34">Dierking and Al-Zangana, 2017</xref>; <xref ref-type="bibr" rid="B101">Renner-Rao et al., 2019</xref>; <xref ref-type="bibr" rid="B33">Deng et al., 2021</xref>; <xref ref-type="bibr" rid="B51">Harrington and Fratzl, 2021</xref>; <xref ref-type="bibr" rid="B123">Tortora and Jost, 2021</xref>; <xref ref-type="bibr" rid="B16">Cai et al., 2023</xref>; <xref ref-type="bibr" rid="B142">Zhang et al., 2023</xref>), which reinforces the hypothesis of smecticity enhancing the material&#x2019;s mechanical properties through increased alignment.</p>
<p>The computational liquid crystal phase-field methodology used in this paper, largely based on the Landau&#x2013;de Gennes (LdG) models and their many generalizations, has been widely used to simulate and predict self-assembly, self-organization, rheology, bulk, interfacial transport phenomena, and more for both thermotropic and lyotropic LCs (<xref ref-type="bibr" rid="B10">Biscari et al., 2007</xref>; <xref ref-type="bibr" rid="B53">Hormann and Zimmer, 2007</xref>; <xref ref-type="bibr" rid="B96">Popa-Nita and Sluckin, 2007</xref>; <xref ref-type="bibr" rid="B113">Saunders et al., 2007</xref>; <xref ref-type="bibr" rid="B78">Mohieddin Abukhdeir and Rey, 2008b</xref>; <xref ref-type="bibr" rid="B107">Rey, 2010</xref>; <xref ref-type="bibr" rid="B18">Coles and Strazielle, 2011</xref>; <xref ref-type="bibr" rid="B42">Garti et al., 2012</xref>; <xref ref-type="bibr" rid="B80">Mukherjee, 2014</xref>; <xref ref-type="bibr" rid="B49">Han et al., 2015</xref>; <xref ref-type="bibr" rid="B115">Selinger, 2016</xref>; <xref ref-type="bibr" rid="B21">Collings and Hird, 2017</xref>; <xref ref-type="bibr" rid="B128">Vitral et al., 2019</xref>; <xref ref-type="bibr" rid="B22">Copic and Mertelj, 2020</xref>; <xref ref-type="bibr" rid="B129">Vitral et al., 2020</xref>; <xref ref-type="bibr" rid="B114">Schimming et al., 2021</xref>; <xref ref-type="bibr" rid="B14">Bukharina et al., 2022</xref>; <xref ref-type="bibr" rid="B88">Paget et al., 2022</xref>; <xref ref-type="bibr" rid="B140">Zaluzhnyy et al., 2022</xref>). As in other coupled OP transitions, the challenges for a given energy landscape include the following issues:<list list-type="simple">
<list-item>
<p>&#x2022; What is the total number of local maxima, minima, and saddles in the energy surface for a given quench?</p>
</list-item>
<list-item>
<p>&#x2022; Where are the local minima and maxima located for a given quench?</p>
</list-item>
<list-item>
<p>&#x2022; When do we find bi-stability?</p>
</list-item>
<list-item>
<p>&#x2022; What states exist when one of the OPs is zero (e.g., nematic and plastic crystal)?</p>
</list-item>
<list-item>
<p>&#x2022; What are the shortest path directions connecting minima?</p>
</list-item>
</list>
</p>
<p>Previous works have excellently characterized these mesophases and phase transitions, including phase diagrams, orientation distribution function profiles, bifurcation analysis, and the use of imaging techniques, calorimetry characterization, and dynamic simulations, accentuating the thermodynamic I-SmA phase transition perspective (<xref ref-type="bibr" rid="B89">Palffy-Muhoray, 1999</xref>; <xref ref-type="bibr" rid="B37">Dogic and Fraden, 2001</xref>; <xref ref-type="bibr" rid="B82">Mukherjee et al., 2001</xref>; <xref ref-type="bibr" rid="B66">Larin, 2004</xref>; <xref ref-type="bibr" rid="B126">Urban et al., 2005</xref>; <xref ref-type="bibr" rid="B10">Biscari et al., 2007</xref>; <xref ref-type="bibr" rid="B23">Das and Mukherjee, 2009</xref>; <xref ref-type="bibr" rid="B17">Chahine et al., 2010</xref>; <xref ref-type="bibr" rid="B83">Nandi et al., 2012</xref>; <xref ref-type="bibr" rid="B55">Izzo and De Oliveira, 2019</xref>; <xref ref-type="bibr" rid="B46">Gudimalla et al., 2021</xref>; <xref ref-type="bibr" rid="B60">Khan and Mukherjee, 2021</xref>; <xref ref-type="bibr" rid="B81">Mukherjee, 2021</xref>). Given that the first four key issues are essentially anchored in the spatial features of the energy landscape, we develop, implement, and validate a novel geometric methodology. The use of geometric methods to characterize thermodynamics, phase transitions, and energy landscapes has been widely recognized as a useful and complementary set of tools (<xref ref-type="bibr" rid="B76">Miller, 1925</xref>; <xref ref-type="bibr" rid="B53">Hormann and Zimmer, 2007</xref>; <xref ref-type="bibr" rid="B99">Quevedo et al., 2011</xref>; <xref ref-type="bibr" rid="B131">Wales, 2018</xref>; <xref ref-type="bibr" rid="B133">Wang et al., 2020</xref>; <xref ref-type="bibr" rid="B29">Demirci and Holland, 2022</xref>; <xref ref-type="bibr" rid="B70">Liu et al., 2022</xref>; <xref ref-type="bibr" rid="B98">Quevedo et al., 2022</xref>). These approaches rely on establishing a proper thermodynamic surface in terms of variables such as pressure, temperature, and chemical potential. Here, we extend and generalize these geometric-thermodynamic methods for self-assembly in anisotropic soft matter in general and phase ordering in the I-SmA transition. The methodology is summarized in <xref ref-type="fig" rid="F1">Figure 1</xref>. The triangle&#x2019;s center is the key focus of this paper, the direct isotropic-SmA transition, as characterized by an energy landscape given by the Landau free energy <inline-formula id="inf1">
<mml:math id="m1">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> as a function of the positional <inline-formula id="inf2">
<mml:math id="m2">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> and orientational <inline-formula id="inf3">
<mml:math id="m3">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> order parameters for a given temperature <inline-formula id="inf4">
<mml:math id="m4">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. Importantly, the surface parametrization is explicit and known as Monge parameterization, and it is the starting point for the geometric methodology developed in this paper.</p>
<fig id="F1" position="float">
<label>FIGURE 1</label>
<caption>
<p>Methodology of the geometric-thermodynamics formulated (<xref ref-type="sec" rid="s2">Section 2</xref>) and implemented (<xref ref-type="sec" rid="s3">Section 3</xref>) in this paper. The central energy landscape corresponding to the direct I-SmA transition is characterized using three geometric methods (I) level sets/steepest descent/critical points, (II) geodesics and curvature lines, and (III) shape and Casorati curvedness analysis; the bottom left shows changes in shape from cup (left), to rut, saddle, ridge, and cap (right) and changes in the degree of curvedness for a rut patch (left) into a flat plane. The outer arrows show the connection and integration of the three calculations.</p>
</caption>
<graphic xlink:href="frsfm-04-1359128-g001.tif"/>
</fig>
<p>In (I), at the upper vertex from <xref ref-type="fig" rid="F1">Figure 1,</xref> the critical points (dots) of the surface (maxima, minima, and saddles) are determined as a function of changes in temperature using level-set curves and curves of the steepest descent. The closed loops in the former allow the detection of minima/maxima and saddles, and the signs of the gradient curves differentiate the stable from the unstable. In addition, saddles in the level sets are unstable points. These calculations are guided and validated using the powerful index theorem of polynomials, yielding a conservation equation for the number of maxima <inline-formula id="inf5">
<mml:math id="m5">
<mml:mrow>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mtext>maxima</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, the number of minima <inline-formula id="inf6">
<mml:math id="m6">
<mml:mrow>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mtext>minima</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, and the number of saddles <inline-formula id="inf7">
<mml:math id="m7">
<mml:mrow>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mtext>saddles</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>. At a high temperature for a stable isotropic phase, the landscape is simple and <inline-formula id="inf8">
<mml:math id="m8">
<mml:mrow>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mtext>maxima</mml:mtext>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mtext>saddles</mml:mtext>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, but for low quenches with a stable SmA phase, the landscape is complex, as we find <inline-formula id="inf9">
<mml:math id="m9">
<mml:mrow>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mtext>minima</mml:mtext>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mtext>saddles</mml:mtext>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>4</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf10">
<mml:math id="m10">
<mml:mrow>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mtext>maxima</mml:mtext>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>. In (II), at the lower right vertex, the geodesic lines of the energy landscape are found using accurate ordinary differential equation (ODE) solvers. Importantly, the lack of torsion in the geodesic lines correlates with the energy heights of the maxima and minima. In (II), we also compute the lines of curvature (LOC) that define an orthogonal grid on the energy surface and indicate the maximum or minimum directional curvature flow. In (III), the shape coefficient, or shape index <inline-formula id="inf11">
<mml:math id="m11">
<mml:mrow>
<mml:mi>S</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, relates a given value with a shape that comes from the spectrum <inline-formula id="inf12">
<mml:math id="m12">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#xb1;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> for up/down half-spheres (cup and cap, respectively), <inline-formula id="inf13">
<mml:math id="m13">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#xb1;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>/</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> for up/down cylinders (rut and ridge, respectively), and <inline-formula id="inf14">
<mml:math id="m14">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> for saddle-like shapes; the positive scalar curvedness Casorati curvature <inline-formula id="inf15">
<mml:math id="m15">
<mml:mrow>
<mml:mi>C</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> (surface deviation from planarity) of the energy landscape and its critical points are identified to determine aspects of the minima and maxima, such as anisotropy, the presence or absence of umbilic <inline-formula id="inf16">
<mml:math id="m16">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#xb1;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> points, and the degree of curvedness (magnitude of Casorati curvature), and detect maxima/minima/saddles in a fast and efficient way.</p>
<p>Describing the shape using the dimensionless normalized shape coefficient <inline-formula id="inf17">
<mml:math id="m17">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>S</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> avoids co-mingling properties associated with shape with those associated with curvedness, such as when using the classical differential geometry descriptions based on Gaussian <inline-formula id="inf18">
<mml:math id="m18">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>K</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>, mean <inline-formula id="inf19">
<mml:math id="m19">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>H</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>, and deviatoric <inline-formula id="inf20">
<mml:math id="m20">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>D</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> curvatures. The shape coefficient&#x2013;Casorati curvedness <inline-formula id="inf21">
<mml:math id="m21">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>C</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> method has been successfully applied to several soft-matter materials and equilibrium and dissipative processes (<xref ref-type="bibr" rid="B133">Wang et al., 2020</xref>; <xref ref-type="bibr" rid="B137">Wang et al., 2022b</xref>; <xref ref-type="bibr" rid="B134">Wang et al., 2022a</xref>; <xref ref-type="bibr" rid="B136">Wang et al., 2023b</xref>). For example, for a saddle point, the classical approach yields <inline-formula id="inf22">
<mml:math id="m22">
<mml:mrow>
<mml:mi>K</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:msup>
<mml:mi>D</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>. On the other hand, the <inline-formula id="inf23">
<mml:math id="m23">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>C</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> method detects a saddle simply when <inline-formula id="inf24">
<mml:math id="m24">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> and its curvedness is <inline-formula id="inf25">
<mml:math id="m25">
<mml:mrow>
<mml:mi>C</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>D</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:msqrt>
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>K</mml:mi>
</mml:mrow>
</mml:msqrt>
</mml:mrow>
</mml:math>
</inline-formula>. Finally, the arrows on the triangle side and toward the center denote the integration of the methods to shed new light on the I-SmA transition.</p>
<p>This work builds on the fundamental studies on the self-assembled smectic phase transition (<xref ref-type="bibr" rid="B94">Pleiner et al., 2000</xref>; <xref ref-type="bibr" rid="B82">Mukherjee et al., 2001</xref>; <xref ref-type="bibr" rid="B2">Abukhdeir, 2009</xref>; <xref ref-type="bibr" rid="B5">Abukhdeir and Rey, 2009b</xref>; <xref ref-type="bibr" rid="B81">Mukherjee, 2021</xref>). The particular objectives of this paper are the following:<list list-type="simple">
<list-item>
<p>&#x2022; Characterize the energy landscape using a simple Monge parametrization in terms of nematic and smectic order parameters, including the number and type of critical points and characteristic trajectories between stable, metastable, and unstable isotropic, nematic, and smectic states as a function of the changing degrees of quenching from the isotropic state.</p>
</list-item>
<list-item>
<p>&#x2022; Use classical curvatures (Gaussian and mean) and new soft matter geometric methods (shape coefficient and Casorati curvedness) to shed light on where saddles and cusps are located and their curvedness, thus providing a broad picture of the energy landscape.</p>
</list-item>
<list-item>
<p>&#x2022; Integrate thermodynamic stability, polynomial-based charge conservation methods, and geometric methods.</p>
</list-item>
<list-item>
<p>&#x2022; Detect and characterize unstable nematic and metastable plastic crystal states that emerge at medium and large degrees of quenching.</p>
</list-item>
</list>
</p>
<p>The remainder of this paper is organized as follows: <xref ref-type="sec" rid="s2">Section 2</xref> presents the OPs, Landau free energy (<xref ref-type="sec" rid="s2-1">Section 2.1</xref>), geometric thermodynamics (<xref ref-type="sec" rid="s2-2">Section 2.2</xref>), and computational methods (<xref ref-type="sec" rid="s2-3">Section 2.3</xref>). <xref ref-type="sec" rid="s3">Section 3</xref> presents the results and discussion. <xref ref-type="sec" rid="s4">Section 4</xref> summarizes the key findings, their significance, and the novelty of the geometric approach.</p>
</sec>
<sec sec-type="methods" id="s2">
<title>2 Methodology</title>
<sec id="s2-1">
<title>2.1 Order parameters and Landau model</title>
<sec id="s2-1-1">
<title>2.1.1 Nematic and smectic A phases: orientational and positional order parameters</title>
<p>The SmA LC phase has partial 1D positional and orientational order. Two order parameters, the orientational order parameter <bold>Q</bold> and the positional order parameter <inline-formula id="inf26">
<mml:math id="m26">
<mml:mrow>
<mml:mi mathvariant="bold">&#x3a8;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, are used for this mesophase characterization. These parameters capture the average molecular order by the establishment of distinctive moduli at the transition, with the moduli <inline-formula id="inf27">
<mml:math id="m27">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mi>P</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> for the orientation and <inline-formula id="inf28">
<mml:math id="m28">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> for the position (<xref ref-type="bibr" rid="B87">Oswald and Pieranski, 2005b</xref>; <xref ref-type="bibr" rid="B107">Rey, 2010</xref>; <xref ref-type="bibr" rid="B80">Mukherjee, 2014</xref>; <xref ref-type="bibr" rid="B129">Vitral et al., 2020</xref>).</p>
<p>The theoretical characterization of the partial orientational alignment in LCs is described by an orientation distribution function, the tensor order parameter <bold>Q</bold> (<xref ref-type="bibr" rid="B25">De Gennes and Prost, 1993</xref>), which is as follows:<disp-formula id="e2_1">
<mml:math id="m29">
<mml:mrow>
<mml:mi mathvariant="bold">Q</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mtext mathvariant="bold">nn</mml:mtext>
<mml:mo>&#x2212;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mi mathvariant="bold">I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x2b;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:mfrac>
<mml:mi>P</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mtext mathvariant="bold">mm</mml:mtext>
<mml:mo>&#x2212;</mml:mo>
<mml:mtext mathvariant="bold">ll</mml:mtext>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
<label>(2.1)</label>
</disp-formula>where <bold>n</bold>, <bold>m</bold>, and <bold>l</bold> are the molecular unit vectors and <bold>I</bold> is the unit tensor. The <bold>Q</bold> tensor is expressed in terms of the orthonormal director triad (<bold>n</bold>, <bold>m</bold>, and <bold>l</bold>), which are the eigenvectors of <bold>Q</bold> that describe the molecular axes and the scalar moduli <inline-formula id="inf29">
<mml:math id="m30">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mi>P</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> that measure the magnitude of the average molecular orientation (<xref ref-type="bibr" rid="B27">De Luca and Rey, 2006</xref>; <xref ref-type="bibr" rid="B18">Coles and Strazielle, 2011</xref>). The <bold>Q</bold> tensor is symmetric and traceless, i.e., <inline-formula id="inf30">
<mml:math id="m31">
<mml:mrow>
<mml:mi mathvariant="bold">Q</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:msup>
<mml:mi mathvariant="bold">Q</mml:mi>
<mml:mi>T</mml:mi>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf31">
<mml:math id="m32">
<mml:mrow>
<mml:mi mathvariant="bold">Q</mml:mi>
<mml:mo>:</mml:mo>
<mml:mi mathvariant="bold">I</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, and it has five independent components. It is comprised of uniaxial <inline-formula id="inf32">
<mml:math id="m33">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
<mml:mtext>&#x2009;and&#x2009;</mml:mtext>
<mml:mi mathvariant="bold">n</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> and biaxial <inline-formula id="inf33">
<mml:math id="m34">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>P</mml:mi>
<mml:mtext>&#x2009;and&#x2009;</mml:mtext>
<mml:mi mathvariant="bold">m</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi mathvariant="bold">l</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> contributions. The uniaxial contribution <inline-formula id="inf34">
<mml:math id="m35">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:mfrac>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mtext mathvariant="bold">nn</mml:mtext>
<mml:mo>:</mml:mo>
<mml:mi mathvariant="bold">Q</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> corresponds to the major eigenvalue/eigenvector, and the biaxial contribution <inline-formula id="inf35">
<mml:math id="m36">
<mml:mrow>
<mml:mi>P</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:mfrac>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mtext mathvariant="bold">mm</mml:mtext>
<mml:mo>&#x2212;</mml:mo>
<mml:mtext mathvariant="bold">ll</mml:mtext>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>:</mml:mo>
<mml:mi mathvariant="bold">Q</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> corresponds to the minor eigenvalues/eigenvectors. A free-energy expansion near the nematic transition, as per the Landau formalism, was defined, yielding an LdG free-energy expression for nematic orientational alignment <inline-formula id="inf36">
<mml:math id="m37">
<mml:mrow>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mi>N</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> (<xref ref-type="bibr" rid="B25">de Gennes and Prost, 1993</xref>), truncated up to the fourth-order term with respect to <inline-formula id="inf37">
<mml:math id="m38">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> as follows:<disp-formula id="e2_2">
<mml:math id="m39">
<mml:mrow>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mi>N</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:mfrac>
<mml:mi>a</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="bold">Q</mml:mi>
<mml:mo>:</mml:mo>
<mml:mi mathvariant="bold">Q</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:mfrac>
<mml:mi>b</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="bold">Q</mml:mi>
<mml:mo>&#x22c5;</mml:mo>
<mml:mi mathvariant="bold">Q</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>:</mml:mo>
<mml:mi mathvariant="bold">Q</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>4</mml:mn>
</mml:mrow>
</mml:mfrac>
<mml:mi>c</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="bold">Q</mml:mi>
<mml:mo>:</mml:mo>
<mml:mi mathvariant="bold">Q</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo>&#x2b;</mml:mo>
<mml:mo>&#x2026;</mml:mo>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
<label>(2.2)</label>
</disp-formula>where <inline-formula id="inf38">
<mml:math id="m40">
<mml:mrow>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the energy of the isotropic state, <inline-formula id="inf39">
<mml:math id="m41">
<mml:mrow>
<mml:mi>a</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>T</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:msubsup>
<mml:mi>T</mml:mi>
<mml:mrow>
<mml:mi>N</mml:mi>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mi>&#x2a;</mml:mi>
</mml:msubsup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf40">
<mml:math id="m42">
<mml:mrow>
<mml:msubsup>
<mml:mi>T</mml:mi>
<mml:mrow>
<mml:mi>N</mml:mi>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mi>&#x2a;</mml:mi>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> is the critical nematic phase transition temperature, and <inline-formula id="inf41">
<mml:math id="m43">
<mml:mrow>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf42">
<mml:math id="m44">
<mml:mrow>
<mml:mi>b</mml:mi>
<mml:mo>,</mml:mo>
<mml:mtext>and&#x2009;</mml:mtext>
<mml:mi>c</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> are experimentally measured material parameters.</p>
<p>In addition to the orientational organization, due to its lamellar configuration and the periodic structure of the uniaxial SmA phase, a one-dimensional positional order parameter is required, and the complex wave vector <inline-formula id="inf43">
<mml:math id="m45">
<mml:mrow>
<mml:mi mathvariant="bold">&#x3a8;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> has been widely used for this purpose (<xref ref-type="bibr" rid="B82">Mukherjee et al., 2001</xref>; <xref ref-type="bibr" rid="B6">Abukhdeir and Rey, 2009c</xref>; <xref ref-type="bibr" rid="B129">Vitral et al., 2020</xref>). It is typified by the phase <inline-formula id="inf44">
<mml:math id="m46">
<mml:mrow>
<mml:mi>&#x3d5;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, and its modulus <inline-formula id="inf45">
<mml:math id="m47">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> characterizes a one-dimensional density wave used to describe such a layered nature as follows:<disp-formula id="e2_3">
<mml:math id="m48">
<mml:mrow>
<mml:mi mathvariant="bold">&#x3c8;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>&#x3c8;</mml:mi>
<mml:msup>
<mml:mi>e</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>&#x3d5;</mml:mi>
</mml:mrow>
</mml:msup>
<mml:mo>.</mml:mo>
</mml:mrow>
</mml:math>
<label>(2.3)</label>
</disp-formula>
</p>
<p>Then, a free-energy functional of the smectic positional order, <inline-formula id="inf46">
<mml:math id="m49">
<mml:mrow>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mi>S</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, is introduced, accounting for positional ordering, and the material-dependent parameters are <inline-formula id="inf47">
<mml:math id="m50">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>&#x3b1;</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>T</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:msubsup>
<mml:mi>T</mml:mi>
<mml:mrow>
<mml:mi>A</mml:mi>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mi>&#x2a;</mml:mi>
</mml:msubsup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf48">
<mml:math id="m51">
<mml:mrow>
<mml:mi>&#x3b2;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> around the critical smectic transition temperature <inline-formula id="inf49">
<mml:math id="m52">
<mml:mrow>
<mml:msubsup>
<mml:mi>T</mml:mi>
<mml:mrow>
<mml:mi>A</mml:mi>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mi>&#x2a;</mml:mi>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula>. <disp-formula id="e2_4">
<mml:math id="m53">
<mml:mrow>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mi>S</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>&#x3b1;</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mfenced open="|" close="|" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="bold">&#x3c8;</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo>&#x2b;</mml:mo>
<mml:mi>&#x3b2;</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mfenced open="|" close="|" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="bold">&#x3c8;</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mn>4</mml:mn>
</mml:msup>
<mml:mo>&#x2b;</mml:mo>
<mml:mo>&#x22ef;</mml:mo>
<mml:mo>.</mml:mo>
</mml:mrow>
</mml:math>
<label>(2.4)</label>
</disp-formula>
</p>
<p>In the simplest case, the I-SmA transition is captured by the free-energy contributions of the form <inline-formula id="inf50">
<mml:math id="m54">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mi>N</mml:mi>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mi>S</mml:mi>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mi>N</mml:mi>
<mml:mi>S</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, where <inline-formula id="inf51">
<mml:math id="m55">
<mml:mrow>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mi>N</mml:mi>
<mml:mi>S</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the coupling free-energy term, which is given as follows:<disp-formula id="e2_5">
<mml:math id="m56">
<mml:mrow>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mi>N</mml:mi>
<mml:mi>S</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3b4;</mml:mi>
<mml:mrow>
<mml:mfenced open="|" close="|" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="bold">&#x3c8;</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="bold">Q</mml:mi>
<mml:mo>:</mml:mo>
<mml:mi mathvariant="bold">Q</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:mfrac>
<mml:mi>e</mml:mi>
<mml:mi mathvariant="bold">Q</mml:mi>
<mml:mo>:</mml:mo>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mo>&#x2207;</mml:mo>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>&#x2207;</mml:mo>
<mml:mi>&#x3c8;</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x2b;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:mfrac>
<mml:msub>
<mml:mi>b</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:msup>
<mml:mrow>
<mml:mfenced open="|" close="|" separators="&#x7c;">
<mml:mrow>
<mml:mo>&#x2207;</mml:mo>
<mml:mi mathvariant="bold">&#x3c8;</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo>&#x2b;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>4</mml:mn>
</mml:mrow>
</mml:mfrac>
<mml:msub>
<mml:mi>b</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:msup>
<mml:mrow>
<mml:mfenced open="|" close="|" separators="&#x7c;">
<mml:mrow>
<mml:msup>
<mml:mo>&#x2207;</mml:mo>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mi mathvariant="bold">&#x3c8;</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo>.</mml:mo>
</mml:mrow>
</mml:math>
<label>(2.5)</label>
</disp-formula>
</p>
</sec>
<sec id="s2-1-2">
<title>2.1.2 Landau model for the isotropic&#x2013;smectic A transition</title>
<p>The well-established models have been formulated and used to describe the I-SmA transition, including non-direct transitions (<xref ref-type="bibr" rid="B94">Pleiner et al., 2000</xref>; <xref ref-type="bibr" rid="B5">Abukhdeir and Rey, 2009b</xref>; <xref ref-type="bibr" rid="B83">Nandi et al., 2012</xref>; <xref ref-type="bibr" rid="B91">Pevnyi et al., 2014</xref>; <xref ref-type="bibr" rid="B55">Izzo and De Oliveira, 2019</xref>; <xref ref-type="bibr" rid="B81">Mukherjee, 2021</xref>; <xref ref-type="bibr" rid="B88">Paget et al., 2022</xref>). These models include the nematic and smectic contributions (Eqs. <xref ref-type="disp-formula" rid="e2_2">2.2</xref>, <xref ref-type="disp-formula" rid="e2_4">2.4</xref>, and <xref ref-type="disp-formula" rid="e2_5">2.5</xref>) and consolidate the final total free-energy density using coupling terms (<inline-formula id="inf52">
<mml:math id="m57">
<mml:mrow>
<mml:mi>&#x3b4;</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>e</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>) that favor one phase over the other and terms that account for the energy cost from the coexistence of the positional ordering (<inline-formula id="inf53">
<mml:math id="m58">
<mml:mrow>
<mml:msub>
<mml:mi>b</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>b</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>):<disp-formula id="e2_6">
<mml:math id="m59">
<mml:mrow>
<mml:mtable columnalign="left">
<mml:mtr>
<mml:mtd>
<mml:mrow>
<mml:mtext>Total&#x2009;Energy</mml:mtext>
<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#x222b;</mml:mo>
<mml:mi>F</mml:mi>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mi>d</mml:mi>
<mml:mi>V</mml:mi>
</mml:mrow>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:mtable columnalign="center">
<mml:mtr>
<mml:mtd>
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mo>&#x3d;</mml:mo>
</mml:mrow>
</mml:mtd>
<mml:mtd>
<mml:mrow>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:mfrac>
<mml:mi>a</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="bold">Q</mml:mi>
<mml:mo>:</mml:mo>
<mml:mi mathvariant="bold">Q</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:mfrac>
<mml:mi>b</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="bold">Q</mml:mi>
<mml:mo>&#x22c5;</mml:mo>
<mml:mi mathvariant="bold">Q</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>:</mml:mo>
<mml:mi mathvariant="bold">Q</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>4</mml:mn>
</mml:mrow>
</mml:mfrac>
<mml:mi>c</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="bold">Q</mml:mi>
<mml:mo>:</mml:mo>
<mml:mi mathvariant="bold">Q</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo>&#x2b;</mml:mo>
<mml:mi>&#x3b1;</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mfenced open="|" close="|" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="bold">&#x3c8;</mml:mi>
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<mml:mn>4</mml:mn>
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<mml:mi>b</mml:mi>
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<mml:mn>2</mml:mn>
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<label>(2.6)</label>
</disp-formula>
</p>
<p>Considering Equation <xref ref-type="disp-formula" rid="e2_6">2.6</xref>, assuming a spatially homogenous system, and performing reparameterization (see <xref ref-type="sec" rid="s10">Supplementary Appendix A1</xref>), we find the following governing free energy density <inline-formula id="inf54">
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</inline-formula>:<disp-formula id="e2_7">
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<mml:mrow>
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<mml:mi>a</mml:mi>
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<mml:mrow>
<mml:mn>9</mml:mn>
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<mml:mi>b</mml:mi>
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<mml:mo>,</mml:mo>
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<label>(2.7)</label>
</disp-formula>where <inline-formula id="inf55">
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<mml:mrow>
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<mml:mi>F</mml:mi>
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</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the nematic contribution, <inline-formula id="inf56">
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<mml:msub>
<mml:mi>F</mml:mi>
<mml:mi>S</mml:mi>
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</mml:mrow>
</mml:math>
</inline-formula> is the smectic contribution, and <inline-formula id="inf57">
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<mml:mi>F</mml:mi>
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<mml:mi>S</mml:mi>
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</mml:msub>
</mml:mrow>
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</inline-formula> is the crucial coupling contribution between the positional and orientational OPs that are in their expanded form. In this paper, we evaluate the OPs in their extended domain of dependence <inline-formula id="inf58">
<mml:math id="m65">
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</mml:mrow>
<mml:mo>:</mml:mo>
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<mml:mn>1</mml:mn>
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<mml:mn>1</mml:mn>
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</mml:mrow>
<mml:mo>&#xd7;</mml:mo>
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<mml:mn>1</mml:mn>
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</mml:mrow>
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</inline-formula> to fully capture the important phenomena at the nematic axis <inline-formula id="inf59">
<mml:math id="m66">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
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<mml:mn>0</mml:mn>
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</inline-formula> and the smectic axis <inline-formula id="inf60">
<mml:math id="m67">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
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</inline-formula>; we note that negative nematic OP states (molecular alignment normal to the director orientation) are usually considered in nematostatics (<xref ref-type="bibr" rid="B44">Golmohammadi and Rey, 2010</xref>), but in the particular equilibrium spatially homogeneous I-SmA transitions considered in this study, these orientation states play no role.</p>
<p>The possible states are obtained by the minimization of the homogeneous free energy in Equation <xref ref-type="disp-formula" rid="e2_7">2.7</xref> concerning the two non-conserved OPs <inline-formula id="inf61">
<mml:math id="m68">
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</inline-formula>. This yields a system of ODEs (see <xref ref-type="sec" rid="s10">Supplementary Appendix A1</xref>). Then, at a given temperature <inline-formula id="inf62">
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</mml:mrow>
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</inline-formula>, different phases arise according to the ordering contributions. As mentioned in the introduction, positional and orientational ordering define an LC state, which varies in accordance with the combination of these order parameters. In this paper, we consider the following:<disp-formula id="e2_8">
<mml:math id="m70">
<mml:mrow>
<mml:mtable columnalign="left">
<mml:mtr>
<mml:mtd>
<mml:mrow>
<mml:mtext>Isotropic&#xa0;</mml:mtext>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mtext>Iso</mml:mtext>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>:</mml:mo>
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<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
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<mml:mo>&#x3d;</mml:mo>
<mml:mi>&#x3c8;</mml:mi>
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<mml:mn>0</mml:mn>
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</mml:mrow>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
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<mml:mtext>Nematic&#xa0;</mml:mtext>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
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</mml:mrow>
</mml:mfenced>
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<mml:mn>0</mml:mn>
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</mml:mrow>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:mrow>
<mml:mtext>Plastic&#xa0;crystal&#xa0;</mml:mtext>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
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</mml:mrow>
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<mml:mtr>
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<mml:mo>&#x2212;</mml:mo>
<mml:mi mathvariant="normal">A</mml:mi>
<mml:mtext>&#xa0;</mml:mtext>
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</mml:msub>
<mml:mo>&#x3e;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>&#x3e;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>.</mml:mo>
</mml:mrow>
</mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:mrow>
</mml:math>
<label>(2.8)</label>
</disp-formula>
</p>
<p>The isotropic-liquid state is characterized by the absence of positional and orientational orders, the nematic LC possesses only average molecular orientation, the plastic crystal phase (characterized by the density wave) describes a material with positional order and very small-to-none orientational order, and the smectic A LC exhibits positional and orientational orders (<xref ref-type="bibr" rid="B86">Oswald and Pieranski, 2005a</xref>; <xref ref-type="bibr" rid="B87">Oswald and Pieranski, 2005b</xref>; <xref ref-type="bibr" rid="B24">De Gennes, 2007</xref>; <xref ref-type="bibr" rid="B30">Demus et al., 2008a</xref>; <xref ref-type="bibr" rid="B35">DiLisi, 2019</xref>; <xref ref-type="bibr" rid="B81">Mukherjee, 2021</xref>). We note that the density wave behavior, designated as the plastic crystal state in this study, has been reported even for some rod-like systems (<xref ref-type="bibr" rid="B64">Kyrylyuk et al., 2011</xref>; <xref ref-type="bibr" rid="B69">Liu et al., 2014</xref>; <xref ref-type="bibr" rid="B112">Sato et al., 2023</xref>). In this work, the metastable plastic crystal emerges at deep quenches when the isotropic state becomes unstable, the nematic and coupling energies vanish, and the stable phase is SmA. Similarly, since <inline-formula id="inf63">
<mml:math id="m71">
<mml:mrow>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msup>
<mml:mi>e</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>3</mml:mn>
<mml:mi>&#x3b4;</mml:mi>
<mml:msub>
<mml:mi>b</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>/</mml:mo>
<mml:mn>9</mml:mn>
<mml:msub>
<mml:mi>b</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo>&#x3e;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf64">
<mml:math id="m72">
<mml:mrow>
<mml:mrow>
<mml:mfenced open="[" close="]" separators="&#x7c;">
<mml:mrow>
<mml:msubsup>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
<mml:mn>2</mml:mn>
</mml:msubsup>
<mml:msup>
<mml:mi>&#x3c8;</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo>&#x2212;</mml:mo>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mn>3</mml:mn>
<mml:mi>e</mml:mi>
<mml:msub>
<mml:mi>b</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>/</mml:mo>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msup>
<mml:mi>e</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>3</mml:mn>
<mml:mi>&#x3b4;</mml:mi>
<mml:msub>
<mml:mi>b</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
<mml:msup>
<mml:mi>&#x3c8;</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3e;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> for the SmA state in Equation <xref ref-type="disp-formula" rid="e2_7">2.7</xref>, the important coupling term <inline-formula id="inf65">
<mml:math id="m73">
<mml:mrow>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mi>N</mml:mi>
<mml:mi>S</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> promotes the emergence of SmA with the positional and orientational order.</p>
</sec>
</sec>
<sec id="s2-2">
<title>2.2 Geometric thermodynamics for phase ordering in the isotropic&#x2013;smectic A transition</title>
<p>In this section, we investigate the surface geometry of the energy landscape <inline-formula id="inf66">
<mml:math id="m74">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, with a particular emphasis on understanding and characterizing the essential nature of all the critical points. These critical points, derived from Equation <xref ref-type="disp-formula" rid="e2_7">2.7</xref>, include the local maxima, local minima, and saddle points. As mentioned at the end of the introduction, their significance extends across many research fields in liquid crystals, such as self-assembly, kinematics, and thermodynamics of these systems.</p>
<p>Next, we briefly mention the basic argument to keep all critical points, forgoing complex mathematical details. For instance, the time-dependent Ginzburg&#x2013;Landau model (<xref ref-type="bibr" rid="B95">Popa-Nita, 1999</xref>) provides a quantitative study of the spatiotemporal evolution of thermodynamic behaviors on a non-conserved OPs vector <inline-formula id="inf67">
<mml:math id="m75">
<mml:mrow>
<mml:mi mathvariant="bold">p</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>.<disp-formula id="e2_9">
<mml:math id="m76">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:mo>&#x2202;</mml:mo>
<mml:mi mathvariant="bold">p</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2202;</mml:mo>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mo>&#x221d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mo>&#x2202;</mml:mo>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>f</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi>f</mml:mi>
<mml:mi>e</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
<mml:mrow>
<mml:mi>&#x3b4;</mml:mi>
<mml:mi mathvariant="bold">p</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mo>&#x2202;</mml:mo>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mi>e</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2202;</mml:mo>
<mml:mi mathvariant="bold">p</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mo>&#x2212;</mml:mo>
<mml:mo>&#x2207;</mml:mo>
<mml:mo>&#x22c5;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mo>&#x2202;</mml:mo>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mi>e</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2202;</mml:mo>
<mml:mo>&#x2207;</mml:mo>
<mml:mi mathvariant="bold">p</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mo>,</mml:mo>
<mml:mi mathvariant="bold">p</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mrow>
<mml:mfenced open="[" close="]" separators="&#x7c;">
<mml:mrow>
<mml:mtable columnalign="center">
<mml:mtr>
<mml:mtd>
<mml:mi>&#x3c8;</mml:mi>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
<label>(2.9)</label>
</disp-formula>where <inline-formula id="inf68">
<mml:math id="m77">
<mml:mrow>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mi>e</mml:mi>
</mml:msub>
<mml:mo>&#x221d;</mml:mo>
<mml:mo>&#x2207;</mml:mo>
<mml:mi mathvariant="bold">p</mml:mi>
<mml:mo>:</mml:mo>
<mml:mo>&#x2207;</mml:mo>
<mml:mi mathvariant="bold">p</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is the elastic energy dependent on the deformation; note that <inline-formula id="inf69">
<mml:math id="m78">
<mml:mrow>
<mml:mi mathvariant="bold">p</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is the OP vector for the smectic A phase. A special solution of Equation <xref ref-type="disp-formula" rid="e2_9">2.9</xref> describes a front propagation that could describe growing smectic droplets in an isotropic matrix. For example, the wave-like property has been intensively studied by <xref ref-type="bibr" rid="B26">De Luca and Rey (2004)</xref> for the case of chiral nematic fronts propagating into an unstable isotropic phase. In our present smectic model, the wave-like solution <inline-formula id="inf70">
<mml:math id="m79">
<mml:mrow>
<mml:mi mathvariant="bold">p</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mi mathvariant="bold">p</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="bold">x</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi mathvariant="bold">v</mml:mi>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mi mathvariant="bold">p</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mover accent="true">
<mml:mi mathvariant="bold">x</mml:mi>
<mml:mo>&#x223c;</mml:mo>
</mml:mover>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>; <inline-formula id="inf71">
<mml:math id="m80">
<mml:mrow>
<mml:mover accent="true">
<mml:mi mathvariant="bold">x</mml:mi>
<mml:mo>&#x223c;</mml:mo>
</mml:mover>
<mml:mo>&#x3d;</mml:mo>
<mml:mi mathvariant="bold">x</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi mathvariant="bold">v</mml:mi>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> with constant velocity <inline-formula id="inf72">
<mml:math id="m81">
<mml:mrow>
<mml:mi mathvariant="bold">v</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> simplifies Eq. <xref ref-type="disp-formula" rid="e2_9">2.9</xref> to a more compact form, <inline-formula id="inf73">
<mml:math id="m82">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:mo>&#x2202;</mml:mo>
<mml:mi>F</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2202;</mml:mo>
<mml:mi mathvariant="bold">p</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mo>&#x3d;</mml:mo>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="bold">v</mml:mi>
<mml:mo>&#x22c5;</mml:mo>
<mml:mover accent="true">
<mml:mo>&#x2207;</mml:mo>
<mml:mo>&#x223c;</mml:mo>
</mml:mover>
<mml:mo>&#x2b;</mml:mo>
<mml:msup>
<mml:mover accent="true">
<mml:mo>&#x2207;</mml:mo>
<mml:mo>&#x223c;</mml:mo>
</mml:mover>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mi mathvariant="bold">p</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, where all the coefficients are not included for clarity. At <inline-formula id="inf74">
<mml:math id="m83">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:mo>&#x2202;</mml:mo>
<mml:mi>F</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2202;</mml:mo>
<mml:mi mathvariant="bold">p</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, the critical points lay inside the kernel of the linear map defined by the velocity <inline-formula id="inf75">
<mml:math id="m84">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="bold">v</mml:mi>
<mml:mo>&#x22c5;</mml:mo>
<mml:mover accent="true">
<mml:mo>&#x2207;</mml:mo>
<mml:mo>&#x223c;</mml:mo>
</mml:mover>
<mml:mo>&#x2b;</mml:mo>
<mml:msup>
<mml:mover accent="true">
<mml:mo>&#x2207;</mml:mo>
<mml:mo>&#x223c;</mml:mo>
</mml:mover>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>. The polynomial decomposition of <inline-formula id="inf76">
<mml:math id="m85">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> (see Eq. <xref ref-type="disp-formula" rid="e2_7">2.7</xref> for the quartic polynomial expression in two variables <inline-formula id="inf77">
<mml:math id="m86">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>) then gives the governing equation for phase transformation, <inline-formula id="inf78">
<mml:math id="m87">
<mml:mrow>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="bold">v</mml:mi>
<mml:mo>&#x22c5;</mml:mo>
<mml:mover accent="true">
<mml:mo>&#x2207;</mml:mo>
<mml:mo>&#x223c;</mml:mo>
</mml:mover>
<mml:mo>&#x2b;</mml:mo>
<mml:msup>
<mml:mover accent="true">
<mml:mo>&#x2207;</mml:mo>
<mml:mo>&#x223c;</mml:mo>
</mml:mover>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mi mathvariant="bold">p</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:munder>
<mml:mo>&#x220f;</mml:mo>
<mml:mi>i</mml:mi>
</mml:munder>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="bold">p</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mover accent="true">
<mml:mi mathvariant="bold">p</mml:mi>
<mml:mo>&#x5e;</mml:mo>
</mml:mover>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, where <inline-formula id="inf79">
<mml:math id="m88">
<mml:mrow>
<mml:msub>
<mml:mover accent="true">
<mml:mi mathvariant="bold">p</mml:mi>
<mml:mo>&#x5e;</mml:mo>
</mml:mover>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> are the critical points at critical points <inline-formula id="inf80">
<mml:math id="m89">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mo>&#x2202;</mml:mo>
<mml:mi>&#x3c8;</mml:mi>
</mml:msub>
<mml:mi>F</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mo>&#x2202;</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:msub>
<mml:mi>F</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf81">
<mml:math id="m90">
<mml:mrow>
<mml:munder>
<mml:mo>&#x220f;</mml:mo>
<mml:mi>i</mml:mi>
</mml:munder>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="bold">p</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mover accent="true">
<mml:mi mathvariant="bold">p</mml:mi>
<mml:mo>&#x5e;</mml:mo>
</mml:mover>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> is the product function that expresses the polynomial <inline-formula id="inf82">
<mml:math id="m91">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>. The phase transformation depends on the polynomial decomposition of the free energy involving all the critical points. Given the significance of all the critical points on growth, kinematics, and interfaces, we explore their behavior in this section.</p>
<sec id="s2-2-1">
<title>2.2.1 Polynomial index theorem and critical points of the <inline-formula id="inf83">
<mml:math id="m92">
<mml:mrow>
<mml:mi mathvariant="italic">F</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="italic">&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi mathvariant="italic">S</mml:mi>
<mml:mi mathvariant="italic">A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>-energy landscape</title>
<p>Since the two-OP model considered in this study is of quartic order in each of the parameters, a proliferation of critical points and a complex energy landscape are expected. Hence, tools that set upper limits on the number and type of critical points are essential to achieving or enhancing tractability. In this section, we formulate an approach tailored to the I-SmA transition, keeping the complex mathematics to a minimum.</p>
<p>Let <inline-formula id="inf84">
<mml:math id="m93">
<mml:mrow>
<mml:mi mathvariant="normal">f</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>x</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>y</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> be a polynomial of degree <inline-formula id="inf85">
<mml:math id="m94">
<mml:mrow>
<mml:mi>d</mml:mi>
<mml:mo>&#x3e;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>. Then, <inline-formula id="inf86">
<mml:math id="m95">
<mml:mrow>
<mml:mi mathvariant="normal">f</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> has a critical point <inline-formula id="inf87">
<mml:math id="m96">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>, if <inline-formula id="inf88">
<mml:math id="m97">
<mml:mrow>
<mml:msub>
<mml:mo>&#x2202;</mml:mo>
<mml:mi>k</mml:mi>
</mml:msub>
<mml:mi mathvariant="normal">f</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, where the following notation <inline-formula id="inf89">
<mml:math id="m98">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mo>&#x2202;</mml:mo>
<mml:mi>k</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> is adopted for partial derivatives of a given function with respect to <italic>k</italic>. The number of critical points <inline-formula id="inf90">
<mml:math id="m99">
<mml:mrow>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mtext>cp</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is then defined by <inline-formula id="inf91">
<mml:math id="m100">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>d</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>. In general, for a given polynomial <inline-formula id="inf92">
<mml:math id="m101">
<mml:mrow>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> in two variables <inline-formula id="inf93">
<mml:math id="m102">
<mml:mrow>
<mml:mi>x</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>y</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> of degree <inline-formula id="inf94">
<mml:math id="m103">
<mml:mrow>
<mml:msub>
<mml:mi>d</mml:mi>
<mml:mi>x</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf95">
<mml:math id="m104">
<mml:mrow>
<mml:msub>
<mml:mi>d</mml:mi>
<mml:mi>y</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, respectively, we expect at most <inline-formula id="inf96">
<mml:math id="m105">
<mml:mrow>
<mml:msub>
<mml:mi>d</mml:mi>
<mml:mi>x</mml:mi>
</mml:msub>
<mml:msub>
<mml:mi>d</mml:mi>
<mml:mi>y</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> critical points (<xref ref-type="bibr" rid="B39">Durfee et al., 1993</xref>). Thus, the computation of <inline-formula id="inf97">
<mml:math id="m106">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf98">
<mml:math id="m107">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> from the solution of the ODEs (see <xref ref-type="sec" rid="s10">Supplementary Appendix A2</xref>) that minimize the free energy <inline-formula id="inf99">
<mml:math id="m108">
<mml:mrow>
<mml:mi>F</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> will yield at most <inline-formula id="inf100">
<mml:math id="m109">
<mml:mrow>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mtext>cp</mml:mtext>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>9</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> critical points and at least one critical point <inline-formula id="inf101">
<mml:math id="m110">
<mml:mrow>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mtext>cp</mml:mtext>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>. These points include the degenerate and nondegenerate points that follow the well-known nondegeneracy criteria, which are as follows:<disp-formula id="e2_10">
<mml:math id="m111">
<mml:mrow>
<mml:mtable columnalign="center">
<mml:mtr>
<mml:mtd>
<mml:mrow>
<mml:msub>
<mml:mo>&#x2202;</mml:mo>
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mi>&#x3c8;</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3e;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:mtd>
<mml:mtd>
<mml:mrow>
<mml:msub>
<mml:mo>&#x2202;</mml:mo>
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mi>S</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3e;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:mtd>
</mml:mtr>
</mml:mtable>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
<label>(2.10)</label>
</disp-formula>
<disp-formula id="e2_11">
<mml:math id="m112">
<mml:mrow>
<mml:mi>det</mml:mi>
<mml:mrow>
<mml:mfenced open="[" close="]" separators="&#x7c;">
<mml:mrow>
<mml:mtable columnalign="center">
<mml:mtr>
<mml:mtd>
<mml:mrow>
<mml:msub>
<mml:mo>&#x2202;</mml:mo>
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mi>&#x3c8;</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mtd>
<mml:mtd>
<mml:mrow>
<mml:msub>
<mml:mo>&#x2202;</mml:mo>
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mi>S</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:mrow>
<mml:msub>
<mml:mo>&#x2202;</mml:mo>
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mi>&#x3c8;</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mtd>
<mml:mtd>
<mml:mrow>
<mml:msub>
<mml:mo>&#x2202;</mml:mo>
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mi>S</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x2260;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>.</mml:mo>
</mml:mrow>
</mml:math>
<label>(2.11)</label>
</disp-formula>
</p>
<p>The number of critical points <inline-formula id="inf102">
<mml:math id="m113">
<mml:mrow>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mtext>cp</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is bound by the Poincar&#xe9;&#x2013;Hopf index theorem (<xref ref-type="bibr" rid="B62">Knill, 2012</xref>). The index <inline-formula id="inf103">
<mml:math id="m114">
<mml:mrow>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mi mathvariant="normal">f</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> of the gradient vector <inline-formula id="inf104">
<mml:math id="m115">
<mml:mrow>
<mml:mo>&#x2207;</mml:mo>
<mml:mi mathvariant="normal">f</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is computed based on the nondegeneracy of all critical points of <inline-formula id="inf105">
<mml:math id="m116">
<mml:mrow>
<mml:mi mathvariant="normal">f</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, which assigns a value of (&#x2b;1) to a maximum or minimum and a value of (<inline-formula id="inf106">
<mml:math id="m117">
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>) to a saddle (<xref ref-type="bibr" rid="B39">Durfee et al., 1993</xref>) in the following definition:<disp-formula id="e2_12">
<mml:math id="m118">
<mml:mrow>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mi mathvariant="normal">f</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mtext>maxima</mml:mtext>
</mml:msub>
<mml:mo>&#xd7;</mml:mo>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mtext>minima</mml:mtext>
</mml:msub>
<mml:mo>&#xd7;</mml:mo>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mtext>saddles</mml:mtext>
</mml:msub>
<mml:mo>&#xd7;</mml:mo>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mtext>maxima</mml:mtext>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mtext>minima</mml:mtext>
</mml:msub>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mtext>saddles</mml:mtext>
</mml:msub>
<mml:mo>.</mml:mo>
</mml:mrow>
</mml:math>
<label>(2.12)</label>
</disp-formula>
</p>
<p>Here, the index of the free-energy polynomial of Equation <xref ref-type="disp-formula" rid="e2_7">2.7</xref> was computed as <inline-formula id="inf107">
<mml:math id="m119">
<mml:mrow>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mi>F</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> in an area homeomorphic to a disk, which importantly puts a cap on the number of saddles <inline-formula id="inf108">
<mml:math id="m120">
<mml:mrow>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mtext>saddles</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>. Thus, from the index theorem, we conclude that saddles play a crucial role in this transformation across various temperature ranges.</p>
</sec>
<sec id="s2-2-2">
<title>2.2.2 Level-set and steepest descent</title>
<p>In addition to the index <inline-formula id="inf109">
<mml:math id="m121">
<mml:mrow>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mi>F</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, the gradient vector <inline-formula id="inf110">
<mml:math id="m122">
<mml:mrow>
<mml:mo>&#x2207;</mml:mo>
<mml:mi mathvariant="normal">f</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> stores the information required to compute the directional derivative of <inline-formula id="inf111">
<mml:math id="m123">
<mml:mrow>
<mml:mi mathvariant="normal">f</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>x</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>y</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> for any direction at any point <inline-formula id="inf112">
<mml:math id="m124">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>, which provides the rate of change in <inline-formula id="inf113">
<mml:math id="m125">
<mml:mrow>
<mml:mi mathvariant="normal">f</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> as it approaches <inline-formula id="inf114">
<mml:math id="m126">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>. This directional derivative is just the inner product of the gradient and the direction of a certain vector <inline-formula id="inf115">
<mml:math id="m127">
<mml:mrow>
<mml:mi mathvariant="bold">u</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>:<disp-formula id="e2_13">
<mml:math id="m128">
<mml:mrow>
<mml:msub>
<mml:mi>D</mml:mi>
<mml:mi>u</mml:mi>
</mml:msub>
<mml:mi mathvariant="normal">f</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>x</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>y</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#x2207;</mml:mo>
<mml:mi mathvariant="normal">f</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>x</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>y</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x22c5;</mml:mo>
<mml:mi mathvariant="bold">u</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>.</mml:mo>
</mml:mrow>
</mml:math>
<label>(2.13)</label>
</disp-formula>
</p>
<p>The gradient then contains the direction of the greatest change of <inline-formula id="inf116">
<mml:math id="m129">
<mml:mrow>
<mml:mi mathvariant="normal">f</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, known as the steepest descent, or ascent, as the opposite direction that may be computed with <inline-formula id="inf117">
<mml:math id="m130">
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mo>&#x2207;</mml:mo>
<mml:mi mathvariant="normal">f</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. Contrary to this, a vector orthogonal to <inline-formula id="inf118">
<mml:math id="m131">
<mml:mrow>
<mml:mo>&#x2207;</mml:mo>
<mml:mi mathvariant="normal">f</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> will point toward a zero change in <inline-formula id="inf119">
<mml:math id="m132">
<mml:mrow>
<mml:mi mathvariant="normal">f</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. These are vectors that lie on the tangent plane and are normal to a surface that can be constructed by the level set of the scalar-valued function <inline-formula id="inf120">
<mml:math id="m133">
<mml:mrow>
<mml:mi mathvariant="normal">f</mml:mi>
<mml:mo>:</mml:mo>
<mml:msup>
<mml:mi mathvariant="double-struck">R</mml:mi>
<mml:mi>n</mml:mi>
</mml:msup>
<mml:mo>&#x2192;</mml:mo>
<mml:mi mathvariant="double-struck">R</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. These are cross-sections of the <inline-formula id="inf121">
<mml:math id="m134">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>x</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>y</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>-frame, individually representing its different levels <italic>c</italic> and containing any real solution of <inline-formula id="inf122">
<mml:math id="m135">
<mml:mrow>
<mml:mi mathvariant="normal">f</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>x</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>y</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>.<disp-formula id="e2_14">
<mml:math id="m136">
<mml:mrow>
<mml:mrow>
<mml:mfenced open="{" close="}" separators="&#x7c;">
<mml:mrow>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>x</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>y</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x2208;</mml:mo>
<mml:msup>
<mml:mi mathvariant="double-struck">R</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mrow>
<mml:mfenced open="|" close="" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="normal">f</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>x</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>y</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>c</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>.</mml:mo>
</mml:mrow>
</mml:math>
<label>(2.14)</label>
</disp-formula>
</p>
<p>The energy landscape <inline-formula id="inf123">
<mml:math id="m137">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> is a surface whose main features are characterized by the shape of the level-set curves, the direction of the steepest descent curves, and the location and nature of the critical points. For a given set of critical point locations, the level sets and steepest descent indicate how and if local minima can be reached. For example, local minima (maxima) on the energy surface are characterized by ellipses, and the steepest descent curves are converging (diverging) splay curves (see <xref ref-type="fig" rid="F2">Figure 2</xref>). This is similar to the minimum energy path (MEP) approach (<xref ref-type="bibr" rid="B73">Massi and Straub, 2001</xref>; <xref ref-type="bibr" rid="B70">Liu et al., 2022</xref>), which seeks to locate and characterize the conformation changes between chemical states based on their relationship with their characteristic energy hypersurface (<xref ref-type="bibr" rid="B41">Fischer and Karplus, 1992</xref>; <xref ref-type="bibr" rid="B132">Wang et al., 1996</xref>; <xref ref-type="bibr" rid="B70">Liu et al., 2022</xref>) that describes the thermodynamic equilibrium and self-assembly process.</p>
<fig id="F2" position="float">
<label>FIGURE 2</label>
<caption>
<p>Schematic of the first family of the free-energy landscape <inline-formula id="inf124">
<mml:math id="m138">
<mml:mrow>
<mml:mi mathvariant="bold-italic">F</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>: level-sets, steepest descent, and polynomial index (<xref ref-type="sec" rid="s2-2-1">Sections 2.2.1</xref>&#x2013;<xref ref-type="sec" rid="s2">2</xref>) and their relationship. The symbols on the top right are kept throughout the paper to designate the nondegenerate points: maximum, minimum, and saddle. The polynomial index is then computed based on the topological charge assigned to these points. The representation of the local steepest gradient field is included for each point. Here, <inline-formula id="inf125">
<mml:math id="m139">
<mml:mrow>
<mml:mi mathvariant="bold-italic">&#x3c7;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> denotes the Euler characteristic of the area enclosed by the level-set curve.</p>
</caption>
<graphic xlink:href="frsfm-04-1359128-g002.tif"/>
</fig>
<p>
<xref ref-type="fig" rid="F2">Figure 2</xref> presents the connection between the polynomial index and the level set and steepest descent curves. By assigning topological charges to the critical points based on their nondegeneracy, the polynomial index is constructed. Each nondegenerate point is then linked to the expected local behavior of the gradient vector.</p>
</sec>
<sec id="s2-2-3">
<title>2.2.3 Lines of curvature and geodesics</title>
<p>The lines of curvature are computed by solving a set of equations defined by the coefficients of the first (<bold>g)</bold> and second (<bold>b)</bold> fundamental forms, as shown in <xref ref-type="sec" rid="s10">Supplementary Appendix A2</xref> (<xref ref-type="bibr" rid="B71">Maekawa, 1996</xref>; <xref ref-type="bibr" rid="B40">Farouki, 1998</xref>). The LOC have been used to describe the relationship between entropy production in membranes and interfaces (<xref ref-type="bibr" rid="B133">Wang et al., 2020</xref>) and the curvature of isotropic&#x2013;smectic interfaces under self-organization (equilibrium) and self-assembly (dynamic) states (<xref ref-type="bibr" rid="B128">Vitral et al., 2019</xref>) using an orthonormal network. These LOC applied to the free-energy landscape describe the change in the order parameters with respect to the arc-length <inline-formula id="inf126">
<mml:math id="m140">
<mml:mrow>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, following the principal direction of the tangent vectors (Eq. A.6) along <inline-formula id="inf127">
<mml:math id="m141">
<mml:mrow>
<mml:mi>&#x3b3;</mml:mi>
<mml:mo>:</mml:mo>
<mml:mi mathvariant="normal">d</mml:mi>
<mml:mi>&#x3b3;</mml:mi>
<mml:mo>/</mml:mo>
<mml:mi mathvariant="normal">d</mml:mi>
<mml:mi>s</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi mathvariant="bold">r</mml:mi>
<mml:mi>k</mml:mi>
</mml:msub>
<mml:mo>/</mml:mo>
<mml:mrow>
<mml:mfenced open="&#x2016;" close="&#x2016;" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi mathvariant="bold">r</mml:mi>
<mml:mi>k</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, where <inline-formula id="inf128">
<mml:math id="m142">
<mml:mrow>
<mml:mi>&#x3b3;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is a line of curvature. Thus,<disp-formula id="e2_15">
<mml:math id="m143">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:mi mathvariant="normal">d</mml:mi>
<mml:mi>&#x3c8;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="normal">d</mml:mi>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>&#x3b7;</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>M</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3ba;</mml:mi>
<mml:mi>A</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mtext>&#x2003;and&#x2003;</mml:mtext>
<mml:mfrac>
<mml:mrow>
<mml:mi mathvariant="normal">d</mml:mi>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="normal">d</mml:mi>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3b7;</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>L</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3ba;</mml:mi>
<mml:mi>E</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
<label>(2.15)</label>
</disp-formula>or<disp-formula id="e2_16">
<mml:math id="m144">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:mi mathvariant="normal">d</mml:mi>
<mml:mi>&#x3c8;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="normal">d</mml:mi>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>&#x3bc;</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>N</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3ba;</mml:mi>
<mml:mi>G</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mtext>&#x2003;and&#x2003;</mml:mtext>
<mml:mfrac>
<mml:mrow>
<mml:mi mathvariant="normal">d</mml:mi>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="normal">d</mml:mi>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3bc;</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>M</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3ba;</mml:mi>
<mml:mi>A</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
<label>(2.16)</label>
</disp-formula>where <inline-formula id="inf129">
<mml:math id="m145">
<mml:mrow>
<mml:mi>&#x3ba;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> represents the principal curvatures <inline-formula id="inf130">
<mml:math id="m146">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3ba;</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>&#x3ba;</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf131">
<mml:math id="m147">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>E</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>A</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>G</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>, and <inline-formula id="inf132">
<mml:math id="m148">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>L</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>M</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>N</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> are the coefficients of the first and second fundamental forms (see <xref ref-type="sec" rid="s10">Supplementary Appendix A2</xref>) and <inline-formula id="inf133">
<mml:math id="m149">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3b7;</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>&#x3bc;</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> are non-zero coefficients defined by<disp-formula id="e2_17">
<mml:math id="m150">
<mml:mrow>
<mml:mtable columnalign="center">
<mml:mtr>
<mml:mtd>
<mml:mi>&#x3b7;</mml:mi>
</mml:mtd>
<mml:mtd>
<mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>E</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>M</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3ba;</mml:mi>
<mml:mi>A</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>2</mml:mn>
<mml:mi>A</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>M</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3ba;</mml:mi>
<mml:mi>A</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>L</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3ba;</mml:mi>
<mml:mi>E</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x2b;</mml:mo>
<mml:mi>G</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>L</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3ba;</mml:mi>
<mml:mi>E</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>/</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:mi>&#x3bc;</mml:mi>
</mml:mtd>
<mml:mtd>
<mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>E</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>N</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3ba;</mml:mi>
<mml:mi>G</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>2</mml:mn>
<mml:mi>A</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>M</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3ba;</mml:mi>
<mml:mi>A</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>N</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3ba;</mml:mi>
<mml:mi>G</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x2b;</mml:mo>
<mml:mi>G</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>M</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3ba;</mml:mi>
<mml:mi>A</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>/</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:mtd>
</mml:mtr>
</mml:mtable>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
<label>(2.17)</label>
</disp-formula>
<disp-formula id="e2_18">
<mml:math id="m151">
<mml:mrow>
<mml:mi mathvariant="normal">d</mml:mi>
<mml:mi>s</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:msqrt>
<mml:mrow>
<mml:mi>E</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="normal">d</mml:mi>
<mml:mi>&#x3c8;</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>2</mml:mn>
<mml:mi>A</mml:mi>
<mml:mi mathvariant="normal">d</mml:mi>
<mml:mi>&#x3c8;</mml:mi>
<mml:mi mathvariant="normal">d</mml:mi>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:mi>G</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="normal">d</mml:mi>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:mrow>
</mml:msqrt>
<mml:mo>.</mml:mo>
</mml:mrow>
</mml:math>
<label>(2.18)</label>
</disp-formula>
</p>
<p>Under our free-energy framework <inline-formula id="inf134">
<mml:math id="m152">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, the geodesic lines indicate the shortest path between two points in the thermodynamic equilibrium state (<xref ref-type="bibr" rid="B36">Do Carmo, 2016</xref>; <xref ref-type="bibr" rid="B133">Wang et al., 2020</xref>), which can also show the self-assembly path connecting the phases expected in the I-SmA energy landscape. This is similar to what is found in the analysis of the geometry of thermodynamic stable states of ideal gases (<xref ref-type="bibr" rid="B100">Quevedo et al., 2008</xref>). This is described by a curve with the smallest arc length connecting two points on a given surface, and it is given by the following equation (<xref ref-type="bibr" rid="B36">Do Carmo, 2016</xref>; <xref ref-type="bibr" rid="B133">Wang et al., 2020</xref>):<disp-formula id="e2_19">
<mml:math id="m153">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:msup>
<mml:mi>d</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:msup>
<mml:mi>y</mml:mi>
<mml:mi>k</mml:mi>
</mml:msup>
</mml:mrow>
<mml:mrow>
<mml:mi>d</mml:mi>
<mml:msup>
<mml:mi>s</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:mrow>
</mml:mfrac>
<mml:mo>&#x2b;</mml:mo>
<mml:msubsup>
<mml:mi mathvariant="normal">&#x393;</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
<mml:mi>k</mml:mi>
</mml:msubsup>
<mml:mfrac>
<mml:mrow>
<mml:mi>d</mml:mi>
<mml:msup>
<mml:mi>y</mml:mi>
<mml:mi>i</mml:mi>
</mml:msup>
</mml:mrow>
<mml:mrow>
<mml:mi>d</mml:mi>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mfrac>
<mml:mrow>
<mml:mi>d</mml:mi>
<mml:msup>
<mml:mi>y</mml:mi>
<mml:mi>j</mml:mi>
</mml:msup>
</mml:mrow>
<mml:mrow>
<mml:mi>d</mml:mi>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
<label>(2.19)</label>
</disp-formula>where <italic>E</italic>, <italic>A</italic>, and <italic>G</italic> are the first fundamental form coefficients, <inline-formula id="inf135">
<mml:math id="m154">
<mml:mrow>
<mml:msubsup>
<mml:mi mathvariant="normal">&#x393;</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
<mml:mi>k</mml:mi>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> are the Christoffel symbols, and the geodesic curve is <inline-formula id="inf136">
<mml:math id="m155">
<mml:mrow>
<mml:msup>
<mml:mi>y</mml:mi>
<mml:mi>i</mml:mi>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> for the <italic>i</italic>th component of the quantities that define the free-energy parametrized surface <inline-formula id="inf137">
<mml:math id="m156">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> (see <xref ref-type="sec" rid="s10">Supplementary Appendix A3</xref>).</p>
</sec>
<sec id="s2-2-4">
<title>2.2.4 Casorati curvature and shape coefficient</title>
<p>In this section, we provide details of the Casorati curvature <inline-formula id="inf138">
<mml:math id="m157">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>C</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> and shape coefficient <inline-formula id="inf139">
<mml:math id="m158">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>S</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> given in the introduction and <xref ref-type="fig" rid="F1">Figure 1</xref> (lower left vertex). A method presented by <xref ref-type="bibr" rid="B133">Wang et al. (2020)</xref> redefines a thermodynamic hypersurface into a Monge shape-curvedness surface patch for the characterization of entropy production in LC membranes and interfaces. As mentioned above, in this paper, we use this methodology to describe the local geometry of the I-SmA phase transition energy landscape using a normalized shape coefficient <inline-formula id="inf140">
<mml:math id="m159">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>S</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> that distinguishes between three primary shapes: cup/cap (spherical), rut/ridge (cylindrical), and saddle, and the Casorati curvature <inline-formula id="inf141">
<mml:math id="m160">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>C</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> for the curvature magnitude measurement (<xref ref-type="bibr" rid="B63">Koenderink and van Doorn, 1992</xref>; <xref ref-type="bibr" rid="B7">Aguilar Gutierrez and Rey, 2018</xref>). This requires the reparameterization of the energy landscape <inline-formula id="inf142">
<mml:math id="m161">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> into a Monge patch (<xref ref-type="bibr" rid="B1">Abbena et al., 2017</xref>) comprised of the shape coefficient <inline-formula id="inf143">
<mml:math id="m162">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>S</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> and Casorati curvature <inline-formula id="inf144">
<mml:math id="m163">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>C</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>.</p>
<p>Classical curvature concepts, in addition to the Casorati curvature <inline-formula id="inf145">
<mml:math id="m164">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>C</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> and shape coefficient <inline-formula id="inf146">
<mml:math id="m165">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>S</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> (<xref ref-type="bibr" rid="B135">Wang et al., 2023a</xref>), are used in the description of the curvedness and shape, such as the (i) mean <inline-formula id="inf147">
<mml:math id="m166">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>H</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>, (ii) deviatoric <inline-formula id="inf148">
<mml:math id="m167">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>D</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>, and (iii) Gaussian <inline-formula id="inf149">
<mml:math id="m168">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>K</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> curvature (see <xref ref-type="sec" rid="s10">Supplementary Appendix A2</xref>), whose information is stored in the surface gradient of the surface unit normal <inline-formula id="inf150">
<mml:math id="m169">
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mo>&#x2207;</mml:mo>
<mml:mi>s</mml:mi>
</mml:msub>
<mml:mi mathvariant="bold">k</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, where <inline-formula id="inf151">
<mml:math id="m170">
<mml:mrow>
<mml:msub>
<mml:mo>&#x2207;</mml:mo>
<mml:mi>s</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="bold">I</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mtext mathvariant="bold">kk</mml:mtext>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#xb7;</mml:mo>
<mml:mo>&#x2207;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> is the surface gradient and <inline-formula id="inf152">
<mml:math id="m171">
<mml:mrow>
<mml:mo>&#x2207;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> is the gradient operator. From this, the symmetric curvature tensor is defined as <inline-formula id="inf153">
<mml:math id="m172">
<mml:mrow>
<mml:mi mathvariant="bold">b</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mo>&#x2207;</mml:mo>
<mml:mi>s</mml:mi>
</mml:msub>
<mml:mi mathvariant="bold">k</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>&#x3ba;</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:msub>
<mml:mi mathvariant="bold">e</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:msub>
<mml:mi mathvariant="bold">e</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi>&#x3ba;</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:msub>
<mml:mi mathvariant="bold">e</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:msub>
<mml:mi mathvariant="bold">e</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, where <inline-formula id="inf154">
<mml:math id="m173">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3ba;</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>&#x3ba;</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> are its eigenvalues (see <xref ref-type="fig" rid="F1">Figure 1II</xref>) characterizing the principal curvatures (<xref ref-type="bibr" rid="B7">Aguilar Gutierrez and Rey, 2018</xref>). The Casorati curvature is defined by <inline-formula id="inf155">
<mml:math id="m174">
<mml:mrow>
<mml:mi>C</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:msqrt>
<mml:mrow>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msubsup>
<mml:mi>&#x3ba;</mml:mi>
<mml:mn>1</mml:mn>
<mml:mn>2</mml:mn>
</mml:msubsup>
<mml:mo>&#x2b;</mml:mo>
<mml:msubsup>
<mml:mi>&#x3ba;</mml:mi>
<mml:mn>2</mml:mn>
<mml:mn>2</mml:mn>
</mml:msubsup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>/</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msqrt>
</mml:mrow>
</mml:math>
</inline-formula>, the mean curvature by <inline-formula id="inf156">
<mml:math id="m175">
<mml:mrow>
<mml:mi>H</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mrow>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3ba;</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi>&#x3ba;</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>/</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, the deviatoric curvature by <inline-formula id="inf157">
<mml:math id="m176">
<mml:mrow>
<mml:mi>D</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mrow>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3ba;</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>&#x3ba;</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>/</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, and the Gaussian curvature by <inline-formula id="inf158">
<mml:math id="m177">
<mml:mrow>
<mml:mi>K</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>&#x3ba;</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:msub>
<mml:mi>&#x3ba;</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, for which the principal curvatures are assumed to follow <inline-formula id="inf159">
<mml:math id="m178">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3ba;</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>&#x2265;</mml:mo>
<mml:msub>
<mml:mi>&#x3ba;</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> (<xref ref-type="bibr" rid="B133">Wang et al., 2020</xref>).</p>
<p>The non-dimensionality of the shape coefficient condenses information that allows it to classify the local shape into simple geometries within the normalized range <inline-formula id="inf160">
<mml:math id="m179">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mo>&#x2208;</mml:mo>
<mml:mrow>
<mml:mfenced open="[" close="]" separators="&#x7c;">
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>,</mml:mo>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>. The Casorati curvature, however, quantifies how curved a surface is (<xref ref-type="bibr" rid="B7">Aguilar Gutierrez and Rey, 2018</xref>; <xref ref-type="bibr" rid="B133">Wang et al., 2020</xref>) (see <xref ref-type="sec" rid="s10">Supplementary Appendix A2</xref>)<disp-formula id="e2_20">
<mml:math id="m180">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>&#x3c0;</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mi>arctan</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:mi>H</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>D</mml:mi>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mtext>&#x2003;and&#x2003;</mml:mtext>
<mml:mi>C</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:msqrt>
<mml:mrow>
<mml:mn>2</mml:mn>
<mml:msup>
<mml:mi>H</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>K</mml:mi>
</mml:mrow>
</mml:msqrt>
<mml:mo>.</mml:mo>
</mml:mrow>
</mml:math>
<label>(2.20)</label>
</disp-formula>
</p>
<p>The primary fundamental shapes are then generalized with <inline-formula id="inf161">
<mml:math id="m181">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mrow>
<mml:mfenced open="{" close="}" separators="&#x7c;">
<mml:mrow>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
<mml:mo>&#xb1;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>/</mml:mo>
<mml:mn>2</mml:mn>
<mml:mo>,</mml:mo>
<mml:mo>&#xb1;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, assigning the values to a saddle (0), a rut <inline-formula id="inf162">
<mml:math id="m182">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>/</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>, a ridge <inline-formula id="inf163">
<mml:math id="m183">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>/</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>, a cup <inline-formula id="inf164">
<mml:math id="m184">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>, and a cap <inline-formula id="inf165">
<mml:math id="m185">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>, where the <inline-formula id="inf166">
<mml:math id="m186">
<mml:mrow>
<mml:mo>&#xb1;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> sign defines if it is a concave-up (negative) or concave-down (positive) patch (see <xref ref-type="fig" rid="F1">Figure 1</xref>.III). It is important to notice that these are primary shapes and a continuous spectrum is contained within the normalized parameter interval. The Casorati curvature varies within <inline-formula id="inf167">
<mml:math id="m187">
<mml:mrow>
<mml:mfenced open="" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mrow>
<mml:mi>C</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mrow>
<mml:mfenced open="[" close="" separators="&#x7c;">
<mml:mrow>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
<mml:mo>&#x2b;</mml:mo>
<mml:mi>&#x221e;</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>, defining a flat surface with no curvature <inline-formula id="inf168">
<mml:math id="m188">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>C</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> and a curved surface <inline-formula id="inf169">
<mml:math id="m189">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>C</mml:mi>
<mml:mo>&#x3e;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>, respectively. <xref ref-type="fig" rid="F1">Figure 1</xref> (III) shows a schematic representation of the Casorati curvature (upper set) and the shape coefficient spectrum (lower set).</p>
</sec>
</sec>
<sec id="s2-3">
<title>2.3 Computational methods</title>
<p>In this work, we sought detailed information of the phase ordering as the quenching degree increases from the highest possible temperature for the existence of the SmA phase. We found that it is possible to classify the ordering and geometry by defining three quenching regimes: shallow quench, middle quench, and deep quench (with three temperatures corresponding to each of them, as listed in <xref ref-type="table" rid="T2">Table 2</xref>). As the degree of quenching increased, the isotropic phase lost stability, while the SmA gained stability, and a number of saddle nodes and supercritical bifurcations emerged at the boundaries of these quench regimes. Given the nonlinearities and OP couplings in the energy density and the differential geometry quantities, high-performance computational techniques were developed, applied, and tested when exact data were available, and high fidelity was demonstrated. Stability, accuracy, and dispersion criteria were implemented according to the standard numerical methods. The validation of our results was established using previous studies (<xref ref-type="bibr" rid="B126">Urban et al., 2005</xref>; <xref ref-type="bibr" rid="B3">Abukhdeir and Rey, 2008</xref>; <xref ref-type="bibr" rid="B18">Coles and Strazielle, 2011</xref>). As mentioned in the introduction, a well-characterized member of the n-cyanobiphenyl family (12CB) has been chosen as the study case for its I-SmA transition behavior (see <xref ref-type="sec" rid="s10">Supplementary Appendix A4</xref>).</p>
<p>The calculation sequence was as follows: (1) level sets and steepest descent curves were obtained with (i) the complete solution of Equation <xref ref-type="disp-formula" rid="e2_7">2.7</xref> at different temperatures for the phases in Equation <xref ref-type="disp-formula" rid="e2_8">2.8</xref> using the stability criteria in <xref ref-type="sec" rid="s2-2-1">Section 2.2.1,</xref> which categorizes the critical points that are bound by the number <inline-formula id="inf170">
<mml:math id="m190">
<mml:mrow>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mrow>
<mml:mtext>cp</mml:mtext>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and the I-SmA LdG free-energy index <inline-formula id="inf171">
<mml:math id="m191">
<mml:mrow>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mi>F</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> in Equation <xref ref-type="disp-formula" rid="e2_12">2.12</xref> and (ii) the orthogonal pair of steepest descent and level-set curves from <xref ref-type="sec" rid="s2-2-2">Section 2.2.2.</xref> (2) Geodesics/LOC calculations in <xref ref-type="sec" rid="s2-2-3">Section 2.2.3</xref> involved coupled nonlinear second-order stiff ODEs, which are numerically unstable depending on the step size taken, especially the system of equations defined by the discretization of the geodesics (Equation <xref ref-type="disp-formula" rid="e2_19">2.19)</xref> (see <xref ref-type="sec" rid="s10">Supplementary Appendix A2</xref>). Boundary conditions must be provided to solve the geodesic. We used the shooting method, which requires the definition of a starting point, chosen in our case study to be the stable isotropic/smectic A phases. In addition to equation stiffness, it is also worth noting that the model is arc-length parametrized, meaning that the arc length was computed for every step. In the LOC case, the sets of Equations 2.15&#x2013;2.16 while seeming analogous, are in reality distinct instances of the principal directions that depend on the arc length. The sign of the proceeding direction at a given point <inline-formula id="inf172">
<mml:math id="m192">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3c8;</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mrow>
<mml:mi>A</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> must be adjusted according to the local surface geometry, meaning that the system solved was switched from one to another depending on the maximum and minimum principal curvatures <inline-formula id="inf173">
<mml:math id="m193">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3ba;</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>&#x3ba;</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> (<xref ref-type="bibr" rid="B40">Farouki, 1998</xref>; <xref ref-type="bibr" rid="B133">Wang et al., 2020</xref>). For this, we followed a very robust algorithm developed for computing LOC (<xref ref-type="bibr" rid="B71">Maekawa, 1996</xref>; <xref ref-type="bibr" rid="B40">Farouki, 1998</xref>), which generates a pair of orthogonal curves at <inline-formula id="inf174">
<mml:math id="m194">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3c8;</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mrow>
<mml:mi>A</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> following the criterion: Equation <xref ref-type="disp-formula" rid="e2_15">2.15</xref> is solved if <inline-formula id="inf175">
<mml:math id="m195">
<mml:mrow>
<mml:mrow>
<mml:mfenced open="|" close="|" separators="&#x7c;">
<mml:mrow>
<mml:mi>L</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3ba;</mml:mi>
<mml:mi>E</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x2265;</mml:mo>
<mml:mrow>
<mml:mfenced open="|" close="|" separators="&#x7c;">
<mml:mrow>
<mml:mi>N</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3ba;</mml:mi>
<mml:mi>G</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, and Equation <xref ref-type="disp-formula" rid="e2_16">2.16</xref> is used otherwise. Then, a curvature network was constructed with the orthogonal LOC by solving the ODE system at a point <inline-formula id="inf176">
<mml:math id="m196">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3c8;</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mrow>
<mml:mi>A</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> along the length defined by the energy landscape for a sufficiently small step-size that balances out resolution, solution stability, and computational time. (3) For obtaining the Casorati and shape coefficient in <xref ref-type="sec" rid="s2-2-4">Section 2.2.4</xref>, we computed the first fundamental forms and the principal curvatures of the free-energy landscape (see <xref ref-type="sec" rid="s10">Supplementary Appendix A2</xref>).</p>
</sec>
</sec>
<sec sec-type="results|discussion" id="s3">
<title>3 Results and discussion</title>
<p>In this paper, we present a complete description and characterization of the energy landscape of the isotropic&#x2013;smectic A transition using a two-non-conserved order parameter version of the Landau&#x2013;de Gennes model for the following reasons: (i) the kinetics of phase transformations for non-conserved order parameters is dependent on stable, metastable, and unstable critical points of free energy (<xref ref-type="bibr" rid="B124">Tuckerman and Bechhoefer, 1992</xref>; <xref ref-type="bibr" rid="B28">De Luca and Rey, 2003</xref>; <xref ref-type="bibr" rid="B26">De Luca and Rey, 2004</xref>); this point is briefly elaborated at the beginning of <xref ref-type="sec" rid="s2-2">Section 2.2</xref>; (ii) in the case of phase transformation by propagating fronts, where a stable phase replaces an unstable phase, non-monotonic ordering structures appear at the interface due to the presence of various critical points (<xref ref-type="bibr" rid="B124">Tuckerman and Bechhoefer, 1992</xref>); (iii) in the case of drop formation of a stable phase in a metastable matrix, one can expect thin film-like layers with intermediate degrees of order between the droplet phase and matrix (<xref ref-type="bibr" rid="B4">Abukhdeir and Rey, 2009a</xref>); (iv) interfacial processes as in a LC drop couple shape-bulk and surface structure-size due to orientational order (<xref ref-type="bibr" rid="B103">Rey, 2000</xref>; <xref ref-type="bibr" rid="B108">Rey and Denn, 2002</xref>; <xref ref-type="bibr" rid="B104">Rey, 2004a</xref>; <xref ref-type="bibr" rid="B105">Rey, 2004b</xref>; <xref ref-type="bibr" rid="B106">Rey, 2006</xref>). In view of these phenomena, we do not neglect metastable and unstable ordering states, such as nematic or plastic (density wave) phases, as previously suggested (<xref ref-type="bibr" rid="B113">Saunders et al., 2007</xref>). How exactly they will manifest themselves under nucleation and growth and spinodal transformation of the isotropic phase into the SmA phase will be examined in future work and is outside the scope of this paper.</p>
<sec id="s3-1">
<title>3.1 Quench zones and critical points and their stability</title>
<p>
<xref ref-type="fig" rid="F3">Figure 3</xref> presents the orientational and positional order phase diagram as a function of temperature T obtained by solving Equation <xref ref-type="disp-formula" rid="e2_7">2.7</xref> with 12CB material parameters (<xref ref-type="sec" rid="s10">Supplementary Appendix A4</xref>). Subscripts on the OPs denote stable (s), unstable (u), and metastable (m); superscripts denote larger <inline-formula id="inf177">
<mml:math id="m197">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mo>&#x2b;</mml:mo>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> and smaller <inline-formula id="inf178">
<mml:math id="m198">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> values. The line style (full, dashed, and doted) identifies the phase (see Eq. <xref ref-type="disp-formula" rid="e2_8">2.8</xref>). The figure frames the three quenches (see <xref ref-type="table" rid="T2">Table 2</xref>) delimited by key temperatures: the deep quench for <inline-formula id="inf179">
<mml:math id="m199">
<mml:mrow>
<mml:mi>T</mml:mi>
<mml:mo>&#x3c;</mml:mo>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>SD</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> (light blue), the middle quench for <inline-formula id="inf180">
<mml:math id="m200">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>SD</mml:mtext>
</mml:msub>
<mml:mo>&#x3c;</mml:mo>
<mml:mi>T</mml:mi>
<mml:mo>&#x3c;</mml:mo>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>NG</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> (light purple), and the shallow quench for <inline-formula id="inf181">
<mml:math id="m201">
<mml:mrow>
<mml:mi>T</mml:mi>
<mml:mo>&#x3c;</mml:mo>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>IL</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> (light red). Here, <inline-formula id="inf182">
<mml:math id="m202">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>SD</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the spinodal decomposition temperature, <inline-formula id="inf183">
<mml:math id="m203">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>NG</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the nucleation and growth temperature, and <inline-formula id="inf184">
<mml:math id="m204">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>IL</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the maximum temperature for the existence of any smectic order phase, as explained below. Using quenching measures, we can characterize the critical point features and determine whether they are stable, unstable, or metastable, depending on the quench zone. To fully characterize the nature of all the sources and sinks at the nematic axis <inline-formula id="inf185">
<mml:math id="m205">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
<mml:mo>&#x3e;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>, we include <inline-formula id="inf186">
<mml:math id="m206">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>&#x3c;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> solutions. These non-physical solutions <inline-formula id="inf187">
<mml:math id="m207">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
<mml:mo>&#x3e;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>&#x3c;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> arise from the mirror symmetry of the free energy <inline-formula id="inf188">
<mml:math id="m208">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mi>&#x3c8;</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>F</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3c8;</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> but assist in characterizing transitions and bifurcations that occur at the <inline-formula id="inf189">
<mml:math id="m209">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> axis. In describing and classifying results, we focus on the I-SmA transition, and the quench depth refers to a temperature decrease from the highest temperature <inline-formula id="inf190">
<mml:math id="m210">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>IL</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> at which the metastable SmA arises. Thus, reference to nucleation and growth mode, NG, indicates the temperature interval in which the isotropic (SmA) phase is metastable (stable), and when referring to spinodal decomposition, SD, the isotropic (SmA) phase is unstable (stable). The challenges regarding the location of the critical points at a given quench, their stability, and the other possible states when considering the entirety of the points, which were introduced at the beginning of the paper, are addressed below.</p>
<fig id="F3" position="float">
<label>FIGURE 3</label>
<caption>
<p>Positional <inline-formula id="inf191">
<mml:math id="m211">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3c8;</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and orientational <inline-formula id="inf192">
<mml:math id="m212">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mrow>
<mml:mi>A</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> order parameters of the isotropic (<italic>Iso</italic>) to SmA phase transition for 12CB as a function of temperature with all the critical points: sub-index <inline-formula id="inf193">
<mml:math id="m213">
<mml:mrow>
<mml:mi mathvariant="bold-italic">n</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>s</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>m</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>u</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> refers to stable, metastable, and unstable, respectively. The three different zones correspond to three different quenches: <bold>(A)</bold> deep quench (DQ), <bold>(B)</bold> middle quench (MQ), and <bold>(C)</bold> shallow quench (SQ). Derived from the LdG model (Eq. <xref ref-type="disp-formula" rid="e2_7">2.7</xref>). The material parameters used (<xref ref-type="bibr" rid="B126">Urban et al., 2005</xref>; <xref ref-type="bibr" rid="B3">Abukhdeir and Rey, 2008</xref>; <xref ref-type="bibr" rid="B18">Coles and Strazielle, 2011</xref>) are as follows: <inline-formula id="inf194">
<mml:math id="m214">
<mml:mrow>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>2</mml:mn>
<mml:mo>&#x22c5;</mml:mo>
<mml:msup>
<mml:mn>10</mml:mn>
<mml:mn>5</mml:mn>
</mml:msup>
<mml:mrow>
<mml:mi mathvariant="normal">J</mml:mi>
<mml:mo>/</mml:mo>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="normal">K</mml:mi>
<mml:mtext>&#xa0;</mml:mtext>
<mml:msup>
<mml:mi mathvariant="normal">m</mml:mi>
<mml:mn>3</mml:mn>
</mml:msup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf195">
<mml:math id="m215">
<mml:mrow>
<mml:mi mathvariant="italic">b</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>2.823</mml:mn>
<mml:mo>&#x22c5;</mml:mo>
<mml:msup>
<mml:mn>10</mml:mn>
<mml:mn>7</mml:mn>
</mml:msup>
<mml:mi mathvariant="normal">J</mml:mi>
<mml:mo>/</mml:mo>
<mml:msup>
<mml:mi mathvariant="normal">m</mml:mi>
<mml:mn>3</mml:mn>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf196">
<mml:math id="m216">
<mml:mrow>
<mml:mi mathvariant="italic">c</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1.972</mml:mn>
<mml:mo>&#x22c5;</mml:mo>
<mml:msup>
<mml:mn>10</mml:mn>
<mml:mn>7</mml:mn>
</mml:msup>
<mml:mi mathvariant="normal">J</mml:mi>
<mml:mo>/</mml:mo>
<mml:msup>
<mml:mi mathvariant="normal">m</mml:mi>
<mml:mn>3</mml:mn>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf197">
<mml:math id="m217">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3b1;</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1.903</mml:mn>
<mml:mo>&#x22c5;</mml:mo>
<mml:msup>
<mml:mn>10</mml:mn>
<mml:mn>6</mml:mn>
</mml:msup>
<mml:mrow>
<mml:mi mathvariant="normal">J</mml:mi>
<mml:mo>/</mml:mo>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi mathvariant="normal">K</mml:mi>
<mml:mtext>&#xa0;</mml:mtext>
<mml:msup>
<mml:mi mathvariant="normal">m</mml:mi>
<mml:mn>3</mml:mn>
</mml:msup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf198">
<mml:math id="m218">
<mml:mrow>
<mml:mi mathvariant="italic">&#x3b2;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>3.956</mml:mn>
<mml:mo>&#x22c5;</mml:mo>
<mml:msup>
<mml:mn>10</mml:mn>
<mml:mn>8</mml:mn>
</mml:msup>
<mml:mi mathvariant="normal">J</mml:mi>
<mml:mo>/</mml:mo>
<mml:msup>
<mml:mi mathvariant="normal">m</mml:mi>
<mml:mn>3</mml:mn>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf199">
<mml:math id="m219">
<mml:mrow>
<mml:mi mathvariant="italic">&#x3b4;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>9.792</mml:mn>
<mml:mo>&#x22c5;</mml:mo>
<mml:msup>
<mml:mn>10</mml:mn>
<mml:mn>6</mml:mn>
</mml:msup>
<mml:mi mathvariant="normal">J</mml:mi>
<mml:mo>/</mml:mo>
<mml:msup>
<mml:mi mathvariant="normal">m</mml:mi>
<mml:mn>3</mml:mn>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf200">
<mml:math id="m220">
<mml:mrow>
<mml:mi mathvariant="italic">e</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1.938</mml:mn>
<mml:mo>&#x22c5;</mml:mo>
<mml:msup>
<mml:mn>10</mml:mn>
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>11</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mi>J</mml:mi>
<mml:mo>/</mml:mo>
<mml:mtext>m</mml:mtext>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf201">
<mml:math id="m221">
<mml:mrow>
<mml:msub>
<mml:mi>b</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>&#x22c5;</mml:mo>
<mml:msup>
<mml:mn>10</mml:mn>
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>11</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mi mathvariant="italic">J</mml:mi>
<mml:mo>/</mml:mo>
<mml:mtext>m</mml:mtext>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf202">
<mml:math id="m222">
<mml:mrow>
<mml:msub>
<mml:mi>b</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>3.334</mml:mn>
<mml:mo>&#x22c5;</mml:mo>
<mml:msup>
<mml:mn>10</mml:mn>
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>30</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mi mathvariant="italic">J</mml:mi>
<mml:mo>/</mml:mo>
<mml:mtext>m</mml:mtext>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf203">
<mml:math id="m223">
<mml:mrow>
<mml:msub>
<mml:mi>d</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>2</mml:mn>
<mml:mi>&#x3c0;</mml:mi>
<mml:mo>/</mml:mo>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mn>3.9</mml:mn>
<mml:mo>&#x22c5;</mml:mo>
<mml:msup>
<mml:mn>10</mml:mn>
<mml:mn>9</mml:mn>
</mml:msup>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mi mathvariant="normal">m</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf204">
<mml:math id="m224">
<mml:mrow>
<mml:msubsup>
<mml:mi>T</mml:mi>
<mml:mrow>
<mml:mi>N</mml:mi>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mo>&#x2a;</mml:mo>
</mml:msubsup>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>322.85</mml:mn>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mi mathvariant="normal">K</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, and <inline-formula id="inf205">
<mml:math id="m225">
<mml:mrow>
<mml:msubsup>
<mml:mi>T</mml:mi>
<mml:mrow>
<mml:mi>A</mml:mi>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mo>&#x2a;</mml:mo>
</mml:msubsup>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>330.5</mml:mn>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mi mathvariant="normal">K</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. The results are fully consistent with <xref ref-type="fig" rid="F4">Figure 4</xref> of <xref ref-type="bibr" rid="B3">Abukhdeir and Rey (2008</xref>). Key temperatures: <inline-formula id="inf206">
<mml:math id="m226">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>SD</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf207">
<mml:math id="m227">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>NG</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, and <inline-formula id="inf208">
<mml:math id="m228">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>IL</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> are listed in <xref ref-type="table" rid="T2">Table 2</xref>.</p>
</caption>
<graphic xlink:href="frsfm-04-1359128-g003.tif"/>
</fig>
<p>The phases in the deep quench (light blue region in <xref ref-type="fig" rid="F2">Figure 2</xref>) to the spinodal decomposition region are the following:<list list-type="simple">
<list-item>
<p>&#x2022; <inline-formula id="inf209">
<mml:math id="m229">
<mml:mrow>
<mml:msubsup>
<mml:mi>S</mml:mi>
<mml:mrow>
<mml:mi>A</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>s</mml:mi>
</mml:mrow>
<mml:mo>&#x2b;</mml:mo>
</mml:msubsup>
<mml:mo>,</mml:mo>
<mml:msubsup>
<mml:mi>&#x3c8;</mml:mi>
<mml:mi>s</mml:mi>
<mml:mo>&#x2b;</mml:mo>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> stable SmA black, continuous lines.</p>
</list-item>
<list-item>
<p>&#x2022; <inline-formula id="inf210">
<mml:math id="m230">
<mml:mrow>
<mml:mi>I</mml:mi>
<mml:mi>s</mml:mi>
<mml:msub>
<mml:mi>o</mml:mi>
<mml:mi>u</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> unstable isotropic, red-dotted line.</p>
</list-item>
<list-item>
<p>&#x2022; <inline-formula id="inf211">
<mml:math id="m231">
<mml:mrow>
<mml:msubsup>
<mml:mi>S</mml:mi>
<mml:mrow>
<mml:mi>A</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>u</mml:mi>
</mml:mrow>
<mml:mo>&#x2212;</mml:mo>
</mml:msubsup>
<mml:mo>,</mml:mo>
<mml:msubsup>
<mml:mi>&#x3c8;</mml:mi>
<mml:mi>u</mml:mi>
<mml:mo>&#x2212;</mml:mo>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> unstable nematic/smectic, black, dashed lines.</p>
</list-item>
<list-item>
<p>&#x2022; <inline-formula id="inf212">
<mml:math id="m232">
<mml:mrow>
<mml:msubsup>
<mml:mi>N</mml:mi>
<mml:mi>u</mml:mi>
<mml:mo>&#x2b;</mml:mo>
</mml:msubsup>
<mml:mo>,</mml:mo>
<mml:msubsup>
<mml:mi>N</mml:mi>
<mml:mi>u</mml:mi>
<mml:mo>&#x2212;</mml:mo>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> unstable nematic, blue, dashed lines.</p>
</list-item>
<list-item>
<p>&#x2022; <inline-formula id="inf213">
<mml:math id="m233">
<mml:mrow>
<mml:msubsup>
<mml:mi>&#x3c8;</mml:mi>
<mml:mi>u</mml:mi>
<mml:mo>&#x2b;</mml:mo>
</mml:msubsup>
<mml:mo>,</mml:mo>
<mml:msubsup>
<mml:mi>&#x3c8;</mml:mi>
<mml:mi>u</mml:mi>
<mml:mo>&#x2212;</mml:mo>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> unstable smectic, gray, dash-dotted lines.</p>
</list-item>
<list-item>
<p>&#x2022; <inline-formula id="inf214">
<mml:math id="m234">
<mml:mrow>
<mml:msubsup>
<mml:mi>P</mml:mi>
<mml:mi>m</mml:mi>
<mml:mo>&#x2b;</mml:mo>
</mml:msubsup>
<mml:mo>,</mml:mo>
<mml:msubsup>
<mml:mi>P</mml:mi>
<mml:mi>m</mml:mi>
<mml:mo>&#x2212;</mml:mo>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> metastable plastic crystal, purple, dash-dotted lines.</p>
</list-item>
</list>
</p>
<p>Here, the SD region exhibits an unstable isotropic state and a stable SmA. In addition, we find a metastable plastic region. This region exists for <inline-formula id="inf215">
<mml:math id="m235">
<mml:mrow>
<mml:mi>T</mml:mi>
<mml:mo>&#x3c;</mml:mo>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>SD</mml:mtext>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>330.6</mml:mn>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mi mathvariant="normal">K</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. Thus, we expect that quenching an isotopic phase into the spinodal region will transform the phase into a stable SmA phase, but the presence of unstable smectic and metastable plastic crystal states introduces complexities to the energy landscape.</p>
<p>The phases in the middle quench (light purple region in <xref ref-type="fig" rid="F2">Figure 2</xref>) to the nucleation and growth region are the following:<list list-type="simple">
<list-item>
<p>&#x2022; <inline-formula id="inf216">
<mml:math id="m236">
<mml:mrow>
<mml:msubsup>
<mml:mi>S</mml:mi>
<mml:mrow>
<mml:mi>A</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>s</mml:mi>
</mml:mrow>
<mml:mo>&#x2b;</mml:mo>
</mml:msubsup>
<mml:mo>,</mml:mo>
<mml:msubsup>
<mml:mi>&#x3c8;</mml:mi>
<mml:mi>s</mml:mi>
<mml:mo>&#x2b;</mml:mo>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> stable SmA, black, continuous lines.</p>
</list-item>
<list-item>
<p>&#x2022; <inline-formula id="inf217">
<mml:math id="m237">
<mml:mrow>
<mml:mi>I</mml:mi>
<mml:mi>s</mml:mi>
<mml:msub>
<mml:mi>o</mml:mi>
<mml:mi>m</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> metastable isotropic, red-dotted line.</p>
</list-item>
<list-item>
<p>&#x2022; <inline-formula id="inf218">
<mml:math id="m238">
<mml:mrow>
<mml:msubsup>
<mml:mi>S</mml:mi>
<mml:mrow>
<mml:mi>A</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>u</mml:mi>
</mml:mrow>
<mml:mo>&#x2212;</mml:mo>
</mml:msubsup>
<mml:mo>,</mml:mo>
<mml:msubsup>
<mml:mi>&#x3c8;</mml:mi>
<mml:mi>u</mml:mi>
<mml:mo>&#x2212;</mml:mo>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> unstable nematic/smectic, black, dashed lines.</p>
</list-item>
<list-item>
<p>&#x2022; <inline-formula id="inf219">
<mml:math id="m239">
<mml:mrow>
<mml:msubsup>
<mml:mi>N</mml:mi>
<mml:mi>u</mml:mi>
<mml:mo>&#x2b;</mml:mo>
</mml:msubsup>
<mml:mo>,</mml:mo>
<mml:msubsup>
<mml:mi>N</mml:mi>
<mml:mi>u</mml:mi>
<mml:mo>&#x2212;</mml:mo>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> unstable nematic, blue, dashed line.</p>
</list-item>
<list-item>
<p>&#x2022; <inline-formula id="inf220">
<mml:math id="m240">
<mml:mrow>
<mml:msubsup>
<mml:mi>&#x3c8;</mml:mi>
<mml:mi>u</mml:mi>
<mml:mo>&#x2b;</mml:mo>
</mml:msubsup>
<mml:mo>,</mml:mo>
<mml:msubsup>
<mml:mi>&#x3c8;</mml:mi>
<mml:mi>u</mml:mi>
<mml:mo>&#x2212;</mml:mo>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> unstable smectic, gray, dash-dotted line.</p>
</list-item>
</list>
</p>
<p>In the ND region, the isotropic state is metastable and SmA is stable, as in the SD region, in addition to the unstable smectic and nematic phases. However, the density wave is no longer present as it vanishes at the temperature <inline-formula id="inf221">
<mml:math id="m241">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>SD</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, and the isotropic phase becomes metastable. This region exists <inline-formula id="inf222">
<mml:math id="m242">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>SD</mml:mtext>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>330.6</mml:mn>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mi mathvariant="normal">K</mml:mi>
<mml:mo>&#x3c;</mml:mo>
<mml:mi>T</mml:mi>
<mml:mo>&#x3c;</mml:mo>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>NG</mml:mtext>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>331.3</mml:mn>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mi mathvariant="normal">K</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. Thus, we expect that quenching an isotropic phase into the NG region will transform the phase into a stable SmA phase by droplet growth.</p>
<p>The phases in the shallow quench (light red region in <xref ref-type="fig" rid="F2">Figure 2</xref>) are the following:<list list-type="simple">
<list-item>
<p>&#x2022; <inline-formula id="inf223">
<mml:math id="m243">
<mml:mrow>
<mml:msubsup>
<mml:mi>S</mml:mi>
<mml:mrow>
<mml:mi>A</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>m</mml:mi>
</mml:mrow>
<mml:mo>&#x2b;</mml:mo>
</mml:msubsup>
<mml:mo>,</mml:mo>
<mml:msubsup>
<mml:mi>&#x3c8;</mml:mi>
<mml:mi>m</mml:mi>
<mml:mo>&#x2b;</mml:mo>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> metastable SmA, black, continuous lines</p>
</list-item>
<list-item>
<p>&#x2022; <inline-formula id="inf224">
<mml:math id="m244">
<mml:mrow>
<mml:mi>I</mml:mi>
<mml:mi>s</mml:mi>
<mml:msub>
<mml:mi>o</mml:mi>
<mml:mi>s</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> stable isotropic, red-dotted line.</p>
</list-item>
<list-item>
<p>&#x2022; <inline-formula id="inf225">
<mml:math id="m245">
<mml:mrow>
<mml:msubsup>
<mml:mi>S</mml:mi>
<mml:mrow>
<mml:mi>A</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>u</mml:mi>
</mml:mrow>
<mml:mo>&#x2212;</mml:mo>
</mml:msubsup>
<mml:mo>,</mml:mo>
<mml:msubsup>
<mml:mi>&#x3c8;</mml:mi>
<mml:mi>u</mml:mi>
<mml:mo>&#x2212;</mml:mo>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> unstable nematic, black, dashed line.</p>
</list-item>
<list-item>
<p>&#x2022; <inline-formula id="inf226">
<mml:math id="m246">
<mml:mrow>
<mml:msubsup>
<mml:mi>&#x3c8;</mml:mi>
<mml:mi>u</mml:mi>
<mml:mo>&#x2b;</mml:mo>
</mml:msubsup>
<mml:mo>,</mml:mo>
<mml:msubsup>
<mml:mi>&#x3c8;</mml:mi>
<mml:mi>u</mml:mi>
<mml:mo>&#x2212;</mml:mo>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> unstable smectic, gray, dash-dotted line.</p>
</list-item>
</list>
</p>
<p>At shallow quenches, the isotropic phase is now stable, while SmA is only metastable. In addition, the unstable smectic state remains, but the nematic loop closes and vanishes at the temperature T<sub>NG</sub>. This region is then defined by <inline-formula id="inf227">
<mml:math id="m247">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>NG</mml:mtext>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>331.3</mml:mn>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mi mathvariant="normal">K</mml:mi>
<mml:mo>&#x3c;</mml:mo>
<mml:mi>T</mml:mi>
<mml:mo>&#x3c;</mml:mo>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>IL</mml:mtext>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>331.85</mml:mn>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mi mathvariant="normal">K</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. Quenching from the NG triggers a phase transition at temperature <inline-formula id="inf228">
<mml:math id="m248">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, as obtained with the solution of Equation <xref ref-type="disp-formula" rid="e2_7">2.7</xref> for a temperature at which <inline-formula id="inf229">
<mml:math id="m249">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>; thus, a temperature higher than the isotropic limit <inline-formula id="inf230">
<mml:math id="m250">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>IL</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> will lead to a disordered state.</p>
<p>The deep quench is characterized by the strong stability and presence of the expected smectic A phase and by a supercritical bifurcation (<xref ref-type="bibr" rid="B86">Oswald and Pieranski, 2005a</xref>) or plastic loop since it belongs to the metastable plastic crystal phase, where mirror symmetry is broken at the temperature <inline-formula id="inf231">
<mml:math id="m251">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>SD</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>. The nematic order effect is seen in the orientational order parameter diagram, with the presence of the nematic loop in the deep- and middle-quench zones. This marks the entrance of the shallow quench and the stability change of the SmA phase. A summary of these key temperatures and regions is given in <xref ref-type="table" rid="T1">Table 1</xref>.</p>
<table-wrap id="T1" position="float">
<label>TABLE 1</label>
<caption>
<p>Summary of stable, unstable, and metastable states in each quench zone and their key temperatures as presented in <xref ref-type="fig" rid="F2">Figure 2</xref>.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th align="left"/>
<th align="left">Deep Quench (DQ): Spinodal Decomposition for the Isotropic phase.</th>
<th align="left">Middle Quench (MQ): Nucleation and Growth for the Isotropic phase.</th>
<th align="left">Shallow Quench (SQ): Limit for the Isotropic phase.</th>
<th align="left">Zero quench</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="left">Primary roots</td>
<td align="left">Stable smectic A Unstable isotropic</td>
<td align="left">Stable smectic A Metastable isotropic</td>
<td align="left">Metastable smectic A Stable isotropic</td>
<td align="left">Stable isotropic</td>
</tr>
<tr>
<td align="left">Secondary roots</td>
<td align="left">Metastable plastic crystal Unstable nematic Unstable smectic A</td>
<td align="left">Unstable nematic</td>
<td align="left">NA</td>
<td align="left">NA</td>
</tr>
<tr>
<td align="left">Transition temperature</td>
<td align="left">
<inline-formula id="inf232">
<mml:math id="m252">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>SD</mml:mtext>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>330.6</mml:mn>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mi mathvariant="normal">K</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td align="left">
<inline-formula id="inf233">
<mml:math id="m253">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>NG</mml:mtext>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>331.3</mml:mn>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mi mathvariant="normal">K</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td align="left">
<inline-formula id="inf234">
<mml:math id="m254">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>IL</mml:mtext>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>331.85</mml:mn>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mi mathvariant="normal">K</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td align="left"/>
</tr>
</tbody>
</table>
</table-wrap>
<p>The computed energy landscape has a critical root population that decreases exactly as predicted by the polynomial index theorem (Eq. <xref ref-type="disp-formula" rid="e2_12">2.12</xref>; <xref ref-type="fig" rid="F2">Figure 2</xref>) as the temperature increases. This is summarized in <xref ref-type="table" rid="T2">Table 2</xref>, where the number of nondegenerate points is included along with their type and the index value for each quench zone (see (2.12)).</p>
<table-wrap id="T2" position="float">
<label>TABLE 2</label>
<caption>
<p>Number and types of critical points on the phase diagram for different temperatures and their indexes; the three cases presented in <xref ref-type="fig" rid="F2">Figure 2</xref> correspond to the three quench zones; and one for the complete isotropic phase transition. The symbol style used for each of them is kept constant throughout the paper. Numerical results are in exact agreement with the polynomial index theorem (Eq. <xref ref-type="disp-formula" rid="e2_12">2.12</xref>).</p>
</caption>
<table>
<thead valign="top">
<tr>
<th align="left">T [K]</th>
<th align="left">Maxima</th>
<th align="left">Minima</th>
<th align="left">Number of saddles</th>
<th align="left">Index <inline-formula id="inf235">
<mml:math id="m255">
<mml:mrow>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mi>F</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>
</th>
<th align="left">Zone</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="left">A&#x2013;330</td>
<td align="left">1</td>
<td align="left">4</td>
<td align="left">4</td>
<td align="left">1</td>
<td align="left">Deep quench (spinodal decomposition, SD)</td>
</tr>
<tr>
<td align="left">B&#x2013;331</td>
<td align="left">1</td>
<td align="left">3</td>
<td align="left">3</td>
<td align="left">1</td>
<td align="left">Middle quench (nucleation and growth, NG)</td>
</tr>
<tr>
<td align="left">C&#x2013;331.85</td>
<td align="left">0</td>
<td align="left">3</td>
<td align="left">2</td>
<td align="left">1</td>
<td align="left">Shallow quench</td>
</tr>
<tr>
<td align="left">
<inline-formula id="inf236">
<mml:math id="m256">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> &#x3e; <inline-formula id="inf237">
<mml:math id="m257">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>IL</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td align="left">0</td>
<td align="left">1</td>
<td align="left">0</td>
<td align="left">1</td>
<td align="left"/>
</tr>
<tr>
<td align="left">Symbol in <xref ref-type="fig" rid="F3">Figure 3</xref>
</td>
<td align="left">&#x394;</td>
<td align="left">&#x25cb;</td>
<td align="left">&#x25a1;</td>
<td align="left"/>
<td align="left"/>
</tr>
</tbody>
</table>
</table-wrap>
<p>The bounds in Equations 2.10&#x2013;2.11 provide stability boundaries that do not necessarily mean a phase transition line but present the possible real physical phases that can be displayed within that range given the first-order nature of this transition (<xref ref-type="bibr" rid="B82">Mukherjee et al., 2001</xref>). Thus, the phase transition line was found by looking for a metastability&#x2013;stability exchange of the I-SmA phases using level sets and computing the temperature at which both phases present the same energy level. This temperature, <inline-formula id="inf238">
<mml:math id="m258">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, happened to be around the limit at which the nematic unstable loop vanished, going from deep quench (DQ) to middle quench (MQ), as seen in <xref ref-type="fig" rid="F2">Figure 2</xref>, agreeing with the experimental transition temperature of 331.3&#xa0;K (<xref ref-type="bibr" rid="B18">Coles and Strazielle, 2011</xref>).</p>
<p>
<xref ref-type="fig" rid="F4">Figure 4</xref> represents the <inline-formula id="inf239">
<mml:math id="m259">
<mml:mrow>
<mml:mover accent="true">
<mml:mi>F</mml:mi>
<mml:mo>&#x5e;</mml:mo>
</mml:mover>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> dimensionless free-energy landscape 2D projection for the LdG model using the parameters in <xref ref-type="fig" rid="F2">Figure 2</xref>, with its corresponding 3D energy landscape at a given temperature. We have expanded the <inline-formula id="inf240">
<mml:math id="m260">
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>-axis to negative values to emphasize the mirror symmetry along the positional order. The level-set curves have been included with all the critical points at three different temperatures in <xref ref-type="table" rid="T2">Table 2</xref>, which are representative of each quench zone. As quench depth decreases (temperature increases), the critical point population density decreases <inline-formula id="inf241">
<mml:math id="m261">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mtext>cp</mml:mtext>
</mml:msub>
<mml:mo>&#x2193;</mml:mo>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>, which corresponds to the merging and vanishing of the nondegenerate points, as seen in <xref ref-type="fig" rid="F2">Figure 2</xref>. Increasing <inline-formula id="inf242">
<mml:math id="m262">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> from the spinodal region, the plastic loop eventually converges at a supercritical bifurcation (<xref ref-type="bibr" rid="B50">Han and Rey, 1993</xref>; <xref ref-type="bibr" rid="B102">Rey, 1995</xref>) at <inline-formula id="inf243">
<mml:math id="m263">
<mml:mrow>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mtext>SD</mml:mtext>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, which is a process that replaces two minima <inline-formula id="inf244">
<mml:math id="m264">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> and one saddle <inline-formula id="inf245">
<mml:math id="m265">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> with a single minimum <inline-formula id="inf246">
<mml:math id="m266">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>. Entering the NG quench region with three saddles and four nodes (i.e., maximum or minimum), a further increase in <inline-formula id="inf247">
<mml:math id="m267">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> eventually leads to saddle-node bifurcation (<xref ref-type="bibr" rid="B102">Rey, 1995</xref>), with the elimination of a nematic saddle and a node. The shallow quench now has three minima and two saddles, which, after another saddle-node bifurcation of smectic phases, eventually leads to a planar surface with no order. It is noteworthy that the sequence of saddle number elimination as <inline-formula id="inf248">
<mml:math id="m268">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> increases and order decreases is multi-stepwise: <inline-formula id="inf249">
<mml:math id="m269">
<mml:mrow>
<mml:mn>4</mml:mn>
<mml:mo>&#x2192;</mml:mo>
<mml:mn>3</mml:mn>
<mml:mo>&#x2192;</mml:mo>
<mml:mn>2</mml:mn>
<mml:mo>&#x2192;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>. Likewise, the sequence of local minima elimination as <inline-formula id="inf250">
<mml:math id="m270">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> increases and order multi-stepwise decreases is <inline-formula id="inf251">
<mml:math id="m271">
<mml:mrow>
<mml:mn>4</mml:mn>
<mml:mo>&#x2192;</mml:mo>
<mml:mn>3</mml:mn>
<mml:mo>&#x2192;</mml:mo>
<mml:mn>3</mml:mn>
<mml:mo>&#x2192;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>. On the other hand, the elimination of the maxima follows a single step: <inline-formula id="inf252">
<mml:math id="m272">
<mml:mrow>
<mml:mn>1</mml:mn>
<mml:mo>&#x2192;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>&#x2192;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>&#x2192;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>. This shows that for shallow quench, local maxima play no role, and for deep quench, saddles and minima are equal in number.</p>
<fig id="F4" position="float">
<label>FIGURE 4</label>
<caption>
<p>Energy landscape constructed with the positional <inline-formula id="inf253">
<mml:math id="m273">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> and orientational <inline-formula id="inf254">
<mml:math id="m274">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> order parameters of the I-SmA phase transition for 12CB (see <xref ref-type="fig" rid="F3">Figure 3</xref>, and <xref ref-type="sec" rid="s10">Supplementary Appendix A4</xref>) and the dimensionless free energy <inline-formula id="inf255">
<mml:math id="m275">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mover accent="true">
<mml:mi>F</mml:mi>
<mml:mo>&#x5e;</mml:mo>
</mml:mover>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> at the three temperatures <bold>(A&#x2013;C)</bold> included in <xref ref-type="table" rid="T2">Table 2</xref>. The LDG model was nondimensionalized for visualization purposes with the orientational temperature-dependent parameter <italic>a</italic>. The critical points are included for each scenario. The temperatures, which belong to each quench zone, and the markers are listed in <xref ref-type="table" rid="T2">Table 2,</xref> indicating the stability type assigned based on the criteria given by Eqs <xref ref-type="disp-formula" rid="e2_10">2.10</xref>&#x2013;<xref ref-type="disp-formula" rid="e2_11">2.11</xref>. The black and red markers indicate the primary roots: SmA and isotropic state, respectively. <inline-formula id="inf256">
<mml:math id="m276">
<mml:mrow>
<mml:mover accent="true">
<mml:mi>F</mml:mi>
<mml:mo>&#x5e;</mml:mo>
</mml:mover>
</mml:mrow>
</mml:math>
</inline-formula> ranges in the color bar are kept the same throughout the paper. The 3D plots of the same energy landscapes are included right below each one for the same temperatures for a more complete view.</p>
</caption>
<graphic xlink:href="frsfm-04-1359128-g004.tif"/>
</fig>
</sec>
<sec id="s3-2">
<title>3.2 Level-set curves and steepest descent lines</title>
<p>
<xref ref-type="fig" rid="F5">Figure 5</xref> presents the level-set curves and the steepest descent lines (see <xref ref-type="sec" rid="s2-2-2">Section 2.2.2</xref>), including the critical points projected on the dimensionless energy surface <inline-formula id="inf257">
<mml:math id="m277">
<mml:mrow>
<mml:mover accent="true">
<mml:mi>F</mml:mi>
<mml:mo>&#x5e;</mml:mo>
</mml:mover>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> for the three quench regimes listed in <xref ref-type="table" rid="T1">Table 1</xref>. The blue (red) region corresponds to lower (higher) energy. It is seen that the steepest descent and level-set curves are members of an orthogonal family, where the level-set curves indicate a constant free-energy value and the steepest descent presents a path leading to primary roots that come from the minimization of the LdG model. The pair of roots (isotropic and smectic A phases) are divided by a set of maximums, minima, and saddles that discretely disappear as the quench depth decreases. In the deep-quench region, (A) the main feature is the family of elliptical rings around the stable SmA state (black dot), whose largest axes are oriented toward the unstable phases. The level sets identify the nematic saddle and nematic maximum as well as the metastable plastic root. The principal steepest descent line connects the unstable smectic (white square) with the stable isotropic state (black dot) and defines a collecting manifold with nearly horizontal, constant <inline-formula id="inf258">
<mml:math id="m278">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> values. In the middle quench, nucleation, and growth, (B) the region of elliptical trajectories surrounding the SmA phase moves toward the isotropic state, causing the horizontal band of steepest descent lines to narrow. Furthermore, the steepest descent inverted <italic>L</italic> shows how energy states near the energetically high region end at the isotropic state (red dot). In (C), the metastability of SmA is shown by a lack of elliptical trajectories and the stability of the isotropic state.</p>
<fig id="F5" position="float">
<label>FIGURE 5</label>
<caption>
<p>
<inline-formula id="inf259">
<mml:math id="m279">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>-energy landscape with level-set, steepest descent curves, and critical points for <inline-formula id="inf260">
<mml:math id="m280">
<mml:mrow>
<mml:mn>0</mml:mn>
<mml:mo>&#x2264;</mml:mo>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>&#x2264;</mml:mo>
<mml:mn>0.3</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf261">
<mml:math id="m281">
<mml:mrow>
<mml:mn>0</mml:mn>
<mml:mo>&#x2264;</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
<mml:mo>&#x2264;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, and the dimensionless free energy <inline-formula id="inf262">
<mml:math id="m282">
<mml:mrow>
<mml:mover accent="true">
<mml:mi>F</mml:mi>
<mml:mo>&#x5e;</mml:mo>
</mml:mover>
</mml:mrow>
</mml:math>
</inline-formula> levels, at the same parameters from <xref ref-type="fig" rid="F3">FIGURE 3</xref>. The three temperatures used <bold>(A&#x2013;C)</bold> and the marker styles are listed in <xref ref-type="table" rid="T2">Table 2</xref>. The continuous black lines go along the steepest descent and perpendicularly to the level-set curves that mark different free-energy levels. The black- and red-filled markers correspond to the SmA and isotropic states, respectively. A zoomed-in plot around the SmA is included for (b).</p>
</caption>
<graphic xlink:href="frsfm-04-1359128-g005.tif"/>
</fig>
<p>In partial summary, the level-sets/steepest descent lines show the main features of the energy landscape; the number, location, and type of critical points; and the basin of attraction of SmA under spinodal and nucleation and growth conditions.</p>
</sec>
<sec id="s3-3">
<title>3.3 Lines of curvature and geodesics</title>
<p>
<xref ref-type="fig" rid="F6">Figure 6</xref> shows the 2D projection and 3D plot of the LOC network on the energy landscape <inline-formula id="inf263">
<mml:math id="m283">
<mml:mrow>
<mml:mover accent="true">
<mml:mi>F</mml:mi>
<mml:mo>&#x5e;</mml:mo>
</mml:mover>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> generated with the algorithm described in <xref ref-type="sec" rid="s2">Section 2</xref>. It consists of orthogonal curve pairs that follow the minimum and maximum curvatures (cyan and magenta, respectively) at a given point on the surface. It also includes the primary roots shown in <xref ref-type="table" rid="T1">Table 1</xref>.</p>
<fig id="F6" position="float">
<label>FIGURE 6</label>
<caption>
<p>Lines of curvature projected on the <inline-formula id="inf264">
<mml:math id="m284">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>-energy landscape at the same three temperatures <bold>(A&#x2013;C)</bold> and with the marker styles listed in <xref ref-type="table" rid="T2">Table 2</xref>. The parameters used are listed in <xref ref-type="fig" rid="F3">Figure 3</xref>. The circular marker presents the SmA state at those temperatures. The cyan and magenta orthogonal network of LOC corresponds to the minimum <inline-formula id="inf265">
<mml:math id="m285">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3ba;</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and maximum <inline-formula id="inf266">
<mml:math id="m286">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3ba;</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> curvatures, respectively.</p>
</caption>
<graphic xlink:href="frsfm-04-1359128-g006.tif"/>
</fig>
<p>The maximum curvature (magenta) lines on the top left and bottom right closely follow the energy contours, corresponding to high <inline-formula id="inf267">
<mml:math id="m287">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>&#x2013;low <inline-formula id="inf268">
<mml:math id="m288">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> and <italic>vice versa</italic>, while along the downward diagonal, they funnel out consistently in accordance with the energy landscape. The minimum curvature (cyan) lines form a set of nearly parallel <italic>L</italic>-lines, which are nearly vertical along the nematic axis and nearly horizontal close to the smectic axis. This is consistent with the fact that most critical points are around the diagonal region, as shown in <xref ref-type="fig" rid="F5">Figure 5</xref>. Furthermore, since the energy surface envelope is roughly a concave-up expanding cylinder with flat edges, it follows that we must find circular curvature lines (as in the circular lines of a cylinder) and diverging straight lines (like in an expanding cone).</p>
<p>
<xref ref-type="fig" rid="F7">Figure 7</xref> shows the projected geodesic lines of the energy landscape, computed by solving Equation <xref ref-type="disp-formula" rid="e2_19">2.19</xref> for the temperatures belonging to the three quench zones listed in <xref ref-type="table" rid="T1">Table 1</xref> with the method provided in <xref ref-type="sec" rid="s10">Supplementary Appendix A2</xref>. The geodesic family origin is the isotropic state that changes stability from unstable (A) to metastable (B) to stable (C), as seen in <xref ref-type="fig" rid="F4">Figure 4</xref>. The lines minimize the path length and are therefore significant directions for phase changes. These lines show an expanding funnel whose centerline (purple) connects the two primary I-SmA phases. This line, which resembles the MEP introduced in <xref ref-type="sec" rid="s2-2-2">Section 2.2.2</xref>, follows the minimum-curvature tendency designated in <xref ref-type="fig" rid="F6">Figure 6</xref> by the magenta lines. In addition, this phase-connecting geodesic becomes straighter as the depth quench is increased, achieving essentially a straight line in (A), and it starts to bend in the direction of a greater change in energy as the shallow quench (C) is reached. Another important observation is that the change in shape, direction, and bending are reflected in the LOC as the quench regime changes, as opposed to what the geodesics show in this figure.</p>
<fig id="F7" position="float">
<label>FIGURE 7</label>
<caption>
<p>Geodesics superimposed on the <inline-formula id="inf269">
<mml:math id="m289">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>-energy landscape at the same three temperatures <bold>(A&#x2013;C)</bold> and with the marker styles listed in <xref ref-type="table" rid="T2">Table 2</xref>. The parameters used are listed in <xref ref-type="fig" rid="F3">Figure 3</xref>. The circular marker presents the SmA state at those temperatures. All lines, dashed and continuous, are geodesics; however, the continuous line connects both the isotropic and SmA primary roots.</p>
</caption>
<graphic xlink:href="frsfm-04-1359128-g007.tif"/>
</fig>
</sec>
<sec id="s3-4">
<title>3.4 Shape coefficient and Casorati curvature</title>
<p>
<xref ref-type="fig" rid="F8">Figure 8</xref> shows the Casorati <inline-formula id="inf270">
<mml:math id="m290">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>C</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> curvature (top) and shape <inline-formula id="inf271">
<mml:math id="m291">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>S</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> coefficient (bottom) heatmaps as a function of the OPs <inline-formula id="inf272">
<mml:math id="m292">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> coordinates. The Casorati curvature and shape coefficient were computed using Equation <xref ref-type="disp-formula" rid="e2_20">2.20</xref> and the definitions in <xref ref-type="sec" rid="s10">Supplementary Appendix A2</xref> at the temperatures listed in <xref ref-type="table" rid="T1">Table 1</xref> for the three representative quench zones. The primary root that corresponds to the most stable phase at each temperature was included in the bottom-right corner of each plot containing the Casorati curvature and shape coefficient values at those coordinates.</p>
<fig id="F8" position="float">
<label>FIGURE 8</label>
<caption>
<p>Casorati curvature <inline-formula id="inf273">
<mml:math id="m293">
<mml:mrow>
<mml:mi mathvariant="bold-italic">C</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> (upper set) and <bold>s</bold>hape <inline-formula id="inf274">
<mml:math id="m294">
<mml:mrow>
<mml:mi mathvariant="bold-italic">S</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> coefficient (lower set) heatmaps as a function of <inline-formula id="inf275">
<mml:math id="m295">
<mml:mrow>
<mml:mi mathvariant="bold-italic">&#x3c8;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf276">
<mml:math id="m296">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> at the same three temperatures <bold>(A&#x2013;C)</bold> and with the marker styles listed in <xref ref-type="table" rid="T2">Table 2</xref>. The parameters used are listed in <xref ref-type="fig" rid="F3">Figure 3</xref>. The coordinates on the bottom-right corner correspond to the SmA phase (black dot) with their Casorati and shape coefficient values at each temperature.</p>
</caption>
<graphic xlink:href="frsfm-04-1359128-g008.tif"/>
</fig>
<p>The Casorati curvature presents major activity along the zone where the critical points move as the temperature varies. It can be noted that the Casorati curvature decreases as the quench depth decreases, which follows a trend toward the isotropic transition, where both order parameters are zero and the energy surface is planar and, hence, <inline-formula id="inf277">
<mml:math id="m297">
<mml:mrow>
<mml:mi>C</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>. The crucial feature of the computed <inline-formula id="inf278">
<mml:math id="m298">
<mml:mrow>
<mml:mi>C</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>C</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> is the presence of a bent vertical tubular region of higher <inline-formula id="inf279">
<mml:math id="m299">
<mml:mrow>
<mml:mi>C</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> values in a matrix of low <inline-formula id="inf280">
<mml:math id="m300">
<mml:mrow>
<mml:mi>C</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. Within this high <inline-formula id="inf281">
<mml:math id="m301">
<mml:mrow>
<mml:mi>C</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> tube, the curvedness increases as we move toward and beyond the SmA phase in the SD and NG zones. In the shallow quench, the increase is attenuated as the energy surface evolves toward planarity. Interestingly, an approximate scaling for the high <inline-formula id="inf282">
<mml:math id="m302">
<mml:mrow>
<mml:mi>C</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> tube is a power law <inline-formula id="inf283">
<mml:math id="m303">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
<mml:mo>&#x2248;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>0.1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>; <inline-formula id="inf284">
<mml:math id="m304">
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mo>&#x2248;</mml:mo>
<mml:mn>0.01</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>; more accurate fittings require parameters, but the important point is that smectic ordering produces a large increase in orientational ordering along the high <inline-formula id="inf285">
<mml:math id="m305">
<mml:mrow>
<mml:mi>C</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> tube. Furthermore, if we compare <xref ref-type="fig" rid="F8">Figure 8A</xref> (top right) with the energy landscape of <xref ref-type="fig" rid="F5">Figure 5</xref> (bottom left) in the spinodal mode, we see that the axis of the high <inline-formula id="inf286">
<mml:math id="m306">
<mml:mrow>
<mml:mi>C</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> tube follows the steepest descent line that starts at the metastable plastic crystal and traverses the stable SmA phase to end at the higher energy states <inline-formula id="inf287">
<mml:math id="m307">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>&#x2248;</mml:mo>
<mml:mn>0.3</mml:mn>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
<mml:mo>&#x2248;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>. Hence, the <inline-formula id="inf288">
<mml:math id="m308">
<mml:mrow>
<mml:mi>C</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> tube is another distinguishing feature of the energy landscape.</p>
<p>We now search for the distinguished feature(s) of the shape <inline-formula id="inf289">
<mml:math id="m309">
<mml:mrow>
<mml:mi>S</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> coefficient. <xref ref-type="fig" rid="F8">Figure 8</xref> (bottom) shows that the shape coefficient associated with the local minima at each temperature does not reach <inline-formula id="inf290">
<mml:math id="m310">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> or a perfect ideal cup shape, as previously observed from the findings of <xref ref-type="bibr" rid="B133">Wang et al. (2020</xref>). The reason behind this is the energy surface anisotropy that originates from the LdG polynomial structure (Eq. <xref ref-type="disp-formula" rid="e2_7">2.7</xref>), as observed already in non-circular level-set curves (see elliptical curves surrounding minima in <xref ref-type="fig" rid="F4">Figure 4</xref>). In addition, we noticed that for stability, the shape coefficient follows a trend, assigning the local minima to a surface lying between a cup <inline-formula id="inf291">
<mml:math id="m311">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> and a rut <inline-formula id="inf292">
<mml:math id="m312">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>0.5</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> shape; for completeness, we note that the intermediate value <inline-formula id="inf293">
<mml:math id="m313">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>0.75</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> is usually denoted as a trough. This shape condition of local minima <inline-formula id="inf294">
<mml:math id="m314">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>&#x3c;</mml:mo>
<mml:mi>S</mml:mi>
<mml:mo>&#x3c;</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>0.5</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> corresponds to a shape with small and large principal curvatures. This behavior is also supported by the LOC network seen in <xref ref-type="fig" rid="F6">Figure 6</xref>. In more quantitative detail, <xref ref-type="fig" rid="F8">Figure 8</xref> shows that the correspondence between the OPs of the local minimum and the energy surface shape is as follows:<list list-type="simple">
<list-item>
<p>&#x2022; Spinodal decomposition mode: <inline-formula id="inf295">
<mml:math id="m315">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0.2</mml:mn>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0.74</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>0.55</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, and the shape is between a rut and a trough.</p>
</list-item>
<list-item>
<p>&#x2022; Nucleation and growth mode: <inline-formula id="inf296">
<mml:math id="m316">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0.16</mml:mn>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0.66</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>0.54</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, and the shape is between a rut and a trough.</p>
</list-item>
<list-item>
<p>&#x2022; Shallow quench mode: <inline-formula id="inf297">
<mml:math id="m317">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>0.84</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, and the shape is between a trough and a cup.</p>
</list-item>
</list>
</p>
<p>
<xref ref-type="fig" rid="F8">Figure 8</xref> (bottom) shows another distinguishing feature of the shape index, with the blue concave-up (ridge) domain describing a bent channel that narrows and widens as the order increases. The outer red domains indicate unstable or concave-down (cap) states, and the green boundaries are saddle-like shapes. Hence, the shape landscape for smectic phases follows the previously established rules (<xref ref-type="bibr" rid="B133">Wang et al., 2020</xref>) of shape coexistence, where moving from left to right in each panel from <xref ref-type="fig" rid="F8">Figure 8</xref> (bottom), we find the following:<disp-formula id="equ1">
<mml:math id="m318">
<mml:mrow>
<mml:munder>
<mml:munder>
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0.75</mml:mn>
<mml:mo>&#x2192;</mml:mo>
<mml:mi>S</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0.5</mml:mn>
</mml:mrow>
<mml:mo stretchy="true">&#x23df;</mml:mo>
</mml:munder>
<mml:mrow>
<mml:mtext>concave&#xa0;down</mml:mtext>
</mml:mrow>
</mml:munder>
<mml:mo>&#x2192;</mml:mo>
<mml:munder>
<mml:munder>
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:mo stretchy="true">&#x23df;</mml:mo>
</mml:munder>
<mml:mtext>saddle</mml:mtext>
</mml:munder>
<mml:mo>&#x2192;</mml:mo>
<mml:mi>S</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:munder>
<mml:munder>
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>0.5</mml:mn>
<mml:mo>&#x2192;</mml:mo>
<mml:mi>S</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>0.75</mml:mn>
<mml:mo>&#x2192;</mml:mo>
<mml:mi>S</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>0.5</mml:mn>
</mml:mrow>
<mml:mo stretchy="true">&#x23df;</mml:mo>
</mml:munder>
<mml:mrow>
<mml:mtext>concave</mml:mtext>
<mml:mo>&#x2212;</mml:mo>
<mml:mtext>up</mml:mtext>
</mml:mrow>
</mml:munder>
<mml:mo>&#x2192;</mml:mo>
<mml:munder>
<mml:munder>
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:mo stretchy="true">&#x23df;</mml:mo>
</mml:munder>
<mml:mtext>saddle</mml:mtext>
</mml:munder>
<mml:mo>&#x2192;</mml:mo>
<mml:munder>
<mml:munder>
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0.5</mml:mn>
<mml:mo>&#x2192;</mml:mo>
<mml:mi>S</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0.75</mml:mn>
</mml:mrow>
<mml:mo stretchy="true">&#x23df;</mml:mo>
</mml:munder>
<mml:mrow>
<mml:mtext>concave&#xa0;down</mml:mtext>
</mml:mrow>
</mml:munder>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
</disp-formula>where saddles are needed to separate minima from maxima, which is in agreement with the polynomial theorem for critical point index <inline-formula id="inf298">
<mml:math id="m319">
<mml:mrow>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mi>F</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> given in <xref ref-type="table" rid="T2">Table 2</xref> and Equation <xref ref-type="disp-formula" rid="e2_12">2.12</xref>.</p>
<p>
<xref ref-type="fig" rid="F9">Figure 9</xref> takes the geodesics presented in <xref ref-type="fig" rid="F7">Figure 7</xref> and projects them on the Casorati and shape coefficient heatmaps from <xref ref-type="fig" rid="F8">Figure 8</xref> for each temperature according to the three quenching zones listed in <xref ref-type="table" rid="T1">Table 1</xref>.</p>
<fig id="F9" position="float">
<label>FIGURE 9</label>
<caption>
<p>Geodesics projected on the Casorati curvature <inline-formula id="inf303">
<mml:math id="m325">
<mml:mrow>
<mml:mi mathvariant="bold-italic">C</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> (upper set) and shape <inline-formula id="inf304">
<mml:math id="m326">
<mml:mrow>
<mml:mi mathvariant="bold-italic">S</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> coefficient (lower set) heatmaps from <xref ref-type="fig" rid="F8">Figure 8</xref> as a function of <inline-formula id="inf305">
<mml:math id="m327">
<mml:mrow>
<mml:mi mathvariant="bold-italic">&#x3c8;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf306">
<mml:math id="m328">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> at the same three temperatures <bold>(A-C)</bold> and with the marker styles listed in <xref ref-type="table" rid="T2">Table 2</xref>. The parameters used are listed in <xref ref-type="fig" rid="F3">Figure 3</xref>. The circular marker presents the SmA phase at those temperatures. The geodesics are the same from <xref ref-type="fig" rid="F7">Figure 7,</xref> with different color scheme for visualization purposes. The continuous purple lines are geodesics that connect both the isotropic and SmA phases.</p>
</caption>
<graphic xlink:href="frsfm-04-1359128-g009.tif"/>
</fig>
<p>The main features gleaned from the Casorati-I-SmA geodesic correlations from <xref ref-type="fig" rid="F9">Figure 9</xref> (top) are the following:<list list-type="simple">
<list-item>
<p>&#x2022; The intersection of the geodesic with the high curvedness Casorati tube occurs at high <inline-formula id="inf299">
<mml:math id="m320">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> values but is eventually lost because the slope of the geodesic increases with <inline-formula id="inf300">
<mml:math id="m321">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, while the Casorati tube bends to the right. For the intersection of the geodesic and tube, we need a geodesic slope <inline-formula id="inf301">
<mml:math id="m322">
<mml:mrow>
<mml:mi>m</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> given by the following equation:</p>
</list-item>
</list>
<disp-formula id="equ2">
<mml:math id="m323">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mrow>
<mml:mi>A</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>C</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x2248;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>0.1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mn>0.01</mml:mn>
</mml:msup>
<mml:mo>;</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mrow>
<mml:mi>A</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>G</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x2248;</mml:mo>
<mml:mi>m</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>&#x2192;</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mrow>
<mml:mi>A</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>C</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mrow>
<mml:mi>A</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>G</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x2192;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>0.1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mn>0.01</mml:mn>
</mml:msup>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>m</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>&#x2192;</mml:mo>
<mml:mi>m</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>0.1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mn>0.01</mml:mn>
</mml:msup>
</mml:mrow>
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
</disp-formula>where the subscripts <inline-formula id="inf302">
<mml:math id="m324">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>C</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>G</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> denote the Casorati and geodesics. This is only possible in the deep and intermediate quenches.<list list-type="simple">
<list-item>
<p>&#x2022; The I-SmA geodesics for NG and SD modes largely avoid the higher Casorati curvatures, indicating paths of lower curvatures.</p>
</list-item>
</list>
</p>
<p>The main features gleaned from shape coefficient-I-SmA geodesic correlations from <xref ref-type="fig" rid="F9">Figure 9</xref> (bottom) are as follows:<list list-type="simple">
<list-item>
<p>&#x2022; The geodesic path remains well-contained in the shape index channel comprehending ruts and trough concave-up shapes, except at low OP and low-temperature values, where saddle-like (green areas close to the origin) and concave-down (red areas close to the origin) shapes arise.</p>
</list-item>
<list-item>
<p>&#x2022; The development of SmA droplets that may form from an intermediate quench into the NG mode starts with a <inline-formula id="inf307">
<mml:math id="m329">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mo>&#x2248;</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>0.54</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> in the smectic phase and ends with <inline-formula id="inf308">
<mml:math id="m330">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mo>&#x2248;</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>0.84</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> in the isotropic state; therefore, the geodesic path to drop formation involves relatively modest shape configurational changes.</p>
</list-item>
</list>
</p>
<p>
<xref ref-type="fig" rid="F10">Figure 10</xref> integrates the curve families on the energy landscape corresponding to the SD quench regime <inline-formula id="inf309">
<mml:math id="m331">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>T</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>330</mml:mn>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mi mathvariant="normal">K</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>, the stable SmA phase (black dot), and an unstable isotropic phase. It presents the steepest descent lines, LOC, and geodesics on the <inline-formula id="inf310">
<mml:math id="m332">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>-energy landscape. The linear diagonal geodesic connecting the isotropic (unstable)-to-smectic A (stable) phases partitions the rut and trough region and serves as an attracting manifold for maximal LOC and curves of the steepest descent; the congruence of these three lines indicates why, at this temperature, SmA is the attractor. On the bottom right high-energy area, the congruence is now between minimal LOC, curves of steepest descent, and curved geodesics, indicating a repelling landscape.</p>
<fig id="F10" position="float">
<label>FIGURE 10</label>
<caption>
<p>
<inline-formula id="inf311">
<mml:math id="m333">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3c8;</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>A</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>-energy landscape for temperature (A) as listed in <xref ref-type="table" rid="T2">Table 2,</xref> integrating the steepest descent lines (green), LOC (cyan-min and magenta-max), and geodesics (orange) curves. The black dot represents the stable primary root, SmA phase.</p>
</caption>
<graphic xlink:href="frsfm-04-1359128-g010.tif"/>
</fig>
</sec>
</sec>
<sec sec-type="conclusion" id="s4">
<title>4 Conclusion</title>
<p>In this paper, we developed, implemented, and tested a novel computational geometrical method that complements classical liquid crystal phase transition modeling for the complex case of two-order-parameter symmetry breaking. This approach uses complementary geometric schemes to link the thermodynamic energy landscape of the isotropic-to-smectic A liquid crystal direct phase transition with novel soft-matter geometric metrics such as the Casorati curvature and shape coefficient. We summarize the results and their significance as follows:<list list-type="simple">
<list-item>
<p>1. A previously presented and comprehensive study of the Landau&#x2013;de Gennes free-energy model (<xref ref-type="bibr" rid="B25">De Gennes and Prost, 1993</xref>; <xref ref-type="bibr" rid="B94">Pleiner et al., 2000</xref>; <xref ref-type="bibr" rid="B66">Larin, 2004</xref>; <xref ref-type="bibr" rid="B87">Oswald and Pieranski, 2005b</xref>; <xref ref-type="bibr" rid="B38">Donald et al., 2006</xref>; <xref ref-type="bibr" rid="B10">Biscari et al., 2007</xref>; <xref ref-type="bibr" rid="B5">Abukhdeir and Rey, 2009b</xref>; <xref ref-type="bibr" rid="B83">Nandi et al., 2012</xref>; <xref ref-type="bibr" rid="B55">Izzo and De Oliveira, 2019</xref>) for the direct isotropic-to-smectic A transition with well-known material properties (<xref ref-type="bibr" rid="B126">Urban et al., 2005</xref>; <xref ref-type="bibr" rid="B3">Abukhdeir and Rey, 2008</xref>; <xref ref-type="bibr" rid="B18">Coles and Strazielle, 2011</xref>) formed the basis of the theory and computational modeling characterization of phase ordering with two non-conserved order parameters.</p>
</list-item>
<list-item>
<p>2. The Landau free-energy landscape was obtained using explicit Monge surface parametrization as a function of orientational and positional orders, allowing the deployment of, in a simple manner, differential geometry calculations (Eq. <xref ref-type="disp-formula" rid="e2_7">2.7</xref>)</p>
</list-item>
<list-item>
<p>3. The index polynomial theorem (Eq. <xref ref-type="disp-formula" rid="e2_12">2.12</xref>) for the number of critical roots as a function of quench depth revealed the importance of saddle roots in the spinodal and nucleation and growth region; without the knowledge of parametric free-energy coefficient data, the theorem shows that the maximum number of critical roots is <inline-formula id="inf312">
<mml:math id="m334">
<mml:mrow>
<mml:mn>3</mml:mn>
<mml:mi mathvariant="normal">x</mml:mi>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> since the free energy is a quartic polynomial in the two order parameters.</p>
</list-item>
<list-item>
<p>4. Using high-performance computing and high-fidelity numerical methods for nonlinear algebraic and differential equations, the following curve families were calculated (<xref ref-type="sec" rid="s3-2">Sections 3.2</xref> and <xref ref-type="sec" rid="s3-3">3.3</xref>): level-set-steepest descent and geodesic-principal curvatures. These special curves revealed the location of critical points already predicted by the index theorem, directions of small and large curvatures, and minimal length connections between isotropic and smectic roots. In particular, linear geodesics joining isotropic and smectic states in nucleation and growth and spinodal quenches revealed phase transformation paths. The level-set curves around stable roots were elliptical, indicating anisotropy originating from the Landau free-energy polynomial.</p>
</list-item>
<list-item>
<p>5. The emergence of metastable plastic crystals at deep quenches and unstable nematic states at deep and intermediate quenches was characterized, and their annihilation through supercritical and saddle-node bifurcation was captured, re-emphasizing results from the index polynomial theorem. The relevance of the nematic or plastic order at the interfaces of smectic A drops in an isotropic matrix was pointed out.</p>
</list-item>
<list-item>
<p>6. Previously presented measures of shape and curvedness (Casorati) in soft-matter materials were used (<xref ref-type="sec" rid="s3-4">Section 3.4</xref>) to characterize the energy landscape with purely geometric measures instead of order parameter coordinates. The calculations were integrated with the curve families, showing consistency and revealing that the Casorati landscape is a bent, higher-curved tube embedded in a low-curvedness matrix; the tube is well-fitted with a power law function. The smectic A root resides inside this tube and moves downward as the temperature increases. The shape coefficient landscape is characterized by a wide channel of concave-up shapes separated from an area of concave-down shapes by saddle-like interfaces, which is in agreement with shape coexistence phenomena.</p>
</list-item>
<list-item>
<p>7. Plotting all the curve families (point 4 above) in the energy landscape, we find that at large quench, the isotropic-to-smectic A geodesic is an attractor for maximal lines of curvature and curves of the steepest descent, explaining the stability of the smectic A state.</p>
</list-item>
</list>
</p>
<p>The combination of parameter-free predictions from polynomial theorems with the computational geometry of the free-energy landscape contributes to the evolving understanding and characterization of the isotropic-to-smectic A transition, which is of high interest to biological colloidal liquid crystals, such as in the precursors to the mussel byssus (<xref ref-type="bibr" rid="B101">Renner-Rao et al., 2019</xref>; <xref ref-type="bibr" rid="B51">Harrington and Fratzl, 2021</xref>; <xref ref-type="bibr" rid="B58">Jehle et al., 2021</xref>) through droplet nucleation/growth and colloidal impingement. We demonstrated that the presence of two non-conserved order parameters creates challenges in equilibrium spatially homogeneous simulations, but how time-dependent processes such as droplet growth resolve the couplings of shape&#x2013;size&#x2013;structure&#x2013;interface remains to be elucidated in future work by building on the present results and methods.</p>
</sec>
</body>
<back>
<sec sec-type="data-availability" id="s5">
<title>Data availability statement</title>
<p>Data related to this work will be made available by request to the authors. Requests to access the datasets should be directed to ADR, <email>alejandro.rey@mcgill.ca</email>.</p>
</sec>
<sec id="s6">
<title>Author contributions</title>
<p>DZ: writing&#x2013;original draft, conceptualization, data curation, formal analysis, investigation, methodology, software, validation, visualization, and writing&#x2013;review and editing. ZW: conceptualization, writing&#x2013;review and editing, validation, and methodology. ND: conceptualization, supervision, and writing&#x2013;review and editing. MH: conceptualization, supervision, and writing&#x2013;review and editing. AR: conceptualization, validation, writing&#x2013;review and editing, formal analysis, funding acquisition, project administration, resources, and supervision.</p>
</sec>
<sec sec-type="funding-information" id="s7">
<title>Funding</title>
<p>The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. This work was supported by the Fonds de Recherche du Qu&#xe9;bec (FRQNT) (2021-PR-284991) and the Natural Science and Engineering Research Council of Canada (NSERC) (&#x23;223086). Author DUZC thanks Consejo Nacional de Humanidades, Ciencia y Tecnolog&#xed;a (CONAHCYT) (853563), and McGill Engineering Doctoral Award (MEDA) scholarships for financial support.</p>
</sec>
<ack>
<p>The authors acknowledge Digital Research Alliance of Canada (ID 4700) for computational resources and technical support.</p>
</ack>
<sec sec-type="COI-statement" id="s8">
<title>Conflict of interest</title>
<p>The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.</p>
</sec>
<sec sec-type="disclaimer" id="s9">
<title>Publisher&#x2019;s note</title>
<p>All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.</p>
</sec>
<sec id="s10">
<title>Supplementary material</title>
<p>The Supplementary Material for this article can be found online at: <ext-link ext-link-type="uri" xlink:href="https://www.frontiersin.org/articles/10.3389/frsfm.2024.1359128/full#supplementary-material">https://www.frontiersin.org/articles/10.3389/frsfm.2024.1359128/full&#x23;supplementary-material</ext-link>
</p>
<supplementary-material xlink:href="DataSheet1.docx" id="SM1" mimetype="application/docx" xmlns:xlink="http://www.w3.org/1999/xlink"/>
</sec>
<ref-list>
<title>References</title>
<ref id="B1">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Abbena</surname>
<given-names>E.</given-names>
</name>
<name>
<surname>Salamon</surname>
<given-names>S.</given-names>
</name>
<name>
<surname>Gray</surname>
<given-names>A.</given-names>
</name>
</person-group> (<year>2017</year>). <source>Modern differential geometry of curves and surfaces with Mathematica</source>. <publisher-loc>China</publisher-loc>: <publisher-name>CRC Press</publisher-name>.</citation>
</ref>
<ref id="B2">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Abukhdeir</surname>
<given-names>N. M.</given-names>
</name>
</person-group> (<year>2009</year>). <source>
<italic>Growth, dynamics, and texture modeling of the lamellar smectic-A liquid crystalline transition.</italic> Doctor of Philosophy</source>. <publisher-loc>Canada</publisher-loc>: <publisher-name>McGill University</publisher-name>.</citation>
</ref>
<ref id="B3">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Abukhdeir</surname>
<given-names>N. M.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2008</year>). <article-title>Simulation of spherulite growth using a comprehensive approach to modeling the first-order isotropic/smectic-A mesophase transition</article-title>. <source>arXiv Prepr. arXiv:0807.4525</source>. <pub-id pub-id-type="doi">10.48550/arXiv.0807.4525</pub-id>
</citation>
</ref>
<ref id="B4">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Abukhdeir</surname>
<given-names>N. M.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2009a</year>). <article-title>Metastable nematic preordering in smectic liquid crystalline phase transitions</article-title>. <source>Macromolecules</source> <volume>42</volume>, <fpage>3841</fpage>&#x2013;<lpage>3844</lpage>. <pub-id pub-id-type="doi">10.1021/ma900796b</pub-id>
</citation>
</ref>
<ref id="B5">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Abukhdeir</surname>
<given-names>N. M.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2009b</year>). <article-title>Nonisothermal model for the direct isotropic/smectic-A liquid-crystalline transition</article-title>. <source>Langmuir</source> <volume>25</volume>, <fpage>11923</fpage>&#x2013;<lpage>11929</lpage>. <pub-id pub-id-type="doi">10.1021/la9015965</pub-id>
</citation>
</ref>
<ref id="B6">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Abukhdeir</surname>
<given-names>N. M.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2009c</year>). <article-title>Shape-dynamic growth, structure, and elasticity of homogeneously oriented spherulites in an isotropic/smectic-A mesophase transition</article-title>. <source>Liq. Cryst.</source> <volume>36</volume>, <fpage>1125</fpage>&#x2013;<lpage>1137</lpage>. <pub-id pub-id-type="doi">10.1080/02678290902878754</pub-id>
</citation>
</ref>
<ref id="B7">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Aguilar Gutierrez</surname>
<given-names>O. F.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2018</year>). <article-title>Extracting shape from curvature evolution in moving surfaces</article-title>. <source>Soft Matter</source> <volume>14</volume>, <fpage>1465</fpage>&#x2013;<lpage>1473</lpage>. <pub-id pub-id-type="doi">10.1039/c7sm02409f</pub-id>
</citation>
</ref>
<ref id="B8">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Bellini</surname>
<given-names>T.</given-names>
</name>
<name>
<surname>Clark</surname>
<given-names>N. A.</given-names>
</name>
<name>
<surname>Link</surname>
<given-names>D. R.</given-names>
</name>
</person-group> (<year>2002</year>). <article-title>Isotropic to smectic a phase transitions in a porous matrix: a case of multiporous phase coexistence</article-title>. <source>J. Phys. Condens. Matter</source> <volume>15</volume>, <fpage>S175</fpage>&#x2013;<lpage>S182</lpage>. <pub-id pub-id-type="doi">10.1088/0953-8984/15/1/322</pub-id>
</citation>
</ref>
<ref id="B9">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Berent</surname>
<given-names>K.</given-names>
</name>
<name>
<surname>Cartwright</surname>
<given-names>J. H. E.</given-names>
</name>
<name>
<surname>Checa</surname>
<given-names>A. G.</given-names>
</name>
<name>
<surname>Pimentel</surname>
<given-names>C.</given-names>
</name>
<name>
<surname>Ramos-Silva</surname>
<given-names>P.</given-names>
</name>
<name>
<surname>Sainz-D&#xed;az</surname>
<given-names>C. I.</given-names>
</name>
</person-group> (<year>2022</year>). <article-title>Helical microstructures in molluscan biomineralization are a biological example of close packed helices that may form from a colloidal liquid crystal precursor in a twist--bend nematic phase</article-title>. <source>Phys. Rev. Mater.</source> <volume>6</volume>, <fpage>105601</fpage>. <pub-id pub-id-type="doi">10.1103/physrevmaterials.6.105601</pub-id>
</citation>
</ref>
<ref id="B10">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Biscari</surname>
<given-names>P.</given-names>
</name>
<name>
<surname>Calderer</surname>
<given-names>M. C.</given-names>
</name>
<name>
<surname>Terentjev</surname>
<given-names>E. M.</given-names>
</name>
</person-group> (<year>2007</year>). <article-title>Landau-de Gennes theory of isotropic-nematic-smectic liquid crystal transitions</article-title>. <source>Phys. Rev. E Stat. Nonlin Soft Matter Phys.</source> <volume>75</volume>, <fpage>051707</fpage>. <pub-id pub-id-type="doi">10.1103/physreve.75.051707</pub-id>
</citation>
</ref>
<ref id="B11">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Blinov</surname>
<given-names>L. M.</given-names>
</name>
</person-group> (<year>2011</year>). <source>Structure and properties of liquid crystals</source>. <publisher-loc>Dordrecht</publisher-loc>: <publisher-name>Springer</publisher-name>.</citation>
</ref>
<ref id="B12">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Bowick</surname>
<given-names>M. J.</given-names>
</name>
<name>
<surname>Kinderlehrer</surname>
<given-names>D.</given-names>
</name>
<name>
<surname>Menon</surname>
<given-names>G.</given-names>
</name>
<name>
<surname>Radin</surname>
<given-names>C.</given-names>
</name>
</person-group> (<year>2017</year>). <source>
<italic>Mathematics and materials</italic>, American mathematical soc</source>.</citation>
</ref>
<ref id="B13">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Bradley</surname>
<given-names>P. A.</given-names>
</name>
</person-group> (<year>2019</year>). <source>
<italic>On the physicochemical control of collagen fibrilligenesis and biomineralization.</italic> Doctor of Philosophy</source>. <publisher-loc>USA</publisher-loc>: <publisher-name>Northeastern University</publisher-name>.</citation>
</ref>
<ref id="B14">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Bukharina</surname>
<given-names>D.</given-names>
</name>
<name>
<surname>Kim</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Han</surname>
<given-names>M. J.</given-names>
</name>
<name>
<surname>Tsukruk</surname>
<given-names>V. V.</given-names>
</name>
</person-group> (<year>2022</year>). <article-title>Cellulose nanocrystals&#x27; assembly under ionic strength variation: from high orientation ordering to a random orientation</article-title>. <source>Langmuir</source> <volume>38</volume>, <fpage>6363</fpage>&#x2013;<lpage>6375</lpage>. <pub-id pub-id-type="doi">10.1021/acs.langmuir.2c00293</pub-id>
</citation>
</ref>
<ref id="B15">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Bunsell</surname>
<given-names>A. R.</given-names>
</name>
<name>
<surname>Joann&#xe8;s</surname>
<given-names>S.</given-names>
</name>
<name>
<surname>Thionnet</surname>
<given-names>A.</given-names>
</name>
</person-group> (<year>2021</year>). <source>Fundamentals of fibre reinforced composite materials</source>. <publisher-loc>Germany</publisher-loc>: <publisher-name>CRC Press</publisher-name>.</citation>
</ref>
<ref id="B16">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Cai</surname>
<given-names>A.</given-names>
</name>
<name>
<surname>Abdali</surname>
<given-names>Z.</given-names>
</name>
<name>
<surname>Saldanha</surname>
<given-names>D. J.</given-names>
</name>
<name>
<surname>Aminzare</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Dorval Courchesne</surname>
<given-names>N.-M.</given-names>
</name>
</person-group> (<year>2023</year>). <article-title>Endowing textiles with self-repairing ability through the fabrication of composites with a bacterial biofilm</article-title>. <source>Sci. Rep.</source> <volume>13</volume>, <fpage>11389</fpage>. <pub-id pub-id-type="doi">10.1038/s41598-023-38501-2</pub-id>
</citation>
</ref>
<ref id="B17">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Chahine</surname>
<given-names>G.</given-names>
</name>
<name>
<surname>Kityk</surname>
<given-names>A. V.</given-names>
</name>
<name>
<surname>D&#xe9;marest</surname>
<given-names>N.</given-names>
</name>
<name>
<surname>Jean</surname>
<given-names>F.</given-names>
</name>
<name>
<surname>Knorr</surname>
<given-names>K.</given-names>
</name>
<name>
<surname>Huber</surname>
<given-names>P.</given-names>
</name>
<etal/>
</person-group> (<year>2010</year>). <article-title>Collective molecular reorientation of a calamitic liquid crystal (12CB) confined in alumina nanochannels</article-title>. <source>Phys. Rev. E</source> <volume>82</volume>, <fpage>011706</fpage>. <pub-id pub-id-type="doi">10.1103/physreve.82.011706</pub-id>
</citation>
</ref>
<ref id="B18">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Coles</surname>
<given-names>H. J.</given-names>
</name>
<name>
<surname>Strazielle</surname>
<given-names>C.</given-names>
</name>
</person-group> (<year>2011</year>). <article-title>The order-disorder phase transition in liquid crystals as a function of molecular structure. I. The alkyl cyanobiphenyls</article-title>. <source>Mol. Cryst. Liq. Cryst.</source> <volume>55</volume>, <fpage>237</fpage>&#x2013;<lpage>250</lpage>. <pub-id pub-id-type="doi">10.1080/00268947908069805</pub-id>
</citation>
</ref>
<ref id="B19">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Collings</surname>
<given-names>P. J.</given-names>
</name>
</person-group> (<year>1997</year>). <source>Phase structures and transitions in thermotropic liquid crystals handbook of liquid crystal research</source>.</citation>
</ref>
<ref id="B20">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Collings</surname>
<given-names>P. J.</given-names>
</name>
<name>
<surname>Goodby</surname>
<given-names>J. W.</given-names>
</name>
</person-group> (<year>2019</year>). <source>Introduction to liquid crystals: chemistry and physics</source>. <publisher-loc>Germany</publisher-loc>: <publisher-name>Crc Press</publisher-name>.</citation>
</ref>
<ref id="B21">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Collings</surname>
<given-names>P. J.</given-names>
</name>
<name>
<surname>Hird</surname>
<given-names>M.</given-names>
</name>
</person-group> (<year>2017</year>). <source>Introduction to liquid crystals chemistry and physics</source>. <publisher-loc>Germany</publisher-loc>: <publisher-name>CRC Press</publisher-name>.</citation>
</ref>
<ref id="B22">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Copic</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Mertelj</surname>
<given-names>A.</given-names>
</name>
</person-group> (<year>2020</year>). <article-title>Q-tensor model of twist-bend and splay nematic phases</article-title>. <source>Phys. Rev. E</source> <volume>101</volume>, <fpage>022704</fpage>. <pub-id pub-id-type="doi">10.1103/physreve.101.022704</pub-id>
</citation>
</ref>
<ref id="B23">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Das</surname>
<given-names>A. K.</given-names>
</name>
<name>
<surname>Mukherjee</surname>
<given-names>P. K.</given-names>
</name>
</person-group> (<year>2009</year>). <article-title>Phenomenological theory of the direct isotropic to hexatic-B phase transition</article-title>. <source>J. Chem. Phys.</source> <volume>130</volume>, <fpage>054901</fpage>. <pub-id pub-id-type="doi">10.1063/1.3067425</pub-id>
</citation>
</ref>
<ref id="B24">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>de Gennes</surname>
<given-names>P. G.</given-names>
</name>
</person-group> (<year>2007</year>). <article-title>Some remarks on the polymorphism of smectics</article-title>. <source>Mol. Cryst. Liq. Cryst.</source> <volume>21</volume>, <fpage>49</fpage>&#x2013;<lpage>76</lpage>. <pub-id pub-id-type="doi">10.1080/15421407308083313</pub-id>
</citation>
</ref>
<ref id="B25">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>de Gennes</surname>
<given-names>P.-G.</given-names>
</name>
<name>
<surname>Prost</surname>
<given-names>J.</given-names>
</name>
</person-group> (<year>1993</year>). <source>The physics of liquid crystals</source>. <publisher-loc>Oxford</publisher-loc>: <publisher-name>Oxford University Press</publisher-name>.</citation>
</ref>
<ref id="B26">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>de Luca</surname>
<given-names>G.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2004</year>). <article-title>Chiral front propagation in liquid-crystalline materials: formation of the planar monodomain twisted plywood architecture of biological fibrous composites</article-title>. <source>Phys. Rev. E</source> <volume>69</volume>, <fpage>011706</fpage>. <pub-id pub-id-type="doi">10.1103/physreve.69.011706</pub-id>
</citation>
</ref>
<ref id="B27">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>de Luca</surname>
<given-names>G.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2006</year>). <article-title>Dynamic interactions between nematic point defects in the spinning extrusion duct of spiders</article-title>. <source>J. Chem. Phys.</source> <volume>124</volume>, <fpage>144904</fpage>. <pub-id pub-id-type="doi">10.1063/1.2186640</pub-id>
</citation>
</ref>
<ref id="B28">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>de Luca</surname>
<given-names>G.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A.</given-names>
</name>
</person-group> (<year>2003</year>). <article-title>Monodomain and polydomain helicoids in chiral liquid-crystalline phases and their biological analogues</article-title>. <source>Eur. Phys. J. E</source> <volume>12</volume>, <fpage>291</fpage>&#x2013;<lpage>302</lpage>. <pub-id pub-id-type="doi">10.1140/epje/i2002-10164-3</pub-id>
</citation>
</ref>
<ref id="B29">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Demirci</surname>
<given-names>N.</given-names>
</name>
<name>
<surname>Holland</surname>
<given-names>M. A.</given-names>
</name>
</person-group> (<year>2022</year>). <article-title>Cortical thickness systematically varies with curvature and depth in healthy human brains</article-title>. <source>Hum. Brain Mapp.</source> <volume>43</volume>, <fpage>2064</fpage>&#x2013;<lpage>2084</lpage>. <pub-id pub-id-type="doi">10.1002/hbm.25776</pub-id>
</citation>
</ref>
<ref id="B30">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Demus</surname>
<given-names>D.</given-names>
</name>
<name>
<surname>Goodby</surname>
<given-names>J. W.</given-names>
</name>
<name>
<surname>Gray</surname>
<given-names>G. W.</given-names>
</name>
<name>
<surname>Spiess</surname>
<given-names>H. W.</given-names>
</name>
<name>
<surname>Vill</surname>
<given-names>V.</given-names>
</name>
</person-group> (<year>2008a</year>). <source>Handbook of liquid crystals</source>.</citation>
</ref>
<ref id="B31">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Demus</surname>
<given-names>D.</given-names>
</name>
<name>
<surname>Goodby</surname>
<given-names>J. W.</given-names>
</name>
<name>
<surname>Gray</surname>
<given-names>G. W.</given-names>
</name>
<name>
<surname>Spiess</surname>
<given-names>H. W.</given-names>
</name>
<name>
<surname>Vill</surname>
<given-names>V.</given-names>
</name>
</person-group> (<year>2008b</year>). <source>Handbook of liquid crystals, volume 3: high molecular weight liquid crystals</source>. <publisher-loc>USA</publisher-loc>: <publisher-name>John Wiley and Sons</publisher-name>.</citation>
</ref>
<ref id="B32">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Demus</surname>
<given-names>D.</given-names>
</name>
<name>
<surname>Goodby</surname>
<given-names>J. W.</given-names>
</name>
<name>
<surname>Gray</surname>
<given-names>G. W.</given-names>
</name>
<name>
<surname>Spiess</surname>
<given-names>H. W.</given-names>
</name>
<name>
<surname>Vill</surname>
<given-names>V.</given-names>
</name>
</person-group> (<year>2011</year>). <source>Handbook of liquid crystals, volume 2A: low molecular weight liquid crystals I: calamitic liquid crystals</source>. <publisher-loc>USA</publisher-loc>: <publisher-name>John Wiley and Sons</publisher-name>.</citation>
</ref>
<ref id="B33">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Deng</surname>
<given-names>F.</given-names>
</name>
<name>
<surname>Dang</surname>
<given-names>Y.</given-names>
</name>
<name>
<surname>Tang</surname>
<given-names>L.</given-names>
</name>
<name>
<surname>Hu</surname>
<given-names>T.</given-names>
</name>
<name>
<surname>Ding</surname>
<given-names>C.</given-names>
</name>
<name>
<surname>Hu</surname>
<given-names>X.</given-names>
</name>
<etal/>
</person-group> (<year>2021</year>). <article-title>Tendon-inspired fibers from liquid crystalline collagen as the pre-oriented bioink</article-title>. <source>Int. J. Biol. Macromol.</source> <volume>185</volume>, <fpage>739</fpage>&#x2013;<lpage>749</lpage>. <pub-id pub-id-type="doi">10.1016/j.ijbiomac.2021.06.173</pub-id>
</citation>
</ref>
<ref id="B34">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Dierking</surname>
<given-names>I.</given-names>
</name>
<name>
<surname>al-Zangana</surname>
<given-names>S.</given-names>
</name>
</person-group> (<year>2017</year>). <article-title>Lyotropic liquid crystal phases from anisotropic nanomaterials</article-title>. <source>Nanomater. (Basel)</source> <volume>7</volume>, <fpage>305</fpage>. <pub-id pub-id-type="doi">10.3390/nano7100305</pub-id>
</citation>
</ref>
<ref id="B35">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Dilisi</surname>
<given-names>G. A.</given-names>
</name>
</person-group> (<year>2019</year>). in <source>An introduction to liquid crystals</source>. Editor <person-group person-group-type="editor">
<name>
<surname>DELUCA</surname>
<given-names>J. J.</given-names>
</name>
</person-group> (<publisher-loc>New York</publisher-loc>: <publisher-name>Morgan and Claypool Publishers</publisher-name>).</citation>
</ref>
<ref id="B36">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Do Carmo</surname>
<given-names>M. P.</given-names>
</name>
</person-group> (<year>2016</year>). <source>Differential geometry of curves and surfaces: revised and updated</source>. <edition>second edition</edition>. <publisher-loc>New York</publisher-loc>: <publisher-name>Courier Dover Publications</publisher-name>.</citation>
</ref>
<ref id="B37">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Dogic</surname>
<given-names>Z.</given-names>
</name>
<name>
<surname>Fraden</surname>
<given-names>S.</given-names>
</name>
</person-group> (<year>2001</year>). <article-title>Development of model colloidal liquid crystals and the kinetics of the isotropic-smectic transition</article-title>. <source>Philosophical Trans. R. Soc. a-Mathematical Phys. Eng. Sci.</source> <volume>359</volume>, <fpage>997</fpage>&#x2013;<lpage>1015</lpage>. <comment>DOI, M. 1981</comment>. <pub-id pub-id-type="doi">10.1098/rsta.2000.0814</pub-id>
</citation>
</ref>
<ref id="B79">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Doi</surname>
<given-names>M.</given-names>
</name>
</person-group> <year>2022</year> <article-title>Molecular dynamics and rheological properties of concentrated solutions of rodlike polymers in isotropic and liquid crystalline phases</article-title>. <source>J. Polym. Sci. Polym. Phys. Ed.</source> <volume>19</volume>, <fpage>229</fpage>&#x2013;<lpage>243</lpage>. <pub-id pub-id-type="doi">10.1002/pol.1981.180190205</pub-id>
</citation>
</ref>
<ref id="B38">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Donald</surname>
<given-names>A. M.</given-names>
</name>
<name>
<surname>Windle</surname>
<given-names>A. H.</given-names>
</name>
<name>
<surname>Hanna</surname>
<given-names>S.</given-names>
</name>
</person-group> (<year>2006</year>). <source>Liquid crystalline polymers</source>. <publisher-loc>Cambridge</publisher-loc>: <publisher-name>Cambridge University Press</publisher-name>.</citation>
</ref>
<ref id="B39">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Durfee</surname>
<given-names>A.</given-names>
</name>
<name>
<surname>Kronenfeld</surname>
<given-names>N.</given-names>
</name>
<name>
<surname>Munson</surname>
<given-names>H.</given-names>
</name>
<name>
<surname>Roy</surname>
<given-names>J.</given-names>
</name>
<name>
<surname>Westby</surname>
<given-names>I.</given-names>
</name>
</person-group> (<year>1993</year>). <article-title>Counting critical points of real polynomials in two variables</article-title>. <source>Am. Math. Mon.</source> <volume>100</volume>, <fpage>255</fpage>&#x2013;<lpage>271</lpage>. <pub-id pub-id-type="doi">10.2307/2324459</pub-id>
</citation>
</ref>
<ref id="B40">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Farouki</surname>
<given-names>R. T.</given-names>
</name>
</person-group> (<year>1998</year>). <article-title>On integrating lines of curvature</article-title>. <source>Comput. Aided Geom. Des.</source> <volume>15</volume>, <fpage>187</fpage>&#x2013;<lpage>192</lpage>. <pub-id pub-id-type="doi">10.1016/s0167-8396(97)00022-8</pub-id>
</citation>
</ref>
<ref id="B41">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Fischer</surname>
<given-names>S.</given-names>
</name>
<name>
<surname>Karplus</surname>
<given-names>M.</given-names>
</name>
</person-group> (<year>1992</year>). <article-title>Conjugate peak refinement: an algorithm for finding reaction paths and accurate transition states in systems with many degrees of freedom</article-title>. <source>Chem. Phys. Lett.</source> <volume>194</volume>, <fpage>252</fpage>&#x2013;<lpage>261</lpage>. <pub-id pub-id-type="doi">10.1016/0009-2614(92)85543-j</pub-id>
</citation>
</ref>
<ref id="B42">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Garti</surname>
<given-names>N.</given-names>
</name>
<name>
<surname>Somasundaran</surname>
<given-names>P.</given-names>
</name>
<name>
<surname>Mezzenga</surname>
<given-names>R.</given-names>
</name>
</person-group> (<year>2012</year>). <source>Self-assembled supramolecular architectures: lyotropic liquid crystals</source>. <publisher-loc>USA</publisher-loc>: <publisher-name>John Wiley and Sons</publisher-name>.</citation>
</ref>
<ref id="B43">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Golmohammadi</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2009</year>). <article-title>Thermodynamic modelling of carbonaceous mesophase mixtures</article-title>. <source>Liq. Cryst.</source> <volume>36</volume>, <fpage>75</fpage>&#x2013;<lpage>92</lpage>. <pub-id pub-id-type="doi">10.1080/02678290802666218</pub-id>
</citation>
</ref>
<ref id="B44">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Golmohammadi</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2010</year>). <article-title>Structural modeling of carbonaceous mesophase amphotropic mixtures under uniaxial extensional flow</article-title>. <source>J. Chem. Phys.</source> <volume>133</volume>, <fpage>034903</fpage>. <pub-id pub-id-type="doi">10.1063/1.3455505</pub-id>
</citation>
</ref>
<ref id="B45">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Gorkunov</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Osipov</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Lagerwall</surname>
<given-names>J.</given-names>
</name>
<name>
<surname>Giesselmann</surname>
<given-names>F.</given-names>
</name>
</person-group> (<year>2007</year>). <article-title>Order-disorder molecular model of the smectic-A&#x2013;smectic-C phase transition in materials with conventional and anomalously weak layer contraction</article-title>. <source>Phys. Rev. E</source> <volume>76</volume>, <fpage>051706</fpage>. <pub-id pub-id-type="doi">10.1103/physreve.76.051706</pub-id>
</citation>
</ref>
<ref id="B46">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Gudimalla</surname>
<given-names>A.</given-names>
</name>
<name>
<surname>Thomas</surname>
<given-names>S.</given-names>
</name>
<name>
<surname>Zidan&#x161;ek</surname>
<given-names>A.</given-names>
</name>
</person-group> (<year>2021</year>). <article-title>Phase behaviour of n-CB liquid crystals confined to controlled pore glasses</article-title>. <source>J. Mol. Struct.</source> <volume>1235</volume>, <fpage>130217</fpage>. <pub-id pub-id-type="doi">10.1016/j.molstruc.2021.130217</pub-id>
</citation>
</ref>
<ref id="B47">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Gurevich</surname>
<given-names>S.</given-names>
</name>
<name>
<surname>Soule</surname>
<given-names>E.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A.</given-names>
</name>
<name>
<surname>Reven</surname>
<given-names>L.</given-names>
</name>
<name>
<surname>Provatas</surname>
<given-names>N.</given-names>
</name>
</person-group> (<year>2014</year>). <article-title>Self-assembly via branching morphologies in nematic liquid-crystal nanocomposites</article-title>. <source>Phys. Rev. E</source> <volume>90</volume>, <fpage>020501</fpage>. <pub-id pub-id-type="doi">10.1103/physreve.90.020501</pub-id>
</citation>
</ref>
<ref id="B48">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Gurin</surname>
<given-names>P.</given-names>
</name>
<name>
<surname>Odriozola</surname>
<given-names>G.</given-names>
</name>
<name>
<surname>Varga</surname>
<given-names>S.</given-names>
</name>
</person-group> (<year>2021</year>). <article-title>Enhanced two-dimensional nematic order in slit-like pores</article-title>. <source>New J. Phys.</source> <volume>23</volume>, <fpage>063053</fpage>. <pub-id pub-id-type="doi">10.1088/1367-2630/ac05e1</pub-id>
</citation>
</ref>
<ref id="B49">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Han</surname>
<given-names>J. Q.</given-names>
</name>
<name>
<surname>Luo</surname>
<given-names>Y.</given-names>
</name>
<name>
<surname>Wang</surname>
<given-names>W.</given-names>
</name>
<name>
<surname>Zhang</surname>
<given-names>P. W.</given-names>
</name>
<name>
<surname>Zhang</surname>
<given-names>Z. F.</given-names>
</name>
</person-group> (<year>2015</year>). <article-title>From microscopic theory to macroscopic theory: a systematic study on modeling for liquid crystals</article-title>. <source>Archive Ration. Mech. Analysis</source> <volume>215</volume>, <fpage>741</fpage>&#x2013;<lpage>809</lpage>. <pub-id pub-id-type="doi">10.1007/s00205-014-0792-3</pub-id>
</citation>
</ref>
<ref id="B50">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Han</surname>
<given-names>W.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A.</given-names>
</name>
</person-group> (<year>1993</year>). <article-title>Supercritical bifurcations in simple shear flow of a non-aligning nematic: reactive parameter and director anchoring effects</article-title>. <source>J. Newt. fluid Mech.</source> <volume>48</volume>, <fpage>181</fpage>&#x2013;<lpage>210</lpage>. <pub-id pub-id-type="doi">10.1016/0377-0257(93)80070-r</pub-id>
</citation>
</ref>
<ref id="B51">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Harrington</surname>
<given-names>M. J.</given-names>
</name>
<name>
<surname>Fratzl</surname>
<given-names>P.</given-names>
</name>
</person-group> (<year>2021</year>). <article-title>Natural load-bearing protein materials</article-title>. <source>Prog. Mater. Sci.</source> <volume>120</volume>, <fpage>100767</fpage>. <pub-id pub-id-type="doi">10.1016/j.pmatsci.2020.100767</pub-id>
</citation>
</ref>
<ref id="B52">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Hawkins</surname>
<given-names>R. J.</given-names>
</name>
<name>
<surname>April</surname>
<given-names>E. W.</given-names>
</name>
</person-group> (<year>1983</year>). &#x201c;<article-title>Liquid crystals in living tissues</article-title>,&#x201d; in <source>Advances in liquid crystals</source>. Editor <person-group person-group-type="editor">
<name>
<surname>BROWN</surname>
<given-names>G. H.</given-names>
</name>
</person-group> (<publisher-loc>Germany</publisher-loc>: <publisher-name>Elsevier</publisher-name>).</citation>
</ref>
<ref id="B53">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Hormann</surname>
<given-names>K.</given-names>
</name>
<name>
<surname>Zimmer</surname>
<given-names>J.</given-names>
</name>
</person-group> (<year>2007</year>). <article-title>On Landau theory and symmetric energy landscapes for phase transitions</article-title>. <source>J. Mech. Phys. Solids</source> <volume>55</volume>, <fpage>1385</fpage>&#x2013;<lpage>1409</lpage>. <pub-id pub-id-type="doi">10.1016/j.jmps.2007.01.004</pub-id>
</citation>
</ref>
<ref id="B54">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Idziak</surname>
<given-names>S. H. J.</given-names>
</name>
<name>
<surname>Koltover</surname>
<given-names>I.</given-names>
</name>
<name>
<surname>Davidson</surname>
<given-names>P.</given-names>
</name>
<name>
<surname>Ruths</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Li</surname>
<given-names>Y.</given-names>
</name>
<name>
<surname>Israelachvili</surname>
<given-names>J. N.</given-names>
</name>
<etal/>
</person-group> (<year>1996</year>). <article-title>Structure under confinement in a smectic-A and lyotropic surfactant hexagonal phase</article-title>. <source>Phys. B Condens. Matter</source> <volume>221</volume>, <fpage>289</fpage>&#x2013;<lpage>295</lpage>. <pub-id pub-id-type="doi">10.1016/0921-4526(95)00939-6</pub-id>
</citation>
</ref>
<ref id="B55">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Izzo</surname>
<given-names>D.</given-names>
</name>
<name>
<surname>de Oliveira</surname>
<given-names>M. J.</given-names>
</name>
</person-group> (<year>2019</year>). <article-title>Landau theory for isotropic, nematic, smectic-A, and smectic-C phases</article-title>. <source>Liq. Cryst.</source> <volume>47</volume>, <fpage>99</fpage>&#x2013;<lpage>105</lpage>. <pub-id pub-id-type="doi">10.1080/02678292.2019.1631968</pub-id>
</citation>
</ref>
<ref id="B56">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Jackson</surname>
<given-names>K.</given-names>
</name>
<name>
<surname>Peivandi</surname>
<given-names>A.</given-names>
</name>
<name>
<surname>Fogal</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Tian</surname>
<given-names>L.</given-names>
</name>
<name>
<surname>Hosseinidoust</surname>
<given-names>Z.</given-names>
</name>
</person-group> (<year>2021</year>). <article-title>Filamentous phages as building blocks for bioactive hydrogels</article-title>. <source>ACS Appl. Bio Mater.</source> <volume>4</volume>, <fpage>2262</fpage>&#x2013;<lpage>2273</lpage>. <pub-id pub-id-type="doi">10.1021/acsabm.0c01557</pub-id>
</citation>
</ref>
<ref id="B57">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>J&#xe1;kli</surname>
<given-names>A.</given-names>
</name>
<name>
<surname>Saupe</surname>
<given-names>A.</given-names>
</name>
</person-group> (<year>2006</year>). <source>One-and two-dimensional fluids: properties of smectic, lamellar and columnar liquid crystals</source>. <publisher-loc>New York</publisher-loc>: <publisher-name>CRC Press</publisher-name>.</citation>
</ref>
<ref id="B58">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Jehle</surname>
<given-names>F.</given-names>
</name>
<name>
<surname>Priemel</surname>
<given-names>T.</given-names>
</name>
<name>
<surname>Strauss</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Fratzl</surname>
<given-names>P.</given-names>
</name>
<name>
<surname>Bertinetti</surname>
<given-names>L.</given-names>
</name>
<name>
<surname>Harrington</surname>
<given-names>M. J.</given-names>
</name>
</person-group> (<year>2021</year>). <article-title>Collagen pentablock copolymers form smectic liquid crystals as precursors for mussel byssus fabrication</article-title>. <source>ACS Nano</source> <volume>15</volume>, <fpage>6829</fpage>&#x2013;<lpage>6838</lpage>. <pub-id pub-id-type="doi">10.1021/acsnano.0c10457</pub-id>
</citation>
</ref>
<ref id="B59">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Khadem</surname>
<given-names>S. A.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2021</year>). <article-title>Nucleation and growth of cholesteric collagen tactoids: a time-series statistical analysis based on integration of direct numerical simulation (DNS) and long short-term memory recurrent neural network (LSTM-RNN)</article-title>. <source>J. Colloid Interface Sci.</source> <volume>582</volume>, <fpage>859</fpage>&#x2013;<lpage>873</lpage>. <pub-id pub-id-type="doi">10.1016/j.jcis.2020.08.052</pub-id>
</citation>
</ref>
<ref id="B60">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Khan</surname>
<given-names>B. C.</given-names>
</name>
<name>
<surname>Mukherjee</surname>
<given-names>P. K.</given-names>
</name>
</person-group> (<year>2021</year>). <article-title>Isotropic to smectic-A phase transition in taper-shaped liquid crystal</article-title>. <source>J. Mol. Liq.</source> <volume>329</volume>, <fpage>115539</fpage>. <pub-id pub-id-type="doi">10.1016/j.molliq.2021.115539</pub-id>
</citation>
</ref>
<ref id="B61">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Knight</surname>
<given-names>D.</given-names>
</name>
<name>
<surname>Vollrath</surname>
<given-names>F.</given-names>
</name>
</person-group> (<year>1999</year>). <article-title>Hexagonal columnar liquid crystal in the cells secreting spider silk</article-title>. <source>Tissue Cell.</source> <volume>31</volume>, <fpage>617</fpage>&#x2013;<lpage>620</lpage>. <pub-id pub-id-type="doi">10.1054/tice.1999.0076</pub-id>
</citation>
</ref>
<ref id="B62">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Knill</surname>
<given-names>O.</given-names>
</name>
</person-group> (<year>2012</year>). <article-title>A graph theoretical Poincar&#xe9;-Hopf theorem</article-title>. <source>arXiv Prepr. arXiv:1201.1162</source>. <pub-id pub-id-type="doi">10.48550/arXiv.1201.1162</pub-id>
</citation>
</ref>
<ref id="B63">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Koenderink</surname>
<given-names>J. J.</given-names>
</name>
<name>
<surname>van Doorn</surname>
<given-names>A. J.</given-names>
</name>
</person-group> (<year>1992</year>). <article-title>Surface shape and curvature scales</article-title>. <source>Image Vis. Comput.</source> <volume>10</volume>, <fpage>557</fpage>&#x2013;<lpage>564</lpage>. <pub-id pub-id-type="doi">10.1016/0262-8856(92)90076-f</pub-id>
</citation>
</ref>
<ref id="B64">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Kyrylyuk</surname>
<given-names>A. V.</given-names>
</name>
<name>
<surname>Anne van de Haar</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Rossi</surname>
<given-names>L.</given-names>
</name>
<name>
<surname>Wouterse</surname>
<given-names>A.</given-names>
</name>
<name>
<surname>Philipse</surname>
<given-names>A. P.</given-names>
</name>
</person-group> (<year>2011</year>). <article-title>Isochoric ideality in jammed random packings of non-spherical granular matter</article-title>. <source>Soft Matter</source> <volume>7</volume>, <fpage>1671</fpage>&#x2013;<lpage>1674</lpage>. <pub-id pub-id-type="doi">10.1039/c0sm00754d</pub-id>
</citation>
</ref>
<ref id="B65">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Lagerwall</surname>
<given-names>J. P.</given-names>
</name>
</person-group> (<year>2016</year>). <article-title>An introduction to the physics of liquid crystals</article-title>. <source>Fluids, Colloids Soft Mater. Introd. Soft Matter Phys.</source>, <fpage>307</fpage>&#x2013;<lpage>340</lpage>. <pub-id pub-id-type="doi">10.1002/9781119220510.ch16</pub-id>
</citation>
</ref>
<ref id="B66">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Larin</surname>
<given-names>E. S.</given-names>
</name>
</person-group> (<year>2004</year>). <article-title>Phase diagram of transitions from an isotropic phase to nematic and smectic (uniaxial, biaxial) phases in liquid crystals with achiral molecules</article-title>. <source>Phys. Solid State</source> <volume>46</volume>, <fpage>1560</fpage>&#x2013;<lpage>1568</lpage>. <pub-id pub-id-type="doi">10.1134/1.1788795</pub-id>
</citation>
</ref>
<ref id="B67">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Lenoble</surname>
<given-names>J.</given-names>
</name>
<name>
<surname>Campidelli</surname>
<given-names>S.</given-names>
</name>
<name>
<surname>Maringa</surname>
<given-names>N.</given-names>
</name>
<name>
<surname>Donnio</surname>
<given-names>B.</given-names>
</name>
<name>
<surname>Guillon</surname>
<given-names>D.</given-names>
</name>
<name>
<surname>Yevlampieva</surname>
<given-names>N.</given-names>
</name>
<etal/>
</person-group> (<year>2007</year>). <article-title>Liquid&#x2212; crystalline Janus-type fullerodendrimers displaying tunable smectic&#x2212; columnar mesomorphism</article-title>. <source>J. Am. Chem. Soc.</source> <volume>129</volume>, <fpage>9941</fpage>&#x2013;<lpage>9952</lpage>. <pub-id pub-id-type="doi">10.1021/ja071012o</pub-id>
</citation>
</ref>
<ref id="B68">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Li</surname>
<given-names>C.-Z.</given-names>
</name>
<name>
<surname>Matsuo</surname>
<given-names>Y.</given-names>
</name>
<name>
<surname>Nakamura</surname>
<given-names>E.</given-names>
</name>
</person-group> (<year>2009</year>). <article-title>Luminescent bow-tie-shaped decaaryl[60]fullerene mesogens</article-title>. <source>J. Am. Chem. Soc.</source> <volume>131</volume>, <fpage>17058</fpage>&#x2013;<lpage>17059</lpage>. <pub-id pub-id-type="doi">10.1021/ja907908m</pub-id>
</citation>
</ref>
<ref id="B69">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Liu</surname>
<given-names>B.</given-names>
</name>
<name>
<surname>Besseling</surname>
<given-names>T. H.</given-names>
</name>
<name>
<surname>Hermes</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Demir&#xf6;rs</surname>
<given-names>A. F.</given-names>
</name>
<name>
<surname>Imhof</surname>
<given-names>A.</given-names>
</name>
<name>
<surname>van Blaaderen</surname>
<given-names>A.</given-names>
</name>
</person-group> (<year>2014</year>). <article-title>Switching plastic crystals of colloidal rods with electric fields</article-title>. <source>Nat. Commun.</source> <volume>5</volume>, <fpage>3092</fpage>. <pub-id pub-id-type="doi">10.1038/ncomms4092</pub-id>
</citation>
</ref>
<ref id="B70">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Liu</surname>
<given-names>X.</given-names>
</name>
<name>
<surname>Chen</surname>
<given-names>H.</given-names>
</name>
<name>
<surname>Ortner</surname>
<given-names>C.</given-names>
</name>
</person-group> (<year>2022</year>). <article-title>Stability of the minimum energy path</article-title>. <source>arXiv Prepr. arXiv:2204.00984</source>. <pub-id pub-id-type="doi">10.1007/s00211-023-01391-7</pub-id>
</citation>
</ref>
<ref id="B71">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Maekawa</surname>
<given-names>T.</given-names>
</name>
</person-group> (<year>1996</year>). <source>Computation of shortest paths on free-form parametric surfaces</source>.</citation>
</ref>
<ref id="B72">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Manolakis</surname>
<given-names>I.</given-names>
</name>
<name>
<surname>Azhar</surname>
<given-names>U.</given-names>
</name>
</person-group> (<year>2020</year>). <article-title>Recent advances in mussel-inspired synthetic polymers as marine antifouling coatings</article-title>. <source>Coatings</source> <volume>10</volume>, <fpage>653</fpage>. <pub-id pub-id-type="doi">10.3390/coatings10070653</pub-id>
</citation>
</ref>
<ref id="B73">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Massi</surname>
<given-names>F.</given-names>
</name>
<name>
<surname>Straub</surname>
<given-names>J. E.</given-names>
</name>
</person-group> (<year>2001</year>). <article-title>Energy landscape theory for Alzheimer&#x27;s amyloid &#x3b2;-peptide fibril elongation</article-title>. <source>Proteins Struct. Funct. Bioinforma.</source> <volume>42</volume>, <fpage>217</fpage>&#x2013;<lpage>229</lpage>. <pub-id pub-id-type="doi">10.1002/1097-0134(20010201)42:2&#x3c;217::aid-prot90&#x3e;3.0.co;2-n</pub-id>
</citation>
</ref>
<ref id="B74">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Matthews</surname>
<given-names>J. A.</given-names>
</name>
<name>
<surname>Wnek</surname>
<given-names>G. E.</given-names>
</name>
<name>
<surname>Simpson</surname>
<given-names>D. G.</given-names>
</name>
<name>
<surname>Bowlin</surname>
<given-names>G. L.</given-names>
</name>
</person-group> (<year>2002</year>). <article-title>Electrospinning of collagen nanofibers</article-title>. <source>Biomacromolecules</source> <volume>3</volume>, <fpage>232</fpage>&#x2013;<lpage>238</lpage>. <pub-id pub-id-type="doi">10.1021/bm015533u</pub-id>
</citation>
</ref>
<ref id="B75">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Milette</surname>
<given-names>J.</given-names>
</name>
<name>
<surname>Toader</surname>
<given-names>V.</given-names>
</name>
<name>
<surname>Soul&#xe9;</surname>
<given-names>E. R.</given-names>
</name>
<name>
<surname>Lennox</surname>
<given-names>R. B.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
<name>
<surname>Reven</surname>
<given-names>L.</given-names>
</name>
</person-group> (<year>2013</year>). <article-title>A molecular and thermodynamic view of the assembly of gold nanoparticles in nematic liquid crystal</article-title>. <source>Langmuir</source> <volume>29</volume>, <fpage>1258</fpage>&#x2013;<lpage>1263</lpage>. <pub-id pub-id-type="doi">10.1021/la304189n</pub-id>
</citation>
</ref>
<ref id="B76">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Miller</surname>
<given-names>W. L.</given-names>
</name>
</person-group> (<year>1925</year>). <article-title>The method of willard gibbs in chemical thermodynamics</article-title>. <source>Chem. Rev.</source> <volume>1</volume>, <fpage>293</fpage>&#x2013;<lpage>344</lpage>. <pub-id pub-id-type="doi">10.1021/cr60004a001</pub-id>
</citation>
</ref>
<ref id="B77">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Mohieddin Abukhdeir</surname>
<given-names>N.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2008a</year>). <article-title>Defect kinetics and dynamics of pattern coarsening in a two-dimensional smectic-A system</article-title>. <source>New J. Phys.</source> <volume>10</volume>, <fpage>063025</fpage>. <pub-id pub-id-type="doi">10.1088/1367-2630/10/6/063025</pub-id>
</citation>
</ref>
<ref id="B78">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Mohieddin Abukhdeir</surname>
<given-names>N.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> <year>2008b</year>. <article-title>Modeling the isotropic/smectic-C tilted lamellar liquid crystalline transition</article-title>.</citation>
</ref>
<ref id="B80">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Mukherjee</surname>
<given-names>P. K.</given-names>
</name>
</person-group> (<year>2014</year>). <article-title>Isotropic to smectic-A phase transition: a review</article-title>. <source>J. Mol. Liq.</source> <volume>190</volume>, <fpage>99</fpage>&#x2013;<lpage>111</lpage>. <pub-id pub-id-type="doi">10.1016/j.molliq.2013.11.001</pub-id>
</citation>
</ref>
<ref id="B81">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Mukherjee</surname>
<given-names>P. K.</given-names>
</name>
</person-group> (<year>2021</year>). <article-title>Advances of isotropic to smectic phase transitions</article-title>. <source>J. Mol. Liq.</source> <volume>340</volume>, <fpage>117227</fpage>. <pub-id pub-id-type="doi">10.1016/j.molliq.2021.117227</pub-id>
</citation>
</ref>
<ref id="B82">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Mukherjee</surname>
<given-names>P. K.</given-names>
</name>
<name>
<surname>Pleiner</surname>
<given-names>H.</given-names>
</name>
<name>
<surname>Brand</surname>
<given-names>H. R.</given-names>
</name>
</person-group> (<year>2001</year>). <article-title>Simple Landau model of the smectic-A-isotropic phase transition</article-title>. <source>Eur. Phys. J. E</source> <volume>4</volume>, <fpage>293</fpage>&#x2013;<lpage>297</lpage>. <pub-id pub-id-type="doi">10.1007/s101890170111</pub-id>
</citation>
</ref>
<ref id="B83">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Nandi</surname>
<given-names>B.</given-names>
</name>
<name>
<surname>Saha</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Mukherjee</surname>
<given-names>P. K.</given-names>
</name>
</person-group> (<year>2012</year>). <article-title>Landau theory of the direct smectic-A to isotropic phase transition</article-title>. <source>Int. J. Mod. Phys. B</source> <volume>11</volume>, <fpage>2425</fpage>&#x2013;<lpage>2432</lpage>. <pub-id pub-id-type="doi">10.1142/s0217979297001234</pub-id>
</citation>
</ref>
<ref id="B84">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Nesrullajev</surname>
<given-names>A.</given-names>
</name>
</person-group> (<year>2022</year>). <article-title>Optical refracting properties, birefringence and order parameter in mixtures of liquid crystals: direct smectic A &#x2013; Isotropic and reverse isotropic &#x2013; smectic A phase transitions</article-title>. <source>J. Mol. Liq.</source> <volume>345</volume>, <fpage>117716</fpage>. <pub-id pub-id-type="doi">10.1016/j.molliq.2021.117716</pub-id>
</citation>
</ref>
<ref id="B85">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Oh</surname>
<given-names>C. S.</given-names>
</name>
</person-group> (<year>1977</year>). <article-title>Induced smectic mesomorphism by incompatible nematogens</article-title>. <source>Mol. Cryst. Liq. Cryst.</source> <volume>42</volume>, <fpage>1</fpage>&#x2013;<lpage>14</lpage>. <pub-id pub-id-type="doi">10.1080/15421407708084491</pub-id>
</citation>
</ref>
<ref id="B86">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Oswald</surname>
<given-names>P.</given-names>
</name>
<name>
<surname>Pieranski</surname>
<given-names>P.</given-names>
</name>
</person-group> (<year>2005a</year>). <source>Nematic and cholesteric liquid crystals: concepts and physical properties illustrated by experiments</source>. <publisher-loc>China</publisher-loc>: <publisher-name>CRC Press</publisher-name>.</citation>
</ref>
<ref id="B87">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Oswald</surname>
<given-names>P.</given-names>
</name>
<name>
<surname>Pieranski</surname>
<given-names>P.</given-names>
</name>
</person-group> (<year>2005b</year>). <source>Smectic and columnar liquid crystals</source>.</citation>
</ref>
<ref id="B88">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Paget</surname>
<given-names>J.</given-names>
</name>
<name>
<surname>Alberti</surname>
<given-names>U.</given-names>
</name>
<name>
<surname>Mazza</surname>
<given-names>M. G.</given-names>
</name>
<name>
<surname>Archer</surname>
<given-names>A. J.</given-names>
</name>
<name>
<surname>Shendruk</surname>
<given-names>T. N.</given-names>
</name>
</person-group> (<year>2022</year>). <article-title>Smectic layering: Landau theory for a complex-tensor order parameter</article-title>. <source>J. Phys. A Math. Theor.</source> <volume>55</volume>, <fpage>354001</fpage>. <pub-id pub-id-type="doi">10.1088/1751-8121/ac80df</pub-id>
</citation>
</ref>
<ref id="B89">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Palffy-Muhoray</surname>
<given-names>P.</given-names>
</name>
</person-group> (<year>1999</year>). <article-title>Dynamics of filaments during the isotropic-smectic A phase transition</article-title>. <source>J. Nonlinear Sci.</source> <volume>9</volume>, <fpage>417</fpage>&#x2013;<lpage>437</lpage>. <pub-id pub-id-type="doi">10.1007/s003329900075</pub-id>
</citation>
</ref>
<ref id="B90">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Petrov</surname>
<given-names>A. G.</given-names>
</name>
</person-group> (<year>2013</year>). <article-title>Flexoelectricity in lyotropics and in living liquid crystals</article-title>. <source>Flexoelectricity Liq. Cryst. theory, Exp. Appl. World Sci</source>. <pub-id pub-id-type="doi">10.1142/9781848168008_0007</pub-id>
</citation>
</ref>
<ref id="B91">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Pevnyi</surname>
<given-names>M. Y.</given-names>
</name>
<name>
<surname>Selinger</surname>
<given-names>J. V.</given-names>
</name>
<name>
<surname>Sluckin</surname>
<given-names>T. J.</given-names>
</name>
</person-group> (<year>2014</year>). <article-title>Modeling smectic layers in confined geometries: order parameter and defects</article-title>. <source>Phys. Rev. E Stat. Nonlin Soft Matter Phys.</source> <volume>90</volume>, <fpage>032507</fpage>. <pub-id pub-id-type="doi">10.1103/physreve.90.032507</pub-id>
</citation>
</ref>
<ref id="B92">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Picken</surname>
<given-names>S. J.</given-names>
</name>
</person-group> (<year>1990</year>). <article-title>Orientational order in aramid solutions determined by diamagnetic susceptibility and birefringence measurements</article-title>. <source>Macromolecules</source> <volume>23</volume>, <fpage>464</fpage>&#x2013;<lpage>470</lpage>. <pub-id pub-id-type="doi">10.1021/ma00204a019</pub-id>
</citation>
</ref>
<ref id="B93">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Pikin</surname>
<given-names>S. A.</given-names>
</name>
</person-group> (<year>1991</year>). <source>Structural transformations in liquid crystals</source>.</citation>
</ref>
<ref id="B94">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Pleiner</surname>
<given-names>H.</given-names>
</name>
<name>
<surname>Mukherjee</surname>
<given-names>P. K.</given-names>
</name>
<name>
<surname>Brand</surname>
<given-names>H. R.</given-names>
</name>
</person-group> (<year>2000</year>). <article-title>Direct transitions from isotropic to smectic phases</article-title>. <source>Proc. Freiburger Arbeitstagung Flussigkristalle</source>, <fpage>P59</fpage>.</citation>
</ref>
<ref id="B95">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Popa-Nita</surname>
<given-names>V.</given-names>
</name>
</person-group> (<year>1999</year>). <article-title>Statics and kinetics at the nematic-isotropic interface in porous media</article-title>. <source>Eur. Phys. J. B-Condensed Matter Complex Syst.</source> <volume>12</volume>, <fpage>83</fpage>&#x2013;<lpage>90</lpage>. <pub-id pub-id-type="doi">10.1007/s100510050981</pub-id>
</citation>
</ref>
<ref id="B96">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Popa-Nita</surname>
<given-names>V.</given-names>
</name>
<name>
<surname>Sluckin</surname>
<given-names>T. J.</given-names>
</name>
</person-group> (<year>2007</year>). <source>Waves at the nematic-isotropic interface: nematic-non-nematic and polymer-nematic mixtures</source>. <publisher-loc>Netherlands</publisher-loc>: <publisher-name>Springer</publisher-name>, <fpage>253</fpage>&#x2013;<lpage>267</lpage>.</citation>
</ref>
<ref id="B97">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Pouget</surname>
<given-names>E.</given-names>
</name>
<name>
<surname>Grelet</surname>
<given-names>E.</given-names>
</name>
<name>
<surname>Lettinga</surname>
<given-names>M. P.</given-names>
</name>
</person-group> (<year>2011</year>). <article-title>Dynamics in the smectic phase of stiff viral rods</article-title>. <source>Phys. Rev. E Stat. Nonlin Soft Matter Phys.</source> <volume>84</volume>, <fpage>041704</fpage>. <pub-id pub-id-type="doi">10.1103/physreve.84.041704</pub-id>
</citation>
</ref>
<ref id="B98">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Quevedo</surname>
<given-names>H.</given-names>
</name>
<name>
<surname>Quevedo</surname>
<given-names>M. N.</given-names>
</name>
<name>
<surname>S&#xe1;nchez</surname>
<given-names>A.</given-names>
</name>
</person-group> (<year>2022</year>). <article-title>Geometrothermodynamics of van der Waals systems</article-title>. <source>J. Geometry Phys.</source> <volume>176</volume>, <fpage>104495</fpage>. <pub-id pub-id-type="doi">10.1016/j.geomphys.2022.104495</pub-id>
</citation>
</ref>
<ref id="B99">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Quevedo</surname>
<given-names>H.</given-names>
</name>
<name>
<surname>S&#xe1;nchez</surname>
<given-names>A.</given-names>
</name>
<name>
<surname>Taj</surname>
<given-names>S.</given-names>
</name>
<name>
<surname>V&#xe1;zquez</surname>
<given-names>A.</given-names>
</name>
</person-group> (<year>2011</year>). <article-title>Phase transitions in geometrothermodynamics</article-title>. <source>General Relativ. Gravit.</source> <volume>43</volume>, <fpage>1153</fpage>&#x2013;<lpage>1165</lpage>. <pub-id pub-id-type="doi">10.1007/s10714-010-0996-2</pub-id>
</citation>
</ref>
<ref id="B100">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Quevedo</surname>
<given-names>H.</given-names>
</name>
<name>
<surname>S&#xe1;nchez</surname>
<given-names>A.</given-names>
</name>
<name>
<surname>V&#xe1;zquez</surname>
<given-names>A.</given-names>
</name>
</person-group> (<year>2008</year>). <article-title>Invariant geometry of the ideal gas</article-title>. <source>arXiv Prepr. arXiv:0811.0222</source>.</citation>
</ref>
<ref id="B101">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Renner-Rao</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Clark</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Harrington</surname>
<given-names>M. J.</given-names>
</name>
</person-group> (<year>2019</year>). <article-title>Fiber Formation from liquid crystalline collagen vesicles isolated from mussels</article-title>. <source>Langmuir</source> <volume>35</volume>, <fpage>15992</fpage>&#x2013;<lpage>16001</lpage>. <pub-id pub-id-type="doi">10.1021/acs.langmuir.9b01932</pub-id>
</citation>
</ref>
<ref id="B102">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>1995</year>). <article-title>Bifurcational analysis of the isotropic-discotic nematic phase transition in the presence of extensional flow</article-title>. <source>Liq. Cryst.</source> <volume>19</volume>, <fpage>325</fpage>&#x2013;<lpage>331</lpage>. <pub-id pub-id-type="doi">10.1080/02678299508031988</pub-id>
</citation>
</ref>
<ref id="B103">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2000</year>). <article-title>Viscoelastic theory for nematic interfaces</article-title>. <source>Phys. Rev. E</source> <volume>61</volume>, <fpage>1540</fpage>&#x2013;<lpage>1549</lpage>. <pub-id pub-id-type="doi">10.1103/physreve.61.1540</pub-id>
</citation>
</ref>
<ref id="B104">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2004a</year>). <article-title>Interfacial thermodynamics of polymeric mesophases</article-title>. <source>Macromol. theory simulations</source> <volume>13</volume>, <fpage>686</fpage>&#x2013;<lpage>696</lpage>. <pub-id pub-id-type="doi">10.1002/mats.200400030</pub-id>
</citation>
</ref>
<ref id="B105">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2004b</year>). <article-title>Thermodynamics of soft anisotropic interfaces</article-title>. <source>J. Chem. Phys.</source> <volume>120</volume>, <fpage>2010</fpage>&#x2013;<lpage>2019</lpage>. <pub-id pub-id-type="doi">10.1063/1.1635357</pub-id>
</citation>
</ref>
<ref id="B106">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2006</year>). <article-title>Mechanical model for anisotropic curved interfaces with applications to surfactant-laden Liquid&#x2212; liquid crystal interfaces</article-title>. <source>Langmuir</source> <volume>22</volume>, <fpage>219</fpage>&#x2013;<lpage>228</lpage>. <pub-id pub-id-type="doi">10.1021/la051974d</pub-id>
</citation>
</ref>
<ref id="B107">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2010</year>). <article-title>Liquid crystal models of biological materials and processes</article-title>. <source>Soft Matter</source> <volume>6</volume>, <fpage>3402</fpage>&#x2013;<lpage>3429</lpage>. <pub-id pub-id-type="doi">10.1039/b921576j</pub-id>
</citation>
</ref>
<ref id="B108">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
<name>
<surname>Denn</surname>
<given-names>M. M.</given-names>
</name>
</person-group> (<year>2002</year>). <article-title>Dynamical phenomena in liquid-crystalline materials</article-title>. <source>Annu. Rev. Fluid Mech.</source> <volume>34</volume>, <fpage>233</fpage>&#x2013;<lpage>266</lpage>. <pub-id pub-id-type="doi">10.1146/annurev.fluid.34.082401.191847</pub-id>
</citation>
</ref>
<ref id="B109">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
<name>
<surname>Herrera-Valencia</surname>
<given-names>E. E.</given-names>
</name>
</person-group> (<year>2012</year>). <article-title>Liquid crystal models of biological materials and silk spinning</article-title>. <source>Biopolymers</source> <volume>97</volume>, <fpage>374</fpage>&#x2013;<lpage>396</lpage>. <pub-id pub-id-type="doi">10.1002/bip.21723</pub-id>
</citation>
</ref>
<ref id="B110">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
<name>
<surname>Herrera-Valencia</surname>
<given-names>E. E.</given-names>
</name>
<name>
<surname>Murugesan</surname>
<given-names>Y. K.</given-names>
</name>
</person-group> (<year>2013</year>). <article-title>Structure and dynamics of biological liquid crystals</article-title>. <source>Liq. Cryst.</source> <volume>41</volume>, <fpage>430</fpage>&#x2013;<lpage>451</lpage>. <pub-id pub-id-type="doi">10.1080/02678292.2013.845698</pub-id>
</citation>
</ref>
<ref id="B111">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Salamonczyk</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Zhang</surname>
<given-names>J.</given-names>
</name>
<name>
<surname>Portale</surname>
<given-names>G.</given-names>
</name>
<name>
<surname>Zhu</surname>
<given-names>C.</given-names>
</name>
<name>
<surname>Kentzinger</surname>
<given-names>E.</given-names>
</name>
<name>
<surname>Gleeson</surname>
<given-names>J. T.</given-names>
</name>
<etal/>
</person-group> (<year>2016</year>). <article-title>Smectic phase in suspensions of gapped DNA duplexes</article-title>. <source>Nat. Commun.</source> <volume>7</volume>, <fpage>13358</fpage>. <pub-id pub-id-type="doi">10.1038/ncomms13358</pub-id>
</citation>
</ref>
<ref id="B112">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Sato</surname>
<given-names>C.</given-names>
</name>
<name>
<surname>Takeda</surname>
<given-names>T.</given-names>
</name>
<name>
<surname>Dekura</surname>
<given-names>S.</given-names>
</name>
<name>
<surname>Suzuki</surname>
<given-names>Y.</given-names>
</name>
<name>
<surname>Kawamata</surname>
<given-names>J.</given-names>
</name>
<name>
<surname>Akutagawa</surname>
<given-names>T.</given-names>
</name>
</person-group> (<year>2023</year>). <article-title>Chiral plastic crystal of solid-state dual rotators</article-title>. <source>Cryst. Growth and Des.</source> <volume>23</volume>, <fpage>5889</fpage>&#x2013;<lpage>5898</lpage>. <pub-id pub-id-type="doi">10.1021/acs.cgd.3c00495</pub-id>
</citation>
</ref>
<ref id="B113">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Saunders</surname>
<given-names>K.</given-names>
</name>
<name>
<surname>Hernandez</surname>
<given-names>D.</given-names>
</name>
<name>
<surname>Pearson</surname>
<given-names>S.</given-names>
</name>
<name>
<surname>Toner</surname>
<given-names>J.</given-names>
</name>
</person-group> (<year>2007</year>). <article-title>Disordering to order: de Vries behavior from a Landau theory for smectic phases</article-title>. <source>Phys. Rev. Lett.</source> <volume>98</volume>, <fpage>197801</fpage>. <pub-id pub-id-type="doi">10.1103/physrevlett.98.197801</pub-id>
</citation>
</ref>
<ref id="B114">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Schimming</surname>
<given-names>C. D.</given-names>
</name>
<name>
<surname>Vi&#xf1;als</surname>
<given-names>J.</given-names>
</name>
<name>
<surname>Walker</surname>
<given-names>S. W.</given-names>
</name>
</person-group> (<year>2021</year>). <article-title>Numerical method for the equilibrium configurations of a Maier-Saupe bulk potential in a Q-tensor model of an anisotropic nematic liquid crystal</article-title>. <source>J. Comput. Phys.</source> <volume>441</volume>, <fpage>110441</fpage>. <pub-id pub-id-type="doi">10.1016/j.jcp.2021.110441</pub-id>
</citation>
</ref>
<ref id="B115">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Selinger</surname>
<given-names>J. V.</given-names>
</name>
</person-group> (<year>2016</year>). <source>Introduction to the theory of soft matter: from ideal gases to liquid crystals</source>. <publisher-loc>Germany</publisher-loc>: <publisher-name>Springer</publisher-name>.</citation>
</ref>
<ref id="B116">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Sonnet</surname>
<given-names>A. M.</given-names>
</name>
<name>
<surname>Virga</surname>
<given-names>E. G.</given-names>
</name>
</person-group> (<year>2012</year>). <source>Dissipative ordered fluids: theories for liquid crystals</source>. <publisher-loc>Germany</publisher-loc>: <publisher-name>Springer Science and Business Media</publisher-name>.</citation>
</ref>
<ref id="B117">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Soul&#xe9;</surname>
<given-names>E. R.</given-names>
</name>
<name>
<surname>Lavigne</surname>
<given-names>C.</given-names>
</name>
<name>
<surname>Reven</surname>
<given-names>L.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2012a</year>). <article-title>Multiple interfaces in diffusional phase transitions in binary mesogen-nonmesogen mixtures undergoing metastable phase separations</article-title>. <source>Phys. Rev. E</source> <volume>86</volume>, <fpage>011605</fpage>. <pub-id pub-id-type="doi">10.1103/physreve.86.011605</pub-id>
</citation>
</ref>
<ref id="B118">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Soul&#xe9;</surname>
<given-names>E. R.</given-names>
</name>
<name>
<surname>Milette</surname>
<given-names>J.</given-names>
</name>
<name>
<surname>Reven</surname>
<given-names>L.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2012b</year>). <article-title>Phase equilibrium and structure formation in gold nanoparticles&#x2014;nematic liquid crystal composites: experiments and theory</article-title>. <source>Soft Matter</source> <volume>8</volume>, <fpage>2860</fpage>&#x2013;<lpage>2866</lpage>. <pub-id pub-id-type="doi">10.1039/c2sm07091j</pub-id>
</citation>
</ref>
<ref id="B119">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Soul&#xe9;</surname>
<given-names>E. R.</given-names>
</name>
<name>
<surname>Reven</surname>
<given-names>L.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2012c</year>). <article-title>Thermodynamic modelling of phase equilibrium in nanoparticles&#x2013;nematic liquid crystals composites</article-title>. <source>Mol. Cryst. Liq. Cryst.</source> <volume>553</volume>, <fpage>118</fpage>&#x2013;<lpage>126</lpage>. <pub-id pub-id-type="doi">10.1080/15421406.2011.609447</pub-id>
</citation>
</ref>
<ref id="B120">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Soule</surname>
<given-names>E. R.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2011</year>). <article-title>A good and computationally efficient polynomial approximation to the Maier&#x2013;Saupe nematic free energy</article-title>. <source>Liq. Cryst.</source> <volume>38</volume>, <fpage>201</fpage>&#x2013;<lpage>205</lpage>. <pub-id pub-id-type="doi">10.1080/02678292.2010.539303</pub-id>
</citation>
</ref>
<ref id="B121">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Soule</surname>
<given-names>E. R.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2012</year>). <article-title>Modelling complex liquid crystal mixtures: from polymer dispersed mesophase to nematic nanocolloids</article-title>. <source>Mol. Simul.</source> <volume>38</volume>, <fpage>735</fpage>&#x2013;<lpage>750</lpage>. <pub-id pub-id-type="doi">10.1080/08927022.2012.669478</pub-id>
</citation>
</ref>
<ref id="B122">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Stewart</surname>
<given-names>I. W.</given-names>
</name>
</person-group> (<year>2019</year>). <source>The static and dynamic continuum theory of liquid crystals: a mathematical introduction</source>. <publisher-loc>Germany</publisher-loc>: <publisher-name>Crc Press</publisher-name>.</citation>
</ref>
<ref id="B123">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Tortora</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Jost</surname>
<given-names>D.</given-names>
</name>
</person-group> (<year>2021</year>). <article-title>Morphogenesis and self-organization of persistent filaments confined within flexible biopolymeric shells</article-title>. <source>arXiv Prepr.</source> <pub-id pub-id-type="doi">10.48550/arXiv.2107.02598</pub-id>
</citation>
</ref>
<ref id="B124">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Tuckerman</surname>
<given-names>L. S.</given-names>
</name>
<name>
<surname>Bechhoefer</surname>
<given-names>J.</given-names>
</name>
</person-group> (<year>1992</year>). <article-title>Dynamical mechanism for the formation of metastable phases: the case of two nonconserved order parameters</article-title>. <source>Phys. Rev. A</source> <volume>46</volume>, <fpage>3178</fpage>&#x2013;<lpage>3192</lpage>. <pub-id pub-id-type="doi">10.1103/physreva.46.3178</pub-id>
</citation>
</ref>
<ref id="B125">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Turek</surname>
<given-names>D. E.</given-names>
</name>
<name>
<surname>Simon</surname>
<given-names>G. P.</given-names>
</name>
<name>
<surname>Tiu</surname>
<given-names>C.</given-names>
</name>
</person-group> (<year>2020</year>). &#x201c;<article-title>Relationships among rheology, morphology, and solid-state properties in thermotropic liquid-crystalline polymers</article-title>,&#x201d; in <source>Handbook of applied polymer processing technology</source> (<publisher-loc>Germany</publisher-loc>: <publisher-name>CRC Press</publisher-name>).</citation>
</ref>
<ref id="B126">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Urban</surname>
<given-names>S.</given-names>
</name>
<name>
<surname>Przedmojski</surname>
<given-names>J.</given-names>
</name>
<name>
<surname>Czub</surname>
<given-names>J.</given-names>
</name>
</person-group> (<year>2005</year>). <article-title>X-ray studies of the layer thickness in smectic phases</article-title>. <source>Liq. Cryst.</source> <volume>32</volume>, <fpage>619</fpage>&#x2013;<lpage>624</lpage>. <pub-id pub-id-type="doi">10.1080/02678290500116920</pub-id>
</citation>
</ref>
<ref id="B127">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Viney</surname>
<given-names>C.</given-names>
</name>
</person-group> (<year>2004</year>). <article-title>Self-assembly as a route to fibrous materials: concepts, opportunities and challenges</article-title>. <source>Curr. Opin. Solid State and Mater. Sci.</source> <volume>8</volume>, <fpage>95</fpage>&#x2013;<lpage>101</lpage>. <pub-id pub-id-type="doi">10.1016/j.cossms.2004.04.001</pub-id>
</citation>
</ref>
<ref id="B128">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Vitral</surname>
<given-names>E.</given-names>
</name>
<name>
<surname>Leo</surname>
<given-names>P. H.</given-names>
</name>
<name>
<surname>Vinals</surname>
<given-names>J.</given-names>
</name>
</person-group> (<year>2019</year>). <article-title>Role of Gaussian curvature on local equilibrium and dynamics of smectic-isotropic interfaces</article-title>. <source>Phys. Rev. E</source> <volume>100</volume>, <fpage>032805</fpage>. <pub-id pub-id-type="doi">10.1103/physreve.100.032805</pub-id>
</citation>
</ref>
<ref id="B129">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Vitral</surname>
<given-names>E.</given-names>
</name>
<name>
<surname>Leo</surname>
<given-names>P. H.</given-names>
</name>
<name>
<surname>Vinals</surname>
<given-names>J.</given-names>
</name>
</person-group> (<year>2020</year>). <article-title>Model of the dynamics of an interface between a smectic phase and an isotropic phase of different density</article-title>. <source>Phys. Rev. Fluids</source> <volume>5</volume>, <fpage>073302</fpage>. <pub-id pub-id-type="doi">10.1103/physrevfluids.5.073302</pub-id>
</citation>
</ref>
<ref id="B130">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Waite</surname>
<given-names>J. H.</given-names>
</name>
<name>
<surname>Harrington</surname>
<given-names>M. J.</given-names>
</name>
</person-group> (<year>2022</year>). <article-title>Following the thread: Mytilus mussel byssus as an inspired multi-functional biomaterial</article-title>. <source>Can. J. Chem.</source> <volume>100</volume>, <fpage>197</fpage>&#x2013;<lpage>211</lpage>. <pub-id pub-id-type="doi">10.1139/cjc-2021-0191</pub-id>
</citation>
</ref>
<ref id="B131">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Wales</surname>
<given-names>D. J.</given-names>
</name>
</person-group> (<year>2018</year>). <article-title>Exploring energy landscapes</article-title>. <source>Annu. Rev. Phys. Chem.</source> <volume>69</volume>, <fpage>401</fpage>&#x2013;<lpage>425</lpage>. <pub-id pub-id-type="doi">10.1146/annurev-physchem-050317-021219</pub-id>
</citation>
</ref>
<ref id="B132">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Wang</surname>
<given-names>H. Y.</given-names>
</name>
<name>
<surname>Wang</surname>
<given-names>Y. Z.</given-names>
</name>
<name>
<surname>Tsakalakos</surname>
<given-names>T.</given-names>
</name>
<name>
<surname>Semenovskaya</surname>
<given-names>S.</given-names>
</name>
<name>
<surname>Khachaturyan</surname>
<given-names>A. G.</given-names>
</name>
</person-group> (<year>1996</year>). <article-title>Indirect nucleation in phase transformations with symmetry reduction</article-title>. <source>Philosophical Mag. a-Physics Condens. Matter Struct. Defects Mech. Prop.</source> <volume>74</volume>, <fpage>1407</fpage>&#x2013;<lpage>1420</lpage>. <pub-id pub-id-type="doi">10.1080/01418619608240732</pub-id>
</citation>
</ref>
<ref id="B133">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Wang</surname>
<given-names>Z.</given-names>
</name>
<name>
<surname>Servio</surname>
<given-names>P.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2020</year>). <article-title>Rate of entropy production in evolving interfaces and membranes under astigmatic kinematics: shape evolution in geometric-dissipation landscapes</article-title>. <source>Entropy</source> <volume>22</volume>, <fpage>909</fpage>. <pub-id pub-id-type="doi">10.3390/e22090909</pub-id>
</citation>
</ref>
<ref id="B134">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Wang</surname>
<given-names>Z.</given-names>
</name>
<name>
<surname>Servio</surname>
<given-names>P.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2022a</year>). <article-title>Complex nanowrinkling in chiral liquid crystal surfaces: from shaping mechanisms to geometric statistics</article-title>. <source>Nanomaterials</source> <volume>12</volume>, <fpage>1555</fpage>. <pub-id pub-id-type="doi">10.3390/nano12091555</pub-id>
</citation>
</ref>
<ref id="B135">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Wang</surname>
<given-names>Z.</given-names>
</name>
<name>
<surname>Servio</surname>
<given-names>P.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2023a</year>). <article-title>Geometry-structure models for liquid crystal interfaces, drops and membranes: wrinkling, shape selection and dissipative shape evolution</article-title>. <source>Soft Matter</source> <volume>19</volume>, <fpage>9344</fpage>&#x2013;<lpage>9364</lpage>. <pub-id pub-id-type="doi">10.1039/d3sm01164j</pub-id>
</citation>
</ref>
<ref id="B136">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Wang</surname>
<given-names>Z.</given-names>
</name>
<name>
<surname>Servio</surname>
<given-names>P.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A. D.</given-names>
</name>
</person-group> (<year>2023b</year>). <article-title>Pattern formation, structure and functionalities of wrinkled liquid crystal surfaces: a soft matter biomimicry platform</article-title>. <source>Front. Soft Matter</source> <volume>3</volume>, <fpage>1123324</fpage>. <pub-id pub-id-type="doi">10.3389/frsfm.2023.1123324</pub-id>
</citation>
</ref>
<ref id="B137">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Wang</surname>
<given-names>Z.</given-names>
</name>
<name>
<surname>Servio</surname>
<given-names>P.</given-names>
</name>
<name>
<surname>Rey</surname>
<given-names>A.</given-names>
</name>
</person-group> (<year>2022b</year>). <article-title>Wrinkling pattern formation with periodic nematic orientation: from egg cartons to corrugated surfaces</article-title>. <source>Phys. Rev. E</source> <volume>105</volume>, <fpage>034702</fpage>. <pub-id pub-id-type="doi">10.1103/physreve.105.034702</pub-id>
</citation>
</ref>
<ref id="B138">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Ward</surname>
<given-names>I.</given-names>
</name>
</person-group> (<year>1993</year>). <source>New developments in the production of high modulus and high strength flexible polymers. Orientational Phenomena in Polymers</source>. <publisher-loc>Germany</publisher-loc>: <publisher-name>Springer</publisher-name>, <fpage>103</fpage>&#x2013;<lpage>110</lpage>.</citation>
</ref>
<ref id="B139">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Wojcik</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Lewandowski</surname>
<given-names>W.</given-names>
</name>
<name>
<surname>Matraszek</surname>
<given-names>J.</given-names>
</name>
<name>
<surname>Mieczkowski</surname>
<given-names>J.</given-names>
</name>
<name>
<surname>Borysiuk</surname>
<given-names>J.</given-names>
</name>
<name>
<surname>Pociecha</surname>
<given-names>D.</given-names>
</name>
<etal/>
</person-group> (<year>2009</year>). <article-title>Liquid-crystalline phases made of gold nanoparticles</article-title>. <source>Angew. Chem. Int. Ed.</source> <volume>48</volume>, <fpage>5167</fpage>&#x2013;<lpage>5169</lpage>. <pub-id pub-id-type="doi">10.1002/anie.200901206</pub-id>
</citation>
</ref>
<ref id="B140">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Zaluzhnyy</surname>
<given-names>I. A.</given-names>
</name>
<name>
<surname>Kurta</surname>
<given-names>R.</given-names>
</name>
<name>
<surname>Sprung</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Vartanyants</surname>
<given-names>I. A.</given-names>
</name>
<name>
<surname>Ostrovskii</surname>
<given-names>B. I.</given-names>
</name>
</person-group> (<year>2022</year>). <article-title>Angular structure factor of the hexatic-B liquid crystals: bridging theory and experiment</article-title>. <source>Soft Matter</source> <volume>18</volume>, <fpage>783</fpage>&#x2013;<lpage>792</lpage>. <pub-id pub-id-type="doi">10.1039/d1sm01446c</pub-id>
</citation>
</ref>
<ref id="B141">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Zannoni</surname>
<given-names>C.</given-names>
</name>
</person-group> (<year>2022</year>). <source>Liquid crystals and their computer simulations</source>. <publisher-loc>Germany</publisher-loc>: <publisher-name>Cambridge University Press</publisher-name>.</citation>
</ref>
<ref id="B142">
<citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname>Zhang</surname>
<given-names>Z.</given-names>
</name>
<name>
<surname>Yang</surname>
<given-names>X.</given-names>
</name>
<name>
<surname>Zhao</surname>
<given-names>Y.</given-names>
</name>
<name>
<surname>Ye</surname>
<given-names>F.</given-names>
</name>
<name>
<surname>Shang</surname>
<given-names>L.</given-names>
</name>
</person-group> (<year>2023</year>). <article-title>Liquid crystal materials for biomedical applications</article-title>. <source>Adv. Mater.</source> <volume>35</volume>, <fpage>2300220</fpage>. <pub-id pub-id-type="doi">10.1002/adma.202300220</pub-id>
</citation>
</ref>
<ref id="B143">
<citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname>Ziabicki</surname>
<given-names>A.</given-names>
</name>
</person-group> (<year>1993</year>). <source>Orientation mechanisms in the development of high-performance fibers Orientational Phenomena in Polymers</source>. <publisher-loc>Germany</publisher-loc>: <publisher-name>Springer</publisher-name>, <fpage>1</fpage>&#x2013;<lpage>7</lpage>.</citation>
</ref>
</ref-list>
</back>
</article>