Structure and Pattern Formation in Biological Liquid Crystals: Insights From Theory and Simulation of Self-Assembly and Self-Organization

1 Introduction

Rod-like lyotropic polymer and colloidal liquid crystals are viscoelastic multifunctional materials that undergo phase ordering transitions with increasing concentration mainly due to hardcore excluded volume interactions, giving rise to mesophases with intermediate degrees of orientational and positional order (Tsuji and Rey, 1997; Lhuillier and Rey, 2004; Donald et al., 2006; Mottram and Newton, 2014; Zhao et al., 2017). The simplest phase ordering transition is the isotropic-to-nematic N transition, predicted by the Onsager model (Lekkerkerker and Vroege, 1993; Selinger, 2016), where the isotropic I phase undergoes an orientational ordering transition characterized by the emergence of molecular or fibrillar alignment (non-zero scalar order parameter S) along an anisotropic axis or director n; this vector is of unit length (n ⋅n = 1) (Donald et al., 2006). In the uniaxial nematic phase (Donald et al., 2006), the characterizing non-conserved order parameter (NCOP) is a symmetric traceless uniaxial tensor Q = S (nn − I/3), where S controls fast time scales and short nano-length scales, and n is the slow variable associated with micron range length scales and unique direction of material property anisotropy (transverse isotropy) (Tsuji and Rey, 1997; Liu and Calderer, 2000; Rey, 2009). Since the uniaxial N phase has no positional order (P₀), anisotropic viscosity is possible and since orientation gradients due to spatial gradients of n (∇n ≠ 0) store energy, these materials are also elastic. The elasticity is known as Frank elasticity and the elastic modes are splay, bend, and twist (Barbero and Evangelista, 2000).

The presence of chemical, geometric and electrostatic chirality gives rise to chiral nematic N* phases, widely observed in in vitro bio macromolecular solutions [DNA, collagen, in in vivo secretions (silk, sperm)], and in liquid crystal solid analogues as collagen in bone, cellulose in plant cell walls, and chitin in insects, which are the focus on this review (Neville and Caveney, 1969; Neville, 1993; Belamie et al., 2006; Bouligand, 2008; Rey, 2010; Rey and Herrera-Valencia, 2012; Lagerwall et al., 2014; Rey et al., 2014; Sharma et al., 2014; Canejo et al., 2017; Mitov, 2017; Priemel et al., 2017; Almeida et al., 2018; Harrington et al., 2018; Khadem et al., 2020; Casado et al., 2021; Harrington and Fratzl, 2021). In these anisotropic viscoelastic materials, chirality introduces spatial periodicity along the helix axis h and an intrinsic periodic length scale or pitch P₀, and hence at the simplest level they can behave as highly viscous along the h direction. For relative long pitch (mm-range) biaxiality can be neglected and the Q-tensor is uniaxial and spatially periodic along the helix direction. The scalar helicity of the N* is H = 4π/P₀ = Q: (∇ ×Q)/(Q: Q), defined by the curl of Q (Khadem et al., 2020).

Figure 1 shows a schematic of the isotropic phase 1), uniaxial nematic N 2) and cholesteric N* 3) local organization and the non-conserved tensor order parameter Q for each case, neglecting fluctuations and assuming that S is the equilibrium value (Donald et al., 2006). The point to emphasize when going from left to right in Figure 1, is that the symmetry breaking increases and the number of distinguishing directions increase from zero to two (n, h).

FIGURE 1

FIGURE 1. Schematics of (A) isotropic, (B) uniaxial nematic, and (C) chiral nematic liquid crystal organization of rod-like fibrillar materials and their uniaxial tensor order parameter Q. The isotropic state is a viscous liquid, the uniaxial nematic N and chiral nematic N* are anisotropic viscoelastic materials. The uniaxial nematic is distinguished by a uniaxial director n₀. The chiral nematic Q-tensor is periodic in the helix h direction. The pitch P₀ of a biological cholesteric liquid crystal is usually in the micron-range.

Representative examples that reflect the abundance and ubiquity of LC order in biological-related materials in vivo and in vitro solutions, as well as frozen-in in the case of biological analogues are given (Rey, 2010; Mitov, 2017). We note that biological analogues are fibrillar composite solids displaying mainly the cholesteric organization, known as the Bouligand architecture (Ritchie, 2014; Suksangpanya et al., 2017; Yang et al., 2017; Natarajan and Gilman, 2018). The fibrous composite Bouligand plywood architecture has a number of advanced functionalities including water sensing (Rofouie et al., 2015) and optimal mechanical fibre reinforcement abilities under uniform planar stress loading normal to the helix direction. In the former case, certain coniferous plants exhibit quantitative blue-to-green color changes according to material water content; the connection with structural colour has been reviewed (Vignolini et al., 2013). In the latter case the reinforcement efficiency of a Bouligand architecture is constant at 0.375 while for a simple nematic order is varies between 0 and 1 (Aguilar Gutierrez and Rey, 2016b). A comprehensive toolbox for structure reconstruction and pitch determinations for the Bouligand and related cornea-like organizations from 2D cross-sectional micrographs has been developed using differential geometry and automatized by visualization software (Gutierrez and Rey, 2014a; Gutierrez and Rey, 2014b; Gutierrez and Rey, 2016). Hence the interest of focusing on biological LCs materials is to use LC science to contribute to the development of biological material science and to use the biological systems as a source of inspiration to new biomimetic and green material engineering processes.

Following the widely accepted terminology for equilibrium self-assembly in closed systems and dissipative self-organization in open systems, we find many important fundamental and practical processes in LC material science and physics where phase ordering occurs under minimization of energy (closed system) or it is coupled to transport of mass, thermal energy and/or linear momentum (open system). These two phenomena are highlighted in Figure 2, which also gives the organization of this review, as discussed in what follows. On the left of Figure 2, we find phase ordering in closed systems and two representative examples of increased complexity: 1) 2D Nematic N phase ordering in curved deformable elastic membranes (Murugesan and Rey, 2010a; Murugesan and Rey, 2010b) and 2) Cholesteric N* phase ordering in LC-colloid composites (filled cholesterics with second phase inclusions) (Rey et al., 2016). On the right of Figure 2, we find phase ordering in open systems and two representative examples of biomimetic practical utility: 3) Paranematic PN phase ordering under co-existing shear and extensional kinematics (Rey and Murugesan, 2011), and 4) Cholesteric N* phase ordering under directed water evaporation in dilute aqueous acidic triple-helix collagen solutions (Murugesan et al., 2011).

FIGURE 2

FIGURE 2. Organization of this review paper. Structure and pattern formation in biological liquid crystal through phase ordering in close (self-assembly on the left) and open (self-organization on the right) systems. (1) Nematic phase ordering on elastic deformable membranes (cellulose fibrils in plant cell walls); (2) cholesteric phase ordering in the presence of second phase inclusions (defects in plant cell walls); (3) Paranematic phase ordering under shear/extensional flow of dilute collagen; (4) Cholesteric phase ordering under directed dehydration. Lastly in (5) we present computational platforms for self-assembly and self-organization in liquid crystals. (1) is adapted from dissertation (Murugesan, 2012). (2) is republished with permission of Royal Society of Chemistry, from (Rey, 2010); permission conveyed through Copyright Clearance Center, Inc. (3) is reprinted with permission from Aguilar Gutierrez and Rey, (2016a). Copyright (2016) American Chemical Society. (4) is republished with permission of Royal Society of Chemistry, from (Gutierrez and Rey, 2017); permission conveyed through Copyright Clearance Center, Inc.

1.1 Nematic Phase Ordering in Curved Elastic Membranes

The nematic and chiral nematic phase ordering in a closed system is well understood, characterized and presented in many monographs. In particular nucleation and growth, defect generation mechanisms, and effect of electromagnetic fields (Selinger, 2016; Rey, 2010). More recently, driven by questions arising from biological material synthesis, such as plant cell walls with a basic fibrous composite architecture, models that capture the interaction between orientational ordering and elastic membrane curvature are being developed and refined (Murugesan and Rey, 2010a; Murugesan and Rey, 2010b; Rey et al., 2016; Rey and Murugesan, 2011; Murugesan et al., 2011; Murugesan and Rey, 2010c). Even with the simplest tubular elastic membranes with zero Gaussian curvature (K = 0) the director field n that emerges from PO can display many possible modes, including helical and ring-like orientation of fibrils, leading to new curvophobic and curvophilic mechanisms in 2D nematic ordering on curved elastic membranes. In this paper, we will highlight and summarize the new mechanism and validate it with observations and measurements from plant-based cellulosic materials.

1.2 Cholesteric N* Phase Ordering in LC-Colloid Composites (Filled Cholesterics)

Following the closed system phase ordering approach and again motivated from plant-based facts, we focus on colloidal particle-cholesteric liquid crystals and establish what defect organization emerges as a function of the local particle arrangements, such as triangular and square. This organizational mechanism is known as Zimmers’ rule (Murugesan et al., 2013; Phillips et al., 2011a; Phillips et al., 2011b; Rey et al., 2011; Shams et al., 2015; Soule and Rey, 2012). Since in the N* various singular and non-singular disclinations are possible, the number and type of these defect provides semi- quantitative information on the degree of anisotropy in elastic material properties. Equally significant, the defects in the phase ordering model in the presence of colloidal inclusion (pit canals of plants) provide proof and certainty that many solid analogues displaying the Bouligand architecture are formed through a liquid crystalline phase.

1.3 Paranematic Phase Ordering Under Shear and Extensional Kinematics

Cholesteric rheology is highly complex due to the addition of chirality, exhibiting uncoiling, orientational wave- propagation, disclination nucleation which generate films and bulk materials with many unavoidable defects when applications require defect-free monodomains (Rey, 1996a; Rey, 1996b; Rey, 2000b; Rey, 2002b; Rey, 2002c; Cui et al., 2006; Marenduzzo et al., 2006; Marenduzzo et al., 2004; Cui and Wang, 2011; Venhaus et al., 2013; Noroozi et al., 2014; Echeverria et al., 2015; Echeverria et al., 2017). One approach to achieve monodomains is to process the precursor materials in the dilute regime using the principle of flow-induced-orientation (FIO), widely used in polymer processing (Aguilar Gutierrez and Rey, 2016a). In actual film processing, the presence of shear and extensional deformation rates are always present. In this section, we show how to select the intensity and type of flow deformation mix that leads to a well-organized paranematic (PN) organization with a preselected director n and scalar order parameter. The state is paranematic because the equilibrium concentration used to flow-process the dilute solution is below the Onsager threshold and if the deformation rate introduced by the flow is removed, Q eventually relaxes to the zero tensor state: Q = 0.

1.4 Cholesteric Phase Ordering Under Directed Evaporation

Directed evaporation, especially of isotropic or colloidal droplet phases generates phase ordering of various defect content. Hence starting from a paranematic well-oriented phase with an order parameter S close to the target one, one can in principle obtain defect-free films with directed evaporation, where the mass transfer drives the order in a controlled way (Gutierrez and Rey, 2017). Hence if we look at the open system examples, we find evaporation examples such as in (Chu et al., 2018; Pospisil et al., 2020).

In the next sections of this review, we focus on the salient features of nematic and cholesteric phase ordering under close and open conditions and in bulk and membrane geometries. The emphasis is on highlighting mechanisms and applications in these phase ordering processes, while mathematical derivations (included in the references) are kept at a minimum. We follow the numbering of Figure 2.

2 Self-Assembly and Self-Organization in Biological Liquid Crystalline Materials

2.1 Closed System Phase Ordering Models, Predictions and Validations

2.1.1 Nematic Phase Ordering in Curved Elastic Membranes

The plant cell wall (PCW) is nature’s most abundant biological fibrous composite. In order to demonstrate the universality of nematic phase ordering mechanisms in nature, we focus on the uniqueness of PCW and base on 2D phase ordering on elastic membranes (Murugesan and Rey, 2010a; Rey et al., 2016). Recent work on 2D isotropic-nematic transition includes the recent work of Zonta and Soule (Zonta and Soulé, 2019).

To operate as a supporting mechanical structure and at the same time allow for cell growth through expansion, plant cells are encompassed by a stiff, multilayered, viscoelastic-viscoplastic composite wall (Brett, 1985). In terms of geometric dimensionality, the typical thickness of plant cell walls generally varies from 0.1 to 0.3 µm, about one hundred times thinner than the size of the plant cell itself. It is made of sequentially deposited layers of different thicknesses, chemical composition and structural organization. At the appropriate scale, the PCW can be considered as a reinforced biological fibre-laden composite membrane consisting of well-aligned cellulose microfibrils (CMFs) of high tensile strength coated with hemicellulose and embedded in a matrix of polysaccharides (pectin/lignin) and structural glycoproteins (Neville, 1993). Different plant species synthesize cell walls with a wide range of mechanical properties by tuning the arrangement of these four material components (cellulose, lignin, hemicellulose and pectin) in the cell wall and/or cellular structure.

Thus understanding the process through which the CMFs are oriented in a specific direction is crucial in unravelling the mechanisms underlying nature’s material synthesis. Figure 3 shows the basic plant cell wall textures, and the microfibrils observed microfibril angles in PCW’s, where the CMFs are oriented in strategic directions to generate commonly observed textures such as line, helix, ring, crossed helix and helicoid (Emons and Mulder, 2000). In the line (ring) mode, the CMF’s are aligned parallel (perpendicular) to the major axis of the cell. In the helical mode, the CMF’s are oriented obliquely to the major axis of the cell and in the ring mode they are in the azimuthal direction. In the helicoid, the CMF gradually rotates and in the crossed helix the rotation is discontinuous Here we discuss (a)-(b) only.

FIGURE 3

FIGURE 3. Basic plant cell wall textures: (A) line, (B) helix, (C) ring, (D) crossed helix and (E) helicoid. The black lines indicate the CMFs and the dashed line represents the major axis of the cell. Adapted from dissertation (Murugesan, 2012).

The Onsager 3D isotropic-nematic model predicts that when the rod volume fraction ϕ is less than ϕ_I = 3.34D/L, the phase is isotropic and when it is larger than ϕ_n = 4.48D/L, it is nematic (Donald et al., 2006). External fields such as magnetic, shear and extensional flow fields can shift these predictions, such that in the dilute regime the isotropic phase is ordered and oriented by the external field. This field-induced phase is denoted paranematic phase, such that when the external field is turned off, the paranematic phase relaxes to the isotropic state. To predict plant cell textures, we introduce the concept of curvature-induced paranematic ordering and consider the nematic tensor order parameter Q as two-dimensional since the fibrils are tangential to the curved elastic substrate. The elastic substrate is described by the classical membrane Helfrich elasticity due to bending (changes in mean curvature) and torsion (changes in deviatoric curvature) elastic modes. The total free energy minimization process that drives the alignment of molecules on a 2D surface is called “planar self-assembly.” We only consider dilute CFM concentration regimes that correspond to isotropic states on flat surfaces but become nematic under curved membranes. Nematic LCs exhibit anchoring energy at surfaces, which in the simplest case involve Q: kk couplings contributions. Here k is the surface unit normal. The next higher order contribution associated with a surface is Q: ∇_sk; b = −∇_sk = H (e₁e₁ + e₂e₂) + D (e₁e₁ − e₂e₂), where b is the symmetric curvature tensor. H(D) is the mean (deviatoric) surface curvature given by 2H = κ₁ + κ₂; 2D = κ₁ − κ₂; κ₁ > κ₂ and {e_i} are the principal curvatures directions. Since we only consider surface ordering, the proper biaxial 2D tensor is Q = S (nn − I_s/2). The schematic of the proposed curvature-driven planar self-assembly model is presented in Figure 4. Cell walls with unidirectional CMF alignment can be considered as a two dimensional fibre reinforced membrane. An integrated mechanical model for curvature driven planar self-assembly of rigid rods on an arbitrarily curved fluid membrane through novel surface phenomena such as curvophobic and curvophilic effects, has been applied to predict the uniaxial modes in plant cell walls. Curvophilic (curvophobic) effects seeks to align the CMFs along high (zero) curvature directions. The ratio of these two effects introduces an internal length scale l.

FIGURE 4

FIGURE 4. Schematic of curvature-induced 2D nematic LC self-assembly on elastic membranes. At dilute CMF concentrations below Onsager’s critical limit, the isotropic Q = 0 state observed on a flat membrane (left) may lead to a nematic state Q > 0 (right) due to curvature-mediated interactions that include couplings between Q and the membrane curvature tensor b.

The total free energy of the fibre-laden membrane system per unit area $\widehat{F}$ is posited to be

$\widehat{F}={\widehat{F}}_{\text{membrane}}\left(\mathbf{b}\right)+{\widehat{F}}_{\text{fibre}}\left(\mathbf{b}\right)+{\widehat{F}}_{\text{coupling}}\left(\mathbf{b}\cdot \left(\mathbf{Q}+\frac{1}{2}{\mathbf{I}}_s\right)\right)(1)$

When the free energy of the ensemble is minimized on a given cylinder of radius R (H = D = −1/R), the equilibrium fibre alignment angle cos α = (n ⋅e₁) and order parameter S = 2Q: nn are obtained as a function of dimensionless radius R* = R/l of the cylindrical membrane. Figure 5 depicts the fibre orientation modes in terms of cosine of equilibrium fibre orientation angle with respect to the azimuthal coordinate (cos α) and equilibrium fibre order in terms of scalar order parameter (S), obtained by minimizing Eq. 1 with typical parameters’ values. At low dimensionless membrane radius, for 0 < R* < 0.1847, the curvophilic free energy is negligible and the curvophobic free energy is minimized, thus the fibres are aligned in the axial direction resulting in the line mode. At intermediate membrane radius, for 0.1847 < R* < 0.3867, the competition between the fibre interactions and curvophilic free energy aligns the fibres at an angle between 0 and 90°, resulting in the helical mode. At high membrane radius, for R* > 0.3867, the curvophobic free energy is negligible and the curvophilic free energy is minimized, thus the fibres are aligned in the azimuthal direction resulting in the ring mode. At low membrane radius, the fibres are more uniformly aligned in the axial direction. As the membrane radius increases, the fibre order S decreases until a local minimum (cusp) is reached, due to the cancellation of free energy contributions from curvophobic and curvophilic interactions. At the onset of the ring mode, the fibre order S starts increasing until a local maximum is reached promoted by curvophilic interactions. At high membrane radii, the fibre order vanishes.

FIGURE 5

FIGURE 5. Fibre orientation cos α = (n ⋅e₁) for cylindrical membranes of circular cross-section. The three possible modes are: line mode (l), helical mode (h), ring mode (r) that correspond to alignment along the cylinder axis, oblique to the axis and in the azimuthal direction, respectively. The order parameter S exhibits a typical V-shape and the orientation a step-like response as the radius increases. The inset semi-log plot shows the decrease of S towards zero as R* → ∞ (Murugesan et al., 2015). Republished with permission of Royal Society of Chemistry, from (Murugesan et al., 2011); permission conveyed through Copyright Clearance Center, Inc.

Despite the fact that the present 2D phase ordering model does not take into account temporal variation of cellulose microfibril concentration during different stages of cell wall deposition, the predicted fibre orientation (Figure 5) is in good qualitative agreement with the experimental observations for the orientation of cellulose microfibrils in cell walls of tracheids of conifers based on field emission scanning microscopy (Abe and Funada, 2005). Future opportunities in this area include self-assembly in evolving surface shapes.

2.1.2 Cholesteric N* Phase Ordering in LC-Colloid Composites

Cholesteric (N*) phase ordering is widely observed in solid analogues. A representative case is the extracellular domain of a plant cell in which chiral self-assembly takes place in a 3D space that is rich in secondary inclusions such as pit canals and other plant cells themselves. The presence of these second phases has an impact on the self-organization of defects such as the type (singular or non-singular cores), topological charge (±1/2, ±1, … ), number density and possible positional order of disclinations. To affect the director field n, the inclusion’s characteristic size L should be greater than the extrapolation length or ratio between bulk Frank orientational elasticity K and the anchoring surface energy W: L > K/W. For curved inclusions such as cylindrical fibres with mean curvature −1/R, the strong anchoring introduces radial, oblique or azimuthal surface orientation that propagates into the bulk and interacts with orientation fields from neighboring inclusions leading to defects. For nematics and cholesterics the defect charge C in a LC matrix that contains N inclusions is C = −(N −2)/2, where the minus sign corresponds to tangential anchoring conditions. The inclusion of these secondary phases induces confinement of the cholesteric material between them, and the resulting frustration in the mesophase is relieved through nucleation of defects and distortion of the pitch of the mesophase as shown in Figure 6A. There is strong evidence that the solid helicoidal plywood architecture is formed through LC self-assembly by the existence of specific defect patterns observed in the presence of these secondary phases in the domain of self-assembly (Reis et al., 1992).

FIGURE 6

FIGURE 6. Helicoidal plywoods in plant cell walls with secondary phases. (A) Adaptation of helicoidal plywood to embedded pit canals (L inside the gray circles) by dilation of layers (shown by width changes in arc-patterns) and bending of layers (bending of arced patterns) and a − 1 saddle defect (denoted by *) in a natural helicoidal plywood due to inclusion of four (L) lumens in walnut (Gutierrez and Rey, 2017). (B) A self-assembly simulation of a −1 saddle defect. For filled LCs, a defect of strength −(N −2)/2 is expected (Rey et al., 2016). For the four (L) lumens, N = 4 and the plywood has a –1 saddle defect (*). Republished with permission of Royal Society of Chemistry, from (Rey, 2010); permission conveyed through Copyright Clearance Center, Inc.

Figure 6B shows a 2D visualization of the director field during N* (out-of-plane n_z-component) phase ordering computed by minimizing the total free energy in a rectangular domain with four inclusions. In this case the free energy has local (Landau-de Gennes), elastic (Frank elasticity), and chiral contributions: F = F_fibre(Q) + F_elastic (∇Q) + F_chiral (∇ ×Q). At the center of the N = 4 arrangement, there is a non-singular − 1 saddle defect as in the biological analogue. The colored ring spacing is related to the pitch P₀. Matching the main features of the experimental texture one is then able to extract the concentration of rods (terms in F_fibre(Q)), the elastic anisotropy [terms in F_elastic (∇Q)] and pitch effects [terms in F_chiral (∇ ×Q)] and check for consistency with characteristic values of biological LCP. This representative example shows how the liquid crystal modeling platform provides quantitative information of biological self-assembled structures.

2.2 Open System Phase Ordering Models, Predictions and Validations

2.2.1 Paranematic Phase Ordering Under Shear and Extensional Kinematics

Coupling phase ordering and transport phenomena lead to new self-organization mechanism by interactions between the spatio-temporal behavior of Q with symmetry- breaking fields, such as flow (Rey, 1995a; Rey, 1995b; Rey, 1995c; Rey and Tsuji, 1998; Pujolle-Robic et al., 2002; Forest et al., 2004a; Forest et al., 2004b; Hess and Ilg, 2005; Ji et al., 2006; Mendil-Jakani et al., 2009; Golmohammadi and Rey, 2010a; Golmohammadi and Rey, 2010b; Xu et al., 2011), heat (Huisman and Fasolino, 2007; Abukhdeir et al., 2008; Abukhdeir and Rey, 2009; Yang et al., 2009; Abukhdeir, 2016) and mass fluxes (Rey, 1997; Rey, 2000a; Bedolla Pantoja et al., 2019). For example, in (Rey, 1995a; Rey, 1995b), it is shown that subjecting isotropic phases to extensional flows is rich in symmetry breaking bifurcations, since the equivalence condition

$\mathbf{T}\cdot \mathbf{f}\left(\mathbf{v},\text{Pe},\varphi \right)=\mathbf{f}\left(\mathbf{T}\cdot \mathbf{v},\text{Pe},\varphi \right)(2)$

is satisfied for rotational and reflectional transformation matrix T, here f is the system of nematodynamic equations, v is the solution vector in terms of eigenvectors of Q, Pe is the Péclet number, and ϕ is the rod concentration.

In the present representative open system, we discuss the injection of momentum by shear and extension flow into isotropic states of dilute collagen solutions to created paranematic states. The phase diagram of aqueous acidic triple helix collagen solutions has been formulated using phase ordering processes that include Onsager hard core interaction, electrostatic repulsion, chirality and attractive interactions of the Maier-Saupe type (Khadem and Rey, 2019a; Khadem and Rey, 2019b; Khadem and Rey, 2021). The predictions are in excellent agreement with experimental data (Gobeaux et al., 2007). A significant effect of these additional interactions is to bend the biphasic I-N* region in an acidic-collagen concentration diagram towards higher collagen concentrations than Onsager’s predictions (Donald et al., 2006). Since processing macroscopic defect-free collagen films is not generally possible, film casting flows of dilute isotropic triple helix collagen solutions are used (Rey, 2010). These flows usually contain mixtures of vorticity and deformation rates and flow-align the rods according to the balance of nematodynamic torques and thermodynamic torques. An approach that sheds important quantitative information is to perform characterization studies based on the four-roll mill apparatus, which provides all the possible flow configurations, from pure vortex to simple shear, and to uniaxial extensional flow. The equations of Q-tensor nematodynamics were solved using experimental parametric data for collagen solutions. The key issue was to determine how the different couplings between vorticity and order parameter (W ⋅Q), and deformation rate and order parameter (A ⋅Q) generate orientation (n) and order (S), to create a collagen paranematic phase that relaxes back to isotropic (Q = 0) if the flow is removed. One important output of the study is what is the optimal vorticity-deformation rate mixture and their intensities to target well oriented samples with sufficient scalar order parameter. Figure 7 in terms of director angle with respect to the flow direction and scalar order parameter S. The ellipsoids represent Q + I/3, the colour intensity is the Deborah ($\text{De}=\left(\sqrt{\mathbf{A}:\mathbf{A}}+\sqrt{\mathbf{W}:\mathbf{W}}\right)/(8.4D_r)$) or molecular time scale/flow time scale, where D_r is the rotational diffusion constant). The curved line trajectories’ parameter number κ is the flow type ($\kappa =\left(\sqrt{\mathbf{A}:\mathbf{A}}-\sqrt{\mathbf{W}:\mathbf{W}}\right)/\left(\sqrt{\mathbf{A}:\mathbf{A}}+\sqrt{\mathbf{W}:\mathbf{W}}\right)$), and the white dash line triangle represents the target processing space for good orientation and good order.

FIGURE 7

FIGURE 7. Computed flow induced structuring in terms of the scalar order parameter S and the director angle θ with respect to the flows’ direction. The color intensity is the Deborah number De and the colour curved lines correspond to the flow type $\kappa =\left(\sqrt{\mathbf{A}:\mathbf{A}}-\sqrt{\mathbf{W}:\mathbf{W}}\right)/\left(\sqrt{\mathbf{A}:\mathbf{A}}+\sqrt{\mathbf{W}:\mathbf{W}}\right)$, where A is the deformation rate and W the vorticity. The white dashed line triangle specifies the kinematic values that lead to a paranematic phase with optimal structure. Reprinted with permission from (Aguilar Gutierrez and Rey, 2016a). Copyright (2016) American Chemical Society.

As expect extensional flow (κ = ±1) is required as it is the strongest orienting field with De less than 0.7 corresponding to deformation rates of the order of 700 s⁻¹. Pure vorticity (κ = −1) does not create phase ordering and S = 0. These results expand results of shear flow effects on the I-N transition and highlight the importance of extensional (potential) flow. Since the generated paranematic PN phase is characterized by an order parameter S_PN(De, κ) (y-axis in Figure 7), one can then estimate the equivalent equilibrium value in terms of concentration, pitch and pH: S_PN(De, κ) = S_eq (ϕ, P₀, pH, …).

2.2.2 Cholesteric Phase Ordering Under Directed Evaporation

The coupling of phase ordering and phase separation has been widely studied in various mesophase-solvent-nanoparticle blends and self-selected textures and defects in spinodal and nucleation and growth modes have been characterized and validated (Das and Rey, 2004; Das and Rey, 2005; Das and Rey, 2006; Matsuyama, 2008; Soule and Rey, 2012; Milette et al., 2013; Soulé and Rey, 2013; Gurevich et al., 2014; Soule and Rey, 2019; Khadem and Rey, 2021). In contrast with this case, in open systems loss of solvent, as in dehydration, is dictated or coupled to external sinks and not to equilibrium thermodynamic driving forces. As in the case of undercooling in solidification, when the temperature is below the thermodynamic liquidus value and instability sets in, we will see below a representative corresponding soft matter example of dehydration of collagen solutions, when the externally controlled evaporation creates a water concentration profile below its equilibrium value (Kirkwood and Fuller, 2009; Gutierrez and Rey, 2017).

In this section we start with the dehydration analysis of the PN material produced in the previously discussed step (Section 2.2.1). After imposing shear/extensional flow to an isotropic aqueous acidic triple collagen solution to create a PN state, one can then use directed evaporation of the PN phase to create a stable, defect-free cholesteric N* phase (Figure 8):

$\text{Isotropic}\left(S=0\right)\stackrel{\text{De},\kappa }{\to }\text{Paranematic}\left({\mathbf{Q}}_{\text{PN}}\right)\stackrel{{\mathbf{W}}_{{\text{H}}_2\text{O}}}{\to }\text{Cholesteric}\left({\mathbf{Q}}_{\text{Ch}}\left(\mathbf{x}\right)\right)(3)$

where ${\mathbf{W}}_{{\text{H}}_2\text{O}}$ is the water mass flux vector, whose intensity and directionality need to be 2 controlled to create defect-free collagen films, and the PN Q-tensor order parameter depends on the sought after cholesteric tensor order parameter has to have the proper helicity: Q_Ch(x) = S_Ch(n(x)n(x) − I/3); Q: ∇ ×Q_Ch ≠ 0. The challenge is that by removing the flow, the Q_PN seeks to relax to Q = 0 under thermodynamic driving forces and this has to be counterbalanced by ${\mathbf{W}}_{{\text{H}}_2\text{O}}$.

FIGURE 8

FIGURE 8. Schematic representation of the drying process used by Kirkwood and Fuller (Kirkwood and Fuller, 2009) to produce defect-free collagen films (A) and a well aligned cholesteric helix after drying (B). In this case water leaves the film along the horizontal direction and the helix forms along the water flux. Republished with permission of Royal Society of Chemistry, from (Gutierrez and Rey, 2017); permission conveyed through Copyright Clearance Center, Inc.

The optimal design and operation match the scalar order parameter produced by the shear/extensional flow with the order parameter of the final equilibrium cholesteric dehydrated films: S_PN(De, κ) ≈ S_eq (ϕ, P₀, pH, …) = S_Ch(ϕ, P₀, pH, …), so that there is minimal adjustment in order throughout the drying process. Simultaneously, the helix (h, P₀) forms by the torques generated by the water concentration gradients, to eventually achieve a perfectly aligned helix with uniform pitch. The synchronicity in a film of size L, is obtained when the ratio of the collagen rotational time scale 1/(6D_r) water diffusion time scale L²/D achieves an optimal value (Π = Π_op):

$\mathrm{\Pi }=\frac{\text{phase\hspace{0.17em}ordering\hspace{0.17em}time\hspace{0.17em}scale}}{\text{dehydration\hspace{0.17em}time\hspace{0.17em}scale}}=\frac{D}{6D_r{L}^2}(4)$

This novel phase ordering/mass transfer coupling is shown in Figure 9. The figure shows the integrated phase ordering/directed evaporation cycle in an open system self-organization process. The yellow central arrow indicates how starting with concentration fluctuations we create order (S), torques and helicity (P₀). Fluctuation in concentration (mass transfer) creates nematic order (S) which creates torques to form the helix (h). Mass transfer is controlled by water diffusion constant D (right box) and phase ordering by the collagen rotational diffusion D_r (left box).

FIGURE 9

FIGURE 9. Integrated phase ordering/directed evaporation cycle in an open system self-organization process. The arrow indicates how starting with concentration fluctuations we create order, torques and helicity. Fluctuation in concentration (mass transfer) creates nematic order (S) which creates torques to form the helix (h) Mass transfer is controlled by water diffusion constant D (right box) and phase ordering by the collagen rotational diffusion D_r (left box). Redrew with permission of Royal Society of Chemistry, from (Gutierrez and Rey, 2017) for higher figure resolution; permission conveyed through Copyright Clearance Center, Inc.

Using an integrated mass transfer and Q-tensor nematodynamics with moving boundaries representative results of self-organization were obtained for almost perfectly synchronized (Π = Π_c, left figure in Figure 10) and largely unsynchronized (Π < Π_c, right figure in Figure 10) phase ordering/evaporation. The PN-to-N* transformation under mass transfer coupling to phase ordering offers new routes to material processing not available to phase ordering-phase separation closed system process since driving forces are related to the rate of entropy production and not to equilibrium free energy. Additional process variables that can be explored to target surface wrinkling functionalities are airflow intensity, directionality and flow type (tangential shear, stagnation flows).

FIGURE 10

FIGURE 10. (A) Nearly synchronized (Π ≈ Π_op) phase ordering/mass transfer produces an almost defect-free film, and 1 disclination is present; (B) Fast evaporation (Π < Π_op) creates a central defect network with many disclinations and rod-polygon shape textures. Republished with permission of Royal Society of Chemistry, from (Gutierrez and Rey, 2017); permission conveyed through Copyright Clearance Center, Inc.

At the end of the drying process the collagen film surface has characteristic nano-wrinkles, which are ubiquitous in many cholesteric free surfaces (Terris et al., 1992; Fernandes et al., 2013; Rofouie et al., 2014; Rofouie et al., 2017a; Rofouie et al., 2017b; Almeida et al., 2018; Wang et al., 2019; Wang et al., 2020a). The wavelength is usually of the order of the pitch P₀. The amplitude of these wrinkles is in the nano-scale and not in the micron scale, as when driven by Frank elasticity in micron range confinements. The origin of these nano-wrinkles is the nematic liquid crystal capillary pressure p_c:

$p_c={\nabla }_s\cdot \mathit{\xi },\mathit{\xi }=\gamma \mathbf{k}+\left(\mathbf{I}-\mathbf{k}\mathbf{k}\right)\cdot \frac{\partial \gamma }{\partial \mathbf{k}},\gamma ={\gamma }_0+\frac{W}{2}\left(\mathbf{n}\mathbf{n}:\mathbf{k}\mathbf{k}\right)(5)$

where ξ is the Cahn-Hoffman capillary vector, γ is the surface tension, and W is the quadratic the anchoring coefficient (Cheong and Rey, 2002a; Cheong and Rey, 2002b; Cheong and Rey, 2002c; Rey, 2002a; Rey, 2004a; Rey, 2004b; Rey, 2005; Rey, 2006; Wincure and Rey, 2007). Performing the surface gradient of the capillary vector one finds that the easy shape condition leads to:

$\kappa =\text{Curvature}=\frac{\text{driving\hspace{0.17em}director\hspace{0.17em}pressure}}{\text{effective\hspace{0.17em}tension}}=\frac{2W\left[\mathbf{n}{\partial }_s\mathbf{n}:\left(\mathbf{t}\mathbf{k}+\mathbf{k}\mathbf{t}\right)\right]}{{\gamma }_0+4W\mathbf{n}\mathbf{n}:\mathbf{t}\mathbf{t}}(6)$

The brackets in the numerator of Eq. 6 can be shown to represent a 1-D splay-bend director distortion and is the source of a periodic director capillary pressure.

Figure 11 shows a representative calculation of the surface relief profile for W/γ₀ = 0.002 to 0.004.

FIGURE 11

FIGURE 11. Surface relief of a cholesteric surface displaying characteristic nanowrinkling with a wavelength equal to the pitch P₀. Here B = W/γ₀ is the scaled anchoring coefficient. Reprinted from Rofouie et al. (2014), Copyright (2014), with permission from Elsevier.

A key finding of this surface pattern formation model is that the amplitude h_max and wavelength λ does scale linearly with the P₀ as observed in experiments:

$h_{\text{max}}=\frac{1}{4\pi }\frac{W}{{\gamma }_0}{P}_0,\lambda =P_0,h_{\text{max}}=\frac{1}{4\pi }\frac{W}{{\gamma }_0}\lambda (7)$

Extensions of this model to higher order anchoring functions predict two-wavelength patterns (Wang et al., 2020a) and two dimensional director fields are expected to generate egg cartons (Wang et al., 2021). Coupling the liquid crystal shape equation with interfacial mass transfer leads to new Marangoni flows (Rey, 1999; Yue et al., 2005; Herminghaus et al., 2014; Choi and Takezoe, 2016) that can be explored in the presence of wrinkling. Introducing these nanoscale effects to wetting and film phenomena (Kondic and Cummings, 2021) in liquid crystals may provide new routes to multiscale pattern formation.

3 Computational Platforms

We end this review with a discussion of some computational platforms that are used in liquid crystal self-assembly and self-organization that may also be applied to areas such as active liquid crystal matter (Forest et al., 2013; Yang et al., 2016; Wang, 2021; Zhang et al., 2021). Some of the challenges arise because of the multiscale 19 nature of the tensor order parameter, the presence of orientational boundary layers, nucleation and annihilation of disclinations under flow, need to resolve disclination cores, the presence of interfaces and contact lines, nucleation-growth-impingement of colloidal drops, presence of faceted particles, 3D disclination line textures, complex 3D director fields, and more. For example, to simulate real liquid crystal polymer textured flows with Q-tensor nematodynamics, the energy ratio R and Ericksen number Er need to be specified (Grecov and Rey, 2003):

$R=\frac{{\left(\text{geometric\hspace{0.17em}scale}\right)}^2}{{\left(\text{defect\hspace{0.17em}core\hspace{0.17em}size}\right)}^2}\approx 10^{10},1<\text{Er}=\frac{{\left(\text{geometric\hspace{0.17em}scale}\right)}^2}{{\left(\text{flow\hspace{0.17em}scale}\right)}^2}<10^7(8)$

Here the energy ratio is expressed in terms of geometric (gap used in a rheometer) and defect radius length scales. So, unless high-performance computing and adaptive methods are used no resolution of defect nucleation, and defect annihilation is possible, as required by experiments (Zhu and Farrell, 2002). Even in the absence of flow, it was shown that to resolve defects in monomeric LCs a mesh smaller than 3 nm is needed (Chen et al., 2021). The platforms include:

(a) Geometric modeling. i) It has been used to resolve the arc-patterns observed in cross-sectional projections of cholesteric and orthogonal plywood. The projected director field is converted to planar curves whose curvature provides information on the pitch of the material using the inverse problem techniques. ii) Characterization of nanowrinkling can be assisted by computation of higher order moments of the surface relief distribution using accurate integration methods. iii) Interfacial and membrane geometries (Gutierrez and Rey, 2018; Wang et al., 2020b) can be better resolved by decoupling shape from curvedness instead of the usual mean and Gaussian curvature that comingle these two quantities. Recasting the equilibrium and dissipative shape equations in terms of shape and curvedness and their rates provides an additional phase space where trajectories detect the birth, death and transformations of shapes.

(b) Phase field modeling (Yang et al., 2006; Yang et al., 2013; Zhao et al., 2016; Qiu et al., 2021). i) It has been used to simulate near equilibrium and flow phenomena, using stable variational-informed schemes that provide consistent description of the Q-tensor, composition and accurately resolve interfacial physics (Zhao et al., 2016; Qiu et al., 2021). ii) Phase ordering and phase separation in liquid crystal nanocomposites (Gurevich et al., 2014; Gurevich et al., 2017) leading to dendritic patterns have been resolved accurately using adaptive schemes that resolve nematic order, concentration, as well as nanoparticle positional ordering.

(c) AI-based modeling. i) Neural networks have been used to predict various material properties with high accuracy, based on direct optical images (Sigaki et al., 2019; Sigaki et al., 2020). ii) The nearly intractable problem of nucleation, growth and impingement at a large population level has been resolved using AI techniques, to show that the limits of direct numerical simulation can be largely extended (Khadem and Rey, 2021).

(d) Defect detection and tensor visualization methods. Singular and non-singular defects nucleate, annihilate, and act as defect sources. For singular defects the normalized biaxiality index is a useful index to detect disclinations since cores are biaxial (Zhu and Farrell, 2002; Jankun-Kelly et al., 2009; Fu and Abukhdeir, 2014).

(e) Optical methods. Resolving textures and defects using FDTD methods (Han and Rey, 1995; Hwang and Rey, 2005; Hwang et al., 2007) is a useful tool in characterization methods, such rheo-optics, when one can follow the action of deformation rates on Q-tensor response (Hwang et al., 2007).

(f) Bifurcation methods and flow solvers. Liquid cyrstals are rarely flow-aligning under shearing conditions and usually display solution multiplicities, periodic and oscillatory modes, and flow-induced transitions that require the use of bifurcation methods, and adaptive schemes for torque balances (Farhoudi and Rey, 1993; Han and Rey, 1993a; Han and Rey, 1993b; Han and Rey, 1994; Singh and Rey, 1998; de Andrade Lima and Rey, 2004; Forest et al., 2004b; Fox et al., 2020; Chillingworth et al., 2021).

4 Conclusion, Summary and Outlook

Table 1 summarizes the key finding and identifies opportunities for future work in liquid crystal self-assembly and self-organization, as follows. Liquid crystal phase ordering describes many processes and structures observed in nature and can be described and quantified by non-conserved order parameter models such as the Q-tensor equations for self-assembly. Here defect generation and identification, provide an important tool to describe processes that started with liquid crystalline precursors and end with solid liquid crystal analogues. A material system that offers fertile inspirational ground is the plant cell wall where the orientation of cellulose microfibrils displays typical nematic and cholesteric architectures. Here we demonstrated that a 2D phase ordering process on an elastic membrane serves as a good model for self-assembly. By reducing the physical and phase space dimensionality to 2D we find couplings that guide the fibres towards and away from principal curvatures, augmenting the number of mechanisms beyond the excluded volume interactions that create nematic phases. In the presence of second phases, colloidal particles, inclusions, liquid crystal organization accommodates to the geometric constraints by creating defects of various kinds with positional order that reflect the inserted particles arrangements. Consistency between Q-tensor models a variety of actual systems including plant cell walls and synthetic materials like carbon-carbon composites and polymer dispersed liquid crystals has been presented. Future directions include shape-changing inclusions with translational and rotational degrees of freedom.

TABLE 1

TABLE 1. Summary of self-organization and self-assembly in open and closed systems, with paranematic, nematic and cholesteric order in complex physical spaces, observed experimentally and predicted by theory and simulation.

In the presence of transport processes including heat, mass, and momentum, couplings with conserved order parameters provide new mechanisms of self-organization (Park et al., 2021). The film processing of aqueous acidic triple helix collagen for tissue engineering applications is an example where sequential transport processes, flow and then directed dehydration, can create an essentially defect-free cholesteric thin (typically in the micron range) film. Subjecting a dilute isotropic collagen solution to a shear/extensional flow produces a paranematic phase that can be dehydrated at a rate that retains the flow-induced order but that creates a moving front that converts the nearly uniform director to a helix. Future directions include the use of transient flows and film casting flows as well as different evaporation processes and confinement (Srinivasarao, 2021).

In closing, liquid crystal-based self-assembly and self-organization research is a driving force for new biomimetic principles for green manufacturing of advanced and multifunctional materials and devices, as recent publications show.

Author Contributions

ZW, PS, and AR contributed to conception of the study. ZW prepared the figures and bibliography. AR wrote the first draft of the manuscript. ZW, PS and AR revised the manuscript. All authors contributed to manuscript revision, read, and approved the submitted version.

Funding

This work is supported by the Natural Science and Engineering Research Council of Canada (NSERC) (Grant number: 223086).

Conflict of Interest

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.

Publisher’s Note

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.

Acknowledgments

AR and PS are thankful for financial support from the Natural Science and Engineering Research Council of Canada (NSERC), and to Compute Canada and CLUMEQ for support and access to high performance computing. AR is thankful to McGill University for financial support through the James McGill Professor appointment. ZW is grateful to the Faculty of Engineering for support through the MEDA scholarship program. AR wishes to acknowledge the fundamental contributions of Gino de Luca, Ahmad Sayyed Khadem, Oscar Aguilar Gutierrez, Yogesh Murugesan, Oscar Matus Rivas, E.E. Herrera Valencia, Ziheng Wang, Pardis Rofouie, Yun Yan, Gaurav Gupta, Dana Grecov, Won Hee Han, Philip K. Chan, Ezequiel Soule, Nasser Mohieddin Abukhdeir, Susanta K Da, Samir Mushrif, Paul M. Phillips, Mojdeh Golmohammadi, Luiz R. P de Andrade Lima, Alireza Shams, Dae Kun Hwang, Benjamin Wincure, Ae-Gyeong Cheong, Hang Hu, Mohan Srinivasarao, Linda Reven, and Raffaele Mezzenga.

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