New challenges for the pressure evolution of the glass temperature

The ways of portrayal of the pressure evolution of the glass temperature (Tg) beyond the dominated Simon-Glatzel-like pattern are discussed. This includes the possible common description of Tg(P) dependences in systems described by dtg/dP>0 and dTg/dP<0. The latter is associated with the maximum of Tg(P) curve hidden in the negative pressures domain. The issue of volume and density changes along the vitrification curve is also noted. Finally, the universal pattern of vitrification associated with the crossover from the low density (isotropic stretching) to the high density (isotropic compression) systems is proposed. Hypothetically, it may obey any glass former, from molecular liquids to colloids.


Introduction
Liquids on cooling solidify in the ordered crystalline state when passing the melting temperature ( m T ). However, the fluidity can be also preserved below melting, down to the glass temperature m g T T  , where the solidification from the metastable ultraviscous/ultraslowing liquid to the solid amorphous glass state occurs (Berthier and Ediger, 2016;Rzoska et al., 2010;Donth, 2000). There are also numerous semi-crystalline systems where the vitrification is related to the solidification of one or few elements of symmetry: as examples can serve orientationally disordered crystals (ODICs, plastic crystals) (Drozd- Rzoska et al., 2006a and2006b) or liquid crystals (Drozd- Rzoska, 2006; Drozd- Rzoska, 2009). For many systems passing m T without crystallization is associated with extreme temperature quench (Donth, 2000). However, there are also numerous glass formers where entering the metastable ultraviscous/ultraslowing domain is possible at any practical experimental cooling rate (Berthier and Ediger, 2016;Rzoska et al., 2010;Donth, 2000). Turnbull (Turnbull, 1969;Angell, 2008) (Böhmer, 1993). The pressure counterpart of the VFT equation was first proposed for the analysis of viscosity changes in glycerol by Johari and Whalley (1972) and later for the primary relaxation time in dibutyl phthalate (Paluch et al., 1996): where: const T  , P 0  and P o  denote prefactors, the amplitude const A  and g P P  0 is the "VFT-like" singular pressure. However, eqs. (2) can reliably portray experimental data only for 'strong' (weakly non-Arrhenius) glass formers, assuming that measurements terminates at 0 max P P  . In ref. (Paluch, Rzoska et al., 1998) the relation able to portray previtreous 'dynamic effects' for arbitrary glass formers and ranges of pressures was proposed: The comparison of eqs. (3) and (4) Rzoska, Rzoska and Roland, 2007). All these show that the reliable and effective portrayal of the pressure evolution of the glass temperature can constitute one of milestones in dealing with the glass transition. This report presents the resume of this issue, supplemented by possible extensions beyond the current state-of-the art.

Parameterization of the pressure evolution of melting and glass temperatures
There are several relations for describing the pressure evolution of melting temperature: the most popular is the Simon-Glatzel (SG) equation, due its simple form and the limited number of fitted parameters (Simon and Glatzel, 1929;Skripov and Faizulin, 2006): where 0 T , a and b are adjustable parameters. It can be derived from the Clausius-Clapeyron (C-C) equation  (Skripov and Faizulin, 2006). This relation describes melting, where the 'sudden and sharp' change of volume or density ( V  ,   ) and the heat capacity takes place. However, the C-C equation can be linked to any fusion phenomenon, provided it is associated with detectable changes in heat capacity and volume/density. This occurs also for the glass transition temperature, although the transformation is 'stretched' in temperature or pressure and occurs between the disordered (ultravisous) liquid and the disordered solid (glass), as exemplified in Fig. 1.   Fig. 1 are for 10 K/min. cooling / heating rate.
As mentioned above the 'reasonable' metric of the glass transition is the isochronal or isoviscous condition   Poise (Donth, 2000). Generally, such condition is absent along the melting curve within the P-T plane (Skripov and Faizulin, 2006). However, the isochronal condition for   P T m is clearly fulfilled if melting is associated with only one element of symmetry, as for the isotropicnematic transition in liquid crystals (Roland, Bogoslovov et al., 2008). Heuristic similarities between melting and vitrification can (elongated molecules) (Donth, 2000;Turnbull, 1969;Angell, 2008). Consequently, one can expect that the pressure dependence of m T are paralleled by   P T g evolution. Regarding the vitrification, S. Peter Andersson and Ove Andersson (AA) introduced the SG-type relation for describing the pressure evolution of the glass temperature in poly(propylene) glycol (Andersson and Andersson, 1998): where k1, k2 and k3 are empirical, adjustable parameters.
The AA equation has become the key tool for describing   . This success was notably strengthen by its derivation within the Avramov-Milchev (AM) phenomenological model for vitrification (Avramov and Milchev, 1988;Roland and Casalini, 2003): Notwithstanding, there is a discrepancy between eqs.  Table I.
can be the general phenomenon. The description of the reversal melting was first proposed by Rein and Demus (RD) (Rein and Demus, 1993;Demus and Pelzl, 1988) and subsequently by Kechin (K) (Kechin, 1995;Kechin 2001): , with the pressure dependence given as where the index '0' is related to the reference point ( 0 0 ,

P T
). Hence, taking as the reference the atmospheric pressure as the reference one can indicate the following meaning of parameters in eqs. (6) -(9) and the power exponent . For SG and AA eqs. (6) and (7), as well as K&RD eq. (8), the reference has to be taken . Other selections of 0 T yields non-optimal and effective values of fitted coefficients. In ref. (Skripov and Faizulin, 2006) the following relation was derived (Drozd- Rzoska, 2005): Eq. (11) is able to portray systems with the maximum of melting or vitrification curve, even if they are hidden in the negative pressures domain. It can be also applied for systems were and then . The latter equation is in agreement with the empirical relation for the pressure evolution of the bulk modulus recalled above (Murnaghan, 1944).
There are few other approaches which starting from the C-C or related Lindemann relation (Skripov and Faizulin, 2006), developed for melting. They are briefly presented below, with indications of their applicability for the glass formation. All these is supplemented by few new formulas, resulted from such reasoning. Schlosser et al. (Schlosser et al., 1989) starting from the Lindemann relation (Lindemann, 1910;Skripove and Faizuli, 2006) and the definition of the Grüneisen parameter as (Grüneisen, 1913) obtained the relation focusing on the volume dependence of the melting temperature. Generalizing this dependence for the arbitrary fusion process one obtains: where the index '0' is for the zero-pressure (~atmospheric pressure) reference. Assuming for the the following relation was derived (originally for melting): One may expect that it is able to portray systems described both by 0 ,  dP dT m g and 0 ,  dP dT m g . For small/moderate pressures eq. (14) can be reduced to the Kraut-Kennedy relation (Schlosser et al., 1989, Kraut andKennedy, 1966) , originally developed for melting: It can be converted to the density related dependence along melting or vitrification curves: Linking eqs. (12) and (15) one obtains the relation for pressure induced volume changes along melting or vitrification curve: This relation is in fair agreement with the Murnaghan equation, broadly used is earth sciences (Murnaghan, 1944;Skripov and Faizulin, 2006). Recalling the dependence eq. (15) can be converted to the SG-or AA-type equation (Schlosser et al, 1989): It this relation the SG exponent , i.e. it differs from Burakovsky [Burakovsky et al., 2003) predictions.
Kumari and Dass (Kumari and Dass, 1988;Dass, 1995) also applied the framework of the Lindemann criterion (Lindemann, 1910) and workout the relation originally focused on the pressure evolution of the melting temperature, focusing on alkali metals: where     (19) and can be extended to the negative pressures domain when introducing the reference related to the absolute stability limit in the negative pressures domain:

The analysis of experimental data
When considering the parameterization of    (22) It is visible that the description via DR and SG/AA relations overlaps and both can be extended into the negative pressures domain. However, such possibility for the AA and SG relation may be casual since it does not takes place for Rein&Demus and Kechin eq. (9), for Kumari&Dass eq. (19) or for pressure counterparts of the VFT relation (eqs. (2) and (3)).
Regarding the 'general' DR eq. (11), the following transformation of experimental data was proposed to test the domain of its validity (Drozd- Rzoska  Concluding, equations (22) and (23) Table I).    Pressure dependence of melting temperature of germanium (based on data from ref. (Vaidya, 1969;Porowski, 2015). Experimental data are portrayed by DR eq. (11), with the support of the preliminary derivative-based analysis (eq. (23)) yielding also optimal values of parameters: this is shown in the inset.   , i.e. the system becomes extremely good glass former.  The pressure evolution of the glass temperature for glycerol. The solid blue curve, with 'dotted' and 'dashed' parts is related to DR eq. (11) and the preliminary analysis via eq. (23). Experimental data are from author's measurements [60]    Albite dT g / dP < 0 Fig. 6 The pressure evolution of the glass temperature in albite ( NaAlSi3O8 ), the component of magmatic, metamorphic rocks. The plot bases on experimental data from ref. (Bagdassarov, 2004). The solid curve is related to eq. (11).
Generally, the experimental evidence of glass formers characterized by 0  dP dT g is very limited (see Table I). Such behavior seems to be characteristic for some strongly bonded systems. Fig. 7 shows results of such studies for albite, geophysically important material, which can be well portrayed by eq. (11), revealing the maximum of   P T g curve 'hidden' in negative pressures domain.

Universal aspects of the pressure evolution of the glass temperature
The above discussion indicated the possible common phenomenological description of   P T g evolution in glass formers described by The question arises of the more microscopic insight. In ref. (Voigtmann, 2006a) analysed the vitrification within frames of the square-well (SW) model associated with the relatively simple 13 potential:     r U for distances d r  supplemented with an SW attraction within the range and U(r) = 0 beyond was analyzed. The SW approach proved its superior ability for describing colloidal glass formers, which can be thus considered as a kind of archetypical experimental glass forming model system. In ref. (Voigtmann, 2006a) the possibility of the common description of glass forming molecular liquids and colloids was shown, using the plot ). In ref. [64] only glycerol was discussed, for the clarity of reasoning. This report also focuses on glycerol, but for the notably enhanced range of pressures, basing on data from Fig. 5. This is supplemented by experimental data for albite, where 0  dP dT g (Fig. 6). In ref. ( : numbers are given for glycerol. In ref. (Voigtmann, 2006b) the partial agreement between predictions of SW and LJ model was obtained after ad hoc shifting . It is notable that so far experiments in colloids are carried out under atmospheric pressure and obtained phase diagrams are presented using the volume fraction ( ) -interaction strength or temperature axes. Such data were model-mapped into the pressuretemperature plane in ref. (Voigtmann, 2006a).
: glass formers approach the hard-sphere limit. Following ref. (Voigtmann, 2006) in this domain: : there is a universal 'generic steep' anomaly and this regime is characteristic for molecular glass formers. Regime III " for 0   g T the low density and weak interactions domain occurs. It is available for colloidal glass formers and does not accessible for molecular ones. In refs. (Voigtmann 2006a and 2006b) glass forming systems for which 0  dP dT g were not discussed.  The pressure dependence of the glass temperature, summarizing the model discussion (Voirtmann 2005 and2006): SW is for the square-well potential model, LJthe Lennard-Jones potential model and HS is for the hard spheres model. For details see the text of the given paragraph and refs. (Voigtmann, 2006a). Experimental data for glycerol are taken from Fig. 6: they are present in the 'natural scaled" units. Data for albite are from Fig. 7. Note that for open green diamonds (glycerol) and open circles (albite) the reference pressure was takes into account: . Data for the polymer mediated colloid are from refs. (Voigtmann, 2006a;Pham, 2002). For details see comments in the given paragraph. Note the disappearance of the 'generic steep' anomaly (indicated by the vertical arrow) and the ability for describing arbitrary glass former.
One of the most striking features of refs. (Voigtmann 2006a and2006b) is the 'generic steep' anomaly, presumably occurrying only for molecular glass formers. However, this unique phenomenon has few surprising features. First, it is very strong and associated with exactly the same 'singular' value of 23 . 0   g T for arbitrary molecular glass former. Well above the singularity experimental data for all molecular glass formers overlaps. Second, the 'generic' anomaly appears in the loglog scale but no hallmarks of such behavior appears in the linear scale for any 'native'    Andersson, 1998;Roland, Hensel-Bielowka, et al., 2005). Third, although real high pressure results for colloidal glass formers are still not available, one can easily show that such data also will follow the same 'generic steep anomaly' pattern, in disagreement with 'recalculated' data shown in Fig. 7 (stars). Following all these, one can conclude that the 'generic steep" anomaly is the consequence of Consequently, the "generic steep" anomaly disappears and   P T g experimental data for molecular glass formers can be mapped also to the low density ( 0   T ) domain. When linking such analysis with eq. (11) one also obtains the possibility of describing systems characterized by 0  dP dT g , as shown for the extrapolated behavior for glycerol and for albite in Fig. 7. Fig. 7 also shows that the re-entrant glass forming colloids mapped from experimental studies under atmospheric pressure to the P-T plane are related to the case 0  dP dT g . For glycerol, for very high pressures, the behavior described by 5 4 g g P T  emerges and the evolution approaches the hard sphere limit pattern (Voigtmann, 2006a). One of arguments for the generic importance of the 'steepness' anomaly in refs. (Voigtmann 2006a and2006b) was the possibility of it reproduction by the model-fluid with LJ potential containing properly adjusted attraction term. However, for the analysis of   in such model-fluid the absolute stability limit have to be taken into account: after the transformation P P   the 'generic steep anomaly' disappears also for the LJ model fluid.
Concluding, the plot   g P 10 log vs. in such plot. This is the key feature of the intermediate regime II. There are no unique 'generic' steep anomalies. Finally, worth indicating is the general difference between  g P vs.  g T data taken from concentrational experiment under atmospheric pressure (1) and from the real high pressure experiment (2) for colloidal glass formers. The case (1) for re-entrant colloidal glass former can be linked to the group of systems where 0  dP dT g . The characterization of the solvent is constant but the number of colloidal particles and distances between them can change when 'decreasing pressure" ( 0   ). For such system the problem of the absolute stability limit is absent: it is naturally related to 0   g P and the negative pressures domain does not exist. For the case (2), compressing changes notably not only not only distances between colloidal particles but also properties of the solvent. Changes of density of the solvent (typically ~ 30 % for GPa P 1  ) are associated with very strong changes in dynamics, particularly near the glass temperature. In this case 'rarefication' associated with the isotropic stretching and entering pressures domain can yield even stronger changes for the solvent. Stretching is terminated by the absolute stability limit spinodal in negative pressures domain. All these show that for the case (1) properties of the colloidal glass former are dominated almost exclusively by colloidal particles. In the case (2) at least equally important is the impact of the solvent. Fig. 7 indicates the clear link between molecular and colloidal glass formers: they follow the same patter the plot   g P 10 log vs. . Model fluids based on SW and LJ potentials offer the nice frame for getting the fundamental insight into experimental data within such presentation.

Concluding remarks
This report presents proposals of few equations for describing the pressure evolution of the glass temperature beyond the dominated SG/AA pattern. They make the description of glass forming systems where both 0  dP dT g and 0  dP dT g possible. The ways of portrayal were extended also for the evolution of    , V T g and    , V P g . The basic relevance of including into the analysis negative pressures and the preliminary derivative-based and distortionssensitive 16 analysis has been shown. From results presented the possible general pattern for   P T g evolution for glass forming systems ranging from low molecular weight liquids, resins, polymer melt, liquid crystals to colloidal fluids emerges. In the low density region the extended SG-type equation can describe experimental data. On increasing pressures, for intermediate densities, the gradual inclusion of the 'damping term' can lead to the reversal (re-entrant, 0  dP dT g ) vitrification. However, for strongly compressed and high density systems the crossover to the second, HS-type, dependence   such general characterization is manifested as the less or more marked S-shape. It is notable that this picture may be valid both for molecular and colloidal glass formers, although for the latter real high pressure experiments are still required. ) dependences are used. The discussion for the latter (Poirier, 2000;Skripov and Faizulin, 2006) indicate that notable distortions appears for . Taking into account the compressibility of typical molecular liquids such domain starts for GPa P 5 . 1 . In the opinion of the authors, equally important can be the distance of the reference point from the possible maximum of   P T g , even if it is 'hidden' by a phase transition or crossover to another form of vitrification.
Finally, we would like to stress the significance of the above discussion for the glass transition physics, material engineering and geophysical and planetary studies.

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.

Funding
This report was prepared due to the support of the National Centre for Science (Narodowe Centrum Nauki (NCN), Poland) grant ref. UMO-2016/21/B/ST3/02203.

Figure 2
Pressure dependence of melting temperature of germanium (based on data from ref. (Vaidya, 1969;Porowski, 2015). Experimental data are portrayed by DR eq. (11), with the support of the preliminary derivative-based analysis (eq. (23)) yielding also optimal values of parameters: this is shown in the inset.

Figure 4
The pressure evolution of melting and glass temperature for selenium. The change of m g T T value is indicated. Solid curves are described by DR eq. (11): parameters were derived from the preliminary analysis based on eq. (23). Experimental data were taken from refs. (Deaton and Blum, 1965;Katayama et al., 2000;Ford et al., 1988;Tanaka, 1984;Caprion and Schober, 2002).

Figure 5
The pressure evolution of the glass temperature for glycerol. The solid blue curve, with 'dotted' and 'dashed' parts is related to DR eq. (11) and the preliminary analysis via eq. (23). Experimental data are from author's measurements [60] and from refs. (Drozd- Rzoska, 2005; Drozd- Rzoska et al., 2007, Cook, et al. 1994Pronin et al. 2010). The dashed line and stars (in magenta) in the negative pressures domain denotes the possible absolute stability limit location: this was determined from the analysis of   P  experimental data via eq. (5). The inset shows the pressure evolution of dP dT g coefficient.

Figure 6
The pressure evolution of the glass temperature in albite ( NaAlSi3O8 ), the component of magmatic, metamorphic rocks. The plot bases on experimental data from ref. (Bagdassarov, 2004). The solid curve is related to eq. (11).

Figure 7
The pressure dependence of the glass temperature, summarizing the model discussion (Voirtmann 2005 and2006): SW is for the square-well potential model, LJthe Lennard-Jones potential model and HS is for the hard spheres model. For details see the text of the given paragraph and refs. (Voigtmann, 2006a). Experimental data for glycerol are taken from Fig. 6: they are present in the 'natural scaled" units. Data for albite are from Fig. 7. Note that for open green diamonds (glycerol) and open circles (albite) the reference pressure was takes into account: . Data for the polymer mediated colloid are from refs. (Voigtmann, 2006a;Pham, 2002). For details see comments in the given paragraph. Note the disappearance of the 'generic steep' anomaly (indicated by the vertical arrow) and the ability for describing arbitrary glass former.