Activation volume of selected liquid crystals in the density scaling regime

In this paper, we demonstrate and thoroughly analyze the activation volumetric properties of selected liquid crystals in the nematic and crystalline E phases in comparison with those reported for glass-forming liquids. In the analysis, we have employed and evaluated two entropic models (based on either total or configurational entropies) to describe the longitudinal relaxation times of the liquid crystals in the density scaling regime. In this study, we have also exploited two equations of state: volumetric and activation volumetric ones. As a result, we have established that the activation volumetric properties of the selected liquid crystals are quite opposite to such typical properties of glass-forming materials, i.e., the activation volume decreases and the isothermal bulk modulus increases when a liquid crystal is isothermally compressed. Using the model based on the configurational entropy, we suggest that the increasing pressure dependences of the activation volume in isothermal conditions and the negative curvature of the pressure dependences of isothermal longitudinal relaxation times can be related to the formation of antiparallel doublets in the examined liquid crystals. A similar pressure effect on relaxation dynamics may be also observed for other material groups in case of systems, the molecules of which form some supramolecular structures.


S1. Model-dependent relations between parameters of the volumetric and activation volumetric equations of state
To conduct a comparative study of the activation volumetric properties of the selected LC systems in terms of the Avramov and MYEGA models, we have made an attempt to formulate the relationships relied on the T-V MYEGA models between the exponents  EOS and  act as well as between the isothermal bulk modulus B act and B T . It has turned out that such a derivation is only possible in an approximate way due to the morphology of the T-V in the volumetric EOS (Eq. (12) in the main part), one can employ the first-order Taylor series expansion of the quotient in Eq. (S3). Since one may also apply the first-order Taylor series expansion to the exponent The exponent in the latter equation can be transformed to a power form by the next first-order Taylor series expansion as follows . In this case, we arrive at an isothermal auxiliary equation at different temperatures, the value of the parameter  0 should remain unchanged for a given material at least to a good approximation.

S2. Results of the analysis based on the T-V Avramov model
As an example, in Fig making the assumption that the reference state (T 0 ,p 0 ) in the volumetric EOS is fixed at the crystal -Cr E transition temperature T 0 =301.5K at ambient pressure p 0 =0.1MPa. Then, we can directly describe the dependence  || (T,p) as shown in Fig. S1(a) by using Eq. (4), with the specific volume V(T,p) expressed by the volumetric EOS with the values of its parameters found by fitting pVT measurement data to this EOS (see Fig. S1(b) in this document and Table 1 in the main part). A good quality of the fitting procedure to Eq. (S10) is confirmed by the adjusted measure R 2 =0.9997 for this LC system (see Table S1 for other 5 tested LCs) and also reflected in the density scaling of the longitudinal relaxation times of 8BT, which is well satisfied (see Fig. S1(c)) with the value of the scaling exponent, =4.59±0.03, evaluated by fitting to the Avramov model (Eq. (S10)). Subsequently, we have established ( Fig. S1(d)) the activation volumes V act from Eq. (6) with Eq. (7) in the main part, i.e.,  Table S1 for the values of its parameters) with the specific volume expressed as the function V(T,p) by the volumetric EOS given by Eq. (12) with Eqs.
(13) and (14) in the main part. (b) Pressure dependences of specific volumes measured along different isotherms and their fits to the volumetric EOS (see Table 1 Table S2 for the values of its parameters).

S3. Comparison of the density scaling behavior evaluated in terms of the T-V Avramov and MYEGA models
In Fig. S3, the fitted curves generated from the Avramov and MYEGA density scaling models are compared as functions of the scaling variable =T . This comparison shows that the curves fitted to the MYEGA model reveal a stronger negative curvature than those generated from the Avramov model, especially in case of the LC systems investigated in the nematic phase (Fig. S3(b)). Since the fitted values of the scaling exponents  and  M are nearly the same for a given tested material, i.e., the scaling variables  determined from the Avramov and MYEGA model are only slightly different for this material, Fig. S3 reliably illustrates the small differences in the density scaling curves fitted to the Avramov and MYEGA model, which are caused by the other parameters of the models except for the scaling exponent. For instance, one can notice that the slower convergence and the weaker negative curvature of the curves fitted to the Avramov model applied to the examined LC systems in the nematic phase are additionally associated with the extremely small values of the preexponential factor  0 , which are physically much more reasonable in case of the MYEGA model, although the latter model easier and better reflects the negative curvature of the dependences log 10  || ().
Nevertheless, the preexponential factor  0 does not affect the activation volume defined by Eq.
(3) in the main part via the derivative of ln || with respect to pressure. Thus, we need to look deeper into what determines the activation volume in both the Avramov and MYEGA models, which is done in the section Results and Discussion in the main part.

Fig. S3
Plot of the fitted curves in terms of the T-V Avramov and MYEGA models (i.e., Eqs. (4) and (5) in the main part, also shown in this document as Eqs. (S10) and (S1)) with the values of their parameters collected respectively in Tables S1 in this document and Table 2 in the main part for all the LC systems examined in the Cr E phase (a) and in the nematic phase (b) in the dielectric measurement ranges.

S4. Physical meaning of the fitting parameters in the T-V Avramov and MYEGA models in case of GF systems
It is worth noting that not only the density scaling exponent but also the other fitting parameters have well-defined physical meanings in both the entropic models. The preexponential factor in each model is typically interpreted as a high-temperature limiting value of the considered relaxation times. However, the parameters A, D, A M , and D M result from specific assumptions underlying the entropic models.
In the T-V Avramov model, the parameter A has been argued 27 (S11) In case of GF, the reference state (T r ,V r ) is usually defined at the glass transition at ambient pressure, although one can choose another reference state, e.g., the melting point at ambient pressure. In case of LC systems, the reference state can be assumed at the transition to another LC phase at ambient pressure. The reference parameters ) , ( (S12) and the total system entropy, 26 with the temperature-volume dependent energy barrier distribution width where R is the gas constant, and the Grüneisen parameter can be evaluated for a supercooled liquid near the glass transition from the isobaric thermal expansivity α p , the isothermal compressibility κ T , and the isochoric heat capacity C V , whereas 11 Z has been interpreted to be proportional to the number of available pathways for local motions of a molecule or polymer segment (and this number has been also assumed to be roughly proportional to the coordination number of the liquid lattice), 26,29 The parameter D comes from the original derivation 26 of Eq. (S10), i.e., Eq. (4) in the main part, which results in (S16) In the T-V MYEGA model, the fitting parameters A M and D M are related respectively to some volume-dependent energetic parameters, 23 is an effective activation barrier for the considered relaxation process and ) (V H M is the energy difference between two states (on the assumption that network constraints in a two-state simple system may be intact or broken). 23,32 The detailed equations, In the original Adam-Gibbs model, 33 the parameter M B was assumed to be a constant dependent on the product of the critical configurational entropy * c s (i.e., the configurational entropy of the smallest cooperative subsystem that can perform a rearrangement into another configuration) and the potential energy barrier  hindering the transition to a new configuration, where  is related to a difference in the Gibbs free energies of the rearrangeable subsystems and the others. However, further investigations have shown 21,23,25 that the dependence of the parameter M B on thermodynamic conditions cannot be neglected.
For our discussion (presented in the main part) on two entropic models applied to study the density scaling in LC systems, it is important that the parameter A M is embedded in the equation formulated for the configurational entropy within the T-V MYEGA model as 23 which can be also expressed as follows ) ) ( exp (  ln  3  ) , ( (S22)