Free energy calculations liquid crystals

The quality of the mean-field approximation can be tested in simulations of the same lattice model [13]. Ideally, direct free-energy calculations of the liquid and solid phases would allow us to locate the point where the two phases coexist. However, in the present studies we followed a less accurate, but simpler approach we observed the onset of freezing in a simulation where the system was slowly cooled. To diminish the effect of supercooling at the freezing point, we introduced a terraced substrate into the system to act as a crystallization seed [14]. We verified that this seed had little effect on the phase coexistence temperature. For details, see Sect. A.3. At freezing, we have... 

In a self-assembling soft matter system, the composition also fluctuates little around the ideally ordered value however, the molecules are in a liquid state, that is, they diffuse and are not tethered to ideal positions. Therefore, there is no simple reference state of the particle-based model with a known free energy, and previous simulation techniques for calculating the absolute free energy of hard crystals do not straightforward carry over to self-assembling soft matter systems. 

Consider the three typical aligned nematic liquid crystal cells as depicted in Figures 6.6a-6.6c corresponding to planar, homeotropic, and twisted NLCs of positive anisotropy. With the applied electric field shown, they correspond to the splay, bend, and twist deformations in nematic liquid crystals. Strictly speaking, it is only in the third case that we have an example of pure twist deformation, so that only one elastic constant K22 enters into the free-energy calculation. In the first and second cases, in general, substantial director axis reorientation involves some combination of splay (S) and bend (B) deformations pure S and B deformations, characterized by elastic constants Kii and respectively, occur only for small reorientation. 

The parameters of the potential function are adjusted to give the best description of bulk properties of the ionic crystal. However, different parameters confribute differently to the surface properties as compared to the bulk properties and the choice of one parameter set might be good for bulk, but questionable for surface properties. Therefore, the results of the calculations of the surface free energies and of the surface stresses in section 4.4.7.S in Table 5 should be understood merely as estimates, although the agreement with the scarce experimental data on surface free energy of liquid salts is within 10%. 

Table 4. Excess free energy per particle for the different bulk structures and the liquid state calculated via a thermodynamic intergration in the limit of infinite number of particles [102]. The reference state for the free energy calculation of the liquid was the hard-sphere fluid, and for the bulk solid structures we used an Einstein-crystal. In some cases we also used the hard-sphere system as a reference state for the sohd structures. We found that the solid free energies obtained via these two distinct routes agreed to within 0.005kgT, which corresponds to our estimate of the statistical error in this calculation. The statistical accuracy of the computed free energy of the liquid is estimated to be iO.Olfc T. In the table, the values in brackets indicate the volume fraction at which the excess free energy was calculated. The calculated excess free energies for the fee and the hep structures can be compared directly, as they were calculated at the same pressure, whereas the others are not. The fcc-hcp free energy difference is always smaller than (1 x Q kgT)...

The domains of existence of the lamellar liquid crystals can be in principle identified by calculating the free energies of the various possible phases. Because of the difficulty in performing such a calculation, a first purpose of this paper is to employ the thermodynamic formalism developed by Ruckenstein10 to extract some information about the domain of stability of the lamellar phase. 

In solutions of water and surfactant, the surfactant monolayers can join, tail side against tail side, to form bilayers, which form lamellar liquid crystals whose bilayers are planar and are arrayed periodically in the direction normal to the bilayer surface. The bilayer thickens upon addition of oil, and the distance between bilayers can be changed by adding salts or other solutes. In the oil-free case, the hydrocarbon tails can be fluidlike (La) lamellar liquid crystal or can be solidlike (Lp) lamellar liquid crystal. There also occurs another phase, Pp, called the modulated or rippled phase, in which the bilayer thickness varies chaotically in place of the lamellae. Assuming lamellar liquid crystalline symmetry, Goldstein and Leibler [19] have constructed a Hamiltonian in which (1) the intrabilayer energy is calculated... 

Worked Example 10.1 shows how to calculate the Frank-Oseen free energy and use it to predict the response of a liquid crystal to a magnetic or electric field. Such calculations are used to design practical liquid-crystal display devices. They also can be used to determine the values of the elastic constants. 

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