Elastic constants order parameter dependence

In rigid-rod mesophases where interactions are dominated by excluded-volume interactions, the contributions of elasticity to the free energy are enfropic in nature. The elastic constants ku are proportional to and order-parameter-dependent... 

J.K. Kruger No, not necessarily. There are many transitions, strong transitions of the first order in the solid state, where you don t see jumps in the elastic constants. It really depends on how strongly the density couples to your measured test of stability, say, to the elastic constants. The thermal behavior of the mode Gruneisen parameters, for example, may reduce the influence of the density on the sound velocity [J.K. Kruger, R. Roberts, H.-G. Unruh, K.-P. Fruhauf, J. Helwig, H.E. Muser, Progr. Colloid Polym. Sci.. 71, 77-85 (1985)]. 

Therefore, switch-off times are independent of the field strength and directly dependent on material parameters, such as viscosity coefficients and elastic constants, and the cell configuration. Therefore, they are often three or four orders of magnitude larger than the switch-on times. However, sophisticated addressing techniques can produce much shorter combined response times ( on + off The nematic director should be inclined, e.g. 1° pretilt,... 

Figure 3. Spontaneous strains and elastic properties at the 422 < i> 222 transition in Te02. (a) Spontaneous strain data extracted from the lattice-parameter data of Worlton and Beyerlein (1975). The linear pressure dependence of (e - (filled circles) is consistent with second-order character for the transition. Other data are for non-symmetiy-breaking strains (e + 62) (open circles), 63 (crosses), (b) Variation of the symmetry-adapted elastic constant (Cn - Cu) at room temperature (after Peercy et al. 1975). The ratio of slopes above and below Po is 3 1 and deviates from 2 1 due to the contribution of the non-symmetry-breaking strains. (After Carpenter and Salje 1998).

Elastic constants depend on pressure and temperature because of the anharmonicity of the interatomic potentials. From the dependence of bulk and shear moduli on hydrostatic and uniaxial pressure, third order elastic constants and Griineisen parameters may be determined. Griineisen parameter shows the effect of changing volume, V, on the phonon mode frequencies, co. 

The order parameter S is a very important quantity in a partially ordered system. It is the measure of the extent of the anisotropy of the liquid crystal physical properties, e.g., elastic constants, viscosity coefficients, dielectric anisotropy, birefringence, and so on. S is temperature dependent and decreases as the temperature increases. The typical temperature dependence of S is shown in Figure 1.16. 

As for elastic constants, the CEF effects in thermal expansion and the Schottky specific heat are especially pronounced in TmSb, see fig. 18. Equations (45) and (46) give a good fit to the data, (Ott and Liithi 1976, 1977) with y, = yj = -1.2 as seen from fig. 18. The CEF Griineisen parameters measuring the volume dependence of the CEF levels turn out to be of the order of one. In this way a number of cubic compounds have been studied so far TmSb, PrSb, SmSb, ErSb, CeTe and TmTe (Ott and Liithi 1977), and TmCu, TmZn and TmCd (Morin and Williamson 1984). 

The elastic constants depend on the product of the order parameters of two neighboring molecules. If one of the molecules had the order of 0, the second molecule can orient along any direction with the same inter-molecular interaction energy even if it has non-zero order parameter. Therefore the elastic constants are proportional to S. When the temperature changes, the order parameter will change and so will the elastic constants. 

In order to have the transition from the first state to the second in all those cases, the apphed field must be sufficiently high so that the decrease of the electric energy can compensate for the increase of the elastic energy and surface energy. The threshold field, above which the applied field can produce the transition, depends on (1) droplet size, (2) droplet shape, (3) anchoring condition, and (4) the material parameters, such as elastic constants and dielectric anisotropy, of the liquid crystal [3,24,25]. 

Recently Tao et al. extended the MS theory by adding to Eq. (3) the isotropic, density-dependent component of the molecular interactions (/o(r) in the form of the Lennard-Jones potential (/o(r) = 4e [(o-/r) -(o-/r) ]. As a result they obtained a better agreement of the calculated and experimental quantities characterizing the nematic-isotropic transition, for example, volume change at and the values of dT ldp. Chrzanowska and Sokalski considered the case when the parameter Lennard-Jones potential is dependent on the orientation of molecules that allows one to predict properly for MBBA such properties as order parameters, elastic constants, and rotational viscosity coefficients. 

It has been assumed that molecular properties contribute additively to the macroscopic tensor components, which are consequently proportional to the number density. If intermolecular interactions contribute to the physical property, then deviations from a linear dependence of the property on density are expected. Also the contribution of orientational order will be more complex, since the properties will depend on the degree of order of interacting molecules. Effects of molecular interactions contribute to the dielectric properties of polar mesogens, and are particularly important for elastic and visoelastic properties. Molecular mean field theories of elastic properties predict that elastic constants should be proportional to the square of the order parameter this result highlights the significance of pairwise interactions. 

To proceed we need to know how the functional A[f(fi-, 22 Ri 2)] varies when the equilibrium state of the liquid crystal is elastically distorted. A macroscopic strain will not influence Mi 2 or g, p. since these are dependent only on molecular parameters of the model the free energy changes because the single particle distribution functions change. We assume that for the small distortions described by the Frank elastic constants, the single particle orientational distribution function, defined with respect to a local director axis, is also independent of strain i. e. elastic torques do not change the molecular order parameters. The product of distribution functions /(i2j, R ) f, Q2i R2) will change with strain because the director orientations at R1 and R2 will differ, and the evaluation of the strain dependence of the... 

The statistical theories of elasticity have shown that the principal elastic constants depend on the single particle distribution functions and the intermolecular forces. The former can be accounted for in terms of order parameters, but intermolecular parameters are more diffieult to interpret in terms of molecular properties. Results for hard particle potentials relate the elastic constants to particle dimensions, but the depen-... 

The main temperature dependence of elastic constants is due to the order parameter. Following the predictions of mean field theory Eq. (96), reduced elastic constants c, have been introduced, which should be independent of temperature, defined by ... 

As explained in Sec. 6.1.7.4 of this chapter, the compressional elastic constant B vanishes at the SmA-SmA critical point so that the system is very close to a nematic. An interesting consequence is that the energy of dislocations becomes very weak and their proliferation may lead to a destruction of smectic order. Prost and Toner [109] have shown that depending on bare parameters of a particular system, either the nematic bubble or the critical point could be observed. 

It is clear that the transition from a nematic (or cholesteric) to a smectic phase will result in the divergence of certain of the elastic constants. In particular, k22 and 1 33 will diverge at the N-SmA phase transition. The type of divergence observed will depend on the nature of the phase transition, which can be either first or second order [143, 144]. The transition is second order if the nematic phase is sufficiently wide, such that the nematic order parameter is saturated at the transition. Both de Gennes [145] and McMillan [146] developed theories of the SmA-N phase transition that have implications for light scattering. The form of the divergence of the twist and bend elastic constants can be written as ... 

Alexe-Ionescu and co-workers [229,230] have analysed the temperature dependence of Ki2 by theoretical considerations. They introduced a term that is proportional to S and which is not present in the other elastic constants. At small S in the vicinity of TVj.i, the linear order parameter term should dominate the temperature dependance of the splay-bend elastic constant, and the ratio to the bulk elastic constant Ky lK should be proportional to 1/5. 

The first theoretical dicussion of the temperature dependenee of elastic constants within the framework of the molecular-statistical Maier-Saupe theory was given by Saupe [241]. He attributed their temperature dependence to changes in the order parameter S and the molar volume with temperature, and he introduced reduced elastic constants... 

A theoretical relation between the nematic elastic constants and the order parameter, without the need for a molecular interpretation, can be established by a Landau-de Gennes expansion of the free energy and comparison with the Frank-Oseen elastic energy expression. While the Frank theory describes the free energy in terms of derivatives of the director field in terms of symmetries and completely disregards the nematic order parameter. The Landau-de Gennes expansion expresses the free energy in terms of the tensor order parameter 0,-, and its derivatives (see e.g. [287,288]). For uniaxial nematics, this spatially dependent tensor order parameter is... 

Monseiesan and Trebin [292] have made an attempt to predict theoretically the temperature dependence of the elastic constants of orthorhombic nematics on the basis of an extension of the Landau-Ginzburg-de Gennes theory, where the free energy is minimized by the order parameter ... 

The hydrostatic pressure dependence of C33 was also studied by Klimker and Rosen (1973) (see fig. 32). The increase of C33 under pressure is ascribed to the variation of 033(0) due to the higher-order elastic constants. The observed increase of the spin-reorientation anomaly dip under pressure was explained by the pressure dependence of magnetoelastic coupling parameters and changes in magnetic ordering due to the shifts of Tc and Tsr-... 

The anharmonicity of the crystal lattice exhibits itself in hydrostatic pressure dependence of the elastic constants, moduli, lattice parameters, thermal e7q)ansion, temperature variation of the volume compressibility and in the existence of phonon-phonon interactions. The anharmonic properties of the crystal can be characterized by higher-order elastic constants. 

The analysis of the simulations of crystalline Si 2 has focused on a study of the lattice parameters and their temperatures dependence, on the elastic constants (which can be gotten from the Parrinello-Rahman fluctuation relations [45]), and quantities characterizing the local aspects of the structure (probability distributions to find an atom a distance r away from its equilibrium position, partial radial pair distributions, average bond angles and their distributions, etc.). Of particular interest is also the global order parameter ()) of the phase transition between a-quartz and 5-quartz, which measures the rotation of the (distorted) tetrahedra about the [100] axis,... 

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