Crystalline elastic constants

Equation (5.2) also implies that a crystalline solid becomes mechanically unstable when an elastic constant vanishes. Explicitly, for a three-dimensional cubic solid the stability conditions can be expressed in terms of the elastic stiffness coefficients of the substance [9] as... 

Hur S-T, Gim M-J, Yoo H-J, Choi S-W, Takezoe H (2011) Enhanced thermal stability of liquid crystalline blue phase I with decreasing bend and splay elastic constant ratio K33/K11. Soft Matter, in press (DOI 10.1039/clsm06046e)... 

The big difference between normal isotropic liquids and nematic liquids is the effect of anisotropy on the viscous and elastic properties of the material. Liquid crystals of low molecular weight can be Newtonian anisotropic fluids, whereas liquid crystalline polymers can be rate and strain dependent anisotropic non-Newtonian fluids. The anisotropy gives rise to 5 viscosities and 3 elastic constants. In addition, the effective flow properties are determined by the flow dependent and history dependent texture. This all makes the rheology of LCPs extremely complicated. 

Figure 17. Variations of bulk properties derived from variations of the individual elastic constants of stishovite. (a) Bulk modulus (K) and shear modulus (G). Solid lines are the average of Reuss and Voigt limits the latter are shown as dotted lines (Voigt limit > Reuss limit), (b) Velocities of P (top) and S (lower) waves through a poly crystalline aggregate of stishovite. Circles indicate experimental values obtained by Li et al. (1996) at room pressure and 3GPa. After Carpenter et al. (2000a).

Hill R (1952) The elastic behaviour of a crystalline aggregate. Proc Phys Soc Lond A 65 349-354 Hochli UT (1972) Elastic constants and soft optical modes in gadolinium molybdate. Phys Rev B 6 1814-1823... 

There have been many efforts for combining the atomistic and continuum levels, as mentioned in Sect. 1. Recently, Santos et al. [11] proposed an atomistic-continuum model. In this model, the three-dimensional system is composed of a matrix, described as a continuum and an inclusion, embedded in the continuum, where the inclusion is described by an atomistic model. The model is validated for homogeneous materials (an fee argon crystal and an amorphous polymer). Yang et al. [96] have applied the atomistic-continuum model to the plastic deformation of Bisphenol-A polycarbonate where an inclusion deforms plastically in an elastic medium under uniaxial extension and pure shear. Here the atomistic-continuum model is validated for a heterogeneous material and elastic constant of semi crystalline poly( trimethylene terephthalate) (PTT) is predicted. 

The whole simulations are performed according to the following procedure. First, elastic constants of PTT in amorphous phase Camor are calculated using the atomistic modeling, which will be used as input values for the matrix CP in the atomistic-continuum model. Second, elastic constants of PTT in crystalline phase CCTSt are also evaluated in the same manner as those of amorphous PTT. Third, the atomistic-continuum model is validated for heterogeneous material by comparing the calculated elastic constant for the system of infinite lamellas with its exact solution. Finally, elastic constants of semicrystalline PTT with dif-... 

Thus one would expect from a (6x6) matrix of the elastic stiffness coefficients (c,y) or compliance coefficients (sy) that there are 36 elastic constants. By the application of thermodynamic equilibrium criteria, cy (or Sjj) matrix can be shown to be symmetrical cy =cji and sy=Sji). Therefore there can be only 21 independent elastic constants for a completely anisotropic solid. These are known as first order elastic constants. For a crystalline material, periodicity brings in elements of symmetry. Therefore symmetry operation on a given crystal must be consistent with the representation of the elastic quantities. Thus for example in a cubic crystal the existence of 3C4 and 4C3 axes makes several of the elastic constants equal to each other or zero (zero when under symmetry operation cy becomes -cy,). As a result, cubic crystal has only three independent elastic constants (cu== C22=C33, C44= css= and Ci2=ci3= C2i=C23=C3i=C32). Cubic Symmetry is the highest that can be attained in a crystalline solid but a glass is even more symmetrical in the sense that it is completely isotropic. Therefore the independent elastic constants reduce further to only two, because C44=( c - C i)l2. 

J.H. Gieske and G.R. Barsch, Pressure dependence of the elastic constants of single crystalline aluminum oxide, Phys. Status Solidi 29, 121-131 (1967). 

According to the elastic continuum theory of liquid crystals which was introduced in Chapter 1, the three kinds of deformations can be described by three elastic constants, An(splay), / (twist) and / (bend). In the case of small molecular mass liquid crystals, the three constants are mainly determined by the chemical composition of the liquid crystalline molecules. Among them, K22 is the smallest while the other two are approximately close. All three elastic constants are of the order of 10 12 N. The elastic constants of some important liquid crystals are listed in Table 6.1. Each kind of liquid crystals is a mixture of R5-pentyl and R6-hexyl homologues in the ratio of 40 60. The data are obtained at the temperature of T = Tc — 10 °C where Tc is the clear temperature. 

For liquid crystalline polymers, the elastic constants are determined not only by the chemical composition but also by the degree of polymerization, i.e., the length of the molecular chain. One main aim of this section is to address the effects of molecular chain length on the elastic constants of liquid crystalline polymers. Figure 6.1 shows the three typical deformations of nematic liquid crystalline polymers. The length and flexibility of liquid crystalline polymers make the elastic constants of liquid crystalline polymers quite different from those of monomer liquid crystals. 

The above qualitative description illustrates that the molecular length has an important effect on the elastic constants of liquid crystalline polymers. 

Priest (1973) and Straley (1973), in terms of the classical virial expansion, the Onsager theory (referred to in Section 2.1) and the curvature moduli theory, derived the elastic constants of rigid liquid crystalline polymers. The free energy varies according to the change of the excluded volume of the rods due to the deformation. The numerical calculation of elastic constants (Lee, 1987) are shown in Table 6.2. 

The three elastic constants are reduced by ksT/D (D is the diameter of the rod). They are the functions of the dimensionless parameter Q = cf>L/D with cf) the volume fraction of the liquid crystalline polymers and L the molecular length. It is shown in the Table 6.2 that the three elastic constants increase as Q and the order parameter S increase. Among them the bend elastic constant K33 varies dramatically, and finally becomes infinite as S approaches unity in the perfectly ordered state. 

Assume that the degree of the ordering of liquid crystalline polymers is high and the orientational distribution function is simply Gaussian, Odijk (1986) developed the analytical formulae for elastic constants... 

If the semi-flexible liquid crystalline polymers are well stretched along the director and thus the order parameter is high, the underlying equations are valid and can demonstrate qualitatively the effect of chain flexibility on the splay elastic constant. 

In fact, the flexibility (or rigidity) of polymer chains affects the elastic constants. The molecular bend elasticity is a measure of the flexibility of liquid crystalline polymers and the persistence length is associated with the chain flexibility by... 

To strictly analyze the conformational distribution of liquid crystalline polymers and accordingly, to calculate the elastic constants are very difficult. Odijk (1985) derived the elastic constants for semi-flexible... 

The elastic constants of liquid crystalline polymers can be measured in terms of the Frederiks transitions under the presence of a magnetic or electric field. Raleigh light scattering is also a method for measuring the elastic constants. Those techniques successfully applied to small molecular mass liquid crystals may not be applicable to liquid crystalline polymers. This is why very few experimental data of elastic constants are available for liquid crystalline polymers. 

The Raleigh light scattering, another important approach to measure elastic constants and viscosities simultaneously, is applicable to liquid crystalline polymers as well. 

In analog to the approach used by Odijk when dealing with elastic constants, Lee (1988) took the orientation distribution function approximately as Gaussian. When the system is highly ordered, the asymptotic expression can be deduced for viscosities of liquid crystalline polymers, e.g., the Miesowicz viscosities (in the unit of fj) are expressed by... 

In a more recent study, Baraille et a/.88 compared Hartree-Fock and density-functional calculations on crystalline Mg. They found results similar to those above as well as a tendency for the Hartree-Fock calculations to yield a too small cohesive energy i.e., also for the solids the Hartree-Fock approximation underbinds. Equivalently with the tendency of Hartree-Fock calculations to produce too large vibrational frequencies for finite molecules, the elastic constants for this material were overestimated by the Hartree-Fock calculations, whereas there was some scatter for the density-functional results. 

All these results apply to a completely general triclinic crystal system whose elastic properties are expressed by the twenty-one independent quantities Cy or Jy. For crystals of higher symmetry there are further relations between the Cy or 5y which reduce their number still further. For the hexagonal and cubic systems these relations are illustrated in fig. 8.1, together with similar relations for a completely isotropic, non-crystalline material. It can be seen that for a hexagonal crystal like ice there are only five non-zero independent elastic constants Jn, i3> % and 44 or the corresponding Cy. 

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