Polymer crystals elastic constants

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. 

Most recent estimates of the crystal elastic constants for polymers follow the method of calculating the second derivatives of the Helmholtz Free Energy A with respect to deformation of a small volume element Vo [62,63]. The stiffness constants Ctmnk are defined as... 

The sets of equations are solved by the assumption of periodic waves and, by expansion in powers of the wave number, a relation is found for the limiting case of long waves so that the elements of the dynamical matrix elastic constants of the continuum. It is also possible to derive the Raman frequencies from the lattice dynamics analysis but this does not seem to have been done for polymer crystals, though they have been derived for example, for NaCl and for diamond. 

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. 

Fig. 12. Dependence of the elastic constant C22 in chain direction on the conversion in partly polymerized PTS crystals. The black square is the theoretically expected value for the pure polymer. The curves marked (V) and (R) are calculated according to Eq. (3) and (4), respectively...

In microscale models the explicit chain nature has generally been integrated out completely. Polymers are often described by variants of models, which were primarily developed for small molecular weight materials. Examples include the Avrami model of crystallization,- and the director model for liquid crystal polymer texture. Polymeric characteristics appear via the values of certain constants, i.e. different Frank elastic constant for liquid crystal polymers rather than via explicit chain simulations. While models such as the liquid crystal director model are based on continuum theory, they typically capture spatiotemporal interactions, which demand modelling on a very fine scale to capture the essential effects. It is not always clearly defined over which range of scales this approach can be applied. 

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 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. 

Next we note that there are two physieally different sources of temperature and pressure dependence of the elastic constants of polymers. One, in common with that exhibited by all inorganic crystals, arises from anharmonic effects in the interatomic or intermolecular interactions. The second is due to the temperature-assisted reversible shear and volumetric relaxations under stress that are particularly prominent in glassy polymers or in the amorphous components of semi-crystalline polymers. The latter are characterized by dynamic relaxation spectra incorporating specific features for different polymers that play a central role in their linear viscoelastic response, which we discuss in more detail in Chapter 5. 

With few exceptions, we shall idealize the elasticity of solids as isotropic, as stated earlier, so as not to burden the discussion of the physical mechanisms with inessential operational detail. We note here, however, that many cubic crystals are quite anisotropic. Tungsten, W, which is often cited as being isotropic, is so only at room temperature. Thus, we shall make use principally of the elastic relations in eqs. (4.15) and (4.16), unless we are specifically interested in anisotropic solids such as some polymer product that had undergone deformation processing. The relationships among various combinations of elastic constants of isotropic elasticity are listed in Table 4.1 for ready reference. 

The perfect-crystal model of Karasawa et al. considers a basic force-field approach for the analysis of crystal properties. The model contains covalent-bonded interactions along the polymer chain as well as non-bonded van der Waals interactions between molecules and Coulombic interactions when relevant, all with appropriate temperature dependences. Table 4.3 lists some of the temperature-dependent elastic moduli and some elastic constants cy of ideal polyethylene, determined by Karasawa et al. (1991), which will be of interest to us in later chapters. These are shown also in Fig. 4.1. Of these Ec (= l/ssj) gives directly the main-chain Young s modulus of polyethylene. Also listed is the transverse shear elastic constant c e, which can be considered to be a good measure of... 

Preedericksz transition in planar geometry is uniform in the plane of the layer and varies only in the z direction. However, in some exceptional cases, when the splay elastic constant Ki is much larger than the twist elastic constant K2 (e.g., in liquid crystal polymers), a spatially periodic out-of-plane director distortion becomes energetically favourable. The resulting splay-twist (ST) Freedericksz state is manifested in experiments in the form of a longitudinal stripe pattern running parallel to the initial director alignment no x. 

A procedure is described that can measure optical phase retardations of birefrigent materials with a resolution 2x10-4 radians. The method relies on phase modulation with alternate right and left handed circularly polarized light. Phase sensitive detection is employed to reduce noise and thermal fluctuations in the optics and light source. The method is useful in Frederik s transition measurements to determine the elastic constants of weakly birefringent polymer liquid crystals with long equilibration times. 

The three elastic constants of a liquid crystal are important physical parameters which depend on the interaction between the molecules in the liquid crystalline state. While a large number of theoretical and experimental investigations on the elastic constants are contained in the literature for thermotropic liquid crystals, very little is known about them in the case of lyotropic polymer liquid crystals such as those formed by poly-Y-benzyl-L-glutamate (PBLG) in various organic solvents. Some theoretical investigations have been carried out 3 the experimental data is limited largely to measurements of the twist elastic constant and a few recent measurements of the bend and splay constants. ... 

For a thermotropic liquid crystal, its physical properties, such as birefringence, viscosity, dielectric anisotropy, and elastic constant, are all dependent on the operation temperature -except at different rates. Polymer-stabilized BPLC is no exception [45]. Figure 14.10 shows... 

In principle, atomistic studies with good quality force fields should be sufficient to represent liquid crystal phases or polymer melts to a high level of accuracy and most material properties (order parameters, densities, viscosities elastic constants etc.) should be available from such simulations. In practise, this is rarely (if ever) the case. For example, using molecular dynamics, the computational cost of atomistic simulations is such that it is rarely possible to simulate for longer than a few tens of nanoseconds for (say) 10000 atoms. Even these modest times often require several months of CPU time on todays fastest processors. 

The preconditions for the use of polymer liquid crystals in display applications are that they exhibit bulk optical properties dependent on the molecular orientation in the mesophase and that this orientation may be altered on application of an external field. In this chapter we shall be concerned with electric or optical fields only. The particular optical property, i.e. (a) the birefringence, (b) the dichroism or (c) the scattering power, defines the display construction in terms of the use of polarized (a and b) or non-polarized (b and c) light, whereas the ability to switch from one orientation to another depends on the anisotropic electric permittivity and the orientational elastic constants. The dynamics of the induced orientation will depend, additionally, on the viscosity constants of the material. 

The initial research on electro-optic phenomena in side-chain polymer liquid crystals concentrated on systems that exhibited nematic phases so that a ready comparison could be made with low molar mass mesogens. Such measurements have established that electro-optic devices are feasible and have allowed elastic constants to be deduced from applications of the continuum theory. This theory, originally derived for low molar mass nematic liquid crystals, defines a relationship for the free energy density F in terms of the elastic constants (/ ) and the director n such that ... 

In recent years there has been increased interest in the theoretical calculation of the elastic constants for ideal and fully oriented polymers based on knowledge of their crystal structures. This increased interest arises from the major developments in computational methods and from the success achieved in producing very highly oriented polymers with reasonably high stiffness. 

The moduli were calculated from the threshold of the Frederiks transition ((4.9) induced by a magnetic (Ax > 0) and electric (Ae < 0)) field in homeotropically oriented liquid crystal layers. The same order of magnitude (10 -10 dyn), which is typical of conventional nematics, has been found for elastic moduli Kn and for other nematic polymers [233, 234]. Unwinding of the helical structure of chiral nematic polymers allowed the elastic constant K22 to be calculated K22 10" dyn for an arylic comb-like copolymer with cholesterol and cyanobiphenyl side-chair mesogens [229]). 

In Part VII, Greg Rutledge discusses the modeling and simulation of polymer crystals. He uses this as an excellent opportunity to introduce principles and techniques of solid-state physics useful in the study of polymers. The mathematical description of polymer helices and the calculation of X-ray diffraction patterns from crystals are explained. Both optimization (energy minimization, lattice dynamics) and sampling (MC, MD) methods for the simulation of polymer crystals are then discussed. Applications are presented from the calculation of thermal expansion, elastic coefficients, and even activation energies and rate constants for defect migration by TST methods. 

From these considerations on elastic constants of the graphite single crystal we can derive the structural precondition for this technically interesting anisotropic polymer carbon. 

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