Molecular orientation equilibrium distribution

We must point out two related limitations of the LE theory. First, it applies to small-molecule LCs and to LCPs in the limit of vanishing strain rate. This is because the LE theory uses a vector n to represent the orientation state of the fluid, tacitly assuming that the molecular orientation distribution stays at its equilibrium state. This is reasonable when the molecular relaxation time is much shorter than the characteristic time of the flow. Second, the theory does not allow orientational defects, which would be singularities in the n field. In reality, LCs and LCPs tend to have a high density of defects. ° Near the defect core, large spatial gradients distort the molecular orientation distribution, thus invalidating the LE theory. 

The origin of the difficulty is the assumption of an equilibrium orientation distribution. In reality, the large spatial gradients at the defect would distort the molecular orientation distribution and severely reduce the local order parameter. 

In general the net macroscopic pressure tensor is determined by two different molecular effects One pressure tensor component associated with the pressure and a second one associated with the viscous stresses. For a fluid at rest, the system is in an equilibrium static state containing no velocity or pressure gradients so the average pressure equals the static pressure everywhere in the system. The static pressure is thus always acting normal to any control volume surface area in the fluid independent of its orientation. For a compressible fluid at rest, the static pressure may be identified with the pressure of classical thermodynamics as may be derived from the diagonal elements of the pressure tensor expression (2.189) when the equilibrium distribution function is known. On the assumption that there is local thermodynamic equilibrium even when the fluid is in motion this concept of stress is retained at the macroscopic level. For an incompressible fluid the thermodynamic, or more correctly thermostatic, pressure cannot be deflned except as the limit of pressure in a sequence of compressible fluids. In this case the pressure has to be taken as an independent dynamical variable [2] (Sects. 5.13-5.24). 

In response to a sudden deformation, the tube is deformed, i.e., the distribution of orientations of the chain segments is shifted from its equilibrium distribution, and the relaxation of a molecule back to its undeformed configuration is constrained by its confinement in the tube. When the imposed deformation is very small, the first relaxation process that occurs is equilibration within the tube, as mentioned briefly in Section 6.3.5. Equilibration involves the redistribution of stress along the chain within the tube. Further relaxation can only occur as a result of the molecule escaping the constraints of the tube, and this requires it to slither along or reptate out of its tube. This is a much slower process and is the reason that there is a plateau in the relaxation modulus for polymers with a narrow molecular weight distribution. In other words, in the plateau region of relaxation times, there is no available mechanism of... 

Let us now briefly review the essential features of the efiect under consideration. Without an applied external fidd we have a random distribution of molecular axes as well as a certain state of chemical equilibrium regarding the mutual interconversion of Ai and Ag. Application of an external field determines some preferential orientation of the molecular dipoles of Ag. The system tends to establish this as rapidly as possible. Usually, this is achieved by means of rotational diffusion. In principle, however, the chemical process provides an alternative means of orienting dipoles without actually rotating them. The equilibrium constant jST(0) of those molecules which have an angle 0 with regard to the applied field will be changed according to... 

Since either type of distribution can be present without association, dielectric loss measurements do not provide a diagnosis of H bonding. Nevertheless, dispersion behavior often can be explained in terms of rearrangements dependent on H bonding. Type I curves may result from the various molecular sizes and shapes of the H bonded polymers. Type II curves may occur when a H bond equilibrium has a relaxation time considerably different from the orientation time giving the main peak. This happens, for example, in supercooled liquid n-propanol (416). Consequently, this tool deserves more detailed attention. 

As alluded to in the introduction to this entry, the LE theory was conceived for small molecule LCs while molecular theories are intended for LCPs. LC molecules retain their equilibrium orientation distribution. LCPs are susceptible to disturbances to their distribution function T (m) its temporal relaxation gives rise to molecular viscoelasticity, while its spatial gradient produces distortional elasticity. A natural question is whether the molecular theories reduce properly to the continuum LE theory in the limiting case of an undisturbed orientation distribution. This situation arises in the weak flow limit where the flow is weak De < . 1) and spatial distortions are small ([V l [Pg.2962]

---