Athermal systems

Figure 2. Theoretical displacement isotherms for various values, calculated from the Roe theory Athermal systems (Ax = 0, Xpd = 0). The dotted line is the extrapolation of the isotherm into the physically inaccessible region...

In general, it can be very difficult to determine the nature of the boundary terms. A specific result in an exactly solvable case is discussed in Section IV.A.2. Equation (55) is the Gallavotti-Cohen FT derived in the context of deterministic Anosov systems [28]. In that case, Sp stands for the so-called phase space compression factor. It has been experimentally tested by Ciliberto and co-workers in Rayleigh-Bemard convection [52] and turbulent flows [53]. Similar relations have also been tested in athermal systems, for example, in fluidized granular media [54] or the case of two-level systems in fluorescent diamond defects excited by light [55]. 

A polymer solution consists of a solvent 1 (usually monomeric) and a solute 2 (polymeric). Since in this section we consider athermal systems, there is no mixing energy and the fields only contain the adsorption energies A. U and... 

The first term in Eq. (24) represents the ideal term. If we consider an athermal system, it is sufficient to take into account the entropy mixing alone so that = 0 And we have an important implication with the second virial coefficient A2l ... 

The solvent activities in athermal polymer solutions are systematically nnderestimated, often by 10% (in the case of Entropic-FV) or more (for UNIFAC-FV). For athermal systems, the residual term is zero. Such an error cannot be entirely attribnted to the small interaction effects present in such systems. 

The UNIFAC-FV expression (Equation 16.49), the first FV equation proposed, which is derived from the theory of Flory, is not as successful for athermal systems compared to more receut simpler equatious. This may be due to the values of the parameters h and c employed in this model. Fitting these parameters may improve the performance of the UNIFAC-FV term. The results with this model seem particularly seusitive to the deusity values employed (Figure 16.3). 

First we introduce the isotropic interaction which results from the mixing of rods and solvents. The previous theory, without the mixing contribution to entropy, is only applicable to an athermal system. If there is a mixing entropy contribution, the free energy in Equation 2.39 is implemented by a term x4>ns, where x is the Flory-Huggins interaction parameter. Flory called the mixing term the isotropic soft interaction to distinguish it from the steric interaction of rods. 

For nearly athermal systems, the proportionality factors, S and S, are taken as equal 1. Thus, for the systems without strong interactions the binary parameters were well approximated by the geometric and algebraic averages. For example, for PS/PVME blends the assumption = 1... 

Systems that are miscible with one another without any change in temperature are termed athermic systems [14.18], [14.19] they include the following solvent pairs ... 

Flory, P. J., and Ronca, G., Theory of systems of rodlike particles I. Athermal systems. Mol. Cryst. Liq. Cryst., 54, 289-310 (1979a). 

For nearly athermal systems, the proportionality factors, S. and Sy, are taken as equal to 1. Thus, for the systems without strong interactions, the binary parameters are weU approximated by the geometric and algebraic averages. For example, for PS/PVME blends, the assumption 5 = 5v = 1 resulted in 0.1 % deviation for the experimental values of the cross-parameters (Xie et al. 1992 Xie and Simha, 1997, private communication ). In contrast, it is to be expected that for systems with strong intermolecular interactions such mixture rules may fail and experimental values for the cross-factors may have to be found. However, least squares lit of Eqs. 2.42 and 2.43 to experimental values of CO2 miscibilities in PS (in a wide range of P and T) yielded values for and Sy close to 1 (Xie et al. 1997). 

The Doi and Edwards [67] reptation model provides simple mixing rules for miscible systems without the thermodynamic interactions. For athermal systems Tsenoglou [16, 191] proposed the double reptation model ... 

Within the HTA scheme, the liquid structure of the mixture is approximated by the structure of the athermal system, that is, gay(f)== gly r). Thus to first order, Eq. (5.3) can be approximated as... 

One extra approximation must be invoked for the block copolymer case relative to the blend. The reference athermal system is not a mixture of A and B chains, but a connected block copolymer of A and B segments. Analytic solution of the thread PRISM equations has not been achieved for this case. Thus, as a technical approximation the reference hard-core direct correlations functions have been approximated by their blend values given in Section IV,D. Such an approximation should be excellent for large N and the diblock architecture, but will deteriorate in accuracy for multiblock architectures. 

Although the potential only interacts directly with the A s, there are, as we shall see, more effects which contribute to a. a% represents a collective response coefficient . The upper index 0 is meant to indicate, that we are dealing with an athermal system where x = 0. 

The same problem can be viewed in a different way. According to the results of computer simulations [45] the nematic ordering in an athermal system of elongated rigid particles is formed only if the axial ratio is more than three. At the same time, the effective value of LID for typical mesogenic mole-... 

TRANSITIONS TO THE LIQUID-CRYSTALLINE STATE IN ATHERMAL SYSTEMS INVOLVING RIGID-CHAIN POLYMERS... 

It should be emphasized that this expression of the entropy of mixing is applicable only to athermic systems or to mixtures exhibiting only weak interactions between molecules—that is, solutions with low enthalpy of mixing. Deviations from ideality could arise in particular in the following situations, which will be... 

N. Xu, C. S. O Hern, and L. Kondic. Stabilization of nonlinear velocity profiles in athermal systems undergoing planar shear flow. Phys. Rev. E, 72 041504, 2005. 

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