Intermolecular interactions excluded volume

Orientation in crosslinked elastomers primarily reflects the configurational entropy and intramolecular conformational energy of the chains. However, as first shown by deuterium NMR experiments on silicone rubber (Deloche and Samulski, 1981 Sotta et al., 1987), unattached probe molecules and chains become oriented by virtue of their presence in a deformed network. This nematic coupling effect is brought about intermolecular interactions (excluded volume interactions and anisotropic forces) which can cause nematic coupling (Zemel and Roland, 1992a Tassin et al., 1990). The orientation is only locally effective, so it makes a negligible conttibution to the stress (Doi and Watanabe, 1991), and the chains retain their isotropic dimensions (Sotta et al., 1987). 

We begin by formulating the free energy of liquid-crystalline polymer solutions using the wormlike hard spherocylinder model, a cylinder with hemispheres at both ends. This model allows the intermolecular excluded volume to be expressed more simply than a hard cylinder. It is characterized by the length of the cylinder part Lc( 3 L - d), the Kuhn segment number N, and the hard-core diameter d. We assume that the interaction potential between them is given by... 

The very fact that the vapor phase of many substances can condense to form a liquid is a consequence of the existence of attractive van der Waals forces between atoms or molecules. An attractive intermolecular force is not needed for a gas to condense into a solid solidification can occur purely as a result of excluded-volume interactions among the molecules at sufficiently large densities. The pressure in a fluid, the cohesion between materials, and the existence of surface energy or surface tension all result, partially or wholly, from van der Waals forces. 

Changes in excluded volume and in intramolecular hydrodynamic interaction appear at the present time to be the only acceptable explanations for the onset of shear rate dependence in systems without appreciable intermolecular interactions. It seems likely that both internal viscosity and finite extensibility would assume importance only at much higher shear rates. 

Intcrmolecular Contributions. Increasing concentration reduces the effects of excluded volume and intramolecular, hydrodynamic on viscoelastic properties (Section 5). Internal viscosity and finite extensibilty have already been eliminated as primary causes of shear rate dependence in the viscosity. Thus, none of the intramolecular mechanisms, even abetted by an increased effective viscosity in the molecular environment, can account for the increase in shear rate dependence with concentration, e.g., the dependence of power-law exponent on coil overlap c[r/] (Fig. 8.9). Changes in intermolecular interaction with increased shear rate seems to be the only reasonable source of enhanced shear rate dependence, at least with respect to the early deviations from Newtonian behavior and through a substantial portion of the power law regime. 

The justification for using the combining rule for the a-parameter is that this parameter is related to the attractive forces, and from intermolecular potential theory the attractive parameter in the intermolecular potential for the interaction between an unlike pair of molecules is given by a relationship similar to eq. (42). Similarly, the excluded volume or repulsive parameter b for an unlike pair would be given by eq. (43) if molecules were hard spheres. Most of the molecules are non-spherical, and do not have only hard-body interactions. Also there is not a one-to-one relationship between the attractive part of the intermolecular potential and a parameter in an equation of state. Consequently, these combining rules do not have a rigorous basis, and others have been proposed. 

For ease of calculation, we make a number of simplifying assumptions. These are relaxed in advanced treatments of the subject. First, rather than requiring tetrahedral bonds at each vertex of the chain, we allow all bond angles and assume that these are randomly distributed. Second, we ignore any excluded volumes or interactions between the segments of the chain. In this sense, our calculation is similar to the Bernoulli model of the ideal gas, which neglects intermolecular interactions. Our approximation is called the freely jointed chain model. 

As temperature increases, the layer expands and the orientation of endbeads smears. This is shown in Figure 1.46. The observed expansion of layer thickness is attributed mainly to the temperature dependence of the intermolecular interaction and excluded volume effect not due to bond stretching. Temperature dependence, which is pertinent to the annealing process of thin polymeric lubricant films, has been carefully examined by Hsia et al. [171]. 

In dilute solution, all macromolecular chains undergo interactions with each other resulting in the so-called intermolecular excluded volume effect, corresponding to the intermolecular potential. This effect is also observed if one does not assume particular cohesive forces to occur between the macromolecular chains. Under these conditions, the second virial coefficient is calculated from the equation1,2) ... 

In the context of van der Waals theory, a and b are positive parameters characterizing, respectively, the magnitude of the attractive and repulsive (excluded volume) intermolecular interactions. Use this partition function to derive an expression for the excess chemical potential of a distinguished molecule (the solute) in its pure fluid. Note that specific terms in this expression can be related to contributions from either the attractive or excluded-volume interactions. Use the Tpp data given in Table 3.3 for liquid n-heptane along its saturation curve to evaluate the influence of these separate contributions on test-particle insertions of a single n-heptane molecule in liquid n-heptane as a function of density. In light of your results, comment on the statement made in the discussion above that the use of the potential distribution theorem to evaluate pff depends on primarily local interactions between the solute and the solvent. 

Improvements upon the theories require more detailed treatment of intramolecular as well as intermolecular interactions. As we mentioned in the previous section, use of Mayer /-functions has been made by Yamakawa and others to take intramolecular excluded volume effects into consid ation. However, in their calculation, parts of macromolecules between two consecutive contact points of two molecules are replaced always by Gaussian-free chains. While this approximation may be correct for a small number of contact points for very long molecules, it certainly invalidates the /-function expansion itself for higher orders. On the other hand, our results in the previous section indicate clearly that collective interactions of two macromolecules play important roles in explaining the molecular weight dependence of Ag. 

There is only a qualitative justification for eqns, (3.3.6 and 3.3.7). The a parameter is related to attractive forces, and, from intermolecular potential theory, the parameter in the attractive part of the intermolecular potential for a mixed interaction is given by a relation like eqn. (3.3.6). Similarly, the excluded volume parameter b would be given by eqn. (3.3.7) if the molecules were hard spheres. However, there is no direct relation between the attractive part of the intermolecular potential and the a parameter in a cubic EOS, and real molecules are not hard spheres. 

If the solvent is good, the only important intermolecular interactions are then steric short range excluded volume repulsions. 

The free energy is separable into two contributions (1) the intra-molecular interactions associated with the helix-coil transition (2) the intermolecular excluded volume interactions which are responsible for any nematic order. The difference in free energy per chain between a helical and randomly coiled molecule arising from intramolecular interactions is -N n s. The intermolecular excluded volume interactions in the mean field Meier-Saupe like approximation [Eqs. (III.2) - (III.6) with V = N] gives a contribution to the free energy difference, (AF) which is sketched in... 

On going from the isotropic to the anisotropic LC state, the orientation-dependent attractive interactions come into play [125,126] while the steric interactions (the excluded-volume effect) between the mesogenic rods are relaxed [127]. In the LC state, all molecules are required to take an asymmetric shape. Accordingly, chain segments adopt a unique conformer distribution called a nematic conformation [26,93-95,102,103,105]. The Y-Vsp relation determined in the aforementioned studies may be used to examine the mean-field potentials effective in nematic as well as isotropic hquids. After Frank [128] and HUdebrand and Scott [2,129], the intermolecular interaction potentials such as... 

Models of polymer dynamics are also partitioned by their assumptions as to the dominant forces in solution, these assumptions being totally independent of the assumed concentration dependence. In some models, excluded-volume forces (topological constrmnts) dominate, while hydrodynamic interactions dress the monomer diffusion coefficient. In other models, hydrodjmamic interactions dominate, while chcun-crossing constriunts cure secondary. Experimentally, Dg c) is directly accessible, but the intermolecular forces Ccm at best only be inferred from numerical coefficients D, a, and so forth. 

The second correction accounts for weak interactions between the molecules. It is done with the constant, a, modifying the measured pressure, p. If one wants to describe a state where molecules are condensed, they must be held together by some interactions which cannot be neglected. Interactions between pairs of molecules are always proportional to the square of concentration, so it is best to give the correction term the form a(nA ) and add it to the pressure. The reduction of pressure when molecules attract each other is perhaps obvious if one considers that the molecules that collide with the surface are being attracted away from the surface by the other molecules of the gas. The resulting equation correcting for both the excluded volume and the intermolecular interaction is the van der Waals equation, the boxed equation of Fig. 2.99. The constants, a and b, are different for every gas. 

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