Excluded volume equation

Counterion condensation on a point charge in three dimensions for the excluded volume equation (1.68). It is expected that in contrast to the previous case (Problem 4) the limiting equilibrium solution will exist for the appropriate version of (1.68) and the corresponding limiting singularity in the electric potential should be studied. 

Several methods have been proposed in the literature for this purpose. The first method, which may be called the Flory-Fox-Schaefgen method, is based on the combination of Flory and Fox s viscosity equation (4) and Flory s excluded volume equation (8) [see, Flory and Fox (103)]. The substitution of Eqs. (5) and (6) into Eq. (8) yields... 

The problem of the second virial coeffident of flexible chain polymers has three phases. In the first place, there is the problem of interchain interactions in the second, the problem of the inteochain interactions and finally, the problem of the coupling of the inter- and intra-chain interactions. Different approximations are possible for each phase of the problem, various combinations of such approximations are also possible, and consequently we are confronted by a wide variety of possible theories. For example, the present development of a new excluded-volume equation for the intrachain problem almost doubles the variety of Az-theories, since most existing theories have been developed on the basis of the Flory expression for the excluded volume effect and the new theory can be combined, in prindple, with all of these theories. On the other hand, we have as yet no computational data for A2 and furthermore, the experimental accuracy of A2 measurements is still rather poor, as illustrated in Fig. 2. It is therefore unlikely that one particular combination of several approximations can be properly chosen from among the... 

Flere b corresponds to the repulsive part of the potential, which is equivalent to the excluded volume due to the finite atomic size, and a/v corresponds to the attractive part of the potential. The van der Waals equation... 

The nth virial coefficient = < is independent of the temperature. It is tempting to assume that the pressure of hard spheres in tln-ee dimensions is given by a similar expression, with d replaced by the excluded volume b, but this is clearly an approximation as shown by our previous discussion of the virial series for hard spheres. This is the excluded volume correction used in van der Waals equation, which is discussed next. Other ID models have been solved exactly in [14, 15 and 16]. ... 

The major deficiency of the equation as written is that there is no excluded volume, a deficiency DFl could rectify for the central ion, but not for all ions around the central ion. This deficiency has been addressed within the DFl framework by Outhwaite [9]. 

The simplest extension to the DH equation that does at least allow the qualitative trends at higher concentrations to be examined is to treat the excluded volume rationally. This model, in which the ion of charge z-Cq is given an ionic radius d- is temied the primitive model. If we assume an essentially spherical equation for the u. . 

Equation (8.97) shows that the second virial coefficient is a measure of the excluded volume of the solute according to the model we have considered. From the assumption that solute molecules come into surface contact in defining the excluded volume, it is apparent that this concept is easier to apply to, say, compact protein molecules in which hydrogen bonding and disulfide bridges maintain the tertiary structure (see Sec. 1.4) than to random coils. We shall return to the latter presently, but for now let us consider the application of Eq. (8.97) to a globular protein. This is the objective of the following example. 

Equation (23) predicts a dependence of xR on M2. Experimentally, it was found that the relaxation time for flexible polymer chains in dilute solutions obeys a different scaling law, i.e. t M3/2. The Rouse model does not consider excluded volume effects or polymer-solvent interactions, it assumes a Gaussian behavior for the chain conformation even when distorted by the flow. Its domain of validity is therefore limited to modest deformations under 0-conditions. The weakest point, however, was neglecting hydrodynamic interaction which will now be discussed. 

Both of the above approaches rely in most cases on classical ideas that picture the atoms and molecules in the system interacting via ordinary electrical and steric forces. These interactions between the species are expressed in terms of force fields, i.e., sets of mathematical equations that describe the attractions and repulsions between the atomic charges, the forces needed to stretch or compress the chemical bonds, repulsions between the atoms due to then-excluded volumes, etc. A variety of different force fields have been developed by different workers to represent the forces present in chemical systems, and although these differ in their details, they generally tend to include the same aspects of the molecular interactions. Some are directed more specifically at the forces important for, say, protein structure, while others focus more on features important in liquids. With time more and more sophisticated force fields are continually being introduced to include additional aspects of the interatomic interactions, e.g., polarizations of the atomic charge clouds and more subtle effects associated with quantum chemical effects. Naturally, inclusion of these additional features requires greater computational effort, so that a compromise between sophistication and practicality is required. 

In it the pressure P is found in terms of temperature T and volume per particle v. The interactions enter through two parameters uq, the excluded volume per particle, and a, the binding energy per particle at unity density in a bulk fluid. The equation of state arose as an extension of the ideal gas law,... 

In order to accurately describe such oscillations, which have been the center of attention of modern liquid state theory, two major requirements need be fulfilled. The first has already been discussed above, i.e., the need to accurately resolve the nonlocal interactions, in particular the repulsive interactions. The second is the need to accurately resolve the mechanisms of the equation of state of the bulk fluid. Thus we need a mechanistically accurate bulk equation of state in order to create a free energy functional which can accurately resolve nonuniform fluid phenomena related to the nonlocality of interactions. So far we have only discussed the original van der Waals form of equation of state and its slight modification by choosing a high-density estimate for the excluded volume, vq = for a fluid with effective hard sphere diameter a, instead of the low-density estimate vq = suggested by van der Waals. These two estimates really suggest... 

The GvdW theory has been applied also to mixtures of Leonard-Jones fluids [15,16], The extension to mixtures is straightforward with respect to the binding energy and interaction with an external field but not quite so straightforward with respect to the excluded volume effect. The GvdW(S) theory produces for a mixture of c components an equation of state of the form... 

Such weaknesses of the present implementation include the lack of an explicit inclusion of intermolecular forces other than excluded volume, resulting in a qualitatively inaccurate description of the equation of state. Another weakness is that the model shows lattice artefacts when dealing with problems of polymer crystallization or liquid-cristalline order only rather flexible poly-... 

Some years ago, on the basis of the excluded-volume interaction of chains, Hess [49] presented a generalized Rouse model in order to treat consistently the dynamics of entangled polymeric liquids. The theory treats a generalized Langevin equation where the entanglement friction function appears as a kernel... 

A second problem with the random walk model concerns the interaction between segments far apart along the contour of the chain but which are close together in space. This is the so-called "excluded volume" effect. The inclusion of this effect gives rise to an expansion of the chain, and in three-dimensions, 2 a, r3/5 (9), rather than the r dependence given in equation (I). 

These models retain the form of the nonbonded interaction used in the chemically realistic modeling, i.e., they use either an interaction of the Lennard-Jones or of the exponential-6 type. The repulsive parts of these potentials generate the necessary local excluded volume, whereas the attractive long-range parts can be used to model varying solvent quality for dilute or semi-dilute solutions and to generate a reasonable equation-of-state behavior for polymeric melts. 

As a consequence of the excluded volume associated with the core, interior and surface branch cells, steric congestion is expected to occur due to tethered connectivity to the core. Furthermore, the number of dendrimer surface groups, Z, amplifies with each subsequent generation (G). This occurs according to geometric branching laws, which are related to core multiplicity (iVc) and branch cell multiplicity (iVb). These values are defined by the following equation ... 

There are numerous equations in the literature describing the concentration dependence of the viscosity of dispersions. Some are from curve fitting whilst others are based on a model of the flow. A common theme is to start with a dilute dispersion, for which we may define the viscosity from the hydrodynamic analysis, and then to consider what occurs when more particles are added to replace some of the continuous phase. The best analysis of this situation is due to Dougherty and Krieger18 and the analysis presented here, due to Ball and Richmond,19 is particularly transparent and emphasises the problem of excluded volume. The starting point is the differentiation of Equation (3.42) to give the initial rate of change of viscosity with concentration ... 

Where (pm is the maximum concentration at which flow is possible -above this solid-like behaviour will occur. q>/(pm is the volume effectively occupied by particles in unit volume of the suspension and therefore is not just the geometric volume but is the excluded volume. This is an important point that will have increasing relevance later. Now integration of Equation (3.53) with the boundary condition that as... 

The effective hard sphere diameter may be used to estimate the excluded volume of the particles, and hence the low shear limiting viscosity by modifying Equation (3.56). The liquid/solid transition of these charged particles will occur at... 

Having dealt with the excluded volume effect arising from the first term, AG, on the rhs of Eq. (9.4.2), we now examine a few other solvent effects associated with the remaining terms on the rhs of this equation. These are referred... 

This is certainly true of the simple properties, such as the molecular weight of a mixture. For the van der Waals equation of state, the parameter b stands for the excluded volume due to the molecule, which suggests that the linear additive relation, or the arithmetic mean, may be appropriate for an ideal mixture ... 

It can be shown that the < 2> of an interrupted helical polypeptide is expressed by Eq. (C-3) for mean-square dipole moment of a random-coil unit. Precisely, this replacement is permissible if we neglect excluded-volume effects. Nagai (107) has shown theoretically that these effects on < 2> are virtually absent in randomly coiled macromolecules, even when they are appreciable on the molecular dimensions. It is our belief that Nagai s conclusion may apply to interrupted helical polypeptides as well. 

To a first approximation, one can assume that the excluded volume term (A/xc) is determined only by the physical volume occupied by the biopolymer molecules/particles (see equations (5.32), (5.34) and (5.35)). The electrostatic term, A if1 = A V i T2, is determined by the ideal Dorman contribution, A corrected by factors to take into account the electrostatic interactions amongst the ions (Ti) and the chain-like character of the polyelectrolyte (T2) (Nagasawa and Takahashi, 1972). 

Under the conditions of screening of electrostatic interactions between polyions, as occurs at high ionic strength (say, / > 0.1 mol dm- ), or in solutions containing neutral (non-ionic) polymers, the excluded volume term is the leading term in the theoretical equation for the second virial coefficient. In this latter type of situation, the sizes and conformation/ architecture of the biopolymer molecules/particles become of substantial importance. 

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