Effect of excluded volume

Chan H S and Dill K A 1989 Intrachain loops in polymers effects of excluded volume J. Chem. Phys. 90 493-509... 

A second approach [7] allows for the effects of excluded volume correlations and self-avoidance by use of scaling arguments. In this picture, the layer is viewed... 

Of course, the effect of excluded volume is opposite and greatly exceeds that shown in Fig. 1.10, which is produced by uncorrelated collective interaction. Unfortunately, neither of them results in sign-alternating behaviour of angular or translational momentum correlation functions. This does not have a simple explanation either in gas-like or solid-like models of liquids. As is clearly seen from MD calculations, even in... 

The above observation is significant, Theoretical considerations 102,103,104), as well as some experimental studies 105,106), revealed an effect of excluded volume on the rates and equilibria of polymeric reagents. For example, the equilibrium constant of dissociation of high molecular weight aggregates (MW > 1(f) such as... 

The present work demonstrates that (1) conclusions reached earlier for model chains also apply to realistic chains, (2) effects seen with finite chains may survive in extremely long chains, (3) the limit at large n for (o, 2 - 1) / (ar2 - 1) provides little information about the effect of excluded volume on the dipole moment of infinitely long chains, and (4) an alternative relationship between a 2 and ar2 may provide useful information on the relationship of and 0 for long chains. 

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. 

As a contribution to the study of these problems, stochastic models are here developed for two cases a freely-jointed chain in any number of dimensions, and a one-dimensional chain with nearest-neighbor correlations. Our work has been directly inspired by two different sources the Monte Carlo studies by Verdier23,24 of the dynamics of chains confined to simple cubic lattices, and the analytical treatment by Glauber25 of the dynamics of linear Ising models. No attempt is made in this work to introduce the effects of excluded volume or hydrodynamic interactions. 

Fig. 28 shows a typical example of experimental data plotted according to Eq. (IV-25). From the slope a value of 1/30 is calculated for the quantity y>j3N (1 — ip). Since in this example N < 50, this would mean that the free volume 1 y> amounts to 0.16 which seems unreasonably high. A direct comparison of Eq. (IV-23) with DiMarzio s Eq. (IV-22a) shows that the free volume would amount to 1 — y> = 0.25. It seems, likely, therefore, that Jackson c.s. overestimate the effect of excluded volume obstruction, and that DiMarzio s and Khasanovich s treatments are more nearly correct. 

The effect of excluded volume on g for linear chains has been calculated, first more qualitatively by Weill and des Cloiseaux193 on the basis of scaling arguments, then by Akcasu and Benmouna202 quantitatively on the basis of the blob-model. The result is as follows... 

The catalyst density increase, a , associated with the effect of excluded volume, is... 

At the comparison of concentration dependencies of the characteristic quantities (6.61) with experimental determinations, one has to remember that effect of excluded volume was not taken into account in equations (6.61), which allow us to say only about qualitative correspondence. The behaviour of the initial viscosity is the most widely studied (Poh and Ong 1984, Takahashi et al. 1985). The concentration dependence of the viscosity coefficient in the melt-like region can be represented by a power law (Phillies 1995). The index can be found to be approximately 25 + 1, in accordance with (6.61). There are some differences in the behaviour of polymer solutions, which are connected with different behaviour of macromolecular coils at dilution. 

Comparison with experimental data demonstrates that the bead-spring model allows one to describe correctly linear viscoelastic behaviour of dilute polymer solutions in wide range of frequencies (see Section 6.2.2), if the effects of excluded volume, hydrodynamic interaction, and internal viscosity are taken into account. The validity of the theory for non-linear region is restricted by the terms of the second power with respect to velocity gradient for non-steady-state flow and by the terms of the third order for steady-state flow due to approximations taken in Chapter 2, when relaxation modes of macromolecule were being determined. 

Note that increases with decreasing electrolyte concentration, whereas x decreases, so that the expansion factor Increases due to the combined effects of excluded volume and stiffening. 

Flory, P. j. Treatment of the effect of excluded volume and deduction of unperturbed dimensions of polymer chains. Configurational parameter for cellulose derivatives. Makromol. Chem. 98, 128 (1966). 

Kim, Y.H., Stites, W.E. Effects of excluded volume upon protein stability in covalently cross-linked proteins with variable linker lengths. Biochemistry 2008, 47, 8804-14. 

We may now consider the effect of excluded volume on dimensions of polymer coils in solution. It may be recalled that the mathematical model for evaluating is that of a series of connected vectors (representing the... 

The so-called 0 (theta) conditions, in marginally weak solvents, are usually preferred in solution viscosity measurements. Polymer chains are believed to manifest their "unperturbed dimensions" under 0 conditions. This is a result of the nearly perfect balancing of the effects of "excluded volume" (a consequence of the self-avoidance of the random walk path of a polymer chain in a random coil configuration) by unfavorable interactions with the solvent molecules. 

The effect of excluded volume as predicted by the Tschoegl equation (39) is to shift the dynamic properties to more Rouse-like behavior. If one keeps h at infinity and takes = 1/3, then one obtains curves for [G ju and [G"]k which lie between the corresponding quantities for h = 1 and 25 in Fig. 2.2. Thus the effect of increasing e is qualitatively equivalent to decreasing h in the original Zimm theory (29). This effect of g diminishes as h decreases, and disappears at h = 0. 

For example, one can define a chain expansion parameter at which characterizes the effect of excluded volume on the Stokes radius Rg by ... 

Statistical fractals are generated by disordered (random) processes. An element of disorder is typical of most real physical phenomena and objects. The fact that disorder, i.e., the absence of any spatial correlation, is a sufficient condition for the formation of fractals was first noted by Mandelbrot [1]. A typical example of this type of fractal is the random-walk path. However, real physical systems are often inadequately described by purely statistical models. Among other reasons, this is due to the effect of excluded volume. The essence of this effect lies in the geometric restriction that forbids two different elements of a system to occupy the same volume in space. This restriction is to be taken into account in the corresponding modelling [10, 11]. The best-known examples of this type of models are self-avoiding random walk, lattice animals and statistical percolation. 

The fractal dimension of purely statistical models, i.e., models without the effect of excluded volume, can be determined accurately [see Equation (11.9a)]. For linear polymers, this model corresponds to phantom random-walk. In the case of branched statistical fractals, the corresponding model is a statistical branched cluster, whose branching obeys the random-walk statistics. Since the root-mean-square distance between the random-walk ends is proportional to the number of walk steps N, then D = 2 irrespective of the space dimension. These types of structures have been studied [61, 75-77]. The value D = 4 irrespective of d was obtained for a branched fractal. Unlike ideal statistical models, models with excluded volume, i.e., those involving correlations, cannot be accurately solved in the general case. The Df values for these systems are usually found either using numerical methods such as the Monte Carlo method or taking into account the spatial position of a renormalisation group. 

Equation (11.18) has two peculiar features firstly, D depends appreciably on d and, secondly, there exists a critical dimension of Euclidean space d = % for which D = 4 in accordance with the ideal statistical model, i.e., the model withont correlations. At d > 8, the correlations cansed by the effect of excluded volume are no longer significant and Df does not change. The value d = % was found in studies of branched polymers and lattice animals [79]. Calcnlations using formula (11.18) are in good agreement with the known results for lattice animals. ... 

Let us consider conditions under which d = 2. For a real polymer, with allowance for the effect of excluded volume, one can write that [57] ... 

The osmotic virial coefficienL B (see eqn S.41), arises largely from the effect of excluded volume. If we imagine a solution of a macromolecule being built by the successive addition of macromolecules to the solvent, each one being excluded by the ones that preceded it, then the value of B turns out to be (PI9.18)... 

This first part of this article deals only with treatment of bonded interactions of polymer chains, appropriate only for modeling chains under -point conditions. Problems connected with effects of excluded volume are presented at the end of this chapter. The excluded volume effect for chains in good solvents are also presented in Chaps. IIB [10] and HID [11] of this handbook and in books by Freed [12], de Gennes [13], des Cloizeaux and Jannink [14], and... 

In real polymer network the effects of excluded volume and chain entanglements should be taken into account. In 1977 Hory [26] formulated the constrained junction model of real networks. According to this theory fluctuations of junctiOTs are affected by chains interpenetration, and as the result the elastic free energy is a sum of the elastic free energy of the phantom network AAph (given by Eq. (5.78)) and the free energy of constraints AA ... 

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