Particle excluded volume

A relation between the mean end-to-end distance of the entire chain (R2) and the mean end-to-end distance of a subchain b can be found from simple speculation. This relation includes temperature T, mean distance b between the nearest along chain particles, excluded volume parameter v and the number of particles on the chain N. When dimensional considerations are taken into account, the relation can be written in the form... 

The most widely used statistical model for fluids is that due to van der Waals, which includes a mean attractive potential with a hard particle excluded volume. For such a model the Helmholtz free energy can be written as [41,42] ... 

Cast films of emulsion blends comprised of non-fluorinated acrylics and fluorinated acryhcs exhibited surface enrichment of the fluorinated acryhc upon exposure of the film above the film formation temperature of the fluorinated acryhc [546]. Similar results were observed for smaU particle size fluorinated acryhc copolymer emulsion blends with large particle size S-nBA copolymers [547]. The combination of low free energy and smaller size of the fluorinated acryhc particles (excluded volume/percolation) yielded much higher concentration of the fluorinated polymer at the surface than in the bulk film. 

It is convenient to begin by backtracking to a discussion of AS for an athermal mixture. We shall consider a dilute solution containing N2 solute molecules, each of which has an excluded volume u. The excluded volume of a particle is that volume for which the center of mass of a second particle is excluded from entering. Although we assume no specific geometry for the molecules at this time, Fig. 8.10 shows how the excluded volume is defined for two spheres of radius a. The two spheres are in surface contact when their centers are separated by a distance 2a. The excluded volume for the pair has the volume (4/3)7r(2a), or eight times the volume of one sphere. This volume is indicated by the broken line in Fig. 8.10. Since this volume is associated with the interaction of two spheres, the excluded volume per sphere is... 

Regardless of the particle geometry, the excluded volume exceeds the actual volume of the molecules by a factor which depends on the shape of the particles. 

Rigid particles other than unsolvated spheres. It is easy to conclude qualitatively that either solvation or ellipticity (or both) produces a friction factor which is larger than that obtained for a nonsolvated sphere of the same mass. This conclusion is illustrated in Fig. 9.10, which shows the swelling of a sphere due to solvation and also the spherical excluded volume that an ellipsoidal particle requires to rotate through all possible orientations. 

Another interesting version of the MM model considers a variable excluded-volume interaction between same species particles [92]. In the absence of interactions the system is mapped on the standard MM model which has a first-order IPT between A- and B-saturated phases. On increasing the strength of the interaction the first-order transition line, observed for weak interactions, terminates at a tricritical point where two second-order transitions meet. These transitions, which separate the A-saturated, reactive, and B-saturated phases, belong to the same universality class as directed percolation, as follows from the value of critical exponents calculated by means of time-dependent Monte Carlo simulations and series expansions [92]. 

Suppose we have a physical system with small rigid particles immersed in an atomic solvent. We assume that the densities of the solvent and the colloid material are roughly equal. Then the particles will not settle to the bottom of their container due to gravity. As theorists, we have to model the interactions present in the system. The obvious interaction is the excluded-volume effect caused by the finite volume of the particles. Experimental realizations are suspensions of sterically stabilized PMMA particles, (Fig. 4). Formally, the interaction potential can be written as... 

The analogy with the virial expansion of PF for a real gas in powers of 1/F, where the excluded volume occupies an equivalent role, is obvious. If the gas molecules can be regarded as point particles which exert no forces on one another, u = 0, the second and higher virial coefficients (42, Azy etc.) vanish, and the gas behaves ideally. Similarly in the dilute polymer solutions when w = 0, i.e., at 1 = , Eqs. (70), (71), and (72) reduce to vanT Hoff s law... 

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,... 

The second generalization is the reinterpretation of the excluded volume per particle V(). Realizing that only binary collisions are likely in a low-density gas, van der Waals suggested the value Ina /I for hard spheres of diameter a and for particles which were modeled as hard spheres with attractive tails. Thus, for the Lennard-Jones fluid where the pair potential actually is... 

This form is particularly simple, since /(r) is independent of the type of species i. It behaves as if each particle excluded a volume equal to the smallest cube into which its effective hard sphere shape could be placed. The alternative form... 

Monte Carlo computer simulations were also carried out on filled networks [50,61-63] in an attempt to obtain a better molecular interpretation of how such dispersed fillers reinforce elastomeric materials. The approach taken enabled estimation of the effect of the excluded volume of the filler particles on the network chains and on the elastic properties of the networks. In the first step, distribution functions for the end-to-end vectors of the chains were obtained by applying Monte Carlo methods to rotational isomeric state representations of the chains [64], Conformations of chains that overlapped with any filler particle during the simulation were rejected. The resulting perturbed distributions were then used in the three-chain elasticity model [16] to obtain the desired stress-strain isotherms in elongation. 

The influence of K on g is caused by the so-called wall effect and an increase of void volume that is accessible for small particles with K. Both of these effects correspond to the appearance of the excluded volume in the places of contact of rigid particles between themselves (Figure 9.18) or a rigid wall (Figure 9.20). 

The relativistic EOS of nuclear matter for supernova explosions was investigated recently [11], To include bound states such as a-particlcs, medium modifications of the few-body states have to be taken into account. Simple concepts used there such as the excluded volume should be replaced by more rigorous treatments based on a systematic many-particle approach. We will report on results including two-particle correlations into the nuclear matter EOS. New results are presented calculating the effects of three and four-particle correlations. 

In [11] the alpha-particles were included into the EOS, and detailed comparisons of the outcome with respect to the alpha-particle contribution has been made. We will elaborate this item further, first by using a systematic quantum statistical treatment instead of the simplifying concept of excluded volume, second by including also other (two- and three-particle) correlations. 

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 secondary electroviscous effect is the enhancement of the viscosity due to particle-particle interactions, and this of course will control the excluded volume of the particles. The most complete analysis is that due to Russel30 and we may take this analysis for pair interactions as the starting point. Russel s result gives the viscosity 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... 

Semi-dilute solution p2 - As the concentration rises the rods will begin to interact and their diffusion will become restricted. However, we have not allowed for the excluded volume of the rod and have treated it as a line with no thickness. The excluded volume is of the order of bL2 and until the concentration of rods is such that the particles overlap into this excluded volume the spatial distribution of rods is relatively unaffected ... 

Tables of this sort are valid for Gaussian coils only. In thermodynamically good solvents the Gaussian behaviour of chain molecules is perturbed by what is called the excluded volume effect 30. The P/(0) function depends on the distribution of mass within the particle and this, in turn, is changed if the volume effect is operative.

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