Excluded volume forces, models

Crystalline or orientational orderings are mostly controlled by repulsive forces, such as excluded-volume forces. The crystalline transitions in ideal hard-sphere fluids and the nematic liquid crystalline transitions in hard-rod suspensions are convenient simple models of corresponding transitions in fluids composed of uncharged spherical or elongated molecules or particles. The transition from the isotropic to the nematic state can be described theoretically using the Onsager, Maier-Saupe, or Rory theories. 

In the case of higher charge, Z 30 on Fig. 8a, both models result that the like-charged particles being confined to a film that has a thickness around H/D = 7.5 tend to be organized into four particle layers. For the middle-film layers formed with and without excluded volume forces, only some quantitative differences in the particle local density distribution are observed. The main difference introduced by excluded volume forces is found in the surface layers. Taking into account the discrete nature of the solvent results that the surface layers themselves show a structuring with respect to the film surfaces. 

This results in each surface layer consisting of a well-defined sublayers in an immediate vicinity of the film surface the shape of the density profiles of the surface sublayers has a 5-like form indicating that surface sublayers are the quasi-two-dimensional monolayers. The surface layers formed within the DLVO-like model being thinner than the middle-film layers still are far from to be monolayers. As a result, the segregation of the middle-film layers from the surface layers is not so evident in this case. As expected, the difference between models with and without excluded volume forces increases when the macroion charge becomes smaller (Fig. 9b). 

A snapshot of the representative configurations of the macroions in a three-layer film obtained from the simulations with and without excluded volume forces is presented on Fig. 10. The simulations without excluded volume forces (Fig. 10b) serve as a methodological example which illustrates that an adequate modelling of complex colloidal suspension should necessarily take into account the discrete nature of a primary suspending fluid. In the case of a three-layer film, the excluded volume forces play an important role in the organization of both the surface and middle-film layers. In general, the excluded volume forces become more important with a decrease of the interparticle distances this is the case for particle layers, both near the film surfaces and in the middle of the film. 

Three models of excluded volume forces are considered In the first model, called the sphere-sphere model, the proteins and polymers are modeled as rigid spheres of radii, R4 and Rj respectively. In this case, Aj4 is given by... 

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. 

When we expressed Eq. (2.18) for the distribution function p R), we did this based on the assumption that the random walk carried out by a chain of freely jointed segments should be equivalent to the motion of a Brownian particle. We pointed out that the equivalence is lost in the presence of excluded volume forces, however, this is not the only possible deficiency in the treatment. Checking the properties for large values of R we find that the Gaussian function never vanishes and actually extends to infinity. For the model chain, on the other hand, an upper limit exists, and it is reached for... 

In the presence of excluded volume forces the q dependence may be altered in a similar way as for linear chains.There is no exact solution to the dynamics of chains in the presence of excluded volume forces (see Doi and Edwards book), " but the dynamics can be modelled by an effective equation of motion. Excluded volume forces change the exponent in equation (56) if applied to the cluster, likewise in a quasi-particle approximation ... 

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

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-... 

The strong point of molecular dynamic simulations is that, for the particular model, the results are (nearly) exact. In particular, the simulations take all necessary excluded-volume correlations into account. However, still it is not advisable to have blind confidence in the predictions of MD. The simulations typically treat the system classically, many parameters that together define the force field are subject to fine-tuning, and one always should be cautious about the statistical certainty. In passing, we will touch upon some more limitations when we discuss more details of MD simulation of lipid systems. We will not go into all the details here, because the use of MD simulation to study the lipid bilayer has recently been reviewed by other authors already [31,32]. Our idea is to present sufficient information to allow a critical evaluation of the method, and to set the stage for comparison with alternative approaches. 

In Section 3.4a we examine a model for the second virial coefficient that is based on the concept of the excluded volume of the solute particles. A solute-solute interaction arising from the spatial extension of particles is the premise of this model. Therefore the potential exists for learning something about this extension (i.e., particle dimension) for systems for which the model is applicable. In Section 3.4b we consider a model that considers the second virial coefficient in terms of solute-solvent interaction. This approach offers a quantitative measure of such interactions through B. In both instances we only outline the pertinent statistical thermodynamics a somewhat fuller development of these ideas is given in Flory (1953). Finally, we should note that some of the ideas of this section are going to reappear in Chapter 13 in our discussions of polymer-induced forces in colloidal dispersions and of coagulation or steric stabilization (Sections 13.6 and 13.7). 

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