Excluded volume forces

The serious problems encountered if we aim at a microscopically realistic calculation of the excluded volume forces have clearly been noted in the literature. See, for instance, [DLR96],... 

In essence, the ability to maximize entropy by sorting different-sized objects creates a kind of attractive force, called a depletion, or excluded-volume, force. These entropic forces operate for objects in the size range of approximately 10-8 to approximately 10 6 m. For entropy-induced ordering to occur, the particles must be constantly jostling each other and must be constantly agitated by solvent molecules, thus making gravity unimportant. 

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. 

Figure 6. MC data for the radial distribution function between macroions in a bulk solution at high surfactant concentration and different macroion charge numbers as depicted in the figure. The thin solid line shows MC data obtained with excluded volume forces only, i.e. Z = 0. ...

Figure 8. MC data for the normalized local density distribution of the macroions in a film formed at macroion bulk volume fraction rj = 0.05. The macroion charge is fixed at Z = 30. (a) The film has thickness H/D = 7.5 and contains four particle layers two surface monolayers and two middle-film layers. The dashed line shows MC data obtained from the simulation without excluded volume forces, (b) The film has thickness H/D = 5 and contains three particle layers two surface monolayers and one middle-film layer, (c) The film has thickness H/D = 3.5 and contains two surface monolayers only.

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). 

Figure 10. Snapshots of the representative macroion configurations from a MC runs for a film shown in Fig. 8b. (a) The case when excluded volume forces are taken into account (b) The case without excluded volume forces.

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... 

Polymer chains, in the melt, behave as if they are in the 0-condition, so the dimensions are those in the unperturbed state. This argument was put forward by Flory on energetic grounds and has been confirmed by neutron scattering (Strobl, 1996). The consideration begins with an analysis of the excluded-volume forces on an ideal chain. These arise from non-uniform density distributions in the system of an ideal chain in solution as shown in Figure 1.3. 

This shows the way that the local monomer concenfration, c, varies from the centre of the chain (x = 0) to either end. The excluded-volume forces on the chain create a potential energy y/ sensed by each repeat unit, which depends on and on a volume parameter Vg that controls their magnitude ... 

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