Hard-sphere model excluded volume

The solvent-excluded volume is a molecular volume calculation that finds the volume of space which a given solvent cannot reach. This is done by determining the surface created by running a spherical probe over a hard sphere model of molecule. The size of the probe sphere is based on the size of the solvent molecule. 

If we take the billiard-ball, or hard-sphere, model literally, we can calculate the excluded volume constant, b, from the diameter of the molecular billiard balls, ct. The centers of two billiard balls, each of radius a, can come no closer than r = a. Therefore, we can consider that around each molecule there is a... 

Figure 8.1 The process of computing the incremental chemical potential involves adding one extra segment to an M - 1 segment chain moving in the solvent. The tangent hard sphere model of a (M — l)-mer (M = 5) is shown here. The dashed circles enclose the volume excluded to the centers of the solvent spheres.

The Pratt-Chandler theory has been extended to consider complex molecules. For example, the hard-sphere model of -butane may have an excluded volume Av(f, X), which is a function of the torsion angle (j) and depends on the exclusion radius X of the methylene spheres. Then the part of the PMF (the potential of mean force) arising from the solute-solvent interaction can be related to the reversible work required to create a cavity with the shape and excluded volume Av((/>, X) of the -butane molecule. 

Repulsive interactions are important when molecules are close to each other. They result from the overlap of electrons when atoms approach one another. As molecules move very close to each other the potential energy rises steeply, due partly to repulsive interactions between electrons, but also due to forces with a quantum mechanical origin in the Pauli exclusion principle. Repulsive interactions effectively correspond to steric or excluded volume interactions. Because a molecule cannot come into contact with other molecules, it effectively excludes volume to these other molecules. The simplest model for an excluded volume interaction is the hard sphere model. The hard sphere model has direct application to one class of soft materials, namely sterically stabilized colloidal dispersions. These are described in Section 3.6. It is also used as a reference system for modelling the behaviour of simple fluids. The hard sphere potential, V(r), has a particularly simple form ... 

The hard sphere model is based on the excluded volume of spherical particles. An excluded volume theory has been developed to account for the orientational ordering of liquid crystal molecules, assuming them to be hard rods. This is the Onsager theory and its variants, outlined in Section 5.5.2. Excluded volume interactions influence the conformation of polymer chains. The conformation of an ideal chain is described by a random walk. However,... 

To interpret die first correction term we start, in the spirit of the hard-sphere model, by subtracting from the voliune F of the container an excluded volume, Fe, that is not available for the molecules so that P(F — F ) = RT. Then, by... 

First, let s consider the size of the molecules based on the hard sphere model. The entire volume of the system will no longer be available to the molecules. We can account for this effect by replacing the volume term in the ideal gas model with one for available volume. Recall that in the hard sphere model, the molecules have a diameter (J. Thus, the center of one molecule cannot approach another molecule closer than a distance (j. The excluded volume of the two molecules is then (4/3) jrcr. Dividing by 2 and multiplying by Avogadro s number, we get one mole of molecules occupying a volume b = (2/3) 7Tcr lVA- To correct for size, we modify the ideal gas model to include only the unoccupied molar volume, v — h). Hence, we get ... 

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 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... 

The hole correction of the electrostatic energy is a nonlocal mechanism just like the excluded volume effect in the GvdW theory of simple fluids. We shall now consider the charge density around a hard sphere ion in an electrolyte solution still represented in the symmetrical primitive model. In order to account for this fact in the simplest way we shall assume that the charge density p,(r) around an ion of type i maintains its simple exponential form as obtained in the usual Debye-Hiickel theory, i.e.,... 

No real system is fully random. Random systems are over-simplified ideal models similar to those of strictly regular structures. Most relevant is the effect of the finite volume of the monomer units which implies that two units can approach each other only up to their diameter. Thus a certain volume is forbidden or excluded for the individual repeating units. For hard sphere monomers this excluded volume is just eight times the monomer volume. This excluded volume... 

The generator matrix treatment of simple chains with excluded volume described earlier S 010) properly reproduces the known chain length dependence of the mean-square dimensions in the limit of infinite chains. The purpose of this paper is to compare the behaviour of finite generator matrix chains with that of Monte-Carlo chains in which atoms participating in long-range interactions behave as hard spheres. The model for the unperturbed chain is that developed by Flory et at. for PE (S 027). 

Response of the mean square dipole moment, < J2>, to excluded volume is evaluated for several chains via Monte-Carlo methods. The chains differ in the manner in which dipolar moment vectors are attached to the local coordinate systems for the skeletal bonds. In the unperturbed state, configurational statistics are those specified by the usual RIS model for linear PE chains. Excluded volume is introduced by requiring chain atoms participating in long-range interactions to behave as hard spheres. 

One model which has been extensively used to model polymers in the continuum is the bead-spring model. In this model a polymer chain consists of Nbeads (mers) connected by a spring. The easiest way to include excluded volume interactions is to represent the beads as spheres centered at each connection point on the chain. The spheres can either be hard or soft. For soft spheres, a Lennard-Jones interaction is often used, where the interaction between monomers is... 

We now consider packing issues associated with the reference term of van der Waals approaches exemplified by Eq. (4.1), p. 61. As noted there, the simplest interaction model appropriate for those terms is a hard-core model. The distinguished molecule considered will perfectly repel solvent molecules from an overlap, or excluded, volume. The general issues we develop will apply to such molecules in general solvents, i.e., the solvent need not be simple in the same sense as the hard-core solute we treat. But we will exemplify our general conclusions with results on the hard-sphere solvent system. The notation here, however, continues to use the tilde, as /i (i "), to indicate that the distinguished solute might serve as a reference case for subsequent treatment of other interactions. 

Typical MC examples include nonintersecting random walks on a lattice or off-lattice necklaces of hard spheres jointed freely, The latter case involves a polymer chain lattice model in which the real chain is substituted by a selfavoiding random walk model on a periodic lattice. The excluded volume effect is taken into account by the condition that no site may be occupied more than once. 

The freely jointed chain model may also be used with MC simulations. The excluded volume effect is taken into account by putting a hard sphere on... 

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. 

The volume of space bounded by the solvent-accessible molecular surface is called the solvent-excluded volume because it is the volume of space from which solvent is excluded by the presence of the molecule when the solvent molecule is also modelled as a hard sphere. Moreover, the interstitial volume is the volume consisting of packing defects between the atoms that are too small to admit a probe sphere of a given radius in practice, it is calculated as the difference between the solvent-excluded volume and the van der Waals volume. An analytical method was developed by Connolly able to calculate the solvent-excluded volume [Connolly, 1983b] several other numerical and analytical approaches have been proposed. 

Figure 1. The mapping of the core softened potential (5) on the hard sphere diameter. The temperature (r = kT/Uj ) - density (y = b/fi) dependence of the excluded volume (b) for model parameters set dp, =2.27, Ur/Ua=2, ds=10.29...

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