Hard spheres excluded volume

Thus we can see that a combination of van der Waals treatment of hard sphere excluded volume and Debye-Huckel treatment of screening with a minor generalization to account for hole correction of electrostatic interactions yields quite accurate bulk thermodynamic data for symmetrical salt solutions. 

We recently performed calculations of the partition function for a randomly jointed chain with hard-sphere excluded-volume interactions [21], namely, the same model for which the swelling factor was calculated in Section IV.A. The effective coupling function for the two-bond K = 2) Kuhn segment is displayed in Figure 5.8. It is apparent that the spectrum of fixed points becomes quasicontinuous for g > 2. (An examination of the roots of the equation 3Q K" g) — [3Q(K = 0 confirms this... 

The first term in Eq. [429] is the hard-sphere exclusion volume term, which decreases the counterion concentration at the surface the second and third terms respectively represent the fluctuation potential and increase the surface concentration of ions. These terms can be included as an activity coefficient in a general-purpose algorithm through the exponential term in Eq. [378]. For ion distances Ar closer to the surface than three ion radii, Bratko and Vlachy multiply the ion concentrations in Eqs. [430] by an excluded volume correction factor B r) = (Ar- - a)j4a for distances Ar < a, ion concentrations are, of course, zero because of hard-sphere exclusion. 

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 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. 

The above argument shows that complete overlap of coil domains is improbable for large n and hence gives plausibility to the excluded volume concept as applied to random coils. More importantly, however, it introduces the notion that coil interpenetration must be discussed in terms of probability. For hard spheres the probability of interpenetration is zero, but for random coils the boundaries of the domain are softer and the probability for interpenetration must be analyzed in more detail. One method for doing this will be discussed in the next section. Before turning to this, however, we note that the Flory-Huggins theory can also be used to yield a value for the second virial coefficient. 

Before concluding this discussion of the excluded volume, it is desirable to introduce the concept of an equivalent impenetrable sphere having a size chosen to give an excluded volume equal to that of the actual polymer molecule. Two such hard spheres can be brought no closer together than the distance at which their centers are separated by the sphere diameter de. At all greater distances the interaction is considered to be zero. Hence / = for a dey and fa = 0 for a[Pg.529]

This is especially clear in the case of hard spheres each of which excludes a volume (47r/3)dg which is eight times its net volume. Thus, the portions of the spherical domains at distances between de/2 and de of the centers of two nearby molecules may overlap, with the result that the volume actually excluded by these two molecules is less than 2(47r/3)d. ... 

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... 

In order to accurately describe such oscillations, which have been the center of attention of modern liquid state theory, two major requirements need be fulfilled. The first has already been discussed above, i.e., the need to accurately resolve the nonlocal interactions, in particular the repulsive interactions. The second is the need to accurately resolve the mechanisms of the equation of state of the bulk fluid. Thus we need a mechanistically accurate bulk equation of state in order to create a free energy functional which can accurately resolve nonuniform fluid phenomena related to the nonlocality of interactions. So far we have only discussed the original van der Waals form of equation of state and its slight modification by choosing a high-density estimate for the excluded volume, vq = for a fluid with effective hard sphere diameter a, instead of the low-density estimate vq = suggested by van der Waals. These two estimates really suggest... 

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... 

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.,... 

A problem arises, in that the strong r b dependence of My requires that close overlap of spins be prevented. Thus, even though excluded volume interactions have no effect on chain dimensions in the bulk amorphous phase, it is important in the present application to build in an excluded volume effect (simulated with appropriate hard sphere potentials), so that occasional close encounters of the RIS phantom segments do not lead to unrealistically large values of M2. 

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... 

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... 

Figure 20. Steps involved in loop formation, (a) Free evolution of the tube in depletive environment (b) formation of an unstable loop at around 3.4 lp (c) gliding of the loop governed by the positions of the two contact points along the fiber and the entry-exit angle (d) trapping of the loop by local defects. The translucent green surface represents the excluded volume for the fluid of hard spheres in (b,c,d) one sees that some of the excluded volume is reduced from the overlap resulting from formation of the loop. See color insert.

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. 

The justification for using the combining rule for the a-parameter is that this parameter is related to the attractive forces, and from intermolecular potential theory the attractive parameter in the intermolecular potential for the interaction between an unlike pair of molecules is given by a relationship similar to eq. (42). Similarly, the excluded volume or repulsive parameter b for an unlike pair would be given by eq. (43) if molecules were hard spheres. Most of the molecules are non-spherical, and do not have only hard-body interactions. Also there is not a one-to-one relationship between the attractive part of the intermolecular potential and a parameter in an equation of state. Consequently, these combining rules do not have a rigorous basis, and others have been proposed. 

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... 

The ideal gas free energy functional is defined exactly from statistical mechanics, dropping the temperature-dependent terms that do not affect the fluid structure. Free energy functional contribution due to the excluded volume of the segments is calculated from Rosenfeld s (1989) DFT for a mixture of hard spheres. The functional derivatives of these free energy functional contributions, which are actually required to solve the set of Euler-Lagrange equations, are straightforward. 

The volume exclusion effect has received recently renewed attention.5 9 Paunov et al.5 argued that the excluded volume should be taken as eight times the real volume of the ions (this corresponds to the first-order correction in the virial expansion for a hard-sphere fluid)... 

The complete DLVO pair potential, dlvo(/), is the sum of the repulsive electrostatic (uy(r)) and the attractive ( w(r)) pair potentials, plus the excluded volume or hard-sphere term mhs(/)> i-e-... 

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... 

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