Excluded volume model

Here,. Ai(X) is the partial SASA of atom i (which depends on the solute configuration X), and Yi is an atomic free energy per unit area associated with atom i. We refer to those models as full SASA. Because it is so simple, this approach is widely used in computations on biomolecules [96-98]. Variations of the solvent-exposed area models are the shell model of Scheraga [99,100], the excluded-volume model of Colonna-Cesari and Sander [101,102], and the Gaussian model of Lazaridis and Karplus [103]. Full SASA models have been used for investigating the thermal denaturation of proteins [103] and to examine protein-protein association [104]. 

Due to their high aspect ratio, nanocarbons dispersed in a polymer matrix can form a percolating conductive network at very low volume fractions (< 0.1 %). The conductivity of a composite above the transition from an insulator can be described by the statistical percolation using an excluded volume model [22,23] to yield the following expression ... 

Let us now consider some actual numerical data for specific mixed biopolymer systems. Table 5.1 shows a set of examples comparing the values of the cross second virial coefficients obtained experimentally by static laser light scattering with those calculated theoretically on the basis of various simple excluded volume models using equations (5.32) to (5.35). For the purposes of this comparison, the experimental data were obtained under conditions of relatively high ionic strength (/ > 0.1 mol dm- ), i.e., under conditions where the contribution of the electrostatic term (A if1) is expected to be relatively insignificant. 

If we turn from phenomenological thermodynamics to statistical thermodynamics, then we can interpret the second virial coefficient in terms of molecular parameters via a model. We pursue this approach for two different models, namely, the excluded-volume model for solute molecules with rigid structures and the Flory-Huggins model for polymer chains, in Section 3.4. 

On the basis of this model, an expression for ASm can be derived. We will not go through the details of the derivation, but merely note the following similarities and differences between this derivation and the one that leads to Equation (58) for the excluded-volume model ... 

You may recall that the temperature where % 2is what Floiy called the theta tern--perature and can now be seen to describe the situation where the second virial coefficient becomes zero (Figure 12-10). This means that at this point pair-wise interactions cancel and the chain becomes nearly ideal, as we discussed in the section on dilute solutions (Chapter 11), where we referred to the Floiy excluded volume model in which the chain expansion factor is given by Equation 12-18 ... 

Using this approach SANS has been used to measure the dimension of the Gaussian coil structure of a single chain in melts, solution and blends, provided an affirmation of the screened excluded volume model, and a verification of scaling laws in polymer solutions, determined the structure of diblock copolymer aggregates, and established the relationship between the micro and macroscopic deformation in rubber elasticity. 

A theoretical treatment of aqueous two-phase extraction at the isoelectric point is presented. We extend the constant pressure solution theory of Hill to the prediction of the chemical potential of a species in a system containing soivent, two polymers and protein. The theory leads to an osmotic virial-type expansion and gives a fundamentai interpretation of the osmotic viriai coefficients in terms of forces between species. The expansion is identical to the Edmunds-Ogston-type expression oniy when certain assumptions are made — one of which is that the solvent is non-interacting. The coefficients are calculated using simple excluded volume models for polymer-protein interactions and are then inserted into the expansion to predict isoelectric partition coefficients. The results are compared with trends observed experimentally for protein partition coefficients as functions of protein and polymer molecular weights. 

For compact particles of a given shape, the molar excluded volume, uNa-, must be proportional to the molecular weight of the particles, M2- Hence, according to the excluded volume model the second virial coefficient of particles of a given shape decreases with increasing molecular weight, being inversely proportional to the latter. 

Quantum chemistry predicts that the annihilation lifetime of a positron species is generally determined by the degree of overlapping of positron and electron wave functions, which leads e.g. to the intrinsic lifetime of the ortho-Ps of 1.4 x 10 sec. In a condensed matter it is obvious that the electron density at the position of the positron will greatly depend on the macroscopic and microscopic (mass) density and thus on parameters such as phase and temperature. This rather simple approach has led to the development of the "free volume" or "excluded volume" model (20, 28-29), whose basic feature it is that the lifetime of a positron or Ps trapped in such a material will depend on the free volume which it has available. 

So far we have only considered single-chain models, that is, modeled the effects of all other chains by friction and random delta-correlated noise. This is obviously a very simplistic assumption, and it is interesting to model interchain interactions explicitly and compare the results with the single-chain models. The first model we will investigate is a multichain model where all monomers interact with each other via the nonbonded potential (eqn [47]). The bonded potential is the same as in the excluded volume model (eqn [48]). This model was proposed by Kremer and Grest (KG) a a simple and efficient model to investigate effects of entanglements. We will call it the KG MD model. 

Excluded-volume models take into account the fact that the finite size of the ions leads to a lower counterion concentration near a charged surface, and to a weaker Debye screening of the electrostatic field (in comparison with the point-ion model), which results in a stronger repulsion between two charged surfaces at short separations [587,588]. 

Let us consider in more details the coion expulsion model, also called reduced screening model [589], This model was developed to explain the strong short-range repulsion detected in foam films see Figure 4.38. It was found [589] that the excluded volume model [588] cannot explain the observed large deviations from the DLVO theory. Quantitative data interpretation was obtained by... 

A self-consistent field (SCF) is constructed for a polymer chain with excluded volume modeled as a self-avoiding random walk (SAW) of N eps (iV->oo). The SCF requires the introduction of a second exponent 0 in addition to the usual v exponent that dharacterizes the size of a SAW. The SCF equals N times the probability of an interaction of the chain end witb a distant part of the chain. In a-dimensional space scales as (self-consistency of the field yields the relations 0 = (4 - d)/3 and v d + ) =2. It is shown that 0q < < 0-j where 0q is the exponent associated with the probability of a SAW returning to its origin and 0-j is the exponent associated with the probability of a SAW forming a large loop with a long tail. A SCF (4>) is also determined for a semidilute solution of polymer volume fraction The number of binary interactions scales as < > <() - ... 

The excluded volume theory is the most commonly used continuum analytical model of percolation. The excluded volume of an object is defined as the volume around the object into which another identical object cannot enter without contacting the first object as illustrated in Figure 2. ° The principal concept in the excluded volume model is that the percolation threshold of a system is determined by the excluded volume of filler particles, rather than their true volume. This is particularly applicable to asymmetrical, unaligned objects for which the excluded volume can differ significantly from their true volume. Therefore, this model has been applied to describe critical percolation phenomena for a wide variety of filler geometries. In addition, excluded volume arguments provide useful theoretical approximations in many computational stu-dies. Excluded volume solutions were first formulated for soft-core (interpenetrable) fillers, and later extended to core-shell (impenetrable hard-core surrounded by a penetrable shell) fillers. ... 

The excluded volume model predicts a decrease in percolation threshold, 4>a with increasing aspect ratio in an isotropic network, which is consistent with experimental findings and results from simulations of rod networks. Note that the underlying assumption of infinite aspea ratio in the excluded volume theory makes eqn [4] most appropriate for fillers with very high aspect ratio. 

As described in Section 7.17.3.3, the excluded volume model predicts an inverse proportionality between the percolation threshold, and aspect ratio in isotropic networks of both penetrable (soft core) and impenetrable rods, which is consistent with experimental findings and results from other analytical and simulation studies. However, the... 

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