Combinatory partition function

The structure of a simple mixture is dominated by the repulsive forces between the molecules [15]. Any model of a liquid mixture and, a fortiori of a polymer solution, should therefore take proper account of the configurational entropy of the mixture [16-18]. In the standard lattice model of a polymer solution, it is assumed that polymers live on a regular lattice of n sites with coordination number q. If there are n2 polymer chains, each occupying r consecutive sites, then the remaining m single sites are occupied by the solvent. The total volume of the incompressible solution is n = m + m2. In the case r = 1, the combinatorial contribution of two kinds of molecules to the partition function is... 

In the canonical partition function of (5.1), we have for simplicity ignored combinatorial prefactors. Free energy perturbation theory [12] relies on evaluating effectively the ratio of the partition functions to obtain the free energy difference between the initial and final states corresponding to coupling parameters A = 1 and 0 (see also Chap. 2),... 

Within the present approximation the partition function of the mixture has the same structure (8) as a pure component, except for the ideal combinatorial factor ... 

Edwards approach (Ref. [10]) is based on field-theoretic path-integral representation of the partition function Wn(R0, RN, N) defining the probability density of the fact that end points of an JV-link chain are placed at the points R0 and Rn, respectively, and the chain turns n times around the string (the obstacle). The same problem in a slightly different way was considered by Prager and Frisch by using the combinatorial methods [11] and later by Saito and Chen by employing Fourier analysis [12]. 

The problem of determination of the partition function Z(k, N) for the iV-link chain having the fc-step primitive path was at first solved in Ref. [17] for the case a = c by application of rather complicated combinatorial methods. The generalization of the method proposed in Ref. [17] for the case c> a was performed in Refs. [19,23] by means of matrix methods which allow one to determine the value Z(k,N) numerically for the isotropic lattice of obstacles. The basic ideas of the paper [17] were used in Ref. [19] for investigation of the influence of topological effects in the problem of rubber elasticity of polymer networks. The dependence of the strain x on the relative deformation A for the uniaxial tension Ax = Xy = 1/Va, kz = A calculated in this paper is presented in Fig. 6 in Moon-ey-Rivlin coordinates (t/t0, A ), where r0 = vT/V0(k — 1/A2) represents the classical elasticity law [13]. (The direct Edwards approach to this problem was used in Ref. [26].) Within the framework of the theory proposed, the swelling properties of polymer networks were investigated in Refs. [19, 23] and the t(A)-dependence for the partially swollen gels was obtained [23]. In these papers, it was shown that the theory presented can be applied to a quantitative description of the experimental data. 

The steric or combinatory part of the partition function for the system follows as the product of the Vj for each of the Op rodlike molecules incorporated in the lattice i.e.,... 

The combinatorial FV expression of the Entropic-FV model is derived from statistical mechanics, using a suitable form of the generalized van der Waals partition function. ... 

Note that these equations are simply the Combinatorial Equation as applied to these two sets of defects. We can set up a partition function(see the above definition), using equation 2.5.3. We apply this to the Schottky and Frenkel defects as examples ... 

When we calculate the thermod3niamic properties, we need to take the logarithm of the grand partition function. Now, in a way quite similar to that in Section II-A, a simple combinatorial... 

The configurational partition function per chain molecule, q, is expressed in terms of combinatorial factor P, free volume, Vf, and the mean potential energy of a cell, eo ... 

Ch. 6) and its combinatorial part conformational partition function for the C, N, I phases... 

From the late 1950s, many papers in the literature studied this problem [23], Most of them employed either the matrix method or the generating function method to calculate the chain partition function. However, in order to apply the theoretical method directly to many chain problems in solutions and gels, we here reformulate the single chain problem using the combinatorial counting method. 

Hydration of a neutral polymer can roughly be classified into two categories direct hydrogen bonds (referred to as H-bonds) between a polymer chain and water molecules (p-w), and the hydrophobic hydration of water molecules surrounding a hydrophobic group on a chain in a cage structure by water-water (w-w) H-bonds. In this section, we extend the combinatorial method for the partition function presented in the previous section to suit for the problem of solvent adsorption, and study polymer conformation change in aqueous solutions due to the direct p-w H-bonds. 

Qp is the combinatorial term of the partition function for a hypothetical system with a random distribution of the empty and molecular segments Qi p is a correction term for the actnal nonrandom distribution of the empty sites... 

As can been seen from Equation 3.13, the FV definition given by Equation 3.12 is employed. The combinatorial term of Equation 3.13 is very similar to that of FH. However, instead of volume or segment fractions, FV fractions are used. In this way, both combinatorial and FV effects are combined into a single expression. The combinatorial-FV expression of the Entropic-FV model is derived from statistical mechanics, using a suitable form of the generalized van der Waals partition function. 

The lattice partition function reduces to the combinatorial factor g. Let us now calculate the corresponding grand partition function. According to (2.3.4) it is obtained by summation over all values of Na and Nb of the partition function, each term of the sum being multiplied by the factor exp Naiha + Nb/ b) which represents the statistical weight of the configuration Na, Nb) for given value of / a, /ab, T and T. To take account of the incompressibility of the lattice we introduce moreover the function such that... 

We may compare (17.6.2) with the corresponding expression (10.7.4) for monomer mixtures. Apart from the combinatorial entropy and factors qjqA and CA/iA, these formulae are exactly the same, when the mole fractions xa and xb are replaced by Xa and Xb- We must also notice that in our present model the configurational specific heat at constant volume cvA vanishes as a consequence of our assumption that the cell partition functions do not depend on the temperature. Therefore the detailed discussion of the effect of intermolecular forces on excess functions presented in Ch. IX-XI, applies also to polymer mixtures. For example p will again give rise to positive deviations from ideality, positive excess entropy and heat absorption. We shall not go into more detail. 

The partitionfunction of the system contains a Boltzmann lattice-energy factor, a combinatorial factor representing the entropy of segment mixing and empty lattice sites, and a statistical-mechanical free-volume term. From the partition function, the equation of state may be calculated in the form of... 

Notice in particular that there is no need to ask for k > 0, but of course one has to give up the interpretation of K -) as a sub-probability (it would rather be a combinatorial term, see in particular the way the Poland-Scheraga model has been introduced). Regardless of the value of K, we realize that, if c < —1, then exp( A )Z is the partition function of the homogeneous pinning model with polynomial decay of the return times (and critical point (3c)- Therefore there is a transition, between free energy equal to —k to larger than —k, at (3c- And it is... 

Determine the probability density, p, or the partition function as a function of the Hamiltonian and the ensemble constraints. This is a taxing task, requiring elements of probability and combinatorial theory. 

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