Chain structures 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... 

The Kieffer model correctly predicts the systematic change of the reduced partition functions of various minerals with structure, as indicated by Taylor and Epstein (1962). For anhydrous sihcates, the decrease in the sequence framework-chain-orthosilicate reflects the decreasing frequency of antisymmetric Si-O stretching modes. The internal frequencies of the carbonate ion give a high reduced partition function at all T. The value for rutile is low because of the low frequencies of the Ti-0 modes (Kieffer, 1982). 

Structural characteristics of adsorbed chains, such as the mean distance of chain ends from the surface and the fraction of segments in each layer, derive from P(i,s). The partition function and thermodynamic properties depend on the eigenvalues of W (Flory, 1969). In the limit n - oo, calculation of thermodynamic functions simplifies because one eigenvalue, denoted by A, dominates the free energy per segment,... 

The model thus has been defined in terms of polymer chain structure, solution structure, and interactions. Now statistical mechanics can be applied to it. We use the partition function devised by Flory (27, 28). For simplicity, instead of the entire system, we study a subsystem containing one polymer molecule plus its share of N /N2 monomer molecules. The partition function of the subsystem has the same form as that for the entire system (27,28) that is... 

Hence, in the light of our both accounts of causality, the molecular dynamics model represents causal processes or chains of events. But is the derivation of a molecule s structure by a molecular dynamics simulation a causal explanation Here the answer is no. The molecular dynamics model alone is not used to explain a causal story elucidating the time evolution of the molecule s conformations. It is used to find the equilibrium conformation situation that comes about a theoretically infinite time interval. The calculation of a molecule s trajectory is only the first step in deriving any observable structural property of this molecule. After a molecular dynamics search we have to screen its trajectory for the energetic minima. We apply the Boltzmann distribution principle to infer the most probable conformation of this molecule.17 It is not a causal principle at work here. This principle is derived from thermodynamics, and hence is statistical. For example, to derive the expression for the Boltzmann distribution, one crucial step is to determine the number of possible realizations there are for each specific distribution of items over a number of energy levels. There is no existing explanation for something like the molecular partition function for a system in thermodynamic equilibrium solely by means of causal processes or causal stories based on considerations on closest possible worlds. 

For the treatment of chain conformation problems in biomacromolecules such as two-state structural transitions to be discussed in the next section, a partition function can be constructed as follows ... 

As we shall see below, the denominator in Eq. (3.5) is the conformational partition function, Z , for the RIS model of the chain. It is constructed as a sum of Boltzmann factors that depend on T and the energies of the first- and higher-order interactions present in all of the conformations of the chain. Structural information does not appear explicitly in Z . However, a wealth of structural information (/, 6, (f>) can appear in the numerator of Eq. (3.5). The numerator also contains all of the thermal and energetic information from Z . The combination of this information allows a rapid estimation of C , even at large n, because computers can rapidly calculate the serial matrix products that appear in the numerator and denominator of Eq. (3.5). 

The partition function Eq. 6 describes a system of mutually interacting chains. Introducing auxiliary fields, U and W, via a Hubbard-Stratonovich transformation, one can decouple the interaction between the chains and rewrite the Hamiltonian in terms of independent chains in fluctuating fields. Then, one can integrate over the chain conformations and obtain a Hamiltonian which only depends on the auxiliary fields. Thermodynamic averages like density or structure factors can be expressed as averages over the fields, U and W, without approximation. 

The special situation of a homogeneous saddle point, corresponding to a homogeneous disordered phase, is particularly interesting. In that case, ex-phdt analytical relations between the single chain partition function and the fields can be obtained, and one recovers the well-known random phase approximation (RPA). To illustrate this approach, we shall now derive the RPA structure factor for the case of a symmetric binary homopolymer blend. 

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. 

Consider first the dilute limit. Clearly, from the structure [eq. (X.49)] of the grand partition function, this corresponds to a case where H - O, so that the weights for two chains, three chains, etc. are much smaller than the weight for a single chain. Also the dilute limit must correspond to r > Tc... 

The change in the conformational entropy of a chain on fusion, at constant volume, can be evaluated from the partition function of the disordered chain, if it is assumed that there are no contributions from the ordered structure. Thus, the conformational entropy on fusion is identified with the entropy of the isolated chain in the pure melt. This entropy can be written as... 

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