Configuration partition function polymer chain

Let us consider a polymer chain with N->oo identical skeletal atoms, either in solution or in the melt, representing our polymer system. Our reference temperature is T0, i.e., the temperature above which no bundles may effectively contribute to crystallization. At T = T0 the chain is assumed to be unperturbed and its configurational partition function is ZN(T0) = kN (N -> oo) [107] for simplicity we use a reduced form Zn = Z /kN (henceforth simply the partition function) so that Zn(T0) = 1. Only at T < To effective bundles may form, see Fig. 1, and we have ZN(T) = 1 + AZN(T - T0) note that the unit term corresponds to the bundle-free infinite-chain configuration. Each bundle with n chain atoms in -c N) will contribute to AZn... 

The inter- and Intramolecular contributions to the entropy and energy of fusion are calculated for several linear aliphatic polyesters and polyamides assuming the fusion process consists of two Independent contributions the volume expansion (intermolecular contribution) and the increase in the conformational freedom of each polymer chain on melting (intramolecular contribution). The intramolecular entropy and energy contributions are obtained from the configurational partition function and Its temperature coefficient calculated for an isolated, unperturbed polymer chain using the RIS approximation. 

The grand partition function of Eq. (85) for 5- simplex lattice is written in terms of six restricted partition functions shown in Fig 19. Out of six configurations two (A i and B >) represent the sums of weights of configurations of the polymer chain within one r-th order subgraph away from surface, and the remaining four S E and represent the surface functions. As in the case of the 4-simplex, the recursion relations for and do not include other variables. 

Models for Stiff-Chain Polymers.— Flory briefly reviewed some of the consequences of separating the configurational partition function for long chain molecules and their solutions into inter- and intra-molecular parts. In particular, he pointed out that this separation, and hence the partition function derived therefrom, are valid only for sufficiently flexible chains, or when the polymer concentration is sufiiciently low. He indicates how this can be rectified in statistical mechanical models for semi-rigid and rod-like polymer molecules. For the latter case this is pursued in considerable detail in a very recent series of papers by Flory and co-workers. ... 

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... 

This expression accounts for the configurational entropy of an ideal binary mixture with identical molecular sizes, but not for that of a polymer solution, since polymer chains are large and flexible. For that case, more contributions arise from the chain conformational entropy, first considered by Meyer [19] and then derived by Huggins [20] and Flory [21]. In analogy with a nonreversing random walk on a lattice, the conformational contribution of polymer chains to the partition function is given by... 

As an example, we shall apply diagrammatic methods to the study of discrete models on a lattice. The polymer system will consist of N chains with fixed lengths, drawn on a lattice. We assume that each chain, defined by its order j, has a fixed origin of position vector r2j 1 and a fixed end point of position vector r,. By definition, the weight associated with a configuration fl equals exp[/40N — bp (ft)] where N is the total number of links (N = Nt +. . . + Nn) and p (ft) is the number of couples of different points belonging to the same chain or to two chains, and located on the same site. The partition function of the system is defined by... 

Lifshitz represents a polymer containing N monomers as a chain with N independent links subjected to interactions. Let SI be the configuration of such a polymer, °Hr Sl) the weight associated with the free chain and t(Sl) the interaction energy (in thermal units 1 //f). The partition function Z can be written in the following symbolic form... 

The first of these difficulties can be avoided for symmetrical polymer mixtures (Na = Nb = N) by working in the semigrandcanonical ensemble of the polymer mixture [107] rather than keeping the volume fractions < )A, B and hence the numbers of chains nA, nB individually fixed, as one would do in experiment and in the canonical ensemble of statistical thermodynamics, we keep the chemical potential difference Ap = pA — pB between the two types of monomers fixed as the given independent variable. While the total volume fraction 1 — < )v taken by monomers is held constant, the volume fractions < )A, B of each species fluctuate and are not known beforehand, but rather are an output of the simulation. Thus in addition to the moves necessary to equilibrate the coil configuration (Fig. 16, upper part), one allows for moves where an A-chain is taken out of the system and replaced by a B-chain or vice versa. Note that for the symmetrical polymer mixture the term representing the contributions of the chemical potentials pA, pB to the grand-canonical partition function Z... 

As in the standard path integral method for finite temperature systems [19], the above pseudo partition function can be regarded as a configurational integral of classical polymers. However, in the variational path integral, the classical isomorphic systems consist of open-chain polymers. Furthermore, distributions of end-point coordinates at = 0 and M are affected by the trial wavefunction and respectively. 

The quasi-lattice model was developed by Roe (13) and Scheutjens and Fleer (14) (SF theory) The basic analysis considered all chain conformations as step-weighted random walks on a quasi-crystalline lattice that extends in parallel layers from the surface. This is illustrated in Figure 16.2 which shows a possible conformation of a polymer molecule at a flat surface. The partition function was written in terms of the number of chain configurations that were treated as connected sequences of segments. In each layer parallel to the surface, random mixing between the segments and solvent molecules was assumed, i.e. by using... 

By definition, the second virial coefficient comes from the connected two chain partition function with all the ends free. An expansion in terms of the coupling constant would involve polymer configurations as shown in Fig. 5. This is incidentally identical to Eq. (27). Each line represents the probability of free polymer going from r, z to r, z,... 

When polymers self-assemble to give periodic ordered phases, only a fraction of chain configurations that are available in homogeneous phases are permissible. SCFT provides a convenient framework for computing the reduction in entropy due to this effect. In this theory, individual chains are assumed to be affected by the presence of a spatially varying external field w iz). For convenience, we restrict our attention to one-dimensional phases. The partition function of a chain of s monomers of type i constrained so that the sth monomer is held fixed at z, qi, is given by... 

Here, z is the fugacity of the walk and Nb is the number of bends in a given configuration. In two dimensions, the grand canonical partition function defined by Eq. 8.15 may be expressed as a sum of two components of PDWs. The recursion relations for the case of a semi-flexible polymer chain are... 

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