Partition function segments

J. M. H. M. Scheutjens and G. J. Fleer (1979) Statistical-theory of the adsorption of interacting chain molecules. 1. Partition-function, segment density distribution, and adsorption-isotherms. J. Phys. Chem. 83, pp. 1619-1635 ibid. (1980) Statistical-theory of the adsorption of interacting chain molecules. 2. Train, loop, and tail size distribution. 84, pp. 178-190 ibid. (1985) Interaction between 2 adsorbed polymer layers. Macromolecules 18, pp. 1882-1900... 

Scheutjens JMHM, Fleer GJ. Statistical theory of the adsorption of interacting chain molecules. I. Partition function, segment density distribution and adsorption isotherms. J Phys Chem 1979 83 1619-1635. 

Several methods have been developed for the quantitative description of such systems. The partition function of the polymer is computed with the help of statistical thermodynamics which finally permits the computation of the degree of conversion 0. In the simplest case, it corresponds to the linear Ising model according to which only the nearest segments interact cooperatively149. The second possibility is to start from already known equilibrium relations and thus to compute the relevant degree of conversion 0. 

Other thermodynamic functions may be derived from the partition function Q, or from the expression for the osmotic pressure. The chemical potential of the solvent in the solution (not to be confused with the excess chemical potential (mi —within a region of uniform segment expectancy, or density) is given, of course, by ... 

A component pn(N) of P(N) represents the relative probability of an N-segment chain terminating at a plane n with first segment located in any of the M planes. Therefore, the sum of the components of P(N) may be identified with the configurational partition function of the chain confined between 1 and M... 

For the calculation of the step free energy we consider first the partition functions G, and G, of straight segments of steps along the x and y direction. The segments have energies per unit length equal to 6 and 8y. In Fourier space one has ... 

Here we have chosen the Kuhn statistical segment length (=2q) as the unit for measuring length. The partition function Z of the total chain is given by... 

In these expressions, UB is the number of ways of adding (me segment to an essentially infinite chain or one loop in the three-dimensional bulk phase, Us is the corresponding number for a two-dimensional train on the surface, aB and a5 are exduded-volume parameters in the bulk and on the surface, respectively, and yB and ys the probability of transforming a sequence of adsorbed segments to a loop and that of the reverse process, respectively. The partition function q°(P, T) of the system is obtained from Eq. (B-38) as follows ... 

Hesselink51 assumed that the conformation of macromolecules adsorbed on the interface consists of one train and two tails. The number of tails with i segments, n(i), is calculated from the partition function for the tail-train conformation, and the normalized segment density distribution p7(z) in tails is derived by methods analogous to those used in Hesselink s derivation of the segment density distribution in loops. The result is... 

Lattice coordination numbers (2) and the cell volumes (vjj) for both the pure components and mixture lattices are assumed to have the same value. The partition function for this ensemble can be formulated following Equation 2. It is assumed now that the partition function, far from the binary critical point can be approximated by its largest term. Since molecule segments and holes can distribute themselves non-randomly, the partition function must incorporate terms to account for this effect. The nonrandomness correction rjj... 

Another kind of self-consistent theory used a continuum diffusion representation to describe the distribution of segments.21 23 The segments were assumed to be subjected to an external potential of a mean field. An analytical approximation of the latter self-consistent theory was suggested by Milner et al. (the MWC model)24 on the basis of the observation that at high stretching the partition function of the brush is dominated by the classical path as the most probable distribution. Under this assumption, it was found that the self-consistent field is parabolic and leads to a parabolic distribution of the monomer density. Similar theories for polyelectrolyte brushes25 27 also adopted the parabolic distribution approximation. 

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,... 

There has been an attempt by Renuncio and Prausnitz to introduce a partition function by modifying the energy of interaction to take into account the non-random mixing of two components. They introduced two site fractions between segment i and j as 0, and 0. where... 

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