Entropy and energy in a lattice model

Again it is convenient to represent the polymer chains as random walks on a lattice, each lattice site being either occupied by one (and only one) chain monomer or by a solvent molecule, as shown in Fig. III.2. We denote the fraction of sites occupied by monomers as This term is related to the concentration c (number of monomers per cm ) by = ca . 

Square of gyration radius Rc for one labeled polystyrene chain in a polystyrene solution of concentration c. Above c = c, decreases with a well-defined exponent. At high c, Ro returns to the ideal chain value. After Daoud t al.. Macromolecules 8, 804 (1975). 

The free energy F for this model has two components an entropy term describing how many arrangements of chains can exist on the lattice for a given and an energy term describing the interactions between adjacent molecules. 

In the mean field approximation the entropy S has a simple structure  

The first term is related to the translational entropy of the chain ( I /N is the chain concentration in dimensionless units). The second term may be similarly conceived as the translational entropy of the solvent molecules (of volume fraction 1 — ). An interesting ch k on eq. (III.l) is obtained for (V = 1. Then eq. (in.l) becomes the standard entropy for a collection of independent systems, each with two states full (probability I ) or empty (probability 1 - ). 

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