Mean field theoiy

Because of the very simple structure of the theory, the theory is generally regarded as a prototype of a mean field theoiy. It most, however, be realized that it is not a real mean field theory in that no mean field has been calculated. The mean field theory will be discussed later on. 

A critical point of order p is a thermodynamic state in which p phases become identical p = 2 for ordinary critical points, of the kind we have been discussing p = 3 for trkritical points, the subject of 9.5 etc. The mean-field-theoiy value Sd of the exponent 5 for a critical point of order p may be inferred from Fig. 1.8 or 3.4 and their obvious generalizations. When p = 2, as in Hg. 1.8 or 3.4, every isotherm at Tthree intersections with the tieline in the equal-areas construction at T=T those previously distinct intersections coalesce in a sin e, three-fold root, which makes tte critical isotherm cubic at the critical point fid = 3. For p-phase equilibrium, the analogue of the curve in Hg. 1.8 or 3.4 has 2p—2 loops the horirontal line that determines the equilibrium pressure or chemk potential makes 2p -1 intersections with the curve, in such a way that the areas of the 2p-2 loops are equal in pairs and at T-T the 2p—1 roots coalesce, so that the critical isotherm is of degree 8d = 2p-l ( = 5 for a tricritical point). Then from (9.64), the bcndeiiine dimensionality for a critical point of order p is... 

For situations of overlapping chains, where lateral fluctuations in the segment concentration become rather small, mean-field descriptions become appropriate. The most successful of this type of theoiy is the lattice model of Scheutjens and Fleer (SF-theoiy). In chapter II.5 some aspects of this model were discussed. This theory predicts how the adsorbed amount and the concentration profile 0(z) depend on the interaction parameters and x and on the chain length N. From the statistical-thermodynamic treatment the Helmholtz energy and, hence, the surface pressure ti can also be obtcdned. When n is expressed as a function of the profile 0(z), the result may be written as ... 

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