A self-consistent field argument

Consida a daise system of identical chains (Fig. II. 1). Let us focus on one chain, which we shall call the white chain (the other chains are black ), and study the repulsive potential U experienced by one white monomer. This U is essentially proportional to the local concentration c of monomers, which is shown in Fig. II. lb. The concentration c has two components. 

The concentration of white monomers (and the corresponding potential C/ ) is peaked around the center of gravity of the white molecule. On the slopes of this peak there is a force — dU /dx pointing outward. In the 

Therefore, the black potential creates an inward force. This force exactly equals the force caused by the white monomers since Utot (like Ctot) is constant in space dUtot/Bx = 0. The chain experiences no force and remains ideal. 

The above argument is useful and valid in three dimensions. However, it applies only if the whole idea of a self-consistent field makes sense. Clearly, there are cases where it fails for example, in one dimension, the chains must be completely stretched (at all concentrations). What hrqipens then in two dimensions The answer can be derived fiom the detailed (local) discussion below. 

---

The free energy functional F [mean field (Landau) approximation for the free energy in the magnetic problem (for which there are many excellent reviews ). 

---