In this post I will present an detailed (informal) derivation of the self-consistent field theory (SCFT) for many-chain polymer systems. I start from the continuous Gaussian chain model. Then the Gaussian chain in an external field is described. After that, a system of multiple two-bead chains as a simplified model for many-chain model is examined. Then this approach is extended to many-chain A-B block copolymer system. Finally, as an application of this derivation, we derive SCFT for many-chain A-B diblock copolyelectrolyte solutions.
The self-consistent field theory (SCFT) for many-chain systems is obtained by imposing a mean-field approximation to simplify the statistical field theories. The statistical field theories can be constructed from the particle-based model by carrying out a particle-to-field transformation.
The general approach for a particle-to-field transformation is to invoke formal techniques related to Hubbard-Stratonovich transformations, which have the effect of decoupling interactions among particles (or polymer segments) and replacing them with interactions between the particles and one or more auxiliary fields.