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.

1. Introduction

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.

2. Continuous Gaussian Chain Model

3. Single Continuous Gaussian Chain in External Field

4. Many Chain Model for Two-Bead Chains

5. Many Chain Model for A-B Block Copolymers

6. Many Chain for Solutions of A-B Diblock Copolyelectrolytes