This note derives the two-point correlation function of an ideal, non-interacting Gaussian chain in a step-by-step way.
It begins with a discrete Gaussian-chain model with independent bond vectors and defines the segment-pair correlation between monomers (i) and (j). The derivation then rewrites the delta functions in Fourier space, changes variables from monomer positions to bond vectors, integrates out the reference position, and reduces the remaining expression to Gaussian bond integrals.
The result shows that the relative separation between two chain segments is itself described by a Gaussian connector distribution whose width is set by the contour distance between those segments. Using this kernel as the basic building block, the note then constructs second correlation functions for diblock-chain densities, including both same-species correlations and the (A)-(B) cross-correlation.
Finally, the derivation takes the long-chain continuum limit and gives the correlation functions in both real space and Fourier space, ending with a compact dimensionless form for the (A)-(B) correlation kernel. These results are standard ingredients for random phase approximation analysis and polymer field theory.
The full derivation with complete equations is provided below.
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]]>This note documents the Hybrid Conjugate Gradient (HCG) strategy for solving polymer SCFT equations.
The method combines robust gradient-based updates with conjugate-direction acceleration to improve convergence behavior in nonlinear self-consistent iterations.
It is useful for reducing iteration cost and stabilizing difficult SCFT solves in high-dimensional parameter settings.
The full method note is provided below.
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]]>This note presents a canonical-ensemble self-consistent field theory (SCFT) derivation for a linear incompressible AB diblock copolymer melt using a full auxiliary-field formulation.
The derivation follows the field-theoretic route from a microscopic polymer model to the effective Hamiltonian in terms of collective and conjugate fields.
At the saddle-point level, the SCFT equations are obtained in the standard propagator-based framework together with incompressibility and self-consistency relations.
The full derivation with complete equations is provided below.
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]]>This note summarizes a canonical-ensemble self-consistent field theory (SCFT) derivation for an linear incompressible AB diblock copolymer melt.
The derivation starts from a microscopic polymer model with:
It then introduces collective density fields and auxiliary conjugate fields through functional delta identities and Fourier representations. This converts the many-chain partition function into a field-theoretic form and defines an effective Hamiltonian.
At the mean-field saddle point, the SCFT equations are obtained, including:
Finally, the derivation is recast into the common real-field and volume-fraction form, and interpreted through the free-energy decomposition into:
The full derivation with complete equations is provided below.
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