A Functional Field Theorem: An Explicit Proof of Axioms and Equations for Applying iSAFT in Polymer Field Theory

TL;DR

This paper proves the mathematical equivalence between polymer Self Consistent Field Theory (SCFT) and interfacial SAFT-based classical Density Functional Theory (DFT), providing a unified framework for modeling complex polymer systems with association and structure.

Contribution

It offers an explicit Legendre duality proof linking SCFT and iSAFT DFT, enabling modular and quantitative modeling of structured, associating polymers.

Findings

Established the equivalence between SCFT and iSAFT DFT via Legendre duality.

Incorporated Wertheim TPT1 association free energy into the functional framework.

Mapped fluctuation corrections and correlation operators between theories.

Abstract

Our work establishes the mathematical equivalence between polymer Self Consistent Field Theory and interfacial SAFT based classical Density Functional Theory by providing an explicit proof that SCFT is the mean field (MF) or saddle point limit of a Legendre dual formulation of iSAFT CDFT. The polymer CDFT ideal chain term is the Legendre conjugate of the single chain generating functional built from standard SCFT propagators, so the Legendre transform converts the iSAFT CDFT Euler Lagrange equations into SCFT fixed point relations. Within this Functional Field Theory, Wertheim TPT1 association free energy can be incorporated into the excess functional, with the mass action variables providing additive SCFT fields yet leaving the propagators unchanged. Linearization about the MF solution produces a reversible mapping for fluctuation corrections, suggesting a universal response that couples the single chain correlation kernel to the iSAFT direct correlation operator, allowing the inclusion of short range structure and association cooperatively beyond mean field. Long range interactions map similarly, showing how functional derivatives can yield SCFT field contributions, while second derivatives yield the long range blocks of the iSAFT correlation operator, providing a unified treatment of polymers. Mathematically, this framework aims to clarify that SCFT and polymer DFT solvers share similar diffusion based propagators, and how iSAFT hard sphere, attraction, and association modules contribute additional field and hessian terms. Specifically, this work provides a Legendre duality proof that places SAFT CDFTs and SCFT on the same MF foundation, and a reversible theoretical framework that transfers iSAFT correlation information into SCFT, enabling modular, quantitative modeling of associating, structured polymer systems.