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Below, we summarize a few of the working equations underlying the form of self-consistent field theory (SCFT) used by PSCF. This is not intended to be an introduction to the field theory itself, and we do not provide a derivation of the SCFT equations. We assume readers are familiar with the principles of polymer physics and statistical mechanics. Several recommended references for information about SCFT are given here.

Model and notation

PSCF solves the SCFT equations for an incompressible mixture of any number of block polymer species and point-like solvent molecular species.

In what follows, we consider a system with \( C \) distinct monomer types, \( P \) block polymer species, and \( S\) solvent species. We use a convention in which the symbols \( \alpha, \beta = 0, \ldots, C-1\) are used as indices for monomer types, symbols \( a, b \) are used to denote indices for blocks within a specific block polymer species, and symbols \( i, j\) are used to denote molecular species indices. In these notes, we use a convention in which species indices are ordered with all polymeric species listed first, so that species index values in the range \( i, j = 0, \ldots, P-1\) denote polymeric species, and values in the range \( P,\ldots, P+S-1 \) denote solvent species. A zero-based indexing scheme for indices is used throughout the notes to facilitate comparison to C/C++ source code.

To characterize inhomogeneous states, we consider the average number concentration \(\langle c_{\alpha}({\bf r}) \rangle\) for monomers of each type \( \alpha \) at each position \( {\bf r} \). In a model for an effectively incompressible liquid in which each monomer is assumed to occupy a volume \( v \), we may define a corresponding volume fraction

\[ \phi_{\alpha}(\textbf{r}) = v \langle c_{\alpha}({\bf r}) \rangle \quad, \quad\quad\quad\quad\quad (A.1) \]

for each monomer type, where \( v \) is a monomer reference volume that, by convention, has the same value for all monomer types. Incompressibility is imposed by requiring that

\[ 1 = \sum_{\alpha=0}^{C-1} \phi_{\alpha}({\bf r}) \quad\quad\quad\quad\quad (A.2) \]

at every position \( {\bf r} \) within the problem domain.

Self-consistent field approximation

SCFT for a liquid of flexible polymers is based on a mean-field approximation that allows us to approximately predict properties of an interacting liquid by considering the behavior of a corresponding inhomogeneous gas of noninteracting molecules. We refer to this hypothetical non-interacting gas as the ideal-gas reference system. The ideal gas reference system must have the same chemical composition as the real system of interest and an inhomogeneous average concenration \( \langle c_{\alpha}({\bf r}) \rangle \) for each monomer type \( \alpha \) that is equal to that in the real system. Inhomogeneous monomer concentrations are induced in this reference system by a set of potential energy fields that couple to different types of monomer: Each monomer of type \( \alpha \) at position \( {\bf r} \) within this ideal gas is subjected to a potential \( k_{B}Tw_{\alpha}(\textbf{r}) \). In a self-consistent field approximation, this potential represents a free energy cost for placing a monomer of a specified type at a specified position, due to interactions among monomers within an inhomogeneous structure.

In SCFT, the monomer chemical potential fields (or "w fields") \( w_{0}, \ldots, w_{C-1} \) are assumed to depend upon volume fraction fields in a manner that reflects the effects of interaction among species in the mixture. PSCF currently uses a simple approximation for this dependence that is based on an inhomogeneous generalization of the parameterization of excess free energy used in the Flory-Huggins theory of homogeneous polymer mixtures. In this approximation, a solution to the SCFT problem is required to satisfy the self-consistent field (SCF) equations

\[ w_{\alpha}({\bf r}) = \sum_{\beta=0}^{C-1} \chi_{\alpha\beta} \phi_{\beta}({\bf r}) + \xi({\bf r}) \quad\quad\quad\quad\quad (A.3) \]

for all \( \alpha = 0,\ldots,C-1 \), in which \( \chi_{\alpha\beta}\) is a binary Flory-Huggins dimensionless interaction parameter, defined such that \( \chi_{\alpha\beta} = \chi_{\beta\alpha} \). Here, \( \xi({\bf r}) \) is a dimensionless Lagrange multiplier pressure field that must be chosen so that the average monomer concentrations in the ideal gas reference system satisfy the incompressibility constraint given above.

In SCFT, average monomer concentration and volume fraction fields are computed by computing these properties for the ideal gas reference system. An iterative solution of an SCFT problem thus starts with an initial guess for the w fields for all monomer types, followed by iterative adjustment of these fields until the w fields and average monomer concentrations (or volume fractions) computed using these fields as inputs to the ideal gas model satisfy the above SCF equations and the incompressibility constraint.

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