Monomer concentration / volume fraction fields (c fields)

Because PSCF uses a continuous chain model in which there are not actually any discrete monomers, we use the word "monomer" in what follows to refer to the amount of material that corresponds to one unit of contour length within a block of a block polymer. This is equivalent to the length of chain that occupies a volume within an incompressible liquid equal to one monomer reference volume \( v\).

The average volume fraction of monomers of type \( \alpha \) at any position \( {\bf r} \) within an incompressible mixture is denoted by \( \phi_{\alpha}({\bf r}) \). This volume fraction is related to the average concentration of such monomers, denoted by \( \langle c_{\alpha}({\bf r}) \rangle \), by the relation

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

in which \( v \) is the monomer reference volume. Monomer volume fractions are thus equivalent to monomer concentrations that have been non-dimensionalized by a monomer volume \( v \). Reflecting this relationship, volume fraction fields are referred to throughout the PSCF source code and in some command names as "c fields", where "c" denotes concentration.

A converged solution to an SCFT problem for an incompressible system must have average monomer concentrations for different monomer types that add to a total local concentration \( c_{0} \equiv 1/v \), so as to obtain volume fractions that satisfy a constraint

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

at every point in the system.

Monomer chemical potential fields (w fields)

Self-consistent field theory is based on a physical approximation in which monomer concentrations and thermodynamic properties are computed by considering a hypothetical reference system of non-interacting molecules that are subjected to a set of inhomogeneous chemical potential fields. In this ideal gas reference system, every monomer of type \( \alpha \) is subjected to a chemical potential field \( k_{B}T w_{\alpha}({\bf r})\) that is mean-field representation of the effect of interactions with other monomers. The dimensionless monomer fields denoted by \( w_{\alpha}({\bf r}) \) are referred to throughout the PSCF source and in the names of some commands as "w fields".

The volume fraction and chemical potential fields in any solution of an incompressible SCFT problem are related by a self-consistency condition

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

Here, \(\chi_{\alpha\beta}\) is a Flory-Huggins interaction parameter for interactions between monomers of types \(\alpha\) and \(\beta\), while \(\xi({\bf r})\) is a Langrange-multiplier field that is chosen so as to yield volume fraction fields that satisfy the incompressibility constraint given above (i.e., for which the sum of the local monomer volume fractions is equal to 1 at every point in the system).

Before attempting to solve an SCFT problem, each PSCF program must read a set of physical parameters from a parameter file and read in an initial guess for the w fields from an input field file. After a solution is obtained, the resulting converged w and c fields are usually written to output files.

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