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PSCF is a package of several programs that are designed to predict equilibrium properties of inhomogeneous states in polymer liquids. PSCF is designed primarily to predict structures and free energies of inhomogeneous equilibrium states of polymer liquids in which inhomogeneity is driven by a tendency of different constituents to to phase separate. The current version of PSCF can be used to perform either self-consistent field theory calculations or stochastic field-theoretic simulations that rely on a partial-saddle point approximation, as discussed below.

Coarse-grained model

All PSCF programs use a coarse-grained model that treat polymer conformations as biased random walks, and that treats the polymer material as incompressible. Systems modeled by PSCF may be either one-component block polymer melts or mixtures that may contain one or more polymer species. To allow simulation of solutions, such mixtures may also optionally contain small molecule solvent species that are treated as structureless particles that occupy specified volume per molecule. Each polymeric species may have a linear or acyclic branched structure. All molecular species in a system are constructed from a finite list of monomer types, in which each monomer type is assigned a statistical segment length. Interactions among different types of monomer are described by binary Flory-Huggins interaction parameters.

Self-consistent field theory (SCFT)

Self-consistent field theory (SCFT) is a form of mean field theory for polymeric liquids. In SCFT, conformational statistics of individual polymers are approximated by the solution of an auxiliary single-polymer problem in which each molecule is acted upon by a set of monomer chemical potential fields (referred to in PSCF documentation as w fields) that represent average effects of interactions with other molecules. The behavior of a single polymer in a specified set of w fields is described by the solution of a modified diffusion equation (MDE), which is a linear partial differential equation that describes the conformational statistics of a random walk polymer in an inhomogeneous environment. Solutions of the MDE can be used to compute average concentrations for monomers of each type as functions of position, yielding monomer concentration fields that are referred to in the PSCF documentation as "c fields". The solution of an SCFT problem must satisfy a set of nonlinear self-consistent field (SCF) equations that require that the w and c fields be related by an empirical description of the thermodynamics of mixing for the chosen set of monomer types.

We recommend the following references as introductions to the SCFT of inhomogeneous polymer liquids:

Matsen, M. W. "The standard Gaussian model for block copolymer melts," J. Phys.: Condens. Matter*, 2001, 14, R21.
Fredrickson, G. H. "The Equilibrium Theory of Inhomogeneous Polymers", Oxford University Press, 2006.
Arora, A., et al. "Broadly Accessible Self-Consistent Field Theory for Block Polymer Materials Discovery," Macromolecules, 2016, 49, 4675.

The reference by Arora et al. is particularly appropriate as an introduction to the implementation of SCFT in PSCF because it was written, in part, as a description of the earlier Fortran version of PSCF, and so uses conventions and working equations similar to those used in the PSCF documentation and source code.

Field-theoretic simulation (FTS)

Field-theoretic simulation (FTS) methods are a family of closely related stochastic simulation methods that sample an equilibrium ensemble of possible configurations of a set of fluctuating chemical potential fields ("w fields"). In contrast, SCFT instead identifies a single optimal field configuration. The theory underlying these methods is based upon a formally exact transformation of the partition function for a system of interacting particles from an integral over particle positions into a functional integral over configurations of a set of fluctuating auxiliary fields. There are two distinct classes of FTS methods, which we refer to here as fully fluctuating (FF-FTS) and partial-saddle point (PS-FTS) field theoretic simulations. Both are discussed below for context, but only the PS-FTS method is actually implemented in the current version of PSCF.

Fully-fluctuating field theoretic simulation (FF-FTS):

An exact formulation of the particle-to-field transformation leads to an expression for the partition function as a functional integral in which the fields and the integrand of the functional integral are complex-valued quantities. The complex Langevin sampling method is currently the best available algorithm for sampling the resulting ensemble. If complex Langevin simulations of the fully fluctuating theory reach a state of statistical equilibrium, they yield ensemble averages that are equivalent to those which would be obtained from simulations of the corresponding particle based model.

The following references discuss the transformation of the partition function into a functional integral, and the formulation of various types of "fully fluctuating" field thoeretic simulations (FF-FTS) that use this exact formulation:

Fredrickson, G. H. and Ganesan, V. and Drolet, F., "Field-theoretic computer simulations methods for polymers and complex fluids," Macromolecules 2002, 35, 16-39.
Fredrickson, G. H. "The Equilibrium Theory of Inhomogeneous Polymers", Oxford University Press, 2006.
Delaney, K.T. and Fredrickson, G.H., "Recent Developments in Fully Fluctuating Field-Theoretic Simulations of Polymer Melts and Solutions", J. Phys. Chem. B* 2016, 120 7615.

Complex Langevin simulations of the fully fluctuating theory are not yet implemented in PSCF.

Partial saddle-point field theoretic simulation (PS-FTS):

FTS methods that are implemented in the current version of PSCF are instead based on a so-called "partial saddle-point" approximation to the fully fluctuating field theory. This is an approximation in which a field component that enforces the incompressibility constraint, sometimes referred to as a pressure field, is treated at a self-consistent field level, while other field component or components that couple to composition fluctuations are allowed to fluctuate freely. The use of a self-consistent field approximation for the pressure field can be shown to be equivalent to approximating this field by its value at a partial saddle-point configuration, which is a field configuration in which the functional derivative of the field theoretic Hamiltonian with respect to this field component vanishes. We refer to any type of stochastic simulation that relies on this approximation as a partial saddle-point field theoretic simulation (PS-FTS).

When the partial saddle-point approximation is applicable, it leads to an expression for the partition function as a functional integral in which all of quantities of interest have real (rather than complex) values. Simulations that rely on the partial-saddle point approximation are generally substantially more efficient than complex Langevin simulations, at a modest cost in accuracy when used to simulate liquids of sufficiently long, strongly overlapping polymers. PSCF implements several Brownian dynamics and Monte-Carlo algorithms for sampling the ensemble defined by the partial saddle-point approximation. The Brownian dynamics method is generally more efficient that Monte-Carlo for large systems, and is recommended for most applications.

The PS-FTS method cannot be applied to as wide a range of different types of systems as the complex Langevin method. At the time of writing, it has only been applied in published work to systems with two monomer types (AB systems) with a positive Flory-Huggins interaction parameter. The implemention of the partial-saddle point approximation that is used in PSCF is designed to allow application to a slightly wider range of systems, including some systems with three monomer types, but still imposes some constraints on the allowed values of interaction parameters.

Work on the development and application of PS-FTS calculations for copolymer melts and polymer mixtures has been driven in recent years primarily by Mark Matsen and coworkers. The following reference provides an overview of development and application of this method up to 2021, with references to earlier work:

M.W. Matsen and T.M. Beardsley, "Field Theoretic Simulations for Block Copolymer Melts Using the Partial Saddle-Point Approximation", Polymers*, 2021, 13, 2437.

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