Model and Parameters (Prev / Up) Polymer Species (Next)

Physical model

All programs in the PSCF package are designed to treat an incompressible liquid mixture that may contain one or more polymer species and any number of small-molecule solvent species. Each polymer species in such a mixture may be either a linear block polymer or an acyclic branched block polymer. Hompolymers are treated as a special case of a block polymer with only one block. Solvent species, if present, are treated as point-like particles that occupy a specified volume per molecule.

In the pscf_pc and pscf_pg programs for periodic domains, conformations of individual polymers may be modelled using either a continuous "thread" model or a discrete "bead" model. The thread model idealizes blocks within a block polymer as continuous random walks. The bead model instead represents each block as a sequence of a finite number of discrete beads connected by harmonic springs. The pscf_1d program for one-dimensional SCFT problems can only use the continuous thread model.

Each polymer species is constructed from chemically homogeneous blocks that each contain monomers of a specific type. Each block within a block polymer is thus assigned a monomer type (identified by an integer index) and a block length. The block length is given by a real (i.e., floating point) number in the continuous thread model and by an integer number of beads in the discrete bead model. In either model, the volume occupied by a block within an incompressible liquid is given by the product of its length and a monomer reference volume. By convention, PSCF requires that the monomer reference volume be the same for every monomer type.

Each small-molecule solvent species is assigned a monomer type index and a dimensionless "size" parameter. The volume occupied by one molecule of a solvent species is given by the product of the size parameter and the monomer reference volume.

Excess free energies arising from interactions between monomers of different types are described using binary Flory-Huggins interaction parameters.

Input parameters

The physical parameters required to describe a model for either a SCFT or PS-FTS calculation thus include:

The number of different monomer types present in the system, and a value for the statistical segment length of each monomer type.
A description of each polymer species that specifies the monomer type and length of each block, and (for branched polymers) a description of how the blocks are connected.
A specification of the monomer type and size (i.e., volume) of each small-molecule solvent species.
A value for either the volume fraction or chemical potential of each polymer and solvent molecular species.
A value for a binary Flory-Huggins interaction parameter \( \chi_{ij} \) for interactions between monomers of types \( i \) and \( j \) for each distinct pair of monomer types.
Parameters that describe the unit cell (for periodic structures) or spatial domain of interest.

To perform a PS-FTS calculation, a user must also input a value for the monomer reference volume, i.e., the volume per monomer or per unit contour length in an incompressible liquid. This parameter is, however, usually not needed as an input to a SCFT calculation.

In addition to these physical parameters, users of PSCF must provide values for computational parameters required by a variety of different numerical algorithms. Among these are parameters that define a spatial discretization for the unit cell or spatial domain by, e.g., defining the number of grid points along each direction of a computational mesh. Calculations that use the continuous thread model for polymer conformations also require a target value for a contour length step size, denoted by ds, that is used to discretize the contour of each block within algorithms that compute statistical properties for individual polymers (i.e., within the solution of the modified diffusion equation). Specific numerical algorithms, such as iteration algorithms used in SCFT or stochastic sampling algorithms used for PS-FTS, require a variety of other more specialized parameters and options.

Energy units

The PSCF documentation, file formats and source code all use "thermal" units for energy. In these units, we take \( k_{B}T = 1\) to define values of quantities with units that involve energy, where \( k_{B} \) is Boltmann's constant and \( T \) is absolute temperature. For example, the value for the chemical potential \( \mu \) of each molecular species is actually given throughout the source code and input and output files as a value of the ratio \( \mu / k_{B}T \).

Length units

PSCF allows the user to use any unit of length they wish, as long as the same choice of unit is used for all input parameters. Users are not to explicitly declare their choice of a unit of length, but are instead responsible for providing input parameter values that have been assigned using some consistent but unspecified choice of length unit.

The only parameters required for a standard SCFT calculation that have units involving length are the values of the monomer statistical segment lengths, and values (or initial guesses) for the dimensions of a spatial domain or perioidic unit cell for the system of interest. In addition, PS-FTS calculations require a value for the monomer reference volume, denoted by \( v \), which has units of volume, or length cubed.

Users may initialize a calculation using input files in which values of these parameters are all defined either using some standard microscopic length unit, such as Angstroms or nanometers, or using units in which values for these quantities are specified using some user-defined system of non-dimensional units, e.g., by defining the unperturbed root-mean-squared of a specific linear polymer species to be equal to unity. In the continuous chain model, conventions for units of length are related to a separate arbitrary choice regarding how much polymer backbone corresponds to one "monomer", or to one unit of contour length. Some common choices of conventions for units of physical length and polymer contour length are discussed in detail in a separate page.

If a user supplies input parameters values that were defined using a consistent choice of a unit of length, output values will automatically be reported using the same units. For example, in SCFT simulations of periodic microstructures in which one or more dimensions of the crystal unit cell are iteratively ajdusted so as to minimize the free energy density, output values for the optimized unit cell parameters will be reported using the same unit of length as those used to define the monomer statistical segment lengths that were provided as inputs.

Model and Parameters (Prev / Up) Polymer Species (Next)