Appendix: Self-Consistent Field Theory

PSCF solves self-consistent field theory (SCFT) equations for an incompressible mixture of any number of linear block copolymer species and point-like solvent molecular species.

SCFT for a liquid of flexible polymers is based on a mean-field approximation that allows us to predict properties of an interacting liquid by considering considering the behavior of a corresponding gas of noninteracting molecules in a spatially inhomogeneous chemical potential landscape. In what follows, let \(k_{B}T\omega_{\alpha}(\textbf{r})\) denote the chemical potential field for monomers of type \(\alpha\), which gives the free energy cost of placing such a monomer at location \(\textbf{r}\). Let \(\rho_{\alpha}(\textbf{r})\) denote the corresponding average volume fraction of monomers of type \(\alpha\) at position \(\textbf{r}\).

PSCF implements a version of SCFT for incompressible liquids in which each species of molecule occupies a well-defined volume, independent of composition and (modest) changes in pressure. In what follows, \(N_{i}\) is dimensionless measure of the volume per molecule (or size) of species \(i\), which is defined for both polymeric and point-like molecules as the ratio of the volume occupied by one molecule of that species to an monomer reference volume. The length of each block within a block copolymer is specified similarly, as a ratio of the block volume to a monomer reference volume.

In what follows, let \(C\) be the number of distinct monomer or solvent types in the system. Let \(P\) denote the number of polymer species and \(S\) be the number of solvent species. Here, we use a convention in which integer indices \(\alpha, \beta = 1, \ldots, C\) indicate monomer types, and indices \(i, j\) denote molecular species. Species indices are ordered with all polymeric species listed first, so that species index values in the range \(i, j = 1, \ldots, P\) denote polymeric species, and values in the range \(P+1,\ldots, P+S\) denote solvent species.

Polymer Species

Each polymeric species with species index \(i\) and overall chain length \(N_{i}\) is treated as a random walk characterized by a contour \(\textbf{R}(s)\), where \(s\) is a contour variable with a range \(0 \leq s \leq N_{i}\). For each such species, we define a pair of functions \(q_{i}(\textbf{r}, s)\) and \(q^{\dagger}_{i}(\textbf{r}, s)\). The functions \(q_{i}\) and \(q^{\dagger}\) are normalized partition functions for chain segments corresponding to contour variable domains \([0,s]\) and \([s,N_{i}]\), respectively, when the monomer at contour position \(s\) is constrained to position \(\textbf{R}(s) = \textbf{r}\). These functions obey a pair of modified diffusion equations

\[ \begin{align}\begin{aligned}\frac{\partial q_{i}}{\partial s} = -H_{\alpha(s)}q_{i}\\\frac{\partial q_{i}^{\dagger}}{\partial s} = +H_{\alpha(s)}q_{i}^{\dagger}\end{aligned}\end{align} \]

in which \(\alpha(s)\) is the monomer type of the block containing monomer \(s\) of polymer species \(i\), and in which \(H_{\alpha}\) is a linear diferential operator

\[H_{\alpha} = -\frac{b_{\alpha}^{2}}{6}\nabla^{2} + \omega_{\alpha}(\textbf{r})\]

in which \(b_{\alpha}\) is the statistical segment length for monomers of type \(\alpha\), and \(\omega_{\alpha}\) is the corresponding chemical potential field. These equations must be solved for \(0 < s < N_{i}\) subject to an initial condition

\[q_{i}(\textbf{r},s=0) = q^{\dagger}_{i}(\textbf{r},s=N) = 1\]

for all \(\textbf{r}\), and boundary condition requiring that \(q\) and \(q^{\dagger}\) be periodic functions of \(\textbf{r}\) with the periodicity of some specified Bravais lattice.

The quantity \(Q_{i}\) is a normalized overall partition function for chains of species \(i\), given by an integral

\[Q_{i} = \frac{1}{V}\int \! d\textbf{r} \; q(\textbf{r},s=N)\]

in which the integral is taken over one unit cell of a periodic structure, and \(V\) denotes the generalized volume per unit cell of a structure that periodic in \(D\) dimensions (i.e., the volume per unit cell for a 3D crystal, area per 2D unit cell for a 2D structure such as the hexagonal cylinder phase, and the length per unit cell in a 1D lamellar phase).

The probability of finding a specific monomer \(s\) of species \(i\) at position \(\textbf{r}\) is proportional to the product \(q_{i}(\textbf{r},s) q^{\dagger}_{i}(\textbf{r},s)\) of the constrained partition functions for the two chain segments that meet at monomer s. Let \(\rho_{\alpha}^{(i)}\) denote the contribution to the local volume fraction of \(\alpha\) monomers from monomers of a polymer species \(i\) that contains at least one block of monomer type \(\alpha\). This quantity is given by a product

\[\rho_{\alpha}^{(i)}(\textbf{r}) = \frac{\overline{\phi}_{i}}{N_{i}Q_{i}} \int\limits_{\alpha(s)=\alpha} \! ds \; q(\textbf{r},s) q^{\dagger}(\textbf{r},s)\]

in which \(\overline{\phi}_{i}\) is the average overall volume fraction of molecule species \(i\) within the mixture, and the integral with respect to \(s\) is taken only over blocks of monomer type \(\alpha\).

Solvent Species

Each solvent species \(i\) is associated with a specific monomer type \(\alpha\) and a volume \(i\). A “monomer” type that is assigned to a solvent species may or may not also be contained within one or more of the polymeric species.

In the single molecule problem for solvent species, the free energy penalty for a solvent molecule of monomer type \(\alpha\) to be located at position \(\textbf{r}\) is given by \(k_{B}T N_{i}\omega_{\alpha}(\textbf{r})\). This yields a solvent concentration \(\rho_{i}(\textbf{r}) \propto \exp(-N_{i}\omega_{\alpha}(\textbf{r}))\).

The normalized overall partition for such a point-like species is given by an integral

\[Q_{i} = \frac{1}{V}\int \! d\textbf{r} \; \exp(-N_{i}\omega_{\alpha}(\textbf{r}))\]

The contribution of solvent species \(i\) of type \(\alpha\) to the local volume fraction of \(\alpha\) is given by a ratio

\[\rho_{\alpha}^{(i)}(\textbf{r}) = \frac{\overline{\phi}_{i}}{Q_{i}} \exp(-N_{i}\omega_{\alpha}(\textbf{r}))\]

in which \(\overline{\phi}_{i}\) is the overall volume fraction of species \(i\) within the mixture.

The total volume fraction \(\rho_{\alpha}(\textbf{r})\) for each monomer type \(\alpha\) is simply given by the sum of contributions from all polymeric species that contain a block or blocks of type \(\alpha\) plus the contribution of any solvent of type \(\alpha\).

Self-Consistent Field Equations

The monomer chemical potential fields are given, within the standard approximation for excess free energies in terms of binary Flory-Huggins interaction parameters, as functions

\[\omega_{\alpha}(\textbf{r}) = \sum_{\beta = 1}^{C} \chi_{\alpha\beta} \rho_{\beta}(\textbf{r}) + \xi(\textbf{r})\]

in which \(\chi_{\alpha\beta}\) is a binary interaction parameter for interactions between monomers of types \(\alpha\) and \(\beta\), and \(\xi(\textbf{r})\) is a Lagrange multiplier pressure field. The interaction parameters in PSCF satisfy \(\chi_{\alpha\alpha}=0\) and (obviously) \(\chi_{\alpha\beta} = \chi_{\beta\alpha}\).

The field \(\xi(\textbf{r})\) must be chosen such that the monomer concentrations satisfy the incompressibility constraint

\[1 = \sum_{\alpha=1}^{C} \rho_{\alpha}(\textbf{r})\]

Thermodynamic Properties

The Helmholtz free energy \(f\) per monomer reference volume, as given in the output file, is given by a sum

\[\begin{split}\frac{f}{k_{B}T} & = \sum_{i=1}^{P+S} \frac{\overline{\phi}_{i}}{N_{i}} \left [ \ln ( \overline{\phi}_{i} / Q_{i}) - 1 \right ] \\ & - \frac{1}{V} \sum_{\alpha=1}^{C} \int \! d\textbf{r} \; \omega_{\alpha}(\textbf{r}) \rho_{\alpha}(\textbf{r}) \\ & + \frac{1}{2V} \sum_{\alpha, \beta =1}^{C} \chi_{\alpha\beta} \int \! d\textbf{r} \; \rho_{\alpha}(\textbf{r}) \rho_{\beta}(\textbf{r})\end{split}\]

Note that the sum over species in the first line is a sum over all species, including polymeric and solvent species, with different ways of defining \(Q_{i}\) for different types of molecule.

The corresponding chemical potential \(\mu_{i}\) for species \(i\) is given by

\[\frac{\mu_{i}}{k_{B}T} = \ln(\overline{\phi}_{i}/Q_{i})\]

The value given in the output file is \(\mu_{i}/k_{B}T\).

The macroscopic physical pressure \(P\) is computed from the identity

\[P = - \frac{f}{v} + \sum_{i=1}\frac{\mu_{i}\overline{\phi}_{i}}{N_{i}v}\]

in which \(v\) is the monomer reference volume and \(f\) is the Helmholtz free energy per reference volume. Note that \(f/v\) is the Helmholtz free energy per volume and \(\overline{\phi}_{i}/(N_{i}v)\) is the average number of molecules of species \(i\) per unit volume. The value given in the output file is the dimensionless value \(Pv/k_{B}T\).

Ensembles

PSCF can be carry out calculations using either canonical ensemble or grand-canonical ensemble.

In canonical ensemble a value of the overall volume fraction \(\overline{\phi}_{i}\) must be given for each species in the input parameter file, and values of chemical potential are computed from the solution.

In grand canonical ensemble, a value of the normalized chemical potential \(\mu_{i}/k_{B}T\) must be given for each species in the input parameter file, and average volume fractions for each species are computed.

In grand-canonical ensemble, values for the Lagrange multplier field \(\xi(\textbf{r})\) and the macroscopic pressure \(P\) are uniquely determined by the values for the chemical potentials.

In canonical ensemble, the value of the Lagrange multplier field \(\xi(\textbf{r})\) is defined only to within a arbitrary spatially homogeneous constant. As a result, the chemical potentials and the macroscopic pressure \(P\) are also undefined in this ensemble, unless an additional constraint is imposed. PSCF resolves this ambiguity by requiring, as a matter of convention, that the spatial average of \(\xi(\textbf{r})\) vanish. In this ensemble, PSCF also outputs values for the pressure, chemical potentials, and \(\omega\) fields that are all consistent with this convention for the average value of \(\xi\). Values for the Hemholtz free energy density of an incompressible liquid can, however, be shown to be independent of changes in the value of \(\xi\) by a homogeneous constant, and are thus independent of this choice of convention.