Confinement and External Fields

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Overview

The SCFT iterators used by pscf_pc and pscf_pg allow users to define two different types of externally imposed fields to describe situations in which a polymeric material interacts with an inhomogeneous environment. A "mask" field is used to describe geometric confinement to a specified region in a manner that does not distinguish among different monomer types. "External" fields instead specify preferential potential energy biases that can induce preferential accumulation of particular monomer types in specific locations. The two features can be used together to allow SCFT calculations to describe situations in which a material is confined to a region, such a thin film and in which the confining surfaces exhibit preferential wetting, using external fields that are nonzero only in narrow regions near the confining surfaces to induce wetting. As described below, a mask field is implemented as a modification of the incompressibility constraint, whereas the external fields are implemented via addition of an external field term to the self-consistent field equation for each monomer type. The resulting framework is designed to be quite general and extensible by users. The current version of PSCF is distributed with a set tools that use this framework to simulate confinement of polymer to a thin film with chemically homogeneous but selective top and bottom surfaces.

We refer to a mask and a set of external fields collectively as "imposed fields" for a given SCFT calculation. These imposed fields can be defined by either of two methods.

• Mask and external fields can be defined in field files that are created separately by the user, which can be read into PSCF using any of several commands that are designed for this purpose, as described below and in the commands page.
• Mask and external fields may be generated within the PSCF software using an ImposedFieldsGenerator object that reads a few parameters specific to a particular type of calculation from the parameter file (e.g., the thickness of a film and parameters that describe the strength of preferential wetting) and generates an appropriate set of imposed fields.

PSCF is currently distributed with an ImposedFieldGenerator that can generate imposed fields for simulating thin films with preferential wetting. Thin films are simulated using a "supercell" that contains alternating regions of polymer and vaccum, as defined by a periodic "mask" field. This thin film field generator is described in detail on the Thin Films page.

The declared space-group symmetry of an SCFT calculation with imposed fields must be chosen so as to guarantee that the mask and external fields are invariant under all elements of the declared space group. The space group that best describes a given system is that which contains every symmetry operation shared by the mask, the external fields, and the polymer morphology itself; this is typically a lower-symmetry space group than that of the bulk polymer morphology that would be formed in the absence of any imposed fields. The choice of an appropriate space-group for a thin film morphology that is defined in a periodic super-cell is discussed in the Thin Films page.

Mask field

The iterators used in pscf_pc allows users to define a "mask" field that imposes geometric confinement within the unit cell by modifying the incompressibility constraint. A mask field is a dimensionless function of position, denoted here by \( m({\bf r}) \), that has a value very close to 1 throughout a region to which the polymer material is confined and a value very close to 0 throughout a region from which the material is being excluded. The incompressibility constraint for a system that is subjected to a mask is modified so as to require that

\[ v \sum_{\alpha=1} c_\alpha({\bf r}) = m({\bf r}) \]

at every point in the system, where \( m({\bf r}) \) is the mask field, \( c_{\alpha}({\bf r}) \) average is the number concentration of monomers of type \( \alpha \) at point \( {\bf r} \), \( v \) is the monomer reference volume, and the summation is over all distinct monomer types. Introduction of a mask provides a simple way to impose geometric confinement within a program that is designed for periodic systems. The mask can have any functional form as long as it satisfies the periodic boundary condition and the space-group symmetry of the system.

Let \( \phi_{\alpha}({\bf r}) \) denote the product

\[ \phi_{\alpha}({\bf r}) = v c_{\alpha}({\bf r}) \]

of monomer reference volume and average monomer concentration, which we refer to as the monomer volume fraction for monomers of type \( \alpha \). In a system with a mask, the sum of the monomer volume fractions at position \( {\bf r} \) is thus constrained to equal to \( m({\bf r}) \), rather than unity.

Let \( \overline{m} \) denote the spatial average of the mask function \( m({\bf r}) \) over the computational unit cell, given by the ratio

\[ \overline{m} = \frac{1}{V} \int d{\bf r} \; m({\bf r}) \quad, \]

where the integral is taken over the entire computational unit cell of total volume \( V \) that includes regions from which the polymer is excluded. The quantity \( \overline{m} \) is thus the fraction of the unit cell volume that is occupied by polymer material, while the product \( \overline{m} V \) is the total volume with the unit cell occupied by material.

A user-defined mask may be set by using a command to read the mask field from a file. The READ_MASK_BASIS reads a file in which the mask field is specified using the symmetry-adapted basis file format, thus defining a field that is guaranteed to exhibit the declared space group symmetry. The READ_MASK_RGRID command instead reads a file in which a mask is defined using the r-grid format as a list values on nodes of the relevant grid, for which the file format does not impose any symmetry. The READ_MASK_BASIS command must be used to initialize a mask for any SCFT calculation in which the iterator explicitly imposes a space-group symmetry, by using an internal representation of w-fields using a symmetry-adapated basis. Because the mask and external field features are implemented in the current version of PSCF (v1.2) only for the pscf_pc program (and not yet for pscf_pg), and the only iterator that is currently distributed for use with pscf_pc (the default Rpc::AmIteratorBasis class) does impose a space group symmetry, the READ_MASK_RGRID command thus actually cannot be used to set up an SCFT calculation in the current version of PSCF.

External fields

The pscf_pc program also allows users to define an external potential field associated with each type of monomer in the system. In what follows, let \( h_{\alpha}({\bf r}) \) denote the external field imposed on monomers of type \( \alpha \). When nonzero external fields have been set, the self-consistent field equation for monomers of type \( \alpha \) becomes

\[ w_{\alpha}({\bf r}) = \sum_{\beta} \chi_{\alpha\beta} \phi_{\beta}({\bf r}) + \xi({\bf r}) + h_{\alpha}({\bf r}) \quad, \]

in which \( \xi ({\bf r}) \) is the Lagrange multiplier local pressure field that imposes incompressibility, \( \chi_{\alpha\beta} \) is the binary Flory-Huggins interaction parameter for monomers of types \( \alpha \) and \( \beta \), and \( \phi_{\beta}({\bf r}) = v c_{\beta}({\bf r}) \) is the local volume fraction of monomers of type \( \beta \). The external fields can also assume any form as long as they satisfy the periodic boundary condition and the declared space-group symmetry of the calculation.

A list of external fields for all monomer types may be set by using a command to read these fields from a single file. The READ_H_BASIS command reads a file in which these fields are defined using a symmetry-adapted basis, creating fields that guaranteed to be invariant under all elements of a declared space group, whild READ_H_RGRID read a field file that uses the r-grid format, for which the format does not impose any symmetr. External fields for an SCFT calculation in which symmetry is imposed by the iterator must be symmetric, and so must be set using the READ_H_BASIS command. Because the only pscf_pc iterator that is provided with the current version of PSCF (v1.2) does impose a space-group symmetry, the READ_H_RGRID command actually cannot be used to set up an SCFT calculation for this version of PSCF.

ImposedFieldsGenerator objects

When studying a specific class of problems, such as polymer thin films, it is convenient for appropriate mask and external fields to be generated algorithmically within the PSCF software, based on numerical values of a few parameters that are provided by the user within the parameter file.
This can be done by the use of an ImposedFieldGenerator object that is designed such a class of problems.

An ImposedFieldsGenerator object may be created by the user by including including an optional block of the parameter file inside of the Iterator block, after all other Iterator parameters have been read. The first parameter inside of this ImposedFieldsGenerator block is "type". The software uses this "type" keyword to determine which imposed fields should be generated, what their functional form should be, and what parameters are necessary to define the imposed fields. Therefore, the implementation of the ImposedFieldsGenerator feature is entirely different depending on what keyword is provided as the value of the "type" parameter. All of the parameters required for the specified "type" of calculation appear after the type parameter.

An example is provided below for the "film" type, which is currently the only allowed value for the type parameter:

Iterator{
  epsilon      1e-6
  maxItr       100
  maxHist      50
  isFlexible   1
  ImposedFieldsGenerator{
    type         film 
    normalVecId  2
    interfaceThickness 0.2
    excludedThickness  0.4
    chiBottom[
                       0
                       10
    ]
    chiTop[
                       0
                       0
    ]
  }
}

In this example, the value "film" for the "type" parameter is followed by specialized 5 input parameters required to define a mask and external fields appropriate to a polymer thin film problem, named normalVecId, interfaceThickness, excludedThickness, chiBottom and chiTop. The "film" type of ImposedFieldsGenerator is described in detail in the Thin Films page.

One advantage of the use of an ImposedFieldsGenerator to programmatically generate a mask and external fields is that this method allows these imposed fields to be automatically recomputed during a calculation as needed. Regeneration of the fields can sometimes be required by changes in the unit cell parameters. The design of the ImposedFieldsGenerator class also allows some specialized parameters that are defined within the associated parameter file block to be declared to be parameters that can be varied during an SCFT parameter sweep, which generally requires recomputation of the mask or external fields before each SCFT calculation within the sweep. For example, for thin films, the chiBottom and chiTop parameters that control preferential wetting along the bottom and top surfaces to be varied during a linear parameter sweep, which requires repeated recomputation of the associated external fields.

Modified free energy calculation

In the presence of imposed mask and external fields, it is necessary to modify the free energy calculation to account for the fields. The free energy per monomer of material is given by

\[ \tilde{f} = \tilde{f}_{\rm id} + \tilde{f}_{\rm int} + \tilde{f}_{\rm ext} \]

where

\begin{eqnarray*} \tilde{f}_{\rm id} & = & \sum_{i=0}^{P+S-1} \frac{\overline{\phi}_{i}}{\overline{m} L_{i}} \left[ \ln \left( \frac{\overline{\phi}_{i}}{\overline{m} Q_{i}}\right) - 1 \right] - \frac{1}{\overline{m} V} \sum_{\alpha=0}^{C-1} \int \! d\textbf{r} \; w_{\alpha}(\textbf{r}) \phi_{\alpha}(\textbf{r}) \\ \tilde{f}_{\rm int} & = & \frac{1}{2 \overline{m} V} \sum_{\alpha, \beta =0}^{C-1} \chi_{\alpha\beta} \int \! d\textbf{r} \; \phi_{\alpha}(\textbf{r}) \phi_{\beta}(\textbf{r}) \\ \tilde{f}_{\rm ext} & = & \frac{1}{\overline{m}V} \sum_{\alpha=0}^{C-1} \int \! d\textbf{r} \; h_{\alpha}(\textbf{r}) \phi_{\alpha}(\textbf{r}) \quad. \end{eqnarray*}

Here, \( \phi_{\alpha}({\bf r}) = v c_{\alpha}({\bf r}) \) is the local volume fraction of monomer type \( \alpha \) at position r , while \( \overline{\phi_{i}} \) is the spatial average volume fraction of molecular species \( i \) across the entire unit cell. These quantities are defined such that the sum of values of \( \phi_{\alpha}({\bf r}) \) over all monomer types is equal to \( m({\bf r}) \) (as discussed above), while the sum of values of \( \overline{\phi}_{i} \) over all molecular species is equal to \( \overline{m} \).

There are three changes in the above equations relative to those that describe systems in the absence of imposed fields (see here).

• There is an additional term, \( \tilde{f}_{\rm ext} \), that contains the average energy per monomer arising from interaction of monomers with associated external potential fields. In any calculation that uses external fields, this property is output as "fExt" along with the other thermodynamic properties.
• The overall volume fraction \( \overline{\phi_i} \) is replaced by \( \overline{\phi_i}/\overline{m} \) within the expression for \( \tilde{f}_{\rm id} \). In a system containing a mask, in which the sum of species volume fractions \( \overline{\phi}_i \) is equal to \( \overline{m} \) rather than unity, the ratio \( \overline{\phi}_{i}/\overline{m} \) represents the fraction of the material volume that is occupied by molecular species \( i \).
• The volume \( \overline{m}V \) that is occupied by material is used to normalize spatial integrals, rather than the volume \( V \) of the computational unit cell, which includes regions from which polymer material is excluded by the mask. Note that the material volume \( \overline{m}V \) is also used to normalize the spatial integral used to define \( Q_i \) in a system with a mask.

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