Parameter File - pscf_1d

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The pscf_1d program allows the solution of the SCFT field equations for problems in which all fields depend only on a single coordinate in a planar planar (Cartesian), cylindrical or spherical geometry. This program may be used to treat a flat interface or thin film by using a planar geometry, or can treat curved interfaces and a cylindrical or spherical micelles by using a cylindrical or spherical coordinate by using a cylindrical or spherical geometry.

Example

The text shown below is a complete parameter file for a simple 1D SCFT calculation performed using the pscf_1d program:

System{
  Mixture{
     nMonomer   2
     monomers[
                1.0
                1.0
     ]
     nPolymer   2
     Polymer{
        nBlock  2
        blocks[
                0   0.5
                1   0.5
        ]
        phi     0.6
     }
     Polymer{
        nBlock  1
        blocks[
                0   1.0
        ]
        phi     0.4
     }
     ds   0.01
  }
  Interaction{
     chi(
           0    1    10.0
     )
  }
  Domain{
     mode     planar
     xMin     0.0
     xMax     8.0
     nx       401
  }
  Iterator{
     epsilon   0.00000001
  }
}

This particular example gives parameters for a calculation of a binary mixture of a symmetric AB diblock copolymer with a homopolymer of type A in a one-dimensional planar geometry.

File format

The overall format for a pscf_1d parameter file is shown below in skeleton form:

System{
  Mixture{ ...  }
  Interaction{ ...  }
  Domain{ ...  }
  Iterator#{ ... }
  Sweep#*{ ... }
}

The overall structure is similar to that of the example pscf_pc parameter file shown previously as an example, with one important difference: The System block of a pscf_1d parameter file does not contain a polymerModel parameter because the pscf_1d program does not offer a choice of bead or thread models, and instead always uses a thread model.

The purposes of each the the top-level sub-blocks of the System block are similar for different PSCF programs, and have been discussed previously. The same formats for the Mixture and Interaction blocks are used by all PSCF programs, and have also been discussed previously. We thus focus below on the contents of the remaining Domain, Iterator, and Sweep blocks.

Domain block

The domain block for pscf_1d describes the 1D spatial domain over which the SCFT equations should be solved, and the mesh used to discretize that domain. The parameter file syntax is described in detail here, and is also summarized below.

The example parameter file shown above was designed to describe a planar geometry. As another example, here is the Domain block for a simulation of a spherical micelle

Domain{
  mode   spherical
  xMax   8.0
  nx     201
}

This example omits an optional parameter xMin that, if present, would appear immediately before xMax. When xMin is omitted, it is set to 0.0 by default. Omitting the xMin parameter in a calculation that is performed using a spherical or radial coordinate system creates a spherical or cylindrical domain that includes the origin.

The full format for the Domain block is

Domain{
  mode*  enumeration ("planar", "cylindrical" or "spherical", "planar" by default)
  xMin*  real (optional, 0 by default)
  xMax   real
  nx     int
}

The meanings of the various parameters are given in tabular form below.

Label	Description
mode*	enumeration ("planar", "cylindrical" or "spherical", "planar" by default)
xMin*	Lower bound of normal coordinate. Optional and 0.0 by default.
xMax	Upper bound of normal coordinate.
nx	Number of grid points used to discretize the domain [xMin, xMax].

Each parameter is discussed in more detail below:

mode : The value of the optional "mode" parameter is an enumuration that specifies the coordinate system. The allowed input values for this parameter are the strings "planar", "cylindrical" or "spherical". Mode "planar" indicates a Cartesian coordinate system appropriate to membrane or thin film, in which the coordinate is measured normal to the film. Mode "cylindrical" indicates a cylindrical coordinate system for a system with cylindrical rotational symmetry, in which the 1D coordinate represents distance from an axis of rotation. Mode "spherical" indicates a spherical coordinate system for a system with spherical symmetry, in which the coordinate represents distance from the origin.

xMin and xMax : The parameters xMin and xMax specify the minimum and maximum values of the relevant spatial coordinate. The parameter xMin is optional, and is set to 0.0 default when this parameter is omitted. In "planar" mode, xMin and xMax are the minimum and maximum values of a Cartesian coordinate, defining a slit bounded bounded by constant values of this coordinate. In "cylindrical" and "spherical" mode, these are minimum and maximum values of a radial coordinate. In cylindrical or spherical mode, omitting xMin defines a simply connected circular or spherical domain of radius equal to xMax that includes the origin. If the parameter xMin is present and is assigned a positive value, such that \( 0 < xMin < xMax \), the problem will be solved in a cylindrical or spherical annular region of inner radius xMin and outer radius xMax.

nx : The parameter nx specifies the number of equally spaced grid points that will be used to discretize the spatial domain. Because the value of nx includes both end-points, the number of grid points is one greater than the number of spatial steps. The distance between grid points in the underlying finite difference discretization, denoteed by dx, is thus given by dx = (xMax - xMin)/(nx-1).

Boundary Conditions

The pscf_1d program solves the modified diffusion equation subject to von Neumann boundary "no flux" conditions that require derivatives of the propagators and concentration fields with respect to the relevant Cartesian or radial normal coordinate to vanish at the lower and upper bounds, xMin and xMax. In the special case of a cylindrical or spherical geometry that includes the origin, for which xMin = 0.0, the requirement of a vanishing radial derivative at x=0 is necessary to guarantee that the 3D gradient vector is continuous at the origin.

Iterators

The Iterator block provides data required by the iterator used to solve the nonlinear self-consistent field (SCF) equations.

The default iterator for pscf_1d is an Anderson-mixing algorithm that is implemented by a class named AmIterator. Because this is the default iterator, it may be chosen by including an iterator block in which the first line contains either the generic label "Iterator" or the specific class name "AmIterator".

The NrIterator class implements a Newton-Raphson iterator. Under favorable conditions, this may converge in only a few steps, but the cost for step is high, and grows rapidly with increasing number of grid points. This iterator is thus not efficient for large problems (more than approximately 1000 grid points).

The BinaryRelaxiterator class implements a simple relaxation iterator for AB systems with only two types of monomer, as described by F. Drolet and G.H. Fredrickson (Phys. Rev. Lett, vol 83, 4317, 1999).

More detailed documentation of the input format for each available pscf_1d iterator is given in the links in the table below:

Class	Description
AmIterator	Anderson-Mixing iterator (default)
NrIterator	Newton-Raphson iterator
BinaryRelaxIterator	Simple relaxation algorithm for AB (two monomer) systems, as described by Drolet and Fredrickson.

Sweep

The default sweep for pscf_1d is a linear sweep algorithm that can be invoked using a block that starts either with the generic label Sweep or the specific label LinearSweep. The required parameter file format has been described here.

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