Parameter File - Domain Block (Prev/up) Periodic domains (pscf_pc and pscf_pg) (Next)

The pscf_1d program solves a self-consistent field problem in which all fields are functions of a single coordinate, in Cartesian, cylindrical, or spherical coordinates. This coordinate could thus be either a single Cartesian coordinate, to describe a system with a planar geometry, or a single radial coordinate, to describe a system with cylindrical or spherical symmetry. The domain block for the pscf_1d program specifies the range of values of this coordinate over which the SCFT equations should be solved, and the number of nodes of a regular mesh that is used to discretize that domain.

Examples

Here is an example of the Domain block of a parameter file for a problem with a 1D planar geometry:

  Domain{
     mode     planar
     xMin     0.0
     xMax     8.0
     nx       401
  }
}

Here is an Domain block that might be used for a simulation of a spherical geometry, such as that used to simulate 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.

Parameter file format

The full format for the pscf_1d Domain block is

Domain{
  mode*  string ("planar", "cylindrical" or "spherical", "planar" by default)
  xMin*  real (optional, 0.0 by default)
  xMax   real
  nx     int
}

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

Label	Description
mode*	enumeration ("planar", "cylindrical" or "spherical", "planar" by default)
xMin*	Lower bound of coordinate. Optional and 0.0 by default.
xMax	Upper bound of 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 string 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. 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.

The parameter xMin is optional, and is set to 0.0 default if the parameter is omitted. In cylindrical or spherical mode, omitting xMin defines a simply connected circular or spherical domain of radius equal to xMax that includes the origin. In cylindrical or spherical model, if the parameter xMin is present and is assigned a positive value, such that 0 < xMin < xMax, the problem will instead be solved within 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 evenly spaced grid points, denoted by dx, is thus 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 that derivatives of the propagators and concentration fields with respect to the relevant Cartesian or radial normal coordinate must vanish at the lower and upper bounds, xMin and xMax.

In the special case of a cylindrical or spherical domain 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 gradient vector is actually continuous at the origin.

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