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PSCF v1.1
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Commands - pscf_fd (Up/Prev) Commands - pscf_pc (Next)
The COMPARE_HOMOGENEOUS command computes differences between properties of an inhomogeneous system of interest, as predicted by SCFT, and those of an homogeneous reference system. The command takes an integer mode parameter, as discussed below.
The integer mode parameter, which takes allowed values of 0, 1, and 2, determines the nature of the homogeneous reference system to which the system of interest is compared:
The only difference between modes 1 and 2 is how the composition of the reference system is computed. In both cases, the composition of the reference system is computed by iteratively solving for a list of species volume fractions for which the chemical potentials have values equal to those of the actual system. Under some circumstances of interest, the solution to this problem is not unique - in systems that are susceptible to phase separation into two or more phases, there can exist two or more compositions that yield the same list of chemical potentials. Because the composition is computed, when this is true, the calculated composition of the reference system may depend on the choice of an initial guess. The only difference between modes 1 and 2 is a difference in the choice of initial guess of composition used for this iterative solution. In mode 1 the initial guess is chosen so as to give a volume fraction for each equal to the local volume fraction for all mononers of that species at the last grid point in the SCFT solution. In mode 2, the initial guess is instead chosen so as to give a composition equal to that found at the first grid point.
In mode 0, an output report is written to standard output that includes the non-dimensional free energy per monomer "f(homo)" of the homogeneous reference system, difference "delta f" between values of free energy per monomer of the actual and reference system. These an all other free energies are output using energy units with \( k_{B}T = 1 \). Free energies per monomer are computed by multiplying the free energy density (total free energy divided by total system volume) by the monomer volume.
The output for mode 0 also contains values of the chemical potential "mu(homo)" and the species volume fraction "phi(homo)" for each polymer and solvent species in the homogneous reference system.
For calculations that are performed using a Cartesian coordinate system, the value of the total system volume V(tot) is is actually the length xmax - xmin of the calculation domain along a direction perpendicular to the interface, or the volume per unit interfacial area. Correspondingly, the values of "Phi (ex)" and "F (ex)" reported in this case are actually values of free energy per unit interfacial area, in thermal energy units, while values of "delta V" for each species are excess volumes per unit area.
For calculations that are performed using a cylindrical coordinate system, the value of the total system "volume" is actually the area of the domain, while values of excess free energies and excess volumes for individual species are also reported as excess free energy per unit length or excess volume per unit length (i.e., excess area) along the axis of rotation of the cylindrically symmetric system.
In modes 1 and 2, the first part of the output report contains non-dimensional values of the following overall thermodynamic properties:
Total excess properties are computed by taking the difference between the value in the inhomogemeous system of interest and that of a homogeneous reference system with the same volume and equal values for all chemical potentials. The value of F (ex) is equal to the value of "delta f" times "V(tot)/v". The value of Phi (ex) is equal to -1 times "delta p" times "V(tot)/v".
The following properties of specific species are also output for each species:
The total excess volume associated with a species can be interpreted as a spatial integral of an excess volume fraction. In an incompressible system, the sum of values of delta V for different species must add to zero.
Values of system volumes and total excess properties have the same interpretation as for mode 0: In Cartesian coordinates, "V(tot)" is actually a length (volume per area), and excess properties are computed per unit area, whereas in cylindrical coordintes "V(tot)" is actually an area, while excess properties are reported as values per unit length along the axis of symmetry.
Properties of spherical and cylindrical micelles can be computed by performing a SCFT calculation in spherical or cylindrical coordinates for a cylindrical or spherical geometry that includes the origin in which a micelles is surrounded by a solvent rich region is large enough to be homogeneous near the outer edge of the computational domain (i.e., near the last grid point). Excess properties of micelles can be computed by invoking the COMPARE_HOMOGENEOUS command 1 to obtain a homogeneous reference system with a composition similar to that of the solvent-rich region that surrounds the micelle.
Interfacial properties can be computed by simulating a system that contains two bulk-like phases separated by an interface, and using mode 1 or 2 to compare it to a homogeneous reference system of equal chemical potentials.
In the case of a flat interface that is treated in Cartesian coordinates, the interfacial tension is given in thermal energy units by the reported value of "Phi (ex)". For such a flat interface, the same value for this excess grand-canonical free energy should be obtained using either mode 1 or mode 2, because the conditions for mechanical equilibrium require that the same value of pressure (and thus the same grand-canonical free energy density) must be obtained in the two coexisting bulk phases. To compute other excess interfacial properties (e.g., excess Helmholtz free energy and excess volume per interfacial area for each species), however, the user to choose a convention to define a Gibbs dividing surface, and to perform some calculations that are not automatically perfomed by the COMPARE_HOMOGENEOUS command.
Computation of any excess property for a curved cylindrical or spherical interface, including interfacial tension, requires the user to define a Gibbs dividing surface and perform some additional calculations beyond those performed by this command.
As a first step in a computation of excess interfacial properties that require the definition of a Gibbs dividing surface, it is sometimes useful for a user to first invoke COMPARE_HOMOGENEOUS command twice with different mode parameter values of 1 and 2. The resulting outputs provide values of species volume fractions and free energy densities in homogemeous systems with compositions that correspond to those of the two bulk phases that are separated by the interface.
Commands - pscf_fd (Up/Prev) Commands - pscf_pc (Next)
1.9.5