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The thermodynamic properties obtained for a converged SCFT solution are output by PSCF in a standard format that we refer to a thermo file block. This information can be written to a file in any of the following ways:
- A thermo file block appears within the log information that is written to standard output during execution of an ITERATE command.
- A thermo file block is included within each of the state files that are created by a SWEEP command.
- A thermo file block can be explicitly written to a file by the WRITE_THERMO command after using the ITERATE command to obtain a converged field configuration.
In the remainder of this page, we first describes the format of a thermo file block, and then present the mathematical equations used by SCFT to compute thermodynamic properties.
Thermo file block format
An example of the thermo file block format produced by an SCFT calculation for a periodic system is shown below:
The meaning of each entry in this file block are described below:
Helmholtz free energy
The first line of a thermo file block contains the label "fHelmholtz" followed by a value. The value of "fHelmholtz" is a non-dimensional form of the Helmholtz free energy, which is also denoted in equations by the symbol \( f \). Specifically, the value of this quantity is the Helmoltz free energy per monomer reference volume in thermal energy units, given by
\[ f \equiv \frac{F v}{V k_{B}T} \quad, \]
where \( F \) is total Helmholtz free energy, \( V \) is total volume, and \( v \) is monomer reference volume. A more detailed discussion is given below.
Pressure and grand-canonical free energy
The value given for the "pressure" in the second line of this example is a non-dimensionalized thermodynamic pressure that is computed from the value of the grand-canonical free energy density. The reported value, denoted by \( p \) in equations, is given by
\[ p = -\frac{\Phi v}{V k_{B}T} \quad, \]
where \( \Phi \) is the total grand-canonical free energy. In a homogeneous material, \( \Phi = -PV \), where \( P \) is macroscopic pressure, so this also yields \( p = Pv/(k_{B}T) \).
Ideal and interaction free energies
The next two properties, labeled "fIdeal" (also denoted by \( f_{\rm id} \)) and "fInter" (also denoted by \( f_{\rm int} \)) are "ideal" and "interaction" contributions to "fHelmholtz" (or \( f \)). Physically, \( f_{\rm id} \) is a "ideal" contribution that arises from polymer configurational entropy, and \( f_{\rm int} \) is an "interaction" contribution that arises from the excess free energy of interactions between different monomer types, as parameterized in PSCF in terms of binary interaction parameters. Mathematical definitions are given below.
In calculations with an external field, a third term will appear after these two, denoted by a label "fExt". This represents the energetic contribution \( f \) that arises directly from interactions between the monomer species and applied external potential fields. Generally, in a system that is subjected to external potentials \( f = f_{\rm id} + f_{\rm int} + f_{\rm ext} \), but \( f_{\rm ext} = 0 \) in models with no external potentials.
Polymer and solvent species properties
The next two sections of the example given above are labeled "polymers" and "solvents". These are lists with one row per chemical species, in which each row gives the species index, the overall volume fraction "phi" and the chemical potential "mu" of one molecular species, in that order. Chemical potential values are reported using thermal energy units, so the numerical value reported for species \( a \) is actually equal to \( \mu_{a} / k_{B}T \).
Lattice parameters
The final section, labeled "cellParams", lists the lattice parameters for the converged SCFT solution. This section appears in the output of SCFT calculations for periodic systems performed using pscf_rpc or pscf_rpg programs. It does not appear in the output of calculations for one-dimensional systems performed using pscf_r1d, since the domain size is fixed such 1D calculations. In pscf_rpc and pscf_rpg calculations, some or all of the lattice parameters may be flexible, and are adjusted during the SCFT calculation so as to minimize the free energy density. In this case, the converged values of lattice parameters values that are allowed to vary are important results of the calculation.
If an SCFT calculation on a periodic structure is performed using a rigid unit cell, with no adjustable lattice parameters, lattice parameter values are still reported, but these values are simply equal to those provided as input to the calculation. Input values for lattice parameters are normally provided within the header block of the field file used to provide an initial guess for the monomer chemical potential fields (the w fields).
Lattice parameters are reported in a thermo file block using whatever system of length units was used in the input field and parameter files.
State files
A "state" file is a type of file that is created for each state (i.e., each set of input parameters) during a parameter sweep performed by a SWEEP command. Each such state file is comprised of a parameter file block followed by a thermo file block. The parameter file block contains the input parameters used for the SCFT calculation at that state, with the Sweep block excluded, formatted in the nested curly-bracket syntax of a PSCF parameter file. The thermo file block contains thermodynamic properties computed for the converged solution.
Each state file thus gives all of the input and output parameters for a single calculation. Names of state files that are generated by a SWEEP command contain an integer index followed by the file name suffix ".stt".
The contents of a state file can also be generated for an individual SCFT calculation (as opposed to a parameter sweep) by invoking the command WRITE_PARAM followed by the command WRITE_THERMO, while using the same file name argument for both commands. In the command file, this might look like
The WRITE_THERMO command is designed to append to the end of an existing file rather than overwrite the file in order to allow both blocks to be written to a single file.
Helmholtz free energy
The remainder of this page discusses the mathematical formulas used by PSCF to compute SCFT thermodynamic properties, and the relationship to the non-dimensionalized quantities reported in a thermo file file. We focus on this page on models in inhomogeneous structures occur in a homogeneous environment, without either imposed boundaries or external fields. Modifications required to describe the effects of an inhomogeneous environment are described on separate page.
In what follows, let \( F \) denote the total Helhmoltz free energy (or free energy per unit cell for a periodic system), and let \( V \) denote corresponding total volume (or volume per unit cell for a periodic system).
The free energy that is reported as "fHelmholtz" in the PSCF thermo file block format is the free energy per monomer reference volume in thermal energy units, also denoted in equations by the symbol \( f \equiv (F v)( V k_{B} T ) \). For a system in a homogeneous environment (i.e., in a model with no mask or external fields), this quantity can be expressed as a sum
\[ f = f_{\rm id} + f_{\rm int} \quad, \]
in which the ideal and interaction contributions are given by
\begin{eqnarray*} f_{\rm id} & = & \sum_{a=0}^{P+S-1} \frac{\overline{\phi}_{a}}{N_{a}} \left [ \ln \left (\frac{\overline{\phi}_{a}}{Q_{a}} \right ) - 1 \right ] - \frac{1}{V} \sum_{i=0}^{C-1} \int \! d\textbf{r} \; w_{i}(\textbf{r}) \phi_{i}(\textbf{r}) \\ f_{\rm int} & = & \frac{1}{2V} \sum_{i, j =0}^{C-1} \chi_{i j} \int \! d\textbf{r} \; \phi_{i}(\textbf{r}) \phi_{j}(\textbf{r}) \quad. \end{eqnarray*}
The sum over species in the first term in the definition of \( f_{\rm id} \) is a sum over all species, including polymeric and solvent species.
The symbol \( N_{a} \) is used here to denote the number of monomers or monomer volumes per molecule of species \( a \), given by the ratio of total molecular volume to monomer reference volume. For a block polymer species, this quantity is given by the sum of the lengths (in the thread model) or number of beads (in a bead model) of all blocks in a molecule of that species. For a solvent species, this quantity is equal to the "size" parameter given in the parameter file. The overall volume fraction of species \( a \), denoted by \( \overline{\phi}_{a} \) is given by
\[ \overline{\phi}_{a} = \frac{M_{a}N_{a}v}{V} \quad, \]
where \( M_{a} \) is the number of molecules of species \( a \) in volume \( V \).
In the special case of a one-component block polymer melt for which the input parameters are given using the theorists convention that overall chain length is equal to 1.0, the monomer reference volume is equal to the volume per chain. In this case, the value reported for "fHelmholtz" is also the free energy per chain in thermal energy units, with \( k_{B}T = 1 \).
Pressure and grand-canonical free energy
The "pressure" property reported in a PSCF thermo block is a non-dimensionalized thermodynamic pressure that is computed from the value of the grand-canonical free energy density. Consider a system that contains a total of \( M_{a} \) molecules of molecular species \( a \), and let \( \mu_{a} \) denote the chemical potential of species \( a \), for each polymer and solvent species. Let \( \Phi \) denote the so-called "grand" or "grand-canonical" free energy, defined as
\[ \Phi \equiv F - \sum_{a} \mu_{a}M_{a} \quad, \]
in which the sum is taken over all polymer and solvent molecular species in the system. We define a thermodynamic pressure \( P \) via the identity
\[ \Phi = -PV \quad, \]
where \( V \) is the system volume. The property reported by PSCF as "pressure", which is denoted here by the symbol \( p \), is actually the value of the ratio
\[ p = \frac{Pv}{k_{B}T} = -\frac{\Phi v}{V k_{B} T} \quad. \]
The value of pressure reported by PSCF is thus \( -1 \) times the grand-canonical free energy per monomer reference volume, in thermal energy units. This is related to the corresponding non-dimension Helmholtz free energy per monomer by the identity
\[ p \equiv \frac{Pv}{k_{B}T} = - f + \sum_{a} \frac{ \mu_{a} \overline{\phi}_{a}}{N_{a}k_{B}T} \quad, \]
where we have used the identity \( \overline{\phi}_{a}/N_{a} = M_{a}v/V \).
Open and closed ensembles
PSCF allows the user to choose either an open or closed statistical ensemble for each molecular species in the mixture, by allowing a user to enter a value for either the overall volume fraction for that species, denoted here by \( \overline{\phi}_{a} \), or a value for the chemical potential, denoted here by \( \mu_{a} \). Values of species chemical potentials in both input and output files used by PSCF, which are denoted indicated the label "mu", are given in thermal energy units, and so correspond to values of the ratio \( \mu_{a}/k_{B}T \).
The value of the chemical potential is computed for each species that is treated in a closed ensemble (i.e., any species for which a volume fraction "phi" is specified in the parameter file) by computing
\[ \frac{\mu_{a}}{k_{B}T } = \ln \left ( \frac{\overline{\phi}_{a} }{ Q_{a} } \right ) \quad, \]
where \( Q_{a} \) is the molecular partition function obtained with the converged w fields. The value of \( \overline{\phi}_{a} \) is computed for every species that is treated in an open ensemble (by specifying a value for the chemical potential "mu" potential rather than volume fraction in the parameter file) is obtained by solving the same equation for \( \overline{\phi}_{a} \), while treating \( \mu_{a} \) as a constant parameter.
Indeterminancy of pressure in canonical ensemble
In any open ensemble (grand-canonical or mixed), for each choice of values of the species chemical potential fields and other input parameters, there exists a unique Lagrange multiplier field \( \xi(\textbf{r}) \) that the satisfies SCF equations and the incompressibility constraint. In such open and mixed ensembles, there also exists a corresponding unique value for the macroscopic pressure \( P \).
In the special case of a completely closed (canonical) ensemble, however, the value of the Lagrange multplier field \( \xi(\textbf{r}) \) is not uniquely determined by the SCF equations and incompressibility constraint. In PSCF, a calculation is performed in canonical ensemble if the parameter file provides a volume fraction, rather than a chemical potential, as an input parameter for every polymer and solvent species. In this case, the relevant working equations for an incompressible system can be shown to define \( \xi(\textbf{r}) \) only to within shifts by an arbitrary spatially homogeneous constant. That is, for any solution of this system of equations, there exist an infinite number of other solutions that are related by a spatially homogeneous shifts in the \( \xi \) field, of the form
\[ \xi({\bf r}) \rightarrow \xi({\bf r}) + C \]
where \( C \) is any constant. Such a shift corresponds to a shift in all of the w-fields by a constant
\[ w_{i}({\bf r}) \rightarrow w_{i}({\bf r}) + C \]
for all monomer types. Such a homogeneous shift of all of the w fields can be shown to cause no change in the computed values of the monomer concentration fields ("c fields") or the Helmholtz free energy, but does change values of chemical potentials and the non-dimensional pressure \( p \). As a result, values for chemical potential and pressure are not uniquely defined in this ensemble, unless an additional mathematical constraint is on the SCFT equations to obtain a unique solution for the constraint field \( \xi({\bf r}) \).
Different conventions for this additional constraint on \( \xi({\bf r}) \) are used in different PSCF programs. The implementations of the pscf_rpc and pscf_rpg programs for periodic microstructures obtain equations with a unique solution by requiring, as a matter of convention, that the spatial average of \( \xi(\textbf{r}) \) must vanish in canonical ensemble. The pscf_r1d code for one-dimensional problems instead uses a convention that requires that, in canonical ensemble, the value of \( w_{i}({\bf r}) \) must vanish for the last monomer type ( \( \alpha = C-1 \)) at the last grid point of a one-dimensional mesh.
A homogeneous shift in the \( \xi({\bf r}) \) field by a constant \( C \) physically to changes in the hydrostatic pressure to which a closed system is subjected by an amount \( \Delta P = k_{B}T C / v \), or to a change in dimensionless pressure \( p \) of the form
\[ p \rightarrow p + C \]
It is straightforward to show using macroscopic classical thermodynamics that, for an incompressible system in which each species \( a \) occupies a constant molecular volume \( N_{a} v \) (corresponding to a constant partial molar volume), such a change in pressure of a closed, isothermal system causes trivial changes in molecular chemical potentials of the form
\[ \mu_{a} \rightarrow \mu_{a} + N_{a} v C \]
for each species. Such a change in pressure also causes a change in the Gibbs free energy \( G = F + PV \), but can be shown to leave the Helmholtz free energy unchanged, since
\[ \left . \frac{\partial G}{\partial P} \right |_{T} = V \quad\quad \left . \frac{\partial F}{\partial P} \right |_{T} = \left . \frac{\partial (G-PV)}{\partial P} \right |_{T} = - P \left . \frac{\partial V}{\partial P} \right |_{T} = 0 \quad, \]
for an incompressible system of constant composition.
Converting between ensembles
When an SCFT calculation is performed in canonical ensemble, each PSCF program outputs converted values for the pressure, chemical potentials, and \( w \) fields that are all consistent with the arbitrary convention used in that program to define a unique solution for \( \xi \). Numerical results for these quantities are thus sensitive to this choice of convention. Converged results for the Helmholtz free energy and volume fraction fields are, however, independent of this choice of convention.
Results of any calculation that is initially carried out in canonical ensemble can be exactly reproduced in a subsequent calculation that is performed in a mixed or grand-canonical ensemble by using the values of some or all of the chemical potentials that were output by the canonical calculation as inputs to a calculation performed in mixed or grand-canonical ensemble.
Attempting to this reverse procedure, however, yields a more complicated relationship between inputs and outputs. Using the converged species volume fractions output by a calculation that is performed in a mixed or grand-canonical as inputs to a subsequent calculation that is performed in canonical ensemble will generally lead to converged results for values for the \( w \) fields, species chemical potentials and pressure that are different from those obtained in the original calculation. This is because PSCF calculation in canonical ensemble will shift the value of \( \xi(\textbf{r}) \) by a homogeneous constant in order to satisfy the additional requirement on \( \xi \) that is imposed only in the special case of a canonical (closed) ensemble. The magnitude of this shift can be inferred from the resulting change in the value of the non-dimensional pressure \( p \). Equal results for monomer volume fraction fields and Helmholtz free energy should, however, be obtained for physically equivalent states in any ensemble.
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