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This page provides an overview of the input parameters required by all of the programs in the PSCF package to characterize a physical system, as well as a discussion of the system of units and conventions used to define parameter values.

Physical model (overview)

All of the programs in the PSCF software package are designed to compute properties of a system that contains an incompressible liquid mixture of one or more block polymers and zero or more small molecule solvents. Each polymer species in such a mixture may be either a linear block polymer or an acyclic branched block polymer with a user-specified connectivity. Homopolymers are treated as a special case of a block polymer with only one block. Conformations of individual polymers is treated using a model of linear polymer segments as continuous random walks, as discussed below. Solvent species, when present, are treated as structureless point-like particles that occupy a specified volume per particle.

The only model for interactions among molecules that is currently implemented is one that treats the mixture as an incompressible liquid, and that uses a symmetric matrix of Flory-Huggins parameters to characterize binary interactions among polymer blocks and solvent species of different monomer types.

Both polymers and solvent species are constructed from a finite list of monomer types. Each block within a block polymer is assigned a monomer type (identified by an integer index) and a block length (given as a floating point number). The length of a block determines how much volume it occupies within an incompressible liquid, as well as the root mean-square (RMS) end-to-end length of the block in a homogeneous liquid. Each small-molecule solvent species is assigned a monomer type index and a steric volume.

The information required to describe the chemical composition of a model thus includes:

The number of different monomer types present in the system and a value for the statistical segment length of each monomer.
A description of each polymer species that includes the length and monomer type of each block and description of how the blocks are connected at junctions.
A specification of the monomer type and size (i.e., volume) of each solvent species.
A value for either the volume fraction or chemical potential of each polymer and solvent molecular species.
A value for a binary Flory-Huggins interaction parameter \( \chi_{ij} \) for interactions between monomers of types \( i \) and \( j \) for each distinct pair of monomer types.

Other user-provided input parameters required to initialize a SCFT or FTS calculation include:

Parameters that describe the unit cell (for periodic structures) or spatial domain within which the self-consistent field equations will be solved.
Parameters that define a spatial discretization for the unit cell or spatial domain.
A contour length step size for discretization of polymer blocks into steps used for numerical integration.

Particular numerical algorithms may require a variety of other algorithmic parameter and options.

Energy units

In all input parameter and output values with units that involve energy, PSCF uses "thermal" units for energy in which \( k_{B}T = 1\), where \( k_{B} \) is Boltmann's constant and \( T \) is absolute temperature. For instance, the chemical potential values reported in the parameter files are reported as \( \mu / k_{B}T \).

Length units

PSCF allows the user to use any system of units for quantities that involve units of length, as long as the same choice of units is used for all input parameters. The user is not asked to declare a specific choice of a unit of length, but is responsible for providing values that are have been chosen using some consistent choice of unit.

The only required parameters for a standard SCFT calculation that have units involving length are the values of the statistical segment lengths, and values (or initial guesses) for the dimensions of the crystal unit cell or spatial domain of the system of interest. Users may initialize a calculation using input files in which these parameters are define either using some standard microscopic length unit (e.g., Angstroms or nanometers) or in which they are defined using some user-defined system of non-dimensional units. Different common choice of conventions are discussed below.

If user-supplied input parameters are consistently assigned values that were defined using some unit of length, output values will be reported using the same units for length. Thus, in simulations of a periodic microstructures in which one or more dimensions of the crystal unit cell are iteratively ajdusted so as to minimize the free energy density, output values for the optimized unit cell parameters will be reported in the same length units as those used for relevant input parameters.

Most outputs of an SCFT calculation other than the optimal values for unit cell parameters, such as the free energy per chain in a one-component block polymer melt or the compositions associated with phase boundaries in a phase portrait) can be shown to depend on input parameters with units of lengths only through the values of ratios of lengths involving unit cell or domain dimensions and unperturbed coil sizes (i.e., RMS end-to-end lengths or radii of gyration) of polymers or blocks of the polymer species present within a system.

Continuous chain model

PSCF uses a model of polymers as continuous random walks. In this model, a specific configuration of a single block of length \( L \) within a block polymer can be characterized by a continous path \( \textbf{R}(s) \) that gives the position \( {\bf R} \) of a point along the polymer as a function of a dimensionless contour coordinate \( s \), for values of \( s \in [0, L] \). A chain segment between points with contour coordinates \( s \) and \( s' \) within a block of monomer type \( \alpha \) is assumed to exhibit a mean-squared end-to-end length in a homogeneous mixture given by

\[ \langle |{\bf R}(s) - {\bf R}(s')|^{2} \rangle_{0} = |s-s'| b_{\alpha}^{2} \quad, \]

in which \( b_{\alpha} \) is a statistical segment length for chain segments of type \( \alpha \). Here, \( \langle \cdots \rangle_{0} \) denotes a statistical average over random-walk configurations within a homogeneous mixture. In an incompressible mixture, each such segment of contour length \( |s-s'| \) is assumed to occupy a volume \( V(s,s') \) given by

\[ V(s,s') = v|s-s'| \quad, \]

where \( v \) is a constant steric volume per polymer contour length. We refer to the parameter \( v \) as the monomer reference volume.

In the above discussion of the continuous chain model, we introduced the notion of a dimensionless "contour length" of a chemically homogeneous segment of polymer, such as a block within a block polymer. The simplest way one might try to define such a length is to take it equal to the number of chemical monomers within that segment. Even that definition leaves some ambiguity, however, because one always has some freedom to redefine what one means by a monomer by grouping together two or more elementary chemical repeat units into a larger unit that we may then treat as a "monomer". For example, when describing a long linear alkane (i.e., linear polyethylene) one could treat either a 1-carbon or 2-carbon unit as a "monomer". This regrouping operation changes the value one would obtain for the number of monomers in a given chain segment, as well as the required values of the monomer statistical segment length and volume per monomer. For example a 100 carbon alkane segment could be described as having a "length" of 100 units using one-carbon monomers or 50 units for two-carbon monomers, using different values for the statistical segment length \( b \) and the volume \( v \) per monomer in these two cases.

In the continuous chain model used in PSCF, we implicitly assume that the number of elementary chemical repeat units in any chain segment of interest is sufficiently large that we may ignore details of chemical structure and characterize positions along a polymer by some sort of real (rather than integer) contour coordinate \( s \). The contour length \( |s-s'| \) of a segment between two points on the same chain with contour coordinates \( s \) and \( s' \) is then assumed to be proportional, but generally not equal, to the number of elementary chemical repeat units contained in the chain segment between those points. As a result, the dimensionless contour length of a chain segment is also proportional to the molecular weight of that segment. Values for the constants of proportionality that determine the number of elementary chemical repeat units and molecular weight per unit contour length for each type of block are not, however, specified by the equations used within PSCF. The software is instead designed to allow to choose from among a variety of possible conventions for these constants of proportionality. The cost of this flexibility is, of course, that it places responsibility on the user to choose and understand a convention and to make sure that the chosen convention is accurately reflected in the values of user-supplied input parameters.

The only restriction that PSCF places on the definition of polymer contour lengths is that a requirement that, as a matter of convention, chain segments of equal contour length but different monomer type correspond to the same volume within an incompressible melt. The steric volume of any chain segment of contour length \( L \) is thus given by \( v L \) , for segments of any monomer type, using the same value for the monomer reference volume \( v \) for blocks of any monomer type.

The requirement that the same monomer reference volume be used for all monomer types does not cause any loss of generality in a model of polymers as continuous random walks, as long as different values may be assigned for the values of the statistical segment lengths of different monomer types. The statistical segment \( b_{\alpha} \) for monomer type \( \alpha \) is defined to be the root-mean-squared distance between the ends of an unperturbed random walk of a chemically homogeneoues chain segment of monomer type \( \alpha \) with a molecular weight corresponding to one unit of contour length. The resulting value is generally different for different monomer types, even when contour lengths are defined so that one unit of contour length corresponds to the same volume for every monomer type.

There are two common types of convention for block contour lengths that we refer to here as "experimentalist" and "theorist" conventions. Both are described below. Experimentalist conventions are generally the most convenient choice for computational studies that are being used to predict behavior for a particular experimental system with blocks of known chemistry and molecular weight. Theorist conventions are instead more convenient for studies of trends with changes in parameters, and for studies of phase maps for classes of systems for which it is known that which predictions of relevant thermodynamic properties can be parameterized by a small set of dimensionless parameters.

"Experimentalist" Conventions

In an "experimentalist" convention, a monomer (or unit of contour length) is defined for each monomer type so as to correspond to a segment with a molecular weight chosen so that the volume occupied by that segment in a polymer melt is equal to a chosen value for a monomer reference volume \( v \). This is a standard approach used in the literature in studies that compare experimental results to predictions of the Flory-Huggins theory for polymer solutions and mixtures, the random-phase approximation (RPA) theory of small-angle scattering and/or to SCFT predictions for inhomogeneous structures. In such studies, the reference volume \( v \) is usually chosen to have a value of of order 50 - 100 cubic Angstrom that is comparable to the volume of some common hydrocarbon chemical repeat unit, but the choice of an exact value is arbitrary. In a system that contains two or more different types of monomer, it generally not possible to choose a single value for the reference volume that corresponds to the volume per chemical repeat unit of every monomer type, because the volume per repeat unit is generally different for different types of monomer.

The contour length \( L \) of a chemically homogeneous polymer segment of some known molecular weight \( M \) and monomers of type \( a \) may be expressed as a ratio

\[ L = M/m_{\alpha} \quad, \]

in which \( m_{\alpha} \) is a molecular weight per unit contour length (or per monomer reference volume) for monomers of type \( a \). The quantity \( m_{\alpha} \) must be defined for each monomer type such that the volume occupied per chain segment of molecular weight \( m_{\alpha} \) in a polymer melt corresponds to the chosen value of \( v \). If a melt of homopolymers containing monomers of type \( \alpha \) exhibits a density \( \rho_{\alpha} \) in grams per volume at the tempeature and pressure of interest, then the value of \( m_{\alpha} \) for that monomer type is given in grams per mole as

\[ m_{\alpha} = \rho_{\alpha} v N_{\rm A} \quad \]

where \( N_{\rm A} \) is Avogadro's number. If \( \rho_{\alpha} \) is given in grams per cubic centimeter and \( v \) is given in cubic Angstroms, then a numerical conversion factor of \( 10^{-24} \) cubic centimeter per cubic Angstrom is also required to obtain consistent units.

The value for the statistical segment length \( b_{\alpha} \) associated with a specific monomer type is then given by

\[ b^{2}_{\alpha} = m \frac{R_{e}^{2}(M)}{M} \]

where \( R_{e}^{2}(M) \) denotes the mean-squared end-to-end length for a homopolymer chain of molecular weight \( M \) and monomer type \( \alpha \). Note that the ratio \( R_{e}^{2}(M)/M \) is independent of molecular weight for chains with random-walk statistics. If a value for the statistical segment length is known for some other definition of a monomer unit (such as a chemical repeat unit, or an effective monomer defined using a different value for \( v \) ), then the ratio \( R_{e}^{2}/M \) may be replaced by ratio of the square of the statistical segment length for that monomer definition divided by the molecular weight per monomer. In experimentalist conventions, Angstrom or nanometer units are usually used for all lengths.

"Theorist" Conventions

In a "theorist" convention, contour coordinates are scaled so as to define the overall contour length of some reference chain or chain segment to be equal to 1.0 by convention. The reference chain is often chosen to correspond to either an entire molecule of some polymer species that is present in the system, or one block of one such species. This is equivalent to setting the "monomer" reference volume equal to the occupied volume of the entire reference polymer or block. Because the equations implemented by PSCF require that the volume per contour length must be the same for blocks of different monomer types, values for ratios of block or chain lengths must be equal to corresponding ratios of occupied volumes. In this convention, the length assigned to each block of each polymer must thus be equal to the ratio of the actual volume of that block to the volume of the reference chain.

For example, in a system that contains a single linear block polymer species, the length of the entire polymer is often set equal to 1.0 by convention. In this convention, we assign the blocks of that polymer block lengths that are each less than 1 and that add to exactly 1. In the case of a one-component block polymer melt, the value of each such block length is then equal to the volume fraction of that block.

The use of theorist convention for contour lengths is compatible with the use of either physical Angstrom or nanometer length units or with the use of a convenient system of non-dimensionalized units for length. The simplest scheme for definining non-dimensionalized lengths is to define the statistical segment length of one monomer type to be equal to 1.0 by convention. In this case, one must then assign every other monomer type a statistical segment length given by the ratio of its actual statistical segment length (e.g., its statistical segment length defined using an experimentalist convention) to that of the reference monomer type. Note that the values of ratios of statistical segment lengths are unchanged by changes in one's convention for the monomer reference volume.

All of the examples shown in this user's manual use a theorist convention in which the overall length of one type of polymer (i.e., the sum of the length of the blocks for a block polymer) is set to 1.0, and in which the statistical segment length of at least one monomer type is set to 1.0. This type of convention is also used in the input files for all of the examples provided with PSCF.

Solvent species

PSCF allows user to model polymer solutions by including zero or more "solvent" species in addition to one or more polymer species. Each solvent molecule is treated as a mobile but structureless point-like particle that occupies some volume within an incompressible mixture. Solvent species can be used to model either rigid small-molecule solvent molecules or oligomeric chain molecules that much shorter than all of the "polymer" species present in the same system.

Each solvent species is assigned a monomer type that is selected from among the same set of monomer types that can be used to describe polymer blocks. A solvent that has properties that are distinct from those of any of the monomers used to construct the polymer species can be assigned an extra monomer type that is simply not used as the monomer type for any polymer block. A solvent species can instead be assigned a mononer type that is the same as that assigned to one or block polymer blocks if that solvent species represents a small oligomer of the specified monomer type, or another species with physical properties very similar to those of such an oligomer. The input file format for PSCF requires that a monomer type that this is only associated with a solvent species must must still be assigned a statistical segment length value, but in this case that segment length value is simply left unused.

Each solvent species is also assigned a volume per molecule. This is specified in the PSCF input files by a dimensionless parameter labelled "size". The value of the size parameter is equal to the ratio of the solvent volume to that of a the mononer reference volume. The "size" parameter for a solvent species is thus closely analogous to the "length" parameter for a polymer block, since both are defined as volumes measured in units of the monomer reference volume.

If one describes a polymer solution using "theorist" units in which the reference volume it defined to be equal to the total volume of some reference polymer species, then values for the solvent size parameter will usually much less than 1.0. Thus for example, a system with a homopolymer of length 1.0 and a solvent with a size equal to 0.01 could be used to describe a homopolymer solution in which the polymer volume is 100 times greater than that of the solvent. It is also valid, and sometimes more convenient, to describe a solution using a theorist convention in which the volume of the solvent species is taken as the reference volume. In this case, the solvent size wil be set equal to 1.0 by convention, and the lengths for polymer blocks will generally be much greater than 1.

Irrelevance of monomer volume in SCFT

A value for the monomer reference volume \( \) is not required as an input to a standard SCFT calculation. As discussed above, when using a model that idealizes polymers as continuous chains to analyze a specific experimental system, the choice of a convention for the monomer reference volume \( v \) determines the appropriate choice of values for several types of related parameters, such as the block lengths, monomer statistical segment lengths, and interaction parameters. It turns out, however, that once the values of these related parameters are determined, the value of \( v \) itself is not needed as an additional independent input to an SCFT calculation. It can be shown that changes in the value of \( v \) alone, without corresponding changes in the values assigned to related parameters (e.g., statistical segment lengths, block lengths and chi parameters) have no effect on SCFT predictions for values of any of the properties normally output by the PSCF programs. Specifically, changes in the value of \( v \) alone can be shown to have no effect on SCFT predictions for average monomer volume fraction fields, polymer or solvent chemical potential values, or the non-dimensionalized SCFT Helmholtz free energy that is reported by PSCF in its standard format for outputting SCFT thermodynamic properties, which is a dimensionless free energy per monomer.

The role of \( v \) in SCFT is discussed in more detail in a separate discussion of the working equations for SCFT. There we show explicitly that the forms of these equations implemented in PSCF simply do not require a value for \( v \) as an independent input parameter.

A value for \( v \) is, however, always required as an independent input parameter for stochastic FTS calculations. If all other parameters are held fixed, smaller values for \( v \) lead to less overlap among polymers, and weaker field fluctuations. In systems with a single species of chain length \( N \) and a single statistical segment length \( b \), the extent of overlap and the magnitude of fluctuations can be characterized by a dimensionless parameter \( \overline{N} = N (b^{3}/v)^2 \) known as the invariant degree of polymerization. SCFT predictions are recovered in the limit \(\overline{N} \rightarrow \infty\) or, equivalenty, in the limit \( v \rightarrow \infty \).

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