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PSCF is based on a coarse-grained picture of polymer statistics that does not attempt to resolve details of chemical structure. The use of a coarse-grained model inherently introduces some freedom into the definition of what one means by a "monomer", and thus into the choice of values for parameters that depend on this definition, such as the length of a block (in the thread model) or the number of monomers in a block (in the bead model).

What is a "Monomer" ?

When using a bead model, the most conceptually straightforward convention might be to choose the number of beads used in PSCF equal to the number of chemical repeat units within the corresponding block of some real polymer of interest. Even that definition leaves some ambiguity, however, because one can always redefine what one means by a "monomer" by grouping together two or more elementary chemical repeat units into a larger unit. 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, and also changes the values of the monomer statistical segment length and volume per monomer. For example, a 100 carbon alkane segment could be described as either containing 100 units using one-carbon monomers or 50 units for two-carbon monomers, while using larger values for the statistical segment length \( b \) and the volume \( v \) per monomer in the model with fewer, larger monomers.

This flexibility in the definition of a "monomer" becomes much greater in the continous thread model. The thread model allows the use of block lengths that are not integers, and that need not be greater than 1. In the thread model, 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 a real contour coordinate \( s \), rather than by an integer. The contour length \( |s-s'| \) of a segment between two points in the same block with contour coordinates \( s \) and \( s' \) is assumed to be proportional, but generally not equal, to the number of chemical repeat units contained in the chain segment between those points. As a result, the dimensionless contour length of a chemically homogeneous 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 for the thread model. The software is instead designed to allow users to choose any convention they wish for these constants of proportionality. The cost of this flexibility is, of course, that it places responsibility on the user to choose a convention and to make sure that the chosen convention is correctly reflected in the values of all parameters that the user provides as inputs to the program.

The only restriction that PSCF places on the definition of polymer contour lengths in the thread model is a requirement that, as a matter of convention, chain segments of equal contour length but different monomer type must 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 a value for the monomer reference volume \( v \) (i.e., the volume per unit contour length) that is the same for all monomer types. The requirement that the same volume per unit contour length be used for all monomer types does not lead to any loss of generality, as long as different values may be assigned for the statistical segment lengths of different monomer types.

There are two common types of convention for block contour lengths and length units used in the thread model, which we refer to here as "experimentalist" and "theorist" conventions. Both are described below. Experimentalist conventions are sometimes convenient for computational studies that are used to predict behavior for a particular experimental system with blocks of known chemistry and molecular weight. Theorist conventions are instead often 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 predictions of coarse-grained models for relevant thermodynamic properties can be parameterized by a small set of dimensionless parameters.

Experimentalist conventions

In an "experimentalist" convention, a monomer (i.e., a 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 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 order 50 - 100 cubic Angstrom that is comparable to the volume of a common hydrocarbon chemical repeat unit containing a few carbon atoms. Ultimately, however, the choice of a value for \( v \) is arbitrary. In a system that contains two or more different types of monomer, it is 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 actual volumes different types of repeat units are generally different.

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 \( \alpha \). 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 value chosen for \( v \). 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

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

where \( R_{e}^{2}(M) \) denotes the mean-squared end-to-end length for a homopolymer chain of molecular weight \( M \) and monomer type \( \alpha \) within a homopolymer melt. 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 often scaled so as to define the overall contour length of some reference chain or chain segment to be equal to unity, 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 total steric volume of the entire reference polymer or block. In this convention, the length assigned to each block of each polymer must then 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 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.

If one describes a polymer solution in a thread model 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 any solvent size parameter is normally 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 molecular 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 by convention, and the lengths for polymer blocks will generally be much greater than 1.

All of the SCFT examples shown in this user's manual use a thread model and a theorist convention in which the overall length of one type of polymer (i.e., the sum of the block lengths) is set to 1, and in which the statistical segment length of at least one monomer type is set to 1. This type of convention is also used in the input files for all of the SCFT examples provided with PSCF.

In simulations that use a bead model, the number of monomers in each block is treated as a physical parameter that is usually much greater than 1. In the bead model, one's choice for the level of discretization that is used in underying numerical algorithms thus also restricts the choice of values for block lengths.

Monomer reference volume

A value for the monomer reference volume \( v \) is not required as an input to a standard SCFT calculation. As discussed above, when using a coarse-grained model of a polymer to analyze a specific experimental system, the choice of an arbitrary value 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 values of any of the properties normally output at the end of a PSCF SCFT calculation. 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 in the standard output of SCFT thermodynamic properties, which is reported as free energy per monomer (or per unit contour length) normalized by \( kT \).

A value for \( v \) is, however, always required as an independent input parameter for PS-FTS calculations. If all other parameters are held fixed, smaller values for \( v \) lead to more overlap among polymers, and weaker field fluctuations. In systems with a single block polymers species of total 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

\[ \overline{N} = N (b^{3}/v)^2 \quad, \]

known as the invariant degree of polymerization. SCFT predictions are recovered in the limit \( \overline{N} \rightarrow \infty \) or, equivalenty, in the limit \( v \rightarrow \infty \).

Example (thread model) : When performing a PS-FTS calculation for a one-component diblock polymer melt using the thread model, one could choose a "theorist" convention in which \( N = 1 \) and \( b = 1 \) for both monomers, and then choose a value of \( v \) so as to obtain a desired value for \( \overline{N} \). For example, in this case we could use \( v = 10^{-2} \) to obtain \( \overline{N} = 10^{4} \).

Example (bead model) : When performing a PS-FTS calculation for a one-component diblock polymer melt using a discrete bead model for a chain of \( N = 100 \) beads, one could choose \( b = 1 \) for both monomer types (thus defining the root-mean-squared bond length as the unit of length) and then choose a value of \( v \) so as to obtain a desired value for \( \overline{N} \). For example, in this case, we could set \( v = 0.1 \) to obtain \( \overline{N} = 10^{4} \).

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