This page discusses models of polymeric species in more detail.

Continuous vs discrete chain models

The pscf_pc and pscf_pg programs for systems with periodic boundary conditions can treat polymer conformations using either of two slightly different mathematical models, which we refer to as the continuous "thread" model and the discrete "bead" model.

The thread model idealizes each block of a polymer as a continuous random walk, with no discrete beads or monomers.
The bead model is instead based on an physical model of each block of a block polymer as a sequence of discrete point particles, or "beads", connected by harmonic springs.

The pscf_pc and pscf_pg programs can both perform either SCFT or PS-FTS calculations, and can both use either the thread or bead model for either type of calculation. In both of these programs, the choice of bead or thread model is global - the same model (bead or thread) must be used for all blocks of all polymer species in the system throughout program execution.

The pscf_1d program for one-dimensional systems can only use the thread model, and can only perform SCFT calculations.

Calculations that use the thread model use numerical algorithms that that discretize the contour length of each block into steps of some small length \( ds \) for numerical purposes. In this model, however, users input values for block lengths and \( ds \) separately, and \( ds \) is understood to be a purely algorithmic parameter, rather than a physical parameter. Numerical algorithms used for the thread model are designed to converge rapidly to the continuus thread limit as the contour length step size \( ds \) and the mesh resolution are both decreased while block lengths and other parameters are held fixed.
Algorithms used for the bead model are instead designed to accurately reproduce the properties of a discrete model in which the number of beads in each block is viewed as a physical parameter.

We generally recommend the use of the thread model in pscf_pc and pscf_pg for SCFT calculations, because it provides more universal results that can characterized using a smaller number of physical parameters. Both bead and thread models have been used in recent work on PS-FTS calculations, which is why PSCF provides a choice of either model in programs that can perform such calculations.

Continuous "thread" model

In the thread model, we imagine that the conformation of a single block of contour length \( L \) within a block polymer can be characterized by a continous space arc \( \textbf{R}(s) \) that gives the position \( {\bf R} \) of a point along the polymer as a function of a dimensionless contour coordinate \( s \), for \( s \in [0, L] \).

In this model, a chain segment of between points with contour coordinates \( s \) and \( s' \) within a block of monomer type \( \alpha \) in a homogeneous gas of non-interacting polymers is given by

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

in which \( b_{\alpha} \) is a statistical segment length for chain segments of type \( \alpha \). Here, \( \langle \cdots \rangle \) denotes a statistical average over random-walk configurations within a homogeneous mixture. Each such chain segment is also taken to occupy a volume \( V(s,s') \) within an incompressible mixture that is given by

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

where \( v \) is the so-called monomer reference volume. In the thread model, in which there are actually no discrete monomers, the parameter \( v \) is best understood to represent a steric volume per unit polymer contour length.

By convention, PSCF requires that contour lengths within blocks of different types be defined such that \( v \) has the same value for all monomer types. This convention entails no loss of generality, as long as monomers of different types can be assigned different values for statistical segment lengths.

Discrete "bead" model

In the "bead" model, each block of a block polymer is taken to consist of a sequence of discrete beads connected by harmonic springs. In this model, the user must specify the integer number of beads in each block of the polymer.

Suppose the beads in a block containing \( N \) beads are labelled sequentially by an integer index \( s = 0, \ldots, N-1 \). Let \( {\bf R}(s) \) denote the position of bead number s. In this model, a chain segment connecting beads with indices \( s \) and \( s' \) within a block of monomer type \( \alpha \) within a homogeneous ideal gas would exhibit a mean-squared end-to-end length given by

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

in which \( b_{\alpha} \) is the statistical segment length for a bond that connects two monomers of type \( \alpha \). In this model, neighboring beads within a block of monomer type \( \alpha \) are connected by harmonic springs with a spring constant \( 3 kT / b_{\alpha}^{2} \), and \( b_{\alpha}^{2} \) is the unperturbed mean-squared length of each such bond.

In the bead model for an incompressible system, every bead is assumed to occupy a volume given by the monomer reference volume \( v \), which is required to have the same value for every monomer type.

Polymer connectivity

In every PSCF program, each polymer in a mixture may be either a linear block polymer or an acyclic branched block polymer. Hompolymers are treated as a special case of a linear block polymer with only one block. Because a linear polymer is a special case of an acyclic branched polymer, the data structures used within the PSCF source code are designed for the general case of an acyclic branched polymer. Because linear block polymers are a very common simple case, however, the PSCF parameter file format allows the user to declare each species to be either linear or branched, and allows use of a simplified file format to describe linear polymer species.

Branched polymers

A branched block polymer is described in PSCF using the language of graph theory. The connectivity of each polymer is described as a connected acyclic graph consisting of a set of vertices connected by edges. Each "edge" in this graph corresponds to one block of the polymer. Each vertex corresponds to either a free chain end or a junction at which two or more blocks meet. Every edge (or block) terminates at two distinct vertices.

The number of edges that terminate at a vertex is known as the degree of that vertex. Each chain end is vertex of degree 1. Each junction in a linear block polymer is a vertex of degree 2. Each branch point in a branched polymer is a vertex of degree 3 or greater.

A graph is said to be "connected" if there is exists a continuous path of edges connecting any two vertices in the graph. A connected graph is "acyclic", or a "tree", if there exists only a single path connecting any two distinct vertices. A standard theorem of graph theory states that any connected acyclic graph with N edges must contain exactly N + 1 vertices. For example, a linear block polymer with N blocks contains two chain ends and N - 1 two-fold junctions, giving N + 1 vertices. Similarly, a star with N edges that are connected at a single central junction has N chain ends and one junction of degree N, again giving N + 1 vertices. An H polymer with N=5 edges has 4 chain ends and two vertices of degree three, giving N+1=6 vertices.

As a result of this thereom, we know that any acyclic block polymer with N blocks has exactly N + 1 vertices. In the description of polymer structure used by PSCF, each block in a polymer with N blocks is assigned a unique block index in the range [0, N-1], while each vertex is assigned a unique integer vertex index in the range [0, N]. Each block in such a graph connects a specific pair of different vertices.

In general, the structure of a branched polymer may thus specified in the thread model by giving a list of blocks in order of increasing block index, and providing the following information about each block:

a monomer type index
a block length
the indices of the two attached vertices

The only difference between the information required in the thread and bead model is that each block length is given as a real number in the thread model and as an integer number of beads in the bead model.

Linear polymers

PSCF allows a user to specify declare explicitly that a polymer species is a linear, and can use this information to define relationships among block and vertex indices. Blocks in a linear polymer are numbered sequentially, starting from one end of the chain. By convention, block i of a linear block polymer with N blocks terminates at vertices with indices i and i+1. This convention allows PSCF to use a simplified file format for information about blocks in a linear block copolymer, in which values of the indices of the attached vertices do not need to be explicitly entered by the user.

Vertices in the bead model

A complete definition of the bead model requires the introduction of an additional convention for the treatment of bonds that are connected to vertices of degree greater than 1, i.e., to junction or branch vertices. The bead model used by PSCF treats each such vertex as a "phantom" bead that does not interact with any chemical potential field, like a bead with a monomer type that occupies no volume and has vanishing Flory-Huggins interaction parameters with all other monomer types. Any bond that connects such a phantom vertex bead to a terminal (first or last) real bead of a block of monomers of type i is taken have a statistical segment length, denoted by \( a_{i} \), that satisfies

\[ a_{i}^{2} = b_{i}^{2}/2 \quad, \]

where \( b_{i} \) is the segment length for a standard bond that connects any two beads of type \( i \). If we think of each standard bond between two beads of type \( i \) as corresponding to a contour length s = 1 of an underlying continuous chain, then a bond that connects the terminal bead of an \( i \) block to a vertex thus corresponds in this model to a "half bond" of contour length s = 1/2.

Consider application of this rule to a junction vertex of degree 2 that connects two blocks of types A and B within a linear block polymer. A model that connects the terminal beads of the A and B blocks to a shared phantom vertex by two such "half bonds" is equivalent to a model that directly connects these two terminal beads with a bond that has an effective statistical segment length \( b_{AB} \), with

\[ b_{AB}^{2} = ( b_{A}^{2} + b_{B}^{2} )/2 \quad, \]

where \( b_{A} \) and \( b_{B} \) are statistical segment lengths of standard bonds of type A and B, respectively. In the case of equal statistical segment lengths, \( b_{A} = b_{B} \), this rule yields a connecting bond for which \( b_{AB} = b_{A} = b_{B} \), thus yielding a conformationally homogeneous chain in this simple case.

This convention was chosen primarily for simplicity. Its main virtues are that it yields conformationally homogeneous behavior for the simple case discussed above, and that it does not require the user to input any more information than that required by the thread model.

No such special rule for the treatment of junctions and branch points is required for the continuous thread model.

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