Polymer 
Historically, various latticebased theories have been used to calculate the thermodynamics of polymer solutions and blends, the FloryHuggins theory (Flory 1953) being the simplest and best known of these. In all of these theories, one or more interaction parameters (e.g., or g, depending on the specific theory) determine the phase behavior of the system, i.e., the conditions under which the system is in one phase as opposed to two or more phases. Lattice theories by their nature are very coarsegrained, and tend to treat interactions at the statistical segment level as opposed to the atomic level. In these theories, the interaction parameters are generally taken as given, or are determined empirically from experimental data (Qian et al. 1991). These parameters, in principle, are determined by the microscopic interactions among the various components of the system. However, no clear prescription exists for determining interaction parameters from, say, a set of realistic atomistic potential energy functions.
In addition to being coarsegrained, the simplest lattice models are meanfield approaches, meaning that they usually neglect to explicitly account for one or more of the following effects: chain connectivity, density fluctuations, concentration fluctuations, and compressibility. The Polymer Reference Interaction Site Model, or PRISM, theory, a very recent approach to the problem of melts and blends, extends beyond these limitations.
PRISM theory, developed over the past five years by Schweizer and Curro (1989) and Curro and Schweizer (1987) is an offlattice continuum theory of polymer melts and blends. It is based on the small molecule RISM theory developed by Chandler and Andersen (1972) and formulated for "ring polymers" (in the context of a Feynman path integral representation of delocalized electrons) by Chandler and coworkers (1984). Although in practical calculations, PRISM theory is itself somewhat coarsegrained, dealing explicitly with "sites" rather than atoms, it is, nevertheless, more nearly atomistic than lattice models. PRISM theory also allows for concentration and density fluctuation effects, as well as the effect of chain connectivity, all of which are neglected in the FloryHuggins approach.
PRISM theory is based on statistical mechanical theories of liquids, the basics of which provide enough material to fill a book (Hansen and McDonald 1986). Therefore, this chapter does not present a comprehensive overview and derivation of RISM and PRISM theory. Instead, the goal here is simply to present the final formulation of the theory, along with enough background from liquid state theory to make it understandable. For more detailed treatments and discussion of subtleties that may be omitted from this presentation, please refer to the primary literature (Chandler and Andersen, 1972; Schweizer and Curro, 1989; Lowden and Chandler, 1973, 1974). If you are interested only in the principal assumptions behind the theorywhat is required for a PRISM calculation, and what results are obtained from a PRISM calculationsee the Condensed summary of PRISM theory section.
Apart from the issue of the quantitative accuracy of PRISM predictions is the issue of obtaining solutions to the PRISM equations in a reasonable amount of time. The PRISM equations are highly nonlinear. Finding a solution can sometimes be an extremely slow process, especially for relatively complicated systems. At high densities or within the twophase region of a polymer blend, convergence to a solution might never occur. We have strived to make our implementation of this complicated theory as userfriendly as possible. Even so, if you perform many PRISM calculations, you will no doubt find yourself frustrated at times in your attempts to find solutions. As PRISM theory matures and better solution methods are discovered, this situation will no doubt improve.
For example, real polyethylene (Figure 34) is made of a sequence of methylene groups each containing a carbon atom and two hydrogen atoms, whereas "PRISM polyethylene" is a chain of unitedatom CH_{2} sites, each site being a single sphere characterized by a hard core diameter, . In principle, one could construct a PRISM representation of polyethylene such that there is a site corresponding to each atom in the CH_{2} group, but increasing the number of sites in this manner would cause a significant increase in computation time (among other problems). (The time required to perform a PRISM calculation increases rapidly with the number of types of sites used to describe a polymer.) In practice, it appears that the intramolecular arrangement of backbone atoms is much more important in determining the structure of polyethylene melts than is the slight asphericity of the CH_{2} groups in the backbone. Thus, PRISM calculations based on unitedatom representations of alkanes and polyethylene have been found to give structures in very good agreement with experiment (Honnell et al. 1991). Of course, in more complicated polymers, a greater number of sites may be unavoidable.

Fundamentally, PRISM theory is a theory of the structure of polymer systems in the liquid state. This structure is represented (in Fourier space) in the form of partial scattering functions, S_{}(k), or (in real space) intermolecular pair distribution functions, g_{}(r). The latter functions describe the spatial correlations between sites on different polymer chains. However, more than just the structure is obtainable from a PRISM calculation. A wide variety of thermodynamic properties of the system can be calculated using the pair distribution functions. These include blend phase diagrams, effective parameters as measured by neutron scattering, and equations of state. The system may be a simple polymer melt, a multicomponent blend, a block copolymer melt, or, in principle, any other polymer system in the liquid state.
Every theory has certain quantities (inputs) that you must specify in order to produce results. In the case of PRISM theory, the two most important inputs are

The most common way of representing the structure of an atomic liquid is by means of the pair distribution function, g(r,r'). This function plays a central role in theories of liquids. It specifies the relative probability of finding an atom at position r, given that another atom is at position r'. In an isotropic atomic liquid, g(r,r') is a function of the magnitude rr' only; that is, it has no angular dependence or dependence on the absolute location within the liquid, and is generally written g(r) (see Figure 36 (a)). At very short distances, g(r) vanishes due to the hard core repulsions between pairs of atoms. At intermediate distances, g(r) has a roughly oscillatory shape corresponding to the packing layers immediately surrounding an atom in a liquid. At long distances, g(r) approaches unity, meaning that all correlations between the positions of farseparated atoms are lost. For comparative purposes, pair distribution functions for a gas and a crystal are shown in Figure 36 (b) and (c).

Modern liquid state theory was developed mainly over the period from the 1950's through the early 1970's. Although various different approaches have been taken, all modern theories are based on the OrnsteinZernike (OZ) equation:
The OZ equation may, alternatively, be expressed as follows:

The OZ equation contains two unknown functions: c(r) and h(r). To solve it requires another equation, a closure, relating h(r) and c(r). A closure is an approximation that allows the OZ equation to be solved. Several standard closures exist (Hansen and McDonald 1986); the one used here is known as the mean spherical approximation, or MSA:
When the system of interest is a liquid of hard spheres, the potential, u(r), is zero for r > . In this case, the MSA closure is equivalent to the PercusYevick, or PY, closure. No real liquid is made up of hard spheres, of course, but it has been found that in dense liquids, the structure is predominantly determined by the harsh repulsive forces experienced by atoms at small separations. Consequently, a hard sphere liquid is often a good representation of a real liquid, and deviations from the hard sphere structure may be treated as small perturbations (Hansen and McDonald 1986).
An alternative to explicit treatment of molecular orientations is to regard each molecule in a liquid as consisting of a number of sites, which may be, but do not have to be, atomic centers. The simplification comes from stipulating that the interactions between sites on different molecules are spherically symmetric in form.
The liquid structure is then described, not by means of a complicated, orientation dependent molecular pair distribution function, but by a set of sitesite pair distribution functions, g_{}(r), where the subscripts and label the type of site. This general formalism is known by the name Interaction Site Model, or ISM. (The simplifications arising from the ISM formalism come with a pricethe loss of explicit orientational information in the distribution functions.)
Although an interaction site may correspond to a single atom, this is not essential. Depending on the nature of the molecule, it is often useful to have fewer sites than atoms, such that two or more atoms are represented by a single site. Figure 38 illustrates possible ISMs for a few molecules. The most important criterion in choosing an ISM representation for a molecule is that the ISM capture enough of the molecule's structure and represent its interactions with other molecules sufficiently well for the application of interest.
An interaction site model is called a Reference Interaction Site Model, or RISM, when the sites are treated as hard spheres, possibly with the hard sphere diameters as adjustable parameters. (Here, the term "reference" indicates that the hard sphere system may form a reference system for a perturbation expansion.) Chandler and Andersen (1972) formulated an approximate integral equation theory for RISM molecules. It is based on a molecular generalization of the OZ equation and on an MSAlike closure relation. This RISM theory has proven very successful in calculating the structures of molecular liquids. In its overall structure, RISM theory is the same for both small molecule and polymeric liquids. The details of this theory are discussed in the next section.

In Eq. 352, the site and molecular densities have been absorbed into the various matrices. To be precise, the elements of H(r) are _{a}_{b}h_{b}(r); the elements of (r)_{ }are ^{~}_{i}_{ab}(r)for sites of type and in the same chain (i.e., chain i), and zero for and in different chains; and the elements of C(r) are simply c_{}(r). Here, _{a} and _{b} refer to the number densities of sites of type and , and ~_{i} refers to the number density of sites of all types contained in chains of type i.
The precise definition of the intramolecular distribution function, _{ab}(r), is:
Note that (r) is absent in the OZ equation for atomic liquids (Eq. 348). Its presence in the generalized OZ equation is due to the fact that the intramolecular arrangement of atoms has an effect on the intermolecular correlations and vice versa. Figure 39 illustrates this graphically. For a rigid molecule with fixed bond lengths, the _{ab}(r)_{ }elements of the (r) matrix are delta functions or sums of delta functions. For a flexible polymer, the intramolecular distribution functions are more complicated.

Because the elements in the generalized OZ equation (Eq. 352) are matrices, this "equation" is really a set of coupled equations. The number of distinct equations is m(m + 1) / 2, where m is the rank of the matrices. This rank is equal to the number of distinct site types in the system. In this context, two sites on a chain are considered distinct if their direct correlation functions with sites on other chains differ from each other. That is, sites j and k are distinct if c_{jl}(r) and c_{kl}(r) differ, where l is the index of a third site. Two sites are equivalent (of the same type) if these direct correlation functions (for all l) are the same. Thus, in a monodisperse melt of simple ring polymers, all sites are equivalent; m is equal to 1, and the generalized OZ equation reduces to only a single integral equation, which is easily solvable in the RISM approximation.
In a melt of linear simple chains, the situation is in principle more complicated in that sites near the chain ends are chemically distinct from those in the chain interior. Thus, to be rigorously correct, the chain would have to be treated as consisting of N/2 distinct sites, where N is the degree of polymerization. (The factor of 1/2 comes from the symmetry of the chain.) This would render the OZ equation unsolvable for practical purposes. For long chains, however, it is a good approximation to neglect explicit end effects and treat all sites in the chain as equivalent. This is the essential approximation made by Curro and Schweizer (1987) that makes RISM theory tractable when applied to linear polymers. For chains more complicated than simple chains such as polyethylene, it may be necessary to include more than one site type in a PRISM representation of a polymer. For example, a vinyl polymer may be regarded as being made up of three distinct site types: a methylene site, an alphacarbon site, and a sidegroup site.
Note that the generalized OrnsteinZernike equation (Eq. 352) is not specific to singlecomponent systems but is completely general. For example, the system described by Eq. 352 may be a onecomponent melt, a multicomponent blend, a polymer solution, or a block copolymer melt, etc. Differences among these systems manifest themselves in the structure of the H(r), C(r), and (r) matrices, but the OZ equation retains the same form.
As was the case for atomic liquids, a closure relation is needed to solve (in an approximate way) the generalized OZ equation. The mean spherical approximation, MSA, appropriately generalized to the molecular case, is used:

First, it is convenient to work in Fourier space, in which the OZ equation is simply an algebraic relation:
Using these definitions, Chandler and Andersen (1972) derived the functional:
To numerically solve Eq. 358, the direct correlation functions, c_{}(r), are approximated within the hard core by the following power series expansion:
The LowdenChandler method described in the Solving the PRISM equations section cannot be used for the RMMSA. (However, the LowdenChandler scheme is used to solve the hard core reference system.) Instead, the PRISM/RMMSA equations may be solved using a socalled Picard iteration technique, based on fast Fourier transforms. See Hansen and McDonald (1986, p. 129) for further details on this method of solving integral equations. (For the RMMSA, the iteration is performed in terms of *C*. This differs slightly from the Picard method used for atomic liquids, which performs the iteration in terms of C.)
Two additional closures now available are the PercusYevick (PY) and Reference Molecular PercusYevick (RMPY) closure. The PY closure is used for continuous potentials (e.g., full 612 LennardJones) rather than the hard core potentials used in the MSA and RMMSA. The RMPY closure is a "molecular" version of the PY closure analogous to the RMMSA (Yethiraj and Schweizer 1992, 1993; Schweizer and Yethiraj 1993).
Note that finding a solution to Eq. 358 is equivalent to finding an extremum (in this case, a minimum) of I_{RISM} with respect to the a_{j}^{} coefficients. Accelrys research has found empirically that a downhill simplex minimization scheme applied to I_{RISM} is quite effective in moving the coefficients from their initial guess values to the vicinity of their final values. When the simplex method is used in conjunction with a conjugategradient scheme, convergence is sometimes faster. Thus, the procedure used in Accelrys's PRISM implementation is as follows. Starting from an initial guess for the a_{j}^{} coefficients, the procedure alternates between several simplex and several conjugategradient minimization steps until the derivatives of I_{RISM} with respect to these coefficients are "reasonably small." Then, NewtonRaphson iterations are performed until these derivatives are effectively zero, and Eq. 358 is thus solved.
The above paragraphs describe the LowdenChandler procedure for solving the PRISM equations using the MSA. If the RMMSA or RMPY closure is chosen instead, the above method is used to solve the hard core reference system only (that is, a system equivalent to the system of interest, except with all the potential tails set to zero). This provides C_{0} in Eq. 360 above. Then, Eq. 361 is used in a Picard iteration scheme to solve the PRISM equations. In the case of the PY closure, only a Picard iteration scheme is used to solve the PRISM equations.
After a prospective solution is found by one of the methods outlined above, verification is performed to determine if it is indeed a valid solution by confirming that the argument of the logarithm is positive for all k values in Eq. 357. In some cases, a "solution" is found with a negative argument at k = 0. For a blend, this generally means that the system is inside the spinodal (i.e., in the twophase region) where the PRISM equations, which implicitly assume a homogeneous, onephase system, are no longer valid. In some cases, it may instead imply that the system is in a gasliquid twophase coexistence region, where again the theory no longer applies. These phenomena are mentioned to emphasize that solutions to the PRISM equations cannot necessarily be found for all possible system conditions.
is the matrix of partial scattering functions. These may be combined into total scattering functions if the scattering form factor for each type of site is known. Scattering functions may be compared directly with the results of (static) neutron, Xray, or light scattering experiments. The
matrix is Fourier transformed to yield the pair distribution functions, g_{}(r). Pair distribution functions may be compared with results of computer simulations. The purpose of performing such comparisons is to probe the validity of the PRISM theory in a particular context, the validity of the potentials used, and/or the validity of the intramolecular distribution functions used in the PRISM calculation. It is possible (currently by trialanderror) to work backwards from measured or simulated distribution functions to optimize the parameters used in a PRISM calculation.
Even in the context of PRISM theory, the equations of state computed by the PRISM module are only approximate because approximations beyond those inherent in PRISM theory are necessary if the pressure is to be calculated from only the pair distribution functions and intermolecular potentials. A variant of a virial pressure equation derived by Honnell et al. (1987) is used in the PRISM module to compute the pressure, P:
In the above equation, threebody contributions to the pressure are omitted. This is done for several reasons. First, the inclusion of threebody terms in the context of PRISM theory requires a "superposition approximation" because PRISM produces only twobody correlation functions. Such an approximation is likely to be unreliable. Second, in calculations that do include threebody terms, it has been found empirically that these terms make only a small contribution to the total pressure (Schweizer and Curro 1988). Finally, without adjustable parameters, the pressure is one of the most difficult quantities to determine quantitatively from either theory or atomistic simulation. The known inaccuracies in virial pressures determined from liquid state theory do not appear to warrant the inclusion of complicated threebody terms.
A derivation of the expression for the RPA parameter, in terms of quantities calculated by means of PRISM theory, is presented by Schweizer and Curro (1989). Only the result, slightly rewritten, is presented here as:
Note that the relation in Eq. 364 contains the direct correlation functions rather than the sitesite interaction energies. The latter are the quantities that appear in the "bare" _{0}, which is the FloryHuggins generalized to the offlattice case. Thus, ^{RPA} represents an interaction parameter that is renormalized by the presence of other sites in the system. Unlike the classical FloryHuggins _{0} parameter, which is simply proportional to 1/T, ^{RPA}, in general, has a more complicated temperature dependence. In addition, ^{RPA} may depend on blend composition (i.e., volume fraction) as well as the degree of polymerization.

The spinodal represents a secondorder phase transition, and as such is characterized by long wavelength fluctuations in the system. The signature of the spinodal is a zero wave vector (k = 0) divergence in the partial scattering functions given by Eq. 361. This corresponds to the condition (Schweizer and Curro 1989):
stability. In the two phase spinodal region, PRISM theory breaks down. Thus, in performing a PRISM calculation of the spinodal curve, this region must be approached from the "outside". That is, it is essential that the initial system conditions be in the one phase region, after which the temperature may be varied to approach the spinodal. Further practical considerations are discussed in Methodology, and examples of calculating spinodal curves are given in the online Tutorial Lesson 4. (Refer to the Pilot interface for this tutorial. For more information, see the Tutorial section.)
In addition to the core pulldowns in the top menu bar, the PRISM module adds the PR_System, PR_Compute, Pseudo_Atom, Textfile, Graph, and Background_Job pulldowns to the lower menu bar.
Pseudoatoms are defined as the instantaneous average of the coordinates of a set of real atoms. For example, the centroid of a phenyl group could be displayed as a pseudoatom created from the six carbon atoms in the ring. Pseudoatoms are referenced in the same way as atoms: their names can be explicitly specified at the time of their creation or can be generated automatically.
In the DeCipher and Analysis modules, pseudoatoms can be defined and used to study geometric properties. In the DeCipher module, pseudoatoms can be used to conjunction with the commands in the Functions and Geometrics pulldowns to plot and visualize several userdefinable properties. In the Discover module, pseudoatoms can be defined and used to define NOE constraints.
Pseudoatoms can also be used in the Color, Display, Label, and List commands in the Molecule pulldown; the Distance, Angle, Dihedral, and XYZ commands in the Measure pulldown; the Center, Move, Overlay, and Superimpose commands in the Transform pulldown; and the commands in the Subset pulldown.
Please see the Insight II User Guide and online documentation for details.
Please see Chapter 19, Graph Pulldown for more information.
Please see Chapter 15, Background_Job Pulldown for more information.
If your system can be adequately described by van der Waals forces, then you must next choose a site representation for the molecules in the system. That is, you must determine a reasonable way of describing each molecule as a sequence of connected sites, where each site will probably represent more than one atom. See the Theory section for examples of this process.
An important point to keep in mind is that the computation time required for a PRISM calculation scales roughly as the fourth power of the total number of site types in the system.
Thus, for initial calculations on a system, it is generally advisable to represent the system with a bare minimum of distinct site types. Having obtained preliminary results, if you then think a more sophisticated site representation is required, go ahead and add more sites. For example, if you wish to perform calculations on a vinyl polymer, you might start with a twosite (or even onesite) representation, and only later proceed to a threesite representation, if needed (see Figure 42).
Having chosen a site representation, you must determine the effective interactions between pairs of sites. The PRISM module treats each such interaction as an impenetrable hard core with diameter _{ab}, attached to which is a LennardJones tail, with well depth _{ab} (see Figure 40). Assuming that the sites in your system each represent more than a single atom, these sitesite interactions are unitedatom potentials.

United atom potentials have been published for many small molecules such as methane, but are relatively rare for the groups making up polymers. Ryckaert and Bellemans (1978), as well as others, have published unitedatom potentials appropriate for polyethylene, but little work has been done for more complicated chains. Still less has been done to derive and validate methods of determining unitedatom potentials from, e.g., a fully atomistic forcefield or quantum mechanics.
This section contains some general suggestions for estimating sitesite potentials. One simple and quick approach is to take as the hard core diameter the distance between the center of a site and the outermost point on the van der Waals surface of all the atoms represented by the site. This approach may be useful for determining the relative hard core diameters of different sites, but is likely to overestimate the diameter for any single site. A more systematic approach is to begin with an atomistic forcefield describing the interactions among two sites; determine a spherically averaged (or, alternatively, minimum) energy as a function of separation between the two sites, and determine a best fit of this function to the hard core/LennardJones representation used by the PRISM module. Finally, if experimental datascattering, equation of state, or phase diagramare available for systems containing the molecules of interest, it may be best to work backwards to find interaction parameters that yield PRISM results in agreement with the available data. Then, you may proceed to perform PRISM calculations for conditions where experimental data are lacking.
Having chosen a site representation and potential energy parameters for your system, you must next choose a form for the intramolecular structure factor. Again, several options are available. If you wish to perform a relatively quick calculation or would be satisfied with results of a more qualitative, as opposed to quantitative nature, then a Gaussian or freelyjointed chain representation may be adequate. For more quantitative work, it is possible to use the RIS module to perform RIS Monte Carlo calculations of intramolecular structure factors (here called (k), but referred to as S(k) in the RIS module). These structure factors may then be used as inputs for PRISM calculations.
Before using the Compute_Omega command, you must use the commands in the Pseudo_Atom pulldown to specify the scattering centers (sites) in the molecule. For the purpose of calculating the omega functions, pseudoatoms with identical alphabetic parts of their names are treated as equivalent. For example, pseudoatoms named A, A1, and A23 would be treated as equivalent sites and distinct from pseudoatoms named B2, AB4, or C.
The omega functions are computed with the following formula:
Once the omega functions are calculated, the name of the file containing these data is written to the omega list file (.omlist file). To make use of these data in a PRISM calculation, you should then specify this file when you use the Omegas command. The DP that you specify for the small molecule in the Omegas command should be the same as that listed in the .omlist file.
The chain or chains in the system are built using Polymerizer. You need to build only one chain of each distinct type. PRISM calculations are done for bulk systems, and the chains you build are used only as visualization aids to help you specify the sitesite interaction potentials.
The PR_System/Components command in PRISM is used to specify the number of site types in the chain and the hard core and LennardJones parameters for interactions between equivalent sites, for each polymer in the system. You should use the PR_System/Interactions command to specify interactions between nonequivalent sites.
For a given polymer component, each site type is associated with a distinct atom in the chain. Which atom you use to represent a site is unimportant, so pick the one that you find most convenient. Again, no details of the chemical composition of the molecules on the screen are used in the PRISM calculations. The molecules are used as visualization aids, but you specify explicitly all of the information needed for the PRISM calculation.
The PR_System/Omegas command is used to specify the intramolecular structure factor, for each polymer in the system. The name of the Omega List File (see Appendix C, File Formats) must be specified for chains that are not Gaussian or freelyjointed chains (e.g., RIS chains). The Omega List File lists the names of files containing structure factor data for various temperatures and degrees of polymerization.
The PR_System/Interactions command is used to specify the hard core and LennardJones interaction parameters, for each pair of nonequivalent sites in the system. Rather than specifying each interaction separately, you can tell PRISM to use combining rules [_{AB} = (1/2) (_{AA }+ _{BB}) and _{AB} = (_{AA}_{BB})^{1/2}] to determine these parameters.
The PR_Compute/Process command is used to select among the three general types of calculations:
The PR_Compute/Conditions command is used to specify whether calculations are to be performed at "constant density" (for blends, this is really constant site volume) or constant pressure.
The constant pressure calculations are performed by varying the density until the desired (virial) pressure is reached. Constant density calculations are performed directly and, thus, are more rapid. Therefore, unless there are special considerations (e.g., you wish to account for thermal expansion), it is generally best to perform PRISM calculations at constant density.
If you have trouble getting the PRISM calculations to converge in a reasonable amount of time, the PR_Compute/Strategy command can be used to optimize the parameters that determine how the background job goes about solving the RISM equations.
Next, the PR_Compute/PR_Run command is used to execute the PRISM background job.
While the job is running, the Textfile/Get command can be used to monitor its progress by reading the log file (extension .prlog). When the background job successfully finishes, you are left with one or more .prtbl files that may be plotted using the Graph/Get command. See Appendix C, File Formats, for descriptions of the contents of these files.
At this writing, the existence of a spinodal curve of the LCST type arising from differences in thermal expansion coefficients (or free volume differences) of the two system components has not been investigated in the context of PRISM theory. However, there is a way to mimic an LCST arising from "specific interactions" (Coleman et al. 1991) in a binary blend by judicious choice of and parameters. For example, one may set _{AA} = _{BB} < _{AB}, with _{AA} = _{BB} = 0 and _{AB} > 0. If the difference _{AB}  _{AA} is large enough, then the athermal system (corresponding to the high temperature limit, where the potential tails become irrelevant) will phase separate. Similarly, if _{AB} is sufficiently large, then the AB attraction (mimicking in very simple fashion a specific interaction, such as hydrogen bonding) will cause mixing at low temperature, resulting in LCST behavior. Apart from studies performed at Biosym (now Molecular Simulations) confirming the existence of this phenomenon (Honeycutt 1991), little research has been done in the context of PRISM theory for systems of this type.
If you are unable to obtain a solution at all for a blend system, or a onecomponent system with attractive interactions, then it is possible that the system is in a twophase region. If you expect your system to exhibit UCST behavior, then try raising the temperature. If you expect LCST behavior, try reducing the temperature. You may of course vary other quantities, such as the density (particularly for athermal blends) or volume fraction, in an attempt to find a solution. In the one component case (and sometimes in the two component case), two phase behavior corresponds to liquidvapor phase separation. Try raising the temperature or reducing the depth of the attractive wells.
Another possibility, if no solution is found for a one or multicomponent system, is that the density or the hard core diameters are too large. (In some cases, the PRISM background job can determine that this is so and write a corresponding message to the .prlog file.) Try reducing one or more of these quantities.
n  t(min) 

1  .3 
2  3.9 
3  19.9 
4  169.0 
For a Gaussian or freelyjointed chain containing more than one type of site, the time required for the PRISM program to calculate the intramolecular structure factors scales as the square of the degree of polymerization. For large chains, this step may take up a large fraction of the total computation time. Because of this, an omega list file is automatically written to disk when structure factors of this type are calculated. The name of the omega list file is <run_name>.omlist. In subsequent calculations, you can then use the Other option for the Omega Type parameter of the PR_System/Omegas command, and specify this file as the Omega List File. For multiple PRISM calculations on large chains, this can save a great deal of time.
When calculating spinodal curves for blend systems with deep LennardJones wells (i.e., large, positive values), it is possible to encounter a liquidvapor spinodal rather than the desired liquidliquid spinodal. The PRISM program currently is unable to distinguish between these situations. One signature of a liquidvapor phase separation is a very flat, or even concave upward, apparent "UCST" spinodal curve (when plotted as a function of volume fraction). By performing a spinodal calculation for a single component system containing only one of the components in the original system (e.g., the component with the deepest LennardJones wells), you can generally confirm whether this is the case, because the only (zero wave vector) spinodal possible in a onecomponent PRISM system is a liquidvapor spinodal. Thus, if you get phase separation in the onecomponent system at a temperature near that at which phase separation is predicted in the blend, the blend spinodal you have obtained is probably a liquidvapor spinodal. You can more directly confirm the presence of this phenomenon by looking at the compressibility in the blend's .prlog file. At a liquidvapor transition, the compressibility diverges.
For polymer blends with temperaturedependent interactions (i.e., potential tails), the RMMSA would appear to be the better closure for obtaining quantitative values of the critical demixing temperature, T_{c}, from realistic potential energy functions, and thus is more rigorously valid. However, the Picard iteration scheme is less wellbehaved than the LowdenChandler method used to solve the MSA version of the theory. Also, in spite of its known inadequacies, the MSA is able to reproduce certain experimentally observed trends, such as effects of microstructure on blend miscibility (Honeycutt 1992b). For melts and athermal blends, the MSA appears to be reasonably adequate.
With regard to the relative validity of the two closures now used in the PRISM module, the case of block copolymers is less clear cut than the case of blends. Calculations by Schweizer (1991) indicate that the RMMSA yields a secondorder microphase separation transition (MST), marked by a divergence in the partial scattering functions at nonzero wavevector. Leibler's (1980) seminal mean field treatment of diblocks predicts the same. However, Leibler and others argue that in real diblock systems, fluctuations should destroy the second order transition (Leibler 1980; Fredrickson and Helfand 1987; Brazhovskii 1975), turning it into a firstorder transition. This prediction is consistent with the MSA results, which fail to predict a secondorder transition (Schweizer, 1991; Honeycutt 1992a). Thus, the MSA may be more appropriate in some ways for block copolymer systems than is the RMMSA.
To access the online tutorials for PRISM, click the biplane or mortarboard icon in the Insight interface, or select Online_Tutorials from the Help pulldown.
Then, from the Open Tutorial window, select Polymer Modeling and Property Prediction tutorials, and then select PRISM Module Tutorials from the list of modules. Choose from the list of available lessons:
Lesson 1: Structure of a Polyethylene Melt
Lesson 2: Equation of State Calculation for FreelyJointed Chain of Polyethylene
Lesson 3: Scattering Functions of a Block Copolymer
Lesson 4: Spinodal Curve for a Simple Blend
Lesson 5: Generating an LCST Spinodal Curve
Lesson 6: Estimating PRISM parameters: One Approach
(Lessons 7, 8, and 9 are not included as online tutorials. They are given below.)
You can access the Open Tutorial window at any time by clicking the Open File button in the lower left corner of the Pilot window.
For a more complete description of Pilot and its use, click the onscreen help button in the Pilot interface or refer to the Introduction in the Insight II manual.
Begin by creating a directory called prism_tut and changing to it. To do this enter the following at the prompt in a UNIX window:
> mkdir prism_tut > cd prism_tutYou will need some auxiliary files to perform this tutorial. These files specify single chain structure factors for polyethylene computed using the RIS Monte Carlo technique. Enter the following commands:
> polymer_tutorials > cp $PRISM_EXAMPLES/*.omlist $cwd > ln s $PRISM_EXAMPLES/skgen skgen
This lesson assumes a basic familiarity with the RIS module. If you are not familiar with this module, see the RIS Tutorials section in Chapter 12. Note: the function referred to as(k) or omega in the PRISM module is known as S(k) or S_of_k in the RIS module.
This lesson uses the following commands, accessed from the specified module.
Polymerizer module
 

Homopolymer




Polymerize




RIS module
 

Sequencer

RIS_Monte_Carlo

Graph


S_Run

Scatterers

Get



Range

Put



MC_Run



Viewer module
 

Session




Unix




Quit




1. Create a Poly(vinyl chloride) Molecule
If you are continuing from the previous lesson, make sure that all the objects on the screen have been deleted before continuing. Otherwise, start Insight II by typing insightII at the UNIX prompt.
Click the Accelrys icon and select the Polymerizer module and build a 10mer of poly(vinyl chloride) (PVC) using the Homopolymer/Polymerize command.
3. Define the Scattering Centers
You have just defined scattering centers appropriate for a two site type PRISM representation of PVC. The first site type represents a methylene group. The second, a substituted methylene and chlorine group together, with the site center located halfway between the carbon atom and the chlorine atom.
Depending on the application, two site types may or may not be enough to adequately represent this vinyl chain. It is possible that three site typesmethylene, substituted methylene, and chlorinewill be required. However, due to the slowness of PRISM calculations for systems containing large numbers of site types, it is generally best to start with fewer sites, adding more only as necessary.
4. Specify the Small Wavevector Range
S_of_k Mode specifies the algorithm
used to compute S(k) (known as(k)
in the PRISM context).
Select the RIS_Monte_Carlo/Range command.
Range To Set: S_of_k
Here, N is the degree of polymerization. The order N method for calculating S(k) is more approximate than the order N^{2} method, but is much faster for large chains (containing more than 100 bonds or so). Since the PVC chain used in this lesson is small, the order N^{2 }method is adequate.
You will perform the S(k) calculation in two steps, one for small wavevector and one for larger wavevector. The PRISM module requires structure factors out to k 20 Å^{1}, but because S(k) varies more rapidly at small k, a finer grid is required for smaller k values than for the larger values. In this step, you have specified the small k fine grid for the first calculation.
5. Run the First RIS Monte Carlo Calculation
Select the RIS_Monte_Carlo/MC_Run command. Enter pvc_smallk for Run Name and the PVC molecule name for Assembly/Molecule. Toggle s_of_k to on and select Execute.
This calculation takes about 1.5 minutes on a Silicon Graphics 4D/25. When it is finished, proceed to the next step.
6. Specify the Large Wavevector Range
Select the RIS_Monte_Carlo/Range command.
Range To Set: S_of_k
7. Run the Second RIS Monte Carlo calculation
Select the RIS_Monte_Carlo/MC_Run command. Enter pvc_largek for Run Name and the PVC molecule name for Assembly/Molecule. Toggle S_of_k to on and select Execute.
This calculation takes about 1.5 minutes on a Silicon Graphics 4D/25. When it is finished, proceed to the next step.
8. Plot and Write Out the Data
Use the Graph/Get command to plot the data from the file pvc_smallk_Sk.tbl. Make three plots on the same graph in the following order:
S_CH2CH2(k) versus k
S_CClCCl(k) versus k
S_CH2CCl(k) versus k
In the PRISM module, the DP of a chain is the total number of sites of all types in that chain. Since end groups are not included in this particular scattering calculation, the DP is 19 rather than 21.
Using any text editor (e.g., vi) create a file named pvc.omlist that contains only the following line:
> 19 300.0 pvc_T300a_om.tbl pvc_T300b_om.tbl
Having done this, you may now use the file pvc.omlist as the Omega List File in the Omegas command of the PRISM module. Note that if you wish to perform calculations on PVC for a DP other than 19 or at a temperature other than 300 K, you will have to compute more RIS Monte Carlo scattering functions. However, the PRISM program is capable of interpolating omega data if none are available for the exact DP and temperature of interest. Temperature interpolation tends to be more reliable than DP interpolation. If interpolation has to be done, a warning is included in the PRISM log file. For more details on the formats of omega files and omega list files, see Appendix C, File Formats.
Continue with the next lesson or use the Session/Quit command to exit Insight II.
This lesson uses the following commands, accessed from the specified module:
PRISM
 

PR_System

PR_Compute

Textfile

Background_Job

Components

Process

Get

Kill_Bkgd_Job

Omegas

Conditions




PR_Run



Viewer module
 

Session




Quit




1. Perform a Calculation on a Toodense System
If you are continuing from the previous lesson, delete all the objects on the screen by typing delete *. Otherwise, start Insight II by typing insightII at the UNIX prompt.
Build a polyethylene chain as in step 2 of lesson 1.
Using the PR_System/Components command, define a one component, one site type polyethylene system with Sigma11 equal to 4.5 and Epsilon11 equal to 0.
Using the PR_System/Omegas command, specify a Gaussian distribution with a DP Of Chain of 1000. Select Execute.
Select the PR_Compute/Process command. Specify a Structure_and_Chi calculation at 450 K. Select Execute.
Select the PR_Compute/Conditions command. Set the Condition Mode to Const_Density of 0.8. Note that constant density is set to 0.6. Select Execute.
Finally, start the calculation with PR_Compute/PR_Run.
After a few seconds, the following message:
Runaway solution: system is probably too dense or hard cores too large
appears at the bottom of the screen. The problem is that the hard core diameter is too large for a methylene group, resulting in an artificially large packing fraction at a density of 0.8 g cm^{3}. The solution to the problem is to reduce Sigma11 to a more reasonable number such as 3.9 Å and to repeat the calculation.
2. Perform a Spinodal Calculation with a Poor Initial Temperature
In the Strategy command:
Conjugate: off
Set up a blend system as in steps 1 through 4 of Lesson 4 given in Pilot. In the PR_Compute/Process command, set these parameters:
Compute Mode: Spinodal
Monitor the progress of the calculation by using the Textfile/Get command to read the .prlog file for your job. It takes several minutes to find a PRISM solution at the initial temperature, but calculations proceed more rapidly after that. Note, though, that the SPINODAL criterion (which vanishes at the spinodal) is large, and decreases only slowly as the temperature is dropped. This is because the initial temperature far exceeds the spinodal temperature.
Note also that the SPINODAL criterion does not necessarily vary monotonically with temperature far away from the spinodal. Because of this, the PRISM job may have trouble locating the spinodal if the initial temperature differs too greatly from the spinodal temperature. When you see this occurring, it is often best to kill the background job and restart the calculation with a more suitable initial temperature. Use the Append_Output option in the PR_Compute/PR_Run command, if necessary, to avoid overwriting any spinodal data generated by the initial calculation.
3. Attempt a Calculation within a GasLiquid Spinodal
When performing PRISM calculations on systems with deep attractive tails, you may encounter situations where no convergence occurs in the PRISM calculation. Sometimes, this indicates liquidvapor phase separation. You have already encountered this situation in Lesson 6 given online through Pilot for the 60 K well case at high temperatures. When this is the case, the remedy is to perform the calculations at higher density or pressure, or alternatively, to decrease the depth of the potential wells if these are not already known to be accurate.
In the PR_Compute/Process command:
Tinit: 450
Use the PR_Compute/PR_Run command to start the background job.
Use the Textfile/Get command to monitor its progress.
Note that convergence to a solution occurs very rapidly at the first few temperatures, but that no convergence occurs once the temperature falls to 300 K. This is because the system is within the liquidvapor twophase region. In a constant density calculation, liquidvapor coexistence is encountered when the temperature drops, contrary to the constant pressure case. Liquidvapor coexistence is only possible for a system with attractive interactions. You can verify this by setting Epsilon11 to 0 and noting that you easily find a solution at 300 K and below for this hard sphere system.
Use the Session/Quit command to exit Insight II.
Note: This lesson takes about 40 minutes on an SGI Indigo (R3000, 33MHz), and longer on slower machines such as an SGI IRIS 4D/25. As in the previous lesson, less detail is provided here than in other tutorials because it is assumed that you are now familiar with the procedures for setting up PRISM calculations.
This lesson uses the following commands, accessed from the specified module:
PRISM
 

PR_System

PR_Compute

Graph


Components

Process

Get


Omegas

Conditions



Interactions

Strategy




PR_Run



Viewer module
 

Session


Quit


Start Insight II by typing insightII at the UNIX operating system prompt.
2. Build Two Polyethylene Oligomers
Click the Accelrys icon and select the Polymerizer module. Build two polyethylene oligomers, each with a DP of 6.
3. Set Up the PRISM Calculation
Click the Accelrys icon and select the PRISM module. Select the PR_System/Components command. Add the two molecules to the system, each with Sigma11 equal to 3.7 Angstroms and Epsilon11 equal to 0.
Select the PR_System/Omegas command. Define the Omega functions for each molecule by accepting all default values.
Pick the site atoms on the two different molecules, and specify an EpsilonAB value of 0.1. Select Execute.
While still in the PR_System/Interactions command:
Interaction Param: Sigma
Select the PR_System/Interactions command.
Interaction Param: Epsilon
Select the PR_Compute/Process command.
Compute Mode: Spinodal
Tinit: 300
Note that all computations will be performed at a volume fraction of 0.5. For a symmetrical system such as this one, 0.5 is the critical volume fraction. Thus, the spinodal temperature computed by the PRISM module is also the critical temperature.
Select the PR_Compute/Conditions command to specify a Const_Density calculation.
Select the PR_Compute/PR_Run command. Enter mmsa_dp_dep as the Run Name, and select Execute.
Note: Unless you have a very fast workstation, this calculation will take at least 30 minutes.
5. Plot the Critical Temperature vs. DP
When the calculation is done, select the Graph/Get command, and enter mmsa_dp_dep_sp.prtbl as the File Name.
X Function: DP
Note that the variation of the critical temperature with the DP is almost a perfect straight line. This is consistent with the classical FloryHuggins theory and with recent experimental and simulation results for isotopic blends (Gehlsen et al. 1992; Deutsch and Binder 1992). If you wish, you may perform a similar calculation using the MSA as the closure to see that the MSA predicts a critical temperature proportional to the square root of the DP. A related consequence is that the values of the critical demixing temperatures predicted by the MSA for UCST systems are far lower than those predicted by the RMMSA.
As the results of this tutorial demonstrate, the RMMSA is in principle the preferred closure. However, difficulties in solving the RMMSA equations may make the MSA the preferred closure in many cases. For melts, the MSA should be adequate. For blends in which a trend is of greater interest than the precise value of T_{c}, the MSA also may be useful.
Finally, recall that in the RMMSA, the hard core system is used as a reference system, which is solved with the MSA. Using current methods of solution, it is not possible to use the RMMSA when the reference system is phaseseparated. Thus, computing LCST phase diagrams using the method of Lesson 5 given online through Pilot is not now possible with the RMMSA.
Use the Session/Quit command to exit Insight II.