Polymers are substances composed of many simple molecules that are repeating structural units, called monomers. A single polymer molecule may consist of hundreds to a million monomers and may have a linear, branched, or network structure. Copolymers are polymers composed of two or more different types of monomers.

The complexity of polymer properties is directly related to the complexity of their microscopic structure, and associated with this is a complexity in the theoretical description of their behavior. An adequate description of many polymeric material properties awaits the development of suitable mathematical and statistical tools.

Polymer science is an interdisciplinary field involving the mathematical sciences, physics, chemistry, biology, and engineering. It is an area with a wide range of theoretical and experimental investigations. Polymer science has provided us with a diverse host of new materials, thereby enhancing technological advancement. Meanwhile, the theoretical description of the properties of these materials has involved the use of a wide range of mathematical sciences techniques, which are illustrated below through the use of several examples. References are selective rather than comprehensive.

Polymeric materials are usually shaped in the liquid state, either in the melt or in solution, followed by a solidification process. Typical liquid-state shaping operations include extrusion, injection molding, fiber drawing, and film blowing. The macromolecules orient as a result of the deformations during processing. The properties (mechanical, optical, and so forth) of the solidified object depend in part on the stress and orientation fields developed during processing and solidification, as well as on morphological features of the crystalline phase in semi-crystalline polymers, in addition to the chemical structure and molecular weight of the polymer. (Polyethylene, a high-molecular-weight analog of paraffin wax, is used both for grocery bags and for prosthetic implants.) The strength of an injection-molded part, for example, depends critically on the placement of the ports through which the molten polymer is introduced. Frequently a polymer must flow through tiny channels and surround small and delicate parts. Finally, flow during processing plays a major role in determining multiphase systems' morphological features, such as incompatible blends, block copolymers, and so on.

From the point of view of materials processing, the goal is to take information about the molecular structure and to use computer-aided design methods to predict the structural and stress state, hence the properties, of the solid shaped object. Such a process simulation requires solution of the continuum mass, momentum, and energy balances through all steps of the process, including solidification, as well as solution of molecular-level equations relating macroscopic quantities and structure. The momentum equation requires a relation (a constitutive equation) between the stress and the deformation. A state-of-the-art overview of process simulation can be found in a chapter of an encyclopedia (Denn, 1988), where monograph and textbook references are given.

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Mathematical Research in Materials Science: Opportunities and Perspectives
3
MACROMOLECULAR STRUCTURES
INTRODUCTION
Polymers are substances composed of many simple molecules that are repeating structural units, called monomers. A single polymer molecule may consist of hundreds to a million monomers and may have a linear, branched, or network structure. Copolymers are polymers composed of two or more different types of monomers.
The complexity of polymer properties is directly related to the complexity of their microscopic structure, and associated with this is a complexity in the theoretical description of their behavior. An adequate description of many polymeric material properties awaits the development of suitable mathematical and statistical tools.
Polymer science is an interdisciplinary field involving the mathematical sciences, physics, chemistry, biology, and engineering. It is an area with a wide range of theoretical and experimental investigations. Polymer science has provided us with a diverse host of new materials, thereby enhancing technological advancement. Meanwhile, the theoretical description of the properties of these materials has involved the use of a wide range of mathematical sciences techniques, which are illustrated below through the use of several examples. References are selective rather than comprehensive.
Polymeric materials are usually shaped in the liquid state, either in the melt or in solution, followed by a solidification process. Typical liquid-state shaping operations include extrusion, injection molding, fiber drawing, and film blowing. The macromolecules orient as a result of the deformations during processing. The properties (mechanical, optical, and so forth) of the solidified object depend in part on the stress and orientation fields developed during processing and solidification, as well as on morphological features of the crystalline phase in semi-crystalline polymers, in addition to the chemical structure and molecular weight of the polymer. (Polyethylene, a high-molecular-weight analog of paraffin wax, is used both for grocery bags and for prosthetic implants.) The strength of an injection-molded part, for example, depends critically on the placement of the ports through which the molten polymer is introduced. Frequently a polymer must flow through tiny channels and surround small and delicate parts. Finally, flow during processing plays a major role in determining multiphase systems' morphological features, such as incompatible blends, block copolymers, and so on.
From the point of view of materials processing, the goal is to take information about the molecular structure and to use computer-aided design methods to predict the structural and stress state, hence the properties, of the solid shaped object. Such a process simulation requires solution of the continuum mass, momentum, and energy balances through all steps of the process, including solidification, as well as solution of molecular-level equations relating macroscopic quantities and structure. The momentum equation requires a relation (a constitutive equation) between the stress and the deformation. A state-of-the-art overview of process simulation can be found in a chapter of an encyclopedia (Denn, 1988), where monograph and textbook references are given.

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The first section below on single-chain conformations addresses the case of individual polymers in solution or in interaction with confining surfaces. Many basic concepts of polymer science have arisen in the study of these single-and few-chain systems, and dilute solution experiments are still essential in characterizing the properties of the polymers, so that polymeric materials can be made reproducibly and theories of polymer properties can be tested against carefully controlled experiments. The next section considers protein properties and scientific questions relevant to applications in biotechnology that are associated with predicting the conformation, folding, and long-time dynamics of proteins.
Subsequent sections discuss the flow and equilibrium properties of polymers in the liquid state, the state from which many polymeric materials are fabricated by extrusion, molding, and so forth. Descriptions of the viscoelastic properties of molten polymers often rely on ad hoc phenomenological models. Although these display remarkable agreement with a wide range of experiments, a firm theoretical underpinning for these models is lacking. Mathematical issues arise from this viscoelasticity because the responses typically associated with viscous liquids and elastic solids are combined; the stress at any position depends on the entire deformation history of the material element located at that position. The formulation of viscoelastic constitutive equations that accurately portray material behavior remains a subject of research. Fundamental issues of existence, uniqueness, and qualitative behavior have not been adequately addressed. Computational schemes for flows in complex geometries often fail to converge under conditions of processing interest. Instabilities occur that are absent in low-molecular-weight liquids. Many of the issues discussed here have been reviewed previously (Denn, 1990).
Many theoretical and mathematical sciences issues are involved in the construction of and solutions to models of polymer fluids, which are being developed along three distinct lines. One of the oldest involves the use of an underlying lattice to greatly simplify the mathematics at some concomitant expense in representation of reality. Recent developments in polymer theory build on successes made over the past few decades in the theory of simple molecular fluids, especially in integral equations and density functional methods. Density functional methods are discussed in this chapter in relation to studies of interfaces in polymer systems, followed by a description of problems in block copolymers and in stiff polymers and liquid crystals, as well as mention of other areas; see also Chapter 8.
Problems in polymer science are almost as diverse as those of all of materials science, and are associated with issues of fracture, thermal stability, ageing, transport, adhesion, and so on. Polymers may be used as organic conductors and nonlinear optical materials. Some of the issues are addressed in other chapters in this report. The omission of specific areas is indicative merely of the limited nature of this report.
SINGLE-CHAIN CONFORMATIONS
Very interesting mathematics arises naturally in the study of polymeric materials. The most basic model in this discipline is the Wiener path model of single polymer chains. This picture represents the shapes of polymer chains as being much like the dynamical trajectories swept out by particles undergoing Brownian motion. At the next level of study,

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the self-interactions and mutual interactions between polymer chains are incorporated into the Wiener path construction, where constraints are added to the family of paths describing the polymers. A wide range of phase transitions are encountered in this generalization. Adsorption of polymers onto surfaces, the helix-coil transition, polymer collapse, protein folding, and phase separation of polymeric fluids are examples. Important areas of research are concerned with polymers at interfaces (Douglas et al., 1986a, b) and the dilute solution properties of polymers dissolved in small-molecule solvents (Douglas et al., 1990). Of particular interest is how the topological structure of the polymer is reflected in the bulk properties of polymer solutions. Polymers can be formed in ring shapes, stars, and combs and also in very irregularly branched structures. The properties of interest in characterizing these specialized polymeric fluids include the radius of gyration, hydrodynamic radius, intrinsic viscosity, and virial coefficients. The theoretical description of these different topological classes of interacting polymers relies on the renormalization group (RG) method (see Chapter 8). This nonrigorous mathematical method is closely related to Borel resummation and Euler transform methods for resummation of asymptotic series and allows practical and useful statements to be drawn from rather meager (but messy) perturbative calculations (Freed, 1987). While great success has been achieved in describing the equilibrium properties of dilute and semidilute polymer solutions (Freed, 1987), the dynamical RG theories are in a more primitive state. The convergence of RG expansions appears to be significantly poorer than in the equilibrium case, and dynamics in semidilute solutions are poorly understood. Major questions also remain in problems such as polymer collapse, where several interactions with different critical dimensionalities become important. Although a majority of polymer scientists consider theories of single-chain properties to be in good shape, serious conceptual and mathematical questions abound. Considering that theories of protein structure and dynamics, of molten polymer flow properties, of rubber elasticity and gel swelling, and of many more polymer phenomena have their conceptual underpinnings in single-chain models, further advances in many of these significant single-chain problems will reverberate through polymer science.
The case of a polymer interacting with a surface provides a good example for a study of the RG method from a purely mathematical sciences standpoint, since exact solutions are possible in this instance. Recently, the Wiener integral formulation of interacting polymers has been recast in terms of an equivalent integral equation formulation (Douglas, 1989a). The surface adsorption phase transition can be understood mathematically in terms of the Fredholm alternative, and the RG method can be interpreted naturally in terms of classical integral equation mathematical machinery. The integral equations describing the surface interacting polymers are fractional differential operators, where the order of the fractional differential operators is related to the Hausdorff dimension of the ''local time'' intersections of the polymer path with the interface. Recent work has involved treating polymers in the proximity of fractal surfaces, and this work is based in large part on rigorous results provided by mathematicians (Douglas, 1989b). Quite recently, the partition function of a polymer near a rough surface ("fractal") has been formulated in terms of renewal theory (Feller, 1971), and results the same as those obtained in heuristic calculations (Douglas, 1989a) are rigorously obtained as a corollary of Feller's theory of recurrent events and some established results of probabilistic potential theory (Hawkes, 1971; Taylor and Wenderl, 1966). This

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nicely illustrates the application of ideas from probabilistic geometry to polymer science.
No discussion of polymer conformations would be complete without a brief mention of the rotational isomeric model, which represents the polymer as a discrete chain with fixed discrete values permitted for dihedral angles (Flory, 1969). This model is mathematically isomorphic to a one-dimensional Ising model, but chain conformational properties are represented in three-dimensional space. The model is, in principle, exactly solvable, and considerable ingenuity has been necessary to evaluate numerically the desired chain conformational properties. While it is possible to anticipate several useful applications of this model to describing properties of stiff and helical polymers, it does not appear that additional mathematical advances are necessary in treating the model.
Polymer solutions are routinely employed to separate proteins. Such processes lead to a consideration of the mutual interaction between polymers and particles of dissimilar shape (folded proteins, vesicles, viruses, and so on). Some consideration of the mathematics involved in describing these mutual interactions reveals a connection with the exterior Dirichlet problem (Spitzer, 1964; Kac, 1974). That work is not phrased in terms of polymer science language, but these calculations nonetheless have important practical applications to understanding the stability of suspensions in the presence of dissolved polymers. There is also a need for studies of capacities associated with stable random processes in this connection. Efficient means of calculating the capacity and other shape functionals (Schiffer, 1954; torsional rigidity, fundamental frequency of drums) are needed for general shaped bodies. Another application (Hubbard and Douglas, 1993) relates the friction coefficient of an arbitrary shaped body undergoing Brownian motion to the electrostatic capacity of that body. The "intrinsic viscosity" is another "shape functional" that very badly needs further study. Mathematically this problem is related to determining the polarization of that body (Schiffer and Szego, 1949). Interesting mathematical problems arise in the solution of the interior and exterior Dirichlet problems when the boundary is a "fractal'' such as a random walk shaped polymer coil, an irregular (non-differentiable) surface, or a diffuse aggregate. Other "random" polymer problems are associated with the description of random copolymer or controlled sequences of several monomers (see also Maddox, 1993). Random copolymers are of considerable commercial importance, while proteins may be viewed as examples of how controlled monomer sequences can yield diverse and specialized materials.
A potentially very fruitful area of work is concerned with random surfaces, polymers of sheetlike connectivity. These mathematical objects are important in solid-state physics, biology, and high-energy physics and offer great promise in synthetic polymer applications (Abraham and Nelson, 1990). However, very little has been done to extend the multidimensional time generalizations of Brownian motion to describe these sheetlike polymers (Yoder, 1975). This mathematical sciences area, which is rather mature but is largely unknown to physicists and material scientists, provides a tremendous opportunity for applications of the mathematical sciences to practical materials science problems.
In summary, there are not only many opportunities for the practical applications of the mathematical sciences to questions arising in treating the properties of single-chain polymers, but there are also many new mathematical and statistical questions that are naturally posed in describing these physical systems and that lie at the frontier of available mathematics and statistics. There are mutual benefits to be gained for the mathematical

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sciences and materials science communities through serious involvement of mathematicians and statisticians in this field.
MODELING PROTEIN STRUCTURE AND DYNAMICS
Proteins, which are composed of 20 naturally occurring amino acids, provide a special class of linear polymers. Nineteen of these 20 amino acids have a definite "handedness"; that is, they are not equivalent to their mirror images. The specific sequence of amino acid monomers along the protein backbone controls the manner in which that specific molecule prefers to "fold" into a compact three-dimensional shape, and the folded shape is intimately connected with the biological task discharged by the protein. Point mutations (genetic changes of amino acid species at one location along the protein backbone) occasionally have little effect on folding pattern and biological activity; often they have disastrous effects that halt biological activity and prove fatal to the host organism. "The second half of the genetic code," as Sir Francis Crick has stressed, requires understanding precisely how the amino acid sequence determines folded structure; it is still largely an open problem (Gierasch and King, 1990; Chan and Dill, 1993; National Research Council, 1994) but obviously has major consequences for the management of genetic diseases and for aspects of biotechnology.
The protein folding problem suggests several potentially useful roles for mathematical sciences activity. The first keys on the nonlinear optimization character of folding: in most cases the biologically active folded form of a protein corresponds to the global minimum of a sequence-dependent potential energy function; see Chapter 8; also Berg (1993). Because of the flexibility of the protein molecule, the optimization typically involves 103 to 105 variables and a huge number of local minima, perhaps of the order of 101000 (Chan and Dill, 1991a, b). Principles of chemical structure and bonding provide strong constraints on folding possibilities, and so specially designed search algorithms are likely to be advantageous.
Folded structures are now experimentally known for several hundred proteins. Attempts to infer general principles of folding from this database (comprising a very small fraction of all proteins) have been only very modestly successful. Consequently, theoretical modeling takes on an enhanced importance. Simplified protein models have included both lattice models (Bryngelson and Wolynes, 1990) and continuum versions and have permitted extensive computer simulations. But the problem is still largely an open one that would benefit from innovative mathematical and statistical insight. Very little is yet known concerning the collapsed state of ordinary polymers, and work in this, perhaps simpler, area should have an impact on our understanding of globular proteins.
It should be mentioned in passing that neural networks (Sasai and Wolynes, 1990) have been applied to the protein folding problem, but so far only with modest success. However, the universes of neural network architectures and of representations of the proteins remain largely unexplored and can likely be strongly improved.
Computer packages, selling for on the order of $100,000, are being used by pharmaceutical and chemical companies in attempts to screen potential drugs before synthesis of the most likely candidates. The protein structure packages are mostly concerned with finding the equilibrium (lowest-energy) configuration of a protein molecule with

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thousands of atoms that interact through a complicated set of empirical potential functions. This process involves a huge multidimensional optimization process with many local minima. The sizes of systems considered are limited partly by the efficiency of algorithms for these minimizations. While many computations ignore solvent molecules, an accurate representation requires their inclusion, thereby enormously increasing the number of atoms to be considered and hence the dimensionality of the space in which the optimization is performed; for instance, see Boehncke et al. (1991) and references therein. Advanced optimization schemes will immediately be utilized by the package purveyors. The presence of charged groups in proteins leads to complexities in the computation of electrostatic forces and energies (Harvey, 1989; Schaefer and Froemmel, 1990). Additional advances with protein structure modeling require improvements in the empirical potential functions, especially in describing the influence of the aqueous solvent. One desired approach involves quantum mechanical treatment of the forces within a small region, with empirical force fields used only for interactions with more distant atoms. However, any successful implementation of such a scheme would greatly increase the computational labor, and so the generation of highly efficient algorithms and massively parallel computational schemes becomes a prime necessity.
Some approaches perceive the folding process to be a dynamical event, so that an understanding of protein folding can be obtained only by recourse to theories of the dynamics of proteins in the cellular environment. If protein folding proceeds with the aid of chaperone molecules (Angier, 1992), the complexity of the theoretical description increases significantly because of the requirement to treat the protein, the chaperone, the solvent, and their mutual interactions. Nevertheless, the theoretical and computational challenges are rather similar to those posed by a description of the unassisted protein folding that is discussed here. Thus, an important emerging area of study involves the description of protein dynamics (McCammon and Harvey, 1989). Studies of protein dynamics resort to the use either of simplifying models (Bryngelson and Wolynes, 1990; Sasai and Wolynes, 1990) or of complicated realistic molecular potential functions. Simplified models sacrifice many protein-specific details in order to provide understanding of generic properties of protein dynamics, whereas the potential functions are so complicated and computationally expensive that their use is limited to following the dynamics of rather simple proteins for rather short times. However, many facets of protein dynamics are specific to sequence and structure and occur on time scales that are long compared to those accessible from molecular dynamics simulations. Thus, there is a need for developing new theoretical concepts and methods to enable the treatment of long-time protein dynamics using the most realistic available molecular potential functions (Karplus and McCammon, 1983). By long-time dynamics is meant molecular dynamics calculations involving time scales much longer than current computational facilities allow.
Even if molecular dynamics computations could follow the individual atom (or group) dynamics of a large aqueous protein over long time scales, the amount of information gathered staggers the imagination. This presents us with the classic statistical mechanical problem of having vastly too much information and of needing some reduced description with suitable accuracy (Mori, 1965; Zwanzig, 1961). However, any such theory must be thoroughly tested, and this requires the performance of long-time molecular dynamics

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calculations for suitable, nontrivial test cases. Such a process has recently been begun (Hu et al., 1991) for a six-amino-acid fragment of the ACTH protein in 945 water molecules; experimental data on this system are available for fluorescent depolarization of the tryptophan emission (Chen et al., 1987). However, simulation with a Cray-optimized program (not commercially available) generates 4.3 ps of trajectory in 1 hour of Cray-YMP time, while the experimental correlation time is 250 ps. Hence, simulations of a single trajectory for 1 nsec require about 10 days of Cray-YMP time, but this pales in comparison with the times of roughly 23 Cray-YMP years required for a comparable simulation of the whole ACTH protein with only 39 amino acids. Thus, the severe limitations on molecular dynamics simulations of protein dynamics and the need for reduced descriptions immediately become evident.
Apart from the use of realistic molecular potentials, such as those provided by CHARM23 (Karplus and McCammon, 1983), another useful goal is to develop a reduced theory that requires as inputs only equilibrium information and friction coefficients (Hu et al., 1991). The former is to come from computer simulations, while the latter comes from experimental data or simple models. Thus, once the necessary theory has been developed and has passed all the stringent tests, efficient methods will need to be devised for "clever" Monte Carlo simulations (see Chapter 8) to obtain requisite equilibrium information.
Initial efforts at developing a reduced theory of long-time peptide dynamics (Hu et al., 1990, 1991) begin by decoupling the solvent motion by use of a hydrodynamic model for the solvent. Thus, the influence of solvent enters through the use of friction coefficients for individual residues or groups of atoms, depending on the model used. In effect, the dynamical equations follow from a formalism in which the memory functions are neglected. The "relevant" variables are chosen as bond vectors for flexible portions of the protein and ones sufficient to specify the orientation of aromatic groups. Given these choices, the general principles of statistical mechanics dictate the appropriate dynamical Langevin equations governing the protein dynamics (Hu et al., 1990, 1991). The theory specifies the required equilibrium input information, and the dynamical equations are numerically integrated to solve for the orientational correlation functions. However, as described above, very long simulations are required to check the theoretical predictions for these correlation functions (see Chapter 8).
The following three limitations on the current theory will require further testing and theoretical justification:
The use of a hydrodynamic model for solvent motion,
The use of a limited set of "relevant" variables, and
The neglect of memory functions.
Limitation 1 is not expected to be important for longer-time dynamics, but improved models are necessary for group friction coefficients (Pastor and Karplus, 1988; Venable and Pastor, 1988). The all-bond model used suffices, but further reduction in the number of "relevant" variables is desirable in order to treat larger protein systems. Limitation 3 is more severe, and it will be necessary to develop a theory of the memory functions that adequately describes the influence of local conformational transitions and that likewise uses only input

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equilibrium information (Perico et al., 1993). Benchmark molecular dynamics calculations and further formal theory will be useful in developing adequate descriptions of memory functions. It should be noted that the theoretical and conceptual problems associated with constructing a reduced theory of long-time protein dynamics are also important in studies concerning the local dynamics of polymers, a subject of relevance to understanding mechanical and dielectric loss properties of polymers and undoubtedly similar phenomena in biological materials; for example, see Travis (1993) and Nanavati and Fernandez (1993). Thus, useful insights can come from cross fertilization between the polymer and protein dynamics areas.
Of interest in the theoretical modeling of biopolymers (polymeric substances formed in a biological system), the above description represents a microcosm of the mathematical sciences research opportunities present in other areas of polymer science (cf., for example, Maddox, 1993). Enormous progress has been made in materials science as a result of computer simulations. It is, however, particularly difficult to simulate polymer systems, because, as is illustrated above, many important properties are associated with the molecules being large and the associated time scales being long. However, the molecules do have a smaller-scale structure with associated shorter time processes, and it is these processes that set the scale for numerical trajectory integration. What is needed is a way to overcome this problem with better integration techniques (perhaps with parallel processors) or with better approximation schemes that jump over the short time scale (such as is done with stiff differential equations). The example given for protein dynamics provides a description of one attempt at developing better approximation schemes, while the modeling of entangled polymer systems is still beset with enormous computational difficulties.
ENTANGLEMENTS, REPTATION, AND ELASTICITY
The flow properties of polymers are quite different from those of small-molecule systems. An understanding of these flow properties, the viscoelasticity of high-polymer liquids, is essential to the design of many technologically important polymer-processing operations in which molten polymers are forced through dies or into molds in order to control the shape and orientation of the product. The most coherent modeling thus far is of flow in viscoelastic liquids, and that setting is what this section mainly considers.
Constitutive Equations
Constitutive equations, which relate the local stress and conformational state to the deformation history and flow, are the focus of the field of rheology. Major journals in this field include the Journal of Rheology, Rheologica Acta, and the Journal of Non-Newtonian Fluid Mechanics, as well as the proceedings of the quadrennial International Congress of Rheology. General introductions can be found in the texts; for example, see Larson (1988) and Bird et al. (1987a); also, see Bird (1987) for a summary of some mathematical successes. The development of constitutive equations has followed two parallel routes, one based on

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general principles of continuum mechanics and one based on kinetic theory and molecular models. The former leads either to integral models, in which the dependence of the stress on the deformation history is expressed in terms of one or several integrals, or to differential models, where the stress and velocity gradients are coupled through a set of differential equations. The formalism is well understood. Coupling with the momentum equation thus leads to complicated sets of highly nonlinear differential or integrodifferential equations. Incorporation of energetics, which is essential in the analysis of polymer-processing flows, requires consideration of the temperature history in the constitutive equation as well.
Molecular approaches usually model the polymer molecule as a mechanical object, for example, a sequence of springs (simulating the entropic elasticity of "flexible" chains, which admit many conformations), rigid rods, and so on (see Chapter 8). The physical model is completed by specifying the way in which a polymer molecule interacts with the surrounding medium, both through thermodynamic potentials (which might induce a transition to a liquid crystalline phase, for example) and dynamically. The primary dynamical interactions are friction and topological constraints. The polymer molecules in these systems are intertwined, and the resulting geometrical relationship between the molecules acts as a constraint (called "entanglements") on their motions. From a topological point of view the fact that the molecules can be "disentangled" means that they are not truly entangled. This makes it very difficult to characterize the process mathematically. No microscopic treatment of entanglements in polymer systems has been possible because a complete statistical mechanical treatment requires imposition of the topological constraints introduced by the entanglements; this is something that may benefit from the emerging theory of knots (Delbruck, 1962; Korsaris and Muthukumar, 1991; Jones, 1985; Thirumalai, 1992). The result is a reliance on a phenomenological description and the construction of idealized simple models (Doi and Edwards, 1986; Bird et al., 1987b), picturing the molecules as being effectively confined within a tube so that their primary mode of motion involves moving along the tube, a motion called reptation (de Gennes, 1979). Hence, the many-chain problem is reduced to that of a single chain in a postulated mean field. The theories do not really explain the nature of entanglements and how they arise.
While these phenomenological models are extremely successful in explaining and predicting a wide variety of nontrivial aspects of the nonlinear viscoelasticity of high-polymer liquids, there are several severe limitations (Lodge et al., 1990): "The main weaknesses of reptation and reptation-based models are three. First, the reptation hypothesis is just that, and has no real justification in molecular theory. Second, reptation assumes the phenomena of entanglement, and therefore offers little insight into what entanglements are, or under what molecular conditions they become effective. The description of entanglements in polymer systems may benefit from theories of knots or from topological field theories (Birmingham et al., 1991; Kholodenko, 1989). Third, as a single-chain-in-a-mean-field postulate, it is difficult to begin with reptation and build in the effects of many-chain interactions in anything but an arbitrary fashion, no matter how plausibly it may be done." Much remains to be accomplished in developing a molecular theory for the dynamics and viscoelastic properties of high polymer liquids. A molecular theory must contend with the problem's cooperative many-chain nature and thereby quantitatively describe entanglements and their influence on the polymer motion. Reptation-like or alternative models must

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emerge as a self-consistent approximation, with a mechanism available for delineating the limitations of the models and for incorporating corrections and superior models in a systematic fashion. Mathematical scientists' assistance is desired for this nontrivial, highly complicated, technologically important problem.
The entanglement problem runs considerably deeper. Rubbery materials exhibit their elastic properties because each molecule in them is attached to other molecules, with the whole forming a network. These connections prevent certain relaxations of the molecules when they are stretched. Descriptions of this source for rubber elasticity have been available for a long time, but again they are based on phenomenological single chain type models. The presence in the network of crosslinks between polymer molecules makes it possible for true topological entanglement (knotting) to occur. However, predicting the manner in which entanglements contribute to rubber elasticity is still a distant, important goal, and polymer elasticity theory for gel swelling is far from satisfactory.
Existence, Well-Posedness
The solution of boundary value problems for the flow of viscoelastic liquids in complex geometries has been exceedingly difficult in the strongly nonlinear regime. Numerical issues are addressed subsequently. Some investigators believe that the inability to obtain solutions in regions of processing interest is a consequence of problem formulation through inappropriate constitutive equations. Only in the past 15 years have fundamental analytical questions of existence, uniqueness, and qualitative behavior of solutions been addressed in a systematic manner. A summary of progress in this area, up to roughly 1987, has appeared (Renardy et al., 1987).
Existence results for initial value problems generally fall into two categories. First, there are results on local time existence, and second, there are results on global time existence and asymptotic stability for small perturbations of the rest state. There are now numerous results of both types for differential as well as integral constitutive models, and there are also some recent results on kinetic theory models. A far more difficult question is whether solutions exist globally in time for large data. For one thing, this depends on the type of the equations. Some constitutive equations are hyperbolic, and when coupled with the momentum equation the system can change type within the flow field (Joseph, 1990; Luskin, 1984). Problems with change of type are very difficult, and few rigorous mathematical results of a general nature are known, even in the simpler situation of gas dynamics (see Chapter 3). Equations of hyperbolic type do not have globally smooth solutions due to shock formation. Viscosity terms or singular memory functions can preclude shock formation. Under suitable hypotheses, there are some global existence results for such models in one-dimensional problems.
It has been shown that many constitutive models admit Hadamard instabilities in certain flows. Even models that are well posed for smooth solutions could become ill posed in discretized versions that step into forbidden regions of stress.
Rigorous mathematical existence theorems have been developed for flows that perturb a state of rest or uniform flow. Both differential and integral models have been

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investigated. There are no global existence results such as are known for the Navier-Stokes equations, and there is even reason to think that such results may not hold.
An interesting question emerges in problems with inflow and outflow boundaries, which may arise from the truncation of the flow domain for numerical purposes. The memory of the fluid manifests itself in the need for prescribing boundary conditions at the inflow boundary, in addition to the ones needed in the Newtonian case. For differential constitutive models, some sets of inflow conditions leading to well-posed problems have been identified, but a characterization of all admissible inflow conditions is but a distant vision.
Numerical Methods and Singularities
Numerical simulation of non-Newtonian flow is a flourishing area of research, with frequent international workshops. The field has been surveyed (Keunings, 1989), and more recent papers, including refereed workshop proceedings, are published regularly in the Journal of Non-Newtonian Fluid Mechanics. A convergence problem known as the High Weissenberg Number Problem still limits many calculations to flow regimes that are uninteresting from the point of view of materials processing. The nature of the problem is understood: large-stress boundary layers, which occur for many viscoelastic constitutive equations near corners and stagnation points, are too sharp to be resolved by usual finite-element and finite-difference techniques, and the resulting errors contaminate the solutions. The cause of the problem is not understood, however. Some investigators believe the problem is one of numerical resolution of large-stress gradients, and have emphasized construction of new algorithms—special finite-element techniques, for example. Others believe the cause is associated with the formulation and question whether integrable solutions to the stresses (that is, finite forces) even exist for the constitutive equations in common use.
The velocity and stress field in the neighborhood of a corner is a fully solved problem for a Newtonian fluid, where the stress exhibits a power-law singularity and is integrable. The solution is unknown for all but the most trivial viscoelastic constitutive equations, but nonexistence of a power law is easily established in a number of cases of interest. Solution of the corner singularity for elementary viscoelastic fluids would be a major advance, since it would establish whether the predicted behavior near corners and in other boundary condition changes is physically sound, and it would provide the opportunity for construction of elements for numerical simulations that incorporate properties of the singularity.
There is some experimental evidence that the no-slip boundary condition (in which the fluid takes on the velocity of a solid boundary at the interface) commonly used for low-molecular-weight fluids breaks down for polymeric liquids in regions of high stress. This observation is discussed below in the context of extrusion instabilities. Wall slip would alleviate the stresses near a corner. From the perspective of micromechanical models of polymer solutions, singularities are an artifact of using too coarse grained a description of fluid rheology and boundary conditions. For example, all commonly used constitutive theories assume local homogeneity of the kinematics that lead to polymer deformation. Some recent theories (El-Kareh and Leal, 1989; Bhave et al., 1991) include the physics of

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mutually exclusive theoretical formulations (which nevertheless arrive at similar analytical expressions) have been successful in capturing the observed phenomena, both the onset of the instability and the (average) measured slip velocities, including the effect of upstream pressure. This line of thinking (reviewed in Denn, 1992) suggests that progress in understanding extrusion instabilities will be made by pursuing concepts more commonly used in areas like adhesion and dynamic fracture. While it is possible to construct physical scenarios of chain response that could lead to oscillatory phenomena like those observed experimentally, dynamical interactions between the nonlinear field equations and the nonlinear slip boundary conditions have not been explored except in the simplest (linear) cases, where only steady solutions have been observed.
Another approach is based on the concept of a "constitutive" instability. Many constitutive models based on molecular concepts exhibit a nonmonotonic shear stress/shear rate curve, hence the breakdown of steady shear flow at a critical rate. In some cases there are two extrema, leading to a series of investigations dating to 1966 that suggest that the multivalued stress curve is the source of spurt flow. Recent studies have explored the dynamics of shear flow of fluids with non-monotonic flow curves, which result in well-posed initial and boundary value problems (for example, Malkus et al., 1991). Analysis and numerical simulations show that the system changes state in a thin layer near the wall, giving the appearance of a slip layer and showing spurt in pressure-driven flow and persistent oscillations in piston-driven flow. A singular perturbation formulation yields a one-parameter family of quadratic, planar dynamical systems having a rich structure.
Flow Instabilities
Polymeric liquids are known to exhibit flow instabilities under conditions where the flow of Newtonian fluids would be stable, and in some cases these conditions correspond to the practical regime for processing or property measurement. Taylor-Couette instabilities can occur even in the absence of inertia, for example, and torsional shear flow (which is widely used in rheometry) shows a rich range of unstable behavior; see Petrie and Denn (1976) and Larson (1992) for general reviews. From the point of view of an analyst, one of the major challenges is the rigorous justification of the traditional techniques of studying stability and bifurcation. While the stability (linear and nonlinear) of the rest state has been studied extensively, there are few rigorous results on the far more interesting question of stability of other flows. The issues to be studied are whether linear stability is indeed determined by spectral properties, whether linear stability implies nonlinear stability for small disturbances, and whether the usual reduction procedures of bifurcation theory (for example, center manifolds) can be rigorously applied. Although these issues are well understood for Newtonian fluids, the proofs available in the literature do not extend to the equations governing viscoelastic fluids.

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Micromechanics
Computational and theoretical issues dealing with micromechanics are discussed in Chapter 8 of this report, but they should be mentioned briefly in the context of polymeric materials. There is obvious interest in attacking viscoelastic flow problems for particles in fluids directly at the microscopic level by solving the micromechanical equations together with those describing the motion; insofar as one is interested in materials that match the microscopic description, micromechanical models contain more information than continuum models. In addition to purely macroscopic variables, there is at least a statistical description of the microstructural state. Furthermore, microstructural simulation of extremely simplified flows shows chain behavior that can differ from the dynamics imbedded in continuum formulations (see, for example, Rallison and Hinch, 1988).
Relevant mathematical sciences issues do not seem to have been considered: (1) Given the equations that describe the microdynamics, what can be said about the general types of behavior (that is, the nature of solutions) that are possible? For example, if the motion is unsteady, for what class of motions does the microstructure exhibit globally attracting trajectories in orientation space? (2) How can the micromechanical model equations for nontrivial flow problems be efficiently solved? Consider, for example, a suspension of rodlike particles. Even if the problem of describing rod interactions is ignored and only noninteracting rods are considered, the exact problem solution not only requires the usual continuum velocity, pressure, and stress fields, but also requires keeping track of the statistical distribution of orientations for each material point. This yields a high-dimensional problem at each material point, coupled with a complex macroscopic problem. Can computationally reasonable schemes be developed to solve such problems, and especially can techniques suitable for massively parallel computing be developed? Can approximation schemes be developed for the full problem (which generally involves hierarchies of moment equations or something equivalent, but is bedeviled by closure problems in the approximation scheme)? What particle behaviors may be lost in such schemes?
THEORY OF THE LIQUID STATE OF POLYMERS
One of the earliest and still widely used models for polymer systems is a lattice model in which the polymers are represented as mutually and self-avoiding random walks on a regular lattice. Each lattice site may be occupied by a monomer or a solvent molecule, or it may be vacant. A species-dependent energy εij is assigned to all pairs of monomers and solvent molecules that occupy neighboring lattice sites. The partition function for the system is defined as the sum of the number of possible configurations for the system, as weighted by the Boltzmann factor exp(-βE), where E is the total interaction energy for the configuration and β is the inverse of the thermal energy.
The standard lattice model has led to the development of a variety of counting methods to enumerate approximately the huge number of possible configurations allowed to the system. Work in the 1940s (Flory, 1941, 1942, 1953; Huggins, 1941, 1942;

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Guggenheim, 1944) proceeded by considering the probability that a neighboring site (bond) is vacant for placing a subsequent monomer (bonded pair of monomers) on the lattice. This probability, however, is evaluated by ignoring all correlations present in the system. While the theory has several notable successes and while Flory-Huggins theory is perhaps the most widely used theory in polymer science, there are many glaring deficiencies of these approximate theories and the standard lattice model upon which it is based.
Recent work has developed a Mayer-like cluster expansion of the partition function for this lattice model in which Flory-Huggins theory emerges as the zeroth order approximation (Freed and Bawendi, 1989; Dudowicz and Freed, 1991a). The theory proceeds as a double expansion in z-1, the inverse of the lattice coordination number (effectively an expansion in d-1, the inverse dimensionality of the system), and in a high-temperature expansion in powers of bεij. The expansion introduces corrections to Flory-Huggins theory that arise from the presence of local packing and interaction-induced correlations in the system. The d-1 expansion is clearly asymptotic at low-volume fractions of polymers (Nemirovsky et al., 1992b). Convergence improves at higher-volume fractions (Dudowicz et al., 1990), but nothing is known mathematically about the nature of the series in this case. The development of a low-temperature expansion for Bεij > 1 has to date been elusive. The dilute polymer d-1 expansion has been performed to fifth order using exact counting methods, but the more general many-chain case has only been carried to second order based on analytical computations using a Mayer-like diagrammatic representation of the series expansion (Freed and Bawendi, 1989; Dudowicz and Freed, 1991a). Further computations will benefit from improved computer-assisted enumeration methods and symbolic mathematical packages. Such computer assistance will be necessary, for instance, to account for the fact that the different submonomer units generally have distinct group interaction energies εij.
The cluster expansion method has been accompanied by an extension of the standard lattice model to permit monomers (and solvent molecules) to occupy several lattice sites (Freed and Bawendi, 1989; Nemirovsky et al., 1987). The method can thus more realistically describe the actual monomer chemical structures, and the fact that different monomers (and solvent molecules) have rather different sizes and shapes and cannot be placed on identical lattice sites. Computations to date with this extended lattice model have already explained a wide variety of puzzling experimental data and have predicted several novel phenomena (Dudowicz and Freed, 1991b, 1992a, c; Freed and Dudowicz, 1992; Nemirovsky et al., 1992a), such as re-entrant microphase separation in block copolymers and the pressure dependence of microphase separation. Further generalizations of the lattice model will require significant advances in the mathematical sciences, for instance, to account for the semiflexibility of polymers and to include the presence of rigid, extended units in the polymer. The former generalization is relevant to the study of polymer glasses, while the latter is important for treating liquid crystalline polymers.
Lattice models are known to suffer from inherent deficiencies because the excess free volume is represented by the presence of unoccupied lattice sites (voids) with equal volumes. Although the generalized lattice model assumes that the voids all have the volume of a lattice cell that is smaller than a monomer size, a more realistic description would have voids with an exponential distribution of volumes. The continuum limit of the generalized lattice

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model and the relation of this limit to off-lattice theories of polymer systems are interesting issues.
One promising, more recent approach to describing the properties of polymers in the liquid state involves developing generalizations of successful theories for small-molecule fluids. Equilibrium microscopic statistical mechanical theories are being constructed for the conformation, structure, thermodynamic properties (for example, equation of state), and phase transitions of both bulk polymeric materials and inhomogeneous systems, such as polymers near a surface or interface and in confined spaces. The primary theoretical tools are liquid-state molecular integral equation methods, thermodynamic density functional theory, and Monte Carlo computer simulation (see Chapter 8).
The "integral equation" theories require solving coupled nonlinear integral equations of an unusual form (Hansen and McDonald, 1986) for atomic and small-molecule fluids. Efficient numerical algorithms have been developed over the past 25 years (Hansen and McDonald, 1986; Chandler, 1982). Certain types of approximation approaches ("closure") and simple polymer models provide existing algorithms that are adequate for polymers (Schweizer and Curro, 1990). However, many situations of interest involve mathematical difficulties that are significantly enhanced due primarily to two factors: (1) For a chemically realistic description of molecular structure (which is necessary for quantitative materials science applications) and/or multicomponent polymer mixtures, the number of coupled nonlinear equations for structural correlation becomes relatively large (Curro and Schweizer, 1987). Numerical solution using existing methods, such as Picard iteration, Newton-Raphson techniques, and so on, is very slow (even on powerful state-of-the-art workstations) and very sensitive to initial guesses and the ''mixing of old and new solutions" (Yethiraj and Schweizer, 1992). (2) The problem becomes increasing difficult as the polymer molecules increase in size because of the enormous growth in the real space length scales over which numerical solutions to the equations must be found, length scales far beyond those considered previously with small-molecule fluid theory. This difficulty gets even worse as a phase transition (critical point) is approached (Hansen and McDonald, 1986). A phase transition greatly complicates the numerical determination of phase diagrams of polymer mixtures. The phase transition problem is particularly important in materials science, and existing iterative solution methods based on a specified grid size and fast Fourier transforms can become impractical or unreliable.
Given the above numerical difficulties, the ability to obtain even approximate analytic solutions would be extremely valuable. Unfortunately, rigorous analytic solution of the nonlinear integral equations is generally only possible for simple atomic fluids (Hansen and McDonald, 1986). However, analytic progress may be possible for more simplified models of polymer molecular structure (Schweizer and Curro, 1990), and a serious mathematical study of this problem would be of great interest.
Opportunities exist in the nonequilibrium area for generalizing and applying nonperturbative statistical dynamical methods that have been developed for simple atomic fluids (mode-mode-coupling theory; see Chapter 9 in Hansen and McDonald, 1986) to the complex problem of describing diffusional and viscoelastic properties of entangled polymer fluids (Schweizer, 1989). The challenge for the mathematical sciences is that a very large number of first-order coupled stochastic differential equations must be solved for the

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effective dynamics of a polymer molecule in a dense liquid. In general, the differential equations involve nonlocal kernels ("memory functions") in both the time and spatial variables, and the stochastic "noise" is not "white" but correlated (often called "colored noise''). In order to obtain the physical predictions from such equations, it has been necessary to employ an oversimplified molecular model of the polymer. This allows a reduction of the coupled differential equations to a linear form that can then be solved via normal mode diagonalization methods. Although this scheme does represent a good first approximation for many ''universal" dynamical questions of materials science interest, it has serious limitations with regard to the level of chemical realism that can be incorporated (and hence the range of time and spatial length scales that can be treated). For example, early-time "glassy" polymer dynamics cannot be reliably studied. For many materials-specific mechanical and transport properties, the appropriate stochastic differential equations are nonlinear, and new mathematical methods to handle such problems would be valuable.
INTERFACES IN POLYMER SYSTEMS
Scientific and engineering studies of the dynamics or surface properties of macroscopic systems often require a knowledge of the interface—the boundary region—separating bulk phases. In many interesting situations, little is known about even the equilibrium or static characteristics of such interfaces, to say nothing of their dynamical properties. This motivates the development of mathematical and statistical theories and models that can provide even simple qualitative trends about interfacial properties. Such models can be treated using a formalism known as the theory of inhomogeneous liquids, which is derived from modern statistical mechanics. This theory provides information about the equilibrium (static) properties of matter but also has implications for theories of interfacial dynamics. The chain-like structure of polymer molecules distinguishes polymeric systems from those composed of small molecules and leads to a variety of novel characteristics of polymeric materials. Nevertheless, polymer and small-molecule systems do share certain basic features. Hence, theories of simple liquids and glasses often provide a point of departure for analysis of polymer systems. Modifications of small-molecule theories are then required to describe the chain connectivity, flexibility, and long-range correlations that are present in polymer systems. The properties of phase-separated systems and, in particular, properties of interfaces are one area that displays both the strong similarities between polymers and small molecules and the differences that are imparted by polymer structure. For these reasons, this section begins with general considerations of theories of interfacial properties before turning to the specific challenges posed by theories of polymer interfaces.
Density functional theory focuses on the behavior of distribution functions or probability densities in media (liquids or gases near surfaces, two-phase regions, periodic solids, and so forth) the properties of which vary in space (see Chapters 2 and 8). Of particular interest is the singlet probability distribution function, usually called the density, which gives the probability of observing a particle at a point in space. At equilibrium, this quantity does not depend on time, and variational principles exist for approximately

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determining its functional form. This approach to elucidating the thermodynamics of inhomogeneous fluids is often called density functional theory.
The variational condition for the density functional enables theorists to derive integral equations for the density or to obtain more qualitative information through an algebraic minimization of some functional with respect to an approximate, parametrized density. With the exception of certain one-dimensional problems and the dilute, ideal gas limit in which interactions between atoms do not affect the thermodynamics, approximations must be made for the kernels appearing in the integral equations. Obtaining an accurate approximation for the kernels themselves generally requires as much effort as solving for the density, and so theorists normally assume some simple, approximate form for the kernels and use the resulting integral equation or variational condition to solve for the density. Unfortunately, little is understood about the way such approximations affect the predictions of theory, although approximate density functionals often provide reasonable results for a broad range of approximation schemes without any ostensible justification (Evans, 1979). Some researchers have questioned the true practicality of the entire density functional methodology (Lovett, 1988). While formal considerations indicate a unique solution to the exact equation, approximate equations may possess multiple solutions. Methods are necessary for determining which solutions are relevant and which may correspond to interesting metastable states. There are various other fundamental mathematical questions concerning density functional theory; see, for example, Lovett and Stillinger (1991).
From a formal mathematical point of view, density functional theory remains poorly understood though broadly used. The additional insight offered by trained mathematical researchers, capable of carefully studying the effects of approximations within the density functional theories, could have a profound effect on materials research. It is hoped that such investigations will (1) lead to an understanding of the stability of the methodology, (2) suggest classes of approximation schemes most likely to faithfully predict the behavior of particular models, (3) lead to new models of many-body systems, and (4) provide guidance to the solution of the integral equations that commonly occur in this field.
While the theory of atomic systems or models with spherically symmetric potentials has advanced rapidly during the past few decades (Evans, 1979), much work remains before an understanding of molecular systems is achieved. Molecular systems involve several (as for CO2 or N2) or many (as for polymers) atoms covalently bonded together to form complicated geometrical objects that often frustrate theories primarily designed to treat the atomic case. Important progress in developing new approaches to molecular models has been achieved recently (Wertheim, 1988) by using conventional graph theory specifically tailored to describe the molecular case. It is hoped that future progress in this area will lead to new and more accurate descriptions of molecular fluids and solids. However, relatively few mathematical researchers with the skills necessary to achieve such goals are currently working in this area.
With every improvement in the theory of equilibrium interfaces comes a compelling urge to advance the theory of interfacial dynamics (Gunton et al., 1983). Present theories of interfacial dynamics are phenomenological and based on order parameters (very often Fourier modes of the density) and fluctuation-dissipation theorems. Even at this level, theories seldom allow for the coupling of more than two order parameters because of the

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intractability of the resulting nonlinear partial differential equations. Mathematical researchers could make a grand contribution to this field by developing new computational or other algorithms for solving such equations. Better still, new formulations based on first principles instead of phenomenological considerations would enhance the field.
An increasing number of new composite materials are being fabricated from phase-separated polymers, where the strengths of the materials are often determined by the interfacial strength and morphology. A significant amount of experimental investigation (Anastasiadis et al., 1988, 1990; Bates et al., 1984) and theoretical research (Scheutjens and Fleer, 1979, 1985; Helfand et al., 1989; Hong and Noolandi, 1983; Joanny and Leibler, 1978; Binder and Frisch, 1984; de Gennes, 1980) has focused on interfaces in binary blends. Recent developments of density functional theory have been associated with extensions to polymer systems (Roe, 1986; Broseta et al., 1990; McMullen and Freed, 1990a, b; Tang and Freed, 1991a), where the complexity of polymers compounds the above-described difficulties. However, many opportunities exist for using density functional theory to aid in understanding currently used approximations and in developing useful extensions for polymer systems. Theories for interfacial properties of polymer systems have been developed that are most useful in two limiting situations. Self-consistent field approaches (Helfand et al., 1989; Binder and Frisch, 1984) are generally applied in the strong segregation limit, while the density functional methods (Joanny and Leibler, 1978; McMullen, 1991) are most suitable in the weak segregation limit. Several predictions of the two limiting theories coincide, indicating close interrelations between the apparently extremely different methods. Fundamental studies of the relation between both types of theories could be useful in order to devise better theories for treating the important intermediate segregation regime.
Density functional theories of interfacial widths and tensions in phase-separated binary polymer blends are predicated on a number of simplifying assumptions as follows (Helfand et al., 1989; Joanny and Leibler, 1978; Binder and Frisch, 1984; de Gennes, 1980): (1) the homogeneous component of the free-energy functional is taken as having a Flory-Huggins form; (2) the interaction parameter χ is assumed to be composition-independent; (3) the inhomogeneous portion of the free-energy functional is truncated at the square gradient term (Roe, 1986; Broseta et al., 1990); (4) different numerical coefficients of the square gradient term are employed for the weak (Joanny and Leibler, 1978) and strong (de Gennes, 1980) segregation limits, and the square gradient coefficient is made composition-independent, an ad hoc assumption lying outside the underlying, original Cahn-Hilliard formulation (Roe, 1986; Broseta et al., 1990; see, however, Akcasu and Sanchez, 1988); (5) comparisons with experiment often take the χ parameter from data for the corresponding block copolymer system; and (6) the system is assumed to be incompressible. While there is some agreement between theory and experiment, the origins of observed discrepancies are unclear (Anastasiadis et al., 1988, 1990; Bates et al., 1984), and it is important to assess the implications of lifting the above theoretical assumptions.
Some studies have begun systematically investigating the validity of the above six assumptions for interfaces in phase-separated binary blends (McMullen and Freed, 1990b; Tang and Freed, 1991a, b, d). Use of rigorous density functional methods has enabled the derivation (Tang and Freed, 1991d) of the de Gennes composition-dependent square gradient coefficient (Joanny and Leibler, 1978; Binder and Frisch, 1984) for the weak

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segregation limit and when nonlocal connectivity considerations arise, as they do in self-consistent field formulations (Tang and Freed, 1991a). The Roe density functional (Roe, 1986) has been shown to be correct only in the limit of infinite molecular weights (Tang and Freed, 1991a), although the finite chain corrections are not necessarily of the generally assumed Flory-Huggins form but are nonetheless important to explain real experiments (Tang and Freed, 1991b). Allowing for simple linear or quadratic dependences of χ on composition has been shown (Tang and Freed, 1991b) to introduce, especially near spinodals, substantial deviations from the standard theories of interfacial widths and tensions. Lattice cluster theory computations of χ for diblock copolymers (Dudowicz and Freed, 1992b) indicate that the junction introduces considerable 1/N entropic contributions to χ. Consequently, χ for the lower-molecular-weight diblock copolymers is predicted to depart, often qualitatively, from χ for the corresponding binary blend.
Interesting problems abound. First of all, it remains to test the incompressibility assumption and determine the dependence of interfacial properties on the thermodynamic state (called equation-of-state effects). A compressible theory is complicated by the fact that the Euler-Lagrange equations for the interfacial profile now become numerically unstable coupled nonlinear differential equations, and suitable mathematical approximation schemes must be devised. The development of a compressible theory would enable prediction of the pressure dependence of interfacial properties, something of potential technological importance. Other questions are associated with the derivation of an improved density functional for the intermediate segregation regime and for the treatment of segregation phenomena that occur near free or substrate surfaces. The latter are important in thin polymer films. The dynamics of phase separation in blends, microphase separation in block copolymers, spinodal decomposition, and crystallization are fields with many opportunities. Many features parallel similar processes in other areas of materials science, as discussed in Chapter 4. However, the extremely slow diffusive motion of large polymers enormously slows the dynamics to enable observation of the linear domain, which is unobservable in metallurgical applications, as well as the stages of nonlinear coarsening and ripening.
BLOCK COPOLYMERS
Materials closely related to composites are formed from block copolymers. Individual block copolymer molecules contain long sequences of one type of unit, followed by a long sequence of another type. When the two types of polymer are incompatible, they attempt to phase separate, but because of the unseverable molecular attachment, the different species cannot get very far from each other. Thus, a microdomain structure is formed that has been considered a regular periodic array with periodicities on the scale of polymer chain dimensions (hundreds to thousands of angstroms) (Bates, 1991; Bates and Frederickson, 1990; Hadziioannou and Skoulious, 1982; Almdal et al., 1990; Bates et al., 1990). Recently, it has been discovered that the arrays in this microphase separation transition may be quite complex geometrically. An example is the double-diamond structure, an array of interpenetrating, continuous phases, each with the symmetry of a diamond lattice. One theory holds that the tendency to form such arrays is related to the drive toward formation

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of minimal surfaces between the two materials (Thomas et al., 1988). Considerable experimental (Bates, 1991; Bates and Frederickson, 1990; Hadziioannou and Skoulious, 1982; Almdal et al., 1990; Bates et al., 1990) and theoretical (Leibler, 1980; Ohta and Kawasaki, 1986; Melenkevitz and Muthukumar, 1991; de la Cruz and Sanchez, 1986) efforts have been devoted to elucidating the morphology of microphase separation in block copolymers, partly because of their technological importance but also because of the deep conceptual scientific questions associated with describing phenomena in these fascinating inhomogeneous systems. Other interests in block copolymers arise from their use in the stabilization of blends and their ability to form micelles and other self-assembling structures.
Theories of block copolymer systems (Leibler, 1980; Ohta and Kawasaki, 1986; Melenkevitz and Muthukumar, 1991; de la Cruz and Sanchez, 1986; Frederickson and Helfand, 1987) are replete with either untested or clearly oversimplifying assumptions. The list of largely untested simplifying assumptions includes all those made in theories of interfacial properties in phase-separated polymer blends and many more. For instance, Customary approximations include the neglect of compressibility, the use of a composition-and molecular-weight-independent effective interaction parameter χeff, the interchangeability of χeff between block copolymer melts and the corresponding binary blends, and the use of the incompressible random phase approximation. A dependence of χeff on pressure and composition, in addition to the assumed dependence on temperature, implies that χeff is a function of the thermodynamic state of the block copolymer system, or equivalently, that χeff depends on compressibility or equation-of-state effects. Because of the rather small differences in computed free energies for different morphologies, it is unclear which of these myriad assumptions is responsible for the current lack of a theoretical explanation, for instance, of the circumstances under which the double diamond and other more intricate structures are to be observed.
Other interesting questions involve (1) describing the pressure dependence of the morphology for microphase-separated block copolymer systems, (2) understanding the implications of possible fluctuating ordered domains at temperatures above the microphase separation transition (Hadziioannou and Skoulious, 1982; Almdal et al., 1990; Bates et al., 1990; Fried and Binder, 1991; Hasegawa et al., 1987; Herman et al., 1987; Rosedale and Bates, 1990), and (3) explaining the observed nonuniversality of the phase diagram (Tang and Freed, 1992), a result that, at fixed molecular weight, conflicts with theoretical predictions (Leihler, 1980; Ohta and Kawasaki, 1986; Melenkevitz and Muthukumar, 1991; de Ia Cruz and Sanchez, 1986; Frederickson and Helfand, 1987). The first question requires the development of a theory for compressible systems, which presents many of the mathematical complexities associated with the treatment of interfacial properties for phase-separated binary blends. Lattice cluster theory computations of the microphase separation transition temperature predict a substantial pressure dependence (Dudowicz and Freed, 1992b), a phenomenon that has to our knowledge not been observed but that would be of technological interest. Monte Carlo simulations (Fried and Binder, 1991) and a scaling theory (Tang and Freed, 1992) suggest that resolution of the second and third questions resides in a nonperturbative treatment of the fluctuating domains, something for which mathematical tools are currently unavailable. A possible resolution of the third question emerges from a consideration of a quadratic composition-dependent χ in the Leibler theory

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of block copolymer microphase separation. This simple modification (Tang and Freed, 1991c) leads to the emergence of five different phase diagrams, two of which coincide with the Leibler and the fluctuation theory phase diagrams. Another two of the new phase diagrams agree with experimental observations. This provides one example in which physically significant deviations appear from standard theories when complexities present in realistic systems are taken into consideration. Lifting the customary assumptions sometimes leads to severe mathematical difficulties, and describing the dynamics of microphase separation poses a number of conceptual and mathematical challenges.
STIFF POLYMERS AND LIQUID CRYSTALS
Polymers that are liquid crystalline in the fluid state, such as Dupont's Kevlar® and Hoechst-Celanese's Vectra®, have found use in high-performance applications, and the versatility of possible polymer morphologies and compositions provides the promise for additional novel liquid crystalline polymers in the future. There are many theoretical problems associated with describing the various liquid crystalline phases possible in polymers. Liquid crystalline polymers are discussed in Chapter 5 in the general context of liquid crystals. One basic problem is associated with the description of semiflexible chains that lie intermediate between fully flexible chains and rigid rods. The Kratky-Porod wormlike chain model provides a venerable solution to this problem, but in a form that is much too cumbersome for simple use in theories of liquid crystalline behavior. Thus, many efforts have been devoted to obtaining analytically more tractable models. A recent advancement in this area uses an analogy between the path integral descriptions of polymers and Dirac fermions (Kholodenko, 1992). The new model correctly reproduces the mean square end-to-end distance of the wormlike chain model, and low-order computations provide results that tend toward the Flory-Huggins and Onsager limits for fully flexible chains and rigid rods, respectively. Applications to liquid crystalline polymers would be welcomed. Many interesting experiments involve polymers with both flexible portions and liquid crystal mesogens on the same chain, and thorough theoretical descriptions are lacking.
The presence of topological constraints has been argued to introduce effective stiffness to chains (Kholodenko, 1991), and this effect may influence the viscoelastic properties of concentrated polymer solutions. Another consideration of chain stiffness enters into descriptions of semiflexible elastomers (Warner and Wang, 1992).
The dynamical theory of liquid crystalline polymers is discussed in the Chapter 5 section on liquid crystals. It suffices to note here that solution of the dynamical equations for structural models is particularly delicate in the case of rigid-backbone polymers that form a nematic liquid-crystalline phase. Here, uniform shear can become unstable at very low shear rates because of the tumbling nature of the nematic phase. As a consequence, the system becomes "structured" in space (polydomain) in the sense that the characteristic orientation of the nematic phase changes spatially, passing through lines of orientational discontinuities (defects or disclinations; Marrucci and Maffetone, 1990). Defect theory has become heavily dependent on differential geometry. There are many outstanding problems, and this general question is addressed further in Chapter 5.

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OTHER PROBLEMS
Polymer science presents an enormous array of other problems for which progress may benefit from improved mathematical approaches. The above discussions are illustrative of some of the ways in which mathematicians can contribute to polymer science. There are many more; see, for example, Kroschwitz (1990). Although these cannot all be discussed in detail here, it is perhaps worthwhile to give several examples.
Most of the polymeric materials ubiquitous around us are actually not pure, but rather are combinations of different types of molecules. For example, the reason that polymeric glasses do not shatter is that they are impact modified. That means that they are blended with rubbery materials. The phases do not truly mix, but rather form an intimate composite of small rubbery domains within the glassy matrix. Many other examples of polymeric composites exist, and frequently polymeric materials contain a large fraction of solid filler material. An important problem is to characterize the path of a fracture, and how the rubbery inclusions in a glass cause the fracture to branch and be terminated (Kausch, 1978; Brostow and Corneliussen, 1986). It is also important to be able to predict the properties of composites, particularly what type of averaging is necessary to describe effective properties (for example, effective viscosity, dielectric constant, or elastic constant). These subjects are discussed in Chapters 5 and 6.
The glassy state of matter is a form in which many polymers find use. There appear to he numerous mathematical issues to be sorted out in the development of a theory for this type of material (see Chapter 6). Many of these problems parallel ones associated with the description of protein dynamics and folding. Descriptions of relaxation phenomena in glasses require the solutions of multidimensional integro-(partial-)differential equations with a wide range of relevant time scales. Methods must be developed for isolating the subset of "relevant" degrees of freedom and for extracting the difficult-to-compute, long-time relaxation dynamics.