Mathematical Research in Materials Science Opportunities and Perspectives (1993) / Chapter Skim
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3 MACROMOLECULAR STRUCTURES
3 MACROMOLECULAR STRUCTURES
Pages 11-33
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From page 11... ...
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.
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From page 12... ...
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.
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From page 13... ...
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 singlechain problems will reverberate through polymer science.
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From page 14... ...
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 onnortunitv for r ~ applications of the mathematical sciences to practical materials science problems.
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From page 15... ...
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 S; also Berg (1993~.
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From page 16... ...
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 longtime dynamics is meant molecular dynamics calculations involving time scales much longer than current computational facilities allow.
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From page 17... ...
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)
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From page 18... ...
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)
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From page 19... ...
The physical mode] 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)
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From page 20... ...
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~.
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From page 21... ...
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.
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From page 22... ...
The possibility that sharkskin may be a physical manifestation of a Hadamard instability arising from the real polymer constitutive equation has been suggested, as has a mechanism based on the propagation of shear discontinuities in a nonlinear material. Two lines of thought are receiving current attention.
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From page 23... ...
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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Relevant mathematical sciences issues do not seem to have been considered: (~) Given the equations that describe the microdynamics, what can be said about the general types of behavior (that is, the nature of solutions)
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From page 25... ...
Further generalizations of the lattice model will require significant advances in the mathematical sciences, for instance, to account for the semiflexibiiity 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.
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From page 26... ...
The challenge for the mathematical sciences is that a very large number of first-order coupled stochastic differential equations must be solved for the 26
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From page 27... ...
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.
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From page 28... ...
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.
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From page 29... ...
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 X parameter from data for the corresponding block copolymer system; and (6)
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From page 30... ...
to introduce, especially near spinodals, substantial deviations from the standard theories of interfacial widths and tensions. Lattice cluster theory computations Of X for diblock copolymers (Dudowicz and Freed, 1992b)
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From page 31... ...
For instance, customary approximations include the neglect of compressibility, the use of a compositionand molecular-weight-independent effective interaction parameter Xeff, the interchangeability of Xeff between block copolymer melts and the corresponding binary blends, and the use of the incompressible random phase approximation. A dependence of Xeff on pressure and composition, in addition to the assumed dependence on temperature, implies that Xeff is a function of the thermodynamic state of the block copolymer system, or equivalently, that Xeff depends on compressibility or equation-of-state effects.
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From page 32... ...
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.
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From page 33... ...
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.
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