Mathematical Research in Materials Science Opportunities and Perspectives (1993) / Chapter Skim
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6 AGGREGATES AND DISORDERED MATERIAL
6 AGGREGATES AND DISORDERED MATERIAL
Pages 66-76
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From page 66... ...
The resulting structure is predicted by equilibrium and nonequilibrium statistical mechanics and computer simulations. The potential for advances in materials applications lies in constructing robust structure and properties in the multidimensional parameter space characteristic of complex fluicis.
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From page 67... ...
A second example currently receiving considerable attention is the formulation and application of electrorheological fluids, which consist of polar particles in a nonpolar fluid. These are normally low-viscosity fluids, but a strong electric field induces dipoles in the particles and creates a volume-fi~ling, particulate network with an elastic or pseudoplastic response to an applied stress.
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From page 68... ...
Nonequilibrium Statistical Mechanics Approaches developed for molecular fluids have been applied rather superficially to colloidal problems, generally ignoring hydrodynamic interactions and simply adopting conventional approximations for dynamical couplings (Hess and KIein, 1983~. Recent alternative closures, analogous to well-established equilibrium closures, convert the Smoluchoski equation to an integrodifferential equation for the nonequilibrium structure.
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From page 69... ...
Effects include radical alterations in the phase rule and in the construction of phase diagrams, restricted validity of the common tangent construction, and the presence of multiple stable equilibrium solutions for a given set of experimental parameters. It can be shown (Johnson and Muller, 1991)
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From page 70... ...
Indeed a composite can sometimes exhibit properties completely unlike those of its constituent materials. For example, transparent glass containing a suspension of spherical gold particles is not gold in color, but rather red.
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From page 71... ...
Mathematicians have begun to embark on this program and have achieved considerable success. Not all the structures proposed by mathematicians are realistic: for example, they mar involve microstructure on widely separated length scales that would be difficult for an experimentalist to reproduce.
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From page 72... ...
Hashin and Strikman (1963) have developed a variational principle that enabled them to derive optimal upper and lower bounds on the elastic moduli in terms of the volume fractions of the two phases of the composite.
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From page 73... ...
The new mathematical methods, including G- and H-convergence, compensated compactness, statistical mechanics of distribution functions, percolation theory, and renormalization group methods, have thus far been developed primarily for linear models of material behavior; it is a major challenge for the future to apply them in nonlinear settings as well. Coupled-field problems, for example, piezoelectricity, have only begun to be studied from this viewpoint; further progress in piezoelectricity could lead to the design of better actuators and other practical devices.
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From page 74... ...
But its basis is an application of a materials science concept, namely, composite materials, especially those with optimal microstructures, to the apparently unrelated area of structural optimization. Also, it has led to the development of new methods for bounding effective moduli (see the subsection Future Directions above)
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From page 75... ...
At the other extreme are relaxation processes, apparently involving rearrangements of large numbers of atoms to lower overall energy, that may have characteristic times of days, months, or years. Because the computer simulation processing time scale for molecular dynamics is approximately 1015 slower than "real" time, the observation and description of relaxation processes in glasses via molecular dynamics computer simulation are, unfortunately, far beyond reach at present.
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From page 76... ...
Enumerate the inequivalent potential energy minima, at least in the sense of exponential rise rate in the large system limit, or as rigorous bounds on this rise rate.
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