AGGREGATES AND DISORDERED MATERIAL
Many materials are made up of aggregates with characteristic length scales ranging from 0.1 to 10 µm. These disordered materials can have properties that are not achievable with a homogeneous phase. For example, polymers can exhibit high impact strength by dispersion of a rubbery phase in a glassy matrix. Ceramic parts are fabricated by molding of concentrated suspensions of colloidal particles in an organic carrier fluid followed by removal or reaction of the liquid phase. A fundamental problem is the prediction of bulk properties from the properties and distributions of the individual phases and interphase interactions.
Glasses and other amorphous materials are another class of disordered materials. In contrast to the aggregates just mentioned, they appear to be uniform down to a length scale of nanometers. Many preparation techniques, including melt quenching, vigorous mechanical working, vapor deposition, and reactive component interdiffusion, are available to produce these materials and to frustrate the tendency toward crystallization. Properties depend on the technique selected in a manner that has thus far eluded deep understanding.
Fluids containing aggregates with complex microstructures arise in a variety of materials applications. Typical microstructures are formed from colloidal and noncolloidal dispersions or through the self-assembly of amphiphilic macromolecules. Examples of materials processed from submicron particles or macromolecular aggregates dispersed in a liquid at concentrations ranging from dilute to closely packed include advanced ceramics, water-based automotive and architectural coatings, polymeric composites synthesized via emulsion copolymerization and from block copolymers, electrorheogical fluids, and nanostructured materials. Noncolloidal systems include structural composites composed of dispersed particles in a polymer matrix. The important properties usually depend on interactions among the particles or dispersed domains through a variety of physico-chemical, hydrodynamic, and Brownian forces (Russel et al., 1990). Free polymer in the liquid phase often moderates the direct interactions. The theory requires accurate predictions of the various interparticle forces via low Reynolds number fluid mechanics, quantum mechanics classical electrostatics, and polymer statistical mechanics. 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 fluids. For example, the end use and application requirements for coatings are numerous and conflicting. Consequently, formulations involve multiple components, with each affecting most of the properties. For relatively simple thermodynamic properties, such as solvent
volatility and viscosity, the theoretical basis has been constructed (Wu, 1987); for nonequilibrium behavior of the fluid phase, such as rheology, and the optical and mechanical properties of the final solid, the basic theory is only partially in place.
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-filling, particulate network with an elastic or pseudoplastic response to an applied stress. Here the issue is how to control or optimize the magnitude and dynamics of the response (Gast and Zukoski, 1990). The physical phenomena are reasonably well defined, but many-body interactions dominate and the behavior is highly nonlinear.
The fabrication of ceramics by colloidal processing aims for dense packing without large-scale inhomogeneities (Lange, 1989). Consolidation via centrifugation or filtration permits application of forces strong enough to overcome the short-range attractions inevitable between dense particles of 0.1-to 10.0-micron diameter in water or an organic liquid. Given accurate constitutive relations for the permeability and the stress supported by the particulate network, continuum theory describes the evolution of density profiles during the consolidation process (Buscall and White, 1987; Auzerais et al., 1990). The mathematical problem is to solve the highly nonlinear set of equations accurately enough to capture the propagation of several nearly discontinuous transitions in a one-, two-, or three-dimensional spatial domain.
For composite materials, the challenge is to relate the spatial structure and properties of the individual phases to macroscopic mechanical, optical, and electronic properties. Prediction of equilibrium structure via statistical mechanics is being developed for some classes of systems (Bates and Frederickson, 1990; Monovoukas and Gast, 1989). Considerable progress has been made in predicting linear responses through variational and bounding techniques (Torquato, 1990). However, nonlinear behaviors of considerable interest, such as yielding, fracture, and nonlinear optical properties, are still largely untouched.
One might define the challenge for mathematical theory as developing techniques for accurately handling strong many-body interactions. This challenge is complementary to the challenge to physics of describing interparticle forces, equilibrium phase behavior, and polymer thermodynamics. Many aspects of the subject have advanced to the point that sophisticated mathematical analysis and numerical solution procedures are needed to provide the quantitative predictions necessary to keep up with the physics and experimental investigations in this area. As is pointed out in Chapter 8, this is the case not just for this area but also in many other areas of materials science.
One can identify a number of relatively advanced approaches to this class of problems:
For colloidal problems, significant movement occurs only after many thermal
fluctuations. This permits analytical integration of the equations of motion on the diffusion time scale and produces equations for the displacement of particles due to random Brownian events and deterministic translation under interparticle and external forces. The difficulty is that hydrodynamic interactions essentially couple all particles within a simulation cell. These effects have been captured with a methodology known as Stokesian dynamics at the expense of very large matrix inversions (Brady and Bossis, 1988). An efficient code that runs on a workstation now enables simulations in three dimensions with a reasonable number of particles.
The equations of motion in the absence of inertia (the usual case for materials applications) are fully linear and can be recast as contraction mappings and iterations to a fixed-point solution. The hydrodynamic force is shape dependent, and so the basic problem for a particle suspension is a three-dimensional boundary value problem with complex and evolving geometry. Lorentz formulated the problem more than 100 years ago as a two-dimensional boundary-integral equation; the result is a Fredholm integral equation of the first kind. For the complex geometries of interest, the ill-posed nature of the problem manifests itself as ill-conditioning, and convergence to the numerical solution cannot be obtained merely by refining the discretization.
The existence of a velocity representation that yields a Fredholm equation of the second kind has been established (Power and Miranda, 1987), and a solution methodology, the completed double layer boundary integral method, has been developed (Kim and Karrila, 1991). The iterative strategies map naturally to emerging high-performance parallel computer architectures (Fuentes and Kim, 1992). Work is in progress, for example, to simulate a system with 1 million particles using 80 million boundary elements on the CM5 parallel supercomputer.
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 Klein, 1983). Recent alternative closures, analogous to well-established equilibrium closures, convert the Smoluchoski equation to an integrodifferential equation for the nonequilibrium structure. Dynamical properties follow from the structure via straightforward integration of the appropriate forces or fluxes (Russel, 1992). Work is under way to test the accuracy of this approach against simulations and experimental data for model systems. If the approximations prove accurate, then mathematical analysis and efficient numerical solution of these coupled integrodifferential equations become important.
For materials with static structures partially characterized by statistical measures, mechanical and electromagnetic properties satisfying linear field equations can be bounded rather accurately by variational techniques (Torquato, 1990; Milton and Kohn, 1988). As with the preceding nonequilibrium statistical mechanics case, testing against computer simulations is usually important, but the results here have the potential of greater generality.
Self-Consistent Field Theories
Approaches formulated for polymer statistics by Edwards, Helfand, and de Gennes in the 1970s have been applied with considerable success to a broad range of interesting problems, including the prediction of microphase separation in diblock melts. With the discovery of the strong stretching approximation, apparently by Semenov, the subject has exploded mathematically. A range of classical techniques, involving asymptotic expansions and conformal mappings complemented by computational solutions, have yielded elegant and relatively simple predictions of phase behavior (Ball et al., 1991b; Kawasaki and Kawakatsu, 1990). Many interesting problems remain, such as the incorporation of nanometer-sized particles into these melts to produce nanocomposites with controlled structure.
The thermodynamic equilibrium of a system is described by specifying the number, composition, elastic state, and spatial configuration of the phases present. Equilibrium not only defines the end-state of an evolving system, but is also useful in trying to describe local features of the system during evolution, such as interface intersection angles, bulk compositions at interfaces, and particle shapes. Two examples of challenging equilibrium problems are mentioned in what follows, as well as the problem of inferring three-dimensional configurations from a two-dimensional cross-section.
The characteristics of phase diagrams are qualitatively altered by elastic stresses. 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) that the long-range elastic forces arising from differences in lattice parameters between phases or from an externally applied load permit several stable equilibrium states that depend on the mechanical boundary conditions and system geometry. A catalogue of possible features, similar to what is known for fluid or unstressed systems, is not known. A geometrical description of such possible phase diagram constructions would be most helpful.
In the absence of stress, the equilibrium morphology of a particle in a matrix is simply a minimal surface problem. In contrast, in the presence of elastic stress, the equilibrium
shape is found by minimizing a certain combination of the interfacial and elastic energy of the system. Very few solutions to this variational problem have been found wherein the particle morphologies are not constrained to be of a certain class of geometric shapes. Recently, solutions have been obtained wherein the particle shapes have not been so constrained but have been limited to a numerical determination of the particle morphologies in two dimensions (Johnson and Voorhees, 1988). Much work remains to be done in this area.
It is often not possible to perform structural measurements directly on 3-D microstructures. They are performed on a 2-D section of the specimen and then are reconstructed for 3-D by a set of rules known as stereology. Thus, for cellular structures it is possible to reconstruct (roughly) a 3-D cell size distribution by the size distribution in a 2-D section. The usual assumptions are that the cell shape is close to spherical and the cells are randomly distributed in space (Underwood, 1970; Exner and Hougardy, 1988). Both assumptions fail for densely packed structures. Moreover, for such structures the topological aspects of packing are of interest. There is, at present, no ''topological stereology'' to reconstruct 3-D topological structures from those of 2-D sections. Even for the case of Voronoi mosaics, the problem is open; see Hermann and Lorz (1992) and the references therein. The typical questions are the following: What are the average numbers of faces, edges, and vertices per cell? What are the distributions of the number of faces, edges, and vertices per cell? The problem of topological classification of 3-D cells is also important (Weaire and Rivier, 1984; Rivier, 1985; Fortes and Ferro, 1985).
EFFECTIVE MODULI OF COMPOSITES
Composite materials are interesting precisely because their overall properties are not just a simple average of the properties of the components. 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. In fact, a well-known approximation formula for calculating the optical properties of such suspensions (known as the Maxwell-Garnet formula) led to an understanding of how the Venetians achieved the colors in their glasses; see Landauer (1978), which contains a good historical survey of the field.
Sometimes a composite can combine desired properties of one phase with other attributes of a second phase. The simplest example of this is a laminate of conducting sheets of metal alternating with insulating layers of rubber. The composite combines the conductivity of the metal in directions parallel to the layers with the insulating properties of the rubber in directions perpendicular to the layers. Formulas abound in the literature for calculating effective properties—such as the effective conductivity, elasticity, and piezoelectric and thermoelastic tensors—of simple laminates. But most of these formulas are derived under restrictive assumptions about the isotropy of the phases or their orientation relative to the direction of slicing. It is interesting, and perhaps illustrative of the potential for mathematicians to make significant contributions, that general formulas for calculating the effective properties of simple laminates have been developed only in the past
10 years and that these were developed by mathematicians (Tartar, 1985; Francfort and Murat, 1986); see Chapter 8.
A surprising example of what can be achieved with a careful choice of composite microstructure is elastically isotropic composite materials with negative Poisson's ratios, that is, materials that are easily compressed but that are resistant to shear (Lakes, 1991; Milton, 1992). Contrary to usual experience, such materials get fatter as they are stretched. They are constructed from a stiff phase and a compliant phase and combine the resistance to shear of the rigid phase with the ease of compression of the compliant phase. The existence of such materials had been a subject of debate for nearly a century. The fact that they were discovered only relatively recently points to the need for a systematic approach to finding what sort of effective behavior is possible and to finding microstructures that achieve this behavior. 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 may involve microstructure on widely separated length scales that would be difficult for an experimentalist to reproduce. But the analysis does provide guidelines for making more realistic structures. One area that needs exploration is the effect of adding surface-area penalizing terms. Once these are introduced, the resultant optimal microstructures are bound to be much more realistic.
In other applications, one is given a specific composite with a microstructure that is partially known and the objective is to estimate its properties. For example, one might want to calculate the fluid permeability of a porous rock, a problem that has obvious applications in the oil industry. Or one might want to calculate the elastic constants and hence the acoustic wavespeeds in a polycrystalline aggregate; this is of interest to geophysicists studying earthquakes (Watt et al., 1976). In this context, one needs to identify appropriate geometric quantities that are strongly correlated with the relevant effective property of interest. Correlation functions provide a natural description of composite geometry, but not all information contained in the correlation functions is relevant to the determination of the effective property. One task for mathematicians is to filter out from the correlation functions relevant information. Many effective moduli, such as effective elasticity and conductivity tensors, are insensitive to the size of the microstructure, so that it is natural to extract from the correlation functions quantities or functions that are insensitive to the size of the microstructure. Examples include geometric parameters that enter series expansions (Milton, 1981; Torquato, 1991) and measures defined on surfaces of spheres that are obtained by Radon transformations (Willis, 1981) or by integration of the usual correlation functions along rays in Fourier space (Avellaneda, 1987a; Tartar, 1990).
Other effective properties can be correlated with the effective property of primary interest. For instance a well-known empirical relation provides a correlation between the electrical conductivity of a porous rock and its permeability. This empirical relation is useful because the electrical conductivity is much easier to measure than the fluid permeability. Some bounding methods provide a means whereby one property can be correlated with another through rigorous inequalities (Cherkaev and Gibiansky, 1992).
Experimentalists and scientists working in other fields have often developed approximation formulas for effective properties based on empirical evidence or a heuristic model. Here mathematicians can help and have helped by placing these approximation
formulas on a rigorous basis by showing either that they correspond to the moduli of certain geometries (Milton, 1985; Avellaneda, 1987b) or that they have the correct asymptotic behavior. Moreover, mathematicians have identified cases where these approximation formulas break down.
There are many other areas of practical importance where mathematicians can make a significant impact. To name a few, we need a better understanding of wave propagation in composites, the effective properties of cracked solids and of multiphase fluids and slurries (where the composite geometry changes with time), and the response of composites that display nonlinear effects such as plasticity or fracture. Not only do these problems have interesting applications; they also have an intrinsic mathematical appeal. The surprising aspect of the subject is the breadth of mathematical techniques that have been introduced to solve current problems, and correspondingly the need for new techniques to address unsolved problems (see Chapter 8).
There is a long history to the study of composite materials. By the early 1970s, the field was considered rather mature, at least as far as linear materials were concerned (Willis, 1981). Over the past 10 years, however, the field has seen a renaissance. There are basically three reasons: (1) a focus on new types of questions, motivated by applications to structural optimization; (2) the introduction of new analytical methods for bounding effective moduli; and (3) the development of new computational methods for simulating the behavior of random media.
When the diameter of the particles in a composite shrinks to zero, the properties of the limit material can be analyzed by studying the limit of solutions of the partial differential equations associated with the elastic moduli of the two phases of the composite. The limit concept for the composite material is called G-convergence. A related (and more recent) concept is H-convergence (Tartar, 1990), and one of the basic tools used in developing this approach is compensated compactness; it allows one to go to the limit, say AnBn → AB, even though An and Bn do not converge to, respectively, A and B in the standard sense.
Some of the recent mathematical work has been devoted to "cleaning up" points of confusion. For example, there was controversy in the physics literature over whether one could discuss composites without a separation of scales (microscopic vs. macroscopic); this has been resolved by the theory of G-convergence (Zhikov et al., 1979). Another important development has been the creation of a rigorous theory of random composites (Papanicoloau and Varadhan, 1982).
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. However, when additional information about the composite is known, such as some geometric features of the second phase, or when the elastic moduli of the two phases differ greatly from each other (that is, the composite is very heterogeneous), the Hashin-Strickman bounds are too crude. Some of the mathematical work has explored connections between bounds for effective moduli and classical questions
in the calculus of variations (Kohn and Strang, 1986).
Of more direct interest to materials science, however, are the new methods for bounding effective moduli introduced by mathematicians and mathematically minded scientists from other fields. They use (1) Lagrangians or lower semicontinuous quadratic forms to derive new variational principles from old ones (Avellaneda et al., 1988; Milton, 1990), (2) methods based on function theory (Milton, 1991), and (3) new applications of the Hashin-Strikman variational principle (Avellaneda, 1987a; Allaire and Kohn, 1992). These techniques have led to many new bounds on effective moduli, often bounds that are in fact optimal. It will take much more work, however, to obtain a full understanding of their power and their limitations.
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. Most of the recent mathematical activity has been concerned with bounds in terms of volume fractions only and incorporates no further statistical information about the microstructure. This sort of bounding is natural for structural optimization, but there are other situations where statistical information is available and could improve the bound computations. We must learn to incorporate more statistical information into new bounding procedures.
An important direction of research is the development of improved algorithms for calculating the bulk effective properties of composite media, especially in challenging situations such as when the system is nondilute, when the contrast (that is, the ratio of moduli) is very different from the contrasts of situations already considered (Dal Maso and Dell'Antonio, 1991), and when the microstructure is not completely specified, as in disordered media. A new direction of research that has started developing recently is nonlinear properties of composites. One of the most important types of nonlinear behavior in materials is mechanical or electrical failure or breakdown. Failure is often the result of small, macroscopic defects in an otherwise homogeneous material. Such a material can be considered to be a dilute composite. Despite some progress that has been made, there is a pressing need to develop new, effective mathematical methods for studying and calculating the macroscopic nonlinear behavior of such materials. Some of the more recent relevant references are Bensoussan et al. (1986), Duxbury et al. (1987), Lee and Duxbury (1987), Stroud and Hui (1988), Bergman (1991), Blumenfeld and Bergman (1991), Chakrabarti (1991), Weinan (1992), and Evans (1992). There will be many more interesting new developments in this area in the next few years.
Colloidal suspensions are discussed in the Complex Fluids section of this chapter. Homogenization methods have been applied in recent years to colloidal suspensions (Fleury, 1980; Levy, 1983, 1985; Sancez-Palencia, 1985, 1987). It is clear, however, that the models are too simple and that, for more realistic models, new ideas will be needed to derive good estimates of the effective physical parameters.
Microcomposites are interesting nonlinear materials in which local field effects are
important (Stroud and Hui, 1988; Blumenfeld and Bergman, 1991). Since the field inside a particle, which is small compared to the wavelength, is different from the field in the surrounding medium, if the particle is irregularly shaped, the electric field in it is highly nonuniform and concentrated in regions of large curvature. This electric field strongly influences both linear and nonlinear optical properties. In the mathematical description of the problem, a transcendental equation including both linear and nonlinear susceptibility always results. Thus far it has been solved only approximately and only for the most simple shapes. Local fields in irregular disordered nonlinear media should be analyzed further.
The goal of structural optimization is to choose the shape or composition of a structure so as to optimize some feature of its elastic behavior. A typical goal is to minimize the weight subject to a constraint on the work done by a given load.
This topic has been considered at length in the mechanical engineering community (Pironneau, 1984; Haftka and Grandhi, 1986). There is what might be called a "standard approach": (1) guess an initial design; then (2) discretize the associated elasticity problem; then (3) use sensitivity analysis and front-tracking methods to improve the design incrementally.
An alternative approach emerged in the mathematical literature of the 1970s. The main idea is to admit as design components not only the originally given materials, but also composites that can be made from them. This process, known as relaxation, does not change the essential design problem (Lurie and Cherkaev, 1982; Murat and Tartar, 1985; Kohn and Strang, 1986). The relaxed formulation has several advantages over the more traditional one: it is easier to solve numerically, it tends to have fewer local minima, and there is a satisfactory mathematical existence theory.
Relaxation is now attracting the attention of a growing community of engineers (Bendse and Kikuchi, 1988; Suzuki and Kikuchi, 1991; Bendse and Mota Soares, 1992). For the most part, it is viewed by engineers as a scheme for "topology optimization."
Thus far, most work in optimal composites has been on compliance optimization problems involving a single loading condition. More work (and new methods) will be required to handle design criteria other than compliance and structures subject to multiple loads. The optimal designs obtained this way naturally make use of composite materials, which may be difficult to manufacture. It is therefore natural to seek methods for avoiding the use of composites. Some promising methods have been proposed and explored numerically, although they do not yet have any basis in mathematical theory.
Should this work be viewed as "materials science"? Some might say not, and it is certainly not in the mainstream of conventional materials science. 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) and of connections with the modeling of coherent phase transitions (see the discussion of martensite materials in Chapter 4), both of which are
mainstream topics. The idea that microstructure is important along with shape is by no means new to materials science; see Ashby (1991) for a thoughtful discussion. The essence of the new approach to structural optimization is the recognition that optimizing microstructure and optimizing shape are really the same problem on different length scales.
GLASSES AND OTHER AMORPHOUS SOLIDS
Solid materials that lack periodic or quasiperiodic order but are homogeneous above the nanometer length scale constitute an important class of substances that offer many technological advantages. These advantages include corrosion resistance, modified magnetic and elastic properties, and ease of fabrication. Glasses are a subset of amorphous solids that are formed by supercooling liquids through their glass transition point, below which flow processes become immeasurably slow. Other disordered solids are produced by mechanical working or strong radiation damage of crystalline materials, by interdiffusion of distinct components, and by vapor-phase deposition. In all cases, the product of these processes is a metastable form of matter. As a result of this metastability, the amorphous material has properties that not only depend on the ambient conditions (temperature and pressure), but also can depend on aspects of the history of the material, such as the conditions of formation and subsequent thermal conditions; for background and more details, see, for example, Angell and Goldstein (1986), Phillips (1982), and O'Reilly and Goldstein (1981).
Computer simulation has begun to play an important role for research in glasses and other disordered solids. Computer simulation is now being used to classify the types of short-range atomic order that appear, in contrast to that of the corresponding crystalline substances. However, these simulations have suffered from a severe time-scale problem: glasses and other amorphous solids exhibit dynamical processes over an extremely wide range of characteristic times. At the short end (10-13 to 10-12 s), atoms vibrate about mechanical equilibrium sites; such motions (possibly quite anharmonic) are well represented by present-day computer simulations. 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. Novel conceptual insights are required to accelerate sluggish and rare relaxation processes (Berg, 1993), perhaps by temporarily switching from one potential energy surface to another with suppressed barriers.
It is desirable to develop a rational predictive capability to determine what materials will resist crystal nucleation, supercool easily, and pass into a rigid glassy state at a sharply defined glass transition temperature. Ideally, the only input should be the chemical structure of the substance under consideration. Such predictive capacity is currently lacking.
Empirically, the relaxation of a wide variety of mechanical, thermal, electrical, and spectroscopic properties in glasses can be described by a "stretched exponential" function of time, that is, a negative exponential of time raised to a power between 0 and 1. Some
theoretical models for such stretched exponential behavior have been advanced, but their relation to and possible deduction from atomic-level Hamiltonians continue to remain obscure. Rigorous mathematical analysis of such connections or proving their strict absence would be a valuable contribution.
Mathematicians might be able to supply basic and extremely valuable insights to this field. For example, it would be important to establish or obtain the following results for some nontrivial class of intermolecular potential functions (e.g., pairwise additive spherical interactions):
Prove that the lowest overall potential energy, in the large system limit, is attained for a periodic spatial arrangement of molecules (a crystal).
Prove that if periodicity is strongly disallowed in some suitable sense, that amorphous molecular arrangements still exist as relative minima of the potential energy function.
Develop rigorous bounds on the rise in potential energy that results from imposition of the nonperiodicity constraint in the preceding result.
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.