Haverford College Haverford Scholarship

Faculty Publications Physics

1999

Pattern formation in nonequilibrium physics

Jerry P. Gollub Haverford College

J. S. Langer

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Repository Citation J.P.Gollub and J.S. Langer, "Pattern Formation in Nonequilibrium Physics", in the Centennial Issue of Reviews of Modern Physics, 71, S396-S403 (1999).

This Journal Article is brought to you for free and open access by the Physics at Haverford Scholarship. It has been accepted for inclusion in Faculty Publications by an authorized administrator of Haverford Scholarship. For more information, please contact [email protected] Pattern formation in nonequilibrium physics

J. P. Gollub Haverford College, Haverford, Pennsylvania 19041 and Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104

J. S. Langer Department of Physics, University of California, Santa Barbara, California 93106

Remarkable and varied pattern-forming phenomena occur in ﬂuids and in phase transformations. The authors describe and compare some of these phenomena, offer reﬂections on their similarities and differences, and consider possibilities for the future development of this ﬁeld.

[S0034-6861(99)04702-9]

I. INTRODUCTION for the present purposes are spatially extended dynami- cal systems with many degrees of freedom, for which The complex patterns that appear everywhere in na- partial differential equations are needed. Corresponding ture have been cause for wonder and fascination physical systems include ﬂuids subjected to heating or throughout human history. People have long been rotation, which exhibit sequences of increasingly com- puzzled, for example, about how intricate snowﬂakes plex spatiotemporal patterns as the driving forces can form, literally, out of thin air; and our minds boggle change. These are all purely deterministic pattern- at the elegance of even the simplest living systems. As forming systems. Understanding how they behave has physicists, we have learned much about natural pattern been a crucial step toward understanding deterministic formation in recent years; we have discovered how rich chaos—one of the most intriguing and profound scien- this subject can be, and how very much remains to be tiﬁc concepts to emerge in this century. understood. Our growing understanding of the physics At the center of our modern understanding of pattern of pattern formation has led us to speculate—so far with formation is the concept of instability. It is interesting to only limited success—about a more general science of note that the mathematical description of instabilities is complexity, and to pose deep questions about our ability strikingly similar to the phenomenological theory of to predict and control natural phenomena. phase transitions ﬁrst given by Landau (Landau and Lif- Although pattern formation—i.e., morphogenesis— shitz, 1969). We now know that complex spatial or tem- has always been a central theme in natural philosophy, it poral patterns emerge when relatively simple systems has reemerged in mainstream nonequilibrium physics are driven into unstable states, that is, into states that only in the last quarter of the 20th Century. This has will deform by large amounts in response to inﬁnitesi- happened, in part, as an outgrowth of physicists’ and mally small perturbations. For example, solar heating of materials scientists’ interest in phase transitions. Many the earth’s surface can drive Rayleigh-Be´nard-like con- of the most familiar examples of pattern formation oc- vective instabilities in the lower layer of the atmosphere, cur in situations in which a system is changing from one and the resulting ﬂow patterns produce fairly regular phase to another—from a liquid to a geometrically pat- arrays of clouds. At stronger driving forces, the convec- terned solid, for example, or from a uniform mixture of tion patterns become unstable and turbulence increases. chemical constituents to a phase-separated pattern of Another familiar example is the roughness of fracture precipitates. As scientists have learned more about the surfaces produced by rapidly moving cracks in brittle equilibrium aspects of phase transitions, many have be- solids. When we look in detail, we see that a straight come interested in the non-equilibrium processes that crack, driven to high enough speeds, becomes unstable accompany them. This line of investigation has led di- in such a way that it bends, sends out sidebranching rectly to questions of pattern formation. cracks, and produces damage in the neighboring mate- Another direction from which physicists have ap- rial. In this case, the physics of the instability that leads proached the study of pattern formation has been the to these irregular patterns is not yet known. theory of nonlinear dynamical systems. Mechanical sys- After an instability has produced a growing distur- tems that can be described by ordinary differential equa- bance in a spatially uniform system, the crucial next step tions often undergo changes from simple to complex be- in the pattern-forming process must be some intrinsi- havior in response to changes in their control cally nonlinear mechanism by which the system moves parameters. For example, the periodically forced and toward a new state. That state may resemble the un- damped pendulum shows chaotic motion for certain in- stable deformation of the original state—the convective tervals of the forcing amplitude, as well as periodic win- rolls in the atmosphere have roughly the same spacing as dows within the chaotic domains—a temporal ‘‘pattern’’ the wavelength of the initial instability. However, in with considerable complexity. This is a simple case, how- many other cases, such as the growth of snowﬂakes, the ever, with only a few degrees of freedom. More relevant new patterns look nothing like the linearly unstable de-

S396 Reviews of Modern Physics, Vol. 71, No. 2, Centenary 1999 0034-6861/99/71(2)/396(8)/$16.60 ©1999 The American Physical Society J. P. Gollub and J. S. Langer: Pattern formation in nonequilibrium physics S397 formations from which they started. The system evolves The best way to visualize the solutions of such equa- in entirely new directions as determined by nonlinear tions is to think of them as points in a multidimensional dynamics. We now understand that it is here, in the non- mathematical space spanned by the dynamical variables, linear phase of the process, that the greatest scientiﬁc that is, a ‘‘phase space.’’ The rules that determine how challenges arise. these points move in the phase space constitute what we The inherent difﬁculty of the pattern-selection prob- call a ‘‘dynamical system.’’ One of the most important lem is a direct consequence of the underlying (linear or developments in this ﬁeld has been the recognition that nonlinear) instabilities of the systems in which these dynamical systems with inﬁnitely many degrees of free- phenomena occur. A system that is linearly unstable is dom can often be described by a ﬁnite number of rel- one for which some response function diverges. This evant variables, that is, in ﬁnite-dimensional phase means that pattern-forming behavior is likely to be ex- spaces. For example, the ﬂow ﬁeld for Rayleigh-Be´nard tremely sensitive to small perturbations or small changes convection not too far from threshold can be described in system parameters. For example, many patterns that accurately by just a few time-dependent Fourier ampli- we see in nature, such as snowﬂake-like dendrites in so- tudes. If we think of the partial differential equations as lidifying alloys, are generated by selective ampliﬁcation being equivalent to ﬁnite sets of coupled ordinary differ- of atomic-scale thermal noise. The shapes and speeds of ential equations, then we can bring powerful mathemati- growing dendrites are also exquisitely sensitive to tiny cal concepts to bear on the analysis of their solutions. crystalline anisotropies of surface energies. As we shall emphasize in the next several sections of Some important questions, therefore, are: Which per- this article, dynamical-systems theory provides at best a turbations and parameters are the sensitively controlling qualitative framework on which to build physical models ones? What are the mechanisms by which those small of pattern formation. Nevertheless, it has produced valu- effects govern the dynamics of pattern formation? What able insights and, in some cases, has even led to predic- are the interrelations between physics at different length tion of novel effects. It will be useful, therefore, to sum- scales in pattern-forming systems? When and how do marize some of these general concepts before looking in atomic-scale mechanisms control macroscopic phenom- more detail at speciﬁc examples. An introductory discus- ena? At present, we have no general strategy for an- sion of the role of dynamical systems theory in ﬂuid me- swering these questions. The best we have been able to chanics has been given by Aref and Gollub (1996). do is to treat each case separately and—because of the In dynamical-systems theory, the stable steady solu- remarkable complexity that has emerged in many of tions of the equations of motion are known as ‘‘stable these problems—with great care. ﬁxed points’’ or ‘‘attractors,’’ and the set of points in the In the next several sections of this article we discuss phase space from which trajectories ﬂow to a given ﬁxed the connection between pattern formation and nonlin- point is its ‘‘basin of attraction.’’ As the control param- ear dynamics, and then describe just a few speciﬁc ex- eters are varied, the system typically passes through ‘‘bi- amples that illustrate the roles of instability and sensitiv- furcations’’ in which a ﬁxed point loses its stability and, ity in nonequilibrium pattern formation. Our examples at the same time, one or more new stable attractors ap- are drawn from ﬂuid dynamics, granular materials, and pear. An especially simple example is the ‘‘pitchfork’’ crystal growth, which are topics that we happen to know bifurcation at which a stable ﬁxed point representing a well. We conclude with some brief remarks, mostly in steady ﬂuid ﬂow, for example, gives rise to two the form of questions, about universality, predictability, symmetry-related ﬁxed points describing cellular ﬂows and long-term prospects for this ﬁeld of research. with opposite polarity. Many other types of bifurcation have been identiﬁed in simple models and also have been seen in experiments. II. PATTERN FORMATION AND DYNAMICAL SYSTEMS The theory of bifurcations in dynamical systems helps us understand why it is sometimes reasonable to de- Our understanding of pattern formation has been dra- scribe a system with inﬁnitely many degrees of freedom matically affected by developments in mathematics. De- using only a ﬁnite (or even relatively small) number of terministic pattern-forming systems are generally de- dynamical variables. An important mathematical result scribed by nonlinear partial differential equations, for known as the ‘‘center manifold theorem’’ (Guckenhe- example, the Navier-Stokes equations for ﬂuids, or imer and Holmes, 1983) indicates that, when a bifurca- reaction-diffusion equations for chemical systems. It is tion occurs, the associated unstable trajectories typically characteristic of such nonlinear equations that they can move away from the originally stable ﬁxed point only have multiple steady solutions for a single set of control within a low-dimensional subspace of the full phase parameters such as external driving forces or boundary space. The subspace is ‘‘attracting’’ in the sense that tra- conditions. These solutions might be homogeneous, or jectories starting elsewhere converge to it, so that the patterned, or even more complex. As the control param- degrees of freedom outside the attracting subspace are eters change, the solutions appear and disappear and effectively irrelevant. It is for this reason that we may change their stabilities. In mathematical models of spa- need only a low-dimensional space of dynamical vari- tially extended systems, different steady solutions can ables to describe some pattern-formation problems near coexist in contact with each other, separated by lines of their thresholds of instability—a remarkable physical re- defects or moving fronts. sult.

Rev. Mod. Phys., Vol. 71, No. 2, Centenary 1999 S398 J. P. Gollub and J. S. Langer: Pattern formation in nonequilibrium physics

Time-varying states, such as oscillatory or turbulent A convenient way of exciting nonlinear waves in a ﬂows, are more complex than simple ﬁxed points. Here, manner that does not directly break any spatial symme- some insight also has been gained from considering dy- try is to subject the ﬂuid container to a small-amplitude namical systems. Oscillatory behavior is generally de- vertical excitation. This leads to standing waves at half scribed as a ﬂow on a limit cycle (or closed loop) in the driving frequency, via an instability and an associ- phase space, and chaotic states may be represented by ated bifurcation ﬁrst demonstrated by Faraday (1831), more complex sets called ‘‘strange attractors.’’ The most 68 years before the American Physical Society was characteristic feature of the latter may be understood in founded. The characteristic periodicity of the resulting terms of the Lyapunov exponents that give the local ex- patterns is approximately the same as the wavelength of ponential divergence or convergence rates between two the most rapidly growing linear instability determined nearby trajectories, in the different directions along and by the dispersion relation for capillary-gravity waves, transverse to those trajectories. If at least one of these but the wave patterns themselves are far more complex exponents, when averaged over time, is positive, then and interesting. nearby orbits will separate from each other exponen- All of the regular patterns that can tile the plane have tially in time. Provided that the entire attracting set is been found in this system, including hexagons, squares, bounded, the only possibility is for the set to be fractal. and stripes (Kudrolli and Gollub, 1996). In addition, In the early 1970s, strange attractors were thought by various types of defects that are analogous to crystalline some to be useful models for turbulent ﬂuids. In fact, the defects occur: grain boundaries, dislocations, and the phase-space paradigm can give only a caricature of the like. Which patterns are stable depends on parameters: real physics because of the large number of relevant de- the ﬂuid viscosity, the driving frequency, and the accel- grees of freedom involved in most turbulent ﬂows. eration. Signiﬁcant domains of coexistence between dif- While much less than 6N (the number of degrees of ferent patterns are also known, where patterns with dif- freedom for a system of N molecules), that number still 3/4 ferent symmetry are simultaneously stable. grows in proportion to R , where R is the Reynolds These phenomena have resisted quantitative explana- number. We do not yet know whether weakly turbulent tion for a number of reasons: the difﬁculty of dealing states (‘‘spatiotemporal chaos’’) that sometimes occur with boundary conditions at the moving surface of the near the onset of instability may be viewed usefully us- ﬂuid; the nonlinearity of the hydrodynamic equations; ing the concepts of dynamical systems. and the complex effects of viscosity. However, a suitable mathematical description, consistent with the general framework of dynamical-systems theory, is now avail- III. PATTERNS AND SPATIOTEMPORAL CHAOS able (Chen and Vin˜als, 1997), and it leads to a satisfac- IN FLUIDS: NONLINEAR WAVES tory explanation of these pattern-forming phenomena. The basic idea is to regard the surface as a superposition Pattern formation has been investigated in an im- of interacting waves propagating in different directions. mense variety of hydrodynamic systems. Examples in- Coupled evolution equations can be written for the vari- clude convection in pure ﬂuids and mixtures; rotating ous wave amplitudes. The coupling coefﬁcients depend ﬂuids, sometimes in combination with thermal transport; on the angles between the wave vectors, and these cou- nonlinear surface waves at interfaces; liquid crystals pling functions depend in turn on the imposed param- driven either thermally or by electromagnetic ﬁelds; eters (such as wave frequency). The entire problem is chemically reacting ﬂuids; and falling droplets. Some variational, but only near the threshold of wave forma- similar phenomena occur in nonlinear optics. Many of tion. That is, the preferred pattern near the onset of these cases have been reviewed by Cross and Hohen- instability is the one that minimizes a certain functional berg (1993); there also have been a host of more recent of the wave amplitudes, in much the same way that the developments. Since it is not possible in a brief space to preferred state of a crystal is the one that minimizes its discuss this wide range of phenomena, we focus here on free energy. Away from threshold, on the other hand, no an example that poses interesting questions about the such variational principle exists, and the variety of be- nature of pattern formation: waves on the surfaces of haviors is correspondingly richer. ﬂuids. We shall also make briefer remarks about other These results raise the question of whether other ﬂuid systems that have revealed strikingly novel phe- types of regular patterns can be formed that are not nomena. spatially periodic but do have rotational symmetry, i.e., The surface of a ﬂuid is an extended dynamical system quasicrystalline patterns. In fact they do occur (Chris- for which the natural variables are the amplitudes and tiansen et al., 1992; Edwards and Fauve, 1994), just as phases of the wavelike deformations. These waves were they do in ordinary crystals. The way in which these at one time regarded as being essentially linear at small different patterns become stable or unstable as the pa- amplitudes. However, even weak nonlinear effects cause rameters are varied has now been worked out in some interactions between waves with different wave vectors detail and appears to be in accord with experiment and can be important in determining wave patterns. (Binks and van de Water, 1997). An example of a qua- When the wave amplitudes are large, the nonlinear ef- sicrystalline pattern is shown in Fig. 1. fects lead to chaotic dynamics in which many degrees of When the wave amplitudes are raised sufﬁciently, freedom are active. transitions to spatially and temporally disordered states

Rev. Mod. Phys., Vol. 71, No. 2, Centenary 1999 J. P. Gollub and J. S. Langer: Pattern formation in nonequilibrium physics S399

FIG. 1. (Color) A quasicrystalline wave pattern with 12-fold rotational symmetry. This standing-wave pattern was produced by forcing a layer of silicone oil simultaneously at two frequencies, using a method invented by Edwards and Fauve. The brightest regions are locally horizontal, whereas darker colors indicate inclined regions. From work done at Haverford for an undergraduate thesis by B. Pier. occur (Kudrolli and Gollub, 1996 and references Rayleigh-Be´nard convection in the presence of rotation therein). In the language of dynamical systems, some of about a vertical axis (Hu et al., 1997). This problem is these new states might be called ‘‘strange attractors,’’ relevant to atmospheric dynamics. Though the basic pat- although they are certainly not low-dimensional objects. tern consists of rolls, as shown in Fig. 2, they are un- They are much less well understood than the standing- stable. Patches of rolls at different angles invade each wave states, and the ways in which they form appears to other as time proceeds, and the pattern remains time depend on the ordered states from which they emerge. dependent indeﬁnitely. This phenomenon has been dis- For example, the hexagonal lattice appears to melt con- cussed theoretically (Tu and Cross, 1992) using a two- tinuously, while the striped phase breaks down inhomo- dimensional nonlinear partial differential equation geneously in regions where the stripes are most strongly known as the complex Ginzburg-Landau equation. Simi- curved. The resulting states of spatiotemporal chaos are lar models have been used successfully for treating a not completely disordered; there can be regions of local variety of nonchaotic pattern-forming phenomena. In order. Furthermore, if the ﬂuid is not too viscous, so that this case, the model is able to reproduce the qualitative the correlation length of the pattern is relatively long, behavior of the experiments, but does not successfully then the symmetry imposed by the boundaries can be describe the divergence of the correlation length of the recovered by averaging over a large number of individu- patchy chaotic ﬂuctuations as the transition is ap- ally ﬂuctuating patterns (Gluckman et al., 1995). A case proached. Thus the goal of understanding spatiotempo- of strongly turbulent capillary waves has also been stud- ral chaos has remained elusive. ied experimentally (Wright et al., 1997). There is one area where the macroscopic treatment of Certain other ﬂuid systems have chaotic states that pattern-forming instabilities connects directly to micro- occur closer to the linear threshold of the primary pat- scopic physics: the effects of thermal noise. Macroscopic tern. In these cases, quantitative comparison with theory patterns often emerge from the ampliﬁcation of noise by is sometimes possible. An example is the behavior of instabilities. Therefore, ﬂuctuations induced by thermal

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FIG. 2. Spatiotemporal chaos in rotating Rayleigh-Be´nard convection shown at two different times. Patches of rolls at different angles invade each other as time proceeds. Courtesy of G. Ahlers. noise (as distinguished from chaotic ﬂuctuations pro- duced by nonlinearity) should be observable near the threshold of instability. This remarkable effect has been FIG. 3. Standing-wave patterns in a vibrating layer of granular demonstrated quantitatively in several ﬂuid systems, for material. (a) Stripes; (b) hexagons and defects; (c) disordered ´ example, ordinary Rayleigh-Benard convection (Wu waves; (d) clusters of localized ‘‘oscillons.’’ Courtesy of P. Um- et al., 1995). As we shall see in Sec. V, very similar am- banhowar and H. Swinney. pliﬁcation of thermal noise occurs in dendritic crystal growth. body systems such as ﬂuids. Because they have huge numbers of degrees of freedom, they can only be under- IV. PATTERNS IN GRANULAR MATERIALS stood in statistical terms. However, individual grains of sand are enormously more massive than atoms or even Patterns quite similar to the interfacial waves de- macromolecules; thus thermal kinetic energy is irrel- scribed in the previous section occur when the ﬂuid is evant to them. On the other hand, each individual grain replaced by a layer of granular matter such as sand or, in has an effectively inﬁnite number of internal degrees of well-controlled recent experiments, uniform metallic or freedom; thus the grains are generally inelastic in their glassy spheres (Melo et al., 1995). Depending on the fre- interactions with each other or with boundaries. They quency and amplitude of the oscillation of the container, also may have irregular shapes; arrays of such grains the upper surface of the grains can arrange itself into may achieve mechanical equilibrium in a variety of con- arrays of stripes or hexagons, as shown in Fig. 3. Lines ﬁgurations and packings. It seems possible, therefore, dividing regions differing in their phase of oscillation, that concepts like ‘‘temperature’’ and ‘‘entropy’’ might and disordered patterns, are also evident. be useful for understanding the behavior of these mate- In addition, granular materials can exhibit localized rials (for example, see Campbell, 1990). solitary excitations known as ‘‘oscillons’’ (Umbanhowar In the oscillating-granular-layer experiments, some of et al., 1996). These can in turn organize themselves into the simpler transitions can be explained in terms of tem- clusters, as shown in Fig. 3(d). This striking discovery poral symmetry breaking resulting from the the low- has given rise to a number of competing theories and dimensional dynamics of the particles as they bounce off has been immensely provocative. It is interesting to note the oscillating container surface. The onset of temporal that localized excitations are also found in ﬂuids. For period doubling in the particle dynamics coincides with example, they have been detected in instabilities in- the spatial transition from stripes to hexagons. Both the duced by electric ﬁelds applied across a layer of nematic particle trajectories and the spatial patterns are then dif- liquid crystal (Dennin et al., 1996). All of these localized ferent on successive cycles of the driver. The analogous states are intrinsically nonlinear phenomena, whether hexagonal state for Faraday waves in ﬂuids can be in- they occur in granular materials or in ordinary ﬂuids, duced either by temporal symmetry breaking of the ex- and do not resemble any known linear instability of the ternal forcing, or by the frequency and viscosity depen- uniform system. dence of the coupling between different traveling-wave Granular materials have been studied empirically for components. Both of these mechanisms are quite differ- centuries in civil engineering and geology. Nevertheless, ent microscopically from the single-particle dynamics we still have no fundamental physical understanding of that generates the hexagons in granular materials. their nonequilibrium properties. In fact, to a modern On the other hand, there are substantial similarities physicist, granular materials look like a novel state of between the granular and ﬂuid behaviors. The granular matter. For a review of this ﬁeld, see Jaeger et al. (1996), material expands or dilates as a result of excitation. and references therein. Roughly speaking, dilation of the granular layer reduces There are several clear distinctions between granular the geometrical constraints that limit ﬂow. This corre- materials and other, superﬁcially comparable, many- ponds to lower viscosity of the conventional ﬂuid. Dila-

Rev. Mod. Phys., Vol. 71, No. 2, Centenary 1999 J. P. Gollub and J. S. Langer: Pattern formation in nonequilibrium physics S401 tion accounts in physical terms for the fact that the striped phase that occurs at high ﬂuid viscosity may be found at low acceleration of the granular material. The differences, however, seem to be emerging dramatically as new experiments and numerical simula- tions probe more deeply into granular phenomena. For example, stresses in nearly static granular materials are highly inhomogeneous, forming localized stress chains that are quite unlike anything seen in ordinary liquids and solids. Even in situations involving ﬂow, the behavior of granular materials often seems to be gov- erned by their tendency to ‘‘jam,’’ that is, to get them- selves into local conﬁgurations from which they are tem- porarily unable to escape. This happens during a part of each cycle in the oscillating-layer experiments. (The concept of ‘‘jammed’’ systems was the topic of a Fall 1997 program at the Institute for Theoretical Physics in Santa Barbara. For more information, consult the FIG. 4. Dendritic xenon crystal growing in a supercooled melt. ITP web site: http://www.itp.ucsb.edu/online/jamming2/ Courtesy of J. Bilgram. schedule.html.) has been made on understanding this phenomenon re- V. GROWTH AT INTERFACES: DENDRITIC cently. The free-dendrite problem is most easily deﬁned SOLIDIFICATION by reference to the xenon dendrite shown in Fig. 4. Here, we are looking at a pure single crystal growing We turn ﬁnally to the topic of dendritic pattern for- into its liquid phase. The speed at which the tip is ad- mation. It is here that some of the deepest questions in vancing, the radius of curvature of the tip, and the way this ﬁeld—the mathematical subtlety of the selection in which the sidebranches emerge behind the tip, all are problem and the sensitivity to small perturbations— determined uniquely by the degree of undercooling, i.e., have emerged most clearly in recent research. by the degree to which the liquid is colder than its freez- Dendritic solidiﬁcation, that is, the ‘‘snowﬂake prob- ing temperature. The question is: How? lem,’’ is one of the most thoroughly investigated topics In the most common situations, dendritic growth is in the general area of nonequilibrium pattern formation. controlled by diffusion—either the diffusion of latent It is only in the last few years, however, that we ﬁnally heat away from the growing solidiﬁcation front or the have learned how these elegant dendritic crystals are diffusion of chemical constituents toward and away from formed in the atmosphere, and why they occur with such that front. These diffusion effects very often lead to diversity that no two of them ever seem to be exactly shape instabilities; small bumps grow out into ﬁngers be- alike. Nevertheless, our present understanding is still far cause, like lightning rods, they concentrate the diffusive from good enough for many practical purposes, for ex- ﬂuxes ahead of them and therefore grow out more rap- ample, for predicting the microstructures of multicom- idly than a ﬂat surface. This instability, generally known ponent cast alloys. as the ‘‘Mullins-Sekerka instability,’’ is the trigger for Much of the research on dendritic crystal growth has pattern formation in solidiﬁcation. been driven, not only by our natural curiosity about such Today’s prevailing theory of free dendrites is gener- phenomena, but also by the need to understand and ally known as the ‘‘solvability theory’’ because it relates control metallurgical microstructures. (For example, see the determination of dendritic behavior to the question Kurz and Fisher, 1989) The interior of a grain of a of whether or not there exists a sensible solution for a freshly solidiﬁed alloy, when viewed under a micro- certain diffusion-related equation that contains a singu- scope, often looks like an interlocking network of highly lar perturbation. The term ‘‘singular’’ means that the developed snowﬂakes. Each grain is formed by a den- perturbation, in this case the surface tension at the so- dritic, i.e., treelike, process in which a crystal of the pri- lidiﬁcation front, completely changes the mathematical mary composition grows out rapidly in a cascade of nature of the problem whenever it appears, no matter branches and sidebranches, leaving solute-rich melt to how inﬁnitesimally weak it might be. In the language of solidify more slowly in the interstices. The speed at dynamical systems, the perturbation controls whether or which the dendrites grow and the regularity and spacing not there exists a stable ﬁxed point. Similar situations of their sidebranches determine the observed micro- occur in ﬂuid dynamics, for example, in the ‘‘viscous ﬁn- structure which, in turn, governs many of the properties gering’’ problem, where a mechanism similar to the of the solidiﬁed material such as its mechanical strength Mullin-Sekerka instability destabilizes a moving inter- and its response to heating and deformation. face between ﬂuids of different viscosities, and a solv- The starting point for investigations of metallurgical ability mechanism determines the resulting ﬁngerlike microstructures or snowﬂakes is the study of single, iso- pattern. (See Langer, 1987, for a pedagogical introduc- lated, freely growing dendrites. Remarkable progress tion to solvability theory, and Langer, 1989, for an over-

Rev. Mod. Phys., Vol. 71, No. 2, Centenary 1999 S402 J. P. Gollub and J. S. Langer: Pattern formation in nonequilibrium physics view including the viscous ﬁngering problem.) seen in ﬂuids and granular materials? In short, are there The solvability theory has been worked out in detail useful ‘‘universality classes’’ for which detailed underly- for many relevant situations such as the xenon dendrite ing mechanisms are less important than, for example, shown in Fig. 4. (See Bisang and Bilgram, 1996, for an more general symmetries or conservation laws? Might account of the xenon experiments, and also for refer- we discover some practical guidelines to tell us how to ences to recent theoretical work by Brener and col- construct predictive models of pattern-forming systems, leagues.) The theory predicts how pattern selection is or shall we have to start from the beginning in consider- determined, not just by the surface tension (itself a very ing each problem? small correction in the diffusion equations), but by the These are not purely philosophical questions. Essen- crystalline anisotropy of the surface tension—an even tially all processes for manufacturing industrial materials weaker perturbation in this case. It further predicts that are nonequilibrium phenomena. Most involve, at one the sidebranches are produced by secondary instabilities stage or another, some version of pattern formation. near the tip that are triggered by thermal noise and am- The degree to which we can develop quantitative, pre- pliﬁed in special ways as they grow out along the sides of dictive models of these phenomena will determine the the primary dendrite. The latter prediction is especially degree to which we can control them and perhaps de- remarkable because it relates macroscopic features— velop entirely new technologies. Will we be able, for sidebranches with spacings of order tens of microns—to example, to write computer programs to predict and molecular ﬂuctuations whose characteristic sizes are of control the microstructures that form during the casting order nanometers. of high-performance alloys? Can we hope to predict, Each of those predictions has been tested in the xenon long in advance, mechanical failure of complex struc- experiment, quantitatively and with no adjustable ﬁtting tural materials? Will we ever be able to predict earth- parameters. They have also been checked in less detail quakes? Or, conversely, might we discover that the com- in experiments using other metallurgical analog materi- plexity of many systems imposes intrinsic limits to our als. In addition, the theory has been checked in numeri- ability to predict their behavior? That too would be an cal studies that have probed its nontrivial mathematical interesting and very important outcome of research in aspects (Karma and Rappel, 1996). As a result, although this ﬁeld. we know that there must be other cases (competing thermal and chemical effects, for example, or cases ACKNOWLEDGMENTS where the anisotropy is large enough that it induces faceting), we now have reason for conﬁdence that we J.S.L. acknowledges support from DOE Grant DE- understand at least some of the basic principles cor- FG03-84ER45108. JPG acknowledges support from rectly. NSF Grant DMR-9704301.

VI. REFLECTIONS AND CONCLUSIONS REFERENCES

We have illustrated pattern-forming phenomena Aref, H., and J.P. Gollub, 1996, in Research Trends in Fluid through a few selected examples, each of which has Dynamics: Report From the United States National Committee given rise to a large literature. Many others could be on Theoretical and Applied Mechanics, edited by J.L. Lumley cited. As we have seen, the inherent sensitivity of (American Institute of Physics, Woodbury, NY ), p. 15. pattern-forming mechanisms to small perturbations Binks, D., and W. van de Water, 1997, Phys. Rev. Lett. 78, means that research in this ﬁeld must take into account 4043. physical phenomena across extraordinarily wide ranges Bisang, U., and J.H. Bilgram, 1996, Phys. Rev. E 54, 5309. of length and time scales. Moreover, studies of pattern Campbell, C.S., 1990, Annu. Rev. Fluid Mech. 22, 57. formation increasingly are being extended to materials Chen, P., and J. Vin˜als, 1997, Phys. Rev. Lett. 79, 2670. that are more complex than isotropic classical ﬂuids and Christiansen, B., P. Alstrom, and M.T. Levinsen, 1992, Phys. homogeneous solids. The case of granular matter de- Rev. Lett. 68, 2157. scribed here is one example of this trend. An important Cross, M.C., and P.C. Hohenberg, 1993, Rev. Mod. Phys. 65, example for the future is pattern formation in biological 851. systems, where the interplay between physical effects Dennin, M., G. Ahlers, and D.S. Cannell, 1996, Phys. Rev. and genetic coding leads to striking diversity. Lett. 77, 2475. Edwards, S., and S. Fauve, 1994, J. Fluid Mech. 278, 123. The expanding complexity and importance of this Faraday, M., 1831, Philos. Trans. R. Soc. London 121, 299. ﬁeld brings urgency to a set of deep questions about Gluckman, B.J., C.B. Arnold, and J.P. Gollub, 1995, Phys. theories of pattern formation and, more generally, about Rev. E 51, 1128. the foundations of nonequilibrium statistical physics. Guckenheimer, J., and P. Holmes, 1983, Nonlinear Oscilla- What does sensitivity to noise and delicate perturbations tions, Dynamical Systems, and Bifurcations of Vector Fields imply about the apparent similarities between different (Springer, New York). systems? Are there, for example, deep connections be- Hu, Y., R.E. Ecke, and G. Ahlers, 1997, Phys. Rev. E 55, 6928. tween dendritic sidebranching and fracture, or are the Jaeger, H.M., S.R. Nagel, and R.P Behringer, 1996, Rev. Mod. apparent similarities superﬁcial and unimportant? What Phys. 68, 1259. about the apparent similarities between the patterns Karma, A., and J. Rappel, 1996, Phys. Rev. Lett. 77, 4050.

Rev. Mod. Phys., Vol. 71, No. 2, Centenary 1999 J. P. Gollub and J. S. Langer: Pattern formation in nonequilibrium physics S403

Kudrolli, A., and J.P. Gollub, 1996, Physica D 97, 133. Melo, F., P.B. Umbanhowar, and H.L. Swinney, 1995, Phys. Kurz, W., and D.J. Fisher, 1989, Fundamentals of Solidiﬁcation Rev. Lett. 75, 3838. (Trans Tech Publications, Brookﬁeld, VT). Miles, J., and D. Henderson, 1990, Annu. Rev. Fluid Mech. 22, Landau, L.D., and E.M. Lifshitz, 1969, Statistical Physics (Per- 143. Tu, Y., and M. Cross, 1992, Phys. Rev. Lett. 69, 2515. gamon, Oxford), Chap. XIV. Umbanhowar, P.B., F. Melo, and H.L. Swinney, 1996, Nature Langer, J.S., 1987, in Chance and Matter, Proceedings of the (London) 382, 793. Les Houches Summer School, Session XLVI, edited by J. Wright, W.B., R. Budakian, D.J. Pine, and S.J. Putterman, Souletie, J. Vannimenus, and R. Stora (North Holland, Am- 1997, Science 278, 1609. sterdam), p. 629. Wu, M., G. Ahlers, and D.S. Cannell, 1995, Phys. Rev. Lett. 75, Langer, J.S., 1989, Science 243, 1150. 1743.

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