Pattern Selection in Solidification

Pattern Selection in Solidification

J.S. LANGER

Directional solidification of alloys produces a wide variety of cellular or lamellar structures which, depending upon growth conditions, may be reproducibly regular or may behave chaotically. It is not well understood how these patterns are selected and controlled or even whether there ever exist sharp selection mechanisms. A related phenomenon is the spatial propagation of a pattern into a system which has been caused to become unstable against pattern-forming deformations. This phenomenon has some features in common with the propagation of sidebranching modes in dendritic solidification. In a class of one-dimensional models, the nonlinear system can be shown to select the propagating mode in which the leading edge of the pattern is just marginally stable. This stability principle, when applicable, predicts both the speed of propagation and the geometrical characteristics of the pattern which forms behind the moving front. A boundary-layer model for fully two or three dimensional solidification problems appears to exhibit similar mathematical behavior.

METALLURGICAL microstructures are relatively simple examples of a broad and extremely interesting class of natural pattern-forming phenomena. Other examples involve flow-patterns in hydrodynamic systems, patterns formed by diffusion of chemical reactants, a n d - - at the highest level of complexity--biological systems. These problems are very difficult, and the basic principles required for their solution seem not yet to be understood. What is needed at the moment are models which are sufficiently tractable to allow exploration of difficult mathematical questions, but which are not so simple that they lose all experimental relevance. It seems possible that metallurgical microstructures may be ideal sources of such models.

In this lecture I shall talk principally about patterns formed during directional solidification of alloys: cellular structures, lamellar eutectics, e t c . ; but I shall also be con- cerned in an important way with the growth of free dendrites. The advantage of the directional geometry is that one can perform an exact linear-stability analysis for an initially flat solidification front, 1 and one can carry the nonlinear calculation far enough to see explicitly the existence of the continuous family of stable cellular patterns which may emerge when the plane front is driven beyond its limit of stability. 2'3'4

The general scheme for this kind of calculation is the following. For any given system there exists some dimen- sionless group of parameters, denoted here by u, which serves as a control parameter. In the case of directional solidification of a dilute alloy, u is proportional to the growth rate and the solute concentration and inversely proportional to the thermal gradient at the front. The amplification rate to for infinitesimal sinusoidal deformations of wave number k is shown schematically in Figure 1 for various values of u. For u greater than some critical value u,. , to

becomes positive within some finite band of wave numbers and the system is unstable against the corresponding deformations. To carry the calculation further, one must use some kind of nonlinear technique. 2 This is usually a perturbation method valid only for small values of u - u,.; but more

J. S. LANGER is with the Institute for Theoretical Physics, University of California at Santa Barbara, Santa Barbara, CA 93106.

This paper is based on a presentation made at the symposium "Establishment of Microstructural Spacing during Dendritic and Coopera- tive Growth" held at the annual meeting of the AIME in Atlanta, Georgia on March 7, 1983 under the joint sponsorship of the ASM-MSD Phase Transformations Committee and the TMS-AIME Solidification Committee.

ambitious procedures have been described in the literature. 3'5 It is important at this stage that one not only look for stationary solutions of the nonlinear equations of motion but also test these solutions for stability against deformations which change both their amplitudes and periodicities. The most dangerous instabilities, that is, the deformations which determine the most stringent stability limits for families of cellular states, are generally those which change the periodicity by creating or destroying cells.

The results of calculations of this kind are conveniently summarized in a u, k diagram like that shown in Figure 2. This figure is taken from a recent paper by Dee and Mathur, 5 who have developed an extended amplitude-equation method for studying the dynamics of cellular fronts formed in directional solidification of thin films. The outer curve in this figure describes neutral stability of the planar front, that is, the locus of points for which to(k) vanishes. The inner curve encloses the region of u, k values for which stable, stationary cellular states exist. Unstable stationary solutions generally exist well outside this region, even in some cases outside the planar neutral stability curve. Note that we are using the k-axis to describe two different quantities in this diagram: first, the wave number of infinitesimal deformations of the plane in drawing the neutral stability curve; and second, the wave number of finite-amplitude cellular states.

The crux of the pattern-selection problem is the fact that the inner region of Figure 2 has nonzero width in k for a range of values of u. Because u is the only control parameter available--there are no other externally adjustable degrees of freedom for this sys tem-- one must conclude that steady state theory plus stability considerations alone are insuffi- cient to determine a unique cellular pattern.

At this point there arises an important experimental question: Does nature select a unique pattern? Or can one produce any cellular periodicity within this band by, for example, making slight adjustments in the way the system is set into motion or by perturbing it appropriately? The experimental answers are not clear at present. Probably the most definite case is the free dendrite for which a sharp selection mechanism seems almost certainly to be operative; but, as we shall see, the free dendrite may not fit theoreti- cally into precisely the scheme outlined above. At the other extreme of possibilities, it has been mentioned elsewhere in this Symposium 6 that thin-film lamellar eutectics may

METALLURGICAL TRANSACTIONS A VOLUME 15A, JUNE 1984--961

k >

>

�9 . - - �9 . . . Fig 1 Schematic amphficatmn rate ~o for deformations of wave number k shown for various values of the control parameter ~,.

be grown, at least metastably, with a range of different spac- ings under identical steady-state growth conditions. In this case, more than one state in the stable band seems to be accessible experimentally. To make matters more compli- cated, however, van Suchtelen 7 has grown CuC1-PbC1 films around a circle and has observed that the lamellar widths A decreased smoothly with increasing distance from the center, that is, with increasing growth speed v. His pictures seem to indicate a sharp selection mechanism for this curved geometry, possibly consistent with the conventional law of the form A2v = constant.

From a theoretical point of view, it seems to me that there are four possibilities for pattern selection. There are special circumstances in which each of these possibilities might be valid.

(1) Variational Principles: A recurring theme in the history of the pattern selection problem is the idea that nonequilibrium systems may operate in such a way as to optimize some quantity, for example, the undercooling at the surface, the growth rate, the rate of entropy production, etc. To my knowledge, no useful form of such a principle has yet been proven to be generally valid for any version of the solidification problem. (However, J. Kirkaldy has expressed an opposite opinion.S) A point that I should like to emphasize in what follows is that, even in special cases where a variational principle does exist, that principle may turn out to be irrelevant for the processes one wants to study.

(2) Noise-Driven Selection: Transitions between the various stable steady-states of a pattern forming system may be induced by external perturbations. If these perturbations obey some stationary distribution law, there may be a corresponding stationary distribution of the states of the system. One might then compute, for example, a most probable (but not a unique) spacing of a cellular pattern. Numerical studies of models which behave in this way have appeared recently in the literature. 9J~ However, we do not yet know how to do such calculations analytically. A crucial question is: What is the noise source? Thermal fluctuations seem to be completely ineffective because the length scales in solidification patterns are far too large, microns or greater, to respond to fluctuations on molecular scales. We are left, then, with noise caused by macroscopic inhomogeneities, irregularities in temperature control or driving mechanisms, etc.; but those kinds of noise are very hard to characterize statistically. It is especially important to recognize that such noise sources may not look at all like thermal fluctua-

\ /

\ // \

,.,85 \ ii \\ ELEuLAR I I

\ STATES I I \ \ I /PLANAR \ \ / / NEUTRAL \ \ \ / /'STABILITY

\ Y

i.,7~ \ \ ~ / / I I > 1.35 1.40 k

Fig. 2-- Stability diagram for directional solidification of a thin film binary alloy. The wave number k is shown in units of a diffusion length. A more detailed explanation may be found in Ref. 5.

tions. Even when a system possesses a variational functional that looks like a free energy, nonthermal noise may not drive the system to the minimum of this functional.

(3) Dynamic Selection Mechanisms: The most important point that I want to make in this lecture is that there exist mathematical models, and presumably physical systems, in which sharp selection occurs for purely dynamical reasons. Although a whole family of steady states may exist, only one of these states may be dynamically accessible from physically feasible initial conditions. That is, a selected state must have a dominant basin of attraction. The meaning of this concept is illustrated in the very schematic flow diagram shown in Figure 3. Here, r/1 and r/2 denote two degrees of freedom of some dynamical system, and the directed lines are paths traversed as functions of time in accord with some set of equations of motion. The r/m axis is a line of fixed points. All points on the negative r h axis are stable, and those on the positive r/l axis are unstable. The basin of attraction for any stable fixed point is simply a line in the r/m, */2 plane, whereas the basin of attraction for the origin 0 is the entire half-plane rh > 0. Thus, 0 may be a specially important or overwhelmingly most probable operating point for this system. Note that I have chosen to place the dominant fixed point 0 at the point of marginal stability along the r h axis. Other positions certainly are possible, but this case seems to be of special interest.

(4) No Selection: There are many situations, still within the general framework of the pattern-forming systems described above, in which no sharp selection mechanism is operative. This could happen, for example, if physically feasible initial states are on the left-hand side of the */2 axis in Figure 3. In this case, different initial states produce different steady state patterns; that is, different trajectories reach different points on the negative r/l axis, so that the system is very sensitive to its starting conditions. A second possibility is that no steady state is reached at all. In this case, the system might wander indefinitely, generating per- petually changing and chaotic patterns, never finding any of its stable steady state configurations. Such behavior has been suggested both in real experiments H and in computer simulations ~2 of Rayleigh-B6nard convection.

962--VOLUME 15A, JUNE 1984 METALLURGICAL TRANSACTIONS A

/ Fig. 3 - - Schematic trajectories for a dynamical system with a line of fixed points (the "0~ axis).

In order to explore these possibilities in detail, one needs to look at some specific models and, as mentioned earlier, to reduce these models to the simplest essential features. One particularly unpleasant aspect of most realistic solidification models - -an aspect that I should like to circumvent somehow--is that they are highly nonlocal in space and time. The behavior of any point on the solidification surface depends on the earlier behavior of distant parts of the pattern, a situation which produces difficult mathematical problems even for a computer. The scale of this nonlocality is generally a diffusion length of the form D / v , where D is the relevant thermal or chemical diffusion constant and v is a growth velocity. One may visualize this length as the thickness of the diffusive boundary layer that accompanies an advancing solidification front and controls its motion. This thickness is usually greater than characteristic wavelengths for morphological instabilities; thus, nonlocality is apt to be an essential feature of the problem. There may be physically realizable situations, however, in which the opposite limit is valid, in which the boundary layer is thin compared to distances over which the solidification front departs from planarity. In such cases, the motion of any point on the surface might be determined only by its instan- taneous position and shape. Local models of this kind are very attractive mathematically and do exhibit nontrivial pattern-forming behavior. I am convinced that we have a great deal to learn from such models even if they turn out to be unrealistic for microstructural calculations.

The simplest class of pattern-forming models is described by one-dimensional nonlinear partial differential equations of the form

0S F( s 02S 04S~ {gt = - - , ~X2, ~X4/] Ill

where f ( x , t) is the displacement in the z-direction of a solidification front at position x and time t. As shown in Figure 4, the z axis may be visualized as representing the average growth direction in some sort of directional solidification experiment. For the moment, we shall neglect the possibility of describing complex structures and shall assume thatf is a single-valued function of x. The quantity F on the right-hand side of [1] generally will be a nonlinear,

Z' 0 I

soLiD

> X

Fig. 4 - - Solidification front showing various quantities defined in the text. The shaded region represents a diffusive boundary layer.

algebraic function of its arguments. In order for an equation of this kind to describe nontrivial pattern formation, it turns out to be essential that F contain at least fourth derivatives. For example, fourth derivatives are needed in order that the amplification rate to have a maximum at nonzero wave number k.

Several examples of models of this kind have been described in the literature, u'~4 Perhaps the most rudimentary of these equations is:

Of e - + 1 f _ f3 [2] Ot

which was first introduced by Swift and Hohenberg 15 and then studied in detail by Pomeau and Manneville, ~6 primarily in the context of hydrodynamic phenomena. Here, e is a control parameter analogous to v - vc which has been chosen so that the "planar" statef = 0 becomes unstable for e > 0. Small perturbations of the f o r m f ~ exp(oJt + ikx) grow or decay according to the law

o., (k) = - (k 2 - 1) 2 [ 3 ]

which has the behavior shown in Figure 1. The analog of the neutral stability curve in Figure 2 is e = (k 2 - 1) 2, and the lowest order nonlinear theory predicts that the boundary of the region of stable cellular states lies above this curve, being given by e = 12(k - 1) 2 in the neighborhood of k = 1, e = 0. Equation [2] is specially useful for our purposes because it does derive from a variational principle. It is a simple matter to show that [2] can be written in the form

O 8 ~ --~tf(x, t) - 8f(x , t) [4]

where 8 / 8 f denotes a variational derivative and

�9 {f} = dx ~(1 - e ) f 2 + 4 f4

[5] Then, for all f ,

d ~

dt 0

so that ~ is a Lyapunov function. If we add a thermal noise source to the right-hand side of [2], then we know that there will be a stationary distribution over f of the form exp( - ~{f}/kBT), where kB is Boltzmann's constant and T is the noise temperature. The most probable state f

METALLURGICAL TRANSACTIONS A VOLUME 15A, JUNE 1984--963

will be the one which minimizes r and the selected wave number k according to this criterion will be the one for which r = r is a minimum as a function ofk . Here, j~ denotes the periodic stationary state of fundamental wavelength 27r/k. This is a well-defined selection principle which can be tested numerically. We expect it to be valid under conditions in which T is large enough that the system will come to equilibrium within accessible observation times. On the other hand, T must be small enough that the distribution over j~ is sharp and the most probable k is indistinguishable from its average value.

In the absence of externally applied noise, however, it is not clear what relevance qb might have to pattern selection. To show how other possibilities might arise, I want to dis- cuss next a mechanism of pattern propagation ~?'m which we shall see has some similarity to dendritic processes. I shall describe this mechanism in the context of Eq. [2], but it should be clear that the ideas apply to other models as well.

Consider an initially structureless system, f = 0, which is "quenched" to some e > 0 so that it becomes uniformly unstable against small deformations. A perturbation which at first is confined to a small region will grow locally into a well-developed oscillatory pattern, and this pattern will spread throughout the rest of the space. Analytic arguments and extensive numerical experiments indicate that the pattern spreads by propagating at a well-defined velocity, the front of the pattern looking much like the tip of a dendrite which generates an array of sidebranches behind it as it moves. A picture of such a pattern front is shown in Figure 5; the nodes are stationary in the laboratory frame and new oscillations are emerging at the front.

A remarkable fact about this process is that, so long as the starting perturbation is well-localized, both the propagation speed and the wavelength of the pattern are completely independent of the shape of that perturbation, and the wavelength is not the one which minimizes the Lyapunov function r There are, I believe,19 special initial configurations which can produce patterns that propagate at other speeds, with other periodicities, but these special configurations have exponentially small oscillating "tails" which extend infinitely far into the otherwise unperturbed region of the system. The latter configurations are roughly analogous to trajectories on the left-hand side of Figure 3, each initial state with a special "tail" evolving into its own stable propagating mode on the r h axis, but being very difficult to prepare in any physically sensible way. The physically accessible, localized, initial configurations are analogous to the trajectories on the right-hand side of the figure, all of which lead to a single mode at 0. Thus, a sharp selection is taking place in this system.

A second remarkable fact is that this selected state seems to be consistent with a marginal-stability principle. The term "stability" is used here in just the same sense as it has been used in the theory of dendritic growth. 2~ That is, we look in the frame of reference moving with the front and ask whether an initially localized perturbation, observed at a fixed point in that frame, will grow or decay. A perturbation which decays is considered stable even if it generates a growing disturbance, like a sidebranch, which moves away from its point of origin near the front. No complete stability theory has yet been developed for the propagating solutions of Eq. [2], but a convincing if not completely rigorous analysis ~9 can be carried out in the limit of small, positive

f

hC

- I . 0

1.2 - - - - - - Re k *

I . , . . . . k l

I I I i i ) I O 0 4 0 0 x

AA _

I I I I ;. 2 7 0 2 9 0 3 1 0 x

Fig. 5--Propagating pattern front. See Ref. 17.

e. Instead of describing that analysis here, I should like to introduce an alternative picture 17 which can be shown to be equivalent to stability theory in a limited sense, and which has the advantage of working at arbitrarily large e.

The idea is to look not at the propagation front itself, but at an infinitesimally small deviation from f = 0 initially localized at a position well ahead of the front. This perturbation can be written in the form

6f(x, t) ~- f dk ~(k)e ikx+~(k)t [7]

where oJ(k) is given by [3], and ~(k) is the Fourier transform of the initial disturbance. We shall assume that this disturbance is sharply localized so that we may neglect the k-dependence of~. Suppose that we observe this perturbation at a moving position x = ct. For large t, we have

6f(ct, t) ~- exp[ik*c + 00(k*)]t [8]

where k* is the point of stationary phase in the complex k-plane (see Eq. [10] below). Now look at Figure 6 which shows schematically, not the full wave forms, but only the envelopes of these wave forms at a sequence of increasing times. Figure 6(a) shows these envelopes as they might look in the laboratory frame; the wave packet grows in amplitude and spreads in both directions. Figure 6(b) shows the same sequence in a frame of reference moving with a speed c which is large enough that the observer outruns the perturbation; that is, the exponential on the right-hand side of [8] decays in time. A limited version of the marginal- stability criterion turns out to be equivalent to the hypothesis that the natural velocity of the pattern front, c*, is that for which this exponential neither grows nor decays. The latter situation is shown in Figure 6(c), where the dashed curve suggests that the leading edge of the perturbation 6f might be identical to the leading edge of the propagation front. Mathematically, this selection criterion is simply:

Re[ic*k* + o~(k*)] = 0 [9]

where the complex wave number k* is determined by the stationary-phase condition

dw ic* + dk---- ~ = 0 [10]


3f'

X (a)

x-ct (b)

3f' \ \

x-c*t (c)

Fig. 6--Sequence of envelopes of a small growing disturbance shown (a) in the laboratory frame, (b) in a frame of reference moving faster than the propagation speed c*, and (c) in the frame moving at c*.

If we agree that the leading edge of 6 f looks like the propagation front, then we can make a very plausible calculation of the fundamental wave number k~ of the nonlinear pattern which is formed behind the front. To do this, it is useful to visualize the pattern as moving with velocity - c * in the frame of reference in which its envelope is at rest. Ahead of the front, the pattern has wave number Re k* and oscillates, at a fixed point in the moving frame, with an angular frequency

= lm[ik*c* + w(k*)] [11]

This frequency may be interpreted as a flux of nodes moving in the - x direction relative to the envelope. As long as nodes are not created or destroyed when they pass through the front of the pattern, this flux must be conserved and must be equal to k~c* in the bulk. Thus

k I = ~ ~ / c :~ [12]

The above predictions for c* and kl have been checked numerically throughout the interesting range ~7 0 < e < 1. (For e > 1, stable nonoscillatory states reappear.) Appar- ently, the limited form of the marginal-stability hypothesis is exact for this particular problem. It will not be true in general, however; there exists a counter-example for which marginal stability remains correct but the limited version breaks down ]9 because the propagation front constructed

METALLURGICAL TRANSACTIONS A

from [7] is itself unstable. When that happens, the velocity c* obtained from [9] and [10] ought at least to be a lower bound to the true propagation speed.

Let us turn finally to the question of what relevance these one-dimensional results might have to more interesting phenomena, like dendrites, in which complex surfaces evolve in two or three dimensions. Here I shall be describing new, ongoing research, and my remarks must be understood as being highly speculative. We 2L22 have been looking recently at a class of two-dimensional models in which the solidification front is a one-dimensional "string" whose motion generates patterns in the plane. The configuration of such an object can be described by its curvature K as a function of distance along the string s:

K =-- - o O / O s [ 1 3 ]

where, as shown in Figure 4, 0 is the angle between the normal to the curve and some arbitrarily chosen direction, here taken to be the z axis. Given the function K(s), one can reconstruct the entire curve up to uniform displacements and rotations. In the spirit of the models discussed previously, we might suppose that the motion of this string is entirely determined by letting its normal velocity v,(s) be a function only of the local shape-dependent quantities K(s), ozK/OS 2, etc. (There seems to be no reason to violate reflection symmetry by introducing odd derivatives.) The resulting equation of motion for K has the form:

[14] 0 s 2 . . . . /

where (d/dt), denotes the rate of change along the normal growth direction. Note that [14] has a structure somewhat similar to that of [1]. Interesting and potentially important variations of these models may be constructed by allowing v, to depend explicitly on 0 to simulate crystalline anistropy, or by introducing another field coupled to K(s, t) to simulate dynamical features of the diffusive boundary layer.

The simplest string model that one might imagine is:

v. = K [15]

This has the nice feature that the equation for steady state growth with velocity v0 in the z direction

v, = v0 cos 0 = K [16]

has needle-crystal solutions

7"/" 0 = - - - 2 tan-t(e v~ [17]

2

or, equivalently,

1 z = - - I n cos(v0x) [18]

v0

As in the realistic solidification problem, there is a family of such solutions for which, in a more conventional notation, rovo = 1, where r0 = I/K(0) is the tip radius. Unfor- tunately, all of these solutions are completely unstable. The instability can be seen most easily by looking at small deviationsf from a stationary straight line, say, z = 0. Linearization of [15] yields

Of 02f - - [ 1 9 ]

0t 0z 2

VOLUME 15A, JUNE 1984--965

and the resulting dispersion relation, to = k z, diverges un- physically at large k. Because this instability persists at infinitesimally small wavelengths, it will occur on curved surfaces such as [18] as well as flat ones such as z = 0.

To control this instability, one can add a "capillary" term:

or 232K V n = K "}- OS 2 [20]

where o" provides a natural length-scale that was missing in [15]. Small deviations from a flat surface now obey the equation

Of _ 3 2 f 2 3y - r - - [ 2 1 ] 3t OX 2 3X 4

and the associated dispersion relation, w = k 2 - ty2k 4,

begins to look like [3]. Note that deformations with wave numbers k greater than 1/tr have become stable. Most interestingly, however, the "capillary" term is a singular perturbation which destroys the steady state needle-crystal solutions. Moreover, the "capillary constant" cr can be eliminated from the problem by suitable choice of length and time scales; thus, the model still lacks an adjustable parameter suitable for controlling pattern formation. There are many ways in which such a control parameter can be introduced; for example, one can add higher powers of K with adjustable coefficients to the right-hand side of [20]. Numerical investigations of some of these possibilities are now being carried out, 22 but it is too early to report any of the results or even to speculate about whether this restricted class of string models looks interesting.

The version of the string model which looks most prom- ising to me at the moment is the one in which a new field is introduced to simulate a thin boundary layer. This model is loosely derived from a realistic picture; thus, it is possible to bring some physical intuition to bear on the choice of parameters. A detailed description of this work should be ready for publication in the not too distant future. It may be useful for present purposes to mention that this model recovers in a relatively simple way many of the known features of the full solidification problem, specifically, the steady state and stability properties of simple shapes such as planes and discs. It even predicts the familiar family of unstable parabolic needle-crystals in the isothermal, Ivantsov limit, and it is sufficiently tractable mathematically that one can show, just as in the simpler string models described above, that this family of solutions no longer exists for finite capillarity. This is an important result because it suggests that needle-crystal solutions do not exist for the realistic solidification problem, either. I see nothing in any of the approximate solutions of this problem or in the numerical calculations of Nash and Glicksman 23 that would preclude such a possibility. Of course, nonexistence of the expected steady state solutions would force us to

reinterpret the marginal-stability theory of dendritic growth. But such a result would not seem to be fatal for the stability hypothesis. The propagating patterns generated by Eq. [2] are not time-independent in any frame of reference, but they do seem to be governed by a stability-related selection principle. It will be interesting to see whether the string models or boundary-layer models can actually produce cellular or dendritic structures, and whether the apparent mathematical relationship between these models and the one-dimensional propagating systems is strong enough to imply similar mechanisms for pattern selection.

ACKNOWLEDGMENTS

This work was supported by the Department of Energy, Contract No. DE-AM03-76SF00034 and by the National Science Foundation, Grant No. PHY77-27084, supple- mented by funds from the National Aeronautics and Space Administration.

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Pattern Selection in Solidification