PHYSICAL REVIEW A VOLUME 41, NUMBER 6 15 MARCH 1990 Linear stability of directional solidification cells David A. Kessler Department of Physics, Uniuersity of Michigan, Ann Arbor, Michigan 48109 Herbert Levine Department of Physics and Institute for Nonlinear Science, Uniuersity of California, San Diego, I. a J'olla, California 92093 (Received 16 October 1989) We formulate the problem of finding the stability spectrum of the cellular pattern seen in direc- tional solidification. This leads to a nonlinear eigenvalue problem for an integro-differential opera- tor. We solve this problem numerically and compare our results to those obtained by linearizing the eigenvalue problem by employing the quasistatic approximation. Contrary to some recent claims, we find no evidence for a Hopf bifurcation to a dendritic pattern. I. INTRODUCTION The study of spatial patterns in nonequilibrium systems has recently been receiving considerable attention. Gen- erally, as one drives a system away froin thermodynamic equilibrium one observes a sequence of different spatial structures; these range from quiescent, to ordered, to slightly disordered, to fully turbulent. It is clearly of great interest to be able to understand the dynamical mechanisms responsible for the observed transitions in a variety of systems. One physical setting for the above scenario is the direc- tional solidification of two component mixtures. Here, diffusion of the impurity concentration can lead to an un- stable solidification front. On the other hand, the system is stabilized by an imposed thermal gradient and interfa- cial surface tension. The control parameter is the growth velocity, determined by pulling the sample at a fixed rate with respect to the underlying gradient. At small veloci- ties, a planar front is stable. Eventually, however, we reach a critical velocity above which a cellular structure forms. The cellular structure has been studied in great depth both theoretically and experimentally. There ex- ists a band of allowed wavelengths and some dynamical mechanisms which enable the system to choose particular periodicities. There is certainly much to be understood here, but this is not the purpose of this work. Instead, we concentrate on the transformation of the cells to parallel arrays of dendrites. This occurs at a ve- locity much larger than the initial onset, and may be con- nected with a jump in the dynamically selected wave- length. It is fair to say that little is understood about this transition. The simplest possible mechanism for the cell to den- drite transition would be a Hopf bifurcation. In this scenario, the cellular structure would exhibit a linear in- stability to an oscillatory state, presumably correspond- ing to the periodic emission of sidebranches. Such a pic- ture was recently put forth by Karma and Pelce, based on an approximate computation using an effective inter- face formulation. There are some problems reconciling the aforemen- tioned hypothesis with currently accepted ideas on free space dendrites. ' For free dendrites, a stability analysis (albeit in the quasistatic limit) found that the steady-state needle crystal is linearly stable. " This led' ' to a noise- induction mechanism for sidebranch generation, which is at least in qualitative agreement with the experiments of Dougherty, Kaplan, and Gollub. ' Since the free den- drite should emerge as the limiting case of a dendrite in directional solidification at large velocity, it is hard to see where the instability might disappear to. Furthermore, the dendritic pattern is specifically dependent on the ex- istence of crystalline anisotropy, ' which plays only a minor quantitative role in the cellular structure. Never- theless, the Hopf bifurcation approach is quite attractive, explaining the existence of the transition and the ap- parent correlation between the sidebranches on neighbor- ing cells. In this paper, we approach this topic in a straightfor- ward, although technically complex, manner. We and others have previously shown how one can solve for the cellular shapes by using the boundary integral method. '" Here, we extend this methodology to the linear stability problem. This method leads to a nonlinear eigenvalue problem for an integro-differential operator, which of course depends on the underlying solution of the steady- state equation. The eigenvalue problem is nonlinear be- cause of the inherent memory in the diffusion equation. That is, the initial operator representing the solution of the partial differential equation for the concentration field depends on the past history of the interface and hence on the assumed eigenvalue. We tackle this difficulty by an iterative technique which will be explained in detail later. The conclusions that we have reached are quite simple. We have studied a range of parameter values including those for which the Karma-Pelce approach predicts an instability. We see no evidence for such an instability. Of course, it is always possible that we have not yet reached onset and increasing the velocity (and/or changing some other set of control parameters) would in fact yield the hypothesized bifurcation. But, our results indicate that (i) the Karma-Pelce approach is not quantitatively 41 3197 1990 The American Physical Society
Transcript
PHYSICAL REVIEW A VOLUME 41, NUMBER 6 15 MARCH 1990
Linear stability of directional solidification cells
David A. KesslerDepartment ofPhysics, Uniuersity of Michigan, Ann Arbor, Michigan 48109
Herbert LevineDepartment ofPhysics and Institute for Nonlinear Science, Uniuersity of California, San Diego,
I.a J'olla, California 92093(Received 16 October 1989)
We formulate the problem of finding the stability spectrum of the cellular pattern seen in direc-tional solidification. This leads to a nonlinear eigenvalue problem for an integro-differential opera-tor. We solve this problem numerically and compare our results to those obtained by linearizing theeigenvalue problem by employing the quasistatic approximation. Contrary to some recent claims,we find no evidence for a Hopf bifurcation to a dendritic pattern.
I. INTRODUCTION
The study of spatial patterns in nonequilibrium systemshas recently been receiving considerable attention. Gen-erally, as one drives a system away froin thermodynamicequilibrium one observes a sequence of different spatialstructures; these range from quiescent, to ordered, toslightly disordered, to fully turbulent. It is clearly ofgreat interest to be able to understand the dynamicalmechanisms responsible for the observed transitions in avariety of systems.
One physical setting for the above scenario is the direc-tional solidification of two component mixtures. Here,diffusion of the impurity concentration can lead to an un-stable solidification front. On the other hand, the systemis stabilized by an imposed thermal gradient and interfa-cial surface tension. The control parameter is the growthvelocity, determined by pulling the sample at a fixed ratewith respect to the underlying gradient. At small veloci-ties, a planar front is stable. Eventually, however, wereach a critical velocity above which a cellular structureforms.
The cellular structure has been studied in great depthboth theoretically and experimentally. There ex-ists a band of allowed wavelengths and some dynamicalmechanisms which enable the system to choose particularperiodicities. There is certainly much to be understoodhere, but this is not the purpose of this work.
Instead, we concentrate on the transformation of thecells to parallel arrays of dendrites. This occurs at a ve-locity much larger than the initial onset, and may be con-nected with a jump in the dynamically selected wave-length. It is fair to say that little is understood aboutthis transition.
The simplest possible mechanism for the cell to den-drite transition would be a Hopf bifurcation. In thisscenario, the cellular structure would exhibit a linear in-stability to an oscillatory state, presumably correspond-ing to the periodic emission of sidebranches. Such a pic-ture was recently put forth by Karma and Pelce, basedon an approximate computation using an effective inter-face formulation.
There are some problems reconciling the aforemen-tioned hypothesis with currently accepted ideas on freespace dendrites. ' For free dendrites, a stability analysis(albeit in the quasistatic limit) found that the steady-stateneedle crystal is linearly stable. " This led' ' to a noise-induction mechanism for sidebranch generation, which isat least in qualitative agreement with the experiments ofDougherty, Kaplan, and Gollub. ' Since the free den-drite should emerge as the limiting case of a dendrite indirectional solidification at large velocity, it is hard to seewhere the instability might disappear to. Furthermore,the dendritic pattern is specifically dependent on the ex-istence of crystalline anisotropy, ' which plays only aminor quantitative role in the cellular structure. Never-theless, the Hopf bifurcation approach is quite attractive,explaining the existence of the transition and the ap-parent correlation between the sidebranches on neighbor-ing cells.
In this paper, we approach this topic in a straightfor-ward, although technically complex, manner. We andothers have previously shown how one can solve for thecellular shapes by using the boundary integral method. '"Here, we extend this methodology to the linear stabilityproblem. This method leads to a nonlinear eigenvalueproblem for an integro-differential operator, which ofcourse depends on the underlying solution of the steady-state equation. The eigenvalue problem is nonlinear be-cause of the inherent memory in the diffusion equation.That is, the initial operator representing the solution ofthe partial differential equation for the concentration fielddepends on the past history of the interface and hence onthe assumed eigenvalue. We tackle this difficulty by aniterative technique which will be explained in detail later.
The conclusions that we have reached are quite simple.We have studied a range of parameter values includingthose for which the Karma-Pelce approach predicts aninstability. We see no evidence for such an instability. Ofcourse, it is always possible that we have not yet reachedonset and increasing the velocity (and/or changing someother set of control parameters) would in fact yield thehypothesized bifurcation. But, our results indicate that(i) the Karma-Pelce approach is not quantitatively
41 3197 1990 The American Physical Society
3198 DAVID A. KESSLER AND HERBERT LEVINE 41
II. INTKGRO-DIFFERENTIAL FORMALISM
The equations of motion for the solidification front of adirectionally solidified alloy are well known. FollowingRef. 1, we derive the system
DV c+uo BC0
BC
r)t
correct and (ii) it is quite unlikely that the cell to dendritetransition is in fact caused by a linear instability. This isquite unfortunate; but finite amplitude transitions seem tobe the rule in this type of pattern forming system.
The outline of this paper is as follows. In the next sec-tion, we discuss the boundary integral formulation of thestability problem. Next, we describe our numerical pro-cedures to first find the steady-state solution, and then tofind the linear spectrum. The latter is done in two ways.First, we employ the quasistatic approximation whichlinearizes the eigenvalue problem. This is useful becauseit gives us an indication as to what is the typical range ofeigen values and which modes, if any, are close toRedo=0. The second method treats the problem exactlyby an iterative scheme. Afterwards, we present our nu-merical results and discuss implications for the physicalsystem.
—(von y+u )(n Vc)ll (n V, )l, = ' "
(1—k)cI, (2)
c=—= ———d v.k
Here we have assumed equal diffusivities of the twophases. In the above, the other material parameters arethe partition coefficient k and the capillary length do.The capillary length arises due to the effect of surfacetension and multiplies the curvature of the interface a.The thermal length g is determined by the imposed tem-perature gradient and vo is the pulling velocity. Thephysical interface velocity is v„+von y, where n is the in-terface normal vector. For a steady-state solution v„=O.
We are interested in computing the stability of a cellu-lar solution of the above equations. Let us measurelengths in units of one-half wavelength of this cellularpattern. This rescaling gives rise to the dimensionless pa-rameters y =2do/l, , lT =2)/k, , and p = vol, /4D.
Using standard techniques, we can eliminate the con-centration field by using a diffusive Green's function. Forfuture use, we will do this in two ways. The first employsthe quasistatic approximation to drop the dc/r)t term inthe field equation. Then, we readily derive an equationfor the interface position x(s)
where the subscript means that the singular integral is evaluated on liquid side. Here the Green s function 0 is a solu-tion of the time-independent equation
V C+2p = —5 (x—x'),B(5)
By
where p is the Peclet number. The explicit form of 0 is given in the Appendix. Note that there is no integral over pasthistory and hence the interfacial evolution is local in time. This makes dynamic simulations fairly straightforward. '
But, it is of course necessary to verify that the quasistatic approximation does not miss any essential phenomena.The second and exact approach relies on the usual diffusive Green's function
V G — = —5 (x —x')5(t t') . —BGBt
(6)
For this calculation, it is more convenient to work in the material frame of reference, which accounts for the absence ofthe uniform velocity term in the above equation. Using this function and again following usual methods, we find
Note that the second term has completely dropped out.This is a consequence of the fact that the term givingrise to the discontinuity in the concentration field(dipole layer) has exactly the correct discontinuity in thefield gradient so as to satisfy the Stefan boundary condi-tion. Note that the time dependence of I in the equationnow includes the overall motion at velocity vo =2p.
Let us briefly revie~ the steady-state approach. Note
I
that the second term in Eq. (4) vanishes. Similarly in Eq.(7), if we assume that the only time dependence of x(s', t')is the constant translation at velocity 2p, the time integralcan be done and we recover the previous result. This isnot surprising, since it is manifestly clear that the quasi-static limit is exact for steady-state patterns.
Let us now assume that we have found a periodicsteady-state solution to the above system which we
LINEAR STABILITY OF DIRECTIONAL SOLIDIFICATION CELLS 3199
denote as x(s). We now perturb this solution by adding anormal shift
spectrum as a function of the Bloch wave vector adefined via
where no is the normal vector of the base solution. Be-cause of time translation invariance, we can limit thetime dependence to the form 5(s, t)=5e '. Finally, we
are interested in the possibility of sidebranching emergingas the coherent oscillation of spatially translated cells.This corresponds to an eigenmode which is spatiallyperiodic in A, and reflection symmetric about s=O. Ofcourse, it might also be of interest to study the stability
t
[sT is the total arclength of one cell of xo(s)] and also tostudy antisymmetric modes. Neither of these extensionsis attempted here.
Substituting this expression into either the quasistaticequation or the full equation and linearizing in 5, we findan eigenvalue equation for co. Let us first deal with thequasistatic limit, which has already been discussed else-where' ' for a slightly different system. We obtain
(to is the base solution tangent vector).Aside from its complexity„ the above equation is a normal eigenvalue problem which determines the possible values
of co. We will discuss the numerical implementation of this procedure in the next section.Finally, let us make the same substitution in the full evolution equation. We obtain the simpler looking equation
+y[5"(s)+v05(s)]+fds' (1—k) yvo+ [5(s)no Vn o V'G+5(s')no V'no V'G(co)]T T
G is the quasistatic Green's function; G ( co ) is a newGreen's function which arises from integrating thediffusive Green's function over past time including thefactor e '. Its form is given in the Appendix. Clearly,G =G(a)=0).
The most striking feature of the last equation is thatthe growth rate co only appears as an argument of theGreen's function. That is, co is determined by the re-quirernent that the operator defined by the right-handside of the last equation has a zero eigenvalue. As anequation for ~, this is nonlinear. We will discuss ourmethod of solution after we discuss how one finds the ei-genvalues of these integro-differential operators. That isthe subject of the next section.
III. NUMERICAL PROCEDURE
In our previous work, we have described at greatlength how one discretizes the interface and converts thesteady-state shape equation to a coupled set of nonlinearequations. Basically, we pick points at fixed relative arc-length and solve for the variables y(tip), 0, j =1, . . . , X.
The total arclength is determined by the periodicity con-dition x (E)=1. The result of this solution is an interfacewhich is then reflected and periodically continued to givethe entire cellular pattern.
Once the above is finished, we turn to a numericaldetermination of the stability operator. We first fit aspline function through the points 0; this gives us a con-tinuous interface. Next, we pick a number of points Mwhich will represent the discretization of the integro-differential operators appearing in the stability equation.M has no a priori relationship to X, the number of pointsused in the steady-state calculation.
Once the interface has been discretized, the integralswhich appear in Eq. (10) or (11) are all replaced withsums and the derivatives by differences. This must bedone carefully because of the singularities of the variousGreen's functions at s'=s. There are generally two typesof singularities which occur (see the Appendix). First, interms with at least one derivative of a Green's function,there are 5 function contributions which must be explicit-ly evaluated. The second type of singularity is the loga-rithmic divergence which occurs in G, G(co), and all
3200 DAVID A. KESSLER AND HERBERT LEVINE 41
derivatives thereof. For this, we explicitly subtract offthe singular piece, using, e.g. ,
totds $$ s
tot
St 1+Sds' s, s' s' + ln s —s' sS,+S 4m
[ln(s„, )—1]f(s) .
x10'.50 I I ~ l
I
I I ~ ~
II ~ I
-0.67
-0.83
Ii
I I I \
i~ ~ ~
Note that at point s, all integrals can be evaluated froms st t to s +s„,by making appropriate use of the spatialperiodicity.
After the discretization of the integrals, the equationstake the schematic form
X N
g A, 5, =co+B,,5, (12)j=0 j=O
in the quasistatic limit and
N
g C,,5, =0j=0
(13)
for the exact formulation. For the first case, we just needto find the eigenvalues of B 'A. This is done via theEISPACK routine RG. For the exact evaluation, however,we proceed as follows. We pick a range of co of interestand find the smallest (in absolute magnitude) eigenvalueof C. Note that since co is in general complex, C is a com-plex matrix and the eigenvalue is found by a differentEISPACK routine CG. We then locate an approximate co
near where the smallest eigenvalue changes sign (bothreal and complex parts). Finally, we use a Newton'siteration algorithm to pin down more exactly the correctvalue of co. The laborious procedure generates exactlyone eigenvalue and the process must be repeated withdifferent starting points for additional allowed values ofCO.
We have checked our program in several ways. First,it is possible to compare our results to the exact answersfor a planar interface solution y = —iT/k. Next, the qua-sistatic program can be compared to the independentlyverified steady-state program in which a small shift is in-serted by hand into a steady-state profile. The quasistaticand exact programs must agree if co is set to zero. Final-ly, we have run the program on cellular states immediate-ly above onset where the bifurcation theory approach al-lows us to predict whether a solution is stable or not. Wehave verified that our program does indeed agree with theanalytical results in that regime.
"%.00' 1.20 2.40 3.60
FIG. 1. Cellular interface for the case lT =2.2, p=0.385.
find all eigenvalues of an M XM matrix in one step. It isthen simple to study the dependence of our results on thelevel of discretization. At lT=5.5 and p=0. 165, thevalues of the eigenvalue with the largest real part are—1.62X10 ~, —1.24X10, and —1.15X10 atM=125, 250, and 400, respectively. This is consistentwith an expected 1/M dependence, and suggests thatour answers at M =100 are about 20% accurate. Thishas also been checked at other parameter values. Thiswill be sufficiently accurate for our conclusion below.
In Table I, we present the least stable eigenvalue foundby our quasistatic program for six cases.
As expected, co tends to increase with Peclet number p.Note that all these modes are real. This is due to the lackof any time delay within the quasistatic approximation.Most importantly, there is no sign of any instability.
We now turn to the exact calculation. Based on theabove result, we expect that the relevant scale of possiblevalues for the lowest co=0. 1. We chose therefore toevaluate the eigenvalue closest to zero of the C (co) ma-trix at a grid of points between —0.5 and +0.5 for bothRedo and Imago. Because of computer time considerations,we chose M= 100 for this scan.
Table II represents a typical set of data from such arun, at lr=2. 2 and p=0.385, where Redo ranges from—0.5 to 0 and Imco from 0 to 0.5 by steps of 0.1. Here yis the actual eigenvalue of the C matrix, which is a func-tion of the hypothesized frequency cu', y=0 determinesthe actual correct eigenvalue co'.
Based on Table II, it is possible to identify twodifferent roots. There are zero crossings of both the realand imaginary parts of y near co=( —0.3, +0.3) andcoo-( —0. 1, +0.2). We then take these values as initialguesses for a Newton's iteration scheme, which solves the
IV. RESULTSTABLE I. Growth rate of lowest mode.
We have chosen the following set of parameters for ourstudy; the partition coefficient is set equal to 0.16 and the(dimensionless) surface energy is 0.18. We have studiedIT=2.2 and 5.5. For each value of IT, we study severaldifferent values of the Peclet number, ranging from quitesmall ( =0.1) to moderate ( =0.7). A typical steady-statepattern in this regime is shown in Fig. 1.
Let us first focus on the quasistatic limit. Here, we can
200200200250250250
2.22.22.25.55.55.5
0.1650.3850.7700.1650.3300.688
co {X10 )
—1.70—8.58
—17.53—1.24—3.15—6.76
41 LINEAR STABILITY OF DIRECTIONAL SOLIDIFICATION CELLS 3201
TABLE II. Variation of eigenvalue with imposed frequency. x10-'000 I I I I I v &
equation detC=O for co. The outcome of this is a con-verged value for the root which turns out to beco*=(—0. 110, 0.195) (lower) and co"=(—0.329, 0.254)(higher one). For comparison, the quasistatic resultswere ( —0.086, 0.0) and ( —0.316, 0.0). The real parts areactually quite close, a result which seems to be generallytrue.
What about positive values of Redo? In Fig. 2, we haveplotted the real part of g versus Imcu for the three casesRedo=0. 1, 0.3, and 0.5 (this is for the same parameter setas above). Notice that there is no tendency whatsoever tocross zero and hence there are no roots in this range.This can be verified by starting our iterative program atrandom locations with Redo &0 and watching it convergein every single case to a root with Redo ~0. Hence thereare no unstable modes.
We have performed this type of scan for all six casesgiven in Table I. In no case do we find a mode with posi-tive growth rate. Even with M=100, there is no doubtwhatsoever that the system is linearly stable.
In Figs. 3 and 4, we present the results for the con-verged eigenvalue, as compared to the theoretical predic-
0.10.3
-1.00
-3.- .00 i I I I I i I I I I I I I I I I I I I I I I I ~
0.00 100 2.00 3.00 4.00 5.00x10 '
FIG. 2. Real part of C matrix eigenvalue for positive values
of Redo.
tion of Karma and Pelce discussed in the Introduction.Figure 3 corresponds to the case lT=2.2 and Fig. 4 toIT =5.5; part (a) refers to the real part and part (b) refersto the imaginary part. We also show for comparison amodified version of the Karma-Pelce theory which usesthe aforementioned quasistatic approximation. For thisapproach, Imago is always precisely zero.
The results are, to say the least, striking. Our datapoints always have Reer 0; we never see any sign of a bi-furcation. The Karma-Pelce (KP) theory in its full formpredicts that the pattern is unstable; hence the theory isinconsistent with our exact numerical results. However,as we have mentioned already, our real parts are relative-ly unchanged from their value in the quasistatic calcula-
2.00 r
+ Exact
KP quasistattc
KP full theory1.17
0.33
r/
rr
r~
////
/// /
rr
/ /r
-0.500.00 0.20
* " " ~--.....+I I I
0,40 0.60 0.80
200 I I
I
I I I 'I
I
I ' ~
II I I I
1.17
+ Exact
KP full theory
0.33
~50 i i i i I s
0.00 0.20I I I
0.40 0.60 0.80 1.00
FIG. 3. Exact spectrum vs Karma-Pelce theory at lT=2.2.(a) Real part; (b) imaginary part.
3202 DAVID A. KESSLER AND HERBERT LEVINE 41
2.00
+ Exact
KP quaststatic
KP full theory1.17
jI I I I
jI I 1 I
jI
///
/////////J////
0.33
-0.500.00
I I I I j I I
0.20 040 060 0.80 100
200 r I I
j~ I I I
jI I 1
1.17
+ Exact
KP full theory
0.33
-0.500.00
s s s a I s & a s j i » i j
0.20 0.40 0.60 0.80 1.00
FIG. 4. Exact spectrum vs Karma-Pelce theory at IT=5.5.(a} Real part; (b) imaginary part.
tion. In fact, our results are completely consistent withthe KP quasistatic results. Their approach suggests thatinclusion of non-quasi-static effects have a dramatic effecton the real part of the eigenvalue; our exact analysisdemonstrates that this does not occur.
For completeness, we present in Fig. S a graph of theeigenfunction corresponding to the least stable mode atIT=2.2 and p=0.385. There is no particular sign of spa-tial oscillations which would be expected for asidebranching-type excitation.
V. DISCUSSION
We have presented the results of an exact stabilityanalysis of the cellular steady-state pattern seen in direc-tional solidification. For the range of parameters studied,we find that the pattern is linearly stable. This contra-dicts the approximate calculations of Karma and Pelce,who find instability in the same range.
There are two different issues that must be addressed.The first is the question of what is wrong with the ap-proach used by KP. Their methodology reduces theproblem of interface dynamics to a one-dimensional prob-lern for the evolution of the periodic set of tip regions.They make the explicit assumption that the tail region ofthe (deep) cells adjusts to the motion of the tip withoutafFecting it.
This latter statement is, we feel, highly suspect. It iswe11 known from studies of the stability of the Saffman-Taylor finger' ' and the free space dendrite" ' that thecoupling of the tip to the tail is crucial for obtaining the
x10 '
0,70 ~ 1 f
jt ~ I 1
j1 0 I ~
jI ~ \ I
jI f I
V 0.00
~ t~~l
Q
-0.70
%.00 1.20j, s
2.40 3.60
are length6.00x10'
FIG. 5. Eigenfunction at 1T=2.2, p=0.385 corresponding tothe least stable mode.
correct stability spectrum. Roughly speaking, any at-tempted alteration of the tip region couples to deforma-tions of the long tail which are dynamically suppressed.This seems to be an unavoidable consequence of the sol-vability condition which selects the correct tip structurein the first place. We have argued elsewhere that the cel-lular pattern at Jinxed p is governed by a similar type ofsolvability mechanism. We are therefore not verysurprised that the stability is governed by phenomenawhich are also quite familiar from earlier studies.
Some supporting evidence for this point of view ispresent in the recent simulation study of Ref. 1S. Heredendrites emerge smoothly from the cellular patternwithout any sign of oscillatory instability. This work ismost important in demonstrating that the quasistatic ap-proximation does not seem to preclude the transition todendrites as it would in the Karma-Pelce approach. Thisis consistent with our results regarding the validity of thequasistatic assumption.
A more crucial question is, if the onset of sidebranch-ing is not a linear instability, what is it? The possibilitiesare roughly the same as those for free space dendrites.There might be a subcritical bifurcation to a limit cycle,with the onset taking place only in the limiting case ofzero surface tension. A second possibility is that side-branching is the response to external noise, as has beenproposed in several papers. The latter possibility presum-ably requires that at least close to the tip, the nascent side-branches are not coherent from cell to cell. Visual in-spection of the pattern seems to show that once thebranches become visible, they do in fact seem coherent.A study similar to the one of Dougherty, Kaplan, andGollub' is clearly needed to resolve this issue.
In either of the above scenarios, the importance of afinite perturbation of the tip is crucial. Only by perturb-ing the tip by a finite amount does it seem possible toavoid the conundrum posed by the ultimate dynamic re-stablization due to the tail region. It might eventuallyprove possible to reinterpret the method of effective inter-face dynamics used by KP to apply only to disturbancesthat are larger than some critical threshold. That is, itwould be worthwhile to test whether the theoretical ap-proach makes physically reasonable predictions even if it
LINEAR STABILITY OF DIRFCTIONAL SOLIDIFICATION CELLS
cannot be mathematically a correct description forinfinitesimal perturbations. In other words, a linear sta-bility with a tiny nonlinear threshold might be describedphenomenologically by the KP equation, even though itwould break down for sidebranching amplitudes whichare too small. This idea seems worthy of further investi-gation.
APPENDIX
Here we supply some more details regarding theGreen s function, which enters into the stability analysis.The basic object which we need is G(co), defined via
(Al)
ACKNO%LKDGMKNTS x=0, (A2)
The authors would like to thank the Aspen Center forPhysics, where part of this work was completed. One ofus (D.A.K.) was supported in part by the U.S. Depart-ment of Energy Grant No. DE-FG-02-85ER54189. H.L.was supported in part by the U.S. Defense AdvancedResearch Projects Agency (DARPA) under the Universi-ty Research Initiative, Grant No. N00014-86-K-0758.
G(co'x x')= J dt'e "X G(x —x'+2py(t t '—
); t t ') .—(A3)
The boundary condition at x =+1 arise via the periodici-ty; the integral over past times is weighted by a factore"" " corresponding to the time dependence of the per-turbation.
It is easy to verify that an explicit expression for G is
—[«~] +p +~l) ( 2+ ))/2I y I+ y(y y ) + cos[n1T(x x )]e
4(p2+ ))/2 2[(nn) +p +co]'(A4)
Derivatives of G can be taken explicitly. The quasistatic limit 6 arises from setting co=0, recovering the Green's func-tion used previously from our steady-state calculations.
The method we use for evaluating G and its derivatives is a direct generalization of that described in Ref. 3. We ex-plicitly perform a sufficient number of subtractions to render the sum convergent at all values of x, x', y, and y'. In de-tail, we replace the above sum by
—,' Q cos[k„(x —x '
) ]e
—k, ly—y'l
e
e y' y 'in[1 —2cos[a(x —x')]e "I I+e —Iy
—«'Ij4m
Here k„=n~ and ky=(k„+p +co)' . The last term is clearly the Laplace Green's function (aside from the n=Opiece) and contains a logarithmic divergence; the sum is convergent even at x =x', y =y' and can be easily computed.
We need similar expressions for first and second derivatives of G. Let us define the following objects:
co= e y'y 'in[1 —2cos[n.(x —x')]e 'y y I+e I» y Ij,4m
(A6)
—p[y —y']1
cos[m(x —x')]e1 —2cos[w(x —x )]e "I +e
Icos[m(x —x')]—2me ' I+cos[n(x —x')]e
2 cos[~(x x ) ]e—~ly —y'I +e
—&~I« —y'Ij
2
(A7)
so = e ' 'tan1
2~e '«sin[a(x —x')]
1 —e '«y Icos[a.(x —x '})
sin[m. (x —x ') ]e1 —2 cos[vr(x —x )]e
—Iy
—«'I+e —~ly —«'I
7T ) ') sin[a(x —x'}]e "I I( 1 —e ")s = e p[y y )
2 1 —2 cos[n(x —x')]e "I +e
(A9)
(A10)
(Al 1)
These arise via the sums
—p(y —y'j e ' cos[k„(x —x ') ]—k„ ly
—y'l
g1 —nk X
—s [y —y']
k
—k ly—y'l .
sin[k„(x —x')]1 —n
X
(A13)
Then, an extremely tedious calculation leads to the ex-plicitly computable results
where R is the finite sum given above for G, and the additional convergent sums are
P[yR„= g sin[k„(x —x')]
k„
-k ly-y l
k
2+2k,
(A19)
R2
2+g cos[k„(x —x')] e ' e "
1 —~y
—y'~2k„
X
(A20)
R g cos[k, (x —x')] k e ' —k„e-k ly-y' -k ly
—y'l
k„
2+ 1 +Iy
—y'I+, (p'+~) 1+2k, 4
(A21)
R, = g sin[k„(x —x')] e ' —e "1 —
~y—y'~+ (y —y')p +co, (p +co)2k„gk2
x
(A22)
This completes the formulation of the Green's function.The only remaining subtlety in the calculation is the
evaluation of the integrals in the vicinity of the singularpoint x =x', y =y'. As mentioned in the text, there aretwo types of singularities encountered. First, there are6-function pieces in n VG and in higher derivatives of G.These must combine to give rise to exactly the change in
the discontinuity in the original integral. Also, there arelogarithmic pieces which are handled by subtraction. Fi-nally, there is a finite piece left over which must be ex-plicitly evaluated. A copy of the program implementingthis rather complex formalism is available upon requestfrom the authors.
'For a general introduction to directional solidification, see J. S.Langer, Rev. Mod. Phys. 52, 1 (1980).
-L. H. Ungar and R. A. Brown, Phys. Rev. B 29, 1367 (1984);30, 3993 (1985);31, 5923 (1985);31, 5931 (1985).
D. Kessler and H. Levine, Phys. Rev. A 39, 3041 (1989).4T. Dombre and V. Hakim, Phys. Rev. A 36, 2811 (1987); M.
Ben-Amar and B. Moussallam, Phys. Rev. Lett. 60, 317(1988); P. Pelce and A. Pumir, J. Cryst. Growth 73, 357(1985); M. Maashal, M. Ben-Amar, and V. Hakim (unpub-lished).
5S. deCheveigne, C. Guthmann, and M. M. Lebrun, J. Phys.
(Paris) 47, 2095 (1986).V. Seetharaman, M. A. Eshelman, and R. Trivedi, Acta Metall.
36, 1165 (1988);36, 1175 (1988).7J. Bechoefer and A. Libchaber, Phys. Rev. A 35, 1393 (1986).8This is claimed by Seetharaman, Eshelman, and Trivedi, Ref. 6,
but this does not seem to have been observed elsewhere.A. Karma and P. Pelce (unpublished).
' D. Kessler, K. Koplik, and H. Levine, Adv. Phys. 37, 255(1988); J. Langer, in Chance and Matter, edited by J. Souletieet al. (North-Holland, Amsterdam, 1987); E. Brener and V. I.Melnikov (unpublished).
41 LINEAR STABILITY OF DIRECTIONAL SOLIDIFICATION CELLS 3205
D. Kessler and H. Levine, Phys. Rev. Lett. 57, 3069 (1986).R. Peiters and J. S. Langer, Phys. Rev. Lett. 56, 1948 (1986).D. Kessler and H. Levine, Europhys. Lett. 4, 215 (1987); M.Barber, A. Barbieri, and J. S. Langer, Phys. Rev. A 36, 3340(1987); B. Caroli, C. Caroli, and B. Roulet, Phys. (Paris) 48,1423 (1987).
' A. Dougherty, P. D. Kaplan, and J. P. Gollub, Phys. Rev.Lett. 58, 1652 (1987).
'~For an example of the use of this formalism for dynamic simu-lations, see Y. Saito, C. Misbah, and H. Muller-Krumbhaar(unpublished) ~
D. Kessler and H. Levine, Phys. Fluids 30, 1246 (1987).' D. Bensimon, Phys. Rev. A 32, 1302 (1986); S. Tanveer, Phys.
Fluids 29, 3537 (1987).D. Bensimon, P. Pelce, and B. Shraiman, J. Phys. (Paris) 48,2081 (1987);E. Brener and V. I. Melnikov (unpublished).
