correction to the heat-balance equation and its influence on velocity selection in dendritic growth

PHYSICAL REVIEW A VOLUME 36, NUMBER 10 NOVEMBER 15, 1987

Correction to the heat-balance equation and its in6uenceon velocity selection in dendritic growth

M.-A. Lemieux and G. KotliarDepartment of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

(Received 21 May 1987)

We propose a curvature correction to the heat-balance equation at the solid-melt interface thatshould be included in any realistic model of dendritic growth. Numerical evidences are presentedto show that this term allows for steady-state solutions in the absence of anisotropy in the two-dimensional nonlocal symmetric model of solidification. It is shown that these solutions can beconjectured analytically in the large undercooling limit. Steady-state solutions are also obtainedwhen anisotropy is added.

INTRODUCTION I. THERMODYNAMICS OF THE INTERFACE

It is now generally recognized that microscopic physi-cal quantities can be of crucial importance in the veloci-ty selection of steady-state solutions in dendritic growthvia the microscopic solvability condition. ' The micro-scopic solvability idea forces us to reconsider carefullythe equations used to model dendritic growth since anynew small term can have a drastic inhuence on the selec-tion of the velocity. The first example of such an addi-tional perturbation is surface tension. As is now wellknown, ' it is a singular perturbation that does not al-low for any steady-state solution. The addition of an-isotropy to the surface tension ' then yields a discretefamily of steady-state solutions. Another example thatwas recently investigated is kinetic undercooling, and itwas found that this term also acts as a singular pertur-bation. Anisotropy was again essential in order to getsteady-state solutions.

In Sec. I of this paper we will propose and justify acorrection that should be included in the heat-balanceequation of any realistic model of solidification. Thiscorrection is very similar to the Gibbs-Thomson correc-tion to the temperature of the interface.

In Sec. II we investigate the inAuence of this new termon the microscopic solvability condition. We present theresults of numerical calculations done with the two-dimensional symmetric nonlocal model of solidificationand include the effects of this new correction. It isfound that the curvature correction to the latent heat al-lows for steady-state solutions in the absence of anisotro-py. As will be shown these solutions are, however, notin a regime where the model is expected to be, strictlyspeaking, valid, because the capillary correction to thelatent heat becomes of order unity. It is also possible toget anisotropy-induced solutions; we study in detail thevelocity of these solutions as a function of the new cur-vature correction, of the anisotropy, and of the under-cooling.

In the Appendix we look at the model in the large un-dercooling limit and in the absence of anisotropy andconjecture how these new anisotropy-free solutionswould appear in an analytical treatment.

In this section we will derive the correct boundaryconditions at the solid-liquid interface for the tempera-ture field. We will propose a new correction to theheat-balance equation very similar to the Gibbs-Thomson correction. The physical origin of this newterm is a result of the conservation of energy. As thecrystal grows, the area of the solid-melt interface in-creases. Energy is needed to create this new area, andtherefore a part of the latent heat released in thesolidification process is stored as interfacial energy in theadditional interface. The local change in area beingdirectly proportional to the local curvature, it shouldcome as no surprise that a curvature-dependent correc-tion is needed to conserve energy.

To derive the new boundary condition let us considera spherical solid of radius R immersed in its melt, and as-sume that the interface is molecularly rough. Both thesolid and the liquid are at temperature T. The pressureof the liquid PL is fixed and equal to the pressure of thesurrounding vessel. The pressure of the solid is

P~ ——PI +yK,where ~ is the interface curvature, defined as positivewhen the radius lies in the solid. Equation (1) defines yas the surface tension. The chemical potentials of bothphases are related by '

ps(Ps, T) =pt. (PL, T)+tc(y f ) lp, —

where p is the density of particles in the solid and f isthe surface free energy per unit area. It is often assumedthat y and f are equal; this is true for a fluid-fluid inter-face but not correct in general for a solid-Quid interface,as was stressed by Gibbs and Cahn. The physicaldifference is related to the difference in the work done tostretch the surface, compared to the work done to formthe surface.

The equilibrium temperature is given by Eq. (2), andan approximate expression for T, as is well known, canbe obtained by using the analyticity of the chemical po-tential. One first notes that for a planar interface

36 4975 1987 The American Physical Society

4976 M.-A. LEMIEUX AND G. KOTLIAR 36

Ps =Pz by Eq. (1) and then ps (Pz, TM ) =pz (Pz, TM ) byEq. (2), where T& is by definition the melting tempera-ture of the solid. One expands Eq. (2) in a Taylor seriesaround the planar point (Pz, TM). For simplicity wewill assume that the solid density is equal to the liquiddensity (since in Sec. II we will use a symmetric modelof growth); it is, however, straightforward to write gen-eral expressions following the same line of thought. Us-ing Ps —PL ——y~ and various Maxwell relations one finds

T = TM ( 1 —dpK), dp =f /L

where L = TM(rtz(Pz, TM ) —rts(Pz, TM )) is the latentheat Per unit volume and qz(Pz, TM) rts(P—z, TM ) isthe difference in entropy per unit volume between theliquid and the solid. This is the Gibbs-Thomson condi-tion.

%'e can now compute the amount of heat released inthe liquid if the solid grows infinitesimally and changesits radius by an amount 6R. Let S& be the solid's sur-face before the growth and Sz be the surface afterwards.The change of energy inside Sz is

6E= Es Ps T —EL PLT 6R dS+ m 6R dS,

where cs, cL are the internal energies per unit volume ofthe solid and the liquid and c is the internal energy perunit area of the interface. Here dS is an infinitesimalelement of surface, so that I 5R dS is the change in

volume and J K 5R dS is the change in area.Using the definition of internal energy, we can write

ss(Ps, T) Ez—(Pz, T)

= T( rts (Ps, T) rlz (—Pz, T) ) (Ps —Pz )—

+p(ps(Ps, T) pz(Pz—, T)) .

The last two terms are easy to evaluate given Eqs. (1)and (2). To evaluate T(rts(Ps, T) —rtz(Pz, T)) we useEq. (3) for T and expand the entropies in a Taylor seriesaround the planar point (Pt, TM ). Ignoring terms of or-der ~, we find

T(qs(Ps, T) —rtz(Pz, T))

L+fK—+ TMyK(Brjs IBP)r ~

+ T.' f f(hertz /&T)p ~—(Bqs IBT)t,tv ]IL

(4) where the derivatives are evaluated at (Pz, TM ). Thus

5E= —f L 5R dS+ f Ic+T y(Brt IBP) +T f[(BrI IBT) —(Br) /B)T) ]/L IK6R dS .

Remembering the equal density assumption it is clearthat there is no work done by the world outside S2 dur-ing this transformation; hence, by the first law of ther-modynamics we have 5E=5Q. [If the densities were tobe unequal then there would be some work done associ-ated to the change of specific volume. This would sim-

ply add an extra term to Eq. (8) but would not changeour conclusions. ) If the transformation occurs during atime interval 6t then

5E =5Q = —f q„5t dS, (8)

with

'[E+ TMy(&rts/dP)7;~

+T'f[(a~, /aT), —(a~, /aT), ]/L] .

(9b)

Equations (9a) and (9b) are the important results of thissection. Here cs, cL and Ds, DI are the heat capacitiesper unit volume at constant pressure and the thermaldiffusion constants. 8/Bn is the gradient normal to theinterface.

where q, is the Aux of heat normal to the interface andis directed towards the liquid. Using 6R/6t =U„, thenormal velocity of the interface, we finally find

Csas BTIBn~ s czDz BTIBn

~

=q„—=Lu„(1—d, K),

(9a)

+ TMf (cz —cs)/L —TM(y+ f)a], (9c)

where the same values for o, and k were used for bothphases since we assumed equal densities. The derivationof Eqs. (9) assumed the solid to be spherical; the expres-sions obtained will, however, be valid for solids of gen-eral shape as long as we can assume local equilibrium.

Note finally that the new term d& is expected to in-crease the growth velocity and therefore has a destabiliz-ing effect. The Mullins-Sekerka dispersion relation, '

describing the growth rate of a perturbation of wavevector k on a planar front according to 5$t, (t)= exp(cut )5/k (0), is indeed easily found to be, when

s=DL and cs=cI.cu = uk ( 1+d, k —2dpDk /u) (10)

in the limit 2kD/U &~1 and co &&Dk . The particularquadratic signature of the d

&term should make its ex-

perimental detection possible by following the time de-velopment of instabilities on a planar front. The ampli-tudes of the Fourier components of the front's Fourierspectrum should behave according to Eq. (10), therebymaking the detection of the quadratic term possible.

Since the d& term effectively diminishes the amount ofheat that has to be released into the liquid [Eq. (9a)] we

Using the thermal expansivity a and the isothermalcompressibility k the previous expression may be rewrit-ten as

'[E+ TMykqs

36 CORRECTION TO THE HEAT-BALANCE EQUATION AND ITS. . . 4977

can expect its presence to speed up the interface andfavor the appearance of steady-state solutions.

II. NUMERICAL RESULTS

p is considered as independent of P and T so that thereis no mass transport (we want to ignore convection) andtherefore u=k =0. The complete set of equations isthen

In this section we investigate how the new d& termaffects the solvability condition and the selection of thesteady-state velocity. We present the results of a studyof the two-dimensional symmetric model of solidificationof a pure substance when the d

&term is included in the

heat-balance equation.A. The symmetric model

Let us now precisely state the nature of the model thatwe used. The model is called "symmetric" becauseeq ——cL ——c, Dz ——DL ——D, and p is equal in both phases. "

BT/Bt =D b T,T (interface) = T~(1 —dol~),

(1 la)

(1 lb)

Dc(dT/dn~ s —BT/Bn

~ L )=Lu„(1—d, a.), (llc)

T(far from the solid) = T„&TM . (1 ld)

Here do f /L—, di ——E/L [by Eq. (9c) with cs ——cL anda=k =0]. di &do since a=f +Tg, where iI is the sur-face entropy per unit area.

We transform Eqs. (11) into an integral equation'

b, —doe[/( x}]=a'f . dx'[1 —d', 1~[((x')]Iexp[g(x') —g(x)]KO([(x —x') +[((x)—g(x')] I' ), (12)

where g(x, t}=g(x)+ut is the position of the interfaceand is the sought steady-state solution, E p is a modifiedBessel function, and

b =(TM —T„)c/L =(vrp)' exp(p)erfc(p' ),where p is the Peclet number. dp, d I are the rescaleddp, d ),

do ——TMcfu/2DL, d', =su/2DL . (13)

We will define the ratio R —=d &/dp, which is v in-dependent and is given for any substance, R =cL /fcTM. The parameters used numerically will be do andR. For a given R we will seek a selected value of dp.

In principle, R can take any positive value. Indeed,R =(E/f)(L/TMc) and the only thermodynamical con-straint is that E&f since s=f +Tel In practi. ce onewould expect s to be of the order of magnitude of f sothat, say, 1&v/f &10. The ratio TMc/L is equal to14.3 for succinonitrile' and is about 2 for water, soT~c/L can be believed to be of order 1 —10. R cantherefore be expected to lie somewhere between 0.1 and10.

B. The numerical technique

We study Eq. (12) with the help of a method first pro-posed by Vander-Broeck' and then applied to thesolidification problem by Meiron and Kessler andLevine. The idea is to allow for a possible cusp in g(x)at the tip. A physically acceptable solution must besmooth so the cusp will vanish; this yields a solvabilitycondition. Given dp and R, a solution is numericallycomputed and then the right slope at the tip k is exam-ined. When it vanishes we find a steady-state solutionwith at least a continuous second derivative at x =0.

The integro-differential equation is solved by discretiz-ing the domain of integration and approximating thefunction by parabolic elements. The resulting system ofnonlinear equations is solved by Newton's method. Thediscretization was changed to check convergence to the

I

continuum limit. We used a Control Data CorporationCyber-205 computer at the John von Neumann Comput-er Center (Princeton, NJ) (JVNCC). Let us firstly sum-marize the results that we obtained.

(i} The effect of d i on the solvability curve A, versus u

is qualitatively diff'erent depending on whether R & 1 orR &1. For R ~ 1 the curve crosses the abscissa at onepoint yielding a steady-state solution (Fig. 1). For R & 1

the curve is qualitatively similar to the one at R =0when no anisotropy is present; there are no solutions(the reader can refer ahead to Fig. 3).

(ii) When R & 1, the only zeros are produced by theanisotropy (Fig. 3). The zero induced by di when R & 1

is stable against the addition of anisotropy (the zeromoves towards smaller velocity). In addition, the anisot-ropy induces new zeros in the A, versus v curve nearv =0, just as in the R & 1 case. Let us now present anddiscuss in detail our various results.

1. Steady-state solutions inducedby the latent heat term ( d q )

Consider Fig. 1 where we plot the slope at x =0 as afunction of d p for p =0. 10 and R = 15. The slope iszero at do ——0.0 (it corresponds to the Ivantsov solution)and also at do ——0.0038 (remember that do ——TMcfu/2DL, and so do is proportional to the velocity); there istherefore a nontrivial selected steady state. It waschecked with a larger R and a coarser discretization thatthe curve does not cross the abscissa again for dp large,so we have only one nontrivial steady state.

Let us now check to see if this solution is acceptable,given our model. We know that Eqs. (11b) and (1 lc) arevalid only if dpK((1 and d&~ &&1 everywhere on the in-terface since higher-order terms in ~ were ignored. Asolution to the model will be, strictly speaking, accept-able only if it satisfies these two conditions; if the twoconditions are not satisfied we cannot justify the trunca-tion to O(v) in Eqs. (lib) and (llc) of the model. Asolution that does not satisfy these two conditions is val-


0. U75—

0. 050—

~ ~ ~

0. 025

0. 020—

0. 025—CL

0. 015—

0 — ~ 0. 010—

-0. 025— 0. 005—

-0. 050—0 0. 0025 0. 0050 0. 0075 0. 0100 0. 0125 0. 0150

cj 07. 5 10. 0 12. 5 15. 0 17. 5

FIG. 1. Slope as a function of do for R =d 1/do ——15 and

p =0.10. There is no anisotropy.FICx. 2. Selected do as a function of R. p =0. 10 for the dots

and p =0.12 for the few squares. There is no anisotropy.

id in the case where Eqs. (11b) and (1 lc) are valid for ar-bitrary K. A solution which is not, strictly speaking, ac-ceptable might nevertheless be qualitatively correct pro-vided that the truncation to 0(1~) does not dispense withsome essential physics.

Let us rewrite the conditions dpK((1 and d~K((I interms of the numerical parameters. We checked that thesteady-state solutions obtained numerically were verysimilar to the Ivantsov parabolas with the same veloci-ties; we will use this to approximate the curvature at thetip by p ', where p is the Ivantsov radius of curvaturesatisfying pv /2D =p. The conditions dpp « I andd, p

' «1 become dov/2D «p and d&v/2D «p. Us-ing Eq. (13) for the definition of do and d'i ——Rdo, theconditions become

d p «p TM c /L

dp (&p/R

(14)

(15)

Let us now come back to the solution of Fig. 1. SinceTMc/L is typically of order I —10 we see that the in-equality of Eq. (14) is easily satisfied (and this will betrue for all the cases considered in this paper). The in-

equality of Eq. (15) is not satisfied, however, since

p /R =0.0067, and therefore d p =p /R. This needlecrystal is too sharp to be rigorously described by ourmodel since p=d&. It is a solution that can thus be, atbest, a qualitatively correct manifestation of the exactsolution that one would obtain with a model valid for ar-bitrary large K. Since the condition dpK((1 is satisfied,the typical scale of the crystal is still much larger thanthe capillary length.

The properties of these solutions are very counterin-tuitive. In Fig. 2 we plot the selected value of dp as afunction of R (p =0.10 for the dots and p =0.12 for thefew squares). We see that the velocity decreases as R in-creases. If we look back at Fig. I this means that thecrossing point is shifting towards dp~op and divergesas d1 decreases. When d& ——0 we therefore recover theby now familiar result ' of a slope everywhere negative.The velocity is also decreasing as the undercooling (i.e.,the Peclet number) increases. These results are surpris-

ing since when R increases by increasing the ratio d& /dp

while keeping the dimensionless undercooling TMC/Lconstant, the latent heat released into the liquid dimin-ishes. Then one would expect the growth velocity to in-crease; also, when the dimensionless under coolingA=TMC/L increases the latent heat is evacuated moreefficiently in the liquid so once again one would expectthat the velocity should increase.

This counterintuitive behavior can, however, be ex-plained. Let us make the hypothesis that the tip regionis the region controlling the selection of the velocitysince from far away the shape is almost parabolic andthere is an Ivantsov solution for any velocity. Approxi-mating again the tip by an Ivantsov parabola we see thatby Eq. (11c) the fiux of heat near the tip is

q =Lv (1—d&v/2Dp) which has a maximum atv =Dp /d &. This last equality can be rewritten asdp ——p/2R. For dp &p/2R the dendrite must slow downto increase the heat Aux, so its velocity will decrease ifwe increase the undercooling. When dp ~p/R the heatfiux q is even negative (near the tip only). This meansthat heat is Aowing towards the solid in the tip region.If R (and therefore d&) gets larger then the amount ofheat Rowing towards the solid increases. This has theeffect of slowing down the growth. In Fig. 2 we see thatdo ~p/R (d, i~~ 1) for small R and do =p/2R(d, v= 1/2) for larger R.

The question that we would like to answer is the fol-lowing: Is it possible to find a range of p and R suchthat the selected steady state is in the d&K« I regime?To satisfy Eq. (15) we would like to increase p and hope-fully find solutions with small velocities (do «p/R). Itis, however, not possible to increase p numerically since,as is explained in the Appendix, for large p the slopebehaves as exp( —2ap/ Qdt ), where a is a number oforder unity. For large p we indeed observe numericallythat the slope becomes too small for our numerical reso-lution.

In the Appendix we present an analytical analysisdone in the limit 6~1. It is conjectured that we canfind steady-state solutions in the absence of anisotropyprovided that R & I. The selected velocity diverges as


R ~1 (so as d&~TMcdo/L). These solutions do satisfy

d&K « 1 ' howcvcr' thc sclcctcd velocities ar c still de-

creasing as the undercooling increases. The explanationof this behavior is unclear. For R & 1 there should beno solutions (without anisotropy). All these predictionsare consistent with the previous numerical results.

We also studied the effect of anisotropy on these solu-tions. The addition of anisotropy to both do and d, (inthe way explained below) has the effect of slightly de-creasing the selected velocity.

Q. 025

0. 020—

0. 015—

0. 010—

2. Steady-state solutions induced by anisotropy

In Fig. 3 we plot the slope at x =0 as a function of dpfor p =1.0 and R =1.0. The curve with the dots is theresult when no anisotropy is included. For the curvewith the squares we added some anisotropy to both dpand d| by multiplying them by ( I+a cosOsinO), where 0is the angle between the growth axis and the normal tothe interface. The same anisotropy is added to bothterms since they share the same physical origin[do f /L, d~ —(—f +Tr——l)/L]. a is the anisotropy param-eter and is here equal to 0.2. It is clear from thesecurves that when R & 1 anisotropy is needed to obtain anontrivial steady-state solution; for a =0.2 there isindeed a solution at dp ——0.022. Since p =p/R =1.0clearly the inequalities of Eqs. (14) and (15) are satisfied.

When R ) 1 we checked (for R up to 10) that the ad-dition of anisotropy was inducing new zeros near dp =0in a way completely similar to Fig. 3. The zero obtainedwithout anisotropy (Fig. 1) was stable and slightly shift-ed towards a smaller velocity.

In Fig. 4 we show the selected value for dp as a func-tion of R. We see that the velocity increases with R,and thus increases with d&, as expected. For the dots,p =1.0 and a =0.2. For the squares, we decrease theundercooling; p =0.5 and a =0.2. As expected, the ve-locity decreases. For the triangles, we decrease the an-isotropy; p =1.0 and a =0.1. The velocity decreaseswith respect to the dotted curve. In the next figures we

Q. 005—I

0 0. 6 0. 8

FIG. 4. Selected do as a function of R. For the dots p =1.0and a =0.2; for the triangles p =1.0 and a =0.1; for thesquares p =0.5 and a =0.2.

study in detail the behavior of the velocity as we varythe anisotropy and the undercooling.

In Fig. 5 we plot the selected dp as a function of theanisotropy (as before anisotropy is added to both do andd&). Here p =0.5 and R =0.4. The velocity increaseswith a and saturates to an approximately constant valuefor a &0.6. This is consistent with what has been ob-served in a previous work in the absence of d

&and in

the presence of the kinetic undercooling.In Fig. 6 the selected dp is plotted as a function of the

Peclet number for R =1.0 and a =0.2. As expected, thevelocity increases with the undercooling. A log-log plotof this figure is shown in Fig. 7. From this plot we candeduce that the selected dp is proportional to p' . Forp —+0 we should expect' vaA, and so dpup, and there-fore our result (obtained over the range 0. 1 &p & 1.0) isconsistent with this prediction. This result implies thatas p ~0 the selected dp will still satisfy dp &&p/R sinced 0ay, so these solutions will satisfy d, Ir « 1 (anddoe « 1) even in the 6~0 limit.

0. 010

0. 005— s ~ s ~

0. 0125

0 ~ ~ s ~

-0. 005-

0. 0100—

0. 0075—

-00. 0050—

-0. 010—0. 0025—

-0. 0150 0. 005 Q. 010

j I

0. 015 0. 020 0. 025 0. 030d 0

Q — ~

Q. 2 0. 4 0. 8 l. 0

FIG. 3. Slope as a function of do for R =1.0 and p =1.0.There is no anisotropy for the dots, and a =0.2 for thesquares.

FICx. 5. Selected do as a function of a, for p =0.5 andR =0.4.


0. 025

0. 020—

0. 015—

0. 010—

0. 005—

0—

0. 2 0. 4 0. 6 0. 8 l. 0

these solutions exist only when R & 1, that is, whend]) T~edP/L.

Additional solutions appear for any R when anisotro-py is added to dp and d&. We studied in detail the range0. 1 & R & 1.0. As expected, the selected velocity in-creases as d& increases. It was found that the velocitysaturates to an almost constant value for large anisotro-py. We also showed that the velocity was proportionalto p' =p in the range 0. 1&@&1.0. These solutionswere already well known in the limit d& =0. ' ' ' Theaddition of d

&slightly increases the value of the selected

velocities.

ACKNOWLEDGMENTS

FIG. 6. Selected do as a function of p, for R =1.0 anda =0.2.

CONCLUSION

In Sec. I we introduced a curvature correction to theheat-balance equation of the solid-melt interface. Thiscorrection should be included in any realistic model ofdendritic growth. The correction can be tested by exper-iments measuring precisely the Mullins-Sekerka disper-sion relation.

In Sec. II we showed that this correction, togetherwith the Gibbs-Thomson correction, gives, in the ab-sence of anisotropy, steady-state solutions that do notsatisfy d&~&&1. These solutions behave counterintui-tively since their velocity decreases as d] or p increases.They may actually be qualitatively correct if the trunca-tion of Eq. (9a) to order a does not throw away someessential physics. These solutions could appear in thedynamics as sharp needle crystals that melt back. Theaddition of anisotropy simply decreases slightly the ve-

locity of these solutions. A plausible analytical conjec-ture of these solutions is presented in the limit of largeundercooling in the Appendix. The analytical solutionsdo behave counterintuitively but they satisfy the condi-tion d&~ &&1 so that the reasons explaining their behav-ior are unclear. It is also predicted in the Appendix that

J. Liu provided the version of the program used forthe numerical simulations. He is also thanked for manyuseful discussions. One of us (M.A.L.) is supported inpart by the Natural Sciences and Engineering ResearchCouncil (NSERC) of Canada and in part by the Mas-sachusetts Institute of Technology. The other (G.K.) ac-knowledges support from the Alfred P. Sloan founda-tion. This work is supported by the National ScienceFoundation (NSF), Grant No. DMR-84-18718.

APPENDIX: AN ANALYTICAL CON JECTUREIN THE LIMIT OF LARGE UNDERCOOLING

In this appendix we will apply the method developedin Ref. 18 to study the two-dimensional nonlocal modelof solidification in the limit of large undercooling and inthe presence of d, . [It should be pointed out that in thelimit of small undercooling (p ~0 limit) the d

&term is a

higher-order elfect in terms of p, the Peclet number].Our analysis is very similar to the analysis of Ref. 18and thus our derivation will only be sketched.

We define 6=1—6 and we will take the limit 6«1.In this limit the Peclet number becomes p =1/(25) &) 1.The equation of motion of the interface is

1 —5 —dos[/(x)]

=2 f dr f dx'I 1 RdoK[g(x )]}—

where

G(x, z, r ix', z', 0)

X G(x, g(x), rIx', g(x'), 0), (Al)

-2. 0—

-0O

—-3. 0

-3. 5—

-4. 0—l. 0 -0. 6

log (p)

-0. 4 -0. 2

FIG. 7. Log-log plot of Fig. 6.

=(4~r) 'expI —[(x —x') +(z —z'+2r) ]/4r} .

(A2)

Here lengths have been expressed in units of 1=2D/vand time has been expressed in units of 1 /D. dp and Rare defined as before.

Eq. (Al) can be reduced to a linear, inhomogeneousdi6'erential equation of infinite order in the limit 5 «1.We first do the change of variables X=x' —x and ex-press g(x') as a Taylor expansion around g(x), themotivation being that in our units g"(x) «1. (Here theprimes denote derivatives with respect to x.) TheIvantsov parabola is indeed given in our units by

36 CORRECTION TO THE HEAT-BALANCE EQUATION AND ITS. . .

g"(x)= —25. The nth derivative of g with respect to xcan be written as g'"'=( —25d/dzI)" g", where zi=j'.It will turn out that the natural variable of the equationwill be g, so that we write

where

L o = g a„(zI ) ( —25 d /d zI )",n =0

(A5)

g"= —25[1+h (zI)] (A3)

and we assume that h «1. We take the 6«1 limit ofEq. (Al) and we furthermore linearize in terms of h. Weget

(Lo —26Lz —dolu )h(zI) =do(1 —R)iz +5bo+0(5 ),(A4)

L, = g b„(zi ) ( —25 d /d zl )",n=0

( 1+ 2) —1/2

with

(A6)

(A7)

a„(zI)=f dr f d Xg o( X, r~ zI)I(2r Xzi)X—" +' /[ r(n-+2)!]+2Rd po,'X"/ n!I

0 oc

b„(zI)—:f dr f dXgo(X, r~

zI)X" [(Xzl —2w) /2r —I/r] /[2(n +2)!]

2Rdop—f dr f dXgo(X, r~

zI)IX" (Xzi —2 r)[1/(n +2)!+—,'(n!)]

&&[I/(2r)1+3@'zI(n+2)X" '/[(n+ I)!]]

(A8)

and

go(X, r~

zI) =(4zrr) 'expI —[X'+(Xzi 2r)']/4—rI .

(A10)

We kept the terms such as ( —25d/dzI)" to all orderssince we will now look for a solution which is essentiallysingular in 6, more precisely for a solution of the WKBtype,

h+(zI) =exp(5 'So+ +S,; + ),where

(A16)

r and X to get closed-form expressions for Lo(S„',zi) and

L, (S ,oIz). We can in principle solve (A12) for So andthen (A14) for S', . The expressions are, however, quitecomplicated. They simplify a lot if we look for solutionsin the regime d 0 « 1. We then get two independenthomogeneous solutions,

h (zI)=exp[5 'So(zl)+S, (zI)+0(5)] . (Al 1)

Lo(S-o, zI) =doiz,

where

(A12)

We will first look for homogeneous solutions to (A4).Inserting (All) into (A4) we get to 0(1)

dSO+ /dzI =i (1

ized)

(I+i—zt)'~ /(2+do )

+ (zI —i+iR )/4+0 (+d )o,

Sz+ ———3iarctan(zI )/4

+ —( 1+R ) ln( 1+zI') +0 ( V d 0 ),

(A17)

(A18)

Lo(SO, zI)= g a„{zI)(—2SO)"n=0

and So =dSO/dzi. To 0(5) we get

(A13)

S', r)LO/BSO+(So' /2)B Lo/r)SO' 2L, (So, zl ) =0, —

{A141

where

and So &is the complex conjugate of SO &+.

We will now try to construct a particular solution toEq. (A4). Since Eq. (A4) is of infinite order there is nosure way of constructing its particular solution. Wemake the guess

hp(zI ) = f ds P(s)F (s)

X [h + ( zI ) /h + (s ) —h ( zI ) /h (s ) ], (A 19)

whereL, (S go)= g b„( I){z—2SO)" .

n=0{A15)

$(s)=do(1 —R)p'(s)+0(5) . (A20)

It is possible to perform the sums explicitly in Eqs.(A13) and (A15) and then to carry out the integrals over

To determine F(s) we ask for hp to satisfy Eq. (A4).We obtain to dominant order

(A21)

When g=+ oo we are infinitely far from the tip of thegrowing crystal and therefore the solution must be equalto the Ivantsov solution. This means, by Eq. (A3), thathp(+oo)=0. This boundary condition was already im-

F(s)=25 'So+(s)SO (s)/I [So~(s)—So (s)][1—2do(1 —R)iz'(s)]II

posed at zI= —co on Eq. (A19). Because both h+ andh diverge as g~+ao we cannot add any componentsof A + to A p without disqualifying our solution; we musttherefore check if our solution, as it stands, satisfies the


I&(5,do, R)= f ds P(s)F(s)/h+(s), (A23)

where we used the property So, + (7] ) =So, ( —ri ).Since h+ and h are independent the condition hp( oo )

=0 therefore implies

I, (5,do, R)=0 . (A24)

Using the fact that 6 «1 we can evaluate Ii by usinga saddle-point approximation around g=i. We get

I](5,do, R) =N/(1 —R)(do)'

X5 ' exp[ —a/(5+do )] (A25)

where Ã, is some number and a =0.3079. . . . Equation(A24) can be satisfy if do ——0, which corresponds to theIvantsov solution. For R &] and 5&0 it cannot besatisfied with do&0 ~ It is also apparently satisfied whenR = 1 and we must investigate the meaning of this pecu-liar result.

When R = 1 the inhomogeneous part of Eq. (A4) is oforder 6.

To treat the problem correctly one should thereforedo the following.

Write the differential equation (A4) to order 6 .Compute the WKB solution to order 6, i.e., compute

S2(7]).Compute F(s) to next order in 5.Obtain the particular solution to order 6.

One would then obtain a solvabiiity condition conceptual-ly similar to Eq (325), but .of higher order in 5,

I2(6, do, R =1)=0 . (A26)

This solvability condition would not be satisfied for arbi-trary 6 and d o. Therefore, although Eq. (A24) issatisfied when A =1, we would not get solutions for ar-bitrary 6 and do.

We can now ask the following question: Is it possible,when R &1, to add I] to I2 in such a way that their sumis zero? Let us assume that we can write I2 to leadingorder in 6 and do as

I2(5, do, R)=N2(do) '" 6~' 'exp[ —a/(6"[/do )] .

(A27)

boundary condition hp( co ) =0. This becomes our "sol-vability condition. " For large g we have

hp(ri) =[h+(7])—h (7])]I)(5,do, R ), as 71~ aa (A22)

with

We expect the same exponential dependence sinceexp(6 'So) will again control the saddle point aroundq=i. Adding I] to I2 we get the following solvabilitycondition:

(a & 11/4 —3R /14 —a(R I@CO J

—63/2+3R/7+t3~R~N /[N (R 1)] (A28)

Since I2 must be a correction of higher order in 6 ascompared to I~ we know that 3/2+3R /7+P(R) &0.We will now make the hypothesis that1/4 —3R /14 —a(R) &0. This is really a guess, but if itis true, then Eq. (A28) will explain the existence of solu-tions in the absence of anisotropy. We know that thisequation has no solution when R =0 (as was checkednumerically ' ). This implies that N2/N~ &0 since thenthe right-hand side is negative which ensures that Eq.(A28) is not solvable (for R =0). With these assump-tions we can therefore conclude from Eq. (A28) the fol-lowing.

(1) For R & 1 there are no solutions.(2) For R & 1 the selected velocity behaves like

5 [42+ 12R +28t3(R ) ]/[7 —6R —28a( R ) ] /( R —1 )oa (A29)

d Ir /d B~ s 0 N', ( 1 —R ) ( d o )——

3 /2+3R/7( 2 /Qd ~

) (A30)

The slope at the tip is not related to this quantity in asimple way, but from this equation we can believe that itbehaves to leading order like exp( —2ap/+do ) whichexplains the fact that the slope gets too small for our nu-merical resolution when p is large. This limited thescope of our numerical study.

(3) As R ~1 the selected velocity diverges. Strictlyspeaking, for R =1 Eq. (A28) is not valid (since it wasderived in the limit do « 1), but it certainly hints that atleast a strong increase in the velocity is to be expected.

All these results are in agreement with the numericalresults that are presented in Sec. II of this paper. Inparticular, the selected do given by Eq. (A29) decreasesas the undercooling increases (thus as 6 decreases), andalso as d~ (thus R) increases. It must be noted, however,that Eqs. (14) and (15) of Sec. II are satisfied in this limitand thus that do«&1 and dies&&1. The explanation ofthis counterintuitive behavior is, as of now, unknown.

We may finally note that I, can be shown to be pro-portional to de/dB

~ q o. Using the relation p =1/(25)we get

E. Ben-Jacob, N. Goldenfeld, B. G. Kotliar, and J. S. Langer,Phys. Rev. Lett. 53, 2110 (1984).

~D. Kessler, J. Koplik, and H. Levine, Phys. Rev. A 31, 1712(1985).

D. Kessler, J. Koplik, and H. Levine (unpublished) ~

4D. I. Meiron, Phys. Rev. A 33, 2704 (1986).

5D. A. Kessler and H. Levine, Phys. Rev. B 33, 7867 (1986).6M.-A. Lemieux, J. Liu, and Cz. Kotliar, Phys. Rev. A 36, 1849

(1987).7J. W. Cahn, Acta Metall. 28, 1333 (1980).J. W. Gibbs, Collected Works (Yale University Press, New

Haven, 1957), Vol. 1, pp. 314—331.


Ref. 8, footnote of p. 315.W. W. Mullins and R. F. Sekerka, J. Appl. Phys. 35, 444(1964).J. S. Langer, Rev. Mod. Phys. 52, 1 (1980)~

G. C. Nash and M. E. Glicksman, Acta Metall. 22, 1283(1974).

' M. E. Glicksman, R. J. Schaefer, and J. D. Ayers, Metall.Trans. 7A, 1747 (1976).

J.-M. Vander-Broeck, Phys. Fluids 26, 2033 (1983).P. Pelce and Y. Pomeau, Stud. Appl. Math. 74, 245 (1986).

' A. Barbieri, D. C. Hong, and J. C. Langer, Phys. Rev. A 35,1802 (1987).

D. A. Kessler, J. Koplik, and H. Levine, Phys. Rev. A 34,4980 (1986).

' B. Caroli, C. Caroli, B. Roulet, and J. S. Langer, Phys. Rev.A 33, 442 (1986).

correction to the heat-balance equation and its influence on velocity selection in dendritic growth

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