Velocity selection problem for combined motion of melting and solidification fronts

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by  Efim A. Brener

Velocity selection problem for combined motion of melting and solidification fronts
Efim A. Brener and D. E. Temkin
Institut fu¨r Festk¨orperforschung, Forschungszentrum Ju¨lich, D-52425 Ju¨lich, Germany
We discuss a free boundary problem for two moving solid-liquid interfaces that strongly interact
viathediffusion fieldin theliquidlayerbetween them. Thisproblem arises in thecontextof liquid
5
filmmigration (LFM)duringthepartialmeltingofsolid alloys. IntheLFMmechanismthesystem
0
chooses a more efficient kinetic path which is controlled by diffusion in theliquid film, whereas the
0
process with only one melting front would be controlled by the very slow diffusion in the mother
2
solid phase. The relatively weak coherency strain energy is the effective driving force for LFM. As

n intheclassicaldendriticgrowthproblems,alsointhiscaseanexactfamilyofsteady-statesolutions
a with two parabolic fronts and an arbitrary velocity exists if capillary effects are neglected [5]. We
J
develop a velocity selection theory for this problem, including anisotropic surface tension effects.

1 Thestrongdiffusioninteractionandcoherencystraineffectsinthesolidnearthemeltingfrontlead
2 tosubstantial changes compared to classical dendriticgrowth.

]
i
c The early observation of liquid film migration (LFM) and cellular precipitation [3]. In these processes a rela-
s
weremadeduringsinteringinthepresenceofliquidphase tively fast diffusion along the grain boundaries controls
-
l [1] orduring partialmelting of alloys[2] (see [3] for a re- the kinetics and the coherency strainenergy plays also a
r
t view). NowadaysLFMis awellestablishedphenomenon controlling role.
m
of great practical importance. In LFM one crystal is Thus, a theoretical description of LFM requires the
.
t melted and another one is solidified. Both solid-liquid solution of a free boundary problem for two combined
a
m interfaces move together with the same velocity. In the moving solid-liquid interfaces with a liquid film in be-
investigated alloys systems the migration velocity is of tween. In Ref.[5]this problem wasconsideredfor simpli-
d- the orderof10−6 10−5 cm/sanditis controlledby the fiedboundaryconditions: thetemperatureandthechem-
−
n solute diffusion through a thin liquid layer between the icalcompositionalongeachinterfacewerekeptconstant.
o twointerfaces[4]. Themigrationvelocityismuchsmaller Their values are different for the melting and solidifica-
c
[ than the characteristic velocity of atomic kinetics at the tion fronts and differ from those far from the migrating
interfaces. Therefore,bothsolidsshouldattheinterfaces liquid film. This means that any capillary, kinetic and
1 be locally in thermodynamic equilibrium with the liquid crystallographic effects at the interfaces were neglected.
v
phase. On the other hand, these local equilibrium states It was found that under these simplified boundary con-
1
2 should be different for the two interfaces to provide the ditions two co-focal parabolic fronts can move together
5 driving force for the process. It is by now well accepted withthe samevelocity. The situationisrathersimilarto
1 (see, for example, [3, 4]) that the difference of the equi- asteady-statemotionofoneparabolicsolidificationfront
0 librium states at the melting and solidification fronts is into a supercooled melt [6, 7] or one parabolic melting
5
duetothecoherencystrainenergy,importantonlyatthe frontintoasuperheatedsolid. Inthisapproximationthe
0
/ melting front because of the sharp concentration profile Pecletnumberswerefound,butthesteady-statevelocity
at ahead the moving melting front (diffusion in the solid remainedundeterminedatthatstage. Thus,theproblem
m phaseisveryslowandthecorrespondingdiffusionlength of velocity selection arises.
is very small). Thus, the liquid composition at the melt-
- Solvability theory has been very successful in predict-
d ing front, which depends on the coherency strain energy
ing certain properties of pattern selecting in dendritic
n
and on the curvature of the front, differs from the liquid
o growth and a number of related phenomena (see, for ex-
composition at the unstressed and curved solidification
c ample, [8, 9, 10]). Inthe two-dimensionaldendritic case,
v: front. Thisleadstothenecessarygradientoftheconcen- the basic approachis as follows. One attempts to model
trationacrosstheliquidfilmandtheprocessiscontrolled
i the dendritic tip by a needle crystal, that is, a shape-
X by the diffusion in the film.
preservingsteady-state growthshape which is a solution
r
If only the melting front existed, the melting process oftheequationofmotiongoverningdiffusionintheneigh-
a
would be controlled by the very slow diffusion in the borhood of a solidification front. This needle crystal is
mothersolidphaseandelasticeffectswouldbeirrelevant. assumed to be close in shape to the parabolic Ivantsov
In the LFM mechanism the system chooses a more effi- solution. If anisotropic capillary effects are included, a
cient kinetic path which is controlledby the much faster singledynamically stable solutionis found for anyexter-
diffusion in the liquid film. However, in this case the nalgrowthconditions. Thistheoryhasbeenextendedto
relativelyweak coherency strainenergyis involvedas ef- the three-dimensional case [11, 12]. We note that cap-
fective driving force for this process. In this respect the illarity is a singular perturbation and the anisotropy of
LFM mechanismis similar to otherwell-knownphenom- the surface energy is a prerequisite for the existence of
ena such as diffusion induced grain boundary migration the solution.
2
with the isotropic part of the surface energy γ and the
y
anisotropyparameterα<<1;θ istheanglebetweenthe
S1
normalto the interface n andthe direction of motion, θ
i
is the direction of the minimum of d (θ) for each of the
x i
interfaces.
L We measure all lengths in the units of the radius of
curvature of the Ivantsov parabolic solidification front,
S2 R2. Introducing a parabolic coordinates system,
2 2
y =(η β )/2, x=ηβ, (4)
−
FIG. 1: Schematic presentation of two moving nearly we look for a solution of the Laplace equation for the
parabolic fronts; S1 and S2 are the melting and growing concentration field C(η,β) in the liquid phase with the
solids, and Lis theliquid film. following boundary conditions:
C(η =η2(β),β)=d2(β)K2(β),
ThemainpurposeofthisLetteristodevelopavelocity
selection theory for LFM, including in the consideration 2
C(η =η1(β),β)= b∆ d1(β)K1(β). (5)
anisotropic surface tension effects. We note that this is − −
notajustroutineextensionoftheexistingtheorybecause
The mass balance conditions at the interfaces are:
the diffusion interaction between two interfaces changes
the problem substantially. 2P(η2(β)+βη2′(β))= (∂C/∂η η2′∂C/∂β)
− −
Wediscussthetwo-dimensionalproblemofthesteady-
state motion of a thin liquid film during the process of
′ ′
isothermal melting of a binary alloy, see Fig.1. We as- 2P(1 ∆)(η1(β)+βη1(β))= (∂C/∂η η1∂C/∂β) (6)
− − −
sume that the diffusion in the solid phases is very slow
and the concentration c in the liquid film obeys the where the Peclet number is P = VR2/2D and V is the
Laplace equation. We introduce the normalized concen- steady-state velocity in the y-direction. We present the
tration C = (c c )/(c c ) with c and c being two moving interfaces in the form,
L L S L S
− −
theliquidusandsolidusconcentrationsoftheequilibrium
phasediagramatagiventemperature. Thentheequilib- η2 =1+ǫ2(β), η1 =η0+ǫ1(β) (7)
rium concentration and the mass balance conditions at
and the concentration field in the form,
the solidification front read
C =d2K2, Vn = D∂C/∂n. (1) C(η,β)=C0(η)+u(η,β), (8)
−
At the melting front the equilibrium concentration is whereC0(η)=−b∆2(η−1)/(η0−1)isthesolutionwith-
changed by the presence of the elastic coherency strain out surface tension (u = ǫ1 = ǫ2=0) which satisfies the
energy [3] and also the diffusional flux changes because boundary conditions: C0(1) = 0 and C0(η0) = −b∆2.
The balance equations at the interfaces, 2P = 2P(1
in the solid ahead of the melting front the concentration
−
is c0 which is different from cS: ∆)η0 = −∂C0/∂η lead to the relations for the Peclet
number and η0:
C =−b∆2−d1K1, Vn(1−∆)=−D∂C/∂n. (2) VR2 b∆2 R1 1
P = = , η0 = = . (9)
Here Vn is the normal velocity; D is the diffusion co- 2D 2(η0 1) rR2 1 ∆
− −
efficient in the liquid film; K is the curvature assumed
to be negative for the interfaces in Fig.1; ∆ = (c0 These relations can also be obtained in the proper limit
c )/(c c ) is the dimensionless driving force; b =− of Eqs.(27)-(28) of Ref. [5] and give the expressions for
S L S
YΩ(da/d−c)2/a2f′′ is the dimensionless constant which the radii of curvature for the two interfaces, R1 and R2,
L
describesthecoherencystrainenergy[4],Ωistheatomic for a given velocity V which remains undetermined at
volume, Y is the bulk elastic modulus, a is the atomic this stage.
constant, f (c) is the free energy of the liquid phase per The surface tension plays a crucial role in the velocity
L
atom, f′′ is the second derivative of f (c) at c = c ; d selection problem. In order to find the small corrections
L L L i
are the anisotropic chemical capillary lengths: to these solutions due to the small(but singular)surface
tension effects we should first find the smallcorrectionu
′′ 2
di(θ)=d0[1−αcos4(θ−θi)], d0 =γΩ/fL(cL−cS) to the diffusional field in the linear approximation with
(3) respect to ǫ and dK (see, for example, [10]). In this
3
approximation, the field u satisfies the Laplace equation theirargumentalmosteverywhere. Thisleadstoǫ1 ǫ2,
≈
and the boundary conditions: K1 K2 andthen, fromEq.(15) we find indirect space:
≈
u(η =1,β)=Ψ2(β)=2Pǫ2+d2K2, d (1+β2)ǫ2 = d1+d2βK2 (16)
dβ 2P∆
(cid:2) (cid:3)
u(η =η0,β)=Ψ1(β)=2Pǫ1 d1K1. (10) Afewremarksareinorder. Firstofall,thisissomewhat
− the expected result because for the thin films the result-
The balance equations read: ing equation should be local and, in principal, can be
derived in the framework of boundary layer techniques.
′
2P(ǫ2+βǫ2)= ∂u/∂η η=1 However,the usedapproximationfor smallargumentsof
− |
hyperbolicfunctionsbreaksdowninthesmallvicinity(of
′ the order of ∆) near the singular points in the complex
2P(1−∆)(ǫ1+βǫ1)=−∂u/∂η|η=η0. (11) plane, β = i. We will discuss this point later. Finally,
±
we can return to Cartesian coordinates because the ex-
The Laplace equation with the boundary conditions,
pression for the curvature is much simpler in this repre-
Eqs.(10), is easily solved by Fourier transform, with re-
spect to the variable β, which we define as sentation. Inthelinearapproximation(1+β2)ǫ2 ≈ζ2(x)
where ζ2(x) is the correctionto the parabolic solutionin
1 ∞ the Cartesian coordinates. Then Eq.(16) reads:
g(λ)= dβexp( iλβ)g(β)
√2π Z−∞ − dζ2 d1+d2
= xK2 (17)
dx 2P∆
∞
1
One finds the first regular corrections to the parabolic
g(β)= dλexp(iλβ)g(λ).
√2π Z−∞ shape by replacing d1 = d2 = d0 and K2 = 1/[R2(1+
x2)3/2] for the parabola: −
Then,
ζ2(x)=σ/[∆(1+x2)1/2], ∆<<1, (18)
Ψ1sinh[λ(η 1)] Ψ2sinh[λ(η η0)]
u(η,λ)= − − . (12)
sinh[λ(η0 1)] − sinh[λ(η0 1)] where σ = d0/(PR2) is the usual stability parameter
− −
which appears in such problems.
Finally, Eqs.(11) read In the vicinity of the point x = i where the curva-
−
ture becomes singular and anisotropy, Eq.(3), becomes
dǫ2 Ψ1 Ψ2cosh[λ(η0 1)] important we rescale the variables [10]:
2P = −
dλ sinh[λ(η0 1)] − sinh[λ(η0 1)]
− − x= i(1 √αz), ζ2 =αF, τ =z+dF/dz, (19)
− −
dǫ1 Ψ1cosh[λ(η0 1)] Ψ2 with a small anisotropy parameter, α << 1. Then
2P(1 ∆) = − .
− dλ sinh[λ(η0 1)] − sinh[λ(η0 1)] Eq.(17) reads
− −
(13)
dF σ 1+d2F/dz2
Eliminating ǫ1 from the first of these two equations, = A(τ) (20)
dz −∆α5/4 (2τ)3/2
′
2Pǫ1 =2P(ǫ2) sinh[λ(η0 1)]+2Pǫ2cosh[λ(η0 1)]+
− − where the anisotropy factor is A(τ)=1 [exp( 4iθ1)+
2 − −
exp( 4iθ2)]/τ . This equation is similar to the limit of
−
d1K1+d2K2cosh[λ(η0 1)], (14) the kinetic dendrite in Eq.(9.1) of Ref.[10]. This simi-
− larity is purely formal because the real interface kinetic
we find from the second one, effects are not included in our description. Instead, the
strongdiffusioninteractionleadstosuchaselectionprob-
′′
2P(ǫ2) 2Pǫ2 =d2K2 lem. The same arguments as in [10] give the selection
−
condition
(d1K1)′ ′cosh[λ(η0 1)] σ =σ⋆ ∆α5/4, ∆<<√α. (21)
+(d2K2) − (15) ∼
−(cid:20)sinh[λ(η0 1)] sinh[λ(η0 1)](cid:21)
− − Moreover, the selected growth direction lies in between
′
Here denotes derivatives with respect to λ. thedirectionsoftheminimaofd1(θ)andd2(θ)forthetwo
These equations are convenient for the further anal- interfaces,θ1 = θ2,becausethe coefficientappearingin
−
ysis. We are interested in the motion of the thin film, A(τ) must be real [10].
which means that (η0 1) ∆ << 1. In this case we While the regular correction given by Eq.(18) is valid
− ≈
can expand the hyperbolic functions for small values of for ∆ << 1, the restriction for the validity of inner
4
Eq.(20) and for the scaling relation, Eq.(21), is much front would be controlled by the very slow diffusion in
stronger, ∆ << √α << 1. In the opposite limit, the mother solid phase. As we already noted, with the
∆ >> √α, we can derive another local inner equation help of the LFM mechanism the system chooses a more
where the important scale x+i √α is much smaller efficient kinetic path to relax to the equilibrium state.
| | ∼
then ∆. The argument of the hyperbolic functions be- Finally, we estimate the velocity V from Eq.(25) and
comes large and we can neglect the term K1 in Eq.(15). the thickness of the film R2∆ from Eq.(26) using char-
Then, in the vicinity of the singular point x= i in the acteristic values of the parameters: D 10−5cm2/s,
complex plane, this equation reads: − d0 10−7cm, b 0.05, ∆ 0.05 and σ⋆∼ 10−2. This
lead∼sto V 10−∼5cm/sand∼R2∆ 10−4cm∼, whichqual-
ζ2(x)= d2K2/P, (22) itatively ag∼ree with typical values∼in LFM experiments.
−
which is the usualinner equation for a single dendrite in In conclusion, we developed a selection theory for the
the one-sided model [10]. With the same rescaling as in process of liquid film migration where the strong diffu-
Eq.(19), we find sioninteractionbetweenmelting andsolidificationfronts
plays a crucial role. This process is very important in
2 2
σ 1+d F/dz
F = A(τ) , (23) practical applications, in particular during sintering in
α7/4 (2τ)3/2
the presence of the liquid phase [1, 3]. However, despite
its practical importance, experimental investigations of
where the anisotropy factor is A(τ) = 1
2exp( 4iθ2)/τ2. This corresponds to the selection con−- thisprocessarefarfromthelevelofaccuracyofthemodel
− experiments in classical dendritic growth. Our approach
dition
extends selection theory developed for dendritic growth
σ =σ⋆ α7/4, ∆>>√α (24) to LFM and we hope that our results will stimulate fur-
∼ ther theoretical and experimental investigations in this
andtheselectedgrowthdirectioncorrespondstothemin- very interesting field. From the theoretical side it would
imum of d2(θ), θ2 = 0. Whereas Eq.(23) and the selec- be a challenge to attack this problem by a direct numer-
tion condition Eq. (24) are valid not only for small ∆ ical approach, for example, by means of the phase-field
[10],theregularcorrectionintheformofEq.(18)requires model.
∆<<1.
The selection conditions, Eqs.(21,24), together with We acknowledge the support by the Deutsche
the relations for the Peclet numbers, Eq.(9), solve the Forschungsgemeinschaftunder project SPP 1120.
posed problem of pattern formation for two combined
nearly parabolic fronts:
2D
V = P2(∆)σ⋆ (25) [1] D. N. Yoon, and W. J. Hupmann, Acta Metall 27, 973
d0
(1979).
and [2] T. MuschikT,W.A.Kaysser,andT. Hehenkamp,Acta
Metall 37, 603 (1989).
R2 =d0/[P(∆)σ⋆], R1 =R2/(1 ∆)2. (26) [3] D. N. Yoon,Int.Mater. Rev.40, (1995).
− [4] D.N.Yoon,J.W.Cahn,C.A.Handwerker,J.E.Blendell,
and Y. J. Baik, In: Interface Migration and Control of
Whiletheseresultsareformallysimilartothefreeden-
Microstrucutres.AmSoc.Metals.Park.Ohio(1985),pp.
dritic case, the selected stability parameter σ⋆ scales as
19-31.
α7/4 only for ∆ >> √α. Otherwise, the other scaling
[5] D.E. Temkin , to bepublished in Acta Materialia.
relation, Eq.(21), holds. In principal, it could be con- [6] G. P.Ivantsov,Dokl. Akad.Nauk SSSR58, 567 (1947).
ceivable that the strong diffusional interaction between [7] G. P.Ivantsov,Dokl. Akad.Nauk SSSR83, 573 (1952).
the front leads to the selection even without anisotropy. [8] Y. Saito, Statistical Physics of Crystal Growth, World
However,ouranalysisoftheinnerequationdoesnotsup- Scientific Publishing, Singapore, 1996.
[9] D.Kessler,J.Koplik,andH.Levine,Adv.Phys.37,255
port this hypothesis. The Peclet number is P(∆) b∆
∼ (1988).
which reflects the fact that the coherency strain energy [10] E. A. Brener, and V. I. Mel’nikov, Adv. Phys. 40 53
plays a crucial role in the LFM mechanism. This pa- (1991).
rameterb isusuallysmallbut the meltingprocessbythe [11] M.BenAmar,andE.A.Brener,Phys.Rev.Lett71,589
LFMmechanismiscontrolledbythe fastdiffusioninthe (1993).
liquid, whereas the the process with only one melting [12] E.A. Brener, Phys.Rev. Lett. 71, 3653 (1993).