Morphological instability induced by the interaction of a particle with a solidifying interface

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by  Layachi Hadji

Morphological Instability Induced by the Interaction of a Particle with a Solidifying
Interface
Layachi Hadji
The University of Alabama, Department of Mathematics, Tuscaloosa, Alabama 35487
(Dated: February 2, 2008)
4
We show that the interaction of a particle with a directionally solidified interface induces the
0
onset of morphological instability provided that theparticle-interface distance falls below a critical
0
value. This instability occurs at pulling velocities that are below thethreshold for theonset of the
2
Mullins-Sekerka instability. The expression for the critical distance reveals that this instability is

n manifested only for certain combinations of the physical and processing parameters. Its occurence
a is attributed to the reversal of the thermal gradient in the melt ahead of the interface and behind
J
theparticle.

1
2 PACSnumbers: 81.05.Ni,81.30.Fb,81.10.Mx,81.10.Dn,81.20.Dn

]
i
c The freezing of a liquid with a dispersed phase takes Z
s place in numerous natural and industrial processes. Liquid phase
-
l Some examples include the formation of ice lenses that
r
t result from the freezing of soil water [1], the freezing
m
of biological cell suspensions in a cryopreservation ex- Particle
. periment [2], the decontamination of metallic pollutants a
t
a from soils [3], the growth of Y123 superconductors by
a H
m the undercooling method [4] and the manufacture of V 0 R
- particulatereinforcedmetalmatrixcomposites(PMMC)
d [5]. The properties of these composite materials are h∞
Deformed
n
enhanced by the addition of the dispersed elements.The
o interface
freezing of a liquid suspension is associated with the Solid phase
c
[ interaction of the constituents of the dispersed phase

with a solidifying interface. The first systematic study
1 of this interaction was carried out by Uhlmann et al. FIG. 1: Sketch of a particle of radius a immersed in a melt
v near a deformable solid-liquid interface; aH is the particle-
0
[6]. They demonstrated the existence of a critical value
4 interface distance measured from the particle’s center to the
0 for the growth rate below which the inclusions are planar solid front, V is the interface growth rate, h∞ is the
4 pushed by the moving interface, and above which they gapwidththatseparatesthelowestpointontheparticlefrom
1 are engulfed by the interface and incorporated into the the planar interface, and R represents the radial coordinate
0 solid. Consequently,verylowgrowthratesareconducive taken along theplanar interface.
4
to particles being pushed by the interface, while high
0
growth rates are conducive to particle engulfment.
/
t
a This hand-waving argument is made more precise in
m The presence of an inclusion in the melt near a this paper. The situation considered here is illustrated
solid-liquid interface introduces locally a change, albeit schematically in Fig. 1. An axisymmetric spherical par-
-
d small, in the thermal gradient ahead of the solid front. ticle of radius a is submerged in a binary alloy in such a
n This, in turn, introduces a small deformation in the waythatthe distance fromits centerto the planarsolid-
o
profile of the interface. The difference in the thermal liquid interface that emerged from the directional solidi-
c
conductivities of the melt and particle stands out as the fication of a dilute binary mixture is aH . We assume a
: 0
v cause for this interfacial deflection [7, 8, 9, 10]. Imagine microgravity environment. The problem is described by
i
X a situation wherein a solid is growing antiparallel to the diffusion of solute equation in the melt and the heat
the direction of the heat flux and toward an inclusion conductionequationsinthemelt,solidphaseandparticle
r
a that is less heat conducting than the melt in which it is in a frame that is moving with velocity V in the upward
immersed. Then, as the width of the gap separating the Z−direction; Z is the vertical coordinate (Z > 0 in the
inclusion from the solid front decreases, heat becomes liquid region)and R is the radialcoordinate takenalong
more easily evacuated from the solid phase than from the planar solid-liquid interface. The diffusion of solute
ahead,leadingto a localreversalofthe thermalgradient inthe solidphase is neglected. The boundary conditions
and, consequently, to the local destabilization of the at the solid-liquid interface account for: (i) curvature
interface. As long as the particle remains pushed, and solute undercoolings in the equation describing the
the disturbance grows and propagates radially with equilibrium temperature, (ii) the heat balance, and (iii)
decreasing magnitude. theconservationofmass. Attheparticle-meltboundary,
[Z−(a+h )]2+R2 =a2, we impose the zeromass flux
∞
2
and the continuity of the temperature and heat flux; h curvature lies in the soild phase.
∞
denotes the width of the gap between the particle and
the planar interface. Far away from the solid front, the The above set of equations ( with ǫ = 0) [12] admits
imposed thermalgradientsin the solid andliquid phases a base state with a planar interface growing at constant
are G(L) and G(S), respectively, while the concentration speed. Thedeterminationofthethermalfieldsinaninfi-
of solute is maintained at C . The nondimensionalisa- nite medium in which a sphericalinclusion has been em-
∞
tionthatweadoptmakesuseoftheparticle’sradiusaas bedded has been carried out [13]. Here, we make use of
length scale [11], a/V, the melting point T , the growth the method of images [14] to calculate the thermal fields
m
rate V, and the concentration far away from the front, inthemeltandparticlethatsatisfytheequilibriumtem-
C , as scales for time, temperature, velocity and con- perature condition at the planar interface, Eq. (3), and
∞
centration, respectively. We have the conditions at the particle’s surface, Eq. (6). Then
anexpressionforthe thermalfieldinthesolidisderived.
∂C ∂C ∂2C This expression must satisfy the temperature condition
ǫ( − )=∆ C+ , (1)
∂t ∂z r ∂z2 at the planar interface, Eq. (3), and must also support
a planar interface growing with constant speed, Eq. (4).
We have
∂T(q) ∂T(q) ∂2T(q)
ǫλ(q)( ∂t − ∂z )=∆rT(q)+ ∂z2 , (2) CB(r,z)=1, in the melt, (7)
where C is the solute concentration, T is the temper-
ature, the superscript (q) stands symbolically for par- T(L)(r,z)=1+G z+M +U(r,z), in the melt, (8)
ticle (q = P), solid (q = S) and liquid (q = L), B L
λ(q) = D/D(q), where D and D(q) are the coefficients
of solute and thermal diffusion, respectively, ǫ = aV/D
U(r,z)
iesrathtoerP, e(1cl/ert)n∂u/m∂rb(err∂,/a∂nrd).∆TrhisetchoerrBeesspsoenlddiinffgerbenotuinadlaorpy- TB(S)(r,z)=1+GSz+M + k , in the solid, (9)
conditions are as follows: at the solid-liquid interface,
3G z
T(S) =T(L) =1−σκ+MC, (3) T(P)(r,z)= L +1+M, in the particle. (10)
B 2+α
where
Svn =(k∇T(S)−∇T(L))·ni, (4) 1−α ∞ z+nH0
U(r,z)=G ( ) . (11)
L 2+α [(z+nH )2+r2]3/2
n=X−∞ 0
n6=0
ǫC(1−K)vn =−∇C·ni. (5) The plot of the basic profile for the melt’s temperature,
Eq. (8), is shown in Fig. 2. It pertains to an axisym-
Attheparticle’ssurface,(z−H )2+r2 =1,thecontinuity
0 metric particle of radius a = 10µm whose center is on
of temperature and of the heat flux imply, the z−axis and whose separation from the interface is
0.5µm. It depicts an unstable configuration, wherein
T(L) =T(P), (α∇T(P)−∇T(L))·n =0, (6)
P the thermal gradient in the gap is reversed due to the
while the zero mass flux condition yields ∇C ·n = 0. thermally insulating character of the particle. The
P
The far field conditions are ∂T(L)/∂z → G and configuration is, however, stable for the case of a highly
L
C = 1 as z → ∞, and ∂T(S)/∂z → G as z → −∞. conducting particle.
S
The symbols that appear in the above equations are
In order to examine the stability of the base state,
defined as follows: σ = σ /aL is the surface energy
SL
we first superimpose axisymmetric, time-dependent in-
parameter, where σ is the interface excess free energy
SL finitesimaldisturbancesθ(q),candηuponthebasicstate
and L is the latent heat of fusion per unit volume;
M = mC∞/Tm, is the morphological parameter, where solutionsTB(q),CB,andtheplanarinterface,respectively.
m is the liquidus slope; S =LVa/(T k(L)) is the Stefan We let
m
number; k = k(S)/k(L); α =k(P)/k(L), where k(q) is the
[T(q),C]=[T(q),C ]+ǫ[θ(q),c], (12)
coefficient of thermal conductance; K is the segregation B B
coefficient; v is the normal growth velocity; n and
n i in Eqs. (1)-(6). The resulting equations are then lin-
n are the unit normal vectors pointing into the melt
P earized with respect to the disturbances. The following
at the interface and particle’s surface, respectively;
perturbed problem is obtained,
G = aG(L)/T and G = aG(S)/T , where G(q) is
L m S m
the imposed (dimensional) thermal gradient, and κ is ∂2θ(q) ∂T(q)
the curvature taken to be positive when the center of ∆rθ(q)+ ∂z2 =−λ(q) ∂Bz , (13)
3
x 10−5 ∂2c
4 ∆rc+ ∂z2 =0, (14)
2
z)
(r,B0
T θ(L) =−G η+σ∆ η−F(r)η+Mc, z =0, (15)
−2 L r
−4
4
3 0.06
2 0.04 F(r)
r 1 0.02 z θ(S) =−G η+σ∆ η− η+Mc, z =0, (16)
0 0 S r k
x 10−4
1 ∂c
=K−1, z =0, (17)
0.8 ∂z
z) 0.6
(r,B0.4
T 0.2 ∂η ∂θ(S) ∂θ(L)
S =k − , (18)
0 ∂t ∂z ∂z
4
3 0.05
2 0.03 0.04 wherethesmallslopeapproximationhasbeeninvokedin
r 1 0.01 0.02 z the expression for the curvature, κ ≈ −∆ η, where η is
0 0 r
the perturbation of the planar position, and
FIG. 2: Plot of the basic temperature profile in the melt
(Eq. (8)) for a particle of radius a = 10µm, dimensionless ∂U 1−α ∞ r2−2(nH )2
0
gap thickness h∞/a = 0.05 and thermal conductance ratios F(r)= ∂z (r,0)=2GL(2+α) [r2+(nH )2]5/2.
α = 0 (left) and α = 10 (right). Note the reversal in the nX=1 0
temperature gradient in thegap separating theparticle from (19)
the interface for the insulating particle case. The profile is The far field conditions for the perturbations are zero
evaluated using parameter values for a succinonitrile-acetone concentration and zero thermal gradients in the liquid
system (SCN-ACE) [15, 16]. and solid phases. The stability problem is solved using
the finite Hankel transform over the range [0,ℓ] [17]. On
using the approximation F(r)∼F(0) as r →0, we find
2ζ(3)(α−1) ℓ(1−K)
θˆ(L) = −[G +σω2+ G ]ηˆ+M J (ωℓ)−λ(L)L(ω) e−ωz+λ(L)L(ω), (20)
h L (2+α)H3 L ω2 1 i
0
2ζ(3)(α−1) ℓ(1−K)
θˆ(S) = −[G +σω2+ G ]ηˆ+M J (ωℓ)−λ(S)L(ω) eωz +λ(S)L(ω), (21)
h S k(2+α)H3 L ω2 1 i
0
ℓG J (ωℓ) 1 ℓ
L 1
L(ω)= + F(r)rJ (ωr)dr, (22)
ω3 ω2 Z 0
0
where the hat symbol indicates transformed variables, the perturbation ηˆis obtained. Its solution yields,
ω is the wavenumber, J is the Bessel function of the
m
first kind of order m and ζ stands for the zeta function, ηˆ(t)= g(ω) +[ηˆ(0)− g(ω)]e−Λ(ω)t/S, (23)
i.e. ζ(3)= ∞ n−3. Λ(ω) Λ(ω)
n=1
P
where
On imposing the heat balance equation at the inter- Λ(ω)= ω G+(1+k)σω2+ 4ζ(3)(α−1)GL , (24)
face, Eq. (18), an evolution equation for the growth of Sh (2+α)H3 i
0
4
0.2 3.5
m ) 3 (α = 0.)
0 µ
s ( 2.5
s
e
) n
ω k 2
g( p thic 1.5 (α = 0.2)
a
al g 1 (α = 0.3)
c
Criti 0.5 (α = 0.4)
−10 0.75 ω 1.5 2 0.01 0.0T1h5erma0l. 0g2radie0n.t0 2( 5K / µm0.0)3 0.035
2.6
m )
µs ( 2.2 ( a = 8 µm )
s
0 ne
ω) ck1.8
( hi
g p t
ga1.4 ( a = 5 µm )
al
c
Criti 1
5 ω 10 12 0.01 0T.0h1e5rmal0 g.0r2adien0.t0 2(5 K / µ0m.0 3) 0.035
FIG.3: Thesmallwavenumberbehavior(left)andthelarge FIG. 4: (left) Plot of the dimensional critical gap thickness,
wavenumber behavior (right) of the function g(ω) for a sys- (dc−1)a, as function of the dimensional thermal gradient in
tem consisting of SCN-ACE containing silicon-carbide (SiC) the melt, G(L), in unit of Kelvin per micrometer for selected
particles. The numerical values of the physical constants are valuesofthethermal conductanceratio α,(right) plot ofthe
[15, 16]: liquidus slope m = −2.8K/wt%, segregation coef- critical gap thickness for 2 insulating particles of radii 5µm
ficient K = 1, C∞ = 1.3Wt%, melting point Tm = 328K, and10µmasfunctionoftheimposedthermalgradientinthe
coefficients of thermal conductivity k(P) = 85Wm−1K−1, melt G(L).
k(S) = 0.225Wm−1K−1 and k(L) = 0.223Wm−1K−1 for
the particle, solid and liquid, respectively; thermal gradi-
ents G(S) = 8000Km−1 and G(L) = 10800Km−1 in the
solid and liquid, repsectively; σsl = 0.009Jm−2; H0 and ℓ Equation(26) describes the instability criterioninterms
havebeenarbitrarilychosentoequal1.1and10,respectively; of dimensionless terms. It implies that for prescribed
L=4.6×107Jm−3. thermalgradientsin the liquidandsolid,and fora given
thermalconductanceratioα,instabilityofthe planarin-
terface sets in whenever the dimensionless distance from
the particle’s center to the interface, measured by H ,
0
equals d . From the physical standpoint, the instabil-
J (ωℓ) λωL(ω) c
g(ω)=M(1+k)ℓ(1−K) 1 − , (25) ity is manifested only for parameter values for which d
c
ωS S
exceeds unity and the expression (1−α)G is positive.
L
G =kG +G is the conductivity weightedthermalgra- Figure 4 depicts the range of parameters for which the
S L
dient, and λ =λ(S)+λ(L). The conditions for marginal instability is observable for a system consisting of SCN-
stability are determined by setting the growth rate to ACEthatisimmersedwithparticlesofdifferentsizesand
zero, i.e. Λ(ω) = 0. However, when Λ = 0, the term thermalconductivities. The plotofthe dimensionalcrit-
g(ω)/Λ(ω)inEq. (23)isundefined. Therefore,nomean- ical gap thickness, h∞ = (dc −1)a, as a function of the
ing to the dispersion relation, Λ(ω) = 0, could exsit if thermal gradient in the melt, G(L), for particles of dif-
ω did not satisfy the auxiliary equation g(ω) = 0. The ferent thermal conductivities show that the critical gap
ratio,g′(ω)/Λ′(ω),where′=d/dω ishoweverfinite. The thickness (i) increases with G(L), (ii) decreases with α,
functiong(ω)issingularatω =0andhasinfinitelymany and (iii) particles characterized by high values of α re-
zeros ω , j = 1,2,3,..., on the postive ω−axis (Fig. 3). quire high thermal gradients for the instability to set in.
j
We set ω = ω in the dispersion relation Λ(ω) = 0 and For increasing particle’s size, the critical gap thickness
j
solve for H . The critical value of the particle-interface increases as α and G(L) remain constant. Furthermore,
0
separation,d , is the largestvalue of H and is obtained wenotethattheonsetoftheinstability,beingofthermal
c 0
by setting ω equal to the smallest root ω . We find origin,does not depend explicitely on the concentration.
j 1
However,the magnitude ofd is affectedby the morpho-
c
4ζ(3)(1−α)GL 1/3 logical number due to the dependence of the auxiliary
d = . (26)
c h(2+α)[G+(1+k)σω12]i function g(ω), and thus of the root ω1, on M. Depend-
5
ingonthephysicalprocessunderconsideration,thepres- gravityconditions (see also [19]). These experiments de-
ence orabsence of this instability canbe imposed by the pict in situ the real time evolution of the solid-liquid in-
selection of an appropriate set of physical parameters as terface shape profile in the vicinity of a single particle.
dictatedbyEq. (26)andshowninFig. 4. Theinstability The sequence of photographs depicting the interaction
put forth in this paper also demonstrates the influence a of polystyrene particles in SCN with an initially planar
tiny impurity in the melt can have on the morphological solid-liquid interface, shown on page 101 of Ref. [18],
stability of the interface. provides qualitative confirmation of the features of the
Ahuja [18] has carriedout a detailed experimentalstudy instability put forth in this paper. The quantitative val-
of the influence of foreign particles on the morphology idationofourpredictionrequirescarefully controlledex-
of a slowly growing solid-liquid interface under normal periments in a microgravity environment.
[1] K.A. Jackson and B. Chalmers, J. Appl. Phys. 29, 1178 [12] Typically, the size of the inclusions vary between 5µm
(1958). and 100µm, and D ≈ 10−9m2s−1. We have consid-
[2] H.IshiguroandB.Rubinsky,Cryobiology31,483(1994). ered here very low growth rates so that the onset of the
[3] G. Gay and M.A. Azouni, Crystal Growth & Design 2, Mullins-Sekerkainstability isremoteandtheparticlere-
135 (2002). mainspushedbytheinterface.Thus,forV ≈10−9ms−1,
[4] A. Endo, H.S. Chauhan, T. Egi and Y. Shiohara, J. we have ǫ ≈ 10−5. We assume that the perturbation to
Mater. Sci. 11, 795 (2002). the interface profile is of order (ǫ). The case of a pure
[5] A.R. Kennedy and T.W. Clyne, Cast Metals 4, 160 substancewithzerogrowthratehasbeenanalyzedbyL.
(1991). Hadji, [Scripta Materialia 48, 665 (2003)].
[6] D.R.Uhlmann,B.Chalmers and K.A.Jackson, J. Appl. [13] H.S. Carslaw and J.C. Jaeger, Conduction of Heat in
Phys. 35, 2986 (1964). Solids, second edition (Clarendon Press, Oxford, 1959).
[7] A.M. Zubko, V.G. Lobanov and V.V. Nikonova, Sov. [14] H. Cheng and L. Greengard, SIAM J. Appl. Math,, 58,
Phys. Crystallgr. 18, 239 (1973). pp. 122-141 (1998).
[8] A.A. Chernov, D.E. Temkin and A.M. Mel’kinova, Sov. [15] W. Kurz and D.J. Fisher, Fundamentals of Solidifica-
Phys. Crystallogr. 22 656-658 (1977). tion, third edition (Trans. Tech. Publications, Switzer-
[9] S. Sen, W.F. Faukler, P. Curreri and D.M. Stefanescu, land, 1992).
Met. Trans. A 28, 2129 (1997). [16] D. M. Stefanescu, R.V. Phalnikar, H. Pang, S. Ahuja,
[10] L. Hadji, Phys. Rev. E 64, 051502 (2001). and B. K. Dhindaw,SIJ Int. 35, 300 (1995).
[11] The directional solidification of an alloy is subject to [17] A.D. Poularikas, The Transforms and Applications
a morphological Mullins-Sekerka instability when the Handbook, second edition (CRC Press, Boca Raton,
growthrateV exceedssomecriticalvalue(W.W.Mullins Florida, 1999). The interaction of a single particle with
and R.F. Sekerka, [J. Appl. Phys. 34, 323 (1963)]). the interface sets up a perturbation in the interface due
Theplanarinterfaceundergoesabifurcationtoacellular to the difference in thermal conductivities (α 6= 1). In
statewhosewavelengthisdeterminedfromacompetition case the instability sets in, this disturbance travels radi-
between the destabilizing diffusion length scale and the ally in all directions, decreasing in magnitude as it goes.
stabilizing capillary length scale. The wavelength varies Far away from theparticle, at r=ℓ,theparticle’s effect
roughlybetween10and100µm.(I.Durand,K.Kassner, vanishes. In our calculations, we have arbitrarily chosen
C. Misbah and H. Mu¨ller-Krumbhaar, [Phys. Rev. Lett. ℓ=10.
76, 3013 (1996)]). This wavelength is of the same order [18] S. Ahuja, Ph.D. Dissertation, The University of Al-
of magnitude as the size of the inclusions that are used abama, p. 100 (1992).
in PMMC and found in other processes. In this paper, [19] J.A.SekharandR.Trivedi,Mater. Sci. Eng. A147,pp.
the mere presence of the particle in the melt provides 9-21 (1991). This paper investigates the morphological
the length scale. Furthermore, the growth rates consid- stabilityofasolidifyinginterfaceinthepresenceoflarge
ered here are so low (ǫ ≪ 1) that the Mullins-Sekerka numberof particles.
instability does not arise.