Interfaces for the 21st Century: New Research Directions in Fluid Mechanics and Materials Science : A Collection of Research Papers Dedicated to Steven H. Davis in Commemoration of hi

, in line with our aim of directly extending conventional phase-field models of solidification to account for convection. This has the advantage that we may treat quasi-incompressible systems [45], in which the density field is taken to be a prescribed function of . We sketch how the model may be derived in the setting of irreversible thermodynamics. The quasi-incompressibility assumption restricts the form of the thermodynamic potentials that may be employed [45]. The model comprises the compressible Navier-Stokes equations with a modified stress tensor that includes additional terms related to gradients of , an energy equation, and a phase-field equation involving a material time derivative of (p. We go on to describe computations based on a simplified form of this phase-field model. In particular, we study a configuration in which a dendrite grows into an undercooled liquid between two thermally insulating plates. This allows us to avoid directly solving the generalized compressible Navier-Stokes equations by adopting a Hele-Shaw approximation. The densities of the solid and liquid phases are allowed to differ, and we study numerically the effect of the density-induced flow on the growth of the dendrite. 2

The Model

We consider a non-isothermal system consisting of a pure material that may exist in two distinct phases. We follow the standard phase-field methodology and introduce a phase-field variable, <j>(x,t), whose value indicates the thermodynamic phase of

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D. M. Anderson,

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the system as a function of position, x, and time, t. A solid-liquid interface is represented by a thin layer in which the phase field varies rapidly between zero (liquid) and unity (solid). The governing equations are derived by following the formalism of irreversible thermodynamics [37,46-48] as described below. 2.1

Governing Equations

We assume that the total entropy, S, in a material volume, Q{t), of the system is given by Jn< /fi(t)

1 PS--4T\V4>)

dV,

(4)

where p is the density and s is the entropy per unit mass. The first term in the integrand, ps, is the classical entropy density per unit volume and the second is a nonclassical term associated with spatial gradients of the phase field. Here, the gradient-entropy coefficient es is assumed to be a constant for simplicity, and F is a homogeneous function of degree unity. The function T allows for a general anisotropic surface energy of the solid-liquid interface and allows the Cahn-Hoffman ^-vector formalism for sharp interfaces [49, 50] to be generalized and extended to diffuse-interface models [44,51]. An isotropic surface energy results from the choice

i W ) = |V0|. The total mass, A4, linear momentum, V, and internal energy, £, associated with the material volume are assumed to have the forms: M = / V=

/

pdV,

(5)

pudV,

(6)

JQ(t)

£=

pe+\P\u\^l-elT\V4>)

dV,

(7)

/n(t)

respectively. Here u is the velocity, e is the internal energy density per unit mass, and €E is the gradient energy coefficient, which is assumed to be constant. The thermodynamic relations

de = T ds + 4 dp + 4l # . (r

(8)

ocp

e = Ts-p/p

+ ti,

(9)

are assumed to apply locally, where p is the thermodynamic pressure and p, is the chemical potential (or Gibbs free energy per unit mass). The physical balance laws for mass, linear momentum, and internal energy are given by

dp

f

— = I "*

J6U(t)

n- mdA,

(11)

A Phase-Field

~y~ + / «*

Model with Convection:

qE-ndA=

J6Q(t)

I

Numerical Simulations

n-m-udA,

135

(12)

JgQ(t)

respectively, where h is the outward unit normal to SQ(t), m is the stress tensor, and qE is the internal energy flux. The momentum balance (11) requires that the rate of change of the total momentum of the material volume results from forces acting on its boundary 6£l(t) (for simplicity we neglect body forces such as gravity; their inclusion is straightforward). The energy balance (12) equates the rate of change of the total internal energy of Q(t) plus the energy flux through its boundary to the rate of work of the forces at its boundary. In addition, the entropy balance takes the form —-+

/ qs-ndA= Jsn(t)

at

sproddV,

(13)

Jn{t)

where qs is the entropy flux and sprod is the local rate of entropy production. The second law of thermodynamics is then expressed by the requirement that sprod is non- negative. To proceed, we recast the conservation laws (10-13) as differential equations. These are used to express the local entropy production in terms of the fluxes m, (JE, and qs, as well as Dfi/Dt. We then identify forms for these quantities that ensure that the local entropy production is non-negative. The fluxes that result from this procedure involve both classical contributions and non-classical contributions that depend on V. In addition, we obtain an evolution equation for the phase field. The details of this procedure are given in Ref. [1] and result in the following governing equations: Dp = - p V • u, Dt p

~bl

= v m

l)t

= e2AT)V

M

p

(14) (15)

' > ' [ r ( V ^ "Pc|'

De — = V • [fcVT] + eEV • I W ) £ Dt

(16)

D4> -£ + Dt

ms:Vu,

(17)

where 1/M is a mobility, m is the stress tensor [see Eq. (24)], ms is a modified stress tensor [see Eq. (25)], ep is the Helmholtz gradient energy coefficient given by eF(T) = e\ + Tes, g(T,p,d>) = e — Ts + p/p is the Gibbs free energy per unit mass, and £ is the generalized ^-vector [44] whose components are defined by £j = dT(p)/dpj, where we have written p = V(f>. The density of the two bulk phases may be different, and we will assume that p depends on 4> alone, p((t>) = pSr ()+ PL [ 1 - r (
(18)

where r() is a monotonic increasing function with r(0) = 0 and r(l) = 1; suitable choices include r((f>) = <j> or r(<j>) = <^>2(3-2). This assumption, in which the density does not depend on temperature or pressure, is known as quasi-incompressibility, as it still allows a nonzero divergence of the velocity vector. This assumption places

136

D. M. Anderson,

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Wheeler

a constraint on the form of the underlying thermodynamic potentials [45] that requires the underlying Gibbs free energy (per unit mass) to have the form (P - Po) g(T,p,) =go(T, ) + •

P() where po is a reference pressure. Here, we assume that the function go(T,t the form e0 -

cTM

r{4>)L

cT In

T TMJ

' 4a"

<7o(T,<

1

1 4as

H{4>)

1

T

(19) has (20) (21)

H(),

in which case the corresponding expressions for the internal energy and entropy densities are 1 PO (22) e = e0 + c(T - TM) r^L+^-H^) 1

e0 - r{4>)L -

TM

Aas

H{4>) + cln

(23)

TM r

where l/as — 1/a—l/as- Here, 1/a is the height of the double well of the Gibbs free energy density at T = TM, and 1/CLE and l/as are the heights of the double wells in the internal energy and entropy densities, respectively. The double-well potential H{4>) is a prescribed function of <j> (see [56]). The quantity eo is a constant reference energy and both the heat capacity per unit mass c and the latent heat per unit mass L are assumed to be constant. The temperature TM is the melting-point temperature at the reference pressure po. The tensors m and ms are given by m =

ms

AT)

-p +

r 2 (v
4 (T)r(V<£)£ ® V + r ,

-P + ^ r 2 ( V 0 ) I - T e | r ( V ^ ) ^ V^ + r ,

(24) (25)

where r is the viscous stress tensor, T =H

Vu+(Vu)T

-(V-u)I

(26)

and fi is the viscosity, which is a function of <j>, (27)

K) = M S K 0 ) + ML[1 • r ( 0 ) ] .

By examining the isothermal one-dimensional solution of the governing equations at the melting temperature TM with ps — PL, it may be shown that the surface tension 7, interface thickness I, and interface attachment coefficient p,Q are related to the phase-field parameters by 7(n) = ^ r ( n ) 7 f ,

l(n)

eF{TM)T{n)J—, V PL

/x0(n)

6pLLlT(n) TMM (28)

A Phase-Field

Model with Convection:

Numerical Simulations

137

We will henceforth confine our attention to the case of isotropic surface energies and set r(V) = |V^|; in this case we note that T(n) = 1. It is also convenient to define an associated capillary length by lc — TMJ/(PL[L2/C]). 2.2

Dimensionless Governing Equations

We non-dimensionalize the governing equations by introducing the following dimensionless variables, which we denote with a prime: x = lox",

t = —t',

u

V

m

L

T = TM +

n>

m ,

I'02

fj, = pLp',

PLP

p = po +

P

~f P,

(29)

(30)

k = kLk'.

Here, the reference length scale IQ is a typical length scale associated with the interface shape, such as a dendrite tip radius, the reference time scale is 12/KL, and the reference velocity scale is U = KL/IO, where KL is the thermal diffusivity in the bulk liquid phase. The dimensionless governing equations are dp + V • (pu) = 0, dt Du

(31)

„

p - ^ = V-m, 2 e

M ^ = e 2 ( l + a0)V2<; Dt

(32)

1

£)

-(i + Wfl'W + W\4>) + Ip-zz D<j>_

De

P^=V-[W)W] + eV^V 2

,(33)

(34)

•K

where, for simplicity, we have omitted the primes on the dimensionless variables. The dimensionless internal energy density is given by e

= Q-r^)

+

iH{<j>)-^--, 2 A7 p

(35)

and the dimensionless stress tensors are m = crp + (1 + aO)^

+ Prr,

(36)

ms =
(37) UL

is the kinematic (38) (39)

r = p{4>) Vu + {Vu)T - - ( V • u) I

(40)

138 D. M. Anderson,

G. B. McFadden & A . A . Wheeler

The source term in the energy equation is m

H = Y

• Vw = j j - [- (p + p*) I + a (9 + S'1) a4, + Prr] : W ,

s

(41)

and the dimensionless parameters are given by Le

aL

s

Q =

0 = P

2

CC F(TMY

e=

8=— 2aEV

ascTM'

f.

A

=^> P* = - ^ . 5 = 4 - ,

to „

=

6l0lcL2 2

K LCTM

u=-^~ = ^ PLpL 2aLe2F(TMy

6ZC =

6lo2_

pLKl jtf =

(43)

cTM

*LMTMC 2

PLI^V

(42) ^Z>

£>PLIICL

=

[KL/IC]

(U)

[L/C]H0'

and Q(
- 1) r # ) ,

M) = l + ^ M _ i j r(),

p{4>) = 1 + ^ - 1 ^ r(0).

(45)

We note that the parameter 7 is related to the capillary (or crispation) number Ca by 7 = 6Pr/Ca, where Ca = PL^L/(IOJ)In the absence of flow these equations reduce to the generalized phase-field equations recently studied in Refs. [52] and [53], The leading-order, free boundary problem that emerges from a sharp-interface limit of these equations depends on the distinguished limit that is taken [52,54,55]. The limit in which A = 0(1) as e —> 0 corresponds to the so-called 'thin interface' limit studied by Karma and Rappell [55]. In this analysis when the thermal conductivities of the solid and liquid phases are unequal, the leading-order temperature is discontinuous across the interface and the leading-order modified Gibbs-Thomson equation contains terms dependent on the interfacial temperature gradients. However, if A = 0(e) as e —> 0, the socalled 'classical' limit, the temperature is continuous across the interface at leading order and a nonlinear form of the modified Gibbs-Thomson equation is obtained at leading order. However, if we formally set the coefficients a, (3, S, and v to zero, thereby omitting the nonstandard terms in the generalized phase-field equations, then the classical sharp-interface analysis recovers, at leading order, a standard free boundary problem in which the interfacial temperature is continuous and the conventional modified Gibbs-Thomson equation is obtained. Here, we study a simplified form of our model by setting the constants a, j3, 6, and u to zero, and we neglect the source term H in the energy equation. The dimensionless governing equations of the simplified model are d

JL + V • (pu) = 0,

(46)

Du p

Wt=v'm'

(47)

A Phase-Field

2 £

M^=e2V20-p

Model with Convection:

I^)

+wW +

Numerical Simulations

ipAQ)

139

(48)

/>^ = V-[W)V0],

(49)

where the dimensionless stress tensor is m = crp + a't' + PrT.

(50)

We have recently examined the full system of governing equations, including flow, in the sharp-interface limit [56]. Our investigations reveal that, in the classical sharp-interface limit, the boundary conditions at a sharp interface in equilibrium comprise the normal stress balance including surface tension and the ClausiusClayperon equation all under isothermal conditions. In the nonequilibrium case we find hydrodynamic conditions on the normal and tangential velocities representing the conservation of mass and the no-slip condition. Jumps conditions on the normal and tangential stresses are also obtained. The temperature is found to be continuous across the interface while the jump in heat flux across the interface is modified by nonequilibrium effects. The temperature of the interface is found to obey a nonequilibrium version of the Clausius-Clapeyron relation. 3

Model Computations

We now describe computations, based upon the phase-field model given by Eqs. (46), (47), (48), (49), and (50) that represent the density change flow associated with the growth of a dendrite from an undercooled melt. To proceed we make a number of additional approximations in order to develop a simplified phase-field model that captures the qualitative features of this situation. First, we consider the dendrite to be two-dimensional and growing in a uniform thin gap of width d between two thermally insulated flat plates. This allows us to ignore the effects of inertia and to model the flow using a Hele-Shaw approximation. The momentum equation may then be written as + ^ V 0 | 2 ) - eV • [V ® V] + ^ V •

V (-±p

{LI(
= 0.

(51)

In the absence of flow it is known that it is essential to include surface energy anisotropy in order to compute dendritic structures using a phase-field model [14]. We will accordingly retain anisotropic surface energy terms in the phase-field equation alone. Specifically, an isotropic surface energy term is used in the momentum equation (51), while the phase-field equation (48) is modified to allow for anisotropic surface energy by using the Calm-Hoffman ^-vector formalism e

'

M

^

= e V

'

' [ r ( V ) ^ ~~ P \\H'{(t,)

+

X9r {
'

(52)

so that the direct effect of anisotropy is upon the interfacial surface energy rather than the flow. We note we have also omitted the pressure dependence in the freeenergy term in the phase-field equation. This is a reasonable approximation for

140

D. M. Anderson,

G. B. McFadden & A. A.

Wheeler

density-driven flows, as evidenced by the insignificant variations of the melting temperature due to pressure fluctuations in the Clausius-Clayperon relation under these conditions. In order to simplify the system further we make the approximation V • (V(/> (8> V) w V(|V|2). This approximation is exact in one space dimension, but not higher dimensions. We justify it by observing that the phase-field variable only changes in the thin interfacial regions where it depends primarily on the perpendicular distance through the layer and hence is approximately one dimensional. Using this approximation the momentum equation (51) becomes - V p + y V . W ) T ] = 0,

(53)

P=4p+||V0|2.

(54)

where 7 * We now integrate these equations across the narrow gap of width d « 1 [58] to obtain -3d2 /z\ (d Ca fi((f>)

© ffl *•

<«>

where here and below the operator V acts in the plane parallel to the thin gap. We now apply the continuity equation (46) to find that p satisfies 2_

2Ca

f.y(4>)d4>

p'() V0 • Vp. _ p-{4>) P{4>) . P'(4>)

(56)

In our numerical computations, we solved the energy equation (49), phasefield equation (52), and pressure equation (56) using VLUGR [57]. This freelydistributed package is designed to solve systems of parabolic partial differential equations in which the solution exhibits regions in space with large gradients. It employs a finite-difference discretization allied to local grid refinement and a variable time step integration of the underlying discretized equations. Computations were conducted on the rectangular domain [0,X] x [0, Y]. Neumann boundary conditions were employed on and 6 on all four sides. However, for the pressure, Neumann boundary conditions were only invoked on the sides x = 0 and y = 0, with Dirichlet boundary conditions on the other two sides. The initial condition represented a small circular solid region centered on the origin in a uniformly undercooled melt with dimensionless temperature T = C(TM — TQ)/L, where To is the initial dimensional temperature. The governing equations were solved with p(<j>) given by Eq. (45), r'{(f)) — 30^ 2 (1 — (f>)2, and H(<j>) = (p2(l — 4>)2. The surface energy had a four-fold anisotropy with T(n) = 1 + 0.015 cos(49), where n = V<£/|V<^>| is a unit vector in the (x,y) plane and 6 is the angle between n and the x-axis. The values of the dimensionless parameters used in the computations are given by Ca = 30, ps/l^L = 1, ^sl^L — 1, A = 7.5, and M = 10. In Fig. 1, we display the results of a typical computation in which PS/PL = 0.9, X = 1, and Y = 3. This figure shows the pressure field, the velocity field and

A Phase-Field

Model with Convection:

Numerical Simulations

141

Pressure

0.8

0.7

0.6

0.4

0.3

0.2

0.1

Figure 1. The phase field, pressure field, and velocity field for a computation at t = 0.3 with PS I PL = 0-9 on a 1 x 3 domain with four-fold anisotropy. The grayscale indicates the liquid ((/> = 0) and solid {<j> = 1) regions, the solid curves are the isobars, and the small arrows represent the velocity field.

the phase field at time t = 0.3. The solid curves are isobars, the arrows represent the local velocity, and the shading indicates the phase field. The x- and y-axes represent planes of symmetry in the calculation, but since X ^ Y the resulting shape has two-fold, not four-fold symmetry due to the presence of the sidewalls. The dendrite growing in the x-direction has a blunter tip than the dendrite growing in the y-direction due to its closer proximity to the sidewall, which has a significant effect on the growth dynamics at this stage. Since the density of the solid is less than that of the liquid, a given amount of material will expand upon solidification, which drives a flow away from the interface into the melt. For a sharp-interface model, the conservation of mass boundary condition takes the form Un

^n

^ - 1 PL

(57)

142

D. M. Anderson,

G. B. McFadden & A. A.

Wheeler

t

| —

^^-

11

Pressure | " ; . : > 0.9

• o.8

•0.7

- 0.6

-0.5

- 0.4

0.3

-0.2

-0.1

Jo 0

0.5

1

1.5

2

Figure 2. The phase field, pressure field, and velocity field at t = 0.75 for a computation with Ps I PL = 1.1 on a 2 x 4 domain with four-fold anisotropy. The grayscale indicates the liquid ( = 0) and solid (

where un = ft • u is the normal component of the fluid velocity at the interface, and vn is the normal velocity of the interface. Thus, for solidification with vn > 0, the flow is away from the interface (un > 0) for ps < PL, and is toward the interface (un < 0) for ps > PL- The computation shows that the flow is greatest in the vicinity of the tip of the dendrite. For this calculation with equal viscosities in the solid and liquid phases, there is also a significant flow in the solid region. This artifact is reduced for computations with ^SIP-L 3> 1; here we are illustrating an extreme example of this effect. Fig. 2 shows a similar situation, but with the density in the solid greater than that of the liquid, PS/PL = 1-1, with X — 2 and Y = 4 at a time t = 0.75. In this case, the advection is toward the interface, as expected. Between the two dendrite tips is a narrow liquid intrusion where little solidification is taking place; the flow velocities that are induced by the density change upon solidification are correspondingly small in this region.

A Phase-Field

4

Model with Convection:

Numerical Simulations

143

Conclusions

In this paper we have shown that computations based on a simplified form of a recent phase-field model that includes convection [1] exhibits numerical solutions that show the expected physical behavior. In particular, we considered the growth of a dendrite in a thin gap between two thermally insulated plates and allowed the density of the solid and liquid phases to be different. We found that the flow was directed towards or away from the dendrite depending on whether the density of the solid phase was greater or less than that of the liquid phase, respectively. Our model for solidification with convection is derived using the formalism of irreversible thermodynamics, and allows the systematic incorporation of a consistent thermodynamic description of the two-phase system. It allows a unified treatment of both equilibrium and non-equilibrium effects in a single set of governing equations. Sharp-interface limits of the diffuse-interface description then lead to boundary conditions of the solid-liquid interface that recover the usual conditions at equilibrium, and provide thermodynamically-consistent generalizations of these conditions under non-equilibrium conditions [56]. The detailed nature of the nonequilibrium conditions at the interface can be sensitive to the specific forms that are assumed to describe the variation of the thermophysical parameters through the interfacial region; for example, the non-equilibrium solute-trapping behavior of a diffuse-interface model of a binary alloy depends quantitatively on the exact form that is assumed for the variation of solute diffusivity, D(), near the interface [59]. In this model, the solid is treated as a liquid with high viscosity. This allows residual convection to occur in the solid, with a magnitude that is determined by the viscosity ratio. The consideration of extreme viscosity ratios tends to eliminate velocity gradients in the solid, but allows states of uniform convection that correspond to rigid-body motion. This is an attractive feature for dealing with such issues as fragmentation and subsequent motion of sidearms through the melt. Such transported fragments can serve as sites for the growth of independent grains when the fragments are incorporated into the growing phase, which is a problem of considerable technological importance. It is thus beneficial to have a model that allows for both topological changes in the interface as well as possible rigid-body motion of the solid phase. 5

Acknowledgments

The authors dedicate this work to Professor S. H. Davis in honor of his sixtieth birthday. We would like to express our most sincere gratitude to Steve for the wide range of contributions he has made to our careers as teacher, colleague, and friend. The authors are also grateful for helpful discussions with W. J. Boettinger, R. J. Braun, T. J. Burns, S. R. Coriell, B. T. Murray, and R. F. Sekerka during the preparation of this manuscript. This research was conducted with the support of the Physical Sciences Research Division of NASA.

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References 1. D. M. Anderson, G. B. McFadden, and A. A. Wheeler, Physica D 135, 175 (2000). 2. S. D. Poisson, Paris: Bachelier (1831). 3. J. W. Gibbs, Scientific Papers of J. Willard Gibbs, ed. H. A. Bumstead and R. G. Van Name, Longmans, Green, and Co., London, UK, pp. 55-371 (1928). 4. J. C. Maxwell, in Scientific Papers of James Clerk Maxwell, Vol. 2, Dover, New York, pp. 541-591 (1952). 5. Lord Rayleigh, Phil. Mag. 33, 209 (1892). 6. J. D. van der Waals, Verhandel. Konink. Akad. Weten. Amsterdam, Sec. 1, Vol. 1, No. 8 (1893); Transl. from Dutch: J. S. Rowlinson, J. Stat. Phys. 20, 197 (1979). 7. D. J. Korteweg, Arch. Need. Sci. Exactes Nat. Ser. II 6, 1 (1901). 8. J. S. Rowlinson and B. Widom, Molecular Theory of Capillarity, Clarendon, Oxford, UK (1989). 9. S. H. Davis, J. Appl. Mech. 50, 977 (1983). 10. E. B. Dussan V. and S. H. Davis, J. Fluid Mech. 65, 71 (1974). 11. D. Jacqmin, J. Comput. Phys. 155, 96 (1999). 12. D. Jacqmin, J. Fluid Mech. 402, 57 (2000). 13. P. Seppecher, Int. J. Engng. Sci. 34, 977 (1996). 14. R. Kobayashi, Physica D 63, 410 (1993). 15. A. A. Wheeler, B. T. Murray, and R. J. Schaefer, Physica D 66, 243 (1993). 16. J. A. Warren and W. J. Boettinger, Acta. Metall. Mater. 43, 689 (1995). 17. S.-L. Wang and R. F. Sekerka, Phys. Rev. E 53, 3760 (1996). 18. A. Karma and W. J. Rappel, Phys. Rev. Lett. 77, 4050 (1996). 19. A. Karma and W. J. Rappel, Phys. Rev. E 57, 4323 (1998). 20. N. Provatas, N. Goldenfeld, and J. A. Dantzig, Phys. Rev. Lett. 80, 3308 (1998). 21. N. Provatas, N. Goldenfeld, and J. A. Dantzig, J. Comput. Phys. 148, 265 (1999). 22. V. L. Ginzburg and L. D. Landau, Soviet Phys. JETP 20, 1064 (1950). 23. P. G. de Gennes, Mol. Cryst. Liq. Cryst. 12, 193 (1971). 24. J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 28, 258 (1958). 25. J. W. Cahn, Acta Metall. 9, 795 (1961). 26. J. W. Cahn and S. M. Allen, J. Phys. (Paris) 38, c7 (1977). 27. S. M. Allen and J. W. Cahn, Acta Metall. Mater. 27, 1085 (1979). 28. R. J. Braun, J. W. Cahn, G. B. McFadden, and A. A. Wheeler, Phil. Trans. Roy. Soc. London A 355, 1787 (1997). 29. D. M. Anderson, G. B. McFadden, and A. A. Wheeler, Ann. Rev. Fluid Mech. 30, 139 (1998). 30. J. S. Langer, unpublished notes (1978). 31. J. S. Langer, in Directions in Condensed Matter Physics, ed. G. Grinstein and G. Mazenko, World Scientific, Philadelphia, PA, pp. 165-186 (1986). 32. G. Caginalp, in Applications of Field Theory to Statistical Mechanics, ed. L. Garrido, Springer-Verlag, Berlin, pp. 216-226 (1985).

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33. 34. 35. 36. 37. 38. 39.

40.

41. 42.

43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59.

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Numerical Simulations

145

J. B. Collins and H. Levine, Phys. Rev. B 31, 6119 (1985). G. Caginalp and P. C. Fife, Phys. Rev. B 33, 7792 (1986). G. Caginalp, Arch. Rat. Mech. Anal. 92, 205 (1986). G. Caginalp, Phys. Rev. A 39, 5887 (1989). O. Penrose and P. C. Fife, Physica D 43, 44 (1990). G. Caginalp and J. Jones, Appl. Math. Lett. 4, 97 (1991). G. Caginalp and J. Jones, in On the Evolution of Phase Boundaries, ed. M. E. Gurtin and G. B. McFadden, The IMA Series in Mathematics and Its Applications, Vol. 43, Springer-Verlag, New York, pp. 27-50 (1992). H. J. Diepers, C. Beckermann, and I. Steinbach, in Solidification Processing 1997, ed. J. Beech and H. Jones, Proc. 4th Decennial Int. Conf. on Solid. Process., University of Sheffield, Sheffield, UK, pp. 426-430 (1997). C. Beckermann, H. J. Diepers, I. Steinbach, A. Karma, and X. Tong, J. Cornput. Phys. 154, 468 (1999). R. Tonhardt, Convective Effects on Dendritic Solidification, Ph.D. Thesis, Department of Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden (1998). R. Tonhardt and G. Amberg, J. Cryst. Growth 194, 406 (1998). A. A. Wheeler and G. B. McFadden, Eur. J. Appl. Math. 7, 367 (1996). J. Lowengrub and J. Truskinovsky, Proc. Roy. Soc. Ser. A 454, 2617 (1998). S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, Dover, New York (1984). A. P. Umantsev, J. Chem. Phys. 96, 605 (1992). S.-L. Wang, R. F. Sekerka, A. A. Wheeler, B. T. Murray, S. R. Coriell, R. J. Braun, and G. B. McFadden, Physical) 69, 189 (1993). D. W. Hoffman and J. W. Cahn, Surface Science 31, 368 (1972). J. W. Cahn and D. W. Hoffman, Acta Metall. 22, 1205 (1974). A. A. Wheeler and G. B. McFadden, Proc. Roy. Soc. Lond. A 453, 1611 (1997). G. B. McFadden, A. A. Wheeler, and D. M. Anderson, Physica D 144, 154 (2000). C. Charach and P. C. Fife, Open Sys. and Infor. Dyn. 5, 99 (1998). G. Caginalp, Phys. Rev. A 39, 5887 (1989). A. Karma and W. J. Rappel, Phys. Rev. E 53, 3017 (1996). D. M. Anderson, G. B. McFadden, and A. A. Wheeler, Physica D 151, 305 (2001). J. G. Blom, R. A. Trompert, and J. G. Verwer, ACM Trans. Math. Software 22, 302 (1996). G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, UK (1970). N. A. Ahmad, A. A. Wheeler, W. J. Boettinger, and G. B. McFadden, Phys. Rev. E 58, 3436 (1998).

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P H A S E FIELD MODEL OF M U L T I C O M P O N E N T ALLOY SOLIDIFICATION W I T H H Y D R O D Y N A M I C S ROBERT F. SEKERKA+ AND ZHIQIANG BI Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213-3890, U.S.A. We develop a thermodynamically consistent phase field model for solidification of a multicomponent alloy, including hydrodynamics. The solid is treated as a very viscous fluid. The model is based on an entropy functional that includes gradiententropy corrections for internal-energy density, partial-mass densities, and phase field. It allows for external forces and chemical reactions, and incorporates nonclassical temperature and chemical potentials, defined as functional derivatives. Korteweg stresses related to inhomogeneities in all of these quantities appear in momentum and energy balances.

1

Introduction

The phase field model is based on a diffuse interface model of solidification [1,2] (crystallization from the melt). It was introduced originally [3-6] to study the dynamics of solidification of a pure material. The model incorporates an auxiliary variable, the phase field,
147

148

R. F. Sekerka & Z. Bi

wald ripening and coalescence [53], recoalescence during dendritic solidification [54], and cell to plane front transitions during directional solidification [55]. Phase field models for solidification of eutectic alloys were developed by Karma [56], Wheeler, McFadden, and Boettinger [57], and Elder, Gunton, and Grant [58]. Phase field models that include hydrodynamics have also been formulated. Early versions by Caginalp and Jones [59,60] in which some inviscid hydrodynamic equations were simply added, a finite-element version by Tonhardt and Amberg [61,62], and a hybrid version based on mixture theory by Beckermann et al. [63, 64] incorporated some aspects of fluid flow with phase field models. Gurtin, Polignone, and Vifials [65] have used a phase field model to treat convection in two-phase binary fluids and immiscible fluids. Thermodynamically-consistent diffuse interface models that include hydrodynamics were formulated for a pure material and for a binary alloy by Anderson, McFadden, and Wheeler [66,67]; the reader is referred, to [66] for an excellent review of the literature. Such models include Korteweg stresses [68] that have been related to Noether's theorem [69] in the context of diffuse interface models by Gurtin and Fried [16,17] and Wheeler and McFadden [70]. Subsequently, Anderson, McFadden, and Wheeler [71] developed a phase field model, including convection and anisotropy, for solidification of a pure material and performed numerical computations for dendritic growth [72]. In the present paper, we develop a phase field model for solidification of a multicomponent alloy, including hydrodynamics. The solid is treated as a fluid having high viscosity, as in [71], in order to avoid the more complicated thermodynamics of a crystalline solid [73-76]. This model is based on an entropy functional and includes gradient-entropy corrections in all dependent variables (internal-energy density, partial-mass densities, and phase field). Because of these gradient-entropy corrections, the theory necessarily incorporates non-classical temperature and chemical potentials [40,77-80]. Korteweg stresses related to inhomogeneities in all dependent variables appear, and different contributions of conserved versus non-conserved quantities become evident. In the interest of generality, we include external forces and chemical reactions in order to see how they interact with non-classical temperature and chemical potentials. Our results reduce to those of Anderson, McFadden, and Wheeler for a compatible case [71]. 2

Theory

We treat a multicomponent fluid system of K components. Our model is based on an entropy functional of the form [ sNCd3x, Jv where V is an arbitrary subvolume of the system and S=

SNC = s(u,pi,p2,...pK,
~ - eu\Vu\2 + Y,£i\Vp^\2

(1)

+ e*>|V
(2)

i=l

is a non-classical entropy density. Here, s(u,pi, p2, • • • pK,
Phase Field Model of Multicomponent

Alloy Solidification

149

mass density of component i, ip is the phase-field variable, and eu, e^, and ev are gradient-entropy coefficients. For simplicity, and in order to focus on aspects of a multicomponent system including fluid flow, we assume isotropy and treat the gradient-entropy coefficients as constants. Generalization to the anisotropic case can be accomplished along the lines indicated by Wheeler and McFadden [70] and carried out for a pure material for gradients only in

_ dS_

f

Sprod '- dt+J 'J , A

-hd2x

> 0,

(3)

where A is the bounding area of the fixed but arbitrary volume V, n is the unit outward normal to V, and J s is the total entropy flux due to both diffusive and convective transport. The inequality in Eq. (3) relates to the second law of thermodynamics, which we require to hold in any subvolume of the system as an inequality in order for an irreversible process to occur. Taking the time derivative of Eq. (1) and integrating by parts with the aid of Gauss's theorem, we obtain

Vod = j

(jj

* + S^

" ~ £ (Y)

dAx

K

L

• hd2x,

(4)

1=1

where the dots above u, pi, and ip indicate partial differentiation with respect to time. The functional derivatives NC

-m

ds ou

NC

ds_

dpi

_2 eiV 2 p/,

(5) (6)

define TNC as a non-classical temperature and ^ c as a non-classical chemical potential" of species i, expressed per unit mass. This notation is motivated by the fact that the classical temperature T and the classical chemical potentials /ij are defined by \/T = ds/du and —fii/T = ds/dpi and also because TNC and fi^c are uniform at equilibrium in the absence of external forces, as will be seen subsequently. The functional derivative ds

^_0

(7)

that multiplies

150

R. F. Sekerka & Z. Bi

For the total masses Mi, the total momentum P , and the total internal energy U, we assume that pi d3x,

Mi= P = / Jv

(8)

pvd3x,

(9)

ud x,

(10)

1 u + -pv2] d x,

(11)

U so that the total energy E

where p = Yli=i Pi *s t n e total density and v is the bary centric velocity. We could include excess quantities in the form of gradient corrections to the integrands in Eqs. (8-11), but we omit this complication in the interest of simplicity and also because it would introduce many new and unknown parameters. The reader is referred to the work of Anderson, McFadden, and Wheeler [66,67] for a treatment of a pure material in which the only excess quantity is a gradient correction to the internal-energy density of the form (1/2)KE\'VP\2, where p is the total density, as well as a treatment [71,72] of a pure material in which the only excess quantities are (anisotropic) gradient entropies and energies of the form —(1/2) e | [r(Vy)] 2 and (1/2) e2E [r(V
Equilibrium

Before treating the dynamics of our system, we examine the conditions for equilibrium for a system contained in a rigid volume Vo having area AQ, a fixed total internal energy U, and fixed total masses Mi of each species. We exclude chemical reactions and external forces in this section, but include them in subsequent sections. For equilibrium, we have the variational condition 0=

S(S-\uUNC

6u +

E

Vi\

NC

6pi + Sv6ip > d?x

i=l u6uVu

+X

iSpiVpi + e^S/pX/ip • hd2a

e

(12)

i=\

where the constants \u and A; are Lagrange multipliers. Since the variations 8u, 6Pi, and Sip are independent within the volume Vo and on the surface Ao, we deduce the equilibrium conditions 1\NC f)

=A U )

(13)

Phase Field Model of Multicomponent

(EL Y)

Alloy Solidification

151

= A.

(14)

Sv = 0,

(15)

as well as natural boundary conditions on AQ, such as fixed values of u, Pi, and
equilibrium, no external forces,

(16)

where N =

1 + X,

(17)

X = euVu ® Vu + ^2 ei^Pi ® ^Pi + e v V< £ ® V(/?.

(18)

x

'

i=l

and

i=l

Here, the quantity 1 is the unit tensor and A A denotes a tensor product having Cartesian components Aa and Ap. For the special case of a classical system, N = (p/T)l, where p is the pressure and T is constant, so Eq. (16) becomes Vp = 0. 2.2

Dynamical Balance Laws

We write integral balance laws for species, momentum, and total energy for a fixed but arbitrary subvolume V having area A as follows: —

/ pvd3x+ JV

Pid3x+

piVi-nd2x=

I

/ pvv -hd2x = I &-hd2x+ JA JA

nd3x,

/ V^p,g,d3 ^vi^t

(19)

152

R. F. Sekerka & Z. Bi

— / {u+-pv2)d3x+

{(u+-pv2)v+jh]-nd2x

= / v-&-hd2x+

^pivi-gid3x,

(21) where v* is the velocity of species i, Ti is the rate of production of species i by chemical reaction, gj is the external force per unit mass acting on species i, and & is a stress tensor.6 The quantities ji are diffusive fluxes of the species relative to the local center of mass. Since mass is conserved during chemical reactions, YHt=i r i = 0 and summation of Eq. (19) over all species leads to an overall conservation of mass with the barycentric velocity defined by v := ^ * = 1 ( p i / p ) v j . The quantity j ^ is an energy flux relative to the local center of mass. It is given a subscript h rather than u because in a classical treatment (no gradient corrections to the local densities) it is actually an enthalpy flux given by j / , = q c + X}f=i # t j i i where q c is the heat flux due to conduction and Hi are partial specific enthalpies. 0 The classical internalenergy flux would be j M = q c + Y2i=i Uiiii where Ui are partial specific internal energies. For a pure material, ji = 0, and there is no distinction between j ^ and j u . These integral balance equations can be converted to differential equations in the usual way by taking the time derivatives inside the integrals as partial derivatives, using Gauss's divergence theorem, and noting that the volume is arbitrary. The results, in Eulerian form, are: ^ + V - ( / 9 l v + j I ) = ri)

^) \pv2)

d(u+

di

+ V

.(pvv-
1

+ V'

f t g i

(22)

,

(23)

2^

One can take the dot product of Eq. (23) with v to obtain an equation for the mechanical energy balance that can be used to simplify Eq. (24), leading to the energy balance equation du + V - ( u v + j f t ) = o-:Vv + ^ j i . g i . ~dt

(25)

i=i

The notation for the first term on the right-hand-side of Eq. (25) is defined as

a,/3=l

X0

where a and /? are Cartesian indices. fc The sign convention for a is such that a = —pi, with 1 the unit tensor, for hydrostatic pressure Pc de Groot and Mazur [81] use the notation J 9 for j ^ and call it (page 18) a "heat flow," and then introduce (page 26) an alternative heat flux J'q equal to our q c . Fitts [82] uses the notation j e for jh and q for our q c , which he calls a "second law heat flux." The semantics are very confusing, but there is agreement regarding the relationships among corresponding variables.

Phase Field Model of Multicomponent

2.3

Alloy Solidification

153

Entropy Production

We substitute Eq. (25) for u and Eq. (22) for pi into Eq. (4) and integrate by parts using Gauss's divergence theorem to obtain 6 >
+

+

i[(r(^ &-)-g(?)"'

+

L

J s - euiNu - ^ tipiVpi - e^ipVip

hd2x

•

i=l / 1 \ NC

ft

NC

hd2x.

+/

(27)

This equation can be simplified by using the equations -j

ft

O

(28) <9
j=i

(29) i=l

and

"(?)-&'(f)>^© N

/

w

i=l

that would apply to a homogeneous system. In Eq. (28), T and pi are classical temperature and chemical potentials, respectively, defined as partial derivatives. Eq. (29) is an Euler equation that incorporates the classical pressure p, and Eq. (30) is a Gibbs-Duhem equation that can be obtained by comparing the differential of Eq. (29) with Eq. (30). As in classical non-equilibrium thermodynamics, we assume that Eqs. (28-30) can be applied locally for conditions that are not too far from equilibrium. We could proceed to use Eqs. (28-30) directly to simplify Eq. (27), but it is more expedient and useful for other purposes to transform them to other forms that relate to sNC, (l/T)NC, and (pi/T)NC. This can be done by rewriting them with the differential operator d replaced by the gradient operator V and then adding and subtracting appropriate terms to obtain

£) NC aNC

Vu

E

t=l

-E(f)

/Hi\NC

\~T)

VPl + V^-V-X, P

?

i+

T~

„•>

6u

V""V

u

e

T-r2

i = l iPiV Pi ~ I-*

(31)

154

R. F. Sekerka & Z. Bi

e u |Vu| 2 + J ^ eilVftl 2 + e ^ V y f

(32)

and NC

«v(I) - 5 > v ( f ) " c - W = -v(f) K

K

+ e u uV 2 (Vw) + Y,

£

ift V 2 (VpO - ev(X?2
(33)

i=l

Under equilibrium conditions such that Eqs. (13-15) hold, Eq. (31) becomes simply Eq. (16) obtained from Noether's theorem. Eq. (33) is a non-classical version of the Gibbs-Duhem equation. In Eqs. (32) and (33), we note the asymmetry between the non-classical (e u and ej) terms involving the conserved quantities, u and pi, and the e^ terms containing the nonconserved order parameter ip. This asymmetry persists and is reflected in the structure of the tensor Y (see Eq. (38)) to be defined later. Examination of Eq. (27) shows that parts of the expressions contained in Eqs. (32) and (33) occur there multiplied by v. Simplification of Eq. (27) can be accomplished most easily by reasoning that the total entropy flux will be given by

NC

s ^(~)

NC

i*-E(f)

NC

Du„ 3i + eu — V u

Dt

ft

+

1

DPi Vft + Dt

^Wtv^ (34)

where D

d

D~t

•v,

(35)

is the substantial derivative. This form was motivated in part by noting that it reduces to the classical result K

Js = sv + 7:7 + 2 , SiJi

classical.

(36)

Here, we have used j ^ = q c + Yli=i -^*J* an<^ Hi = Hi ~ TSi, where Si are partial specific entropies. The purely convective term s v is the obvious generalization of sv, and the terms involving D/Dt are generalizations of the non-classical surface terms containing d/dt in Eq. (27). This last generalization was suggested by the work of Anderson, McFadden, and Wheeler [66,71] who used a similar identification in order to obtain an entropy flux that had Galilean invariance in the presence of fluid flow. The physical interpretation of Wang et al. [19] of a non-classical entropy flux e^ifVtp for a rigid material was related to excess entropy carried into the control volume by motion of a diffuse interface with respect to that volume. With fluid flow, such entropy motion with respect to the control volume would require replacement by ev(Dp/Dt)Vif. We substitute Eq. (34) into Eq. (27), convert the remaining surface terms to volume terms by Gauss's theorem, and simplify (which requires considerable algebra)

Phase Field Model of Multicomponent

Alloy Solidification

155

to obtain the entropy production NC

v

+

NC

+ £;

\k-y[~

S,prod

+

XM? ®

tfx *)

NC

NC

Y: Vv

o?x,

(37)

where NC

1

Y :=

NC


f

P + < - - euuV2u - ] P tiPiV2Pi i=i

{

e u |Vu| 2 + j S i | V , 0 i | 2 + ejV¥>| 2

(38)

In the classical limit, this entropy production is in agreement with that calculated by de Groot and Mazur [81], Fitts [82], and Landau and Lifshitz [83]. Note in Eq. (38) the asymmetry in the non-classical terms involving u and pi (which relate to conserved quantities) as opposed to ip which is essentially a non-conserved order parameter. 2-4

Constitutive Relations and Equilibrium

The terms in the first line of Eq. (37) represent the entropy production due to the energy flux j ^ and the mass fluxes j , . Since the mass fluxes obey 52i=i Ji = 0> w e can eliminate one of them, say j K , and write the volumetric entropy production due to fluxes in the form NC

Jh-V

re-1

+ i=l£

T

_ It

-V

JVC

/ ,

\NC

+ (yj

(gi-g«)

(39)

According to Curie's theorem [82], these vector fluxes jo := jh and jj can couple to the vector driving forces f0 := V(1/T) J V C and f; := -V[(/z» - [iK)/T}NC + (l/T)NC(gi — gK). If we assume linear isotropic constitutive relations, then «-i

i = 0 , 1 , 2 , . . . ,K — 1.

(40)

3=0

Here, B is a positive definite matrix with elements Bij, where i,j = 0 , 1 , 2 , . . . , K—1. For the classical case, this matrix has been shown [84] by Onsager's method to be symmetric. At equilibrium, these driving forces must vanish, which requires TNC to be uniform throughout the system, in agreement with Eq. (13), and

c c c wc -v(§r fir ^-v(|f (i) + gK rV + T

(41)

156

R. F. Sekerka & Z. Bi

Further simplification of Eq. (41) can be made after we consider mechanical equilibrium. The terms in the second line of Eq. (37) represent the entropy production due to phase transformation, viscous dissipation, and chemical reaction, and Curie's theorem allows their constitutive laws to be coupled [82]. For simplicity, we shall assume them to be uncoupled. The chemical reaction term is the same as in the classical case except that (pi/T)NC replaces pi/T. It can be further rewritten in terms of progress variables for chemical reactions [81] but we omit details because they are the same as in the classical case. If the phase field is assumed to obey a linear constitutive equation, it evolves according to Dip

-^

= MVSV,

(42)

where Mv is positive. For equilibrium, Sv = 0, in agreement with Eq. (15). This leaves the entropy production due to viscous dissipation, which can be made positive by choosing Y to have the form

^-(fe + S) + ( K -l"'^ V v '

(43)

where p > 0 is the shear viscosity and K > 0 is the bulk viscosity [85], which is frequently ignored. Then

a.0

'dva dxfj

dvp\ dxa)

n

_ 2 3

+

K(V-v)z,

(44)

which is a sum of squares, and therefore positive. In equilibrium, Y = 0 and from the momentum equation (23), V<x =

-^pigi.

(45)

i=l

The quantity Yli=i PiSi —'• f i s the total external force per unit volume. Thus, for equilibrium, the divergence of Eq. (38) leads to NC

v(f)+v. / 'it \

z=l

v

'

V ^ euuV2u + jNiftVV i=l K

K

V ( I ) - euuV\Vu)

- ^ e i P i V 2 ( V p 8 ) + e „ ( V V ) V ^ (46) i=i

Moreover, for equilibrium, Eq. (33) becomes NC

I > v ( ^ ) i=l

= - v ( | ) + e „ U V 2 ( V U ) + E ^ V 2 ( V P l ) - e v ( V V ) V ^ . (47) t=l

Phase Field Model of Multicomponent Alloy Solidification 157

Adding Eqs. (46) and (47), we obtain JVC

(~)

n^NC + ( 7F I Si = 0. T

(48)

In view of Eq. (41), every coefficient of pi in Eq. (48) is the same and can be factored out of the sum; then, since YLl=i Pi = P 7^ 0, we obtain

-V(|)

+^-J

g, = 0

* = 1,2

«.

(49)

Thus, for equilibrium, the individual diffusion driving forces f\ = 0, for all i — 1,2,..., K, even though their conjugate fluxes ji are not independent. This is the generalization of Eq. (14) when external forces are included. Finally, Eq. (46) for equilibrium can be written

(

1 \

NC

K

r)

ftot + euUV2(Vu) + ^ ejPi V 2 (V Pl ) + ~VV,

i J

(50)

where we have used S^ = 0 to eliminate e^V 2 ^. For no external forces, eu = 0 and 6; = 0, T is constant and Eq. (50) reduces to Vp = Tds/dtp = —p(dum/d
Discussion

The driving forces for diffusive transport of energy and mass associated with Eqs. (39) and (40) are f0 = V(1/T) J V C ,

(51) NC

fi = - V [ ( M i - ^)/T}

NC

+ (l/T) (gi

- g K ).

(52)

These resemble the classical driving forces except that the temperature and chemical potentials are now functional derivatives instead of ordinary derivatives. This correspondence becomes even more interesting when the species specific external forces per unit mass, gi are conservative and can be expressed as gradients of a potential Vi(x)> i-e-i Si = ~^i}i- Then Eq. (49) for equilibrium becomes (since TNC is constant) -V(/xf C + Vi) = 0

i = l,2,...,K,

(53)

which integrates to give ^TOT

;= rfc + ipi = constant

i = 1,2,..., K. OT

(54)

Thus, one can define total chemical potentials, nf , that are uniform at equilibrium; they are the sum of the non-classical intrinsic chemical potential and the external potential. The equilibrium condition (54) can also be obtained directly from a variational principle, as in section 2.1, by considering the problem 6[S — \U{U + W) - Y11=\ \pi\ — 0) where the variation in the work is 8W = J ipi 8pi d3x. For a uniform gravitational field, ipi = gz independent of i,

158

R. F. Sekerka

& Z.

Bi

where z is measured vertically upward. In this case, g; — gK = 0 so that gravity does not appear explicitly in the forces f^ given by Eq. (52). It does, however, appear explicitly in the equilibrium conditions expressed by Eq. (54). This paradox is resolved by noting that gravity affects the pressure (see Eq. (50) for equilibrium) on which the p^c depend. For electrical forces due to an external electric field E = — V, we would have ipi = — {zi/m^J7^, where J- = 96500 coulombs/mol is the Faraday constant, Zi is the valence (signed number of electronic charges |e|) for the species i, and mi is the molecular weight of species i. In this case, gj — gK would usually be nonzero for different ions, and there would be a direct effect of external forces on diffusion. These considerations can be used to extend the Nernst equation [86], used extensively in the neurobiological study of membranes, to the realm of phase field models. We can partially isolate the effect of TNC on diffusive transport by rewriting the associated entropy production given by Eq. (39) in the form t , \ JVC

qs y

' {f)

,

+

1

{f)

v NC

K-1

EM- v (^-^) T O r ]>

(55)

where K

NC

qs=h-J2v ii>

(56)

is a heat flux [83]. Then linear constitutive relations, similar to Eq. (40), could be postulated to relate qs and jj to the driving forces fo and — V(/ij — pK)TOT. In the classical limit, q s = q c + TSiji, which relates closely to the diffusive part of the classical entropy flux given by Eq. (36). Note, however, that the chemical potentials in Eq. (56) still depend on temperature. For our multicomponent model, is also possible to identify the surface free energy 7 by considering variations of a system having a planar interface. This is taken up in Appendix B where it is shown (see Eq. (85)) that 7 is the surface excess of the Kramers potential, in agreement with its classical definition. This allows one to obtain a simple general formula (see Eq. (88)) for 7 in terms of integrals over the squares of the gradients of interface profiles. In comparing the present model to other alloy phase field models, it is important to bear in mind that even without anisotropy, the form of the gradient entropies can lead to subtle differences in results. In a binary alloy, for example, the portion of our gradient entropy associated with the partial-mass densities contains e1\VPl\2 + e2\Vp2\2.

(57)

If we express this in terms of the mass fractions ui and w2 we obtain (ei + e 2 )p 2 |Vw 2 | 2 + 2/o(e2w2 - eiWi)Vp • Vw2 + (eiw2 + e 2 o;|)|Vp| 2 .

(58)

If p were a constant, the latter two terms would vanish and we would just have a constant times | Vw 2 | 2 , as we used in [40] (although there in terms of mole fractions). But if p is not a constant, as in the binary-alloy theory of Anderson, McFadden, and Wheeler [66], we can eliminate the cross term in Eq. (58) only by allowing e\

Phase Field Model of Multicomponent

and e2 to be variables. Indeed, if we choose t\ = u2e/p2 Eq. (58) becomes e|V W2 | 2

+

^|Vp| P

2

Alloy Solidification

159

and e2 = cuie/p2, then

,

(59)

which contains a mandatory coefficient of |Vp| 2 . Therefore, one cannot pass easily from one theory to the other unless care is taken about the precise form of the gradient-entropy corrections, including non-constant coefficients. Since phenomenologically there is no way to decide which of the many possibilities is correct or more meaningful (e.g., constant coefficients on a per unit mass basis versus a per unit volume basis, partial densities versus compositions, cross terms or not) one must ultimately appeal to microscopic considerations or to experiments. Acknowledgments The authors are grateful for discussions with G. B. McFadden. This work was done under support from NSF grant DMR9634056. Appendix A We proceed to derive Eqs. (16-18), which are a consequence of Noether's theorem [69] in three spatial dimensions (three independent variables) but for several dependent variables. In the interest of a compact notation, we denote the spatial variables x, y, z by xa, where a = 1,2,3, and the independent variables u, pi, ip by u1, where i = 0,1, • • • K, K + 1. Then the equilibrium problem of Eq. (12) can be written in the schematic form 8 J = 0, where J=

/ I{xa,u\1—). JV

OX

(60) a

In our particular case, K

I = SNC -XuU-^XiPJ, i=\

(61)

and there is no explicit dependence on xa, but we retain the more general notation until we specialize to this case. We employ the summation convention on repeated indices until further notice. The variational problem becomes d 9I dI f dI , u 3 , f x i J2 dul — dxp dul \r- bu a x + / —T r-dung ax. / TT-^ — Q (7 Jv du% dx0 Q (dv>\ JA d (&ui) The Euler-Lagrange equations are therefore

(62)

d{§£)

dl du

l

d

dl

dxg a I diS \

(63)

160 R. F. Sekerka & Z. Bi

Multiplying Eq. (63) by dul/dxa,

summing on i, and simplifying the result by using

dl —

=

)

. 9xa I ,

dl dv} •

dxa

dl

d2u

d

dxadxp'

•

l

du

dxa

(%)

:

(64)

where the partial derivative on the left is with respect to the variable set x, y, z, we obtain dl dxa

( dl \dxa

dul dl dxn« / d \dx0

_8_ dxp

If I is explicitly independent of Eq. (65) vanishes and we obtain

(65) )

the case for Eq. (61), the first term in

^—Nap dxp

(66)

= 0,

where the tensor K+l

Nap — I5ap - 22

dv?

dl

^o^xjg^y

(67)

and we have restored the summation sign for emphasis. Eqs. (66) and (67) can also be written in the form V • N = 0,

(68)

where K+ l

N = Jl-J^A* i=0

dl dAi

(69)

where 1 is the unit tensor, ® denotes the tensor product, and A 1 = V u \ Eqs. (68) and (69) are the desired generalization of Noether's theorem to several dependent variables. For i" given by Eq. (61) we obtain N

sNC - Xuu - ^

XiPi 1 + X,

(70)

i=l

where X = e u Vu ® Vu + ^2 CiVpi <S> Vft + evVy ® Vtp.

(71)

Replacing the Lagrange multipliers by their values given by Eqs. (13) and (14), we obtain Eqs. (16-18) of the text. The tensor

X s " := X - 1 I eu\Vu\2 + ^ £ l | V f t | 2 + ^|V^| 2 J 1

(72)

is traceless and so represents a pure shear, so X itself contains a non-classical hydrostatic pressure.

Phase Field Model of Multicomponent

Alloy Solidification

161

Appendix B In this Appendix, we examine variations of a planar system at equilibrium to identify and obtain a formula for 7, the surface free energy per unit area (surface tension) associated with a diffuse interface. We exclude chemical reactions and external forces and focus on a system with properties that depend on a single spatial variable x, where — L < x < L. We assume that the system is in equilibrium and contains a diffuse interface of width ~ w « L, localized near x = 0. We therefore focus on the limit L —* 00, but retain finite L for intermediate steps. We begin with the entropy functional given by Eq. (1) and with total masses and total internal energy given by Eqs. (8) and (10), but we allow the volume, which we now call V, to vary in a manner such as to preserve the planar nature of the system. These variations include: t y p e 1 The cross section of the system having area A remains unchanged and the positions of its ends, originally at x = ±L, are varied. type 2 The ends remain fixed at x = ±L and A is changed to A + 6A uniformly in every cross section, independent of x. In a variation of type 1, the system can do work by changing its volume; whereas, in a variation of type 2, it can do work by changing its volume and by changing its cross sectional area, A. The equilibrium condition for such variations is K

6{S -\UU-Y,

KMi - XVV - XAA) = 0,

(73)

i=l

where Au, A^, Ay, and A^ are Lagrange multipliers. As we saw in connection with Eq. (12), \ u = (l/T)NC and A; = (p-i/T)NC will be uniform at equilibrium, and Sv = 0, as given by Eqs (13-15). We shall relate Ay to pressure and XA to 7. Under the above equilibrium conditions, 6S = ( I ) N° Jv Sud3x - £

(^)NC

^ 6Pl d3x

(74)

i=l

+ f sNC6Nd2x-

J eu6uVu + ^2 CiSpiVpi + e^SipVf hd x, JA JA »=i where 6N is the distance of variation along the outward normal and where we have factored out the constants Au = (1/T)NC and Aj = (fii/T)NC. For variations of type 1 and type 2, the second surface integral vanishes; for type 1, the ends of the system are so far from the diffuse interface that inhomogeneities in u, pi, and ip are negligible (this becomes rigorous as L —> 00) and for type 2 variations, Vw, Vp», and V

I u6Nd2x,

(75)

JA

and 6Mi = I 6pid3x+ I pt8Nd2x, Jv JA

(76)

162

R. F. Sekerka & Z. Bi

Eq. (74) becomes (77) 8Nd2i v

'

i=i

Eq. (73) therefore becomes JVC

1*°-$)

^%{%r^

SNd2x-XASA

= 0.

(78)

For a variation of type 1, 6A = 0 and Eq. (78) yields 7VC

WC

(79)

= AV. x=±L

From Eq. (32) for a planar system, we obtain NC

d2u dx2

NC

T

2=1

du

v^ i=i

d2p. dx 2

(80)

2

dpi

+£< i=l

• ^

As L —• oo, only the term p/T on the right-hand-side of Eq. (80) survives. We denote this pressure by p^, and noting also that T —» TNC as L —* oo, Eq. (79) for L —> oo yields NC

\v = (4

(81)

Poo-

We now identify the remaining Lagrange multiplier as XA — -ry/TNC, Eq. (73) to be rewritten in the familiar form SU = TNC 8S-Voo8V

+ Yl ^ °

6M

i +7

8A

-

which allows

(82)

Then for a variation of type 2, Eq. (78) for L —> oo yields

c

/:Hr-ip~ *-(r+-(r^ (83) Eq. (83) can be recast into a more familiar form by defining a non-classical Kramers potential per unit volume .NC ..

u-TNCsNC-Y.^CPi=\

(84)

Phase Field Model of Multicomponent

Alloy Solidification

163

Then [coNC - ^c]

7 =

dx,

(85)

where we have used Eqs. (80) and (84) to deduce that w ^ c = UJNG{X —> ±oo) = —Poo. Eq. (85) resembles its classical counterpart, i.e., 7 is the surface excess Kramers potential. A more useful formula for evaluating 7 can be obtained by using the equation for mechanical equilibrium which we can get from Eq. (38) by setting Y — 0, going to one dimension," and noting that axx must be uniform throughout x, and therefore a constant. We obtain JVC

IV rp J

V_- e„,u^^ T "uddx*

°~XX

+

-

1

+

Tx)

(86)

^^iP

^

dpi dx

(

Mt

where the sign reversal on the second line, relative to Eq. (80), arises from the xx component of X. By means of Eqs. (80), (84), and (86), we obtain -.JVC

NC

du dx

+£•

dpi

'dip

dx

^^Tx

(87)

Substitution of Eq. (87) into Eq. (85) then yields 2

7

K

/

,

"°Q-.{£)+p@

= r-

\ 2

rfpiV

J

[dip +€lp

\dx

dx.

Eq. (88) shows that for the present model, 7 is entirely entropic in origin and that the gradient entropies contribute additively. It turns out to be -TNC times twice the surface excess of S. References 1. 2. 3. 4. 5. 6. 7. 8.

J. W. Cahn, Acta. Met. 8, 554 (1960). B. I. Halperin, P. C. Hohenberg, and S.-K. Ma, Phys. Rev. B 10, 139 (1974). J. S. Langer, unpublished notes (August 1978). G. J. Fix, in Free Boundary Problems: Theory and Applications, Vol. II, eds. A. Fasano and M. Primicerio, Pitman, Boston, p. 580 (1983). J. B. Collins and H. Levine, Phys. Rev. B 31, 6119 (1985). J. S. Langer, Directions in Condensed Matter Physics, eds. G. Grinstein and G. Mazenko, World Scientific, Singapore, p. 165 (1986). G. Caginalp, in Applications of Field Theory to Statistical Mechanics, ed. L. Garrido, Lecture Notes in Physics, Vol. 216, Springer, Berlin, p. 216 (1985). G. Caginalp, in Material Instabilities in Continuum Problems and Related Mathematical Problems, ed. J. M. Ball, Oxford University Press, Oxford, p. 35 (1988).

164

9. 10. 11. 12. 13. 14. 15.

16. 17. 18. 19. 20. 21. 22. 23. 24.

25. 26. 27. 28. 29. 30. 31. 32.

33. 34. 35. 36.

37. 38.

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G. Caginalp and P. C. Fife, Phys. Rev. B 33, 7792 (1986). G. Caginalp, Arch. Rat. Mech. Anal. 92, 205 (1986). G. Caginalp, Ann. Physics (NY) 172, 136 (1986). G. Caginalp and P. C. Fife, SIAM J. Appl. Math. 48, 506 (1988). G. Caginalp, Phys. Rev. A 39, 5887 (1989). G. Caginalp and E. A. Socolovsky, J. Comp. Phys. 95, 85 (1991). G. Caginalp and X. Chen, in On the Evolution of Phase Boundaries, eds. M. E. Gurtin and G. B. McFadden, The IMA Volumes in Mathematics and Its Applications, Vol. 43, Springer-Verlag, Berlin, p. 1 (1992). E. Fried and M. E. Gurtin, Physica D 68, 326 (1994). E. Fried and M. E. Gurtin, Physica D 72, 287 (1994). 0 . Penrose and P. C. Fife, Physica D 43, 44 (1990). S.-L. Wang, R. F. Sekerka, A. A. Wheeler, B. T. Murray, S. R. Coriell, R. J. Braun, and G. B. McFadden, Physica D 69, 189 (1993). R. Kobayashi, Bull. Jpn. Soc. Ind. Appl. Math. 1, 22 (1991). R. Kobayashi, Physica D 63, 410 (1993). R. Kobayashi, in Pattern Formation in Complex Dissipative Systems, ed. S. Kai, World Scientific, Singapore, p. 121 (1992). A. A. Wheeler, B. T. Murray, and R. J. Schaefer, Physica D 66, 243 (1993). B. T. Murray, W. J. Boettinger, G. B. McFadden, and A. A. Wheeler, in Heat Transfer in Melting, Solidification, and Crystal Growth, eds. I. S. Habib and S. Thynell, ASME HTD-234, New York, p. 67 (1993). B. T. Murray, A. A. Wheeler, and M. E. Glicksman, J. Cryst. Growth 154, 386 (1995). S.-L. Wang and R. F. Sekerka, Phys. Rev. E 53, 3760 (1996). S.-L. Wang and R. F. Sekerka, J. Comp. Phys. 127, 110 (1996). N. Provatas, N. Goldenfeld, and J. A. Dantzig, Phys. Rev. Lett. 80, 3306 (1998). N. Provatas, N. Goldenfeld, and J. A. Dantzig, J. Comp. Phys. 148, 265 (1999). Y. T. Kim, N. Provatas, N. Goldenfeld, and J. A. Dantzig, Phys. Rev. E 59, 2549 (1999). N. Provatas, N. Goldenfeld, J. A. Dantzig, J. C. LaCombe, A. Lupulescu, M. B. Koss, M. E. Glicksman, and R. Almgren, Phys. Rev. Lett. 82, 4496 (1999). N. Provatas, N. Goldenfeld, and J. A. Dantzig, in Modeling of Casting, Welding, and Advanced Solidification Processes, eds. B. G. Thomas and C. Beckermann, TMS, San Diego, p. 533 (1998). A. Karma and W.-J. Rappel, Phys. Rev. E 53, 3017 (1996). A. Karma and W.-J. Rappel, Phys. Rev. Lett. 78, 4050 (1996). • A. Karma and W.-J. Rappel, Phys. Rev. E 57, 4323 (1998). A. A. Wheeler and W. J. Boettinger, in On the Evolution of Phase Boundaries, eds. M. E. Gurtin and G. B. McFadden, The IMA Volumes in Mathematics and Its Applications, Vol. 43, Springer-Verlag, Berlin, p. 127 (1992). A. A. Wheeler, W. J. Boettinger, and G. B. McFadden, Phys. Rev. A 45, 7424 (1992). A. A. Wheeler, W. J. Boettinger, and G. B. McFadden, Phys. Rev. E 47, 1893

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(1993). 39. W. J. Boettinger, A. A. Wheeler, G. B. McFadden, and R. Kobayashi, in Modeling of Coarsening and Grain Growth, eds. C. S. Pande and S. P. Marsh, TMS, Warrendale, PA, p. 45 (1993). 40. Z. Bi and R. F. Sekerka, Physica A 261, 95 (1998). 41. Ch. Charach and P. C. Fife, SIAM J. Appl. Math. 58, 1826 (1998). 42. Ch. Charach and P. C. Fife, Open Systems, Information Dynamics 5, 99 (1998). 43. W. J. Boettinger, A. A. Wheeler, B. T. Murray, and G. B. McFadden, Mater. Sci. and Eng. A 178, 217 (1994). 44. M. Conti, Phys. Rev. E 55, 701 (1997). 45. M. Conti, Phys. Rev. E 55, 765 (1997). 46. N. A. Ahamad, A. A. Wheeler, B. J. Boettinger, and G. B. Mcfadden, Phys. Rev. E 58, 3436 (1998). 47. Ch. Charach, C. K. Chen, and P. C. Fife, J. Stat. Phys. 95, 1141 (1999). 48. Ch. Charach and P. C. Fife, J. Cryst. Growth 198/199, 1267 (1999). 49. J. A. Warren and W. J. Boettinger, Acta Metall. Mater. 43, 689 (1995). 50. M.. Conti, Phys. Rev. E 56, 3197 (1997). 51. J. A. Warren and W. J. Boettinger, in Modeling of Casting, Welding, and Advanced Solidification Processes VII, eds. M. Cross and J. Campbell, TMS, Warrendale, PA, p. 601 (1995). 52. J. A. Warren and W. J. Boettinger, in Solidification Processing 1997, eds. J. Beach and H. Jones, Department of Engineering Materials, University of Sheffeld, UK, p. 422 (1997). 53. J. A. Warren and B. T. Murray, Mod. Simul. Mater. Sci. Eng. 4, 215 (1996). 54. W. J. Boettinger and J. A. Warren, Met. Trans. A 27, 657 (1996). 55. W. J. Boettinger and J. A. Warren, J. Cryst. Growth 200, 583 (1999). 56. A. Karma, Phys. Rev. E 49, 2245 (1993). 57. A. A. Wheeler, G. B. McFadden, and W. J. Boettinger, Proc. R. Soc. Lond. A 452, 495 (1996). 58. K. R. Elder, J. D. Gunton, and M. Grant, Phys. Rev. E 54, 6476 (1996). 59. G. Caginalp and J. Jones, App. Math Lett. 4, 97 (1991). 60. G. Caginalp and J. Jones, in On the Evolution of Phase Boundaries, eds. M. E. Gurtin and G. B. McFadden, The IMA Volumes in Mathematics and Its Applications, Vol. 43, Springer-Verlag, Berlin, p. 27 (1992). 61. R. Tonhardt, Doctoral Thesis, Department of Mechanics, Royal Institute of Technology, Stockholm, Sweden (1998). 62. R. Tonhardt and G. Amberg, J. Cryst. Growth 194, 406 (1998). 63. C. Beckermann, H.-J. Diepers, I. Steinbach, A. Karma, and X. Tong, J. Comp. Phys. 154, 468 (1999). 64. H. J. Diepers, C. Beckermann, and I. Steinbach, in Solidification Processing 1997, eds. J. Beach and H. Jones, Department of Engineering Materials, University of Sheffeld, UK, p. 426 (1997). 65. M. E. Gurtin, D. Polignone, and J. Vihals, Math. Models. Methods Appl. Sci. 6, 815 (1996). 66. D. M. Anderson, G. B. McFadden, and A. A. Wheeler, Ann. Rev. Fluid Mech. 30, 139 (1998).

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67. D. M. Anderson and G. B. McFadden, Phys. Fluids 9, 1870 (1997). 68. D. J. Korteweg, Arch. Need. Sci. Exactes Nat. Ser. II 6, 1 (1901). 69. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, Interscience, New York, p. 262 (1953). 70. A. A. Wheeler and G. B. McFadden, Proc. R. Soc. Lond. A 453, 1611 (1997). 71. D. M. Anderson, G. B. McFadden, and A. A. Wheeler, Physica D 135, 175 (2000). 72. D. M. Anderson, G. B. McFadden, and A. A. Wheeler, "A Phase-Field Model of Solidification with Convection: Numerical Simulations," in this volume. 73. F. C. Larche and J. W. Cahn, Acta. Metall. 21, 1051 (1973). 74. F. C. Larche and J. W. Cahn, Acta. Metall. 26, 53 (1978). 75. F. C. Larche and J. W. Cahn, Acta. Metall. 26, 1579 (1978). 76. W. W. Mullins and R. F. Sekerka, J. Chem. Phys. 82, 5192 (1985). 77. Z. Bi and R. F. Sekerka, "Model for Solidification of a Pure Material with Energy as the Phase-Field Variable," unpublished notes based on seminars at Philadelphia and Edinburgh (1997). 78. H. W. Alt and I. Pawlow, in International Series on Applied Mathematics, Vol. 95, Birkhauser Verlag, Basel, p. 1 (1990). 79. H. W. Alt and I. Pawlow, Physica D 59, 389 (1992). 80. H. W. Alt and I. Pawlow, Adv. Math. Sci. Appl. 6, 291 (1996). 81. S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, Dover Edition, New York (1984). 82. D. D. Fitts, Non-Equilibrium Thermodynamics, McGraw-Hill, New York (1962). 83. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press, London (1959). 84. R. F. Sekerka and W. W. Mullins, J. Chem. Phys. 73, 1413 (1980). 85. R. B. Bird, W. E. Stewart, and E. W. Lightfoot, Transport Phenomena, John Wiley & Sons, New York, pp. 79-82 (1960). 86. G. C. Matthews, Cellular Physiology of Nerve and Muscle, Blackwell Scientific, Boston, p. 27 (1991).

T H E EFFECTS OF A S T R E S S - D E P E N D E N T MOBILITY ON INTERFACIAL STABILITY P. W . V O O R H E E S Department Northwestern

of Materials Science and Engineering, University, Evanston, IL 60093, U.S.A. M. J. AZIZ

Division of Engineering and Applied Sciences, Harvard University, Cambridge MA, 02138, U.S.A. A linear morphological stability analysis of a planar interface in interface reactioncontrolled growth is presented. We allow the mobility of the interface to be a function of the elastic stress, and find that the stress-dependent mobility can be either stabilizing or destabilizing. We find that a stress-dependent mobility can be the dominant factor leading to instability at small stresses. The predictions are compared to the recent experiments by Barvosa-Carter et al.ln agreement with their conclusions, we find that the experiments were performed in a parameter range where the stress-dependent mobility is the dominant cause of instability. The prediction of the critical wavenumber and amplification rate of the instability is consistent with the experimental results.

1

Introduction

The stability of interfaces in elastically-stressed solids has been the subject of great interest recently. A reason for this is the connection between the stability of a surface of a thin film that is deposited on a substrate and the subsequent evolution of the film morphology. It is clear that an elastic-stress-driven morphological instability results in a planar film undergoing a transition to islands; for a review, see [1]. Controlling this instability is thus central to the production of novel electronic and optoelectronic devices because the properties of such devices depend upon the morphology of the film. For example, under some circumstances nanoscale islands called "quantum dots" form that can exhibit unique electronic properties as a result of carrier confinement within the small volume of these dots. However, devices with well-characterized properties require that the size of the dots be relatively uniform. To date, much of the work has been focused on the interfacial instability that is driven by the elastic energy in the system. This instability, identified by Asaro and Tiller [2] (AT) and many others [3-6], follows from the dependence of the chemical potential on the elastic energy density. Consider a solid under uniaxial stress in contact with a fluid. Although the stress in the unperturbed or basic state is uniform, this is not the case when the interface is nonplanar. When the surface is perturbed the stress is concentrated at the troughs, in analogy with the stress concentration of a crack, and relieved at the peaks. Because the chemical potential is linearly proportional to the elastic energy density, the chemical potential of an atom is higher at the trough than at the peak and atoms thus move from the trough to the peak tending to make the peaks higher and the troughs deeper. Opposing this, of course, is the surface energy that tends to move atoms in the opposite direction and, thus, leads to a planar surface. The stress concentration in the

167

168

P. W. Voorhees & M. J. Aziz

trough appears in both compression and tension, and thus the AT instability is independent of the sign of the applied stress. Given the basic mechanism, the development of the elastic-energy-driven, or AT, instability must also depend on the mode of mass transport. Thus, the elastic-energy-driven instability has been examined in the context of bulk diffusion [2], surface diffusion [4-6], and evaporation-condensation [4, 7]. The evaporationcondensation model is, essentially, a completely kinetically-controlled interface in which the rate-limiting mass transport process is atom motion across an interface between a solid and another solid, a vapor, or even a liquid. In this case the velocity of the interface is linearly proportional to the difference between the chemical potential at the surface and that in the parent phase. The type of mass transport process does not alter the critical wavenumber of the instability, it alters only the dependence of the growth rate of the instability on the wavenumber. The elastic-energy-driven instability in alloys has also been examined. In an alloy the chemical potential couples to both the stress and the composition. Moreover, as stresses can result from gradients in composition in the solid, the composition and stress are also coupled through the continuum mechanics. As before, however, the evolution of the surface occurs by gradients in the chemical potential for surfacediffusion-driven mass transport [8-11], or by the difference between the chemical potential and that in a fluid for evaporation and condensation mass transport [12]. These compositionally-generated stresses can act to either enhance or prevent the usual AT instability [9,11,12]. In addition, compositionally-generated stresses can cause an instability even in the absence of an applied stress [10-13]. All of this work, however, assumes that the relevant transport coefficient, the surface diffusion coefficient for surface-diffusion-limited interface motion, or the interface mobility for interface-reaction-limited interface motion, is independent of elastic stress. Atomistic simulations have shown that the surface diffusion coefficient can be linearly proportional to the elastic stress on the surface. However, recent experiments by Barvosa-Carter et al. (BC) have shown that a stress-dependent interfacial mobility alone can generate an interfacial instability when the transformation is interface limited [14]. The mechanism for the kinetic instability follows from the observation that the change in the mobility with stress in certain cases can be linearly proportional to the stress. Consider a basic state consisting of a moving planar interface under uniform uniaxial stress. Assume that the stress is compressive, i.e., negative, and that the mobility increases with increasingly positive stress. Then at the troughs, where the stress is more negative than that of the basic state, the mobility is smaller than that of the planar interface. Conversely at the peaks, where the stress is larger than that of the basic state, the mobility is larger than that of the planar interface. Thus, troughs grow slower and peaks grow faster than the moving planar interface. In this case the stress-dependent mobility is destabilizing. Moreover, experiments have shown that the amorphous to crystalline (a-c) transformation in Si is an ideal system in which to explore this kinetic instability. The stress dependence of the interfacial mobility has been measured [15]. The mobility, M, was found to depend upon stress, er, as

M(aij) =

Mexp(Vi*aij/kT),

(1)

The Effects of a Stress-Dependent

Mobility on Interfacial

Stability

169

where V* is the activation strain [15], M is the mobility of the interface in the absence of stress, k is Boltzmann's constant and T is temperature. For the Si[001] a-c interface, V^ = V2*2 = 0.14Q and V33 = — 0.35f2, where fi is the atomic volume, and the off-diagonal elements are zero. The transformation is completely interface-reaction limited, and the other relevant material parameters have been measured. It is possible to carefully control the morphology of the initial interface using lithography. Finally, because, V^ > 0 a compressive uniaxial stress <m < 0, should result in an interfacial instability. Such an instability was observed by BC. BC analyzed the instability using a fully nonlinear numerical approach. This approach has the advantage of being able to follow the instability into the strongly nonlinear, large-amplitude regime and confirmed that the instability was, in fact, responsible for the nonplanar morphologies observed in the experiments. However, because the analysis was numerical, it was difficult to establish the relationship between the various material and processing parameters that control the instability. Grinstein, Tu, and Tersoff [16] examined analytically the effects of a stressdependent mobility in the context of thin-film oxidation. The motion of the interface in this case is controlled by both interface mobility and long-range diffusion. Given the complexity of the problem, however, they did not focus on the reactionlimited case, and their treatment of long-range diffusion is controversial [17,18]. Yu and Suo have recently considered the reaction-limited case as well, focusing on the two-dimensional patterns that might be formed on the interface [19]. We present a linear stability analysis of a planar interface where the motion of the interface is interface-reaction limited. We allow for a stress-dependent interfacial mobility and examine its effect on the stability of the interface. Using the material and processing parameters of the BC experiments, we make a comparison between theory and experiment with no adjustable parameters. However, we assume that the amorphous phase into which the crystal grows is stress free. Extrapolation of the known [20] viscous-blow behavior of a-Si indicates that although there is a significant amount of relaxation in the BC experiments, it does not go to completion in the experimental time scales. We also assume M to be independent of crystallographic orientation. Although the mobility of the [001] interface is known to be 20 times faster on [001] than on [111], M at [001] is an extremum due to crystal symmetry. Nevertheless, we shall neglect this orientation dependence. The effects of growth kinetic anisotropy and flow in the a-Si layer on the instability are currently under investigation. 2

Theory

The crystal grows into the amorphous phase at a constant velocity v in the positive ^-direction, where X3 is parallel with the [001] direction. All quantities except stress and activation strain are assumed to be isotropic. The motion of the a-c interface is kinetically limited, thus in the limit of small driving forces, v is given by [14,15] v = -M

(a)

[AG°V + WpV

+ W (e) +

7K]

,

(2)

where M (er) is the stress-dependent interface mobility, cr is the stress in the small-

170

P. W. Voorhees & M. J. Aziz

strain approximation, the chemical driving force for crystallization AG° is the Gibbs-free-energy change (AG° < 0) per unit volume of growing phase upon crystallization of stress-free material at a planar interface, Wpv is the pV work done on the system by external forces per unit volume crystallized, W (e) is the elastic strain energy, e is the small-strain tensor, 7 is the interfacial energy, and K is the mean curvature of the interface (reckoned positive for a spherical crystal). The elastic strain energy is assumed to be negligible in the parent phase and we will neglect Wpv- Since the transformation is isothermal and occurs in a pure material, the only field equation that must be solved is that for the elastic stress. The stress field is found by requiring that the crystal is in elastic equilibrium, ^ijd

(3)

~ ^!

where summation of repeated indices from 1 to 3 is assumed. The stress is linearly related to the strain, E £kk&ij%] +' 1 + v \ 1 - 2v

et j j

: ,

(4)

where E is Young's modulus and v is Poisson's ratio. The strain is related to gradients in the displacement Uj, e

ij = iui,j + u3,i) / 2 -

(5)

Using Eqs. (4) and (5) in Eq. (3) yields Navier's equations for the displacement field in the crystal, (1 - 2v) uiM + ukM = 0.

(6)

The a-phase is assumed to be stress-free, and thus along the a-c interface, (TijUj = 0,

(7)

where rij is the normal to the interface pointing into the a-phase. Finally, the elastic energy density is given by, W =

aijCij/2.

(8)

In the unperturbed or basic state, the interface is planar and moving at a constant velocity into the amorphous phase. The applied stress in the crystal is biaxial and of magnitude cr°x = a\2 = a a- The stress-free boundary condition (7), implies CTJ3 = 0 throughout.

To examine the linear stability of the moving planar interface located at £3 = 0, we perturb the height of the interface in the a?3-direction, h = h(x\,X2), and all other quantities in normal modes,

I 0 \ v Ui

M W K

\ J

M° W°

V 0 /

( h \ v1

+ 6 M1 l

W 1

V* /

*,

(9)

The Effects of a Stress-Dependent

Mobility on Interfacial Stability

171

where $ = exp (iq -x + rt), q and x are the wavevector and position vector, respectively, in the plane of the interface, <5 0 the perturbation grows, and if r < 0 the perturbation decays. If r is imaginary then an oscillatory instability is possible. Using Eq. (9) in Eq. (2) and setting 6 = 0, we obtain the basic-state, interfacial velocity, v° = -M°

(AG°V + W°) .

(10)

vl = -{AG°v + W°)Ml-M°Wl-M^^K1.

(11)

The perturbed velocity is

1

1

l

1

We now need to find M°, M , W°, W ,V , and K . The elastic-strain-energy density in the basic state follows from the stresses given above, W° = al/Y,

(12)

where Y = E/(l — v). The strain-energy density in the perturbed state is determined by solving Eq. (6) with the boundary condition (7), and the condition that the «H X 3) ""* 0 as x 3 —> —oo [21]. Using these displacements in Eq. (5), substituting into Eq. (4), and using Eq. (8) yields the magnitude of the perturbed elastic-strain-energy density along the interface [12], W1 = -2h{l

+ v)qal/Y,

(13)

where q = |q|. In addition, the magnitude of the trace of the stress is 4k

= -2h(l

+ u)qaa.

(14)

The mobility of the interface in the basic and perturbed states, M° and M1 respectively, can be found by using Eq. (1), M(tr)=M(l + ^ y

(15)

where we have assumed that aijV*j/kT <^C 1. Thus,

M«^( 1 + f2), crl.V* Ml = - ^ ,

(1„ (17)

where of- and a\, are the stresses in the basic and perturbed states, respectively. To determine a\^V*^ we first rewrite it as,

4 - ^ i = olkVn + (VS - Vx\) 4 , ,

(18)

because for an [001] surface of a cubic crystal, V{x = V2*2 [15]. From Eq. (7), we findCT33= 0. Using this and Eq. (14) in Eq. (18) yields o\jV% = -2h(l

+ u)V?1(raq.

(19)

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The stress in a system with a planar interface is 4Vii

=

2 y

( 2 °)

i>-

Using these stresses in Eqs. (16) and (17) yields M0 = M ( l + ^

)

,

(21)

and M 1 = -M2h{\

+ v)V^aaq/kT.

(22)

Thus, if V{i > 0, a compressive stress decreases the mobility of the planar interface. For compressive stresses the perturbed mobility M1 is in phase with the interface shape. Therefore, the mobility is the highest at the peaks and the lowest at the troughs. The remaining terms in Eq. (11) are functions only of the shape of the interface. Using the definition of the curvature in Cartesian coordinates, the form of the perturbed interface, and by taking the derivative with respect to time we obtain, K1

= hq2,

v1 = hr.

(23)

Using Eqns. (12-14) and (21-23) in Eq. (11) yields the dispersion relation, j j j = [(1 + ") aAG°vaa + 2pa2a + 3a/3a3a] q -{l

+ aaa)iq2,

(24)

where a = 2V1*1/feT and j3 = (1 + v)/Y. The dispersion relation is linear in r and thus the instability cannot be oscillatory. The wavenumber at which the growth rate of the instability is zero, qct is found by setting r = 0, (l + is)aAG0vaa + 2pa2a + 3aPa3a Qc=

77r ( l + QCT a )7

.

(25)

For q < qc, r > 0; for q > qc, r < 0, see Fig. 1. The wavenumber with the maximum growth rate, qm, is given by qm — qc/2. Thus, qc determines the length scale of the instability. 3

Discussion

The dispersion relation is quadratic in the wavenumber, q, with the terms related to the elastic stress being linear in q and the interfacial-energy-related terms going as q2. Interfacial energy is always stabilizing, because by assumption ctaV^/kT
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0.002

-0.004

0.2

0.4

0.6

1.0

wavenumber (m~ ) Figure 1. Amplification rate of the instability as a function of the wave number of the disturbance for the a-c Si transformation in the BC experiments.

the usual case where the instability develops by surface diffusion. The term that is linear in the applied stress can be either stabilizing or destabilizing depending on the sign of the product aaa. If aaa < 0, as in the a-c-Si BC experiments, the stress dependence of the mobility itself can drive an interfacial instability. The term that is cubic in the applied stress is the AT instability modified by the stress-dependent mobility. Because aaa < 0 under compressive stresses for the Si[001] a-c interface, both the stress-dependent mobility and the elastic energy promote instability. As shown in Eq. (24), both effects have the same scaling with the wavenumber and thus it can be difficult to determine which phenomenon is responsible for an experimentallyobserved interfacial instability. However, the elastic-energy-driven instability scales with stress in a different manner than the mobility-driven instability. It is clear that for a sufficiently small applied stress, the stress-dependent mobility will dominate. When aaa < 0, this will occur when (1 + A g » ! (26)

^

g

Using the values for the a-c Si interface under the conditions employed in the BC experiments we find that the left-hand-side of the inequality (26) is approximately 12, and thus the instability of the interface is predicted to be due to the kinetic instability. Conversely, if aaa > 0, then the stress-dependent mobility is stabilizing. The critical stress, aca below which the stress-dependent mobility will stabilize the AT instability is found by setting the first term in the brackets of Eq. (24) to zero, a\ = - 1 / (3a) + [1 - 3a 2 (1 + u) AG°v/(3]1/2 / (3a).

(27)

For the a-c Si system, this yields a stress of 9.6 GPa. All tensile stresses below this value result in an interface that is stable against the AT instability. Recent exper-

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imental results [22] confirm this prediction. Thus, in the a-c Si system very large (perhaps impossibly large) tensile stresses are required to overcome the stabilization due to the stress-dependent mobility. The dispersion curve for the a-c transformation in Si is shown in Fig. 1, where we have used aa = -0.5 GPa [14], Y = 100 GPa, v = 0.25, T = 520°C, M = 4.52 x 10" 2 0 m 2 / J s [24], AG°V = -6.27 x 108 J / m 3 [25], and 7 = 0.4 J/m 2 [26,27]. The length scale of the interfacial instability is provided by the critical wavenumber, Eq. (25), and can be also be read directly from Fig. 1. The critical wavelength, Ac = 2ir/qc, of the instability is 8.5 nm. Thus, perturbations with wavelengths in excess of 8.5 nm will amplify and those with wavelengths less than 8.5 nm will decay. The critical wavenumber of the AT instability, q^T is, qf

= 2a2J/r

(28)

For the BC experiment, this yields \^T = 2000 nm. It is thus clear that the length scale of the kinetically-driven instability can be quite different from that driven solely by elastic strain energy. The wavelength of the kinetically-driven instability with the maximum growth rate is 17 nm. Unfortunately, the BC experiments were unable to determine qc because, for reasons of experimental practicality, they started not with a planar interface, but rather with an interface with a large initial sinusoidal perturbation with a wavelength of 400 nm. They found that a perturbation of this wavelength is unstable, and thus is consistent with this prediction. The experimentally-observed amplification rate of such a perturbation is 6 x 10~ 5 /s, whereas the predicted amplification rate is 1.6 x 10 _ 4 /s. This is very good agreement considering the uncertainties in the material parameters. It has been suggested that the stress-dependent mobility may also affect the evolution of elastically-stressed interfaces when growth is not fully reaction limited [14]. To investigate this possibility, we consider an interface evolving in a fully surface-diffusion-limited regime in which the atomic mobility for diffusion, Mp, is stress dependent. In this case, the velocity of the interface is related to a surface gradient in the chemical or diffusion potential at the surface, v oc V s • MD (
7K)

,

(29)

where V s denotes the surface gradient operator. Consider the case of a planar surface where the basic-state stress is constant. Perturbing about this state gives v° + Sv1® oc - V s • (M°D + dM1^)

V s (6W1® + 6-yK1®).

(30)

Because the elastic energy density and the chemical driving force in the basic state are constants, they are not present in Eq. (30) and thus the effects of the perturbed atomic mobility, M^, are on the order 62. Therefore, the stress-dependent mobility, or a stress-dependent, surface diffusion coefficient, will not affect the linear stability of a uniformly-stressed planar surface in purely surface-diffusion-limited growth. However, a stress-dependent atomic mobility will clearly affect the nonlinear, or large amplitude, evolution of the surface. In addition, a stress-dependent interfacial mobility will continue to affect the linear stability in a situation of mixed diffusion and interface-reaction, or step-edge attachment, control [23].

The Effects of a Stress-Dependent Mobility on Interfacial Stability

4

175

Conclusions

The linear stability analysis for interface-reaction-limited growth shows that when the mobility depends upon stress it can either lead to morphological instability or act to stabilize the planar interface. In addition, the more classical elasticenergy-driven AT instability is also present. However, because the kinetically-driven instability scales linearly with the applied stress and the AT instability is quadratic in the applied stress, the kinetically-driven instability should dominate at low levels of stress. When the theory is applied to the a-c transformation in Si, we find that for the parameters used in the BC experiments the observed interfacial instability is a result of the stress-dependent mobility. Finally, the predicted critical wavelength and amplification rate of the kinetically-driven instability is consistent with the experiments. Acknowledgments This research was supported by the National Science Foundation through DMR9707073 (PWV) and DMR 98-13803 (MJA). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

V. A. Shchukin and D. Bimberg, Rev. Modern Phys. 71, 1125 (1999). R. J. Asaro and W. A. Tiller, Metall. Trans. 3, 1789 (1972). M. A. Grinfeld, Dot Akad. Nauk SSSR 290, 1358 (1986). D. J. Srolovitz, Acta Metall. 37, 621 (1989). J. Gao, Int. J. Solids Structures 28, 703 (1991). B. J. Spencer, P. W. Voorhees, and S. H. Davis, Phys. Rev. Lett. 67, 3696 (1991). K.-S. Kim, J. A. Hurtado, and H. Tan, Phys. Rev. Lett. 83, 3872 (1999). J. E. Guyer and P. W. Voorhees, Phys. Rev. Lett. 74, 4031 (1995). F. Leonard and R. Desai, Phys. Rev. B 56, 4955 (1997). V. G. Malyshkin and V. A. Shchukin, Semiconductors 27, 1062 (1993). B. J. Spencer, P. W. Voorhees, and J. Tersoff, in press. J. E. Guyer and P. W. Voorhees, Phys. Rev. B 54, 10710 (1996). B. J. Spencer, P. W. Voorhees, S. H. Davis, and G. B. McFadden, Acta Metall. Mater. 40, 1599 (1992). W. Baravosa-Carter and M. J. Aziz, Phys. Rev. Lett. 8 1 , 1445 (1998). M. J. Aziz, P. C. Sabin, and G.-Q. Lu, Phys. Rev. B 44, 9812 (1991). G. Grinstein, Y. Tu, and J. Tersoff, Phys. Rev. Lett. 8 1 , 2490 (1998). V. P. Zhdanov, Phys. Rev. Lett. 83, 656 (1999). G. Grinstein, Y. Tu, and J. Tersoff, Phys. Rev. Lett. 83, 657 (1999). H. H. Yu and Z. Suo, J. Appl. Phys. 87, 1211 (2000). A. Witvrouw and F. Spaepen, J. Appl. Phys. 74, 7154 (1993). B. J. Spencer, P." W. Voorhees, and S. H. Davis, J. Appl. Phys. 73, 4955 (1993). J. F. Sage, W. Barvosa-Carter, and M. J. Aziz, Appl. Phys. Lett. 77, 516

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(2000). 23. M. J. Aziz, Mater. Res. Soc. Symp. Proc. 618, 233 (2000). 24. G. L. Olson and J. A. Roth, Mater. Sci. Rep. 3, 1 (1988). 25. E. P. Donovan, F. Spaepen, D. Turnbull, J. M. Poate, and D. C. Jacobson, J. Appl. Phys. 57, 1795 (1985). 26. N. Bernstein, M. J. Aziz, and E. Kaxiras, Phys. Rev. B 58, 4579 (1998). 27. C. M. Yang, Ph.D. Thesis, California Institute of Technology, Pasadena, CA (1997).

NON-CONSTANT GROWTH CHARACTERISTICS OF PIVALIC ACID D E N D R I T E S IN MICRO G R A V I T Y J. C. L A C O M B E , M. B . K O S S , A. O. L U P U L E S C U , J. E. F R E I , A N D M. E . G L I C K S M A N Materials Science and Engineering Department, Rensselaer Polytechnic Institute, Troy, New York 12180-3590, U.S.A. E-mail: [email protected] Dendritic-growth experiments conducted aboard the space shuttle Columbia's USMP-4 mission as part of the Isothermal Dendritic Growth Experiment (IDGE) are described. Video data reveals that pivalic acid dendrites growing in the diffusion-controlled environment of low-earth orbit exhibit a range of transient or non-steady-state behaviors. The observed transient features of the growth process continue to be studied, with the objective of elucidating the mechanisms responsible for these behaviors. While not yet clear, experimental factors may be responsible such as the long-range thermal interactions between the dendrite and its neighbors or container. It is also considered that the transient behaviors may arise from fundamental characteristics of isothermal, diffusion-limited, dendritic growth.

1

Introduction

The formation of dendritic microstructures is common during the solidification of most metals and alloys. It is important to understand the process by which these dendrites form because their microstructures can persist through subsequent material processing stages and affect the properties of the finished product. Over the past five decades, the scientific community has produced a large body of theoretical and experimental work describing dendritic growth. Glicksman and Marsh [1] discuss this research in their 1993 review article, and Bisang and Bilgram [2] also include a detailed review as part of their 1996 article on the subject. As these reviews and others make clear, the speed of an advancing dendrite tip is a critical characteristic of dendritic growth. In 1947, Ivantsov [3] introduced two simplifying assumptions in the theory describing thermal dendrites growing in a supercooled melt: 1) that a dendrite can be represented as a shape-preserving paraboloidal interface with a tip radius, R; 2) that the dendrite grows at a constant rate, V. The notion that real dendrites (as opposed to paraboloidal, needle crystals) also grow in a steady-state manner, with constant velocity, is generally considered to be supported by numerous experimental observations of isolated dendrites (see Huang [4] for an example). Most experimental and simulation studies of dendritic growth thus attempt to extract a constant velocity measurement as a parameterization of the kinetics, whereas most theoretical studies assume constant-velocity behavior. Recent work has focused on studying the thermal interactions between a dendrite and its surroundings, and quantifying what is required for a dendrite to grow in a truly "isolated" manner. Thermal interactions may exist between a dendrite and its neighboring tips, container walls [5], or even its own initial structure [6]

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and trailing side branches [7,8]. Such interactions are potential explanations for experimentally-observed growth rates that do not match the predictions for an isolated diffusion-limited dendrite, with the thermal boundary conditions set at infinity. If the strength of these thermal interactions changes over time, the velocity of the dendrite should change as well, and in general, there should be deviations from the conventionally-prescribed behavior predicted by an Ivantsov-like solution for an isolated dendrite. In particular, at low supercoolings, the length scale over which the thermal field extends is known to be large compared to the morphological length scales relevant to the dendritic growth process. With such long thermal diffusion lengths, proximate dendrite arms can mutually interact to generate an operating state that is not steady. This type of behavior is evident in the recent phase-field simulations by Provatas et al. [9], which suggest the existence of an early-time transient in two-dimensional dendritic growth prior to approaching a steady-state growth rate. 2

Experiment

The data presented here derives from an analysis of dendritic growth data obtained from the Isothermal Dendritic Growth Experiment (IDGE). This project compiled data on the growth of pivalic acid (PVA) dendrites in microgravity. Additional detailed information describing earlier IDGE experiments can be found elsewhere [10]. 2.1

Apparatus

The experimental apparatus used for the IDGE experiments was located in the payload bay of the space shuttle Columbia. This equipment, depicted in the schematic of Fig. 1, was specially designed for use on earth as well as the environment of lowearth orbit. The sample material was contained by a rigid quartz growth chamber located within a temperature-controlled bath. The growth chamber interior volume measured approximately 31 mm square by 50 mm long. Nucleation was achieved through the use of a hollow stinger tube that penetrated the wall of the growth chamber. The exterior end of the stinger tube was closed and surrounded by a thermoelectric cooler. The interior end was open, allowing the PVA sample material in the chamber to also fill the stinger. The use of a quartz growth chamber was necessitated by the sample material. Pivalic acid was found to be excessively reactive with the stainless steel and borosilicate glass growth chambers used in earlier IDGE space flight experiments where the sample material was succinonitrile [10]. The PVA that filled stainless-steel chambers rapidly degraded in purity and would not supercool enough to conduct the necessary experiments. Additionally, for reasons that are not completely understood, the PVA-filled stainless-steel chambers were also prone to structural failure. During the operation of the experiment, each dendritic growth cycle began by completely melting the PVA, followed by lowering the melt's temperature to the desired supercooling. After the supercooled-melt's temperature reached steady state, the thermoelectric cooler was activated. This nucleated a small crystal in the end of the stinger, which then propagated down the stinger tube to emerge

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Uniform Temperature Bath

35mm Camera

Video Camera

—v

Xenon Flash

Dendrite

Figure 1. A schematic representation of the experimental apparatus. The sample material is located inside the quartz growth chamber. This is placed within a temperature-controlled bath where the experiment is conducted with in-situ monitoring of process parameters (temperature, etc.) and imaging takes place.

into the chamber as a freely growing dendrite. Once one of these "growth cycles" was completed, a new growth cycle was initiated by re-melting the sample and proceeding as described above. This arrangement, combined with the microgravity conditions, produced dendritic crystals grown under diffusion-limited conditions, with the bath temperature controlled to within 0.002 K (spatially and temporally). The same apparatus was also operated on the ground to acquire an appropriate "baseline" data set describing dendritic growth under the convective conditions associated with earth gravity. During the growth cycles, once a crystal emerged from the stinger, images of the dendrites were obtained from two perpendicular views using both electronic and film cameras. While the film images constituted the primary data source for the IDGE experiments, the observations described here are derived primarily from data obtained from the electronic video cameras. These cameras provided the spatial and temporal resolution that is necessary to study the transient aspects of the growth process. Specifically, an imaging chip array was used of 640 x 480 pixels (256 grayscale levels) and an imaging rate of 29.963 frames per second (~ 30 Hz). 2.2

Analysis Techniques

The data presented here are primarily based upon measurements of dendrite-tip locations as a function of time obtained from the 30 fps video cameras. The base optical resolution of the system is related to the size of an individual pixel in the video camera's imaging array. Each pixel, after correcting for the magnification of the optics system, images a region of the chamber that is approximately 22 /xm by 22 fira. These values also constitute the raw measurement precision for the tip position data. It is beneficial to improve upon this precision by applying a sub-pixel

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Figure 2. Video image of a dendrite from Cycle 04 grown at 0.407 K supercooling during the USMP-4 experiment. This and other similar images were used to produce the results presented here.

resolution image analysis method to each image in the growth cycle (see Figs. 2 and 3 for sample images). The method used for the data presented here begins by first examining the first and last image frames of a growth cycle. The dendrite tips are found in these images using a row-by- row search to locate the lowest pixel in the field of view that is darker than a specified threshold value. If there are more than one pixel in a row satisfying this criteria, an average is calculated to obtain the horizontal coordinate. With these crude estimates of the tip positions at the beginning and end of the growth, it is possible to construct a vector used to predict where the tip will be in all frames during the growth, provided the frame number and frame rate are incorporated. This tip-location prediction scheme improves processing efficiency, though it is only capable of resolving information on the order of 22 /xm (one pixel). Using this predicted tip location, a second stage of refinement is added to the tip-location process for each image. This is achieved by overlaying a "sampling line" along the predicted vector and determining the point along this vector where the image intensity crosses a selected threshold value. However, this time the sampling line is "thick" in that it also comprises several pixels on either side of the mathematical line, creating an averaging effect. Additionally, interpolation is applied to determine the threshold location more precisely. By incorporating a statistically larger number of pixels in this second-stage, tip-locating method, resolution is improved to approximately 7 /im (~ 1/3 pixel). However, the horizontal coordinate

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Video

™ R «20/m 1DD tin

1 1

R «20 ^m a 1 pixel Figure 3. Images of the tip region of the dendrite grown in Cycle 04. a) A 35 mm film image with an approximate tip radius of curvature of 20 [im. b) A 30 fps video image, revealing considerably less detail. The centroid of the tip is calculated using the pixel intensities located within the box enclosing the tip.

is constrained to fall along the predicted vector line. In practice, this is not usually a significant issue, since these dendrites tended to grow within about 10 degrees of vertical, and, once started, do not deviate in direction under nominal growth conditions. The final stage of resolution enhancement is achieved by a somewhat unorthodox approach, which has the advantage of incorporating still more (statistically speaking) information concerning the tip location. Fig. 3b shows a typical video image of a dendrite tip. Using the refined tip location (stage 2, described above) as a reference location, a box is created around the tip. Next, the centroid of the pixel intensity within this box is calculated. In practice, the coordinates of this centroid exhibit a resolution of approximately 2 /xm (~ 1/10 pixel). This approach does not actually locate the tip's interface. Instead, it uses more data to obtain a more consistent reference point, which serves to track the tip's movement over time. The size of the sampling box used in the centroid calculation is somewhat arbitrary. The concerns in its selection are primarily in obtaining a balance between the desire to have a large number of data points contributing to the measurement, and avoiding the inclusion of side branches. When information that is obtained from regions further removed from the tip is used, nascent side branches can contribute to the centroid calculation in a periodic manner. Once extracted from the images, the tip positions are then converted into a displacement vs. time data set that is used for subsequent analysis (see Fig. 4). The displacement is calculated relative to the tip position in the first available frame of video data, which is assigned to time t = 0. The measures of resolution quoted herein are characterized by examining the residuals produced by a linear regression of the displacement data (Fig. 5a). The standard deviation of the spread in these residuals is calculated on a moving basis, with a two-second window. These

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100

ISO

T i m e (seconds) Figure 4. Tip displacement vs. time for dendrite Cycle 09, grown at 0.376 K supercooling. Linear regression was performed using data from the more steady-state portion of the growth (after ~ 125 seconds). The growth rate, 22.9 fim/s results from the slope generated by the regression. Note: 1 of every 30 data points plotted.

a)

Time (seconds)

b)

Time (seconds)

Figure 5. a) Residuals resulting from a linear fit to the tip displacement vs. time data in Fig. 4 for the latter growth stage, i.e., after ~ 125 seconds, b) The standard deviation of the spread in the residuals is below 2 (im, representing the uncertainty in the tip-position measurements.

measurements are shown in Fig. 5b, revealing the 2 /im (at most) uncertainty stated above.

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Results

Earlier observations by the authors [11] indicated that dendritic growth rates are not constant over the time scale of observation. Specifically, the dendrite tip, after completing an initial transient, continues to experience a small acceleration. It is not yet clear whether or not the dendrite approaches a truly constant velocity. The displacement data shown in Fig. 4 are derived from a supercooling of 0.376 K. They are qualitatively representative of the entire body of data analyzed to date. The gaps in the data resulted from the hardware necessity to adjust lighting during 35 mm camera exposures. As with the data presented in [11], this cycle shows an initial transient clearly accelerating, and then evolves toward a more constant-slope regime from which one can extract the velocity. Despite the relative linearity of the latter part of the displacement plot, a leastsquare regression of the data from the portion of the growth between the hash-marks of Fig. 4 (after ~ 125 seconds) reveals that the dendrite is still accelerating late in its growth. The residuals resulting from the linear regression show a systematic deviation from steady-state growth (Fig. 5a). A dendrite growing at constant velocity would be expected to exhibit random residuals forming a Gaussian distribution centered about zero. Instead, as shown, a distinct non-random residual occurs, with monotonic upward curvature in the graph. As mentioned earlier, the residuals also quantify the uncertainty in the measurements, because their spread is a good measure of the resolution resulting from the sub-pixel interpolation measurement of the tip location. A convenient measure of the asymptotic behavior of the tip speed, as evidenced by the non-random residual, is the growth-rate exponent, K, calculated by comparing the data to a power law for the displacement proportional to tK. An exponent of K — 1 indicates linear displacement in time, i.e., constant velocity. Although at first this method may seem less intuitive than simply comparing the instantaneous velocity versus time, the slope of which is the acceleration, the power law helps elucidate the variations from, and approach to, constant-velocity (K = 1) behavior. The measured values of K (Fig. 6) calculated after first smoothing the displacement data using a two-second moving average, reveals a slightly different trend than seen for the dendrite growth cycle presented in [11]. In the previous report, K started out large and approached 1 from above. For this starts out close to 1 and increases slightly (still an accelerating trend). Aside from this initial transient, other dendrites have been seen to behave qualitatively similar in terms of K. 4

Discussion

The data presented here indicates that there are transient aspects of the dendritic growth process which steady-state theory does not describe. Examination of these experimental data suggests that within experimental practice, the establishment of strict isothermal conditions at the dendritic interface may be illusory. We will discuss briefly several potential sources for non-steady-state behavior. In doing so, it is beneficial to describe the time-dependent process as occurring in two distinct periods. The first of these is the initial transient, which is most evident in Fig. 6

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Time ( s e c o n d s ) Figure 6. The growth-rate exponent, K, as a function of time for the dendrite data in Figs. 4 and 5. A value of K = 1 corresponds to steady-state, constant-velocity growth.

prior to approximately 50 seconds. The second distinct time-dependent period is beyond ~ 50 seconds, where the growth-rate exponent continues to increase, though more gradually. To explain the early-growth behavior, the authors [11] suggested two related mechanisms that may induce a transient effect. Summarizing, we note that during the early phase of growth, a dendrite evolves from some pre-dendritic morphological structure to eventually develop an interface shape and thermal field which are commensurate with steady- state growth. This developmental stage takes time. Additionally, in the nascent stages of a dendrite's development, neighboring dendrites (or other branches of an equiaxed dendrite) are located close to one another. As long as these separation distances are less than a few thermal lengths, where the characteristic thermal length is defined here as the thermal diffusivity of the liquid (7 x 10 4 /im 2 /s) divided by the growth rate, thermal interactions are expected to occur between neighboring dendrites causing a localized drop in supercooling. As the dendrites grow, their tip-separation increases, creating a progressive increase in the local effective supercooling. This would manifest itself as an accelerating growth rate. Similar interactions also exist for proximate dendrites growing in the same direction. Eventually, if the interactions are strong enough, one of the dendrites will experience favorable thermal conditions and will "pull away" from the slower dendrite. The experimental (and simulated) observation of an initial transient preceding steady-state growth is not new. In fact, it has been generally assumed that this phase, although easily distinguishable from the steady-state that emerges, is less germane and fundamental to the kinetics than the steady-state itself. However,

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as is revealed here, the growth phase following the initial transient might not ever reach a constant velocity. Thus, the more important issue for such investigations is whether the time-dependent state is fundamental to the physics of isothermal dendritic growth, or appears as an artifact of experiments carried out in finite volumes of supercooled melts. The observation of the effect of neighboring dendrites on growth velocity [11] suggests possible long-range thermal interactions with proximate dendrites or arms growing in the same general direction as the tip in question. It is conceivable that similar interactions may contribute to a second stage of time-dependent growth seen after the initial transient is completed. However, many of the analyzed dendrites were isolated growths (noting that 'isolation" is a strong function of time and supercooling), where the closest neighbor was beyond three thermal lengths. In order to conclusively examine this issue, it will be necessary to analyze a significant number of dendrites growing under a variety of conditions. Additionally, one must firmly establish that the thermal interaction between dendrites separated by more than three thermal lengths are still enough to produce the measured changes in growth velocity. The second transient phase, also described in [11], can also potentially be explained by the finite size of the growth chamber. Pines, Chait, and Zlatkowski [5] showed that a dendrite may interact with the walls of a container as growth proceeds. However, it remains to be determined whether the closing rate between the dendrite and the wall is commensurate with the measured time variations in velocity. Given these considerations, it is our hypothesis that it is more likely that the second stage of non-steady- state behavior is fundamental to isothermal dendritic growth. Since dendrites are not truly parabolic bodies of revolution [4,12], there is no compelling phenomenological reason that dendrites should grow strictly at a constant rate. As the data set from which these observations and conclusions were drawn is the only one we know of that is both diffusion-controlled, and measured at the necessary temporal resolution to make the described measurements, we plan additional work in data reduction, analysis, and modeling, to explore this hypothesis further. 5

Conclusions

In summary, a method was developed for evaluating dendritic growth rates that discriminates fine distinctions in the non-steady-state behavior. This method is applied here to PVA dendrites, seen to grow in microgravity at non-steady- state velocities. The mechanism responsible for this behavior may be related to thermal interactions between a dendrite and its surroundings, or it may be intrinsic to the dendritic solidification process. Efforts to further discriminate among the possible causes are currently under way. Acknowledgements This work was supported under NASA Contract Nos. NAS3-25368 and NAG8-1488. The authors wish to thank D. P. Corrigan for discussions about image processing,

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and L. A. Tennenhouse, and the many Rensselaer undergraduate and graduate student volunteers for assistance in operating the flight experiment. Additional thanks to IDGE Project Manager D. C. Malarik and her team at (or associated with) NASA's Glenn Research Center at Lewis Field for their engineering support, and the staff of the POCC/HOSC at NASA's Marshall Space Flight Center for flight support. Finally, we thank the crew of the STS-87 for bringing our experiment to orbit and back, and for their extra effort in providing us with the additional best available quality video data for our post-flight analysis. References 1. M. E. Glicksman and S. P. Marsh, in Handbook of Crystal Growth, edited by D. T. J. Hurle, Elsevier Science Publishers, Amsterdam, p. 1077 (1993). 2. U. Bisang and J. H. Bilgram, Phys. Rev. E 54, 5309 (1996). 3. G. P. Ivantsov, Dokl. Akad. Nauk, SSSR 58, 567 (1947). 4. S.-C. Huang and M. E. Glicksman, Acta Metall. 29, 701 (1981). 5. V. Pines, A. Chait, and M. Zlatkowski, J. Cryst. Growth 167, 383 (1996). 6. V. Pines, A. Chait, and M. Zlatkowski, J. Cryst. Growth 182, 219 (1997). 7. J. C. LaCombe, M. B. Koss, D. C. Corrigan, A. O. Lupulescu, L. A. Tennenhouse, and M. E. Glicksman, J. Cryst. Growth 206, 331 (1999). 8. J. C. LaCombe, M. B. Koss, D. C. Corrigan, A. O. Lupulescu, J. E. Frei, and M. E. Glicksman, in Solidification 1999, edited by W. H. Hofmeister, et al, TMS, Warrendale, PA, p. 121 (1999). 9. N. Provatas, N. Goldenfeld, J. Dantzig, J. C. LaCombe, A. Lupulescu, M. B. Koss, M. E. Glicksman, and R. Almgren, Phys. Rev. Lett. 82, 4496 (1999). 10. M. B. Koss, L. A. Tennenhouse, J. C. LaCombe, M. E. Glicksman, and E. A. Winsa, Met. Trans. A, (2000) (accepted). 11. J. C. LaCombe, M. B. Koss, and M. E. Glicksman, Phys. Rev. Lett. 83, 2997 (1999). 12. J. C. LaCombe, M. B. Koss, V. E. Fradkov, and M. E. Glicksman, Phys. Rev. E 52, 2778 (1995).

INTERFACES ON ALL SCALES D U R I N G SOLIDIFICATION A N D MELTING M. G. W O R S T E R Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Silver Street, Cambridge CBS 9EW, U.K.

Physics,

One of the challenges in predicting the solidification and melting of multicomponent materials is to describe the evolution of mushy layers: regions of mixed phase in which solid and liquid coexist in intimate contact. If mush is treated as a distinct phase of matter then it is necessary to determine the conditions that apply at, and control, its interfaces with adjacent solid and liquid regions. At a mush-liquid interface, different conditions apply depending on the direction of fluid flow, with which solute is advected, relative to the direction of advance or retreat of the phase boundary. A mathematically complete set of interfacial conditions is presented for the case of steady solidification or melting and is used to determine states of buoyancy-driven convection in a mushy layer.

1

Introduction

Solidification, the transformation of liquid into solid, involves the molecular rearrangement of matter, yet has consequences for global-scale phenomena such as the growth of the Earth's inner core and the freezing of the polar oceans. At every level of description, it is convenient to identify interfaces that separate regions of different phase or physical property. At the smallest scale usually considered, there is an interface separating solid from liquid. On planetary scales, it may be convenient to identify the layer of sea ice on the surface of the ocean as an interface between ocean and atmosphere, for example. Let me be more precise at this point and define an interface as a singular surface, a mathematical construction of infinitesimal thickness, separating two regions in which different field equations apply. The task for the mathematical modeller is to obtain or prescribe properties of the interface that relate the field variables on either side of it. Intrinsic or internal to the interface are physical processes and interactions that are unresolved at the level of mathematical description being used but may need to be explicitly investigated in order to derive the appropriate conditions to be applied at or across the interface. A concrete example is provided by the solid-liquid interface of a pure substance. Its temperature is given by T = Tm-7V-n,

(1)

where Tm is the equilibrium freezing temperature of the substance, 7 is related to the surface energy, and n is the unit normal to the interface pointing into the liquid. The temperature fields on either side of the solid-liquid interface are related by kan-VT\a-kin-VT\l=p3LVn,

(2)

where k is the thermal conductivity, p is the density, L is the latent heat of fusion per unit mass, Vn is the local rate of solidification, and subscripts I and s denote

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properties of the liquid and solid respectively. Two of the parameters in these equations, 7 and L, are properties of the interface resulting from molecular processes. The surface energy results from long-range, intermolecular, van-der-Waals forces, which tend to bind molecules more strongly to the solid when the interface is concave towards the liquid. The latent heat parameter is related to the change (decrease) in entropy as molecules become ordered within the crystalline lattice of the solid. Yet the molecular-scale description is unnecessary when calculating the macroscopic evolution of a solid-liquid interface: the continuum equations (1) and (2) suffice, and the parameters can be determined from thermodynamic measurements. Can sea ice be treated as an interface in the sense described above? This is a question to motivate the discussions presented herein, though the theories proposed have much wider application within a range of geophysical and industrial systems. Sea ice occupies a region no more than a few meters thick. Even though, as we shall see, many complex physical interactions take place within the interior of sea ice, its thickness is many orders of magnitude smaller than the scales of atmospheric and oceanic processes on either side of it. It would therefore be convenient in numerical models of ocean-atmosphere interactions, for example, to be able to relate properties of the ocean to those of the atmosphere at just the other side of the layer of sea ice in a simple and predictive way that does not involve detailed evaluation of the internal dynamics of the sea ice. As we try to address the question posed above, we shall be led to consider interfaces at a number of intermediate scales between the simple, solid-liquid interface and the complex sea-ice interface.

2

Sea Ice — Some Experimental Observations

In experiments designed to investigate the formation and growth of sea ice, Wettlaufer et al. [1] cooled aqueous solutions of sodium chloride (NaCl) through the roof of a rectangular container (Fig. la). In consequence of the fact that salt molecules cannot be readily incorporated within the cystalline lattice of solid ice, almost pure ice crystals grown from solution form a matrix whose interstices are filled with brine enriched by salt rejected by the growing ice. The two-phase region of ice crystals and interstitial brine is an example of a mushy layer, which form very commonly during the solidification of multi-component melts. Measurements of the salt concentration in the solution below the mushy layer (Fig. lb) show that the interstitial brine remains within the mushy layer until the layer has reached a critical thickness hc. Thereafter, brine drains from the mushy layer, increasing the concentration of the underlying liquid. The graph in Fig. l b delineates three stages of evolution: a period of relative quiescence when h < hc; a period of active salt transfer from the mushy layer to the liquid region; and a transition between these periods. The initial period and the transition have been studied quite extensively (see reviews [2] and [3]), while the period of active transfer, effected by compositional convection of the dense brine, is the subject of current research.

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(b)

10

12

14

16

Thickness (cm) Figure 1. (a) The experimental apparatus used to investigate the growth of sea ice. (b) The concentration of NaCl in the liquid region C in weight percent NaCl as a function of the thickness of the mushy layer h. The initial concentration is labelled Co. Rejected brine remains within the interstices of the mushy layer until its thickness exceeds a critical value hc •

2.1

The Quiescent Mushy Layer

The thickness of a mushy layer is controlled principally by a balance of heat fluxes [4]. Additionally, the thermal field within a mushy layer can be approximated as having a quasi-steady, linear profile. Grossly, therefore, the evolution of a quiescent mushy layer is governed by the equations

PsL-{H)

FA-F0

TQ-TA

= k{)

h

'

(3)

where To and TA are the temperatures in the ocean and atmosphere, Fo and FA are the heat fluxes from the ocean and into the atmosphere respectively, h is the thickness of the sea ice, and k((j>) is its mean thermal conductivity, which depends significantly on its mean solid fraction 4> (Fig. 2). The mean solid fraction can be determined from the lever rule

T0-TA

^-ibTTV

(4)

provided T is measured in degrees Celsius. Note that, during growth TA < To < 0 and is positive. This rather simple approach needs to be modified in the case of melting. We see, therefore, that during this period sea ice can indeed be treated as an interface: the field variables To,TA, Fo, and FA in the ocean and atmosphere are related by Eqs. (3) in terms of the interfacial parameters h and cf>, which themselves can be determined from values of the field variables. 2.2

The Transition

The onset of significant brine drainage from the mushy layer is a consequence of a buoyancy-driven instability. Linear theory [5] shows that the critical conditions for stability are governed by the magnitude of a Rayleigh number *^m —

p(T0-TA)gIlh mKV

(5)

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FA

Atmosphere TA Sea ice

Thickness h

Ocean

Solid fraction (,> °

i F0

Fr

Figure 2. The gross features of sea ice, treated as an interface between ocean and atmosphere. The aim is to relate the heat fluxes Fo and FA and the salt flux FQ to the external parameters To and T&, the temperatures of the ocean and atmosphere, and the internal parameters of the sea ice h and .

in which (3 is the constant of proportionality between the density of brine and its salinity, m is the slope of the liquidus curve (the concentration-dependent freezing temperature), g is the acceleration due to gravity, K is the thermal diffusivity, v is the kinematic viscosity, and II = II() is the permeability of the mushy layer, treated as a porous medium [6]. The permeability is difficult to measure directly but, given that the transition to brine drainage occurs at a critical value Rc of i? m , Eq. (5) shows that [n^r

1

= ( ~ ^ )

(To - TA)hc ex {T0 - TA)hc .

(6)

The data from laboratory experiments using a wide range of operating conditions [1] collapse to a single curve by relation (6) as shown in Fig. 3. This figure can be used as a phase diagram for brine convection from sea ice according to the external parameters TO,TA, and the internal (interfacial) parameters (h,
Brine Drainage

Once the critical depth hc is exceeded, brine is observed to drain from the mushy layer in the form of isolated plumes emanating from brine channels, which are narrow, essentially vertical, cylindrical channels in the porous matrix. Brine channels, also called chimneys in the metallurgical literature, form by dissolution of the solid phase of a mushy layer whenever there is flow of the interstitial liquid with a component parallel to the temperature gradient that exceeds the speed of the isotherms [9]. It will be convenient in the remainder of this article to consider situations in which a mushy layer grows upwards from a cooled lower boundary and the residual (interstitial) liquid is less dense than the original melt or solution. This is the case in the commonly studied system of ammonium chloride crystals growing from

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191

200 r

150

hc(T0-TA) m

Strong compositional convection through chimneys

100

(cm wt.%) 50 Weak convection :Xi.

0.0

0.2

,». 0.4

0.6

0.8

1.0

<$>C

Figure 3. The critical conditions for brine drainage from sea ice. The crosses are experimental data. The curve is hand-drawn through the data.

aqueous solution, as illustrated in Fig. 4, for example. The solution is depleted of the salt (NH4C1) forming the solid phase of the mushy layer and therefore has a tendency to rise convectively upwards through and out of the mushy layer. Once the convection is sufficiently strong, chimneys form, as shown in Fig. 4. The top of the mushy layer and the chimney wall are examples of mush-liquid interfaces. Our aim is first to determine appropriate conditions to apply at these interfaces and then to explore the nature of convection in mushy layers, through chimneys, with the aim of describing the whole convecting mushy layer as an interface. The convection in this system is similar to that in sea ice, with the sign of gravity reversed. 3

Interfaces W i t h Mushy Layers

Fig. 5 shows a close-up of the top of a mushy layer of ammonium-chloride crystals showing some of the scales at which different features can be resolved. As previously noted, mushy layers are porous media. Typical sizes of the interstices are in the range 6 ~ 0.1 — 1 mm, which are very much less than the macroscopic dimensions of the mushy layer. Theories of mushy layers do not resolve these scales and can be developed formally in the limit 6 —> 0 [10]. The interface between the mushy layer and the liquid region is ambiguous. It could be considered as the envelope of the tips of the crystals, but that leads to fractal uncertainties: the area of such an envelope, for example, is dependent on the yardstick used to define it. It is more convenient to think of the interface initially as a region of finite thickness Si, as shown in Fig. 5, and then let 6j —» 0. Provided 61 » 8, i.e., the limit 6 —* 0 is taken before the limit 8j —> 0, then the ambiguity is removed. We shall not formally take these limits here, but these ideas will direct our thinking. We are concerned with the nature of mush-liquid interfaces in the presence of flow, and it will be convenient to consider two different frames of reference for different purposes. I shall denote the dynamical fluid flow, flow relative to the solid phase

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Fi gure 4. A photograph showing two chimneys in a mushy layer of ammonium-chloride crystals grown from aqueous solution.

Liquid

Figure 5. The interfacial region of a mushy layer of ammonium chloride crystals growing from aqueous solution. The inter-dendritic spacing has a characteristic length scale <5, while the interface has a characteristic thickness <5/, which can be chosen to encompass any boundary layers in the solute or temperature fields, if necessary.

of the mushy layer, by u, and the motion of the liquid relative to the mush-liquid interface by q.

Interfaces on All Scales During Solidification and Melting 193

3.1

Conservation of Heat

Some interfacial conditions can be determined from conservation laws. Mathematically this can be achieved by applying the divergence theorem to the field equations over a control volume that spans the interface. Conservation of heat is governed in the liquid region by the advection-diffusion equation PiCpl(^

+ u-VT]=V-(k^T),

(7)

and in the mushy region by ^C;~+plCplu-VT

= V-(kS/T)+PsL^,

(8)

[2], where Cp is the specific heat capacity and an overbar denotes a volume-fractionweighted average. The solid fraction is now a local field variable of space and time. Eq. (8) applies equally in the liquid region (where it reduces to Eq. (7) as <j) = 0). It can be integrated (in enthalpy form) across the interface to give Pl

Cpln-q

[T}lm +psLVnl4>}lm = [kn. VT}lm ,

(9)

where n is the unit normal to the interface pointing into the liquid, and Vn is the normal rate of advance of the interface. Eq. (9) can be applied across any thermal boundary layer that might exist, which can be particularly convenient when there is vigorous thermal convection in the liquid region [11]. More commonly, the thermal field is fully resolved on the macroscale, in which case the temperature field is continuous and Eq. (9) reduces to Ps Li Ym

Vn = kn-VTm-kin-VTi , (10) which is similar to Eq. (2). But note that the interfacial value of (ft = <j>m is so far undetermined. 3.2

Conservation of Solute

Similar conservation laws apply to the solute field, governed in the mushy layer by ( l - ^ ) ^ + u - V C = V . (DVC) + (C-CS)^,

(11)

where Cs is the concentration of salt in the solid phase and D is the difFusivity of salt. Again, this equation reduces to the more familiar advection-diffusion equation in the liquid region. Integration of this equation across the interface gives - n • q [ C L + (Cm - Cs)4>m Vn = [-Dn • VC]lm .

(12)

At this point there enters a certain amount of controversy or, at least, uncertainty. Whereas the macroscopic dimensions of mushy layers are controlled by thermal balances, the microscale 6 is determined in part by the solute field. One view is that the scale of solute variations 6c, including any variations within the interfacial region of the liquid, is comparable to 6. In this case, the diffusive flux of solute is unresolved when the homogenizing limit 8 —> 0 is taken, and Eq. (12) reduces to -n-q[C}lm

+ (Cm-Cs)4>mVn

= 0.

(13)

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Note that the jump in the macroscopic solute concentration [C] cannot be determined without invoking further physical principles, and may take on any value, just as discontinuous shear layers can exist in an inviscid fluid. An alternative view is that 6c is larger than 6, at least in the interfacial region of the liquid, where a compositional boundary layer may exist of a thickness larger than the inter-crystalline spacing, and perhaps too in the mushy region, where morphological instabilities giving rise to side branches on the primary dendrites are limited only by surface energy on much smaller scales. In this case, and if one chooses to resolve any compositional boundary layer on the macroscale, the macroscopic solute field is continuous and Eq. (12) becomes (Cm - Cs)cj>m Vn = \-Dn • WCfm .

(14)

If we subsequently choose our smallest resolved scale to be the scale of thermal variations ST, then the limit 6c —* 0 can be taken appropriately given that the ratio of diffusivities D/K
Melting Interfaces

The conservation laws described above are sufficient to determine the evolution of mush-liquid interfaces that are melting, V„ < 0. Although the equations are coupled, it is convenient to think of the thermal conservation equation (10) as determining the rate of advance (retreat) of the mush-liquid interface once the interfacial value of the solid fraction is determined by solute conservation. It is instructive to consider separately what happens in two cases: when the flow is from liquid to mush (q • n < 0); and when the flow is from mush to liquid (q • n > 0). In the first case (Fig. 6a) advection balances diffusion in the interfacial region of the liquid, creating a compositional boundary layer there, which is governed locally by dC

nd2C nc\ (15) ^ ^ where q = n • q < 0 and n is the normal coordinate pointing into the liquid. This has the solution q

= D

C = C0 + {Cm-

C0)e«n/D

,

(16)

where Co is the concentration in the liquid far from the interface and Cm is the interfacial value of the liquid concentration. This can be used in Eq. (14) to determine that Co — Cm\ n • q Cm ~ Cs ) Vn

,

The same conclusion is reached from Eq. (13) if one chooses not to resolve the boundary layer, but recognizes that in this case the solute field may be discontinuous: [C] =C0 -CmWhen the flow is from mush to liquid (q • n > 0) no compositional boundary layer can be sustained at the interface (Fig. 6b). Provided that q • n 3> d6c/dt,

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Figure 6. A mushy layer melting into a hot liquid, (a) When flow is from liquid to mush, advection can balance diffusion of solute on the liquid side of the interface and give rise to a compositional boundary layer there, (b) When flow is from mush to liquid, any compositional boundary layer is advected away from the interface.

the compositional gradient at the interface is vanishingly small and in consequence <> / = 0 there. This is typically the case for a melting interface (one at which the phase change is rate limited by heat transfer), but may not be for a dissolving interface (one at which the phase change is rate limited by solute transfer). Note in all these discussions that because the temperature and composition are tied by the liquidus relation T = TL(C) in the interior of a mushy layer, no compositional boundary layer can exist there, the compositional gradient is proportional to the temperature gradient, and the solute flux on the mush side of the interface is negligible when D/K -C 1. 3.4

Solidifying Interfaces

It needs to be recognized that the differential operators acting on the solid fraction in a mushy layer are hyperbolic. During melting the characteristics of the hyperbolic operator flow towards the interface, whereas they flow away from the interface, into the mushy layer, during solidification. Consequently, an additional boundary condition is required during solidification over and above the conservation laws already described. A common suggestion is arbitrarily to impose < 0 in the mushy layer), and we shall see later that in some circumstances this

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iM' 1

&

(a,) 4-

Marginal Equilibrium

Figure 7. If the length scale over which the solute field varies in the liquid is greater than the dendrite spacing (ai), then the solute field is resolved after the homogenizing limit 5/ —> 0 is taken. Marginal equilibrium in this picture requires that the temperature gradient is equal to the gradient of liquidus temperature at the interface (bi). If the limit of negligible solutal diffusivity is subsequently taken then picture (c) is achieved, in which the solute field is continuous across the, now unresolved, compositional boundary layer. Alternatively, if the dendrite spacing is comparable to the thickness of the compositional boundary layer (3,2), then the homogenizing limit leads to picture (b2), in which the boundary layer is unresolved, but the possibility exists of a j u m p in the solute field. The principle of marginal equilibrium applied to this picture leads again to (c).

condition is not independent from the conservation laws and therefore leads to an under-determined system of equations. An alternative is to apply a principle of marginal equilibrium which says simply that the liquid should not be supercooled (its temperature should be at or above its liquidus temperature) adjacent to the interface. This principle has a physical basis in that mushy layers are caused by supercooling and will certainly relieve if not eliminate it [12]. The principle has so far always led to well-posed mathematical problems with physically-allowable solutions, though these properties are unproven in general.

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When the flow is from liquid to mush and the possibility of a compositional boundary layer exists, the principle of marginal equilibrium is expressed by

&T-"^'

(18)

where TL(C) is the local liquidus temperature [12]. This is illustrated on the lefthand side of Fig. 7. If the compositional boundary layer is unresolved, illustrated on the right-hand side of Fig. 7, the principle of marginal equilibrium is expressed simply by T = Ti(C) on the liquid side of the interface. This in turn implies that the potential jump in solute concentrate [C] — 0 across the interface. Once the limit D/K —> 0 is taken in the first case, both approaches lead to the same result, namely that = 0 at the interface. A rather different situation applies when the liquid flows from mush to liquid [13]. Solute is then advected away from the interface into the liquid region, no compositional boundary layer can exist, and [C] = 0. The fact that 0), then solute is conserved along streamlines in the liquid region. Therefore, during steady solidification, such as might occur in a crystal-pulling configuration or during continuous casting, q-VC = 0

(19)

in the liquid region, where q is measured relative to the steady frame of reference. The principle of marginal equilibrium in this case is expressed by q-VT>q-VTL(C)=0

(20)

on the liquid side of the interface, the final equality being a consequence of Eq. (19). On the other hand, on the mush side of the interface, q . VT = m{C - CS)VS • V4> < 0 ,

(21)

where V s is the velocity of the solid matrix in the steady frame. This equation follows from Eq. (11), and the inequality is required in order that <j> be non-negative. Since <> / = 0 at the interface, Eq. (10) shows that the temperature gradient is continuous. On the assumption that the velocity field is also continuous, the two inequalities (20) and (21) combine to show that q • VT = 0

(22)

at the mush-liquid interface. The interfacial conditions that apply between liquid and mushy regions during steady solidification and melting are summarized in Fig. 8. 4

Steady Convection in a Mushy Layer

Armed with these interface conditions we can investigate steady convection in a mushy layer and follow the evolving morphology of the mush-liquid interfaces as the

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Melting Solute conservation Flow from mush to liquid

0=0

Solidifying Solute conservation

Marginal equilibrium

<> / = 0

q • VT = 0

[C] = 0

Flow from liquid to mush

0^0 see equation (17)

0 =0

<=

[C}=0

Figure 8. Interfacial conditions between the mushy and liquid regions during steady melting or solidification. When flow is from mush to liquid there is no compositional boundary layer, the solute field C is continuous, and solute conservation implies that the solid fraction is zero at the interface. Marginal equilibrium at a solidifying interface has a different expression depending on the direction of flow. When flow is from liquid to mush it is this condition in combination with solute conservation that implies that rp is zero.

2.5

2

Streamlines of q Solid-fraction contours

Figure 9. Convection in a mushy layer just after the formation of a region of negative solid fraction, shown shaded near the centre of the figure. The solid curves are streamlines of q. The dashed curves are contours of solid fraction.

amplitude of convection increases and chimneys are formed. At small amplitudes of the dynamic flow u, the streamlines of q all enter the mushy layer from the overlying liquid region. At some larger amplitude, after the vertical component of u has exceeded the solidification rate, an internal region of the mushy layer is completely dissolved to form a liquid inclusion [13], as shown in Fig. 9. Computation of the subsequent evolution of such a liquid inclusion requires the application of all four types of interface condition described above.

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(a)

199

(b)

Figure 10. (a) Streamlines of q (solid curves) and contours of solid fraction (dashed curves) once a chimney has formed. The chimney is modelled as a singular interface (shown with a thick vertical line) along which boundary conditions are applied that are derived from a lubrication analysis of the flow internal to the chimney, (b) The computed width of the chimney a. Note the greatly exaggerated horizontal scale. These calculations were made with a Darcy number of Da = 0.005 and a Rayleigh number of 12.9.

4-1

Singular Chimneys

Recent numerical calculations [14] have followed the evolution of the liquid inclusion until it bursts through the upper mush-liquid interface to form a chimney. Such computations, in which the flow and structure of the chimney are resolved, are timeconsuming. Significant savings can be made by exploiting the fact that chimneys are long and narrow. Lubrication theory can be used to analyse the flow in the chimney [15-18], the results of which provide boundary conditions for the mushy layer at the chimney wall. By this means, the entire chimney is treated as an interface with a single intrinsic parameter (its width, a) that can be determined as part of the solution. In brief, conservation of heat and mass determine the conditions at the chimney wall [18]

i,: RaC

Ipx (23) 20 v 3RaU ~Da where ip is the stream function such that q = (—ipz,ipx ~ l)i To is the temperature at the bottom of the chimney, and Da = HV2/n2 is the Darcy number. These apply along the chimney wall, indicated by a thick vertical line in Fig. 10. Below the chimney, the boundary conditions at x = 0 are simply that tp = Tx — 0. The width of the chimney a can be determined from the interfacial conditions

Tx=lPTz,

q • V T = 0,

where ipz > 0,

(C - Cs)Vn = (d - C)q • n,

where i>z < 0.

(24) (25)

The first of these is simply Eq. (22) applied where the flow enters the chimney. The second is Eq. (17) applied where the flow exits the chimney (if anywhere), recognizing that the solute concentration is constant along streamlines in the chimney, so that C\ = C\ (ip) is the value of the concentration where the exiting streamline entered the chimney. These conditions have recently been applied successfully [19] to determine the structure and flow within a convecting mushy layer including a determination of the width and shape of the chimney (Fig. 10). Note that in this model the chimney width is determined locally by thermodynamic balances.

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Epilogue

The theory of mushy layers allows the gross properties (solid fraction, bulk composition) of a solidifying multi-component melt to be determined without computing the microstructure of the internal solid-liquid interfaces. We have seen how the mush-liquid interface can be determined in a variety of circumstances and, further, how chimneys within a mushy layer can be treated themselves as singular interfaces between a convecting mushy layer and the adjacent liquid region. This approach provides a relatively inexpensive way to compute the state of a convecting mushy layer and the flux of solute emanating from it. The results of such calculations should direct the search for simple ways to predict the convective heat and salt fluxes from mushy layers in terms of their gross properties. Such results will not only be of significant benefit to climate modellers, for whom sea ice remains an important and inadequately understood interface between ocean and atmosphere, it will enable simple predictions to be made of macrosegregation in cast alloys and mineral segregation in igneous rocks. Thus, in terms of both natural philosophy and advanced materials processing, theories of solidification, with all their attendant interfaces, pose important and challenging problems for the 21st century. Acknowledgments The recent advances in our understanding of solidification in mushy layers reported in this paper have been made in collaboration with post-doctoral colleagues D. M. Anderson, T. P. Schulze, and C. A. Chung. The first two are among the many 'academic children' of Stephen Davis. I count myself fortunate to have benefitted thus from Steve's nurture of young scientists and for the inspiration he has given me more directly in the study of solidifying systems. I am grateful to Joseph Chung, Herbert Huppert, and Tim Schulze for their helpful comments on an earlier draft of this article. References 1. J. S. Wettlaufer, M. G. Worster, and H. E. Huppert, J. Fluid Mech. 344, 291 (1997). 2. M. G. Worster, in Interactive Dynamics of Convection and Solidification, eds. S. H. Davis, H. E. Huppert, U. Muller, and M. G. Worster, Kluwer (1992). 3. M. G. Worster, Ann. Rev. Fluid. Mech. 29, 91 (1997). 4. H. E. Huppert and M. G. Worster, Nature 314, 703 (1985). 5. M. G. Worster, J. Fluid Mech. 237, 649 (1992). 6. 0 . E. Phillips, Flow and Reactions in Permeable Rocks, Cambridge University Press, Cambridge, U.K. (1991). 7. G. Amberg and G. M. Homsy, J. Fluid Mech. 252, 79 (1993). 8. D. M. Anderson and M. G. Worster, J. Fluid Mech. 302, 307 (1995). 9. M. C. Flemings, Solidification Processing, McGraw Hill (1974). 10. P. W. Emms, D.Phil. Thesis, Oxford University, Oxford, U.K. (1993). 11. R. C. Kerr, A. W. Woods, M. G. Worster, and H. E. Huppert, J. Fluid Mech.

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216, 323 (1990). 12. M. G. Worster, J. Fluid Mech. 167, 481 (1986). 13. T. P. Schulze and M. G. Worster, J. Fluid Mech. 388, 197 (1999). 14. T. P. Schulze and M. G. Worster, in Interactive Dynamics of Convection and Solidification, eds. P. Ehrhard, D. S. Riley, and P. H. Steen, Kluwer (2001). 15. P. H. Roberts and D. E. Loper, in Stellar and Planetary Magnetism, ed. A. M. Soward, Gordon k Breach (1983). 16. M. G. Worster, J. Fluid Mech. 224, 335 (1991). 17. A. C. Fowler, Mathematical Models in the Applied Sciences, Cambridge University Press, Cambridge, U.K. (1997). 18. T. P. Schulze and M. G. Worster, J. Fluid Mech. 356, 199 (1998). 19. C. A. Chung and M. G. Worster, J. Fluid Mech., in press.

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PHASE A N D MICROSTRUCTURE SELECTION IN PERITECTICS

WILFRIED KURZ AND STEPHANE DOBLER Swiss Federal Institute of Technology Lausanne, 1015 Lausanne EPFL, Switzerland Peritectic alloys show a great variety of solidification microstructures. In recent years analytical theories of dendritic growth and of transient growth with nucleation ahead of the moving interface have been applied to describe the essentials of phase and microstructure selection. Coupled peritectic two-phase growth was also recently observed. Among the manyfold microstructures; banded structures, controlled by nucleation, convection, and solute trapping are very important in peritectics. Finally, consecutive phase transformations (liquid/solid and solid/solid) take place and lead to two transformation fronts, each one having its own growth behaviour. This multitude of structures constitutes a great challenge for theoretical analysis.

1

Introduction

Peritectic alloys are technologically very important. Alloys such as steels, bronzes, Al-alloys, permanent-magnet materials, High-Tc superconductors, etc., play a major role in present and future devices and machines. During the solidification and solid-state transformation of peritectic alloys a great variety of microstructures are formed. These can be steady-state microstructures or oscillating structures, singlephase microstructures or two-phase in-situ composites from stable or metastable phases. In certain compositional ranges, consecutive phase transformations over a large temperature interval, including liquid/solid and solid/solid transformations, can be observed. Despite the complex interface response of this technically very important class of alloys and their great scientific interest, many of the abovementioned phenomena have not received the necessary attention. An overview of the major observations and physical reasons behind the formation of the various interface morphologies will be given. It is hoped that this will stimulate future research into the physics and mathematics of peritectic phase transformations. The following subjects (limited to directional growth) will be treated: growth of steady-state interfaces, such as dendrites, cells, and in-situ composites; phase competition (stable to metastable phase transition); oscillating interfaces (banding controlled by nucleation, convection, and solute trapping); and consecutive liquid/solid and solid/solid transformations. Finally, some open questions will be presented. 2

S t e a d y - S t a t e Interfaces

There are several steady-state interface morphologies that have to be considered for the discussion in the following sections: • single-phase plane fronts, • cells,

203

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W. Kurz & S. Dobler

• dendrites, • in-situ composites. The first three growth morphologies are single phase and are the result of different growth conditions. For a given temperature gradient, G, an increase of the interface velocity, V, will change the growth morphology from planar to cellular to dendritic. At higher rates, the growth morphology will again produce cellular structures and, at rates where the atom attachment kinetics become dominant, high-velocity bands form. At even higher interface velocities, a plane front is stabilized again. The stability of solidification fronts has been analysed in much depth by Davis and co-workers. For example, Huntley and Davis [1] show the limits of morphological and oscillatory instabilities for directionally-solidified Al-Fe alloys (Fig. la). Their results compare well with observations obtained in rapid laser solidification experiments. From this diagram, it can be seen that for one composition, four intersections with the solutions of the stability analysis are produced. These correspond to i) the transition from a morphologically stable to an unstable plane front, Vc, ii) beginning of (not observable) solute-trapping-controlled plane-front oscillation, V , hi) absolute stability, Va, and iv) the upper limit of the oscillating regime, VMIt is interesting to compare these results with dendrite growth theory. Fig. l b shows the interface response in the form of a T\ — V curve as calculated with the IMS (Ivantsov-Marginal-Stability) model [3,4]. In this figure, the above mentioned transition velocities are also indicated. From a comparison of both approaches, stability analysis and dendrite modelling, the complementarity of the results of these models can be seen. The interface response shown in Fig. l b allows semiquantitative modelling of the different phase and microstructure transitions (see below) that are for the moment too difficult to be analysed quantitatively, although Karma and coworkers [5] have produced some interesting first results. Coupled, two-phase growth of in-situ composites is another interesting phenomenon of peritectic alloys (see Fig. 2). This has been clearly shown only recently [6-8] and was a result of the fact that the alloys used had a very narrow solidusliquidus interval and the low-velocity limit of morphological stability was still experimentally attainable. These structures closely resemble eutectic composites. 3

Phase Competition

In peritectic alloys, generally two solid phases, one stable, the other metastable, compete with each other. The transition from one phase to the other is controlled by nucleation and transient growth phenomena. Present day modelling of the complex phase-selection processes uses: (i) steady-state growth theory to determine the interface response; and, (ii) for directional growth, a maximum temperature criterion, assuming that nucleation is easy enough not to limit the process. One proceeds as follows [4,9]. 1. Determine the interface response (Fig. lb) of all possible phases (i = a, /3, 7, ...) for a fixed temperature gradient and composition T*=T(V)i\G,Co-

(1)

Phase and Microstructure

Selection in Peritectics

205

(a) banding - 9xp€rimtnts • (Crimaud tt ul.) 0.01

0.001 L - — — 10* 10"'

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10 - '

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/

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10-5

\<3*

10'3

10'2

10-'

10°

10'

^ m

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102

Velocity (m/s)

C

D

C

Figure 1. Stability ranges for various microstructures (plane front, P; cells/dendrites, C/D; and high-V bands, B) for Al-Fe alloys and a thermal gradient of 5000 K / m m . (a) Instability diagram from Huntley and Davis [1] showing morphological and oscillatory instability windows including results from laser experiments by Gremaud et al. [2]. (b) Interface response for plane front, T P (V), and dendritic/cellular growth, T^{V) for 1 at.% Fe.

2. Apply a selection criterion between the competing phases and growth morphologies assuming abundant nucleation of all phases so that the observed, phase-growth morphology, P, is given by the maximum of its interface temperature. P = max {T?},i =

a,/3,-y,....

(2)

206

W. Kurz & S. Dobler

Ss

•I?

x»

US

&o

u

Figure 2. Longitudinal section of a coupled, lamellar microstructure in Fe-4.22 at.% Ni at the quenched interface, where the S phase is dark and the 7 phase is a light colour (V = 10 /xm/s, G = 1.2 x 10 4 K/m).

Fig. 3 shows the phase selection procedure for the peritectic Fe-4.3 at.% Ni alloy where the stable 6 phase competes with the metastable 7 phase. The corresponding phase diagram is shown in Fig. 3a with stable and metastable phase equilibria. The interface response is shown in Fig. 3b. It can be seen that, depending on the velocity, the phase and the morphology having the maximum growth temperature varies. Considering only the maximum growth temperature criterion, the growing phase/microstructure is 7 plane front at low solidification velocities. At intermediate velocities 6 cells/dendrites are observed, then 7 dendrites/cells are seen at slightly higher velocities. Finally, the 7 plane front returns at the highest solidification velocities. The selected phase is not necessarily the thermodynamically most stable phase, but is a result of both equilibrium and growth kinetics. This approach is however incomplete and does not always lead to the right answer. A Nucleation-Constitutional-Undercooling (NCU) criterion was therefore developed by Hunziker et al. [10]. Due to nucleation of 8 phase ahead of the interface, 7 plane front is not stable at low velocity, even though it has the maximum growth temperature. The NCU criterion considers the solute concentration profile in front of the solidification front and deduces the equilibrium liquidus temperature profile of the metastable phase. The growing phase may be destabilized by nucleation of the metastable phase when there is a constitutionally undercooled zone, and the nucleation undercooling, ATn, is small enough. This may also lead to an oscillating interface (see the next section).

Phase and Microstructure

Selection in Peritectics

207

1794 1792 & 1790 1788

1 7 8 6

„

3

3.5

c* 4

^

cr45

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55

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•i

(b)

1792 & 1790 H 1788 1786 10"*

t0" s

10*4

I0a 102 VJm/s]

10-'

10°

I0 1

Figure 3. Microstructure/phase selection in Fe-4.3 at.% Ni alloy, (a) Fe—Ni phase diagram (solid curves are stable, dashed curves are metastable). (b) Interface temperature as a function of the solidification velocity for 6 and 7 phases for G = 1.75 x 10 4 K / m .

4

Oscillating Interfaces

In peritectic alloys oscillating interfaces often produce alternating microstructures or phases. Three kinds of such interfaces have been observed forming banded structures by control through: nucleation, convection, and solute trapping. 4-1

Nucleation-Controlled

Banding

Such banding can be obtained at low velocities (V < Vc) when the initial composition, Co, is in the range delimited by both solid phases C* and C 7 in Fig. 3a. The 6 phase will nucleate first and develop a plane front. An initial transient is established and the solid-liquid interface temperature approaches progressively T*(Co). Below the peritectic temperature Tp > J"/(Co), the 7 phase will nucleate. Then 7 overgrows 6. During the new transient to T 7 (Co), the solid-liquid interface temperature rises above the peritectic temperature, and the 6 phase may nucleate again [11]. This cycle of alternating 6 and 7 phases forms a banded microstructure that is shown in Fig. 4a [12,13]. These bands appear at low velocities and therefore convection is generally present except in cases where the crucible has a very small

208

W. Kurz & S. Dobler

0.6 mm

(5 mm

(a)

(IJ)

IUU

nm

(C)

Figure 4. Banded microstructures (growth direction up), (a) Nucleation-controlled banding in Sn-0.9 wt.% Cd alloy, V = 3 /im/s and G = 2.3 x 10 4 K / m . Alpha phase is dark and the beta phase is a light colour, (b) Convection-controlled banding in a Sn-0.075 wt% Cd alloy, V = 4 / a n / s and G = 2.3 x 10 4 K / m [13]. (c) Solute-trapping-controlled banding where cellular morphology is dark and plane front is a light colour [2,19].

diameter (as is the case in Fig. 4a) [14].

4-2

Convection-Controlled Banding

In directional solidification experiments with radial heating, convection is induced by lateral thermal gradients. The direction of flow close to the solid-liquid interface depends on the position and the number of convection rolls formed, which is related to the dimension and geometry of the crucible. For the simplest case of two convection rolls, the rejected solute is transported from the center to the edge inducing a lateral chemical gradient. Thus, the liquid in the center becomes poorer in solute whereas the edge is enriched. If 6 phase is growing first, then 7 phase can nucleate at the edge. A tree-like structure is obtained by the repeated lateral movement of the three-phase junction (6, 7, liquid) due to convection effects (see Fig. 4b) [13,14].

Phase and Microstructure

4-3

Solute-Trapping-Controlled

Selection in Peritectics

209

Banding

In this case there is no phase competition, but competition of two growth morphologies such as dendrites/cells and plane front of the same phase. This can happen in all alloys including peritectics. In Fig. lb, the T(V) relation has a positive slope between the interface temperature minimum, Vm and the interface temperature maximum, VM- The positive slope is due to solute trapping. Coriell and Sekerka [15] and later Davis and coworkers [1,16,17] showed that for plane-front growth, solute trapping leads to oscillatory instabilities. Karma and Sarkissian [18] showed that the oscillatory instability induces large-amplitude relaxation oscillations at a critical velocity. These planar perturbations increase the solidification interface velocity as less solute is rejected and therefore the front accelerates to a velocity where the plane front is morphologically stable. To conform with the thermal field the solidification velocity decreases, developing again a cellular morphology. An oscillating structure of plane front and dendrites/cells is then obtained as shown in Fig. 4c [2,19]. 5

Consecutive Transformations

For alloys having a chemical composition lower than C$, two consecutive phase transformations are observed in directional transformation. Fig. 5 shows both transformation fronts where the upper front is formed by S plane-front solidification, and the lower front is formed by cellular 6 to 7 solid-state transformation. As discussed above, the growth morphology for liquid/6 solidification evolves with increasing solidification velocity from plane front, to cells, to dendrites, to cells, and finally again to plane front at very high solidification velocities. Fig. 6 shows the growth temperature for the various structures as a function of the solidification velocity. At the solidification velocity of 5 /zm/s, the expected morphology is plane front, which is actually observed (see Fig. 5). In the case of the 6/7 solid-state transformation, the diffusion coefficient of solute (Ni) is much lower in 7 iron than in 6 iron. This means that during the solidstate transformation there is a solute pile-up at the transformation front similar to solidification, i.e., volume diffusion in the parent phase. As the solute diffusion coefficient is much smaller in S than in liquid, the interface response function for solid-state transformation is displaced to smaller velocities. Therefore a "high velocity" cellular solid-state transformation front is observed behind a low solidification velocity plane front [20]. This is shown in Fig. 6, where the interface responses for solidification and for solid-state transformation are given. Due to the low absolute stability velocity in solid-state transformations, this phenomenon can be easily followed. Comparison with theory shows reasonable agreement [20]. 6

Conclusions

In summary, analysis of solidification in peritectic alloys is difficult due to the fact that two phases compete with each other. Thus, the microstructure selection

210

W. Kurz & S. Dobler

3

Liquid

\

/

^

^

<»)

Composkiois

.L/5 (b) &7

Figure 5. Consecutive transformation, (a) A schematic of the liquid/6 plane front followed by the cellular 6/7 solid-state transformation. The related phase diagram is also shown, (b) The micrograph is for V = 5 /um/s and Co = 2.75 at.% Ni and shows the two transformation fronts growing with G = 1.15 x 10 4 K / m at a distance of 2.25 mm [20].

depends on the phase and the growth morphology. Four cases can be encountered in the phase-selection process: i) single phase, ii) alternately-growing phases, iii) simultaneously-growing phases, and iv) solid-state transformation. • Single phase: The domain of stable growth of a phase has to be treated by considering nucleation in a constitutionally undercooled zone. Single phase bands of two oscillating growth morphologies can appear at high solidification velocities. • Alternating growth of two phases: A nucleation-controlled, banded two-phase microstructure is observed when neither of the two phases are stable relative to the nucleation of the other phase during transient solidification. Each phase nucleates alternately in front of the other growing phase. Convection-controlled banding occurs when large local variations of chemical composition are induced by fluid flow. • Simultaneous two-phase growth: Simultaneous two-phase growth has been observed with different morphologies: isothermal lamellar and fibrous growth, and coupled growth of cells and plane front. This growth requires very high G/V values to stabilize plane front growth of both phases.

Phase and Micro structure Selection in Peritectics

211

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1799

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V [m/s] Figure 6. Interface responses for liquid/6 and 6/7 transformations for Fe-2.9 at.% Ni and (?liqUid/6 = 1.2 x 10 4 K/m, G^z-f = 1.5 x 10 4 K/m. The unstable solidification front forms needle-like cells while the cells in the solid state are plate-like due to crystal anisotropy effects

[20].

• Solid-state transformation: In consecutive transformation experiments two interfaces can be observed. Analysis of cell morphology, cell spacing and of absolute stability have been undertaken and lead to new insight in solid-state transformation mechanisms. Despite recent progress in the analysis of peritectic solidification and solid-state transformation, there are some important open questions that are worth further detailed study. Solidification • Nucleation-controlled phase selection, including convection

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W. Kurz & S. Dobler

• Isothermal and non-isothermal coupled in-situ composite growth • Coupled growth instabilities • Lamella to fiber transition in coupled growth • Transition from banding to coupled growth • Nucleation- and convection-controlled interface oscillations Solid-State Growth • Interface-energy anisotropy effects • Stress effects • Transformation with interface diffusion, with product-phase diffusion • Solute trapping at solid/solid interfaces • Solid-state interface oscillations • Massive transformations References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

D. A. Huntley and S. H. Davis, Acta Metall. Mater. 41, 2025 (1993). M. Gremaud, M. Carrard, and W. Kurz, Acta Metall. Mater. 39, 1431 (1991). W. Kurz, B. Giovanola, and R. Trivedi, Acta Metall. Mater. 34, 823 (1986). R. Trivedi and W. Kurz, Inter. Mater. Rev. 39, 611 (1994). A. Karma and A. Sarkissian, Metall. Mater. Trans. A 27, 635 (1996). J. H. Lee and J. D. Verhoeven, J. Cryst. Growth 144, 353 (1994). M. Vandyoussefi, H. W. Kerr, and W. Kurz, Acta Mater. 48, 2297 (2000). T. Okane and T. Umeda, submitted to Acta Mater. W. Kurz and P. Gilgien, Mater. Sci. Eng. A 178, 171 (1994). O. Hunziker, M. Vansyoussefi, and W. Kurz, Acta Mater. 46, 6325 (1998). R. Trivedi, Metall. Mater. Trans. A 26, 1583 (1995). W. Kurz and R. Trivedi, Metall. Mater. Trans. A 27, 625 (1996). W. J. Boettinger, S. R. Coriell, A. L. Greer, A. Karma, W. Kurz, M. Rappaz, and R. Trivedi, Acta Mater. 48, 43 (2000). A. Karma, W.-J. Rappel, B. C. Fuh, and R. Trivedi, Metall. Mater. Trans. A 29, 1457 (1998). S. R. Coriell and R. F. Sekerka, J. Cryst. Growth 61, 499 (1983). D. A. Huntley and S. H. Davis, J. Cryst. Growth 132, 141 (1993). D. A. Huntley and S. H. Davis, Phys. Rev. B 53, 3132 (1996), A. Karma and A. Sarkissian, Phys. Rev. E 47, 513 (1993). M. Carrard, M. Gremaud, M. Zimmermann, and W. Kurz, Acta Metall. Mater. 40, 983 (1992). M. Vandyoussefi, H. W. Kerr, and W. Kurz, Acta Mater. 45, 4093 (1997).

MODEL P H A S E D I A G R A M S FOR A N FCC ALLOY R. J. BRAUN Department

of Mathematical Sciences, Newark, DE 19716,

University U.S.A.

of

Delaware,

J. Z H A N G Department of Engineering Sciences and Applied Northwestern University, Evanston, IL 60208,

Mathematics, U.S.A.

J. W . C A H N A N D G. B . M C F A D D E N National

Institute of Standards and Technology, Gaithersburg, MD 20899, U.S.A. A. A. W H E E L E R

Faculty

of Mathematical Studies, Highfield, Southampton,

University of Southampton, S017 1BJ, U.K.

We describe a mean field, free-energy model that allows the computation of realistic phase diagrams for a particular set of ordering transitions in face-centered-cubic (FCC) binary alloys. The model is based on a sixth-order Landau expansion of the free-energy function in terms of the three nonconserved order parameters that describe ordering on the underlying FCC lattice. When combined with appropriate gradient-energy terms, this model will allow the self-consistent calculation of energetic and kinetic anisotropies of phase boundaries in future work.

1

Introduction

Equilibrium states of crystalline solids can undergo a variety of phase transitions as the temperature and concentration of the material is varied [1]. In this paper, we will be concerned with order-disorder transitions for a face-centered-cubic (FCC) binary alloy that is composed of A and B atoms. In such a transition, the structure and spacing of the underlying lattice remains unchanged, but the arrangements of the A and B atoms on the lattice sites can vary with temperature and concentration. For this purpose, the FCC lattice can be regarded as consisting of four interpenetrating simple cubic sublattices. The origin of one sublattice can be taken as the corner site of a unit cell and that of each of the other three sublattices is centered on one of the faces (see Fig. 1). At high temperatures, but below the melting point, the equilibrium state is typically a disordered phase, where the ratio of A and B atoms on each sublattice is the same. The structure is FCC and the four sublattices are equivalent by the symmetry of the FCC lattice. At lower temperatures, ordered phases can be favored, in which case there is a symmetry breaking. For the bulk-ordered phases we consider here, the ratio of A and B atoms are uniform on each sublattice, but the ratio varies from sublattice to sublattice. An example of such an ordered phase occurs in the copper-gold system, in which the CU3AU phase at concentrations near 25% Au consists of gold atoms preferentially occupying one sublattice, which for convenience is taken to be the one at the corner of the unit cell. Since the gold enrichment could have occurred on any of the sublattices, there

213

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R. J. Braun et al.

are four variants. The symmetry of the ordered structure has become that of a primitive cubic lattice. Then, the copper atoms preferentially occupy the remaining three sublattices, a lattice complex composed of the cell faces, in which these three sublattices remain equivalent by symmetry. The CU3AU phase is an example of LI2 ordering in the Strukturbericht notation system [2]. Another example that we consider here is the Llo phase, which represents an ordered phase such as CuAu at concentrations near 50% Au; in this case the symmetry breaking splits the four sublattice into two pairs of two equivalent sublattices, and the symmetry of a primitive-tetragonal lattice. The ordering results in alternating planes of Cu and Au normal to one of the original cube axes, with six possible variants. In the original FCC unit cell the Cu preferentially occupies the corners and one of the sets of faces, and Au the other two sets of faces. Interfaces can occur between any two variants of the same phase over the range of temperature and concentration where this phase is stable. Interfaces can also occur between any two phases at temperatures where these phases coexist, with the concentrations of the phases dictated by equilibrium conditions as depicted by phase diagrams. In previous work, we have examined gradient energy formulations of the diffuse interfaces produced by such order-disorder transitions in FCC alloys [3-5] and HCP alloys [6], the interphase boundaries (IPBs) between an LI2 phase and the disordered FCC phase, and antiphase boundaries (APBs) that separate different variants of the same ordered phase. The formulation permitted the examination of the orientation dependences of the free energy of these interfaces and their mobility. The derivation of these models focused on obtaining a continuum, free-energy functional with a form of the gradient-energy terms that is consistent with the underlying symmetry of the atomic lattice. However, we employed a simplified bulk-free-energy density that, while it permits an easy calculation of the interface properties, has long been known to give an unrealistic phase diagram [7]. Efforts to compute a phase diagram for an FCC binary alloy date back to Shockley and coworkers [7]; the diagram that resulted from the Bragg-Williams approximation used there was topologically different from, and a poor approximation to, the experimental phase diagram for the prototype Cu-Au system [8]. Many efforts to improve upon the situation followed [2,9,10]; realistic phase diagrams for the Cu-Au system were obtained by using the cluster variation method (CVM) and asymmetric multiparticle interactions [11]. The CVM and Monte Carlo (MC) methods have had a number of successes modeling alloy systems of varying complexity; the reader is referred to recent reviews [2,12] for discussion of those methods. Thus, in the last twenty years it has been possible to obtain a realistic phase diagram from statistical mechanical models, particularly for the CVM [2,9,11,13]. However, CVM models are cumbersome and their existence is often ignored [14]. They have been used to make discrete calculations of the energy of the interfaces (e.g., [13,15]). But such discrete calculations of interfaces do not lend themselves to studying the orientation-dependent properties. In this paper, we describe an improved model for the bulk-free-energy density that allows a more realistic description of the equilibrium phase diagram of the system, but still simple enough to be used for describing the orientation dependence of the energy and motion kinetics of the interfaces. It is our hope that its simplicity will find a wide variety of

Model Phase Diagrams for an FCC Alloy

215

Figure 1. A schematic diagram of an FCC lattice. To describe ordering in this model there are four distinguished sites corresponding to a corner and one each for the faces intersecting at that corner.

applications. Our method has elements in common with the sublattice models developed by Sundman, Dupin and others [16-19]. These models have been used successfully to produce realistic phase diagrams for FCC alloy systems. Their method features many more parameters than our model; in practice, these parameters are determined by fitting experimental measurements of the phase diagram and other thermodynamic quantities, such as the latent heat and heat capacity. Here, we show that realistic phase diagrams may also be obtained using our simple model; we anticipate that this simplicity will ultimately be advantageous in the self-consistent calculation of energetic and kinetic anisotropies of phase boundaries. The paper is organized as follows. The formulation is given in Sec. 2; the method for finding phase diagrams and some results are given in Sec. 3. Finally, some discussion and conclusions are presented in the last two sections. 2 2.1

Formulation The Concentration and Order Parameters

We briefly recall the mean-field description of the order-disorder transitions on an FCC lattice given in Ref. [4]. A binary alloy (denoted A-B) on an FCC crystal lattice is described geometrically by four interpenetrating cubic sublattices, with sublattice occupation densities pj defined at the lattice points shown in Fig. 1. The four densities represent the local atomic fraction of species A on each sublattice; their specification is assumed to characterize the overall state of the crystal. It is convenient to introduce four new parameters W, X, Y, and Z defined by W

= - ( p i + p2 + P3 + PA) ,

X = ~(pl+P2~P3-

Pi) ,

(1) (2)

216

R. J. Braun et al.

Y=-(Pl-p2+P3-

p4),

Z = - (pi - Pi - P3 + PA) •

(3) (4)

These relations can be inverted to give px = W + X + Y + Z,

(5)

p2 = W + X - Y - Z,

(6)

p3 = W-X

(7)

+ Y-Z,

p4 = W - X - Y + Z,

(8)

which can be interpreted in terms of the expression [14] p = W + X cos 2-Kx/a + Y cos 2iry/a + Z cos 2irz/a,

(9)

which gives the relations (5-8) upon evaluation at the corners and face centers of the conventional unit cell in Fig. 1; here a is the cubic-lattice constant. Thus, parameter W represents the atomic fraction of the system as a whole and is a conserved order parameter. X, Y, and Z are non-conserved order parameters that can vary between plus and minus one half, and are Fourier coefficients representing density variations along the directions of the crystal axes. In this model the disordered state is represented by px = P2 = Pz = PA = W, which implies that X = Y — Z = 0. Ordered states are characterized by non-zero values for the non-conserved order parameters. For instance, the four equivalent variants of A 3 B L l 2 ordering are |X| = \Y\ = \Z\ ^ 0 with XYZ < 0, and the four variants of AB 3 Ll 2 ordering are \X\ = \Y\ = \Z\ ^ 0 with XYZ > 0. The six variants of L l 0 ordering are X = Y = 0 with \Z\ ^ 0, Y = Z = 0 with \X\ ^ 0, and Z — X = 0 with \Y\ 7^ 0, with layering respectively in the xy, yz, and zx planes. A bulk equilibrium state is characterized by constant values of the densities and order parameters. 2.2

The Thermodynamic

Description

A thermodynamic description of the crystal for the case of an isothermal system is based on the bulk Helmholtz free energy density (per mole) F(X,Y,Z,W,T). In Ref. [4], we considered an energy model based on pair-wise interactions that leads to an unrealistic phase diagram with a multiphase critical point. The problem lies in the positive definiteness of the fourth-order terms in the expansion of the entropic part of F. Pair-wise interaction energies make no contribution to these fourth-order terms. In the present work, we develop an improved model for the freeenergy density, by introducing negative fourth-order contributions to the energy to make the fourth-order free-energy terms no longer positive definite. We are then able to obtain a more realistic description of the binary-alloy phase diagram. The idea behind our modification can be motivated by familiar examples from the bifurcation theory of scalar systems [20]. For a fourth-degree, free-energy density at W = 1/2, the Llo ordering is a second-order phase transition in which the order parameter changes continuously from zero to non-zero values as the temperature passes through its critical value Tc. A scalar model with Y — Z — 0 for the

Model Phase Diagrams for an FCC Alloy

(a)

217

(b)

Figure 2. Schematic bifurcation curves for the order parameter X versus the temperature T near a critical point Tc. Stable and unstable solutions are indicated by solid and dashed curves, respectively, (a) A fourth-degree theory results in a second-order phase transition: the disordered state (X = 0) changes continuously into the ordered state (X ^ 0) at Tc. (b) A sixth-degree theory results in a first-order phase transition: locally stable ordered and disordered states exist on either side of To and have equal energies at To (dotted curve). The order parameter X(T) for the minimum energy state jumps discontinuously as the temperature passes through To.

free-energy density of the Llo transition is F(W, X, T) = f0(W, T) + f2(W, T)X2 + f4{W,

T)X\

(10)

with / 4 (1/2,T) > 0 and / 2 (1/2,T) ~ (T - Tc). For T > Tc the system is in stable equilibrium for the disordered state X = 0, whereas for T < Tc the lowest energy state is an ordered phase with X2 = —f2/(2/4)- The ordered and disordered states are never simultaneously stable, as illustrated in Fig. 2a. A first-order phase transition can be obtained from a sixth-order model such as F(W, X, T) = f0(W, T) + f2(W, T)X2 + f4(W, T)X4 + f6(W,

T)Xe

(11)

where / 2 (1/2,T) ~ (T - Tc) as before, but now / 4 (1/2,T) < 0 and / 6 (1/2,T) > 0. In this illustrated in Fig. 2b, the minimum energy state changes discontinuously from a disordered phase to an ordered phase with X2 = —f4/(2fe) a t the temperature T0 > Tc for which f2 = ft/(^fo)A simplified model [4] that includes the LI2 transition with |X| = \Y\ = \Z\ ^ 0 involves, in addition, a cubic term fo(W,T)XYZ. The coefficient fz(W,T) of the third-order term in the free-energy expansion depends strongly on the composition, and in a symmetric model is antisymmetric about W = 1/2 and equals 0 there. At compositions on either side of W = 1/2, the sign of XYZ can always be chosen to make the term fa(W,T)XYZ negative, with X = ±Y = ±Z. Thus, in a fourthdegree theory, LI2 is always favored over Llo near the critical point Tc, except exactly at W = 1/2. In a sixth-degree theory with negative-definite fourth-order terms, the Llo transition is first order and has a critical point at To > Tc. The

218

R. J. Braun et al.

energy differences between the Llo and LI2 phases at the lower temperature Tc are then unimportant, and the Llo phase can be favored over the LI2 phase near W = 1/2 and T = T0. The free-energy model used in our previous work featured a fourth-degree polynomial expansion with positive definite fourth-order coefficients. This expansion was adequate for the description of the LI2 transition near W = 1/4, but necessarily lead to a multicritical point at W = 1/2 and a second-order phase transition for the Llo structure, whose temperature dependence is similar to Fig. 2a. Our improved models in this paper are based on four-atom interactions in the energy which add negative fourth-order terms of a large enough magnitude to make the fourth-order term negative in a polynomial expansion at W = 1/2. As before the sixth-degree terms result entirely from the entropy and are positive. The main goal of the present work is to extend the model to higher degree in order to obtain a more realistic treatment of all the transitions, and to find appropriate choices for the dependence of the free-energy density on temperature and concentration so that realistic phase diagrams for these order-disorder transitions can be obtained. There are enough parameters in a four-atom interaction model to affect the behavior when W ^ 1/2. In the next two subsections, we describe the models of the internal-energy density E and entropy S that are used to obtain the improved free-energy density F — E — TS that we use in this work. 2.3

The Internal Energy

The internal energy per mole, E, of the system is assumed to be characterized by four-atom cluster energies E40, E31, E22, £13, and £04 of the various occupations of the near-neighbor basic tetrahedral form by A4, A3B, A2B2, AB3, and B4, respectively, that allow for four-particle interactions according to the scheme [11]: E = £4O01020304 + E31 [p!p2P3(l - Pi) + 01/02(1 - 03)04 + 0l(l - 02)0304 + (1 - 0l)020304] + #22 [0102(1 - p 3 )(l - p4) + pi(l - 02)03(1 - 04) + 0l(l - 02)(1 - 03)04 + (1 - 0l)(l - 02)0304 + (1 - 0l)02(l - p3)04 + (1 - 0l)0203(l - p4)] + #13 [0l(l " 02)(l - 03)(l " 04) + (1 - 0l)02(l - p 3 )(l " 04) + (1 " 0l)(l - 02)03(1 " 04) + (1 - 0l)(l " 0 2 )(1 - 03)04] + (12)

£O4(l-0l)(l-02)(l-03)(l-04).

By substituting Eqs. (5-8) into the above expression, we obtain an internal energy of the form E = e 0 + e2(X2 + Y2 + Z2) + e3XYZ e42(X2Y2

+ X2Z2 + Y2Z2),

+ e 4 i(X 4 + F 4 + Z4) + (13)

where the e^ are given in the appendix in terms of £ 4 0 , E31, E22, £13, and E04.

Model Phase Diagrams for an FCC Alloy

2-4 Entropy

219

Approximations

The point approximation to the molar entropy S of the system is given by the expression R 4 S(p1,p2,P3,PA} = --^2l{pj),

(14)

3= 1

where J(a;)=xln(x) + ( l - a ; ) l n ( l - a ; ) ,

(15)

and R is the universal gas constant. This expression for the entropy is the sum of the contribution from each sublattice. We explore two choices based on Eq. (14). One option is to use the expression as is, which is somewhat complicated by the presence of the logarithmic terms, which generally forces one to compute numerical solutions to the problem. Another option is to expand this expression in a power series about the points pj = 1/2, and truncate the series after sixth degree. The result allows more progress to be made analytically. We obtain

(16) Inserting the definitions (5-8) into this expression results in polynomial expressions involving U — W — 1/2 and the order parameters X, Y, Z up to sixth degree. 2.5

Free-Energy Approximations

The form of the Helmholtz free-energy density, F = E — TS, follows from Eq. (13) and is given by an expression of the form F(X, Y, Z, W, T) = e0 + e2(X2 + Y2 + Z2) + e3XYZ e42(X2Y2

+ eAl(X4 + YA + ZA) + +X2Z2

2 2

+Y

Z)

+ ^Yjl{p3),

(17)

where we must use Eqs. (5-8) to get the entropy contribution in terms of the order parameters. Using Eq. (16), the resulting approximation for the free energy takes the form F(X, Y, Z, W, T) = f0 + f2(X2 + Y2 + Z2) + f3XYZ + f41(X4 + Y4 + Z4) + f42(X2Y2 + X2Z2 + Y2Z2) + f51XYZ(X2 +Y2 + Z2) + f61(X6 + Y6 + Z6) + 4 2 f62{X (Y + Z2) + Y4(X2 + Z2) + Z4(X2 + Y2)} + f63X2Y2Z2, (18)

220

R. J. Braun et al.

where the coefficients /;, for parameter choices suggested by those of Kikuchi and de Fontaine [11] discussed above, are given in the appendix. In Sec. 4.1 we shall give the results for a phase diagram using this polynomial approximation. 3

Phase Diagrams

A phase diagram consists of curves in the temperature-concentration plane that describe conditions for equilibrium between various bulk phases at the same temperature, but not necessarily at the same concentration. Mathematically, this is a minimization of the free energy subject to a fixed amount of the total amount of concentration W in the system, which results in a common tangent construction in terms of the free energy of the system [21]. The free energy is also an unconstrained minimum with respect to the non-conserved order parameters Xj. This procedure produces several sets of equations to solve for the concentration between stable bulk phases that delineate regions where two or more phases can coexist. To describe the computation of phase diagrams, we first discuss the bulk equilibrium states that are supported by the model. 3.1

Bulk Phases

In the case of the disordered FCC phase, we have X = Y = Z = 0; for high enough temperatures this will occur for any overall concentration W. We denote the resulting free energy for the bulk FCC phase by FFcc(W,T); for the case in Eq. (17), we then have FFcc{W, T) = F(0,0,0, W,T) = e0(W) + RT [1{W)\.

(19)

For the L l 0 variant with Z / 0, the corresponding free energy becomes

FLI0(Z,W,T)

FLl0(Z,W,T)

= F(0,0,Z,W,T) = e0(W) + e2(W)Z2 + e41Z4 R: +^-mW + Z)+I(W-Z)} ~2

(20)

for this phase. Finally, for the Ll 2 variant with X = Y = Z ^ 0 the free energy FLy2 {Z, W, T) becomes FLl2{Z, W,T) = F(Z,Z,Z,W,T) = e0(W)+ 3 e3(W)Z + 3(e4l+2eA2)Z4

3e2(W)Z2 +

^ p [X{W + 3Z) + 3X(W - Z)\. 3,2

FCC-LIQ

+

(21)

Transition

Conditions under which an equilibrium between the disordered FCC phase and the ordered Llo phase can coexist are described by the system of nonlinear equations c)F ^ ( Z 0 , W o ) = 0,

(22)

Model Phase Diagrams for an FCC Alloy 221 d

^ ( Z o , Wo) =

-^{WFCC)

= Mo,

(23)

and FFCC{WFCC)

- FLlo(Z0,

Wo) - no {WFCc - W0) = 0,

(24)

for a given T. Eq. (22) results from minimizing the free energy with respect to Z, and Eq. (23) is the result of minimizing with respect to W subject to a specified overall concentration, leading to the appearance of the Lagrange multiplier /j,0. Eq. (24) completes the common tangent construction and expresses the fact that the energy is also stationary with respect to variations in the interface position. The unknowns in these equations are the concentration of the disordered phase Wpcc a n d the concentration and order parameter of the Llo phase, WQ and Zo, respectively. These equations were solved using DNSQ [22,23] using suitable initial guesses. These guesses were generated from the To curve, which is, in turn, found by setting the free energies of the two phases equal and requiring dF/dZ — 0. This strategy was used in all cases where there were coexisting ordered and disordered states. 3.3

FCC-L12

Transition

In this case, we must satisfy the system of nonlinear equations ^(Z2,W2)

(25)

= 0,

9FLi2

dFFCc

m

,

^

(26)

and FFcc(WFcc)

- FLU (Z2, W2) - ^2

(WFCC

- W2) = 0,

(27)

for a given T. The unknowns are the concentration of the disordered phase WFcc and the concentration and order parameter of the LI2 phase, W2 and Z2 respectively. 3.4

LI0-LI2

Transition

In this final case, we must satisfy the system of nonlinear equations ^L(Z0,Wo)=0,

(28)

^(Z2,W2)=0,

(29)

^(Z0,W0)

= ^(Z2,W2)

= „lf

(30)

and FLl0(Z0,W0)

- FLl2(Z2,W2)

- /*! (Wo - W2) = 0,

(31)

222

R. J. Brawn et al.

for a given T. The unknowns are the concentrations and order parameters for the respective ordered phases. 4

Results

We next present results of numerically-computed phase diagrams using these equations. The results are given in non-dimensional form by using the energy scale — w to nondimensionalize the free energies and the temperature scale —u>/R to nondimensionalize T. 4-1

Results Using the Sixth-Degree Landau Expansion

We first describe a phase diagram based on the free-energy function given in Eq. (18), using the parameters /o(tf,T) = ~ + 6 C / 2 + T ( l n i + 2 i r 2 ) ,

(32)

h (U, T) = -4 + 40t/ 2 + 2T,

(33)

3

f3(U) = -250(U-aU ),

(34)

/4i = - 4 ,

(35)

/42 = 0,

(36)

/ei = 4,

(37)

/62 - - 3 ,

(38)

fes = 6,

(39)

where U = W - 1/2. With the choice a = 2 and (3 « 27.735894, we will have equal dimensionless temperatures of T = 2.5 at the congruent points of the FCC-L1 2 and FCC-Llo order-disorder transitions near W = 1/4 and W = 1/2, respectively. The choice of the last two parameters, f^ and /g3, in the free-energy function (18) was made so that the sixth-degree terms would drop out of the problem when X = Y = Z and leave the analysis of the bulk LI2 phase unaffected. By solving the required nonlinear equations using the sixth-degree polynomial form and these coefficients, we arrive at the phase diagram shown in Fig. 3. This phase diagram is an idealized, symmetric approximation to the Cu-Au binary system [8]. The choice —LO/R « 265 K would locate the congruent points at about the right temperatures for Cu-Au. The qualitative appearance of the phase diagram for temperatures in the vicinity of the congruent points is satisfactory, but the behavior of the system in the dilute limit is unrealistic. The situation is improved by retaining the logarithmic terms in the entropy contribution to the free-energy density, and we focus our attention on this model in the remainder of the paper. 4-2

Results Using the Free Energy with Logarithmic Terms

We next discuss examples of phase diagrams obtained by using the free-energy model given by Eq. (17) with coefficients given in Table 1. The first case is a qualitative model for the Cu-Au system, and the last two are for demonstration

Model Phase Diagrams for an FCC Alloy

2.6

223

1

fee fee /

2.4

L1 0

1*1 2.2

/

2.0 0.0

0.2

0.4

W Figure 3. A model phase diagram for the Cu-Au system from the parameters listed in Eq. (32). Note that it is symmetric about W = 1/2.

Table 1. Coefficients used for the internal-energy contribution to the free-energy density; here U = W- 1/2. Case eo{U) e2{U) e3(U) e-41

e42

I 6(7 2

- 4 + t/2

200£/(l - 2U2) -12 15

II 6(7* -A + U2 200(7(1 - 2U2) -5 10

III W'2 -4 + U2 100(7(1 - 2U2) -12 15

purposes. We note that with these coefficients, the limit of metastability for the disordered phase for W = 1/2 (corresponding to the temperature Tc in Fig. 2b) occurs at a dimensionless temperature T = 2 in all three cases. With the coefficients of case I, the phase diagram in Fig. 4 is obtained. This diagram is qualitatively similar to that for the Cu-Au system [8], but with symmetry about W = 1/2. This model diagram should be sufficient for the purposes of studying the surface-tension anisotropy of IPBs in our future work. a For the parameters of Case II, we find that the phase diagram has a lowered and dramatically smaller region where the FCC-L1 0 transition occurs; see Fig. 5 and compare to Fig. 4 for Case I. As en rises toward the value —8/3, the F C C - L I Q °In order to obtain quantitative agreement with the congruent point of the FCC-LI2 transition, we would choose —ui/R RS 248 K.

224

R. J. Braun et al.

H

2.3

0.10

0.20

0.30 W

0.40

0.50

Figure 4. A model phase diagram for the Cu-Au system for the parameters of Case I. Note that it is symmetric about W = 1/2.

transition region disappears and becomes a multicritical point at e^i = —8/3. The phase diagram for the parameters of case III is given in Fig. 6. In this case, the congruent point for the FCC-LI2 transition is just outside the FCC-L1 0 coexistence region; further decrease in e% will result in a peritectoid phase diagram. We can summarize the effects of the parameters as follows. The size of e% controls the location of the congruent point for the FCC-LI2 transition. If it is sufficiently large, then the congruent point is shifted to values symmetrically located about W — 1/2 with temperature greater than 2. The value of e^x controls the location of the congruent point for the FCC-Llo transition. If e^x < —8/3, the congruent point remains at W = 1/2 but occurs at T > 2. If e 41 > —8/3, the congruent point is at T = 2 and the transition is second order. Using these parameters we have been able to obtain satisfactory phase diagrams that would allow the description of the variation of concentration across both interphase and antiphase boundaries. 5

Discussion

The solutions to the nonlinear equations that give the curves on the phase diagram become difficult to compute when the dimensionless temperature drops below about 2 for the case with logarithmic terms in the entropy. The difficulty appears to be that the values of the order parameters and concentration are approaching the

Model Phase Diagrams for an FCC Alloy

3.0

!

'

225

1

fee 2.5

fee

-

yT

L12

1.0

0.00

V\

/

h- 2.0 -

1.5

•

I

1

0.10

0.20

\

0.30

0.40

0.50

W Figure 5. The phase diagram for the parameters of Case II. The region where Llo is the sole stable phase is near (W,T) = (0.5,2). The multicritical point at e,ji = —8/3 is being approached and the FCC—Llo transition is disappearing.

0.50

Figure 6. The phase diagram for the parameters of Case III.

226

R. J. Braun et al. 0.50

0.40

0.30 N

0.20

0.10

0.00

Figure 7. The dashed lines show the limits of the allowed ranges for the arguments of the In functions. The solid curves are plots of the order parameters found along the coexistence region boundaries for the parameters of Case I. The lower curves near the middle of the plot are for the LI2 boundaries and the curves in the upper right are for the Llo boundaries.

limits of the ranges allowed by the logarithmic terms. For example, in the freeenergy density for the LI2 phase, there is a term in the argument of the In function that requires 1-W-3Z>0;

(40)

solving for Z gives (41) From the Llo phase, we can also conclude that we must have W>Z.

(42)

We may plot the boundaries of these inequalities along with the solutions to the nonlinear equations that give the phase diagram in the (W, Z)-plane; this has been done in Fig. 7. The temperature is decreasing as the solid curves (representing the coexistence-region boundaries) approach the edges of their allowed ranges (dashed lines). The jumps in slope occur at the ends of the tie line at the eutectoid temperature for Case I. We hypothesize that when the solutions get too close to these boundaries, the iteration procedure breaks down because the procedures allow iterates beyond the boundaries. We believe that we can recast the equations to eliminate the logarithmic terms and thus alleviate this problem; this is allowed by the special form of the free-energy density.

Model Phase Diagrams for an FCC Alloy

6

227

Conclusion

We have been able to compute a simple model phase diagram with an approach that is a modification to the quasi-chemical description. The modifications were rooted in the choices of Kikuchi and coworkers [11,15] in their successful CVM approaches, but we have modified the coefficients to suit our needs in drawing the phase diagram. We found that two coefficients (e$ and e 4 i) in the internal-energy contribution to the free energy made the biggest contributions to controlling the resulting phase diagram. The coefficient e 4 i of the terms involving Xf strongly affects the temperature of the congruent point for the FCC-Llo transition; there is a multicritical point at e4i = —8/3, at which the first-order nature of the transition disappears. Elsewhere, we will show that as one approaches the multicritical point for the FCC-Llo phase transition, there is a quadratic decay in the interfacial energy of the interphase boundaries for any orientation [24]. The size of the coefficient e3 plays a crucial role in setting the temperature and concentration of the congruent point in the FCC-L1 2 transition. When it is larger, the congruent point occurs at larger temperatures. When it decreases, the congruent point disappears inside the FCC-Llo coexistence region and a peritectoid phase diagram occurs. We have not yet computed the peritectoid phase diagram due to numerical difficulties with the logarithmic terms; recasting the root-finding procedure to eliminate these terms should alleviate this difficulty. Acknowledgments It is a pleasure to dedicate this paper to S. H. Davis on the occasion of his sixtieth birthday, and to acknowledge the tremendous influence of his research and his friendship on the many students and colleagues that have had the good fortune of associating with him. The authors are also grateful for helpful discussions with S. Langer and W. J. Boettinger during the preparation of this manuscript. This research was conducted with the support of the Physical Sciences Research Division of NASA and the University of Delaware Research Foundation. Appendix If we begin with Eq. (12), and substitute the expressions (5-8) to eliminate the occupation densities pi, we obtain the expression E = ^ [ 4 ^ 3 1 + 6^22 + 4£i3 + £ 0 4 + £4o] + U[E3i - E13 - EM/2 + Em/2\ 2

+

3

[/ [3£ 4 0 /2 - 3-E22 + 3£W2] + t / [ - 4 £ 3 i + 4 £ i 3 - 2E0A + 2E40] + UA[-AEZ1 + 6£ 2 2 - 4 £ i 3 + £ 0 4 + £40] + (X 2 + Y2 + Z2){E22 - £ 4 0 / 2 - £ 0 4 / 2 + U{4E31 - 4£i3 + 2£ 0 4 - 2E40] + U [SE31 — \2E22 + 8E13 — 2E04 — 2.E40]} + XYZ{-8E31 + 8£i3 - 4E 0 4 + 4£ 4 0 +

228

R. J. Braun et al.

U[-2,2E31 + 48£ 2 2 - 32£ 1 3 + &E04 + 8E40)} + (X2Y2 + X2Z2 4

4

+ Y2Z2){8E31

4

(X + Y + Z ){-AE31

- 12E22 + SE13 - 2E04 - 2E40} +

+ 6£ 2 2 - 4E13 + E04 + E40},

(43)

where U = W - 1/2. We note that if E40 = E31 = E22 = E13 = E04 = 1, then E=\. A simple bond counting argument is based on the bond energies EAA, EAB and EBB between the A-A, A-B, and B-B pairs of atoms, respectively. If we assume that EAA = EBB and use a reference energy corresponding to EAA, then the energies of the different configurations according to this scheme are E40 = 0,

E31 = 3w,

£22 = 4w,

E13 = 3w,

E04 = 0,

(44)

where u> = EAB — EAA, SO that the configurational energies are thus characterized in terms of the energy u. Substituting these expressions into Eq. (43) gives E = EQ + AIO[X2+Y2

+ Z2].

(45)

This expression for the internal-energy density is inadequate for the purposes of drawing phase diagrams [7]. A better model for the multiparticle interactions in the internal energy, as given by Kikuchi and de Fontaine [11] and used in Kikuchi and Cahn [15], is to take E40 = 0,

E31 = 3w(l + a),

E22 =

4LJ,

E13 = 3w(l + 6),

EM = 0, (46)

where the constants a, b, and u) were determined from a best fit to the phase diagram for the case of the Cu-Au system: a = 0.01,

b = -0.08,

^ = -663 K.

(47)

With this scheme, we obtain more terms in the internal-energy density of Eq. (13), and the coefficients e* in that expression are given by e 0 = ^to(U2 - 1/4) [4 + a + b + 4(a - 6)17 + 4(o + b)U2} , e2 = 2w + QwU[a -b + 2(a + b)U], e3 = 12u(a + b)[b - a - 4(a + b)U], e4i = -6w(a + b),

e 42 = 12a;(a + b).

(48)

These expressions are used to motivate the choices made for the energy coefficients shown in Table 1. For the polynomial free energy (18), we have the coefficients /o = \w {(U2 - 1/4) [4 + a + b + 4(a - b)U + 4(a + b)U2]} +

RT Uu2 + ^U4 + ^U6 + ln(l/2)^ , h = 2u + toU[a - b + 2(a + 6)17] + RT [2 + 8U2 + 32C/4] , h = 12w[6 - a - 4(a + b)U] + RT [32C7 + 256C/3] ,

Model Phase Diagrams for an FCC Alloy

/ 4 i = -6u)(a + b) + RT

229

| + 32[/ 2

/ 4 2 = 12w(a + b) + RT [8 + 192C/2] , f51=256RTU,

/ 6 1 = 32/2T/15,

f62 = 32RT,

/ 6 3 = 192i?T.

(49)

These expressions are used to motivate the choices made for the energy coefficients shown in Eqs. (32-39). References 1. J. W. Christian, The Theory of Transformations in Metals and Solids, Part I, Pergamon Press, Oxford, U. K. (1975). 2. F. Ducastelle, Order and Phase Stability in Alloys, North-Holland, New York (1991). 3. R. J. Braun, J. W. Cahn, J. Hagedorn, G. B. McFadden, and A. A. Wheeler, in Mathematics of Micro structure, Evolution, ed. L.-Q. Chen et al., TMS/SIAM, Philadelphia, PA, p. 225 (1996). 4. R. J. Braun, J. W. Cahn, G. B. McFadden, and A. A. Wheeler, Trans. Roy. Soc. London A 355, 1787 (1997). 5. R. J. Braun, J. W. Cahn, G. B. McFadden, H. E. Rushmeier, and A. A. Wheeler, Acta Mater. 46, 1 (1997). 6. J. W. Cahn, S. Han, and G. B. McFadden, J. Stat. Phys. 95, 1337 (1999). 7. F. C. Nix and W. Shockley, Rev. Mod. Phys. 10, 1 (1938). 8. H. Okamoto, D. J. Chakrabarti, D. E. Laughlin, and T. B. Massalski, Bull. Alloy Phase Diagrams 8, 454 (1987). 9. R. Kikuchi, Prog. Theo. Phys. Supp. 87, 69 (1986). 10. R. Kikuchi and H. Sato, Acta Metall. 22, 1099 (1974). 11. R. Kikuchi and D. de Fontaine, Application of Phase Diagrams in Metallurgy and Ceramics, ed. by C. G. Carter, NBS Special Publication 496, 967 (1978). 12. D. de Fontaine, Solid State Phys. 47, 33 (1994). 13. A. Finel, in Ordering and Disordering in Alloys, ed. A. Yavari, Elsevier Applied Science, New York, p. 182 (1992). 14. A. G. Khachaturyan, Prog. Mater. Sci. 22, 1 (1978). 15. R. Kikuchi, and J. W. Cahn, Acta Metall. 27, 1337 (1979). 16. I. Ansara, B. Sundman, and P. Wilemin, Acta Metall. Mater. 36, 977 (1988). 17. N. Dupin, Contribution a devaluation thermodynamique des alliages polyconstitues a base de nickel, Ph.D. Thesis, Laboratoire de Thermodynamique et de Physico-Chimie Metallurgiques de Grenoble, Institut National Polytechnique de Grenoble (1995). 18. I. Ansara, N. Dupin, H. L. Lukas, and B. Sundman, J. Alloys and Compounds 247, 20 (1997). 19. B. Sundman, S. G. Fries, and W. A. Oates, Calphad 22, 335 (1998). 20. S.-N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, Berlin (1982). 21. P. Haasen, Physical Metallurgy, 3rd ed, Cambridge University Press, Cambridge, U. K. (1996).

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22. SLATEC Common Math Library, National Energy Software Center, Argonne National Laboratory, Argonne, IL. The program Snsq was written by K. L. Hiebert and is based on an algorithm of Powell [23]. 23. M. J. D. Powell, "A hybrid method for nonlinear equations," in Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed. Gordon and Breach (1988). 24. G. Tanoglu, R. J. Braun, J. W. Cahn, G. B. McFadden, and A. A. Wheeler, in preparation.

PART 2

C O N T R I B U T E D ABSTRACTS

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233

I N F L U E N C E OF CONTACT-ANGLE C O N D I T I O N S O N EVOLUTION OF SOLIDIFICATION F R O N T S VLADIMIR S. AJAEV AND STEPHEN H. DAVIS Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208

We consider contact-line effects in directional solidification of a dilute binary alloy in a Hele-Shaw cell. We find steady basic-state solutions that satisfy the contact-angle conditions at the walls and then study their stability. Perturbation methods are used for values of the contact angle that are close to ir/2, when the front is almost planar, while a numerical boundary integral method is used for arbitrary contact angles. We find that under physically realistic conditions the curved interface is less stable than the planar front, which is in agreement with experimental observations. The numerical solution is in agreement with the perturbation result in the region where the latter is valid. A weakly nonlinear analysis is carried out in the framework of the same perturbation approach. As the deviation from the planar interface becomes more significant, the region of subcritical bifurcation is expanded. For values of thickness near one-half of the wavelength of the Mullins-Sekerka instability the three-dimensional steady-state structures can be observed even when the interface is normal to the wall; these structures are anti-symmetric with respect to the center plane of the Hele-Shaw cell. As the contact angle deviates from 7r/2, these shapes become unstable and the steady-state interface is symmetric with respect to the center plane. For values of gap thickness much larger than the characteristic diffusion length we derive a long-wave evolution equation with appropriate boundary conditions at the walls and use it for numerical studies of strongly nonlinear evolution of the system. We find that formation of deep roots and secondary bifurcations are promoted due to contact-line effects for concave down (toward the solid) interfaces.

234

C R E E P I N G S T E A D Y T H I N FILM ON A N INCLINED P L A N E WITH A N EDGE NURI AKSEL Department of Applied Mechanics and Fluid Dynamics, University of Bayreuth, D-95440 Bayreuth, Germany

Film flows occur in various technical processes, in environmental sciences, and everyday life. Hence, they were the subject of many investigations, e.g., [1-5], to name a few. Here, we study the influence of a sharp edge on the surface shape and the velocity profile of a creeping steady thin film with high surface tension on an inclined plane. Within the framework of the lubrication assumption an asymptotic solution is constructed [6]. A parameter-free direct relation between the surface shape and the wall friction at the bottom is found that gives rise to an indirect detection of the wall friction by measuring the surface shape. It is shown that capillary effects make the surface shape rise to form a maximum in the near front of the edge similar to the problems with moving film fronts. The region of validity and the sensitivity of the asymptotic solution on the upstream and downstream boundary conditions and on the inclination angle are found by comparing with exact numerical results. A numerical parameter study with respect to the capillary number and inclination angle is performed. Depending on the parameter combinations various surface shapes can be obtained. The theoretical (analytical and numerical) results are verified by experiments. References 1. E. B. Dussan V. and S. H. Davis, J. Fluid Mech. 65, 71 (1974). 2. B. G. Higgins, W. J. Silliman, R. A. Brown, and L. E. Scriven, Ind. Eng. Chem. Fundam. 16, 393 (1977). 3. B. G. Higgins and L. E. Scriven, Ind. Eng. Chem. Fundam. 18, 208 (1979). 4. K. N. Christodoulou and L. E. Scriven, J. Fluid Mech. 208, 321 (1989). 5. P. M. Schweizer, J. Fluid Mech. 193, 285 (1988). 6. N. Aksel, Arch. Appl. Mech. 70, 81 (2000).

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PHASE-FIELD SIMULATION OF C O N V E C T I V E EFFECTS ON DENDRITIC GROWTH GUSTAV AMBERG AND ROBERT TONHARDT Department of Mechanics, KTH, S-100 44 Stockholm, Sweden

Phase-field simulations have been used to study the evolution from a small nucleus to a dendrite in an undercooled melt, in the presence of convection. As a generic case for growth from solid boundaries, the nucleus is in one case assumed to be attached to a solid wall, and a shear flow parallel to the wall is assumed in the melt. The interaction between the melt flow and the growing dendrite results in a changed growth rate and morphology. One aspect that has been investigated is how the effective tilting of the main stem of the dendrite depends on the flow strength. It is shown that the major factor that determines the tilt angle is the selection of a nucleus with an optimal initial crystal orientation. We have also studied natural convection around a nucleus held in an undercooled melt. The phase-field equations are solved together with the equations for viscous fluid flow using an adaptive finite element method. This method allows a large computional domain, so that a single dendrite in an effectively unbounded region can be studied.

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A MODEL FOR A S P R E A D I N G A N D MELTING D R O P L E T ON A HEATED S U B S T R A T E DANIEL MANDERSON Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030 M. GREGORY FOREST Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599 RICHARD SUPERFINE Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599

We develop a model that describes the dynamics of a spreading and melting droplet on a heated substrate. The model, developed in the capillary-dominated limit, is geometrical in nature and couples the contact line, tri-junction, and phase-change dynamics. In this model the competition between spreading and melting is characterized by a single parameter KT that represents the ratio of the characteristic contact-line velocity to the characteristic melting velocity. A key component of the model for the spreading and melting droplet is an equation of motion for the solid. This equation of motion, which accounts for global effects through a balance of forces over the entire solid-liquid interface, including capillary forces at the trijunction, acts in a natural way as the tri-junction condition. This is in contrast to models of tri-junction dynamics during solidification, where it is common to specify a tri-junction condition based on local physics alone. The tri-junction dynamics, as well as the contact angle, contact-line position, and other dynamic quantities for the spreading and melting droplet are predicted by the model and are compared to an isothermally spreading liquid droplet whose dynamics are controlled exclusively by the contact line. Our interest in this comparison has been motivated by experiments performed by Glick (1998) in which polystyrene spheres show different spreading characteristics when subject to these two different thermal configurations. We find that in general the differences between the dynamics of a spreading and melting droplet compared with that of an isothermally spreading droplet are increased as KT increases, that is, when the characteristic contact-line speed is greater than the characteristic melting speed. The presence of the solid phase in the spreading and melting configuration inhibits spreading relative to an isothermally spreading drop of the same initial geometry. Finally, we find that increasing the effect of spreading promotes melting.

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INSTABILITIES OF A THREE-DIMENSIONAL LIQUID D R O P L E T ON A HEATED SOLID SURFACE STEVEN W. BENINTENDI Department of Mechanical and Aerospace Engineering, University of Dayton, Dayton, OH 45469 MARC K. SMITH The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405

The behavior of thin liquid films and liquid droplets is of fundamental importance in many natural and technological processes, such as the coating of surfaces, the recovery of oil from porous media, the development of microfluidic devices, and a host of applications in material processing. It is well known that the behavior of fluid systems can be greatly affected by forces such as capillarity, thermocapillarity, and the wetting characteristics of the material system when in a negligible or a reduced gravitational environment. Although the control of liquid droplets and films has been attained through many methods, we seek an approach whereby thermocapillarity effects are used to prescribe the motion of liquid droplets. We examine the behavior of a three-dimensional liquid droplet on a nonuniformly heated or cooled horizontal solid surface. For a thin viscous droplet, lubrication theory was used to derive a two-dimensional evolution equation that includes the effects of viscosity, gravity, surface tension, slip at the contact line, and thermocapillarity. The evolution equation was coupled to a dynamic contactline condition, relating the contact-line speed to the apparent contact angles, to describe the bulk motion of the droplet. The behavior of the droplet was examined in terms of the imposed thermal field on the solid surface, which includes a mean temperature and a constant temperature gradient superimposed on the mean value. This non-uniformity in the thermal conditions initiates a thermocapillary flow within the droplet that drives bulk migration down the temperature gradient. However, for certain values of the heating parameters, the system exhibits an instability characterized by a slight elongation of the contact line and highlighted by the formation of a dimple at the center of the droplet. When both thermal parameters are active, we show that the instability can be stabilized by a sufficiently large imposed thermal gradient. The critical value at which the interface shape transitions back to a non-dimpled steady state increases with uniform heating of the solid surface. Hysteresis behavior in the critical values in forward and backward traces along the steady-state solution branches is demonstrated and suggests a subcritical instability. The instability also appears under axisymmetric or uniform heating conditions. The results indicate that a critical uniform heating value exists above which the droplet transitions from an axisymmetric configuration to a non-axisymmetric state with the interface shape again characterized by a dimple at the droplet center. For the axisymmetric instability, there was no transition back to a non-dimpled

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state at larger heating values. A stability analysis for the axisymmetric and threedimensional droplet will be conducted to confirm the behavior observed in this study.

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A LABORATORY MODEL FOR T H E SOLIDIFICATION OF T H E EARTH'S I N N E R CORE A N D T H E I N N E R CORE'S SEISMIC A N I S O T R O P Y MICHAEL I. BERGMAN Physics Department, Simon's Rock College, Great Barrington, MA 01230 and Department of Earth and Planetary Sciences, Harvard University, Cambridge, MA 02138

Although seismologists do not yet agree on the details of the depth dependence, the longitudinal variations, and perhaps even the strength, to a first approximation the inner core has been inferred to be elastically anisotropic, with the direction parallel to the rotation axis fast. Many hypotheses have been suggested as a cause for the anisotropy, all involving a lattice-preferred orientation, but the physical reality of many of them has not yet been demonstrated. Using ice as an analog material for high-pressure, hexagonal-closest-packed iron, a series of laboratory experiments have been carried out in which salt-water is solidified from the center of a rotating, hemispherical shell. The experiments reveal more rapid growth of the solid near the pole than the equator, presumably because the Coriolis force inhibits convection within the tangent cylinder parallel to the rotation axis and circumscribing the inner core. Unlike in the Earth where pressure effects are important, in the experiments this leads to colder temperatures within the tangent cylinder. Because of the small length and time scales of the experiments it is not possible to study subsequent solid-state flow, but the experiments do demonstrate that the equilibrium solidification surface is not spherical. Using polarized light on thin sections of the ice, the columnar nature of the dendritic crystals is apparent, with the a-axes lying in the direction of growth. The experiments show how fluid flow in the melt may affect the solidification texture. The laboratory model suggests how a solidification texture is frozen in, from which further textural changes may develop.

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S U P P R E S S I O N OF C H A N N E L C O N V E C T I O N I N SOLIDIFYING P B - S N ALLOYS V I A A N APPLIED M A G N E T I C FIELD MICHAEL I. BERGMAN Physics Department, Simon's Rock College, Great Barrinton, MA 01230 and Department of Earth and Planetary Sciences, Harvard Cambridge, MA 02138

University,

DAVID R. FEARN Department of Mathematics, University of Glasgow, Glasgow G12 8QQ, Scotland, UK JEREMY BLOXHAM Department of Earth and Planetary Sciences, Harvard Cambridge, MA 02138

University,

Channel convection through the porous, dendritic mushy zone in solidifying alloys results from a nonlinear focusing mechanism, whereby liquid enriched in the solute melts dendrites as it convects away from the solid. The local melting reduces the Darcy friction and increases the flow speed to form a convective channel. However, it has been predicted that an applied magnetic field might prevent channels from forming because, as the Lorentz force replaces the Darcy friction as the primary resistance to flow, the focusing mechanism no longer operates. We find that an applied magnetic field can suppress channel convection when Qm, the mushy zone Chandrasekhar number, exceeds order one. Qm is a measure of the ratio of the Lorentz force to the Darcy friction. The longitudinal macrosegregation is not affected by the absence of channels, suggesting such channels are not always primarily responsible for the mass flux between the mushy zone and the melt and/or that convection in the mushy zone can occur without channels forming.

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DYNAMICS AND STABILITY OF VAN-DER-WAALSDRIVEN THIN FILM RUPTURE ANDREW J. BERNOFF Department of Mathematics, Harvey Mudd College, Claremont, CA 91711 THOMAS P. WITELSKI Department of Mathematics, Duke University, Durham, NC 27708-0320

Williams and Davis (1982) derived a lubrication model for the evolution of thin films under the influence of van der Waals forces and surface tension. More recently, Zhang and Lister (1999) showed that this model has an infinite set of similarity solutions corresponding to finite-time film rupture. In this poster we discuss the dynamics and stability of thin-film rupture in planar and axisymmetric geometries. As the film thickness decreases, the planar state loses stability; we discuss the structure of the bifurcating solutions in both geometries. Loss of stability can lead to finite-time rupture of the thin film. We have developed a systematic technique for calculating self-similar rupture solutions and determining their linear stability. The dynamically stable similarity solutions produce observable rupture behavior as localized, finite-time singularities in the models of the flow. For the problem of axisymmetric van-der-Waals-driven rupture, we identify a unique stable similarity solution for point rupture of a thin film and identify a new mode of rupture - annular "ring rupture."

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N E W APPROACHES TO FRONT-TRACKING A N D FRONT-CAPTURING M E T H O D S JERRY BRACKBILL Los Alamos National Laboratory, Los Alamos, NM 87545 DIDIER JAMET Los Alamos National Laboratory, Los Alamos, NM 87545 and Commissariat a lEnergie Atomique, France OLIVIER LEBAIGUE Commissariat a lEnergie Atomique, France DAVID TORRES Los Alamos National Laboratory, Los Alamos, NM 87545

A front-capturing method for liquid-vapor flows with phase-change. The second-gradient theory (also called the Cahn-Hilliard theory or the van der Waals theory) is a thermodynamic model of three-dimensional continuous liquidvapor interfaces. The surface tension is therefore the integral of an energy concentrated in the interfacial zone, proportional to (Vp)2 in the classical form of the theory. From the second-gradient theory, one can derive the equations of motion, which are valid everywhere. Therefore, one solves one system of three-dimensional partial differential equations everywhere: the interfacial motion and the phase-change at the interfaces are just a part of the solution. Moreover, a boundary condition for the density gradient appears naturally: n • Vp must be imposed (n is the unit normal to the boundary), whose value is directly related to the contact angle. The motion of the contact line is therefore a natural part of the solution of the equations of motion of such a fluid. This particularly allows one to account for the contact-angle hysteresis phenomenon is a very simple way: n • Vp has to vary spatially. Simulations of film boiling are performed using this model. Front-tracking without connectivity. Front-tracking is a powerful technique for modeling surface tension. However, it has difficulty in handling topological changes in an interface. This defect is due to the current reliance of front tracking on the connectivity between interfacial points. We have developed a technique in which the surface-tension force can be calculated from interfacial points without connectivity, thus allowing front tracking to model topological changes in an interface naturally much like front-capturing models. The technique is based on determining an indicator function I(x) for the interface. The interfacial normals and curvatures can then be calculated by dif-

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ferentiating the indicator function. We have also developed a projection technique that allows us to dramatically reduce parasitic currents. We present numerical simulations of coalescence and parasitic-current reduction in 2D and 3D.

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M A N I P U L A T I O N OF INTRAVASCULAR GAS EMBOLISM D Y N A M I C S W I T H EXOGENOUS SURFACTANTS A. B. BRANGER Department of Biomedical Engineering, Northwestern University, Evanston, IL 60208 DAVID M. ECKMANN Department of Anesthesia and Institute for Medicine and Engineering, University of Pennsylvania, Philadelphia, PA 19104-4283

B A C K G R O U N D : Intravascular gas embolism (IGE) is the entrapment of gas bubbles in the circulation and can occur in decompression sickness or during cardiopulmonary bypass. These IGE are harmful because they temporarily block blood flow causing local ischemia in the surrounding tissue. Altering the interfacial tension and contact-line mechanics at the bubble/blood/vessel interface with a surfaceactive agent changes the bubble conformation and dynamics. To determine the potential therapeutic benefit of manipulating interfacial forces, we investigated the effects of the surfactant, Antifoam 1510-US, on the dynamics of bubble entrapment and break-up in the intact rat cremaster circulation. METHODS: The animals used in the experiments were handled according to NIH guidelines. Adult male Wistar rats (n = 6) were anesthetized, instrumented, and divided evenly into two groups, control, and those receiving a 2.5 ml pretreatment bolus of diluted Antifoam, raising the surfactant concentration to 1.5% of the total blood volume. Air bubbles (4 (A) were injected through the femoral artery and observed in the cremaster circulation over time with video-microscopy. Dimensions of the bubbles (0.2-10.0 nl), as well as other parameters relating to embolism dynamics, were measured from the video-taped recording. RESULTS: The arterial vessels in the cremaster circulation of rats receiving the pre-treatment of Antifoam showed a much greater degree of constriction (<~ 65%) than the arterioles of the control rats. This increased vasoconstriction in the pretreated animals, coupled with surfactant at the interface, lead to bubble elongation and caused bubble pinch-off into several smaller bubbles, not seen in the controls. CONCLUSIONS: Addition of exogenous surfactant increased deformability of the IGE. This permitted a smaller radius of curvature at the ends of the bubble and allowed for bubble narrowing, elongation, and eventual bubble pinch-off. The pinch-off phenomena induced by surfactant may clinically create multiple smaller bubbles more apt to traverse the microcirculation and lodge in smaller vessels. This might maintain vital organ blood flow and reduce tissue damage. This work was supported by NIH Grant R01 HL60230.

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ADIABATIC HYPERCOOLING OF B I N A R Y MELTS K. BRATTKUS Department of Mathematics, Southern Methodist University, Dallas, TX 75275

A binary melt is hypercooled when it is cooled to a temperature below its solidus. In the isothermal limit planar solidification fronts propagate at a constant velocity determined by the kinetic undercooling and are subject to a long-wavelength morphological instability if speeds fall below a critical value. In this letter we examine the adiabatic limit where the accumulation of a small latent heat release causes the velocity of the interface to slowly decrease through its critical value. The evolution of the hypercooled interface is governed by a damped Kuramoto-Sivashinsky (dKS) equation with coefficients that vary as the interface decelerates. Using this equation we show that morphological transitions are delayed by an amount that reflects both the time the system spends in a stable state and the magnitude of the damping. For a sufficiently large latent heat of fusion the long-wavelength morphological instability is annihilated. Finally, the adiabatic dKS equation predicts late-stage coarsening of the microstructure with length scales that increase as i 1//2 . In finite systems this coarsening removes the morphological instability.

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A N INSOLUBLE SURFACTANT MODEL FOR A D R A I N I N G VERTICAL LIQUID FILM RICHARD J. BRAUN Department of Mathematical Sciences, University of Delaware, Newark, DE 19716

The drainage of a thick non-aqueous film with a surfactant will be studied theoretically and compared with experimental results. Although the film is thin physically, the film is thick enough so that intermolecular forces acting across the film are not important. In the experiment, a thin film is suspended vertically from a wire frame over a bath and gravity drives the drainage of the film back into the bath. Lubrication theory is applied to the situation and the thin film is patched to the bath. Surfactant-transport, Marangoni, and surface-viscous effects are included in theoretical models. Models have been formulated that span the range of drainage regimes, which correspond to the range from rigid to mobile films. Computed and analytical results show that films may be made rigid by surface-viscous or Marangoni effects. Similarity solutions are found in some cases. This is joint work with S. A. Snow and U. C. Pernisz of Dow Corning Corporation and S. Naire of the University of Delaware.

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T H E EFFECT OF TIME-PERIODIC AIRWAY WALL S T R E T C H ON SURFACTANT A N D LIQUID T R A N S P O R T IN T H E L U N G J. L. BULL Biomedical Engineering Department, The University of Michigan, Ann Arbor, MI 48109 D. HALPERN Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350 J. B. GROTBERG Biomedical Engineering Department, The University of Michigan, Ann Arbor, MI 48109 Understanding the transport of surfactant and liquid in the lung is of great importance in designing treatment for respiratory distress syndrome (RDS), and in understanding clearance from the lung. The standard method, surfactant replacement therapy (SRT), for treating this deficiency is to instill exogenous surfactant into the lung. It spreads in the small airways due to Marangoni flows. After the initial transient spreading, the surfactant concentrations at generation 7 (the approximate start of the Marangoni region) and generation 18 (the approximate start of the alveolated region) are essentially constant, so that a steady background surface tension gradient is established. Most of the surfactant transport in SRT takes place once this background gradient is established. On the other hand, liquid and surfactant clearance in normal lungs occur because production of surfactant in the alveoli results in lower surface tensions there than in the airways. This surface tension gradient causes flows towards the trachea. We consider a thin film lining a stretchable tube with a radius that depends on axial position to model airway branching and examine the effects of axial and radial wall oscillations on surfactant and liquid transport. Evolution equations for the film thickness, surface-surfactant concentration, and bulk-surfactant concentration are derived from conservation of momentum, conservation of fluid mass, and conservation of surfactant mass. The model is general and allows for choices of (1) interfacial linear sorption kinetics or squeeze-out phenomena; (2) uniform or linear membrane wall strain; (3) higher (SRT) or lower (liquid and surfactant clearance) concentration at the proximal end of the domain; (4) various strain amplitudes; and (5) various breathing-cycle periods. The evolution equations are solved using the methods of lines, and the cycleaverage transport of both surfactant and liquid is computed. While the transport of surfactant on the surface and in the bulk varies greatly with strain amplitude, the transport does not vary as much with cycling period at a given strain amplitude in our model of SRT. As the transport of surfactant into the alveolar region is considerably higher when the strain amplitude is large, it may be advantageous to use large tidal volumes and large breathing-cycle periods with SRT in order to gain larger surfactant (both surface and bulk) transport. Our model of liquid and surfactant clearance predicts that large breathing-cycle periods and strain amplitudes result in greater clearance.

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T H E D Y N A M I C EFFECTS OF SURFACTANTS O N STATIONARY GAS B U B B L E S IN LIQUID FLOWS DANIEL P. CAVANAGH Department of Chemical Engineering, Bucknell University, Lewisburg, PA 17837 DAVID M. ECKMANN Department of Anesthesia and Institute for Medicine and Engineering, University of Pennsylvania, Philadelphia, PA 19104-^283

We have experimentally examined the effects of common soluble surfactants on gas bubbles in liquid flows in inclined tubes. Air bubbles of known size (A = 0.8, 1.0, 1.5) are held stationary under minimum flow conditions in tubes oriented at specific inclination angles (a = 45°, 90°). Sodium Dodecyl Sulfate (SDS) or Triton X-100 (TX100) is infused into the bulk flow at specific bulk concentrations (T = 10% CMC, 100% CMC). In addition to recording pressure and flow waveforms, we capture images of the bubbles before and during exposure to the surfactant. The modification of the interfacial properties by the surfactant results in bubble behavior ranging from interfacial deformation to deformation plus axial translation to bubble detachment from the wall plus axial translation. Correspondingly, we observe modification of the measured pressure gradient within the test section of the tube. These surfactant-mediated responses are dependent upon the interrelated effects of r , A, a, and the surfactant characteristics. Specifically, a high bulk concentration of surfactant produces more rapid bubble modification and increases the potential for bubble detachment. The potential for detachment increases further as bubble volume is increased. In both vertical tubes where contact forces are not important and in non-vertical tubes the infusion of surfactant may lead to axial translation either with or against the bulk flow. Finally, surfactants that are more effective at modifying contact properties produce bubble-shape and pressure-gradient changes more rapidly. This investigation demonstrates the ability of surfactants to potentially dislodge dried gas bubbles through the manipulation of interfacial properties. This work was supported by the Whitaker Foundation and NIH Grant R01 HL60230.

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B U C K L I N G INSTABILITIES IN THIN VISCOUS SHEETS SAHRAOUI CHAIEB, R. DA SILIVEIRA, L. MAHADEVAN, AND G. H. MCKINLEY Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139

Buckling instabilities are not restricted to solids; they can occur in thin layers of Newtonian fluids. Here we investigate two types of buckling. The first experiment is an instability in a thin viscous layer that lies in a cylindrical annular geometry. As the shear rate becomes larger than a critical threshold, the viscous film deforms sinuously without changing its thickness leading to ripples arranged like the spoke of a bicycle wheel. The ripples appear because the shear gives rise to tensile and compressive stress; the latter are responsible for out-of-plane motions of the sheet. Linear stability analysis predicts the critical shear rate at which the instability occurs as well as the wavelength and angular frequency of the instability. This analysis is compared to the results of our experiments. The second scenario of a buckling of a viscous sheet occurs when a bubble of air rises to the top of a highly viscous liquid, it forms a dome-shaped protuberance on the free surface. Unlike a soap bubble it bursts so slowly as to collapse under its own weight simultaneously, and folds into a wavy structure. This rippling effect occurs for both elastic and viscous sheets, and a theory for its onset is formulated. The growth of the corrugation is governed by the competition between gravitational and bending forces (shearing).

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FLUID-FLUID INTERFACE E X P E R I M E N T S AT T H E U N I V E R S I T Y OF CHICAGO ITAI COHEN, SIDNEY R. NAGEL James Franck Institute, University of Chicago, Chicago IL 60637 MICHAEL P. BRENNER Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 JENS EGGERS Department of Theoretical Physics, University of Essen, Essen, Germany ROMAN O. GRIGORIEV School of Physics, Georgia Institute of Technology, Atlanta, GA 30332-0430 TODD F. DUPONT James Franck Institute, University of Chicago, Chicago IL 60637

Understanding and controlling how a liquid interface changes its topology from being singly connected to being multiply connected (as is the case with a drop dripping from a faucet) or from being bounded to being unbounded in a particular direction (as is the case in the selective-withdrawal problem) is crucial for the control of many manufacturing processes including the creation of emulsions and mono-dispersed sprays. Furthermore, the mathematics describing these types of topological changes is not understood very well. In my poster I will present two beautiful phenomena that explore and shed light on these issues. First, I will show results from experiments on the two fluid drop snap-off problem where fluid is dripped through an outside liquid medium that is viscous. I will then show results from experiments on the selective-withdrawal problem. Here, we lower a straw so that its orifice rests above a water-oil interface. We then withdraw the oil through the straw. By changing the rate of withdrawal we control a transition between having only oil being withdrawn and having water being entrained along with the oil.

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A S Y M P T O T I C ESTIMATES FOR 2-D SLOSHING MODES: THEORY A N D E X P E R I M E N T ANTHONY M. J. DAVIS Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350 PATRICK D. WEIDMAN Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309-0427

A classic problem in surface gravity waves is that of two-dimensional sloshing modes in a channel of arbitrary shape. Exact solutions are known for vertical walls and for triangular containers with a, the angle subtended at the free surface, equal to 45° or 30°. At higher frequencies, the wave motion is confined near the free surface and only the shape of the container at the corners is important. Fox and Kuttler (ZAMP, 1983) conjectured the asymptotic form 1 2*

n+i(l-M)

»=Ta

for the dimensionless frequency parameter. This estimate can be anticipated by simple use of sloping-beach potentials. The next correction for straight-sided containers (fi = 1,2,3) is exponentially small in the exact solutions, 0(l/n) for the infinite dock with gap (a — n), and is now shown to be 0(n~2^) in general. Modifications for asymmetric containers are included. Experiments designed to verify the theory have been successfully completed for odd values of n up to 13. Five geometries, including angles 60° and 135° for which fi is not an integer, were used and surface tension in the fluid minimized. The measured data was adjusted to account for the remaining surface tension and good agreement with the theory was obtained.

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B U O Y A N C Y - D R I V E N I N T E R A C T I O N S OF VISCOUS D R O P S W I T H DEFORMING INTERFACES ROBERT H. DAVIS Department of Chemical Engineering, University of Colorado, Boulder, CO 80309-0424 JOSEPH KUSHNER Department of Chemical Engineering, Arizona State Tempe, AZ 85287-6006

University,

MICHAEL A. ROTHER Department of Chemical Engineering, University of Colorado, Boulder, CO 80309-0424

Two different-sized drops of one liquid immersed in a second, immiscible liquid will move relative to one another under the influence of gravity. When the drops come close together, they interact owing to hydrodynamic disturbances, and various outcomes are possible: separation of the drops, capture with or without coalescence of the drops, breakup of the smaller drop into two or more drops, or even a combination of capture and breakup phenomena. Interfacial deformation may promote any of the results, depending on the drop-to-medium viscosity ratio, the ratio of the smaller drop radius to the larger drop radius, the gravitational Bond number, which is a measure of how much the drops will deform, and the initial horizontal offset between the drops. This poster presents an experimental investigation into the parameter space governing two-drop interactions to compare with theoretical results obtained previously (Zinchenko, Rother, and Davis, J. Fluid Mech. 391, 249 (1999); Davis, Phys. Fluids 11, 1016 (1999)). Trajectories of two drops consisting of a mixture of glycerol and water moving through an external medium of castor oil are photographed and analyzed to check quantitative agreement with computer simulations.

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A N O M A L Y A N D U N C E R T A I N T Y W H E N LIQUID FILMS FLOW OVER SOLID SURFACES WALTER DEBLER Department of Mechanical Engineering, The University of Michigan, Ann Arbor, MI 48109-2125

Previous work on the flow of thin liquid films over inclined solid surfaces has shown the existence of two forms for the advancing leading edge. In an investigation to determine the speed of advance of the chevron-like form of this disturbance, it was observed that in a single experiment the other pattern, rivulets with sides parallel to the down-slope direction, occurred when using silicone oil on a glass surface. Subsequently an extensive series of tests could not replicate this single result on clean glass. Various contaminants were then applied to the glass, but only chevrons appeared when the silicone oil film flowed. Finally, a stain inhibitor for fabrics, Scotch-gard, was sprayed onto the glass and the silicone film produced rivulets at its leading edge. Hence, it was concluded that some unknown substance that yielded a similar effect must have been deposited on the glass plate that had produced, with but one exception, chevrons. The present tests also showed that clear Teflon film will also form rivulets with silicone oil. Moreover, the author obtained rivulets with glycerin on glass, in contrast to previously-reported results of others. The ultimate lesson that can be drawn is that even when the solid surface is. diligently and well cleaned, molecular thin films might be present to alter the contact angle of the liquid at the solid boundary and, thereby, affect the type of leading-edge pattern that is observed in an experiment.

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FLOW BEHAVIOR OF L A N G M U I R MONOLAYERS MICHAEL DENNIN AND R. S. GHASKADVI Department of Physics and Astronomy, University of California, Irvine, Irvine, CA 92697-4575

We report on measurements of the viscosity of Langmuir monolayers. Langmuir monolayers are comprised of amphiphilic molecules that are confined to the airwater interface. They exhibit a rich two-dimensional phase behavior that includes the usual gas, liquid, and solid phases, as well as a number of phases that are analogs of three-dimensional smectic liquid crystals. We have built a Couette viscometer that is capable of measuring velocity profiles, viscosity, and the complex shear modulus of monolayer films. The velocity profiles are obtained by imaging the monolayer using a Brewster-angle microscope. The apparatus has been tested on both Newtonian and non-Newtonian phases, and the results for the velocity profiles are consistent with theoretical calculations. It is known that the addition of divalent ions to the water will "stiffen" monolayers of fatty acids. This is due to the binding of two fatty acids by a single divalent molecule. The influence of divalent ions on the phase behavior has been studied, but the effects on the viscosity have not been studied in detail. We have made measurements of both the viscosity and the shear modulus and find that they increase by three orders of magnitude over a 10-hour time scale. Further, the evolution appears to occur with three distinct time scales. We directly measured the percentage of C a + + ions bound to the monolayer as a function of time by repeatedly transferring the monolayer onto a solid substrate. The multilayer formed by this process was rinsed in an acid to release the C a + + ions, and their concentration was measured with an ion meter. We found that the fraction of bound C a + + had a similar time evolution to the viscosity. Also, a simple ad-hoc model for the viscosity in terms of the percentage of bound C a + + is able to explain the time evolution. A more detailed study of this process will provide important insights into the fundamental nature of viscosity.

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SOLUBLE SURFACTANTS A N D CONTACT-ANGLE D Y N A M I C S DAVID M. ECKMANN Department of Anesthesia and Institute for Medicine and Engineering, University of Pennsylvania, Philadelphi, PA 1910^-4283 Little is detailed about surfactant effects on the contact angle, contact-angle hysteresis, and contact-line motion. Ultimately we are interested in wetting/dewetting phenomena related to intravascular gas embolism and its treatment with exogenous surfactants. We have therefore begun to characterize the relationships between contact angle, contact-angle hysteresis, spreading parameters, and surfactant moieties on various solid substrates. Using a Wilhelmy-plate surface tensiometer, we measured the air-liquid surface tension, a, of aqueous solutions of sodium dodecyl sulfate, (SDS) with and without 5% bovine serum albumin (BSA) present. By video image analysis, we evaluated a drop interfacial shape on a stationary flat horizontal substrate and on a flat slowly-rotating substrate (< l°/second). Drop volumes were 40-80 /A and all measurements were recorded at room temperature in a sealed, humidified chamber 30 minutes after solution instillation into the experimental apparatus. Solid substrates included glass, acrylic, and stainless steel. We measured static, advancing, and receding contact angles, a, and 0 r , respectively, of drops of the same aqueous SDS ± BSA dilutions. We calculated S, the spreading parameter, based on g. The parameter a cos <j) incorporates both the surface tension and contact-angle effects needed to calculate a spreading coefficient. It is a measure of the adhesion force acting at the contact line. On glass, as SDS concentration increases, a cos s increases to a local maximum and then decreases to its plateau value (~33-34 dyne/cm) at and above the CMC. The variation of a cos (ps with concentration of SDS in the presence of BSA (which is surface active) maintains substrate dependence with a more narrow range of values. Competition between BSA and SDS for interfacial sites likely accounts for this. The contact-angle hysteresis (a — T) is nearly constant (~ 26°-30°) over the range of SDS concentrations. There is an 8-fold change in S, which tends from -19.2 to -2.4 as the SDS bulk concentration increases from 0 to the CMC. The surfactant studied lowered interfacial tension, and reduced contact angles and the spreading coefficient with substrate dependence. These experiments help to explain a potential role that exogenous surfactants may have in the treatment of intravascular gas embolism. A particular surfactant delivered to the interface may significantly change the wetting potential of the liquid onto the solid and thereby increase or decrease the adhesion force acting on a trapped bubble. The exact influence of interactions of individual species (surfactant, protein, substrate) on the interfacial quantities measured in these experiments are not easily inferred. They must be derived by continued careful experimental analysis. This work was supported by NIH Grant ROl HL60230.

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E X P E R I M E N T A L STUDIES OF T H E H Y D R O D Y N A M I C S N E A R MOVING CONTACT LINES S. GAROFF Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 E. RAME National Center for Microgravity Research, NASA Glenn Research Center, Cleveland, OH 44135 Dynamic wetting, spreading of a fluid over a solid surface, controls many natural phenomena and technological processes. When surface tension forces are important, the interface shape and local flow field very near the moving contact line control the macroscopic configuration of the fluid body. However, identifying the correct assumptions needed for predictive models of spreading is not trivial. Theoretically, the central difficulty is the unphysical stress singularity at the contact line arising when classical hydrodynamic assumptions are applied up to and including the moving contact line. The singularity suggests that very near the contact line (in an "inner" region), unique microphysical processes other than the classical hydrodynamic assumptions control the fluid motion. Thus, we must explore: what are these unique processes operating on the microscopic scale? And how can we find a legitimate boundary condition for the macroscopic problem? Asymptotic expansions provide a description of the hydrodynamics near moving contact lines in a limited parameter range. These expansions and very accurate experiments provide a tool for answering the questions central to understanding the moving-contact-line problem. We have found that for a set of Newtonian polymer fluids (PDMS, polydimethylsiloxanes) on various surfaces, these asymptotic expansions describe the hydrodynamics at small capillary number, Ca < 0.1, and negligible Reynolds number, Re <~ 0. At higher Ca (with Re ~ 0) and moderate Re ~ 0.1 (with Ca < 104), our experiments find that the curvature of the interface near the contact line is less than that predicted by the asymptotic expansions. Having established the validity of the asymptotic analysis, we may now use it to probe the microphysical properties controlling the fluid motion on the inner scale. Although many processes have been suggested, no direct experimental identification of these processes exists. We must carefully use the tool provided by the asymptotic expansions to indirectly recover information on the inner scale hydrodynamics from our experiments. Use of static approximations or Tanner's Law in the analysis of experimental data leads to systematic errors that obscure the correct characterization of the inner hydrodynamics. We find that for PDMS, the parameters describing the inner hydrodynamics to 0(1) in Ca, as Ca —> 0, must vary with contact-line speed. This variation is complex and suggests that different mechanisms operate at different contact-line speeds. Further, the inner hydrodynamics vary, even for very closely-related systems. Use of Ca to scale the velocity does not unify the data for the different systems, implying that this is not the correct scaling of the fluid velocity in the inner region.

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A X I S Y M M E T R Y - B R E A K I N G INSTABILITIES IN A X I S Y M M E T R I C FREEZING OF ICE ALEXANDER YU. GELFGAT, PINHAS Z. BAR-YOSEPH, ALEXANDER SOLAN Computational Mechanics Laboratory, Faculty of Mechanical Engineering, Technion - Israel Institute of Technology, Haifa 32000, Israel TOMASZ A. KOWALEWSKI Polish Academy of Sciences, IPPT PAN, Swietokrzyska 21, PL 00-049, Warszawa, Poland

This work is devoted to a theoretical explanation of an effect observed during the study of freezing of ice in an axisymmetric container [1]. The experimental setup consists of a cylindrical glass container filled with water, which is immersed in a bath with hot water and is closed by a cold steel cover. After the temperature of the cover is reduced below 0°C, an ice front is formed. The solidification front observed on the ice had an asymmetric, circumferentially-periodic surface shape. Likewise, measurements of the temperature field showed a periodic structure. It was assumed that the effect was due to an axisymmetry-breaking instability of the convective flow of water. Detailed numerical analysis of the stability of the basic axisymmetric flow with respect to all possible three-dimensional perturbations showed that an instability sets in with a relatively high azimuthal wavenumber k, which varies between 7 and 10. Details can be found in [2]. The corresponding pattern of the most unstable three-dimensional perturbation of the temperature is similar to the experimentally-observed temperature distribution. It is shown that the calculated instability is caused by the Rayleigh-Benard mechanism, which leads to the appearance of a system of convective rolls distributed along the azimuthal direction inside a relatively thin convective layer. References 1. T. A. Kowalewski and A. Cybulski, IPPT PAN Report 8/97, Warsaw (1997) (in Polish). 2. A. Yu. Gelfgat, P. Z. Bar-Yoseph, A. Solan, and T. A. Kowalewski, I. J. Trans. Phenomena 1, 173 (1999).

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SEPARATION M E C H A N I C S OF THIN INTERFACIAL LIQUID LAYERS: T H E ROLE OF VISCOUS F I N G E R I N G ASHOK GOPINATH Department of Mechanical Engineering, Naval Postgraduate School, Monterey, CA 93943-5146

The mechanics of separation of a thin interfacial liquid layer trapped between two parallel surfaces was studied in a controlled manner. The liquid of choice was a silicone oil with constant surface tension, but variable viscosity. Different viscosities, layer thicknesses, and separation velocities were used to determine the separation behavior and its dependence on viscous fingering and capillary number. Force, displacement, and time data were recorded for all experimental runs and the plots used to gain preliminary insight into this process. Qualitative flow-visualization data has also been recorded to corroborate the trends in the onset of viscous fingering with the predictions of a simple interfacial stability analysis.

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LARGE FINITE-ELEMENT MODELING OF AXIALLY S Y M M E T R I C FREE-SURFACE FLOWS ROMAN GRIGORIEV School of Physics, Georgia Institute of Technology, Atlanta, GA 30332-0430 TODD DUPONT James Franck Institute, University of Chicago, Chicago, IL 60637

The dynamics of free-surface flows and in particular the mechanisms for singularity formation at the interface of fluids with different physical properties constitute a problem of high theoretical as well as practical interest. The applications are abundant and include the functioning of such commonplace devices as ink-jet printers and fuel injectors, and processes such as oil extraction and fiber spinning. While considerable theoretical and computational advances have been achieved in our understanding of the problem in the inviscid and Stokes limits, the analyses at intermediate Reynolds numbers remain scarce and inconclusive. We present a general computational algorithm able to describe different kinds of axially symmetric, free-surface flows for arbitrary Reynolds numbers. The distinctive feature of the proposed algorithm is that the interface is treated as a mathematical singularity corresponding to the discontinuous change in the fluid properties, rather than being artificially smeared over a finite region, as is usually done. This should allow one to numerically simulate the singularity formation with precision adequate for comparison with experimental data. The idea of the algorithm is to replace the mathematical problem involving multi-fluid flow with complicated boundary conditions imposed at the free interface^) with a simpler problem involving a single-fluid flow with spatially (and temporally) varying viscosity and density, and with a surface-tension force acting at the interface(s), thus effectively getting rid of the boundary conditions. This is achieved by writing the Navier-Stokes equation and the incompressibility condition in the weak form by integrating these equations with an appropriately chosen weight. The obvious advantage of the finite-element formulation is that it contains no singularities associated with the jump in viscosity at the interface. The viscosity jump can be removed by performing integration by parts, effectively reducing the order of the equations by one. Furthermore, the singularity produced by the jump in pressure can also be removed by using weight functions that are themselves discontinuous at the interface. Present implementation assumes no angular dependence and zero angular velocity, but the algorithm allows one to lift these assumptions at the expense of a slight complication in the formulas. The interface is advanced using a version of the interface-tracking technique based on the advection of massless markers with the velocity field computed on a fixed grid, although other thechniques such as the volume-of-fluid or the level-set method can be easily incorporated. Our preliminary results for the two-fluid, selective-withdrawal problem provide an illustration of the computational method.

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MOLECULAR SIMULATIONS OF INTERFACE P H E N O M E N A : A N ALTERNATIVE A P P R O A C H NICOLAS HADJICONSTANTINOU Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139

We present molecular simulations of two classical interfacial phenomena in fluid mechanics. The first is the Rayleigh-Taylor instability, arising when a heavier fluid is situated on top (with respect to a gravity field) of a lighter fluid. Our molecular techniques allow a novel approach to this problem. The instability is triggered by molecular thermal fluctuations instead of the artificial perturbations imposed in continuum simulations. Additionally, a direct comparison between miscible and immiscible Rayleigh-Taylor mixing is possible. The second problem presented is the motion of a contact line in two dimensions. Molecular simulations of a fluid displacing a second immiscible fluid in a twodimensional channel are performed. A continuum simulation of the same problem is employed to answer the following question: what continuum boundary conditions are required to reproduce the molecular interface shape? We found that by using the actual slip profile observed in our molecular simulations as a slip boundary condition and by setting the dynamic contact angle equal to the static value, that we could reproduce the molecular results satisfactorily in the range 0.059 < Ca < 0.072 [Phys. Rev. E 59, 2475 (1999)].

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E X P E R I M E N T A L INVESTIGATION OF E N V I R O N M E N T O X Y G E N C O N T E N T IN SOLDER-JET T E C H N O L O G Y E. HOWELL, S.-Y. LEE, C. M. MEGARIDIS Department of Mechanical Engineering, University of Illinois at Chicago, Chicago, IL 60607 M. MCNALLAN Department of Civil and Materials Engineering, University of Illinois at Chicago, Chicago, IL 60607 D. WALLACE MicroFab Technologies, Piano, TX 75074

Solder-jet technology is targeted for use in high-density-electronics-assembly applications where 63Sn/37Pb solder is utilized as an attachment and/or structural material. Solder jetting relies on ink-jet printing technology to create and precisely place miniature molten-solder droplets (50-100 /xm in diameter) on substrates or pads. These droplets solidify after impact, forming bump deposits that are subsequently used for the flip-chip bonding of electronic components on the substrate. Ink-jet-based solder deposition is low cost (no masks or screens are required), flexible, easily automated using digital technology, suitable for a manufacturing environment (does not operate in a vacuum), highly repeatable, and offers high resolution. Even though solder jetting shows great promise to achieve solder deposition at pitch geometries well below what is feasible with existing techniques, there exists a few formidable challenges that must be overcome before commercialization. Because maximum throughput is an important requirement in electronic assembly, solder jetting is performed openly, but with local environmental control. A sheathing nitrogen-ring flow surrounds the solder jet to minimize solder oxidation that can drastically delay the atomization process and degrade the effective adherence of the solder deposits on the targets. Therefore, successful commercial implementation requires quantification of the influence of the ambient oxygen content on the surface properties of the molten solder jet. The method employed to measure the surface tension of the molten solder jet is the oscillating-jet technique. When a liquid is forced through an elliptical capillary (aspect ratio « 2), an oscillating jet forms. The stationary shape of this jet (wavelength and both semiaxes) is affected by the liquid/gas interface properties and is used to determine the surface tension as a function of distance from the orifice (surface age) using the analysis of Bechtel et al. [J. Fluid Mech. 293, 379 (1995)]. The jetting is performed in a controlled environment containing a known nitrogen/oxygen mixture. A CCD camera with a microscope lens is used to obtain images of the oscillating solder jet along both major axes. From these images, the wavelength and the semiaxes are determined at the beginning, middle, and end of successive wavelengths. This data along with other material properties is used to determine the dynamic surface tension of the molten solder along the direc-

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tion of jet propagation. The experiment is repeated at oxygen mole concentrations ranging from 5ppm to 21% (air). In addition to the oscillating-jet experiments, a maximum-bubble-pressure apparatus is employed to determine the equilibrium values of surface tension of molten solder in controlled nitrogen/oxygen mixtures.

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DROPLET SPREADING WITH SURFACTANT: MODELING AND SIMULATION JOHN HUNTER Department of Mathematics, University of California, Davis, Davis, CA 95616 ZHILIN LI Department of Mathematics, North Carolina State Raleigh, NC 27695-8205 HONGKAI ZHAO Department of Computer Science, Stanford Stanford, CA 94305-9040

University,

University,

W h e n a liquid droplet is placed on a flat solid surface, we may observe partial wetting or complete wetting of the surface depending on the static and dynamic contact angles. We present a simple mathematical model for this free b o u n d a r y problem and a new numerical method to simulate this model. Our numerical algorithm is based on the modified immersed-interface method and the level-set formulation. Our numerical results agree with those experiments and analytic results in the literature. Topological changes are handled easily using our methods.

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AIR E N T R A I N M E N T AT LOW VISCOSITIES ALEXANDRA INDEIKINA, IGOR VERETENNIKOV Department of Physics, University of Notre Dame, Notre Dame, IN 46556 HSUEH-CHIA CHANG Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556

We examine experimentally and theoretically the breakdown of a straight contact line on a roller that rotates into a liquid. This breakdown occurs at a critical dimensionless speed (capillary number Cac) and precedes the industrially important air-entrainment phenomenon. A corn-syrup/water mixture with relatively low viscosity (0.3-0.7 Poise) is used as the test fluid. Our flow-visualization experiments indicate that the liquid interfacial velocity at the nearly critical air cusp above the contact line decelerates rapidly (by two orders of magnitude) over the short length of the cusp (a few capillary lengths). Moreover, Cac is found to be insensitive to agitation in the liquid phase, but extremely sensitive to obstacles placed above the cusp in the gas phase. The evolution of the air flow coupled with changes in the shape of the air-liquid interface is of paramount importance in the stability of the two-dimensional air cusp, since air is dragged into the cusp by both the solid roller and the liquid phase. Both the injection drag forces and the length of the cusp increase with increasing roller speed. The build-up of stagnation pressure, however, saturates as the cusp flattens. Consequently, at a critical Ca, the steady-flow balance breaks down. The required pressure gradient for air ejection becomes impossible in two dimensions — a larger free-surface-curvature gradient is needed to push the air out of the longer flat cusp. Hence, the third dimension should be invoked and the formation of triangular air pockets along the contact line is initiated. We support this physical mechanism for air entrainment via matched asymptotics. Momentum boundary layers on the gas side within the cusp and on the roller and below the cusp interface in the liquid are connected to each other and to inviscid outer solutions. The matching at the contact line requires a molecular model for the gas viscosity when the air-cusp width approaches the mean-free path of gas molecules. Of particular interest is the dependence of Cac on the inception length of the gas momentum boundary layer as specified by the position of a scraper — a potentially useful means of delaying air entrainment in industry.

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NUMERICAL SIMULATIONS OF VIBRATION-INDUCED DROPLET EJECTION ASHLEY JAMES Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455 MARC K. SMITH AND ARI GLEZER The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405

Vibration-induced droplet ejection occurs when a liquid drop is placed on a vertically-vibrating surface. T h e vibration leads to the formation of capillary waves on the free surface of this primary drop. W h e n the forcing amplitude is large enough, secondary droplets are ejected from the wave crests. For low-frequency forcing a low-order, axisymmetric wave mode is excited in the primary drop. For large enough forcing amplitudes, a steep depression or crater forms in the center of the drop during the downward phase of the oscillation cycle. During the next phase the crater collapses and liquid flowing inward forms an upward, high-momentum jet at the center of t h e crater. One or more small droplets may fragment from the end of this jet to produce secondary droplet ejection. For large forcing frequencies the wave motion is chaotic. Secondary droplets are ejected from multiple locations on the free surface of the primary drop. At each ejection site, a crater forms before droplet ejection. As in the low-frequency case, the crater then collapses and a jet is formed at the center of the crater. One or more droplets may then be ejected from the end of the jet. Under certain conditions the entire primary drop will rapidly atomize. An axisymmetric, Navier-Stokes solver, based on the M A C method, is used to simulate the low-frequency process. T h e solver includes a piecewise linear, volumeof-fluid m e t h o d and a continuum-surface-force implementation of surface tension. T h e simulations capture the sequence of events discussed above: a crater is formed, which collapses to form a jet, from which secondary droplets are ejected. Comparison of a time sequence of calculated interface shapes to an experimental sequence shows good agreement. T h e simulations illustrate the large velocities t h a t develop in the jet. T h e simulations will be used to investigate the physical mechanism responsible for droplet ejection. T h e effect of the driving parameters and physical parameters on the ejection process, including the size and velocity of the secondary droplets, will be evaluated. This will be used to gain an understanding of high-frequency atomization which, in turn, will be applied to the design of systems involving spray formation. This technology has application in a wide range of aerospace, industrial, and biomedical applications. A heat transfer cell t h a t utilizes this technology is under development for microelectronics-cooling applications. This cell is similar t o a heat pipe, but has a higher liquid flow rate t h a t results in a substantially higher heat transfer rate.

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INTERFACIAL D Y N A M I C S ASSOCIATED W I T H EVAPORATION OF LNG IN A STORAGE T A N K SANG W. J 0 0 School of Mechanical Engineering, Yeungnam University, Gyongsan 712-749, Korea CHULSOO PARK AND SEONGHO HONG KOGAS, Korea

Liquefied natural gas (LNG) is stored in an insulated tank where heat transfer throught the tank wall into the liquid is released by the evaporative heat loss at the liquid/vapor interface inside the tank. Excess vapor is transported out and the vapor pressure is usually kept constant. When the tank is refilled (usually bottom filled) with new cargo, it has different constituent concentrations from the existing LNG in part due to the preferential evaporation of constituents. As a result the new LNG usually has higher density, and a stable stratification results. The new LNG in the bottom, however, does not release heat efficiently due to the lack of an evaporating interface. As the heat builds up, the density approaches that of the upper layer and a reversal of the layers can occur. In the present study, we examine this multiphase flow of liquid and gas mixtures by stability analysis, a dynamical-systems approach, and finite-difference computations. In particular, the conditions for the reversal and the elapsed time are presented.

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OSCILLATORY THERMOCAPILLARY C O N V E C T I O N GENERATED B Y A B U B B L E M. KASSEMI AND N. RASHIDNIA National Center for Microgravity Research, NASA Glenn Research Center, Cleveland, OH 44135

In this work, we study steady and oscillatory thermocapillary and natural convective flows generated by a bubble on a heated solid surface. The dynamic characteristics of the time-dependent convection are captured using a combined numericalexperimental approach. The index of refraction fringe distribution patterns constructed numerically by taking an inverse Abel transform of the computed temperature fields are compared directly to the experimental Wollaston Prism (WP) interferograms for both steady state and oscillatory convection. The agreement between numerical predictions and experimental measurements is excellent in all cases. It is shown that below the critical Marangoni number, steady-state conditions are attainable. With increasing Ma number, there is a complete transition from steady state up to a final non-periodic fluctuating flow regime through several complicated symmetric and asymmetric oscillatory states. The most prevalent oscillatory mode corresponds to a symmetric up and down fluctuation of the temperature and flow fields associated with an axially travelling wave. Careful examination of the numerical results reveals that the origin of this class of convective instability is closely related to an intricate temporal coupling between large-scale thermal structures that develop in the fluid in the form of a cold finger and the temperature-sensitive surface of the bubble. Gravity and natural convection play an important role in the formation of these thermal structures and the initiation of the oscillatory convection. Consequently, at low-g, the time evolution of the temperature and flow fields around the bubble are very different from their 1-g counterparts for all Ma numbers.

268 A B O U T COMPUTATIONS OF THIN-FILM FLOWS

L. KONDIC Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NY 07102 J. DIEZ AND A. BERTOZZI Department of Mathematics, Duke University, Durham, NC 27708-0320

We explore the computational performance of different models used to compute the flow of a thin film down an inclined plane, within the framework of the lubrication approximation. Our fD computations show t h a t the results and computational efficiency strongly depend on the model t h a t is used to deal with the contact-line singularity. This study allowed us to develop efficient fully nonlinear 2D simulations. Preliminary 2D computational results concerning contact-line instability will be presented.

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VISUALIZATION OF C O N V E C T I O N IN LIQUID METALS JEAN N. KOSTER Department of Aerospace Engineering Sciences, University of Colorado, Boulder, CO 80309-0429

A flow-visualization technique for density fields in liquid and solidifying alloyed metals has been developed. Density fields have been visualized similar to interferometry conventions. Density fields include information on temperature and concentration. In alloys, concentration is the dominant signal, thus chemical segregation in the melt can be revealed. Natural convection and double-diffusive flows have been visualized along with solidification and melting. Some visualized results are challenging theoretical analysis.

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TIME-EVOLVING INTERFACES IN VISCOUS FLOWS MARY CATHERINE A. KROPINSKI Department of Mathematics and Statistics, Simon Praser University, Burnaby, British Columbia V5A 1S6, Canada

We present numerical methods for computing the motion of two-dimensional bubbles or drops in a slow viscous flow. New methods are presented for both the solution of the governing fluid equations and for the time integration of the evolving interfaces. The interfacial velocity of the bubble or drop is found through the solution to an integral equation whose analytic formulation is based on complexvariable theory for the Stokes equations. The numerical methods are spectrally accurate and employ a fast multipole-based iterative solution procedure, which requires only 0{N) operations where N is the number of nodes in the discretisation of the boundary. It is known that the dynamic equations for the interfacial motion become stiff as the curvature of the interface increases. A small-scale decomposition is performed to extract the dominant term driving the stiffness. By introducing an appropriate tangential velocity into the dynamics, this dominant term becomes linear. This leads to implicit time-integration schemes that are explicit in Fourier space. Examples will include the deformation of drops in an extensional flow and the viscous sintering of glass.

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I N F L U E N C E OF A N O N L I N E A R EQUATION OF STATE O N C O N T A M I N A T I O N FRONTS AT A I R / W A T E R INTERFACES J. M. LOPEZ Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804 A. H. HIRSA Department of Mechanical Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590 The presence of surface-active materials, ubiquitous at most gas/liquid interfaces and particularly at the air/water interface, has a pronounced effect on the stress balance at the interface, and this in turn is nonlinearly coupled to the bulk flow. Even with minute amounts of surfactants on a liquid free surface, the boundary conditions for the Navier-Stokes equations are functions of the interfacial viscoelastic properties. The Boussinesq-Scriven constitutive relation for stress at a Newtonian interface consists of three intrinsic properties of the interface. These are the surface tension, er, the surface shear viscosity, fist and the surface dilatational viscosity, KS; consistent measurements of the latter have not yet been reported. These viscoelastic properties vary with the surfactant concentration at the interface, c. Here, we present a fundamental description of the interface and its coupling to the bulk flow and develop an axisymmetric Navier-Stokes numerical model. We perform both numerical studies using this model and experimental measurements of the liquid flow in an annular region bounded by stationary inner and outer cylinders and driven by a rotating floor. The difficulties in obtaining a clean free surface even when triply distilled water is used will be reported. DPIV measurements with a clean free surface will be presented. Computational results from our Navier-Stokes model that incorporates measured a(c) and f/,s(c), and modeled KS(C) for hemicyanine (an insoluble surfactant) will be presented. The computations for surfactant cases are compared with the hydrodynamics of a clean interface and a no-slip rigid surface. These computations provide insight into the dynamics that result from the surfactant with a nonlinear equation of state and finite surface viscosity. In this presentation, we focus on low initial surfactant concentration cases where a contamination front is observed and find that its location varies linearly with the initial concentration. Typically, when the Marangoni stress (due to surface-tension gradients) is dominating, the interface is thought of as immobile, acting as a no-slip surface. However, this is only true for the velocity components in the directions that would have lead to surfactant-concentration gradients. In the present axisymmetric swirling flow, this direction is radial. In the azimuthal direction, since the flow is axisymmetric, there are no azimuthal gradients and so there are no Marangoni stresses acting in that direction. Thus, in the radial direction, the Marangoni stress makes the interface act like a no-slip surface, but in the azimuthal direction it is essentially stress-free. This has fundamental consequences for models of contaminated interfaces that are not planar two-dimensional, where surfactant coverage does not simply mean that the interface is no-slip.

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STABILIZATION OF A N ELECTRICALLY C O N D U C T I N G CAPILLARY B R I D G E FAR B E Y O N D THE RAYLEIGH-PLATEAU LIMIT USING F E E D B A C K CONTROL OF ELECTROSTATIC STRESSES MARK J. MARR-LYON, DAVID B. THIESSEN, FLORIAN J. BLONIGEN, AND PHILIP L. MARSTON Department of Physics, Washington State University, Pullman, WA 99164-2814

A liquid bridge between two solid surfaces is known as a capillary bridge and the stability of such bridges is relevant to materials processing as well as to other aspects of the management of liquids. For a cylindrical bridge in low gravity of radius R and length L, the slenderness S = L/2R has a natural (Rayleigh-Plateau) limit of •K beyond which the bridge breaks. We demonstrate a novel method of suppressing the growth of this mode on an electrically conducting bridge surrounded by an insulating liquid of the same density in a Plateau tank. The shape of the bridge is optically sensed as in our related demonstration of acoustic stabilization [M. J. Marr-Lyon et al., J. Fluid Mech. 351, 345 (1997)]. In the present stabilization method the optical information is used to control the potentials on a pair of ring electrodes concentric with the bridge. Slenderness values can be made to reach 4.46 when the generalized feedback force for the mode of interest is taken to be proportional to the modal amplitude. At S = 4.49, the next higher mode (which is not controlled in our current experiments) is predicted to become unstable. The electrical conductivity of the bridge liquid need not be large. Supported by NASA.

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STABILIZATION OF CAPILLARY B R I D G E S IN AIR FAR B E Y O N D THE RAYLEIGH-PLATEAU LIMIT IN LOW G R A V I T Y USING ACOUSTIC RADIATION P R E S S U R E MARK J. MARR-LYON, DAVID B. THIESSEN, AND PHILIP L. MARSTON Department of Physics, Washington State University, Pullman, WA 99164-2814

In the absence of gravity, cylindrical capillary bridges consisting of liquid between two circular supports naturally become unstable and break when the length L of the bridge exceeds its circumference. This is the Ray leigh-Plateau limit where the slenderness S = L/2R is -K and R is the bridge radius. In experiments performed aboard NASA's low-gravity KC-135 aircraft, it was found that acoustic radiation pressure can be used to stabilize capillary bridges against breakup. Capillary bridges composed of a mixture of water and glycerol were deployed in a 21 kHz ultrasonic standing wave in air. The bridges were extended to a slenderness as great as S = 4.1 prior to breaking. Bridges extended beyond n broke immediately when the ultrasound was turned off. In contrast with previous work [M. J. Marr-Lyon et al, J. Fluid Mech. 351, 345 (1997)], this stabilization method does not use active feedback; the stabilization is a passive effect of the sound field. The acoustic wavelength is chosen such that the average radiation pressure due to the sound field is a function of the local bridge radius, so that areas of larger radius are squeezed, and areas of smaller radius are expanded. The KC-135 environment, however, is poorly suited for exploring the limitations of this method and the passive acoustic stabilization method cannot be simulated in Plateau tanks. Supported by NASA.

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I N T E R A C T I O N S B E T W E E N HELE-SHAW FLOWS A N D DIRECTIONAL SOLIDIFICATION: N U M E R I C A L SIMULATIONS ECKART MEIBURG Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, Santa Barbara, CA 93106

High-accuracy numerical simulations are presented for the nonlinear evolution of directional solidification processes, both in the absence and in the presence of potential flow in the melt. To this end, a numerical method is employed that utilizes an analytical boundary-fitted coordinate transformation in conjunction with a combination of spectral and finite-difference discretizations. The flow field is computed by a boundary-element technique. The accuracy of the computational scheme is demonstrated by comparing the growth rates at small interfacial amplitudes with linear stability results. In calculations with random initial perturbations, we observe the emergence of a large-amplitude dominant wavelength in agreement with the predictions by linear stability theory. Furthermore, the simulations demonstrate the stabilization of the solidification process by a uniform flow parallel to the interface. Results are presented as a function of wavenumber, morphological parameter, and parallel-flow velocity.

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MODELLING T H E CONTACT REGION OF A N E V A P O R A T I N G M E N I S C U S W I T H A V I E W TO APPLICATIONS S. J. S. MORRIS Department of Mechanical Engineering, University of California, Berkeley, Berkeley, CA 94720-1740

The evaporating meniscus of a perfectly wetting liquid exhibits an apparent contact angle 0 owing to small-scale, evaporatively driven flow in the contact region. Existing theory by Wayner predicts © and the heat flow as the solution of a freeboundary problem. Published simulations use that theory to model the behaviour of grooved heat pipes. In that application, the heat flow across the meniscus in a millimetre-sized groove is found by dividing the meniscus in two. The heat flow from the inner contact region is found from the theory of the small-scale flow. That across the outer visible meniscus is found by solving the two-dimensional conduction equation. The simulations cover scales from nanometres to millimetres. At the top end of this range, they include the geometry of the solid from which liquid evaporates. At the other end, they use the existing theory for the small-scale flow. This covers a continuum of behaviours. The extremes include the isothermal meniscus that does not show an apparent contact angle; and two forms of strongly evaporating menisci that do. The first of these occurs in the limit of vanishing flow resistance to evaporation. All evaporation then occurs from a quasi-parallel film at dimensions smaller than those on which 0 is established [1]. The second occurs in the limit of vanishing thermal resistance to evaporation. The contact region is then isothermal, and all heat flow occurs at dimensions larger than those on which 0 is established [2]. As there is no previous analysis of the existing theory, these possibilities are not recognised in the literature. Yet the case of vanishing thermal resistance is common in applications. By emphasising it, one obtains insight, and useful formulae for the heat-pipe problem. Morris [3] simplifies the heat-pipe problem by dividing it into several pieces. To begin, scaling is used to find conditions under which there is negligible heat flow at the dimensions on which 0 is determined. The heat flow is then found from a linear conduction problem in which 0 appears as a parameter. The geometrical complications associated with heat pipes are added to the conduction model in two stages. A simplified treatment of these effects is possible as two conditions typically apply. The solid is highly conductive and 0 is small. Analysis of a model problem (linear meniscus on a conductive slab) shows that for vanishing liquid-solid conductivity ratio, the heat flow is asymptotically independent of solid shape. This is plausible as the heat flow would be independent of the shape of a perfectly conducting, and so isothermal, solid. The predictions of the model thus apply to more complex solid geometries. Next, meniscus curvature is incorporated for it influences the heat flow. Existing simulations include the meniscus shape by solving the two-dimensional conduction equation for the heat flow across the visible meniscus. In that method, the heat flow is a functional of meniscus shape. The new method simplifies the picture. It is shown that for small 0 , the heat flow across

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an arbitrary meniscus can be found by approximating the curve by its osculating parabola at the contact line. Since this parabola is specified by G and the curvature at the contact line, the heat flow is a function of certain parameters rather than a functional of meniscus shape. The new method gives the heat flow explicitly. The predictions are confirmed by published simulations. Morris [2] extends those results by using the existing theory of the contact region to obtain a formula for G. It is shown that for vanishing thermal resistance, the flow has an inner and outer structure. The inner contact region transfers negligible heat, but defines 0 . Heat is transported by pure conduction through the outer region in a geometry imposed by 0 . This part of the analysis overlaps that in the previous paragraph. Formulae are given for G and the total heat flow q as functions of the model parameters. The predictions are confirmed numerically. The phenomenological variables 0 and q depend weakly on atomic-level wetting physics, whose overall role is to impose a scale, namely the film thickness seen at the apparent contact line by the distant flow. 0 depends weakly on this scale, and q depends on wetting physics only through 0 . The poster presented at this meeting covers this analysis of the small-scale flow. References 1. S. J. S. Morris, "Effect of flow resistance in the contact region of an evaporating meniscus," in prep. (2001). 2. S. J. S. Morris, J. Fluid Mech. 432, 1 (2001). 3. S. J. S. Morris, J. Fluid Mech. 411, 59 (2000).

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A S P E C T S OF VORTEX D Y N A M I C S AT A FREE SURFACE BILL PECK, LORENZ SIGURDSON University of Alberta, Edmonton, Alberta, Canada PETROS KOUMOUTSAKOS, AND JENS WALTHER ETH Zurich, Switzerland

In our present work we are investigating the formation and subsequent instability of vortex rings formed by drops impacting a pool. We are especially interested in two aspects of the problem: the vorticity-generation mechanism at the free surface and the instability of the vortex ring at early times after impact. This instability leads to shedding of large-scale structures into the vortex-ring wake that we refer to as petals. In some cases, the petals pinch at their tips forming small vortex rings that travel away from the central axis of the primary vortex ring. We refer to this structure as the blooming vortex ring. For a more thorough discussion of the vortex structure, see [1]. The intensity, or level, of tangential vorticity u5t at a free surface can be predicted with the expression ut = 2n x n, where ft is the free-surface normal vector and an overdot denotes the material derivative [2]. After the drop and pool coalesce, the free-surface normal vector of a particle embedded in the free surface rapidly rotates and vorticity must be present to satisfy the shear-free boundary condition. This argument predicts the sign of vorticity observed in experiments. This argument does not explain however, the means by which torques are applied to fluid elements near the interface to create vorticity. An analysis of the vorticity equation accounting for steep viscosity and density gradients across the interface shows that several mechanisms are likely responsible. These include the well-known baroclinic torque, as well as viscous torques and torques that result from terms dependent on viscosity- and density-gradient coupling. Also, surface tension will lead to vorticity creation through surface gradients in the mean curvature. We are also interested in the mechanism that leads to the instability of the vortex ring. A high wave number instability appears on the vortex ring at early times (5-10 ms) after impact. The wavenumber of the instability decreases with time from about 20 to three, four, or five. This instability does not appear to be a straining instability such as the Widnall instability, but rather a secondary instability of filaments on the primary core. The unstable filaments then wrap around the primary core to form a series of counter-rotating pairs. These interact with one another until a preferred wavenumber is found. These larger pairs convect away from the primary core under their mutual self induction and escape the trapped streamlines of the primary vortex ring. This leads to the formation of petals that are seen in the wake. We are presently investigating this instability with a vortex-in-cell algorithm as used by [3] to study the formation of the Widnall instability on an initially perturbed vortex ring. In that case, the peaks of the waves eventually break leading to vortex filaments being wrapped around the primary core and eventually being

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ejected into the wake. The drop-formed ring instability seems to be substantially different and we hope that the numerical analysis will lead to useful insights. We also hope to determine what part the presence of the interface plays in the onset of this instability. Acknowledgments We thank Karim Shariff for many interesting discussion on vortex rings and acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada. Also, BP thanks the participant support fund for their assistance in attending this conference. References 1. B. Peck and L. Sigurdson, Phys. Fluids 6, 564 (1994). 2. B. Peck and L. Sigurdson, IMA J. of Appl. Math 6 1 , 1 (1998). 3. P. Koumoutsakos, J. H. Walther, and J. B. Sagredo, "DNS of vortex rings using vortex methods," Phys. Fluids, to appear (2001).

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INSTABILITIES AT THE "INTERFACE" B E T W E E N MISCIBLE FLUIDS — E M E R G E N C E OF A N EFFECTIVE SURFACE TENSION PHILIPPE PETITJEANS, PASCAL KUROWSKI, AND JUAN FERNANDEZ Laboratoire de Physique et Mecanique des Milieux Heterogenes, UMR CNRS 7636, Ecole Superieure de Physique et de Chimie Industrielles (ESPCI), 10, rue Vauquelin, 75005 Paris, France

Instabilities can develop at the "interface" between two miscible fluids that exhibit patterns very similar to those observed between immiscible fluids. For example, a layer of colored glycerine hanged-up below a glass plate and placed into a water tank gives rise to organized patterns. Also, a thin layer of glycerine falling into water is unstable and produces falling columns with a regular distance between each other. These similarities between miscible and immiscible fluids lead to the possible existance of an "effective surface tension" between miscible fluids. This effective tension has a meaning for short times only, and should tend to zero with time. It can be explained by the strong concentration gradient at the "interface."

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S E C O N D A R Y INSTABILITIES OF FALLING FILMS USING MODELS CHRISTIAN RUYER-QUIL AND PAUL MANNEVILLE Laboratoire dHydrodynamique (LadHyX), Ecole Polytechnique, F-91128 Palaiseau Cedex, France

Viscous fluids flowing down inclined planes are of special interest in the study of pattern formation and the transition to spatio-temporal chaos. At moderate Reynolds numbers, their dynamics is controlled by surface-tension effects and viscous dissipation. Making use of the slaving principle, one can eliminate most local internal flow variables that are essentially bound to follow the slow evolution of the film thickness h. Combining a systematic expansion of the Navier-Stokes equation in powers of a small dimensionless parameter measuring the amplitude of thickness gradients to a Galerkin approximation method involving a functional basis made of polynoms, we have derived several models of increasing accuracy and complexity In the two-dimensional case, our "best model" is consistent to second order in the gradient expansion and involves four slowly varying quantities: h(x, t), the flow rate q(x,t), and two supplementary fields describing local corrections to the basic semi-parabolic flow profile. Though yielding results in excellent agreement with direct numerical simulations [2] and experiments [3], it is somewhat difficult to use. A simplified version involving just h and q has been obtained from the secondorder equations by neglecting the corrections to the flow profile. Slightly more complicated than Shkadov's model [4], it contains all relevant physical processes, and especially the effects of viscous dispersion. Quantitatively acceptable results are obtained with the simplified model in a domain of flow parameters extending from the onset of the free-surface instability to deep in the strongly nonlinear regime of large-amplitude solitary waves. The critical Reynolds number is correctly predicted, the main characteristics of the waves are faithfully reproduced, e.g., velocity vs. wavelength in the saturated-wave regime, and the simulated film evolution is apparently free of finite-time blow-up. The simplified model can be straightforwardly extended to deal with threedimensional flows by including spanwise modulations. This allows us to tackle the wave-stability problem in the most general context. Floquet analysis in the moving frame has been developed yielding results in general agreement with those obtained by the same method within the boundary-layer approximation [5]. The agreement with experimental results [6] is less good. However, since the extension of the complete model consistent to second order is too complicated to be of use, for the moment it seems difficult to attribute the observed discrepancy to a limitation of the simplified model that would show up only in three dimensions rather than to some sensitivity to the precise choice of physical parameters fitting the experiments. To conclude, the work done "by hand" in deriving models of film flows seems rewarding when compared to computationally expensive direct numerical simulations. The enslaving of hydrodynamic fields to the local thickness and flow rate

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seems to be an important factor explaining the success of a low-order truncation of the Galerkin approximation mostly relying on the functional dependence of the basic-flow profile. Our results support the hope to "understand" the growth of disorder and the transition to turbulence in open flows. References 1. Ch. Ruyer-Quil and P. Manneville, Eur. Phys. J. B 15, 357 (2000). 2. T. R. Salamon, R. C. Armstrong, and R. A. Brown, Phys. Fluids 6, 2202 (1994). 3. (a) J. Liu, J. D. Paul, and J. P. Gollub, J. Fluid Mech. 250, 69 (1993). (b) J. Liu and J. P. Gollub, Phys. Fluids 6, 1702 (1994). 4. V. Ya. Shkadov, Izv. At Nauk SSSR, Mekh. Zhi Gaza 2, 43 (1967). 5. H. C. Chang, E. A. Demekhin, and D. I. Kopelevitch, J. Fluid Mech. 250, 433 (1993). 6. J. Liu, B. Schneider, and J. P. Gollub, Phys. Fluids 7, 55 (1995).

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INTERFACIAL P H E N O M E N A IN S U S P E N S I O N S UWE SCHAFLINGER AND GUNTHER MACHU Institut fur Sromungslehre und Warmeubertragung, Technical University Graz, Inffeldgasse 25, A-8010 Graz, Austria

Recent resuspension experiments suggest that a possible interfacial tension exists between pure fluid and a suspension consisting of the same fluid and heavy, small, solid beads of identical size and density. Since little is known about interfacial phenomena in suspensions we experimentally investigated the formation and expansion of a suspension drop in the same fluid. To our surprise, the motion of the droplet exhibits all phenomena demonstrated by the classical experiments in which vortex rings of one liquid are created in another from drops falling from rest under gravity. Membranes formed even when the concentration of particles was smaller than 5%. We also observed the breaking of the torus by the Rayleigh-Taylor instability and the formation of a cascade of new rings. Two drops falling one behind the other penetrate but do not lose their distinct nature for several moments until they finally mix and move as a single drop. Larger drops typically create a long cylindrical tail of particles throughout the vessel that contains the clear liquid. This column is unstable and it breaks and forms small 'capillary droplets.' By making use of earlier investigations we were able to estimate the interfacial tension from the growth of the torus.

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SESSILE DROP SOLIDIFICATION WILLIAM W. SCHULTZ Department of Mechanical Engineering, The University of Michigan, Ann Arbor, MI 48109-2125 M. GRAE WORSTER Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CBS 9EW, England, UK DANIEL M. ANDERSON Department of Mathematical Sciences, George Mason Fairfax, VA 22030

University,

Containerless solidification, confining its melt by surface tension, is an important technique to produce very pure materials. T h e form of the solidified product is sensitive to conditions at the tri-junction between the solid, the melt, and the surrounding vapor. We reconsider the analysis of Anderson, Worster, and Davis (1996) to test whether a more simple tri-junction condition can model experimental behaviour when the flat solidifying interface is no longer imposed. We find t h a t the simpler condition can describe the inflection point and cusp of the axisymmetric drop if the solid-liquid interface is no longer assumed to be flat. T h e solution is now checked asymptotically for large and small time and for short distances from the tri-junction condition.

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A HOMOGENIZED MONTE-CARLO MODEL FOR FILM G R O W T H TIMOTHY SCHULZE Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300

We derive a fully continuous, macroscopic model for epitaxial film growth from an analogous, but microscopic, Monte-Carlo procedure. Our approach features an explicit adatom density that evolves subject to deposition, diffusion, nucleation, and edge-adsorption of adatoms and is coupled to a separate height evolution equation. A specific aim of this approach is to incorporate multi-species effects, which are important in many systems (e.g., YBCO). To this end, it is natural to make a distinction between an atom on the film surface but not yet incorporated into the film (i.e., in the surface-density field), and an atom that has become part of a completed unit cell (i.e., in the film-height field). Such a distinction is familiar in mesoscopic models involving diffusion on terraces and explicitly represented ledges, but is absent from existing continuum models. Thus, the multi-species version of this model has an evolution equation for the density of each species on the surface of the film.

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I N F L U E N C E OF T H E S U R R O U N D I N G C O N D I T I O N S N E A R T H E INTERFACE ON T H E STABILITY OF LIQUID B R I D G E S V. M. SHEVTSOVA, M. MOJAHED, AND J. C. LEGROS MRC, Universite Libre de Bruxelles, CP-165, 50 av. F. D. Roosevelt, 1050 Brussels, Belgium

Time-dependent thermal convection has been investigated experimentally in deformed liquid bridges of silicone oil with a Prandtl number Pr — 105. The temperature oscillations were measured by five thermocouples placed through the upper rod into different azimuthal positions in the liquid bridge some distance from the free surface. The signals from the thermocouples were taken for Fourier analysis and determination of their phase shift. The experimentally obtained stability diagram (ATcr,V) shows that the transition from axisymmetric steady flow to an oscillatory flow is very sensitive to variations in the volume. It consists of two different oscillatory instability branches belonging to different azimuthal wave numbers. Between them, there is a small range of volumes for which the steady flow is stable up to very high values of AT. For high Prandtl fluids, the branch on which ATcr increases with increasing volume has an azimuthal wave number m = 1, and the descending branch has an azimuthal wave number m = 2. It was found that the beginning of the gap is linked to the upper contact angle a. The physical explanation of the stability diagram follows the arguments proposed by Shevtsova and Legros [Phys. Fluids 10, 1621 (1998)]. A detailed analysis of the power spectrum and phase shift of the thermocouple signals reveals that the instability begins as a mixed mode, with wave numbers m = 0 and m — 1, within a narrow interval of AT. This is followed by a nearly standing wave with wave number m = 1 that changes to an m = 1 travelling wave when AT > 1.2ATcr. It has been proven experimentally that if standing (or travelling) waves are established in the liquid bridge, they can exist indefinitely as long as experimental conditions are maintained constant. To the best of our knowledge, there has been no previous experimental study of the influence of the average temperature inside the liquid bridge (the temperature of the cold rod) on the onset of instability. It is shown that the critical Marangoni number and critical wave number are very sensitive to the average temperature in the liquid bridge (e.g., to the temperature of the cold rod). Is this only an effect of temperature-dependent viscosity, or is it some other effect? The reason for such behaviour remains obscure. We have also discovered the possibility of changing the most dangerous mode at the threshold of instability. By surrounding the liquid bridge with another cylindrical volume with a larger internal diameter and kept at a constant temperature, the critical mode m = 2 can be switched to m = 1.

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CONVECTIVE-DIFFUSIVE LATTICE MODELS OF INTERFACIAL A N D W E T T I N G D Y N A M I C S YITZHAK SHNIDMAN Department of Chemical Engineering and Chemistry, Polytechnic University, Six MetroTech Center, Brooklyn, NY 11201

Many molecular and colloidal systems exhibit coexistence of different phases, separated by interfacial regions with characteristic density/concentration profiles and curvatures. External stresses, temperature gradients, and chemical potentials drive such systems out of thermodynamic equilibrium, giving rise to distinct deformations and flows within and across interfacial boundaries that are coupled to those in the bulk. Dynamic interfacial phenomena occurring in multiphase and thermocapillary flows, and in processes such as wetting, coating, adhesion, friction, and lubrication, play a very important role in many physicochemical, biological, and industrial processes. Molecular dynamics is a realistic tool for simulating these phenomena, but it is limited to small systems and short simulation times, in which statistical fluctuations mask local mean densities and flows. A surface-thermodynamic level of modeling is another approach that has been very useful in the past. It postulates separate constitutive relations between strains and dissipative contributions to the stresses (rheology) at Gibbsian (infinitesimally thin) dividing surfaces. However, these constitutive relations proliferate for interfacial and non-Newtonian systems, and are hard to parametrize experimentally. In addition, the Gibbsian approximation leads to unphysical singularities in some important dynamic processes, such as wetting. We present a novel, convective-diffusive lattice-gas approach for modeling interfacial dynamics in multiphase flows. The equations of motion are expressed in terms of probabilities over microscopic lattice degrees of freedom consisting of molecular occupancies and Eulerian velocities at discrete lattice sites. In contrast to hydrodynamic lattice-gas and lattice-Boltzmann approaches, these velocities are not limited to discrete directions and magnitudes. Our method is based on a local self-consistent mean-field approximation for these probabilities and their moments, which are evolved by alternating discrete convective and dissipative timesteps, both subject to local conservation laws in discrete form. We model dissipation as a Markov-chain process corresponding to diffusion of the molecules and their momenta by short-range hopping that depends explicitly on molecular interactions, as well as on site occupancies and velocities. We demonstrate applications of the isothermal version of this method to computational modeling of interfacial dynamics, wetting, and coating phenomena occurring in multiphase flows that are in contact with moving solid walls, as well as the use of a nonisothermal version of this method for modeling thermocapillary flows.

287 A LEVEL-SET A P P R O A C H TO D O M A I N G R O W T H IN M U L T I C O M P O N E N T FLUIDS KURT A. SMITH, FRANCISCO J. SOLIS, AND MONICA OLVERA DE LA CRUZ Department of Chemical Engineering, Northwestern University, Evanston, IL 60208

During the late stages of spinodal decomposition, domains reach their equilibrium concentrations and sharp interfaces develop. In fluids, surface-tension-induced flow creates a hydrodynamic coarsening mechanism assuming domains are interconnected. For symmetric binary fluids, scaling arguments predict a growth rate of length ~ time at low Reynolds number and length ~ time^1!^ at high Reynolds number. Using a level-set method, we find evidence of a universal curve approaching the length ~ time^2'3' limit at large lengths (high Reynolds number). We find similar behavior in ternary systems. Growth behavior in ternary systems is rich due to the various possible topologies. For example, we find that growth of a disconnected phase may be promoted in the presence of two interconnected phases.

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MORPHOLOGICAL INSTABILITY IN S T R A I N E D ALLOY FILMS

B. J. SPENCER Department of Mathematics, University at Buffalo, Buffalo, NY 14260-2900 P. W. VOORHEES Department of Materials Science and Engineering, Northwestern Evanston, IL 60208

University,

J. TERSOFF IBM Research Division

We analyze the development of compositional and surface nonuniformities during the growth of strained alloy films. A continuum model is derived for the evolution of the film surface and composition due to the processes of deposition and surface diffusion. T h e compositional stresses arising from compositional variations of different-size alloy components is incorporated into the elasticity problem for the epitaxial film, and the thermodynamics of stressed solids is used t o derive the chemical potentials for surface diffusion. From this coupled free-boundary problem for the morphology, composition, and stress state, the stability characteristics of planar film growth is determined from a linear stability analysis. For the case of equal surface mobilities of the alloy components, we find t h a t compositional stresses make the film more unstable to the formation of stress-driven surface undulations. T h e destabilization is greatest over a moderate range of deposition rates, b u t weak at fast and slow deposition rates. For the case of different surface mobilities of the alloy components, we find t h a t the difference in surface mobilities can completely suppress the stress-driven morphological instability. T h e stabilization is not symmetric with respect to the compositional and misfit strains: it occurs in films with compressive (tensile) misfit, when one of the atomic species in the film is b o t h large and slow (large and fast) relative to the other component. A feature of the stabilization due to mobility differences is the existence of a critical deposition rate for stabilization, above which the film is stable. T h e existence of a critical deposition rate suggests the possibility of growing controlled lateral composition modulations using the small-amplitude steady-state behavior of the system at deposition rates slightly below the critical condition.

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STABILITY ISSUES IN SPIN-CASTING MOLTEN METALS PAUL H. STEEN School of Chemical Engineering, Cornell Ithaca, NY 14853-5201

University,

Planar-flow casting (PFC) (i.e., melt-spinning or spin-casting) produces flat product by forcing molten metal onto a moving substrate (i.e., a spinning wheel) where it solidifies and leaves as a ribbon, strip, or foil. Heat and momentum fluxes dominate in the gap between the nozzle and the wheel. Success of the planar-flow process depends on the dynamics and stability of a group of two- and three-phase contact regions. The stability of these contacts is crucial in spin-casting, impact-forming, die-casting, spray-forming, extrusion, or any casting process where bulk properties and surface quality are important. They must also be understood in the design of new processes that depend on rapid phase change and high throughput, whether to produce innovative materials or conventional materials more efficiently. The solidifying flow is distinguished by the following features: • Meniscus: deformable molten-metal/gas interface, • Tri-junction: 3-phase common-line with phase change, • Dynamic contact-line: 3-phase common-line with slip, • Solidification front: surface of solid/liquid phase change. Results: Recent experimental results are based on casting aluminum/silicon (Al/Si) alloys with Si content ranging from 0% (0.999 pure Al) to about 12% Si, depending on the cast. • Experiment: The movement of the upstream meniscus is coordinated with the contact exposure of the wheel, suggesting that the puddle length (or contact length) can respond to the heat-up of the wheel [1]. • Analysis: The flow structure within the puddle, predicted by a previous adhoc analysis [2,3] is rederived in a systematic way to clarify the appropriate scalings [4]. The resulting pressure field is tested against observation [5]. • Theory: Vorticity transport in solidification boundary layers has been elucidated [6]. Forming singularities are likely to be important in PFC where 'fresh' surface is created at high rates; contacting and forming singularities are illustrated in a simple context [7]. References 1. T. Ibaraki, Planar-Flow Melt-Spinning: Experimental Investigation on Solidification, Dynamics of the Liquid Puddle, and Process Operability, M.S. Thesis, Cornell University, Ithaca, NY (1996).

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2. C. Karcher and P. H. Steen, Phys. Fluids 13, 826 (2001). 3. C. Karcher and P. H. Steen, Phys. Fluids 13, 834 (2001). 4. B. L. Reed, Planar-Flow Spin Casting: Momentum Transport, Vorticity Transport, and Texture Formation, Ph.D. Dissertation, Cornell University, Ithaca, NY (2001). 5. P. H. Steen, B. L. Reed, and M. B. Kahn, Solidification-Induced Secondary Flows in Spin-Casting, in Interactive Dynamics of Convection and Solidification, eds. P. Ehrhard, D. Riley, and P. H. Steen, pp. 145-154, Kluwer (2001). 6. B. L. Reed, A. H. Hirsa, and P. H. Steen, J. Fluid Mech. 426, 397 (2001). 7. P. H. Steen and Y.-J. Chen, Chaos 9, 164 (1999).

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THE STABILITY OF THERMOCAPILLARY CONVECTION IN HALF ZONES WITH DEFORMED FREE-SURFACE PROFILES L. B. S. SUMNER Mercer University, School of Engineering, Macon, GA 31207 G. P. NEITZEL The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405

Several theoretical analyses have examined the stability of half zones, always assuming that the free surface of the liquid bridge is cylindrical, as it would be in a microgravity environment. However, recent experiments by different groups have shown that significant free-surface deformation can have a pronounced effect on the stability boundary. Here, experiments and a theoretical stability analysis consider the onset of oscillatory thermocapillary convection in half zones with significantly deformed free-surface profiles. Free-surface deformation is quantified by a volume ratio representing the volume of liquid in the bridge relative to the volume of a right circular cylinder of equal height and endwall radius. The experiments mark the boundary for the onset of oscillatory flow in half zones with volume ratios ranging from 0.5 to 1.6. The theoretical analysis applies energy stability theory to steady, axisymmetric flow in the half zone under the conditions of the experiments. A comparison between the theoretical and experimental results show that the energy limits are above the experimental stability boundaries in clear contradiction to theory. An explanation for the exaggerated stability limits has not been determined although the results may provide new insights about the already well-researched instability mechanisms.

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D I R E C T W R I T E OF PASSIVE CIRCUITRY USING INK-JET T E C H N O L O G Y J. SZCZECH, C. M. MEGARIDIS Department of Mechanical Engineering, University of Illinois at Chicago, Chicago, IL 60607 D. GAMOTA, AND J. ZHANG Motorola Advanced Technology Center, Schaumburg, IL 60196

Ink-jet technology is targeted for use in conformal electronics manufacturing where a homogeneous nano-particle suspension is utilized as the structural material. The suspension is dispensed onto flexible substrates by means of a piezoelectric droplet generator device capable of producing droplets on the order of 30-60 /im in diameter. The droplet generator is driven by a square-pulse waveform in which the rise/fall voltage and dwell time are variable. The substrates containing the dispensed material are then cured at 300°C for 15 minutes to yield a finished product. This method of circuit production is attractive to the electronics industry for it: 1) minimizes material usage and chemical waste production, 2) reduces turnover time, and 3) is capable of rapid prototyping. Ink-jet printing technology offers the potential for highly compact, ultra-lightweight assemblies and, especially, for rapid product introductions and enhancements, which are becoming increasingly critical in today's competitive wireless communications market. Current attempts in producing circuit interconnections at Motorola have proven to be successful. Although improved component tolerances and unique patterning capabilities have been verified using drop-on-demand ink-jet techniques at Motorola, it remains to be demonstrated whether it is capable of the reproducibility, robustness, and accuracy necessary for reliable production of commercial products. The repeatability of the process relies heavily on steady, satellite-free, droplet generations. To attain such optimum droplet generations it is essential to investigate the relation between the device excitation parameters/design and fluid properties. Previous studies show surface tension to be a critical fluid property in the generation of repeatable droplet formations. Therefore, quantification of the influence of surface tension in correlation with the device excitation parameters is required for successful commercial implementation. To quantify the drop-formation process, a short-duration flash videography technique is employed with time-delay capabilities down to 1 ^s. With this technique, it is possible to observe the droplet ejection and formation process as a function of various impulse excitation parameters. Key metrics, such as droplet size and velocity, are measured, and satellite occurrences are recorded to generate an extensive parameter matrix that assists in the optimization of the dispensing process.

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STABILITY OF THE MENISCOID PARTICLE BAND AT A D V A N C I N G INTERFACES IN HELE-SHAW S U S P E N S I O N FLOWS* H. TANG, W. D. GRIVAS, T. J. SINGLER, J. F. GEER, AND D. HOMENTCOVSCHI Department of Mechanical Engineering, State University of New York, Binghamton, Binghamton, NY 13902-6000 Instability can occur in the displacement of two or more contiguous fluids when a less viscous fluid displaces a more viscous fluid. In Hele-Shaw flow, evolution of the instability leads to fingers of the less viscous fluid growing into the more viscous fluid as displacement proceeds. Viscous fingering has been observed in the displacement of colloidal suspensions. The operative physical principle in suspension fingering is that the effective viscosity of a dense suspension exceeds the viscosity of the pure carrier liquid and is a monotonically increasing function of the local particle volume fraction. In natural or industrial processes, and in laboratory experiments which seek to model such processes, the unfavorable viscosity contrast necessary for viscous fingering typically exists as an initial condition in the displacement process. Here, however, we discuss a physical system in which the conditions for viscous fingering are not initially present, but which evolve concomitantly with displacement. We observe the formation of an effectively more viscous region in Hele-Shaw, free-surface suspension flow. The increase in effective viscosity derives from the autogenous accretion of particles in a region adjacent to the free surface between the suspension and the fluid being displaced. This region can itself be unstable to displacement by the trailing suspension, which is effectively less viscous because of its smaller particle volume fraction. It is well known that the volume fraction of particles adjacent to a meniscus formed by a suspension in contact with another immiscible fluid increases if the meniscus is advancing. The mechanisms responsible for particle accumulation are complex, but a simple kinematic argument shows that a particle can enter the meniscus region along centerline-region streamlines faster than it can exit along wall-region streamlines because of the finite diameter of the particle. The cumulative effect of this differential kinematic particle flux is meniscoid banding. When the band achieves a critical thickness and particle volume fraction, fingers of the less viscous trailing suspension may be observed growing into the band. Observations show that band suspension is shed into finger sidebands, which due to their higher effective viscosity move more slowly than the suspension/air interface and tend to elongate with time. Because the band has finite radial extent, there is limited space for finger evolution and the complex evolution patterns observed in nonlinear growth into unbounded domains cannot occur here. However, sufficient band thickness is maintained to allow simple tip-splitting of the fingers and an increase in wavenumber with displacement. The suppression of viscous fingering in connection with recovery and regeneration processes has been a subject of considerable interest, and success in minimizing growth rates of instability for unfavorable viscosity contrasts has been achieved by

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injection of a solute resulting in a graded viscosity in the interfacial zone. The account above pertains to the development of conditions favorable to instability of an initially stable flow. Conversely, we have also observed that, for initially unstable conditions, e.g., when an initially uniform suspension displaces a more viscous fluid, the evolution of an increased effective-viscosity band region can stabilize the interface between the suspension and displaced fluid. A weak local fingering instability is still observed in the narrow band region, but the global flow is stable. Thus, at least for length scales accessible by our apparatus, we conclude that autogenous band growth at the interface of an unstable couple leads to delayed growth of global instability, i.e., the evolution of the band region represents a natural stabilization mechanism of an otherwise very unstable interface, with the small price of a weak internal mode disturbance within the band. This research was supported by DARPA, N00164-96-C-0074, SUNY at Binghamton.

and the IEEC at

* Full paper appearing as "Stability considerations associated with the meniscoid particle band at advancing interfaces in Hele-Shaw suspension flows," H. Tang, W. Grivas, D. Homentcovschi, J. Geer, and T. Singler, Phys. Rev. Lett. 85, 2112 (2000).

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INSTABILITIES IN T H I N FLUID SHEETS B. S. TILLEY Olin College of Engineering, 1735 Great Plains Avenue, Needham, MA 02492-1245 D. T. PAPAGEORGIOU AND R. V. SAMULYAK Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NY 07102

We investigate the interfacial dynamics of potential flow in a thin fluid sheet for varicose long-wave disturbances when surface forces and inertia are the dominant physical effects. A coupled set of evolutions equations is derived for the interfacial deflection and axial velocity. For sufficiently small amplitudes, spatially periodic disturbances on the film lead to counter-propagating waves whose wavespeeds depend on the initial data. For larger-amplitude initial data below a threshold, these standing-wave-type solutions persist. In this regime, general initial conditions lead to quasi-periodic solutions in time. However, there is a class of initial data for which the transients are simply travelling-wave solutions. Beyond this threshold, the film ruptures according to the similarity scales reported in Pugh and Shelley (1998), independent of the amplitude of the initial data above threshold. Near the rupture criterion, transients suggest a similarity solution that has a bounded velocity but is singular in its spatial gradient. Further, a similarity property of the evolution equations predicts the threshold criterion up to the evaluation of a single constant, which is obtained numerically.

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T H E R M A L EFFECTS OF I N T E R N A L INTERFACES: EQUILIBRIUM M I C R O S T R U C T U R E A N D KINETICS A. UMANTSEV Department of Chemistry and Physical Sciences, Saint-Xavier University, Chicago, IL 60655

The thermal effects of internal interfaces, like interphase, antiphase, and domain boundaries, stem from their possession of an internal-energy excess. In my presentation, I will analyze the influence of the interfacial thermal effects on the formation of microstructures. These effects have received very little attention in the literature; their study is long overdue. Thermal effects significantly change the phase diagrams of one-component materials, in particular, those of thin films. For very thin films with thicknesses only 5-20 times greater than the interfacial thickness, phase separation does not occur and equilibrium is achieved with homogeneous transition states that can never be obtained in bulk samples because of their absolute instability. The thermodynamic and dynamic explanations will be presented. This type of a thin-film phase diagram is important for electronic devices and integrated circuits, for the theory of amorphization, and for nanophase composite materials where small particles or thin whiskers capable of undergoing a transition are immersed into a poorly conducting matrix. As large systems evolve far from equilibrium, they produce complicated fractal structures with high interfacial-area density, like dendrites during solidification, highly dispersed coexisting phases during martensitic transformation, or selfentangled domains of opposite ordering after continuous symmetry-breaking ordering below the critical temperature. I will elucidate the influence of different thermal effects on the general properties of evolving microstructures. One of them (the heattrapping effect) is caused by temperature gradients across an interphase boundary that change the course of dynamics and allow the emergence of a metastable (superheated) phase in the growth stage of the transformation. Conditions for such a regime may be met in many organic materials, including polymers. Another thermal effect of interphase boundary motion that will be discussed in conjunction with structure formation is the surface creation and dissipation effect. The temperature gradients across antiphase-domain or grain boundaries stem from the transmission of energy across the interface and cause a drag effect, which changes the course of structural coarsening for the later stages of material evolution. An experimental setup designed to reveal this thermal drag is suggested.

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T H E ROLE OF L O N G - R A N G E FORCES IN T H E STABILITY OF NEMATIC FILMS M. P. VALIGNAT, F. VANDENBROUCK, AND A. M. CAZABAT Laboratoire de Physique de la Matiere Condensee, College de France, Paris, 75231 Cedex 05, France

We present an experimental study of spontaneous and forced wetting of 5CB in the nematic phase on silicon wafers. At room temperature, a droplet of 5CB completely wets the surface and the ellipsometric profile reveals the existence of a characteristic thickness (~ 200 A). As the temperature is increased, this thickness increases and "diverges" at the nematic/isotropic transition. This is the signature of a wetting transition: a 5CB droplet in the isotropic phase does not wet the surface. The question that arises is: why does the wetting transition takes place at the nematic/isotropic transition? In the discussion, we propose a simple model that takes into account long-range interactions and we show that the balance between elastic, anchoring, and van der Waals energies may fix the wetting condition. The knowledge of the energy per unit surface area allows us to predict the stability of a thin nematic film. For 5CB spun-cast onto silicon wafers, thin films are unstable while thick ones are metastable. We then observe experimentally the characteristic spinodal or nucleation dewetting in the two cases.

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V I B R A T I O N - I N D U C E D D R O P ATOMIZATION BOJAN VUKASINOVIC, MARC K. SMITH, AND ARI GLEZER The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405

The subject of this research is a process we call vibration-induced drop atomization (VIDA). The process starts with a small water drop (~ 1 cm in diameter) placed at the center of a horizontal, circular metal diaphragm. The diaphragm is fastened to a holder at its periphery and a piezoelectric disk is centrally mounted to its bottom side. Excitation of the piezoelectric disk by a sinusoidal voltage causes the diaphragm to vibrate in the vertical direction in its fundamental axisymmetric mode. When the excitation signal is held at a fixed frequency and the amplitude is slowly increased, different stages of free-surface motion on the drop are clearly observed. First, axisymmetric standing waves appear on the free surface for small values of the excitation amplitude. These waves have the same frequency as the excitation and are present at even very small excitation amplitudes. At a first critical excitation amplitude, an azimuthal instability is triggered along the contact line. It couples with the existing axisymmetric waves so that the free surface undergoes a transition to a non-axisymmetric wave form. Here, the most energetic spatial mode has a frequency that is the first subharmonic of the excitation frequency. This is a characteristic of the parametric instability seen in a standard Faraday-wave experiment. As the excitation amplitude increases further, the three-dimensional, free-surface waves increase in magnitude, spread over the entire free surface of the drop, and exhibit very complex, nonlinear spatial behaviors. At a second critical excitation amplitude, small secondary droplets begin to be ejected from the freesurface wave crests. When droplet ejection occurs, the ejection sites appear in a central region that covers about two-thirds of the free surface. After droplet ejection begins and for large enough excitation frequencies, complete atomization of the drop will suddenly occur as the result of a rapid increase in the rate of secondary droplet ejection — an event we call drop bursting. If the excitation amplitude is large enough and modulated by a step function, atomization occurs immediately. Even so, we still observe all of the different freesurface motions of the drop described earlier. We see the axisymmetric waves, then the growth of the azimuthal instability, and the development of complex wave patterns primarily near the contact line. Droplet ejection begins around the circumference of the primary drop near the contact line due to the large-amplitude growth of the azimuthal instability. The time from the start of the excitation until the first secondary droplets are ejected is about 15 forcing periods. These secondary droplet ejection sites then develop over the entire central part of the free surface of the drop. For this case, the entire primary drop is atomized in less than a second after the initiation of the excitation. The rate of droplet ejection depends on the excitation amplitude, but more interesting is that for a fixed excitation amplitude and frequency, the rate of droplet ejection can increase or decrease with time. This behavior depends entirely on

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the driving frequency. A droplet can burst or atomize almost instantly, or the bursting can be delayed on the order of seconds to maybe a minute or more after the excitation is applied. In some instances, bursting does not occur at all, even though initial droplet ejection is present; the droplet-ejection process just dies out. All of these facts suggest that the evolution and the rate of droplet ejection depends on an interplay between ejection from the primary drop and the vibrating diaphragm. This dependence is fully understood and experimentally tested. The VIDA process is currently being utilized in a microelectronics cooling application using a self-contained heat transfer cell. It has also proven to be effective in other applications like spray coating, emulsification, encapsulation, and mixing enhancement.

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CAPILLARITY-DRIVEN INSTABILITIES A N D T H E EVOLUTION OF SOLID T H I N FILMS HARRIS WONG Department of Mechanical Engineering, Louisiana State Baton Rouge, LA 70803

University,

MICHAEL J. MIKSIS Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208 PETER W. VOORHEES Department of Material Science and Engineering, Northwestern Evanston, IL 60208

University,

STEPHEN H. DAVIS Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208

Thin solid films are the basic structure in most microelectronic and optoelectronic devices. This is a direct result of the layer-by-layer manufacturing technique, which involves roughly a hundred steps of successive material deposition and patterning. During these processes, the device is often heated to high temperatures, and the solid components inside the device can deform. The deformation becomes more disruptive as the size of integrated circuits decreases. To continue this trend of miniaturization, the stability and evolution of thin solid films needs to be understood. In this poster, we will present the morphological instabilities and evolution of solid-film shapes commonly encountered in micro-devices, such as holes, wedges, rods, and steps.

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INTERFACIAL WAVE THEORY FOR D E N D R I T E G R O W T H JIAN-JUN XU Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 2K6, Canada

During the last decade two analytical theories for free-dendrite growth, the microscopic solvability condition (MSC) theory and the interfacial wave (IFW) theory, were proposed in the areas of condensed-matter physics and material science. These theories were attempts to resolve the problem of selection in dendrite growth, and to explain the essence of pattern formation. This article attempts to clarify the differences and commonalities between these two theories and to compare the predictions of these theories with some of the latest numerical evidence and experimental data. Since the MSC theory is the most well-developed for the two-dimensional case, the comparison of the theories with numerical simulations is made mainly by using, but is not restricted to, two-dimensional, numerical solutions for dendrite growth with anisotropy of surface tension. Such kinds of numerical simulations have been carried out by Wheeler et al. (1993), Provatas et al. (1998, 1999), and Karma et al. (1996, 1997) with the phase-field model, and by Ihle and Muller-Krumbhaar (1994) with the free-boundary-problem model. It is seen that if the anisotropy parameter is not too small, the numerical simulations yield steady needle solutions, whose side-branching structures are not selfsustaining. This result supports the conclusions drawn by both the MSC and the IFW theories. However, the numerical simulations also showed that there exists "a smallest value of the anisotropy parameter," less than which "no steady needle solution was found." This numerical evidence appears to be in agreement with the IFW theory and to contradict the MSC theory. The prediction of the IFW theory is also compared with the latest experimental data obtained by Glicksman et al. (private communication) in the microgravity environment and excellent overall agreement between both is found.

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UNIFORMLY VALID A S Y M P T O T I C SOLUTIONS FOR DENDRITE GROWTH WITH CONVECTION JIAN-JUN XU AND DONG-SHENG YU Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 2K6, Canada

The interaction of convection and dendrite growth has been a subject of great interest in the area of material science in recent years. Experimental observations have shown that convective motion in the liquid may have a significant effect on dendrite growth. The existence of convection may significantly change the growth velocity of the tip and the micro-structure pattern. Preliminary investigations of dendrite growth with convection are usually focused on the special case of zero surface tension. The solution for this special case, as in the Ivantsov solution for dendrite growth with no convection, cannot resolve the problem of the selection of the tip velocity of the dendrite or the dynamics of pattern formation on the interface. However, this solution is expected to provide a basis for further investigations in the general case of dendrite growth with non-zero surface tension. In the past several years, theoretical studies of steady dendrite growth in an external flow have been conducted by a number of authors (such as Ananth and Gill (1989, 1991); Benamar, Bouisson, and Pelce (1988); Saville and Beaghton (1988), etc.) both numerically and analytically. An analytical solution was obtained by Ananth and Gill in terms of the Oseen model of fluid dynamics, which led to a paraboloid shape for the interface. Their solution satisfies all the interface conditions, but does not satisfy a proper upstream far-field condition. In fact, their solution yields a non-vanishing perturbed flow in the upstream far field, which is not appropriate from a physical point of view. Xu (1994) made the first attempt to derive an asymptotic-expansion solution in terms of the Navier-Stokes model of fluid dynamics for the case of the Prandtl number, Pr —> oo. Xu derived a matched-asymptotic-expansion solution for the problem. However, for simplicity, the problem formulated in Xu's work neglected the regular condition for the velocity field at the symmetry axis. Furthermore, the same as in Ananth and Gill's work, the upstream far-field condition for the stream function in the formulation was weakened. In this paper, we attempt to reconsider this problem with a fully justified mathematical formulation. We assume that a dendrite is growing with a constant tip velocity U against the flow, which has a uniform velocity U„> in the far field. We study two limiting cases: • the weak-flow case: Uoo U, and derive uniformly valid asymptotic solutions.

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COMPARISON OF A S Y M P T O T I C SOLUTIONS OF PHASE-FIELD MODELS TO A SHARP-INTERFACE MODEL G. W. YOUNG AND S. I. HARIHARAN Division of Applied Mathematics, Department of Mathematics and Computer Science, The University of Akron, Akron, OH 44325-4002

A one-dimensional directional solidification problem is considered for the purpose of analyzing the relationship between the solution resulting from a phase-field model to that from a sharp-interface model. An asymptotic analysis based upon a large Stefan number is performed on the sharp-interface model. In the phase-field case, the large-Stefan-number expansion is coupled with a small-interface-thickness boundary-layer expansion. The results show agreement at leading order between the two models for the location of the solidification front and the temperature profiles in the solid and liquid phases. However, due to the non-zero interface thickness in the phase-field model, corrections to the sharp-interface location and temperature profiles develop. These corrections result from the conduction of latent heat over the diffuse interface. The magnitude of these corrections increases with the speed of the front due to the corresponding increase in the release of latent heat. By properly selecting the potential coupling the order parameter and temperature in the phase-field model, and by tuning the kinetic parameter, we are able to eliminate the corrections to the outer temperature profiles in the solid and liquid phases of the phase-field model. This in turn eliminates the correction to the location of the solidification front. Hence, the phase-field temperature profiles agree with the sharp-interface profiles, except near the solidification front, where there is smoothing over the diffuse interface and no jump in the temperature gradients.

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A N E W A P P R O A C H TO M E A S U R E T H E C O N T A C T A N G L E A N D T H E EVAPORATION R A T E W I T H FLOW VISUALIZATION IN A SESSILE D R O P NENGLI ZHANG AND DAVID F. CHAO NASA Glenn Research Center, Cleveland, OH 4-4135

The contact angle and the spreading process of a sessile drop are very crucial in many technological processes, such as painting and coating, material processing, film-cooling applications, lubrication, and boiling. Additionally,' it is well known that the surface free energy of polymers cannot be directly measured for their elastic and viscous restraints. Thus, measurements of the liquid contact angle on polymer surfaces become extremely important in order to evaluate the surface free energy of polymers through indirect methods linked with the contact-angle data. Since the occurrence of liquid evaporation is inevitable, the effects of evaporation on the contact angle and spreading become very important for a more complete understanding of these processes. It is of interest to note that evaporation can induce Marangoni-Benard convection in sessile drops [1]. However, the impact of convection inside the drop on the wetting and spreading processes is not clear. The experimental methods used by previous investigators cannot simultaneously measure the spreading process and visualize the convection inside the drop. Based on the laser shadowgraphic system used by the present authors [2,3], a very simple optical procedure has been developed to measure the contact angle, the spreading speed, the evaporation rate, and to visualize the convection inside of a sessile drop simultaneously. Two CCD cameras were used to synchronously record the realtime diameter of the sessile drop, which is essential for determination of both the spreading speed and the evaporation rate, and the shadowgraphic image magnified by the sessile drop acting as a thin plano-convex lens. From the shadowgraph, the convection inside the drop can be observed, if any, and the image outer diameter, which links to the drop profile, can be measured. Simple equations have been derived to calculate the drop profile, including the instantaneous contact angle, height, and volume of the sessile drop, as well as the evaporation rate. The influence of convection inside the drop on the wetting and spreading processes can be figured out through comparison of the drop profiles with and without convection when the sessile drop is placed at different evaporation conditions. References 1. N. Zhang and W. J. Yang, ASME J. Heat Trans. 105, 908 (1983). 2. N. Zhang and W. J. Yang, ASME J. Heat Trans. 104, 656 (1982). 3. N. Zhang and W. J. Yang, Rev. Sci. Instrum. 54, 93 (1983).

PART 3

PANEL DISCUSSION SESSION

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N E W RESEARCH DIRECTIONS IN INTERFACIAL SCIENCE M A R C K. S M I T H The George W. Woodruff School of Mechanical Georgia Institute of Technology, Atlanta, GA 30332-0405 U.S.A., E-mail: [email protected]

Engineering,

The last event of the conference was a panel discussion session chaired by one of the conference organizers, Michael J. Miksis [1]. The panelists were distinguished scientists from the fields of fluid mechanics and materials science and each one is a major contributor to the study of interfaces in his respective fields of research. The panel members were Stephen H. Davis (fluid mechanics and materials science) [2], Joel Koplik (fluid mechanics) [3], Wilfried Kurz (materials science) [4], Tony Maxworthy (fluid mechanics) [5], Robert F. Sekerka (materials science) [6], and Gretar Tryggvason (fluid mechanics) [7]. The panelists were chosen to strike a balance between the fields of fluid mechanics and materials science and also between the activities of theoretical, experimental, and numerical research. Professor Miksis asked each panelist to give a short statement on the general problem or problems he would like to see solved or worked on in the future. The intent of the discussion session was to bring out the similarities between interfacial dynamics in fluid mechanics and materials science and to suggest future avenues for research and cooperation. This account of the discussion is paraphrased from the complete transcript and was written by the author to clearly and smoothly describe the main themes raised during the session. A complete transcript of the discussion session was sent to the National Science Foundation in fulfillment of their contract-support requirements [8].

1

Introduction

There have been tremendous successes in understanding the essential physics of interfaces in the fields of fluid mechanics and materials science. These successes have often been made with similar techniques, such as stability theory and lubrication theory, and common numerical techniques such as phase-field, front-tracking, and level-set methods. The talks and poster presentations given during this conference have shown the similarities between the two research areas with respect to the progress that has been made to date. The job of the panel members in this discussion session was to stimulate discussion and further personal interactions in an attempt to help chart an appropriate course for future exploration in interfacial science. This paper is organized as follows. Section 2 is a paraphrased and rewritten account of the entire discussion session. It provides the background and context for the main themes summarized in Section 3. If desired, Section 3 can be read independently of Section 2. 2

The Flow and Micro structure

of the Discussion

Professor Davis began the discussion by saying, "I think predicting the future is hopeless, " Several individuals on the panel and in the audience expressed immediate agreement with this statement. However, the purpose of this discussion

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Smith

was not to predict.specific future events, but rather to chart a reasonable path for interfacial research in the future. Professor Davis continued by describing one example of the kind of work that he thought was important and necessary in order to make progress in understanding the behavior of complicated systems in the next century. He mentioned the biological problem presented by Gallez [9] in which she modeled in a general way the adhesion of a biological cell or vesicle to a solid substrate. In her model, the biological membrane was modeled as an interface between two liquid films of different viscosities. Charged, mobile surfactants populate this interface. Her interest was in the general deformation of the interface on length scales much larger than either film thickness. To obtain this, she used lubrication theory to develop a set of evolution equations for the behavior of the system that effectively reduced the microscale structure and effects of the membrane into a set of averaged "properties." Each property is expressed as a term in an evolution equation with an associated parameter or coefficient that governs the magnitude of that property. These equations work remarkably well in predicting the macro-scale dynamics of the system. This idea could be taken much further and more of the microscale features of the membrane could be added to the modeling, such as bending stresses, dilatational stresses, etc. In doing so, one could obtain a better model for the behavior of the cell or vesicle; a model that may truly predict the overall system responses on the larger length scales of interest. This is one avenue for the kind of research that can be done in the next century. There are structures, such as cell membranes, that are inherently thin entities when viewed on the larger length scale of the associated global system. Such structures may be modeled as an interface. This kind of modeling works very well when the global system responds to the presence of the interface in such a way that inessential details of the interfacial structure are forgotten. In effect, the global system responds to an averaged effect of the interface. This marriage of interfacial modeling and classical methods like lubrication theory may not work in every case, but when it does the results often prove to be very useful. [Author's note: Another example of this kind of modeling is the contribution to this volume of Worster [10]. In this work, a sea ice layer was modeled as an interface between the ocean and the atmosphere. By examining the solidification processes occurring in sea ice, including the mushy layer, Worster obtained a set of interfacial conditions that could be used to couple the global physical processes occurring on either side of the ice layer.) Professor Sekerka entered the discussion at this point by bringing up a different theme that was underlying the comments of Prof. Davis. The type of interfacial problems that will be studied in the future will be very complex. There will be problems containing fluid mechanics and materials science coupled with complicated biological, chemical, electrical, and certainly other more esoteric physical phenomena. In such problems, one quickly runs out of the expertise needed to do the necessary modeling, experimentation, and computations. So, this next century presents a great opportunity to create multi-disciplinary teams to work on problems of common interest. Such teams would include a modeler, an experimentalist, and an expert in numerical computations who may not necessarily be at the same

New Research Directions in Interfacial

Science

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institution, but who get together to solve a real problem cooperatively, in a realistic time frame. Professor Davis agreed with this comment and added that the physical and chemical processes in biology, for instance, were so broad that no one could possibly do realistic modeling without some guidance from experiments. Thus, the use of cooperative teams will be essential in the next century to solve problems of real interest. Anyone involved in computing over the past few decades has seen the rapid increase in speed and capability of computers and a corresponding decrease in their cost. Professor Tryggvason echoed this trend when he said that in the near future it will be a routine matter to do computations of three-dimensional physical problems in complicated geometries that include strong couplings between various physical effects, such as fluid mechanics, solidification, surfactants, etc. All that will be required is to have the correct constitutive relations for the system. This is a very important and sometimes elusive requirement. To get it right, Prof. Tryggvason agreed with Profs. Davis and Sekerka that we will have to change the way we do business and team up with scientists in other fields in order to get a more informed look at the important problems they face. Continuing, Prof. Tryggvason stated two requirements on theory that he felt needed to be met in the next century. The first is to investigate complicated systems at molecular scales and build the information obtained from these studies into useful constitutive relations for continuum models. The second requirement is to go beyond the detailed continuum models that we will be able to compute and build useful engineering models. As an example, he cited a first principles simulation of the solidification of a binary alloy that includes a forest of dendrites with fluid flow and species segregation all coupled together, as would happen in a casting process. For engineering purposes, we must go beyond the details of the simulation and produce a realistic reduced-order model that is able to predict the average physical properties of the final product. This requirement echoes the statement of Prof. Davis when he suggested the use of classical methods like lubrication theory to average out the inessential details of the structure of an interface to produce a model that can faithfully predict the behavior of the corresponding system on the macroscale. In Prof. Tryggvason's requirement, we take the details from a complex and complete simulation of a physical system and find a way to produce a simple reduced-order model that can accurately predict the global properties of interest. At this point Prof. Timothy Pedley [11] commented from the audience that this idea of going from the microscale to the macroscale is especially important in the field of biology. Here, the challenge is first to find out how cells behave on an individual level, and then to use this information to construct an appropriate model for the function of a tissue or an organ in its in vivo environment, for example. In a brief exchange with the panel and the audience, Prof. Grigory Barenblatt [12] posed a question that can be paraphrased as follows. Suppose that you had the capability to determine the behavior of any physical system based on first principles, say, by using an extremely advanced computing system or a new mathematical technique. Then, what would you do with this capability? What questions would you ask? Professor Sekerka commented that this was indeed the critical

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question. How can we determine the truly important scientific questions or frontiers that need to be addressed? Professor Stephen Morris [13] replied from the audience that this is a question that can not really be answered. If one looks back on the previous century, an important driver for posing scientific questions was technology. Without knowing the technology, some questions would not even exist. For example, the investigation of compressible flow in the Twentieth century would have been rather pointless if humans had not been able to develop a means to travel faster than the speed of sound. {Author's note: While this is certainly a valid argument, we must realize that scientific knowledge also enables technology. An example here would be the development of the first solid-state transistor. That scientific achievement has driven technological development throughout the course of the computer age, and it is still an active driver today. In the end, it will come down to individual human interests. People will examine ideas that are important to them, either from a technological standpoint or purely from scientific curiosity. After examination, our task is to determine what ideas are worth pursuing in order to advance technological development or to promote economic activity.) Professor Joel Koplik changed the direction of the discussion by making some remarks about the treatment of complex fluids. Current continuum models of such fluids leave a lot of information out. Thus, it is extremely important to coordinate continuum models with molecular information obtained from molecular simulations and microscopic measurements from experiments, if possible. This is particularly important at the interface between two complex fluids. One area where improvements would be very useful is in the matching of molecular and continuum simulations. He cited Hadjiconstantinou [14] as one example of such work. Here, Hajiconstantinou simulated the moving contact-line problem by treating the flow in the contact-line region with a molecular model and the flow away from the contact line with the traditional Navier-Stokes continuum model. He developed a good way to match the information from the molecular and the finite-element fluid flow simulations so that they could be properly coupled. The reason this is important is that many large molecular simulations of a moving contact line, for example, use many molecules to simulate the solid surfaces and the liquid far away from the contact line where nothing much happens. The action occurs near the contact line and this region may contain only a small fraction of the total molecules in the simulation. This technique is obviously an inefficient use of available resources. In another remark, Prof. Koplik said that there are not many people doing molecular simulations because of the mistaken belief that they are very difficult computations requiring huge computer resources. This is not true. One major simplification that can be employed is similarity. All Newtonian fluids are similar so the simplest molecular-interaction potential, like Lennard-Jones, is a sufficient model. For surfactants, one need only use a molecule that remains on the interface between two fluids to model the effect of the surfactant on that interface. More complicated surfactant molecules just do the same thing and so their complexity can often be ignored. Molecular simulations of non-Newtonian fluids appear to be another story, because of the great variety of different fluids. However, Prof. Koplik expressed the opinion that there probably exist only a few canonical classes of non-Newtonian fluids and that each of these would have a relatively simple molec-

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ular model. If this is true, then molecular models properly coupled to continuum calculations would become a very powerful technique for the future of interfacial research. Thus, more researchers need to become involved in this work in order to provide some healthy competition for future progress. Similar molecular (or atomic) computations can be done in the field of materials science. Professor Martin Glicksman [15] commented from the audience that he has seen some truly amazing progress in this area of ab initio computations taking place in some national laboratories. In these computations, one begins with only a description of the atoms or molecules involved and the algorithms will calculate the molecular structure of a material with details like annealed grain boundaries and interphasic dislocations. From this information, it has been claimed that one can determine bulk material properties even more accurately than they can be measured. Using this work in continuum-based computations of materials is obviously very attractive since material properties are often a source of contention when experimental and computational results disagree. Professor Glicksman suggested that one agenda for the future should be better communication and cooperation between this group of ab initio scientists and people in the area of computational materials science; for example, developers of new phase-field methods. Professor Wifried Kurz agreed that obtaining precise material properties is crucial to the experimental validation of new continuum theories and computational methods. It was here, in the area of new work, that he was very concerned. There are many practical problems that have received little attention, but that are critical to understanding and predicting phase transformations in materials science. In heterogeneous system, interfaces occur between the different materials, and phase transformations imply deviations from equilibrium. He then presented several topics for future work that he considered were of fundamental importance to the determination of the structure and properties of materials. Nucleation: Heterogeneous nucleation is at the core of phase transformation in materials. Here, systems develop moving material interfaces with boundary layers at the interfaces, and the whole system evolves to form a material with complex microstructure. To predict this microstructure, arid hence the properties of the bulk material, good models of heterogeneous nucleation and growth of the second phase are needed. This is particularly important for directionally solidified peritectic materials. Cellular and Dendritic Growth: A better understanding of the transition from cells to dendrites during solidification is needed. Computational models need to do a better job of coupling the solutal and thermal fields during cellular and dendritic growth. In addition, the directional solidification of dendrites involves a coupling to the mechanical stress field in the solid phase. These solid stresses may be the origin of the loss of texture in materials because of their coupling to the formation of inhomogeneous solutal and thermal fields. Convection: Future work on convection in material growth should involve the influence of convection on the three-dimensional growth of dendrites. The flow of the liquid phase through arrays of dendrites with numerous side branches

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causes the fragmentation of dendrites into small crystals that in turn affect the final microstructure of the material. Accurate predictions of microstructure must deal with this highly coupled fluid-flow and material-growth problem. Multiphase Interfaces: The growth of eutectic and peritectic materials involves multiphase interfaces that exhibit morphological instabilities. These instabilities in isothermal and non-isothermal systems produce interfaces with very complex structure. A better understanding of these instabilities and their coupling to solutal and/or thermal fields is needed, especially for peritectic materials. In addition, the growth of composite material in the coupled zone is a problem needing more work in the future. Solid-State Transformations: Solid-state transformations are extremely important in materials science. In many materials, the structure produced during the liquid-solid transformation is altered or replaced by the structure produced during the solid-state transformation. The final properties of the material can only be determined by an understanding of this final structure. There are interesting aspects to cell and dendrite growth in the solid state. In many materials, the product phase has a higher rate of diffusion resulting in lamellar growth, rather than fibrous (or dendritic) growth. Mechanical stresses can build up in the solid state, disturb the local equilibrium, and change the local atomic attachment kinetics. This kind of interaction is extremely important in order to gain an understanding of martinsite transformations and other transformations in which the solid phase is completely supersaturated. This kind of problem affects the everyday life of a materials scientist interested in designing new materials for industrial or commercial use. Professor Miksis now invited the last member of the panel, Prof. Tony Maxworthy, to make a few remarks. Professor Maxworthy was happy to hear comments from the other panelists that experiments will be a vital part of research in the future. In particular, future experimental research needs to be directed towards the development of finer instrumentation, instruments capable of looking into a fluid-flow or material-transformation process and measuring the fine-scale structure. The development and use of such instrumentation would also benefit collaboration between experimentalists and computational physicists because the scale of the data obtained from each technique would be comparable. This kind of collaboration would in turn stimulate the development of even better theories, numerical schemes, and instrumentation. In terms of collaboration, Prof. Maxworthy was more in favor of an interaction between scientists and engineers in one institution, rather than in geographically dispersed locations. The ease of personal communication in these kinds of collaborations is a great asset and it helps to maintain the vitality of the interaction. Professor Sekerka spoke again saying that this was not always the case. Depending on the people involved, and the ease of travel, long distance collaborations are possible and can be tremendously productive. Professors Sekerka and Maxworthy agreed that collaborations are a highly personal endeavor and either mode is possible and can be productive. It just depends on the personal dynamics of the individuals involved.

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Speaking from the audience, Prof. Robert Kohn [16] raised a concern over the tendency of the materials science community to think of microstructure in terms of multi-point correlations functions. Has this representation of microstructure been pushed to its limits? Does it really do what is needed? Has it in some way limited research? Perhaps, he suggested, a new mode of thinking is needed. As an example, he cited the computer-vision community, in which complex structure is being represented using wavelets as basis functions. Here, the randomness in a structure is included in the model by using joint-distribution coefficients for the wavelet basis functions. Examining this idea and seeing how it has performed for the computer-vision community may stimulate a new mode of thinking in materials science that could take the representation of microstructure in materials to new levels of success and prediction. In a question that addressed many of the ideas presented at this conference, Prof. Yitzhak Shnidman [17] asked what really happens when a system is said to breakdown at the microscale? Is it the physics or the mathematical model of the physics that is in trouble? How can a model based on physical principles of conservation ever really breakdown? He sees the challenge as devising a way to represent these conservation laws correctly and hierarchically from the microscale, through the mesoscale, and into the macroscale. Wavelet methods are one way to proceed; adaptive gridding in a numerical computation is another. But the real problem is to start with a probabilistic description of these conservation laws on a molecular or atomic scale, to add forces and moments on this level, and then to formulate evolution equations that represent the effect of such microscale phenomena up to the mesoscale, and then onward to the macroscale to produce the standard continuum equations that are used so widely today. Professor Sekerka responded to this question with an example from his experience with a colleague who developed a correction to the continuity equation in a multiphase flow problem. Professor Sekerka thought his colleague was simply wrong at first, but a closer inspection revealed that the nature of the flow and its representation did indeed lead to an additional term in the continuity equation. So we are the victims of our own mindset. Broadening our vision and opening our minds to new ideas used in other areas may indeed lead to progress in otherwise unconnected fields of science and engineering. To support this view, Prof. Barenblatt commented again that asking very sharp questions about the applicability and/or limitations of an idea is a very effective method to expand the frontiers of science in any discipline. The problem is to formulate the correct questions. In response, Prof. Sandra Troian [18] said that the success of our predecessors in formulating such questions and answering them was partly due to the fact that they had the time to invest in this activity. One of her fears and concerns is that the present-day system of funding for academic research does not really allow people to ask such questions. The pressure in academics of publishing and obtaining funding is preventing many people from having the time and energy to pose such questions and to work on them. She hoped that one of the important trends in academics in the next century would be to reverse this trend. Academic researchers are highly trained people and they need the time to use the tools they have acquired to pose

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and answer these kinds of fundamental questions. As this point, Prof. Davis took the opportunity to have the last word. He commented on the notion of a linking of scales. Physical systems range in scale from larger than light years to smaller than angstroms. The prospect of a single unified theory that can explain all of the phenomena that may occur on any of these scales is very unlikely. Therefore, theories have evolved to explain phenomena at a particular length scale. They do link to theories at a smaller scale, but mostly in the form of averages. The larger-scale theory "forgets," in a sense, most of the details found at the smaller scale. For example, a microscopic theory of a fluid would be able to compute a great deal about the motion of individual atoms or molecules. However, the information transmitted to a larger-scale continuum theory takes the form of averaged quantities like velocity and pressure, and bulk properties like density and viscosity. The interesting thing is that the continuum theory of fluids was formulated and proved to be very successful years before a microscopic theory of fluids was developed. It was successful because it was expressed in terms of averaged quantities that were measurable at the time. This example shows that the linkage of scales between various theories goes both ways. Small-scale theories can compute quantities of interest for a theory at the next level in scale. Simultaneously, largescale theories can determine and describe the truly important information that is needed from a smaller-scale theory. Noting that the hour had drawn to a close, Prof. Miksis thanked the panel for their contributions, thanked the audience for their attention, and closed the discussion session. 3

Summary

Predicting the future is hopeless. This statement began the panel discussion. After reflecting on it for a few moments, it is amusing to note that our job as scientists and engineers is to do the impossible — predict the future. We do not use a crystal ball for this task, but rather the three tools of the scientific method — experimentation, mathematical analysis, and numerical computation. We use these tools to devise and analyze mathematical models of physical systems and use the results to predict the future for those systems. Throughout the discussion, several comments were made that addressed the use of the scientific method at one level or another. To be successful with this method, one must put forward a clear, testable hypothesis for analysis. In other words, simple, clear, and precise questions need to be asked. There are several motivators that are used to formulate these questions. One is just plain curiosity about an observed phenomenon that may, in turn, lead to the formation of new technology. Another is the development of new technology, or societal demands to improve existing technology. Examples here are micro-electromechanical systems, nano-devices, biotechnology, and ecologically clean technology. All of these technologies need careful development before they can provide any benefit to society in the next century. Successful development will involve asking simple and direct questions with clear answers. These answers will assist in the design, production, and use of these new technologies to the physical and economic benefit of us all.

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To help out in this task, the panel offered some clear directions for interfacial research in fluid mechanics and materials science in the next century. These are summarized below. Collaboration: The new technologies of the future will be so complex and interdisciplinary that no individual could possibly ask all of the right questions. Thus, collaborations involving individuals from several disciplines will be needed. There are several models for collaborations of this kind and the choice belongs to the individual; a single center of excellence, a multi-university center, a loose collection of several individuals in one or several places, etc. Collaborations between people from very different disciplines is difficult, but it will be essential for progress in existing and new fields of science and technology. Instrumentation: Today, we are dealing with smaller and more complex systems than ever before. Micron-sized devices are common and nanometer-sized devices are being developed. New materials are very complex. Systems or processes under development use non-Newtonian fluids, closely packed suspensions, or substances with complex microstructures resulting from the solidification of mixtures of several materials. Observations in these systems will demand new and more sophisticated instrumentation. Measurements will need to resolve smaller length scales, faster time scales, and provide much more detailed information. The development of such instruments will be essential for future progress. The resulting measurements will help guide the development of new models and then verify and support theoretical and numerical predictions. Material Properties: The correct prediction of a physical system's behavior by a model depends on the values of the material properties that are used. This is indeed a critical aspect of comparing and validating a model with its related experiments. Thus, accurate property measurements or direct numerical computations of properties based on numerical models will be essential to the development and use of systems containing complex materials. Modeling: There will probably never be a universal model of a physical system that is valid at all scales. Even if one existed, it would likely be too unwieldy to use effectively. Thus, the modeling of a complex physical system must use the idea of a linkage of scales to good effect. We must formulate and use large-scale theories to determine what is needed from smaller-scale theories. One can use the properties needed in a macroscale model and the mathematical breakdown of the theory as a guide to what to look for and where to look in a microscale model. We also need to learn how to effectively link microscale models with macroscale models. Mathematical Methods: Another critical aspect of modeling is the question of deciding the most useful mathematical framework with which to pose and solve the model equations. People tend to get wedded to a set of techniques that they are comfortable with and fail to see when a new path is needed. We need to be aware of alternate methods of analysis and be willing to borrow

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successful mathematical methods from other disciplines that may prove useful in explaining complex structures in fluids and materials science. Molecular Simulations: Molecular models are becoming increasingly important in fluid mechanics (e.g., for the contact-line problem and the interfacial dynamics of complex fluids) and in materials science (e.g., phase transformations in a wide range of material systems leading to complex microstructure). Critical needs for the future will involve the development of techniques to link the results from such simulations to the material properties and other constitutive relations used in continuum models. In particular, where does a particular molecular phenomenon govern or modify the behavior of a macroscale system? How does one construct an effective boundary condition or constitutive relation for use in the related continuum theory? What ways can we devise to simultaneously link numerical simulations done at the molecular scale to those done at the continuum scale? Reduced-Order Models: Future work will be needed on the techniques of building simplified reduced-order engineering models suitable for the design of complex systems. Formulating such models and determining what kind of information is really important will involve asking clear questions about what is truly needed at this larger scale. We will be able to compute giga-bytes or tera-bytes of data related to all of the field variables in a continuum or macroscale model. How do we visualize these data effectively? How do we average these data to produce the quantities of interest for a reduced-order model? How do we integrate these reduced-order models with one another to produce a system model that can be used to design a complex process or device? Complex Interfaces: Interfacial science in the next century will involve the investigation of thin boundaries with complex structure between two different phases of possibly complex materials. Examples of such boundaries include cell and tissue membranes in biotechnology applications, two-fluid interfaces loaded with surfactants for chemical and biological processing, and material interfaces associated with materials-processing technology in which the solidifying liquids themselves have complex structure and the solidified materials undergo complex phase transformations in either the liquid or solid states. The development of appropriate models of these interfaces that include the essential physics will be a challenge. Work will be needed to develop theoretical and numerical tools to analyze these models and extract the relevant data necessary to make accurate predictions. Finally, experimental tools need to be developed to observe these interfaces on the small scale. These data will be used to guide the development of proper models and to validate their predictions so that these tools can be used for the development and design of new technologies and materials in the future. The path into the future for interfacial science in fluid mechanics and materials science, and indeed in any field of science, depends on how we formulate good questions and use our time and resources to answer them. There may be no way to

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predict that future or to know what we will find, but we do know how to lay down the bricks that pave this research path into the Twenty-First century. References 1. Michael J. Miksis, Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60088, U.S.A. 2. Stephen H. Davis, Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60088, U.S.A. 3. Joel Koplik, Benjamin Levich Institute and Department of Physics, City College of the City University of New York, New York, NY 10031, U.S.A., E-mail: [email protected]. cuny. edu. 4. Wilfried Kurz, Physical Metallurgy Laboratory, Swiss Federal Institute of Technology, 1015 Lausanne EPFL, Switzerland. 5. Tony Maxworthy, Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089-1191, U.S.A. 6. Robert F. Sekerka, Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213-3890, U.S.A. 7. Gretar Tryggvason, Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, MA 01609-2280, U.S.A., E-mail: [email protected]. 8. "Interfaces for the Twenty-First Century," NSF Grant No. DMS-9974551. 9. D. Gallez and E. Ramos de Souza, in Interfaces for the Twenty-First Century, eds. M. K. Smith, M. J. Miksis, G. B. McFadden, G. P. Neitzel, and D. R. Canright, Imperial College Press, p. 73 (2002). 10. M. G. Worster, in Interfaces for the Twenty-First Century, eds. M. K. Smith, M. J. Miksis, G. B. McFadden, G. P. Neitzel, and D. R. Canright, Imperial College Press, p. 187 (2002). 11. Timothy J. Pedley, Department of Applied Mathematics and Theoretical Physics, Silver Street, University of Cambridge, Cambridge, CB3 9EW, U.K. 12. Grigory Barenblatt, Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720-3840, U.S.A. 13. Stephen J. S. Morris, Department of Mechanical Engineering, University of California, Berkeley, Berkeley, CA 94720-1740, U.S.A. 14. N. Hadjiconstantinou, in Interfaces for the Twenty-First Century, eds. M. K. Smith, M. J. Miksis, G. B. McFadden, G. P. Neitzel, and D. R. Canright, Imperial College Press, p. 260 (2002). 15. Martin E. Glicksman, Materials Science and Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, U.S.A., E-mail: [email protected]. 16. Robert V. Kohn, Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, U.S.A. 17. Yitzhak Shnidman, Department of Chemical Engineering and Chemistry, Polytechnic University, 6 MetroTech Center, Brooklyn, NY 11201, U.S.A. 18. Sandra M. Troian, Department of Chemical Engineering, Princeton University, Princeton, NJ 08544-5263, U.S.A.

Interfaces

for

t h e 21 s t Century New Research Directions in Fluid Mechanics

and Materials Science

''"'1's book highlights some recent advances in interfacial research in the fields of fluid mechanics and materials science at the beginning of the twenty-first century. It is an .extension of the presentations made during the conference "Interfaces for the I \\ eniy-First Century," held on August 16-18, 1999, in Monterey, California. It includes papers by sixteen renowned experts in the field of interfacial mechanics, abstracts contributed by research scientists, and a summary of a panel discussion on future research directions. The book covers experimental and theoretical approaches, With the unifying philosophy being the investigation of new techniques for modeling the dynamics of interfaces. A number of new and ex( iting solution methods and experimental studies. as well a.s the physical problems that initiated them, are presented.

'

P259 he ISBN 1-86094-319-5

Imperial College Press www.icpress.co.uk