G.B. McFadden and S.R. Coriell, NIST

and

R.F. Sekerka, CMU

Analytic Solution of

Non-Axisymmetric Isothermal Dendrites

NASA Microgravity Research Program, NSF DMR

•Introduction

•Ivantsov solution

•Horvay-Cahn 2-fold solution

•Small-amplitude 4-fold solution

•Estimate of shape parameter

•Summary

Dendritic Growth

Peclet number: Stefan number:

Ivantsov solution [1947]:

Experimental Check of Ivantsov RelationM.E. Glicksman, M.B. Koss, J.C. LaCombe, et al.

There is a systematic 10% - 15% deviation.

Experimental Check of Ivantsov Relation

“… the diffusion field described by [the Ivantsov solution] is based on a dendrite tip which is a parabolic body of revolution, which is true only near the tip itself.” [Glicksman et al. (1995)]

•Proximity of sidearms or other dendrites (especially at low T)

•Convection driven by density change on solidification

•Residual natural convection in g

•Container size effects

•Non-axisymmetric deviations from Ivantsov solution

Possible reasons for deviation:

Non-Axisymmetric Needle Crystals

Idea: Compute correction to Ivantsov relation S = P eP E1(P) due to 4-fold deviation from a parabola of revolution.

Key ingredients:

• Glicksman et al. have measured the deviation S - P eP E1(P)

• LaCombe et al. have also measured the shape deviation [1995].

• Horvay & Cahn [1961] found an exact needle crystal solution with 2-fold symmetry, exhibiting an amplitude-dependent deviation in S - P eP E1(P) [but wrong sign to account for 4-fold data …]

Non-Axisymmetric Needle Crystals

•Unfortunately, there is no exact generalization of the Horvay- Cahn 2-fold solution to the 4-fold case.

•Instead, we perform an expansion for the 4-fold correction, valid for small-amplitude perturbations to a parabola of revolution.

•Horvay-Cahn solution is written in an ellipsoidal coordinate system. We transform the solution to paraboloidal coordinates, and expand for small eccentricity to find the expansion for a 2-fold solution in paraboloidal coordinates.

•We then generalize the 2-fold solution to the n-fold case (n = 3,4) in paraboloidal coordinates .

Temperature T in the liquid:

2 T + V T/ z = 0

Conservation of energy: Melting temperature:

-LV vn = k T/n T = TM

Far-field boundary condition (bath temperture):

T T = TM - T

Steady-State Isothermal Model of Dendritic Growth

= thermal diffusivity LV = latent heat per unit volume

V = dendrite growth velocity k = thermal conductivity

Characteristic scales: choose T for (T – TM) and 2/V for length.

Note: T/z is a solution if T is.

Ivantsov Solution [1947] (axisymmetric)

Conservation of energy: Temperature field:

Solid-liquid interface:

Parabolic coordinates [, , ] (moving system) :

Horvay-Cahn Solution [1961] (2-fold)

Paraboloids with elliptical cross-section:

Here is the independent variable, and b ≠ 0 generates an elliptical cross section.

Solid-liquid interface is = P, temperature field is T = T():

Conservation of energy:

For b = 0, the axisymmetric Ivantsov solution is recovered.

Expansion of Horvay-Cahn Solution

Procedure:

•Set b = P

•Re-express Horvay-Cahn solution in parabolic coordinates

•Expand in powers of for fixed value of P

Find the thermal field T(,,,), interface shape = f(,,), and Stefan number S() as functions of through 2nd order

Expansion of Horvay-Cahn Solution

At leading order, we recover the Ivantsov solution:

At first order:

S(1) vanishes by symmetry: - corresponds to a rotation, + /2

The solution has 2-fold symmetry in .

Expansion of Horvay-Cahn SolutionAt 2nd order:

where:

exact

2nd order

P = 0.01

Expansion of n-fold Solution

Goal: Find correction S(2) for a solution with n-fold symmetry

where the leading order solution is the Ivantsov solution as before, and the first order solution is given by

Expansion of 4-fold Solution

Key points:

•Fix the tip at z = P/2

•Fix the (average) radius of curvature

•Employ two more diffusion solutions: “anti-derivatives” (method of characteristics)

Expression for S(2)

A symbolic calculation gives the exact result:

Comparison with Shape Measurements

In cylindrical coordinates, our dimensional result is:

LaCombe et al. [1995] fit SCN tip shapes using:

For P 0.004, they find Q() –0.004 cos 4:

Comparison of shapes gives –0.008, and evaluating S(2) for P = 0.004 and = -0.008 then gives

4-Fold Tip Shape

For P = 0.004 and = -0.008:

Huang & Glicksman [1981]

Estimate for Shape ParameterSurface tension anisotropy (n) (cubic crystal):

n = (nx,ny,nz) is the unit normal of the crystal-melt interface.

For SCN, 4 = 0.0055 0.0015 [Glicksman et al. (1986)].

For small anisotropy, the equilibrium shape is geometrically similar to a polar plot of the surface free energy, and we have

Estimate for Shape Parameter

Idea: Dendrite tip is geometrically-similar to the [100]-portion of the equilibrium shape.

For small 4 and r/z ¿ 1, the equilibrium shape is:

Our expansion for the dendrite shape:

From the SCN anisotropy measurement: From the tip shape measurement:

Summary

• Glicksman et al. observe a 10% - 15% discrepancy in the Ivantsov relation for SCN over the range 0.5 K < T < 1.0 K

• Horvay-Cahn exact 2-fold solution gives an amplitude-dependent correction to the Ivantsov relation

• An approximate 4-fold solution can be obtained to second order in , with S = S(0) + 2 S(2)/2 + ...

• LaCombe et al. measure a shape factor -0.008 for P 0.004

• Using = 0.008 gives S/S(0) - 1 = 0.09

• Assuming the dendrite tip is similar to the [001] portion of the anisotropic equilibrium shape gives = - 0.011 0.003

References• M.E. Glicksman and S.P. Marsh, “The Dendrite,” in Handbook of Crystal Growth, ed. D.T.J. Hurle, (Elsevier Science Publishers B.V., Amsterdam, 1993), Vol. 1b, p. 1077.

• M.E. Glicksman, M.B. Koss, L.T. Bushnell, J.C. LaCombe, and E.A. Winsa, ISIJ International 35 (1995) 604.

•S.-C. Huang and M.E. Glicksman, Fundamentals of dendritic solidification – I. Steady-state tip growth, Acta Metall. 29 (1981) 701-715.

•J.C. LaCombe, M.B. Koss, V.E. Fradkov, and M.E. Glicksman, Three-dimensional dendrite-tip morphology, Phys, Rev. E 52 (1995) 2778-2786.

• G.B. McFadden, S.R. Coriell, and R.F. Sekerka, Analytic solution for a non-axisymmetric isothermal dendrite, J. Crystal Growth 208 (2000) 726-745.

•G.B. McFadden, S.R. Coriell, and R.F. Sekerka, Effect of surface free energy anisotropy on dendrite tip shape, Acta Mater. 48 (2000) 3177-3181.

Material Properties of SCN