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 Relation M.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 e P E 1 (P) due to 4-fold deviation from a parabola of revolution. Key ingredients: Glicksman et al. have measured the deviation S - P e P E 1 (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 e P E 1 (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: -L V v n = k  T/  n T = T M Far-field boundary condition (bath temperture): T  T  = T M -  T Steady-State Isothermal Model of Dendritic Growth  = thermal diffusivity L V = latent heat per unit volume V = dendrite growth velocity k = thermal conductivity Characteristic scales: choose  T for (T – T M ) 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 2 nd 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 Solution At 2 nd order: where: exact 2 nd 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 = and  = then gives

4-Fold Tip Shape For P = and  = : Huang & Glicksman [1981]

Estimate for Shape Parameter Surface tension anisotropy  (n) (cubic crystal): n = (n x,n y,n z ) is the unit normal of the crystal-melt interface. For SCN,  4 =  [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) / LaCombe et al. measure a shape factor   for P  Using  = gives S/S (0) - 1 = 0.09 Assuming the dendrite tip is similar to the [001] portion of the anisotropic equilibrium shape gives  =  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 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) J.C. LaCombe, M.B. Koss, V.E. Fradkov, and M.E. Glicksman, Three- dimensional dendrite-tip morphology, Phys, Rev. E 52 (1995) G.B. McFadden, S.R. Coriell, and R.F. Sekerka, Analytic solution for a non- axisymmetric isothermal dendrite, J. Crystal Growth 208 (2000) G.B. McFadden, S.R. Coriell, and R.F. Sekerka, Effect of surface free energy anisotropy on dendrite tip shape, Acta Mater. 48 (2000)

Material Properties of SCN