G.B. McFadden and S.R. Coriell, NIST and R.F. Sekerka, CMU

1. Analytic Solution of Non-Axisymmetric Isothermal Dendrites • Introduction • Ivantsov solution • Horvay-Cahn 2-fold solution • Small-amplitude 4-fold solution • Estimate of shape parameter • Summary G.B. McFadden and S.R. Coriell, NIST and R.F. Sekerka, CMU NASA Microgravity Research Program, NSF DMR

2. Dendritic Growth Peclet number: Stefan number: Ivantsov solution [1947]:

3. Experimental Check of Ivantsov Relation M.E. Glicksman, M.B. Koss, J.C. LaCombe, et al. There is a systematic 10% - 15% deviation.

4. Experimental Check of Ivantsov Relation Possible reasons for deviation: • 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 “… 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)]

5. 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 …]

6. 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 .

7. Steady-State Isothermal Model of Dendritic Growth 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 Note: T/z is a solution if T is.  = 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.

8. Ivantsov Solution [1947] (axisymmetric) Parabolic coordinates [, , ] (moving system) : Solid-liquid interface: Conservation of energy: Temperature field:

9. 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.

10. 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

11. 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 .

12. Expansion of Horvay-Cahn Solution At 2nd order: where: 2nd order exact P = 0.01

13. 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

14. 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)

15. Expression for S(2) A symbolic calculation gives the exact result:

16. 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

17. 4-Fold Tip Shape For P = 0.004 and  = -0.008: Huang & Glicksman [1981]

18. Estimate for Shape Parameter Surface 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

19. 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:

20. 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

21. 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 International35 (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. E52 (1995) 2778-2786. • G.B. McFadden, S.R. Coriell, and R.F. Sekerka, Analytic solution for a non-axisymmetric isothermal dendrite, J. Crystal Growth208 (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.

22. Material Properties of SCN