# Presentation: 320206

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

- 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 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 ...]

**Notes:**
Fern and Antii – matrix plot of sequences up to 10 proteins long – GENBANK
Fern and Tony – reconstructing a genetic sequence from an incomplete sequence with gaps and deletions 

## 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 .

## Steady-State Isothermal Model of Dendritic Growth

-                                 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

-  =  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 Solution

- At 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 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

## 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