Title: G'B' McFadden and S'R' Coriell, NIST
1Analytic Solution of Non-Axisymmetric Isothermal
Dendrites
- G.B. McFadden and S.R. Coriell, NIST
- and
- R.F. Sekerka, CMU
- Introduction
- Ivantsov solution
- Horvay-Cahn 2-fold solution
- Small-amplitude 4-fold solution
- Estimate of shape parameter
- Summary
NASA Microgravity Research Program, NSF DMR
2Dendritic Growth
Peclet number
Stefan number
Ivantsov solution 1947
3Experimental Check of Ivantsov Relation
M.E. Glicksman, M.B. Koss, J.C. LaCombe, et al.
There is a systematic 10 - 15 deviation.
4Experimental 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)
5Non-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
6Non-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
.
7Steady-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.
8Ivantsov Solution 1947 (axisymmetric)
Parabolic coordinates ?, ?, ? (moving system)
Solid-liquid interface
Conservation of energy
Temperature field
9Horvay-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.
10Expansion 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
11Expansion 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 ?.
12Expansion of Horvay-Cahn Solution
At 2nd order
where
2nd order
exact
P 0.01
13Expansion 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
14Expansion 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)
15Expression for S(2)
A symbolic calculation gives the exact result
16Comparison 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
174-Fold Tip Shape
For P 0.004 and ? -0.008
Huang Glicksman 1981
18Estimate 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
19Estimate 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
20Summary
- Glicksman et al. observe a 10 - 15 discrepancy
in the Ivantsov relation for SCN over the range
0.5 K lt ?T lt 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
21References
- 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.
22Material Properties of SCN