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Title: G'B' McFadden and S'R' Coriell, NIST


1
Analytic 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
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 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

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 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.

22
Material Properties of SCN
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