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Dendritic growth with interfacial energy anisotropy
Journal of Crystal Growth 110 (1991) 683—691 North-Holland
683
Dendritic growth with interfacial energy anisotropy Y. Miyata
~,
M.E. Glicksman
* *
and S.H. Tirmizi
Materials Engineering Department, Rensselaer Polytechnic Institute, Troy, New York 12180-3590, USA
Received 13 August 1990; manuscript received in final form 27 November 1990
A theory for dendritic growth of needle shaped crystals growing in supercooled pure melts and dilute alloys is proposed. This theory predicts the growth velocity and tip radii as functions of supercooling and alloy concentrations without introduction of additional conditions such as marginal stability. A key ingredient of the theory is introducing the effect of the interfacial energy anisotropy. Deviation of the shape of the dendrite from a paraboloid of revolution is permitted, consistent with a small interfacial energy anisotropy. The radius of curvature and the growth rate as functions of melt supercooling and alloy concentrations are determined in terms of the anisotropy, and are compared with experimental results. The predictions are in agreement with the experimental results, especially at large supercoolings. The deviation at lower supercoolings can be attributed to the neglect of natural convection in the present theory.
1. Introduction One of the first attempts to predict the growth of smooth needle shaped dendrites was a steadystate diffusion theory by Ivantsov [1]. His solution can be represented as follows: =
~L
(1)
eL Ei(PL).
Eq. (1) relates the growth Peclet number, ~L’ to the dimensionless melt supercooling, ~O, but it fails to predict the unique relationship between growth velocity and tip radius as functions of supercooling. Several attempts have been made to provide an additional relationship, which in conjunction with eq. (1) can predict the growth rate, V. and the radius of curvature, p. Nash and Glicksman [2] found self-consistent corrections to eq. (1) which include the Gibbs—Thomson capillarity condition at the interface. Their work showed, ultimately, that the dendrite selects a tip radius which is not consistent with the maximum *
**
Present address: Mechanical Engineering
Department,
Nagaoka University of Technology, Kamitormoka, Nagaoka 940, Japan. To who all correspondence should be addressed,
0022-0248/91/$03.50
©
1991
—
velocity principle. Langer and Müller-Krumbhaar [3] formulated the marginal stability criterion which determines the dendrite tip radius as the one which is marginally stable to tip-splitting perturbations. Recently, Kessler et a!. [4,51, Saito et at. [6] and Bensimon et at. [71used the concept of microscopic solvability and argued that of all the possible dynamical operating states, the state at which the dendrite grows at the highest velocity is the stable one and is thus the operating mode. The microscopic solvability theory requires a non-zero value of interfacial energy anisotropy (amsotropy in the solid—liquid interfacial energy) for a solution to exist. The theory to be discussed here also requires interfacial anisotropy, but acting in a somewhat different manner. The Ivantsov solution, marginal stability theory, and the microscopic solvability theory, usually employ the assumption that the shape of the dendrite is a paraboloid of revolution. But it has been observed experimentally by Glicksman and co-workers [8] that there always exists near the tip region a small departure of the shape from a perfect paraboloid of revolution. Fig. ~ indicates such a deformed paraboloid. An essential ingredient of the present theory is to take into account small deviations of .
.
Elsevier Science Publishers B.V. (North-Holland)
.
.
684
Y. Miyata et al.
• *
/ Dendritic growth with interfacial
energy anisotropy
ity and tip radii as unique functions of supercooling. An analytical model is proposed here for the growth of a dendrite in a supercooled melt, based solely on the presence of interfacial energy am-
~
I
sotropy and the assumed scaling of dendritic tip shape. 2. Theory 2.1. Interface and anisotropy
The moving dimensionless parabolic coordinates (~,ij, 4)) are adopted, where all coordinates are normalized by the radius of curvature of the dendrite tip. The growth direction is along the z-axis; z (~2 ~2)/2 The diffusion equations are then solved in this coordinate system [10], consistent with the non-paraboloidal shape of the interface. The shape of the actual dendrite is described by the equation =
0,5 mm Fig. 1. Shape of succinonitrile dendrite showing deviation from parabolic shape in the vicinity of the tip. Dots on the photograph represent a parabolic shape.
the dendrite shape which are consistent with the experimentally observed values of the anisotropy of the equilibrium shape of a droplet. Therefore, the theory applies only when the interfacial tension is anisotropic. We assume that the interfacial energy displays an anisotropy, , of the harmonic form [47] (appendix A) y
=
y0[1
+
~m
cos(rnO)],
(2)
where y and y0 are the interfacial energy and the modulus, respectively; 0 is the angle between the growth direction and the normal to the interface, and rn is an even integer index. We then further assume that the dendrites scale with respect to their deformed tip shape (that is, the dimensionless shape of the dendrite is the same for different growth conditions). It is then possible to explain the sharp selection mechanism of the operating state of the dendrite, and determine growth veloc-
2
—
2 =
1 + aq
(3)
‘
where a is a small parameter that accounts for the experimentally observed deviation of the dendrite shape from that of a paraboloid of revolution, ~2 ~ This deviation is related to the strength of the interfacial energy anisotropy, , by the cxpression =
a
(4)
=
when rn-fold interfacial energy amsotropy is assumed (for a detailed derivation, refer to appendix Local equilibrium conditions are then satisfied on the crystal—melt interface. These conditions are: (1) the continuity of the temperature of the liquid and the solid, (2) the heat flux balance and (3) the Gibbs—Thomson (capillary) condition for the interface temperature. .
.
.
2.2. Field equations
By solving the thermal diffusion equations in the solid and liquid phases, the temperature fields
V. Miyata et al.
/
Dendritic growth with interfacial energy anisotropy
for the liquid, TL, and for the solid, T~,can be expressed as follows: TL
=
[e~ ‘I’(n + 1, 1;
T~+ ~
x)1x~pL~2
(5) =
T~+ ~
[e’ L~(x)1x~P,~2 + 1,1; ~
(6)
and solving the solute diffusion equation in the liquid
685
the continuity of temperature, the conservation of the flux, and the capillary condition for the interface temperature. Dimensions of growing dendrites are determined2).byThe the fields behavior fields near the tip canof bethe expanded in powers (small ~j of s~°and ~2; the boundary conditions are then imposed on each corresponding term. The conditions at the interface are: Temperature continuity, TL= T 5.
CL= C~+ ~
[ex
‘I’(n+l, 1;
x)]~=p~2
For the zeroth-order condition in (7)
(8)
~
~O;
where n 0, 1, 2 ~L’ Ps and P are the thermal Peclet numbers for the liquid, and the solid, and the chemical Peclet number, respectively. A~, T0s, B~ and A~ are constants to be determined from the boundary conditions, and T~ and C~ are the far-field temperature and the average concentration of solute in the melt, respectively. 1~(n+ 1, 1; x), ~I’(n + 1, 1; x) and L0 (x) are the confluent hypergeometric polynomials of the first kind, of the second kind, and the Laguerre polynomials, respectively. Fields should generally be expressed as the summation of contributions of solutions with different n values. In order to simplify the mathematics, summations are not taken in this paper and only contributions from n 0 are taken into account. With this approximation, the validity2of is this model is inrestricted to cases wherenumber PLi~ is small; that is, cases where the Peclet small and/or we consider the regions close to the dendrite tip. In order to take into account the contributions of the higher-order terms, and the
i~,
=
For the second-order condition in
[log Ei(PL)1
AoLEi(PL)aPL~
~2;
=
i~,
B~e~’~
[aPs~lo~
~
+ Ps].
(9)
~
[log E~( PL)]
Heat flux balance, 8TL
K 5 ~T5
~L
=
~
~
—
-~—
+ KL
~2
=
—
(
A~E1
PL) ~L
+ .~..B0se~sP5~_[log e”s]. L
=
(10) Solute conservation, P(i k)CL —acL/a~2 —
=
~1° P(1
—
k)[C~ +A 0 E1(P)]
behavior of the fields away from the tip, a perturbative stability theory with higher-order terms needs to be developed, where a more precise shape than the one used in this text would be needed.
Interface temperature,
2.3. Boundary conditions near the tip
TL
=
~jO;
The far-field condition at z oc is already taken into account in the expression of the fields TL and CL. The boundary conditions which must still be considered are those on the mterface, viz.,
=
A0 E1(P)P~-~[log E1(P)].
(11)
TM + K0 + rn OCL; y2 T~+A~El(PL)=TM—~—(1—a)
—~
+ rn0[C~ + AOEI(P)].
(12)
686
V. Miyata et al. AOLEi(PL)aPL~_
~2;
/ Dendritic growth with interfacial energy anisotropy
=
[log Ei(PL)]
~~ineqs. (8)—(13). Eq. (13) gives
y2 ~~(1 —a)(1 +a)2 +m 0A0E1(P)aP~[logE1(P)],
Eqs. (9) and (10) give
=
(13)
where ~H K~ y -b—,
B~=0,
(iSa)
4E1(PL)
L~H —PL-~-—/PL~—[log Ei(PL)].
=
and K0,
-~--~ ~,
(15b) Then, one gets from eq. (12)
with (1_a)[2+(1+a)2~i2j
1
213/2
p
2
K0
=
—
[i
+
(1
+ a)
~
This is the Ivantsov relation between Peclet number and the supercooling, and is identical to eq.
2 —
p(1
—
a)[1 —(1 +
a)2~2]
(16)
~T=4El(PL)=_~_PLePLEi(PL).
‘
are the latent heat of fusion divided by the specific heat, the ratio of thermal conductivities in the solid and liquid, the capillary coefficient, and the mean curvature of the interface, respectively. m0 and k are the liquidus slope and the distribution coefficient of solute, respectively. E1(x) is the integral exponential function. From the above analysis, we see that a non-zero value of the anisotropy parameter, a, which leads to a slightly non-paraboloidal shape, removed the degeneracy of the Peclet number allowing one to solve for V and p as unique functions of supercooling. Thus, a non-zero value of the parameter a, provides sufficient conditions, arising as a result of expanding the fields in powers of ~° and ~2 ~ solve for V and p; no additional physical conditions or assumptions are required. When the parameter a 0 (interfacial energy isotropy), we have only the zeroth-order terms in ~, which alone
(1). 3.2. Prediction for pure rnelts The tip shape of the dendrite is assumed to scale, independently of the supercooling, so that the dimensionless deviation a is constant for all tip radii. Eqs. (12), (10) and (11) then yield, for 4, B~and A0, 4E1(PL) =~T— /
~
~(1 —a),
(17)
H
B~e”~= ~\PL’~~~ + 4E1(PL)
[log E1( PU])
~ ~
-i
x(.~.Ps.~-[1oge1~s])
,
(18)
=
are insufficient to resolve uniquely the Peclet number into V and p independently, thus yielding the classical Ivantsov solution as discussed below.
A0E1(P)
—
(1
+
~
—
k) C~ (19)
=
1
—
k
[log E1 (P)]
theory for pure melts
The tip radius of curvature and the growth rate are then chosen to satisfy eqs. 5. (9) and (13). Eq. (8) is used only to determine T0 The parametric dependence of V and p on the parameter a is shown in fig. 2. Each point on the curve corresponds to a solution of V and p for the
Ivantsov’s theory deals with the situation where interfacial tension is absent. This corresponds to
given supercooling of 1.0 K. The parameter a changes from 1 to 0 (small negative value) as we
3. Predictions 3.1. Ivantsov solution as a special case of the present
—
Y. Miyata et al. / Dendritic growth with interfacial energy anisotropy 102
687
The deformation parameter a is estimated by the strength of the interfacial energy anisotropy using eq. (4) and the anisotropy of the equilibrium shape of a droplet. For succinomtrile, the
..
\lvaritsov
,,,,
.li
>.
magnitude of anisotropy of shape is experimentally obtained to be equilibrium 0.007 [11], measured by4 aofshape analysis of equilibrium an droplet (fig. ref. [9]), corresponding to an interfacial energy anisotropy m 0.049 (appendix, eqs.
iO~~ -
-a=O04
=
0-02
~ i0-~
,
fl3~~te 10.6
the and B.9) above (B.10)). estimate Thereofis an interfacial error of about energy50%aniin
0
-
1O’~
~O-~
“~o
Tip Radius(m) Fig. 2. Relation between the radius of curvature and the growth rate with deformation parameter a.
trace the curve in a clockwise fashion. Each point on the curve is a possible operating state of the dendrite. The value of a chosen as the operating state is a — 0.049, which is consistent with the experimentally observed value of anisotropy of the equilibrium shape for a succinonitrile droplet (see table 1). This operating state is indicated in fig. 2. It is interesting to note that this operating state is different from the state of maximum growth velocity which occurs at a value of a —0.525. The Ivantsov result is also plotted in fig. 2 for comparison, and as a 0, the results asymptotically approach Ivantsov’s solution. Explicit expressions for z~1Tand p are =
=
sotropy; our current estimated value is somewhat different than the one quoted earlier by Singh and Glicksman [15]. The predictions for the succinomtrile dendrite for three different values of the parameter a are shown in fig. 3 and fig. 4 [9,12]. As discussed earlier, choosing any value of a is sufficient to predict V and p as unique functions of supercooling. The theoretical predictions are in reasonable agreement with the experimental results for a value
Heyl I —
2
L
tO
a 049 axO5 ax 2
-~
II
/
I
/
“7
—*
(1+a) ~T=
e~E~(PL)
2
—
/
—
a ______
/
,‘/
ePLEI(PL) a
//
-
(20) —
y
~LXS
e~El(PL)
i1iT
1
—
~“
a
—a
x[(1+a)2_
ePLE 1(PL)]’
(21)
respectively. Eq. (20) reduces to the Ivantsov relation when y a 0. Eq. (21) corresponds to the =
/
0,’
=
1O*r
.
/
/ I
10’ 102
marginal stability relation, because the supercooling L!T is proportional to ~L e”L Ei(PL) by eq. (20).
//
o0
10’
1
Supercooling(K)
Fig. 3. The growth rate with the supercooling for the suecinonitrile. Data are from refs. [9,12].
688
Y Miyata et al.
/ Dendritic growth with
interfacial energy anisotropy
io-’
24
keyl \
\ \
\
10’
I
\,
\
~C
\
\
\Q
0
\
1 I
If
IL
\ \,
I
111111
10.1
IlL
B0
\
\~
I
O~--~
4
\
\, \
-
I
E ~12 >8
\
0.
i 10.2
II
\
\
jQ.5
20
i
\ ~O
a -u
I
.....—1ax05 I ax 2
\
“0
U)
a
1
L~T~Q5K
0
~
02 ‘ 0.4 0.6 ‘ 08 Fig. 5. Growth rate for CACE(rrol’/.) two supercoolings as a function of acetone concentration, CACU, in succinonitrile—acetone alloy. Data are from ref. [13].
\ \.
111111
10
Supercooling (K)
Fig. 4. Variation of tip radius with the supercooling for succinonitrile. Data are from refs. [9,12].
of parameter a = — 0.049, especially at the higher supercoolings. The systematic deviation which get larger at lower supercoolings can be attributed to the neglect of natural convection in the present theory. It is possible to include convective effects in the theory and, using the same concepts of slight deviations in dendrite shape from a paraboloid of revolution, we can predict V and p as functions of supercooling. Future efforts will be aimed at achieving such a solution.
‘
alloy concentrations. Fig. 5 is a plot of growth velocity as a function of solute concentration for dilute SCN—acetone alloys. The open circles are the experimental data points [13]. We see that the theoretical predictions, for a value of anisotropy parameter a = — 0.049, are in substantial agreement with the experimental results. There is a sharp rise in growth velocity as the acetone concentration is increased and the theory predicts a maximum at about 0.09 mol% acetone. Beyond this solute concentration, the growth velocity decreases with concentration, but the drop is not as rapid as observed experimentally [13]. The increase in growth rate is related to a decrease in tip radius (fig. 6), which also reaches a minimum value and then rises very slowly. Again, the rise is
3.3. Prediction for dilute alloy systems 24
The present theory is also capable of predicting V and p as unique functions of supercooling and
20
1,
l6
Table I Physical properties of the succinonitrile
AT=Q5K
U)
~ 12 0
Parameter KS/KL
Value
8
1.14X107m2/s 1.005 23.88 K
6.620X105 KM D 1.27x iO~ m2/s —2.16 K/mol% k 0.10 ____________________________________________________________
~0
~T=09K
~
0
02
0.4
0.6
‘
08
y/~S
C~(moL‘I.) Fig. 6. Dendrite tip radius for two supercoolings as a function of acetone concentration, CACU, in succinonitrile—acetone alloy. Data are from ref. [13].
V. Miyata et at
/
689
Dendritic growth with interfacial energy anisotropy
not as sharp as the experimentally observed values. These predictions for alloys are similar to the ones made by Lipton, Glicksman and Kurz [14], who found a maxima in V and then a slow decrease in V at higher solute concentrations. They arrived at their results using marginal stability considerations and Ivantsov’s diffusion solution for heat and solute.
4. Discussions and conclusions Solid—liquid interfacial energy anisotropy consistent with small deviations in the shape of the dendrite tip could replace marginal stability or the maximum velocity principle in predicting the radius of curvature and the growth rate of dendrites for pure succinonitrile and for succinonitrile—acetone melts. Only small departures from a parabolic shape are needed to be consistent with the experimental value of interfacial energy anisotropy. The theory is not a perturbative stability theory, and the fields are approximated in a twoterm expansion with the first term being the wellknown Ivantsov solution. Therefore, the applicability of the theory is currently restricted to regions near the dendrite tip. In order to take into account the contribution of the higher-order terms, and the behavior of the fields away from the tip, a perturbative stability theory containing higherorder terms needs to be developed; however, a more precise shape than the one used in this text would be needed. The present theoretical predictions are in agreement with experiments at higher supercoolings, for a value of anisotropy parameter a = — 0.049 which seems to be consistent with the latest experimental estimate of the value of interfacial energy ani-
sotropy. The deviations at lower supercoolings can be attributed to the neglect of natural convection under terrestrial conditions. It is possible to indude convective effects in the theory and this will be attempted in future.
Appendix A. Relation between tip deformation and interfacial energy anisotropy The shape of the interface described by eq. (1) in the text can be expressed in dimensionless Cartesian coordinates (x, y, z) as
1
1
/
2
r
—
(i
+ a )2
1
4a
J
—
)
+ a 2
4a (1 + a )2
—
1
}
1 (1
a)2
—
=
1,
—
a)2r2
~
—
4a (A.1)
or (1
=
4az2
—
2(1
+ a)z +
1,
where the z-axis is chosen in the growth direction of the dendrite, and r= (x2 +y2)~2. For regions close to the tip (z 1/2), this equation reduces to (1 — a) r2 = 1 — 2 z. The shape of interface is given corresponding to the value of a, as shown in table 2. We are primarily concerned with small values of a where the shape will deviate from parabolic to The hyperbolic or elliptic according to the sign of a. curvature Ka for a given deformation a is given as 2
+
(1
+ a )2r2
1 (A.2)
K
3/’2
P
0 = —(1 a) [i + (1 + a)2r2] This can be approximated to be —
Table 2 Values of a and corresponding interface form Interface ___________________________________________ a
1 0 to 1 0 —ito 0 —1
Planar H~erbolic Parabolic Elliptic Spherical
Ka =
—
2(1
—
a) [1 — (1 + a )2r21
for small r2 (near the tip). The undercooling due to the finite radius of tip is, therefore, given near the tip (x =y = 0 and z = ~) as —2(1
—
a)[1
—
(1
+ a)2r2] y/~S
(A.3)
690
V. Miyata et at
/
Dendritic growth with interfacial energy anisotropy
The correction for the tip radius of the perfect paraboloid of revolution is, therefore, 1 a. The interfacial energy anisotropy introduces, a correction to the curvature of the perfect paraboloid of revolution [4—7]as
Therefore, the radius of curvature changes linearly with interfacial energy amsotropy. The interfacial energy anisotropy can thus be estimated by measuring the anisotropy of the shape of a droplet.
K [i +
(A.4)
B.2. Anisotropy of the equilibrium shape of a droplet
where 0 is an angle between the growth direction and the normal of the interface. ,,, is the magnitude of rn-fold interfacial energy anisotropy. K is the curvature of the interface. Near the tip of the dendrite,
In a three-dimensional droplet, suppose the shape near its largest diameter is approximated as:
—
m
cos(rnO)],
r,b[1
+~
cos(mO)][1
+2
cos(n4))],
(B.4)
in three-dimensional polar coordinates (r, 0, The radius has a maximum and minimum expressed as ~).
0
=
0;
(A.5)
then, the amsotropy gives to the tip radius of curvature a correction, 1 + ,n~ By comparing the corrections to the tip radius of curvature, a relation a
=
(A.6)
—
rm~= r
=
i)(1
r0(1
—
~)(1
+ 2), —
E2).
To estimate the interfacial energy amsotropy, we calculate the mean curvature at the bulging point, K:
Appendix B. Relationship between interfacial energy anisotropy and the equilibrium shape of a droplet
r0 [1 +
Suppose there is a droplet in equilibrium with rn-fold interfacial energy anisotropy ~m (>0) in a melt. The Gibbs—Thomson relation shows that [1+~cos(rn0)]K=constant. (B.1)
—
(B.5)
rmin
is given between the deformation parameter a and the interfacial energy anisotropy m~ The non-zero deformation parameter a means non-zero anisotropy. They have the same amplitude but different signs. It should be noted that this equality applies only to small deformations of the interface.
B.]. Interfacial energy anisotropy
0(1
K
—
2
—
—
P
—
1
1
p1
P2
—+—,
(B.6)
where p1 and p2 are the principal radii. In cases where the normal to the interface passes through the center of the droplet (this is the case considered in the text), p1 is given by replacing r0 with 2 cos( n 4))] for a fixed 4), and p2 is given by replacing ~ with r0 [1 + ~ cos( m0)] for a fixed 0 in a two-dimensional droplet. A minimum and a maximum mean radius of curvature are given at 0 = = 0 and 0 = 4) = sr/rn. They are:
2
—
Pmax,min
—
1 1 “o (1 ±~ (1 ±~2)
[ 1
1 ±2(1
2) + —
~(1 + m
+
+ n2) 1±2
The radius of curvature, p, has the form p
=
1j
(B.7)
p 0[1 +
m
cos(mO)J,
(B.2)
and it displays maximum and minimum values given by pmaxpo(1
+m),
pmin’po(l
em)’
(B.3)
For an axisymmetric dendrite, we take n
=
m and
= Then, for small amsotropies, these are approximated to be
~2
~
pmax,mini’o[l
±(m2_2)ciI.
(B.8)
Y. Miyata et a!.
/ Dendritic growth with
This means that the interfacial anisotropy, given as, =
2
(rn
—
2)c~,
~m’
is
(B.9)
interfacial energy anisotropy
691
[4] D. Kessler and H. Levine, Phys. Rev. A33 (1986) 2621, 2634. [5] D. Kessler, J. Koplik and H. Levine, Phys. Rev. A33 (1986) 3352, 7867. [6] Y. Saito, G. Goldbeck-Wood and H. Muller-Krumbhaar,
and the corresponding anisotropy of the equilibrium shape of a droplet, i~
[7] D. Bensimon, P. Pelcé and B.I. Shraiman, J. Physique 48
(B.lo)
(i987) 2081. [8] S.C. Huang and M.E. Glicksman, Ada Met. 29 (1981)
~,
=
2~.
Phys. Rev. Letters 58 (1987) 1541.
701.
For m = 2, therefore, both anisotropies have the same value, and for rn = 4 the interfacial energy anisotropy is 7 times larger than that of the equilibrium shape.
References [1] G.P. Ivantsov, DokI. Akad. Nauk SSSR 58 (1947). [2] G.E. Nash and ME. Glicksman, Acta Met. 18 (1970) 287. [3] J.S. Langer and H. MOller-Krumbhaar, Acta Met. 26 (1978) 1681.
[91 S.C. Huang and ME. Glicksman, Ada Met. 29 (1981) 717.
[10] Miyata and T. ME. Suzuki, Met. Trans. A16Tirmizi, (1985) 1807. [ii] Y. ME. Glicksman, Selleck and S.H. unpublished, 1988. [121 ME. Glicksman, R.J. Schaefer and J.D. Ayers, Met. Trans. A7 (1976) 1747. [13] MA. Chopra, M.E. Glicksman and NB. Singh, Met. Trans. A, 19A (1988) 3087. [14] J. Lipton, ME. Glieksman and W. Kurz, Mater. Sci. Eng. 65 (1984) 57. [15] M.E. Glicksman and NB. Singh, J. Crystal Growth 98 (1989) 277.
683
Dendritic growth with interfacial energy anisotropy Y. Miyata
~,
M.E. Glicksman
* *
and S.H. Tirmizi
Materials Engineering Department, Rensselaer Polytechnic Institute, Troy, New York 12180-3590, USA
Received 13 August 1990; manuscript received in final form 27 November 1990
A theory for dendritic growth of needle shaped crystals growing in supercooled pure melts and dilute alloys is proposed. This theory predicts the growth velocity and tip radii as functions of supercooling and alloy concentrations without introduction of additional conditions such as marginal stability. A key ingredient of the theory is introducing the effect of the interfacial energy anisotropy. Deviation of the shape of the dendrite from a paraboloid of revolution is permitted, consistent with a small interfacial energy anisotropy. The radius of curvature and the growth rate as functions of melt supercooling and alloy concentrations are determined in terms of the anisotropy, and are compared with experimental results. The predictions are in agreement with the experimental results, especially at large supercoolings. The deviation at lower supercoolings can be attributed to the neglect of natural convection in the present theory.
1. Introduction One of the first attempts to predict the growth of smooth needle shaped dendrites was a steadystate diffusion theory by Ivantsov [1]. His solution can be represented as follows: =
~L
(1)
eL Ei(PL).
Eq. (1) relates the growth Peclet number, ~L’ to the dimensionless melt supercooling, ~O, but it fails to predict the unique relationship between growth velocity and tip radius as functions of supercooling. Several attempts have been made to provide an additional relationship, which in conjunction with eq. (1) can predict the growth rate, V. and the radius of curvature, p. Nash and Glicksman [2] found self-consistent corrections to eq. (1) which include the Gibbs—Thomson capillarity condition at the interface. Their work showed, ultimately, that the dendrite selects a tip radius which is not consistent with the maximum *
**
Present address: Mechanical Engineering
Department,
Nagaoka University of Technology, Kamitormoka, Nagaoka 940, Japan. To who all correspondence should be addressed,
0022-0248/91/$03.50
©
1991
—
velocity principle. Langer and Müller-Krumbhaar [3] formulated the marginal stability criterion which determines the dendrite tip radius as the one which is marginally stable to tip-splitting perturbations. Recently, Kessler et a!. [4,51, Saito et at. [6] and Bensimon et at. [71used the concept of microscopic solvability and argued that of all the possible dynamical operating states, the state at which the dendrite grows at the highest velocity is the stable one and is thus the operating mode. The microscopic solvability theory requires a non-zero value of interfacial energy anisotropy (amsotropy in the solid—liquid interfacial energy) for a solution to exist. The theory to be discussed here also requires interfacial anisotropy, but acting in a somewhat different manner. The Ivantsov solution, marginal stability theory, and the microscopic solvability theory, usually employ the assumption that the shape of the dendrite is a paraboloid of revolution. But it has been observed experimentally by Glicksman and co-workers [8] that there always exists near the tip region a small departure of the shape from a perfect paraboloid of revolution. Fig. ~ indicates such a deformed paraboloid. An essential ingredient of the present theory is to take into account small deviations of .
.
Elsevier Science Publishers B.V. (North-Holland)
.
.
684
Y. Miyata et al.
• *
/ Dendritic growth with interfacial
energy anisotropy
ity and tip radii as unique functions of supercooling. An analytical model is proposed here for the growth of a dendrite in a supercooled melt, based solely on the presence of interfacial energy am-
~
I
sotropy and the assumed scaling of dendritic tip shape. 2. Theory 2.1. Interface and anisotropy
The moving dimensionless parabolic coordinates (~,ij, 4)) are adopted, where all coordinates are normalized by the radius of curvature of the dendrite tip. The growth direction is along the z-axis; z (~2 ~2)/2 The diffusion equations are then solved in this coordinate system [10], consistent with the non-paraboloidal shape of the interface. The shape of the actual dendrite is described by the equation =
0,5 mm Fig. 1. Shape of succinonitrile dendrite showing deviation from parabolic shape in the vicinity of the tip. Dots on the photograph represent a parabolic shape.
the dendrite shape which are consistent with the experimentally observed values of the anisotropy of the equilibrium shape of a droplet. Therefore, the theory applies only when the interfacial tension is anisotropic. We assume that the interfacial energy displays an anisotropy, , of the harmonic form [47] (appendix A) y
=
y0[1
+
~m
cos(rnO)],
(2)
where y and y0 are the interfacial energy and the modulus, respectively; 0 is the angle between the growth direction and the normal to the interface, and rn is an even integer index. We then further assume that the dendrites scale with respect to their deformed tip shape (that is, the dimensionless shape of the dendrite is the same for different growth conditions). It is then possible to explain the sharp selection mechanism of the operating state of the dendrite, and determine growth veloc-
2
—
2 =
1 + aq
(3)
‘
where a is a small parameter that accounts for the experimentally observed deviation of the dendrite shape from that of a paraboloid of revolution, ~2 ~ This deviation is related to the strength of the interfacial energy anisotropy, , by the cxpression =
a
(4)
=
when rn-fold interfacial energy amsotropy is assumed (for a detailed derivation, refer to appendix Local equilibrium conditions are then satisfied on the crystal—melt interface. These conditions are: (1) the continuity of the temperature of the liquid and the solid, (2) the heat flux balance and (3) the Gibbs—Thomson (capillary) condition for the interface temperature. .
.
.
2.2. Field equations
By solving the thermal diffusion equations in the solid and liquid phases, the temperature fields
V. Miyata et al.
/
Dendritic growth with interfacial energy anisotropy
for the liquid, TL, and for the solid, T~,can be expressed as follows: TL
=
[e~ ‘I’(n + 1, 1;
T~+ ~
x)1x~pL~2
(5) =
T~+ ~
[e’ L~(x)1x~P,~2 + 1,1; ~
(6)
and solving the solute diffusion equation in the liquid
685
the continuity of temperature, the conservation of the flux, and the capillary condition for the interface temperature. Dimensions of growing dendrites are determined2).byThe the fields behavior fields near the tip canof bethe expanded in powers (small ~j of s~°and ~2; the boundary conditions are then imposed on each corresponding term. The conditions at the interface are: Temperature continuity, TL= T 5.
CL= C~+ ~
[ex
‘I’(n+l, 1;
x)]~=p~2
For the zeroth-order condition in (7)
(8)
~
~O;
where n 0, 1, 2 ~L’ Ps and P are the thermal Peclet numbers for the liquid, and the solid, and the chemical Peclet number, respectively. A~, T0s, B~ and A~ are constants to be determined from the boundary conditions, and T~ and C~ are the far-field temperature and the average concentration of solute in the melt, respectively. 1~(n+ 1, 1; x), ~I’(n + 1, 1; x) and L0 (x) are the confluent hypergeometric polynomials of the first kind, of the second kind, and the Laguerre polynomials, respectively. Fields should generally be expressed as the summation of contributions of solutions with different n values. In order to simplify the mathematics, summations are not taken in this paper and only contributions from n 0 are taken into account. With this approximation, the validity2of is this model is inrestricted to cases wherenumber PLi~ is small; that is, cases where the Peclet small and/or we consider the regions close to the dendrite tip. In order to take into account the contributions of the higher-order terms, and the
i~,
=
For the second-order condition in
[log Ei(PL)1
AoLEi(PL)aPL~
~2;
=
i~,
B~e~’~
[aPs~lo~
~
+ Ps].
(9)
~
[log E~( PL)]
Heat flux balance, 8TL
K 5 ~T5
~L
=
~
~
—
-~—
+ KL
~2
=
—
(
A~E1
PL) ~L
+ .~..B0se~sP5~_[log e”s]. L
=
(10) Solute conservation, P(i k)CL —acL/a~2 —
=
~1° P(1
—
k)[C~ +A 0 E1(P)]
behavior of the fields away from the tip, a perturbative stability theory with higher-order terms needs to be developed, where a more precise shape than the one used in this text would be needed.
Interface temperature,
2.3. Boundary conditions near the tip
TL
=
~jO;
The far-field condition at z oc is already taken into account in the expression of the fields TL and CL. The boundary conditions which must still be considered are those on the mterface, viz.,
=
A0 E1(P)P~-~[log E1(P)].
(11)
TM + K0 + rn OCL; y2 T~+A~El(PL)=TM—~—(1—a)
—~
+ rn0[C~ + AOEI(P)].
(12)
686
V. Miyata et al. AOLEi(PL)aPL~_
~2;
/ Dendritic growth with interfacial energy anisotropy
=
[log Ei(PL)]
~~ineqs. (8)—(13). Eq. (13) gives
y2 ~~(1 —a)(1 +a)2 +m 0A0E1(P)aP~[logE1(P)],
Eqs. (9) and (10) give
=
(13)
where ~H K~ y -b—,
B~=0,
(iSa)
4E1(PL)
L~H —PL-~-—/PL~—[log Ei(PL)].
=
and K0,
-~--~ ~,
(15b) Then, one gets from eq. (12)
with (1_a)[2+(1+a)2~i2j
1
213/2
p
2
K0
=
—
[i
+
(1
+ a)
~
This is the Ivantsov relation between Peclet number and the supercooling, and is identical to eq.
2 —
p(1
—
a)[1 —(1 +
a)2~2]
(16)
~T=4El(PL)=_~_PLePLEi(PL).
‘
are the latent heat of fusion divided by the specific heat, the ratio of thermal conductivities in the solid and liquid, the capillary coefficient, and the mean curvature of the interface, respectively. m0 and k are the liquidus slope and the distribution coefficient of solute, respectively. E1(x) is the integral exponential function. From the above analysis, we see that a non-zero value of the anisotropy parameter, a, which leads to a slightly non-paraboloidal shape, removed the degeneracy of the Peclet number allowing one to solve for V and p as unique functions of supercooling. Thus, a non-zero value of the parameter a, provides sufficient conditions, arising as a result of expanding the fields in powers of ~° and ~2 ~ solve for V and p; no additional physical conditions or assumptions are required. When the parameter a 0 (interfacial energy isotropy), we have only the zeroth-order terms in ~, which alone
(1). 3.2. Prediction for pure rnelts The tip shape of the dendrite is assumed to scale, independently of the supercooling, so that the dimensionless deviation a is constant for all tip radii. Eqs. (12), (10) and (11) then yield, for 4, B~and A0, 4E1(PL) =~T— /
~
~(1 —a),
(17)
H
B~e”~= ~\PL’~~~ + 4E1(PL)
[log E1( PU])
~ ~
-i
x(.~.Ps.~-[1oge1~s])
,
(18)
=
are insufficient to resolve uniquely the Peclet number into V and p independently, thus yielding the classical Ivantsov solution as discussed below.
A0E1(P)
—
(1
+
~
—
k) C~ (19)
=
1
—
k
[log E1 (P)]
theory for pure melts
The tip radius of curvature and the growth rate are then chosen to satisfy eqs. 5. (9) and (13). Eq. (8) is used only to determine T0 The parametric dependence of V and p on the parameter a is shown in fig. 2. Each point on the curve corresponds to a solution of V and p for the
Ivantsov’s theory deals with the situation where interfacial tension is absent. This corresponds to
given supercooling of 1.0 K. The parameter a changes from 1 to 0 (small negative value) as we
3. Predictions 3.1. Ivantsov solution as a special case of the present
—
Y. Miyata et al. / Dendritic growth with interfacial energy anisotropy 102
687
The deformation parameter a is estimated by the strength of the interfacial energy anisotropy using eq. (4) and the anisotropy of the equilibrium shape of a droplet. For succinomtrile, the
..
\lvaritsov
,,,,
.li
>.
magnitude of anisotropy of shape is experimentally obtained to be equilibrium 0.007 [11], measured by4 aofshape analysis of equilibrium an droplet (fig. ref. [9]), corresponding to an interfacial energy anisotropy m 0.049 (appendix, eqs.
iO~~ -
-a=O04
=
0-02
~ i0-~
,
fl3~~te 10.6
the and B.9) above (B.10)). estimate Thereofis an interfacial error of about energy50%aniin
0
-
1O’~
~O-~
“~o
Tip Radius(m) Fig. 2. Relation between the radius of curvature and the growth rate with deformation parameter a.
trace the curve in a clockwise fashion. Each point on the curve is a possible operating state of the dendrite. The value of a chosen as the operating state is a — 0.049, which is consistent with the experimentally observed value of anisotropy of the equilibrium shape for a succinonitrile droplet (see table 1). This operating state is indicated in fig. 2. It is interesting to note that this operating state is different from the state of maximum growth velocity which occurs at a value of a —0.525. The Ivantsov result is also plotted in fig. 2 for comparison, and as a 0, the results asymptotically approach Ivantsov’s solution. Explicit expressions for z~1Tand p are =
=
sotropy; our current estimated value is somewhat different than the one quoted earlier by Singh and Glicksman [15]. The predictions for the succinomtrile dendrite for three different values of the parameter a are shown in fig. 3 and fig. 4 [9,12]. As discussed earlier, choosing any value of a is sufficient to predict V and p as unique functions of supercooling. The theoretical predictions are in reasonable agreement with the experimental results for a value
Heyl I —
2
L
tO
a 049 axO5 ax 2
-~
II
/
I
/
“7
—*
(1+a) ~T=
e~E~(PL)
2
—
/
—
a ______
/
,‘/
ePLEI(PL) a
//
-
(20) —
y
~LXS
e~El(PL)
i1iT
1
—
~“
a
—a
x[(1+a)2_
ePLE 1(PL)]’
(21)
respectively. Eq. (20) reduces to the Ivantsov relation when y a 0. Eq. (21) corresponds to the =
/
0,’
=
1O*r
.
/
/ I
10’ 102
marginal stability relation, because the supercooling L!T is proportional to ~L e”L Ei(PL) by eq. (20).
//
o0
10’
1
Supercooling(K)
Fig. 3. The growth rate with the supercooling for the suecinonitrile. Data are from refs. [9,12].
688
Y Miyata et al.
/ Dendritic growth with
interfacial energy anisotropy
io-’
24
keyl \
\ \
\
10’
I
\,
\
~C
\
\
\Q
0
\
1 I
If
IL
\ \,
I
111111
10.1
IlL
B0
\
\~
I
O~--~
4
\
\, \
-
I
E ~12 >8
\
0.
i 10.2
II
\
\
jQ.5
20
i
\ ~O
a -u
I
.....—1ax05 I ax 2
\
“0
U)
a
1
L~T~Q5K
0
~
02 ‘ 0.4 0.6 ‘ 08 Fig. 5. Growth rate for CACE(rrol’/.) two supercoolings as a function of acetone concentration, CACU, in succinonitrile—acetone alloy. Data are from ref. [13].
\ \.
111111
10
Supercooling (K)
Fig. 4. Variation of tip radius with the supercooling for succinonitrile. Data are from refs. [9,12].
of parameter a = — 0.049, especially at the higher supercoolings. The systematic deviation which get larger at lower supercoolings can be attributed to the neglect of natural convection in the present theory. It is possible to include convective effects in the theory and, using the same concepts of slight deviations in dendrite shape from a paraboloid of revolution, we can predict V and p as functions of supercooling. Future efforts will be aimed at achieving such a solution.
‘
alloy concentrations. Fig. 5 is a plot of growth velocity as a function of solute concentration for dilute SCN—acetone alloys. The open circles are the experimental data points [13]. We see that the theoretical predictions, for a value of anisotropy parameter a = — 0.049, are in substantial agreement with the experimental results. There is a sharp rise in growth velocity as the acetone concentration is increased and the theory predicts a maximum at about 0.09 mol% acetone. Beyond this solute concentration, the growth velocity decreases with concentration, but the drop is not as rapid as observed experimentally [13]. The increase in growth rate is related to a decrease in tip radius (fig. 6), which also reaches a minimum value and then rises very slowly. Again, the rise is
3.3. Prediction for dilute alloy systems 24
The present theory is also capable of predicting V and p as unique functions of supercooling and
20
1,
l6
Table I Physical properties of the succinonitrile
AT=Q5K
U)
~ 12 0
Parameter KS/KL
Value
8
1.14X107m2/s 1.005 23.88 K
6.620X105 KM D 1.27x iO~ m2/s —2.16 K/mol% k 0.10 ____________________________________________________________
~0
~T=09K
~
0
02
0.4
0.6
‘
08
y/~S
C~(moL‘I.) Fig. 6. Dendrite tip radius for two supercoolings as a function of acetone concentration, CACU, in succinonitrile—acetone alloy. Data are from ref. [13].
V. Miyata et at
/
689
Dendritic growth with interfacial energy anisotropy
not as sharp as the experimentally observed values. These predictions for alloys are similar to the ones made by Lipton, Glicksman and Kurz [14], who found a maxima in V and then a slow decrease in V at higher solute concentrations. They arrived at their results using marginal stability considerations and Ivantsov’s diffusion solution for heat and solute.
4. Discussions and conclusions Solid—liquid interfacial energy anisotropy consistent with small deviations in the shape of the dendrite tip could replace marginal stability or the maximum velocity principle in predicting the radius of curvature and the growth rate of dendrites for pure succinonitrile and for succinonitrile—acetone melts. Only small departures from a parabolic shape are needed to be consistent with the experimental value of interfacial energy anisotropy. The theory is not a perturbative stability theory, and the fields are approximated in a twoterm expansion with the first term being the wellknown Ivantsov solution. Therefore, the applicability of the theory is currently restricted to regions near the dendrite tip. In order to take into account the contribution of the higher-order terms, and the behavior of the fields away from the tip, a perturbative stability theory containing higherorder terms needs to be developed; however, a more precise shape than the one used in this text would be needed. The present theoretical predictions are in agreement with experiments at higher supercoolings, for a value of anisotropy parameter a = — 0.049 which seems to be consistent with the latest experimental estimate of the value of interfacial energy ani-
sotropy. The deviations at lower supercoolings can be attributed to the neglect of natural convection under terrestrial conditions. It is possible to indude convective effects in the theory and this will be attempted in future.
Appendix A. Relation between tip deformation and interfacial energy anisotropy The shape of the interface described by eq. (1) in the text can be expressed in dimensionless Cartesian coordinates (x, y, z) as
1
1
/
2
r
—
(i
+ a )2
1
4a
J
—
)
+ a 2
4a (1 + a )2
—
1
}
1 (1
a)2
—
=
1,
—
a)2r2
~
—
4a (A.1)
or (1
=
4az2
—
2(1
+ a)z +
1,
where the z-axis is chosen in the growth direction of the dendrite, and r= (x2 +y2)~2. For regions close to the tip (z 1/2), this equation reduces to (1 — a) r2 = 1 — 2 z. The shape of interface is given corresponding to the value of a, as shown in table 2. We are primarily concerned with small values of a where the shape will deviate from parabolic to The hyperbolic or elliptic according to the sign of a. curvature Ka for a given deformation a is given as 2
+
(1
+ a )2r2
1 (A.2)
K
3/’2
P
0 = —(1 a) [i + (1 + a)2r2] This can be approximated to be —
Table 2 Values of a and corresponding interface form Interface ___________________________________________ a
1 0 to 1 0 —ito 0 —1
Planar H~erbolic Parabolic Elliptic Spherical
Ka =
—
2(1
—
a) [1 — (1 + a )2r21
for small r2 (near the tip). The undercooling due to the finite radius of tip is, therefore, given near the tip (x =y = 0 and z = ~) as —2(1
—
a)[1
—
(1
+ a)2r2] y/~S
(A.3)
690
V. Miyata et at
/
Dendritic growth with interfacial energy anisotropy
The correction for the tip radius of the perfect paraboloid of revolution is, therefore, 1 a. The interfacial energy anisotropy introduces, a correction to the curvature of the perfect paraboloid of revolution [4—7]as
Therefore, the radius of curvature changes linearly with interfacial energy amsotropy. The interfacial energy anisotropy can thus be estimated by measuring the anisotropy of the shape of a droplet.
K [i +
(A.4)
B.2. Anisotropy of the equilibrium shape of a droplet
where 0 is an angle between the growth direction and the normal of the interface. ,,, is the magnitude of rn-fold interfacial energy anisotropy. K is the curvature of the interface. Near the tip of the dendrite,
In a three-dimensional droplet, suppose the shape near its largest diameter is approximated as:
—
m
cos(rnO)],
r,b[1
+~
cos(mO)][1
+2
cos(n4))],
(B.4)
in three-dimensional polar coordinates (r, 0, The radius has a maximum and minimum expressed as ~).
0
=
0;
(A.5)
then, the amsotropy gives to the tip radius of curvature a correction, 1 + ,n~ By comparing the corrections to the tip radius of curvature, a relation a
=
(A.6)
—
rm~= r
=
i)(1
r0(1
—
~)(1
+ 2), —
E2).
To estimate the interfacial energy amsotropy, we calculate the mean curvature at the bulging point, K:
Appendix B. Relationship between interfacial energy anisotropy and the equilibrium shape of a droplet
r0 [1 +
Suppose there is a droplet in equilibrium with rn-fold interfacial energy anisotropy ~m (>0) in a melt. The Gibbs—Thomson relation shows that [1+~cos(rn0)]K=constant. (B.1)
—
(B.5)
rmin
is given between the deformation parameter a and the interfacial energy anisotropy m~ The non-zero deformation parameter a means non-zero anisotropy. They have the same amplitude but different signs. It should be noted that this equality applies only to small deformations of the interface.
B.]. Interfacial energy anisotropy
0(1
K
—
2
—
—
P
—
1
1
p1
P2
—+—,
(B.6)
where p1 and p2 are the principal radii. In cases where the normal to the interface passes through the center of the droplet (this is the case considered in the text), p1 is given by replacing r0 with 2 cos( n 4))] for a fixed 4), and p2 is given by replacing ~ with r0 [1 + ~ cos( m0)] for a fixed 0 in a two-dimensional droplet. A minimum and a maximum mean radius of curvature are given at 0 = = 0 and 0 = 4) = sr/rn. They are:
2
—
Pmax,min
—
1 1 “o (1 ±~ (1 ±~2)
[ 1
1 ±2(1
2) + —
~(1 + m
+
+ n2) 1±2
The radius of curvature, p, has the form p
=
1j
(B.7)
p 0[1 +
m
cos(mO)J,
(B.2)
and it displays maximum and minimum values given by pmaxpo(1
+m),
pmin’po(l
em)’
(B.3)
For an axisymmetric dendrite, we take n
=
m and
= Then, for small amsotropies, these are approximated to be
~2
~
pmax,mini’o[l
±(m2_2)ciI.
(B.8)
Y. Miyata et a!.
/ Dendritic growth with
This means that the interfacial anisotropy, given as, =
2
(rn
—
2)c~,
~m’
is
(B.9)
interfacial energy anisotropy
691
[4] D. Kessler and H. Levine, Phys. Rev. A33 (1986) 2621, 2634. [5] D. Kessler, J. Koplik and H. Levine, Phys. Rev. A33 (1986) 3352, 7867. [6] Y. Saito, G. Goldbeck-Wood and H. Muller-Krumbhaar,
and the corresponding anisotropy of the equilibrium shape of a droplet, i~
[7] D. Bensimon, P. Pelcé and B.I. Shraiman, J. Physique 48
(B.lo)
(i987) 2081. [8] S.C. Huang and M.E. Glicksman, Ada Met. 29 (1981)
~,
=
2~.
Phys. Rev. Letters 58 (1987) 1541.
701.
For m = 2, therefore, both anisotropies have the same value, and for rn = 4 the interfacial energy anisotropy is 7 times larger than that of the equilibrium shape.
References [1] G.P. Ivantsov, DokI. Akad. Nauk SSSR 58 (1947). [2] G.E. Nash and ME. Glicksman, Acta Met. 18 (1970) 287. [3] J.S. Langer and H. MOller-Krumbhaar, Acta Met. 26 (1978) 1681.
[91 S.C. Huang and ME. Glicksman, Ada Met. 29 (1981) 717.
[10] Miyata and T. ME. Suzuki, Met. Trans. A16Tirmizi, (1985) 1807. [ii] Y. ME. Glicksman, Selleck and S.H. unpublished, 1988. [121 ME. Glicksman, R.J. Schaefer and J.D. Ayers, Met. Trans. A7 (1976) 1747. [13] MA. Chopra, M.E. Glicksman and NB. Singh, Met. Trans. A, 19A (1988) 3087. [14] J. Lipton, ME. Glieksman and W. Kurz, Mater. Sci. Eng. 65 (1984) 57. [15] M.E. Glicksman and NB. Singh, J. Crystal Growth 98 (1989) 277.