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The validity of steady-state dendrite growth models
Journal of Crystal Growth 43 (1978) 17—20 © North-Holland Publishing Company
THE VALIDITY OF STEADY-STATE DENDRITE GROWTH MODELS R.J. SCHAEFER Material Sciences Division, Naval Research Laboratory, Washington, DC 20375, USA Received 18 May 1977 The thermal interaction between the tip of a dendrite, assumed to be a paraboloid of revolution, and the points along its sides is analyzed. It is found that, except at extreme levels of supercooling, the tip temperature is strongly affected by portions of the paraboloid which experiment reveals to have a well-developed branch structure. It is concluded that a paraboloid or similar steady-state shape is not an adequate model for dendrite heat flow calculations.
Recently, a table has been published [1] listing eleven steady-state theories of dendrite growth in pure supercooled materials and comparing their treatments of the boundary conditions at the solid/liquid interface. None of the theories correctly predicts the velocity supercooling relation for pure materials, leading one to suspect that they all contain some common fault or oversimplification. The theories all assume that the dendrite tip is a paraboloid of revolution or a closely related shape, which advances at steady-state into the supercooled liquid. Thus the theories all carry the implication that the temperature at the tip is not significantly affected by the presence of side branches on the dendrite stem. This note examines the validity of that assumption. During the growth of a crystal, latent heat of fusion is emitted at the solid/liquid interface. The temperature of the interface is thereby elevated above the temperature of the supercooled bath. Ivantsov [2] showed that if the crystal propagates as a paraboloid of revolution growing at steady state along its axis, it has an isothermal surface. Setting this isotherm equal to the equilibrium temperature gives a unique relation between the bath supercooling and the Peclet number defined as
refined Ivantsov’s approach by adding the effects of curvature (the Gibbs Thomson effect) and interfacial kinetics, and by attempting to identify the specific velocity and tip radius of the growing dendrites. It has generally been assumed in these theories, without rigorous justification, that the dendrite grows with a tip radius which gives the maximum possible value of the velocity V. Dendrite growth experiments have generally been designed to measure the rate of growth of the dendrite tip as a function of melt supercooling. Two recent sets of measurements [1,3] gave velocity supercooling relationships which did not agree with any of the theories considered in ref. [1]. Therefore it is reasonable to question how realistically a steadystate model can describe the tip temperature of dendrites, which in reality have branches. For a paraboloid, we can readily calculate the fraction of the temperature rise at the tip which is attributable to latent heat emitted within a given distance of the tip. The paraboloidal model should be expected to work well only if this fraction approaches unity for distances within which branching is imperceptible. The temperature rise at the dendrite tip can be calculated by the theory of moving heat sources [4].
P = yr /2~
When a heat source of strength dq moves at velocity u through a medium of diffusivity a and conductivity
(1)
where v is the velocity of axial growth, r0 is the radius of curvature at the tip of the paraboloid, and a is the thermal diffusivity. Most recent theories of dendritic growth have
k, the temperature rise at any point is given by dT = dq exp [—(v/2a)(E+ R)] 4irk R 17
(2)
18
R.J. Schaefer
/ Validity of steady-state dendrite growth models
where R is the distance from the heat source to the point in question and ~ is the component of R in the direction of motion. The temperature rise at a dendrite tip can be calculated by summing the contribu-
the latent heat times the rate at which material is transforming from liquid to solid: thus dq = Xy2irrdr. Making these substitutions into eq. (3) and using a normalized coordinate system in which y = Y/r0 and
tions from the moving latent heat sources on the paraboloidal surface.to express temperature differences It is convenient dT on a normalized scale by writing dO = CdT/X, where C is the heat capacity and X is the latent heat of fusion. The moving heat source formula then becomes dq exp [ (y/2a)(~+ R)] dO—~~ R (3)
p = r/r0, we obtain2/2) + [p2 + (p4/4)] 1/2}] exp[ P{(p 2 dO = F— i 4, \1 1/2 p dp, (5) + u i4jj where P is the Peclet number as defined by eq. (1). To find the temperature rise at the tip due to heat emitted from that part of the surface from the tip out to radius p = p, we integrate this expression from
—
‘
using the relation a = k/C. Consider the paraboloidal surface
p=Otop=~: -
p
described by the
f
{exp
[
P {(p2/2) +
[p2 + (p4/4)] 1/2 }] }
p-O
equation Y
X{[p2+(p4/4)]”2}
r 2 /2r 0
(4)
,
where r0 is the radius of curvature of the tip. If we wish calculate the temperature rise at due to theto band of moving heat sources in the the tip region between r and r + dr (see fig. 1), we can note that ~ in eq. (3) corresponds to Yin eq. (4), andR in eq. (3) is equal to (Y2 + r2)”2 = [(r4/4r~) + r2] 1/2
‘pdp.
(6)
Letting 2 + (p4/4)] 1/2} U = P{(p/2) 1 + [p this expression+ becomes
}
P~I(l,2/2)+1+I~2÷(~4/4)1 1/2
M
PePf
(eu/u) du. (7)
Using the definition of the exponential integral,
The rate of heat production in this strip is equal to E
~
/c\
(e u/u)~
1(P)f P
we obtain M =Pe~E1(P) 2/2) + 1 + [~2 +(~o4/4)]1/2])} (8) E1(P[(1ô When p = cc, this reduces to the Ivantsov expression M Fe~E 1(P). The fraction of the tip temperature rise due to heat emitted from the surface out to p = ~ is thus
dp P~r/r
f
2/2)+ 1 —
I
y
2
3
4
5
p
Fig. 1. Paraboloid model of a dendrite tip, showing the tip radius r0 and the normalized coordinatesy and p.
—
E1(P) E1(P{co
+
[~2 +~4/4)]h/2})
E 1(P)
This fraction is plotted in fig. several values ofF. The values
2 as a function of p for of M corresponding to
R.J. Schaefer / Validity of steady-state dendrite growth models
Fig. 2. The fraction f of the tip temperature rise which is attributable to heat emitted within the region p <~, as given by eq. (9).
the selected values of P, according to the Ivantsov model, are also indicated in the figure. At small supercoolings, the shape of dendrite tips can be observed directly. The two tin dendrites analyzed by Glicksman and Schaefer [5] had paraboloidal tips, and Peclet numbers of 2.1 X i0~ and 1.0 X io~.These dendrites show a distinct deviation from the paraboloidal shape for p ~ 3. Fig. 2 indicates that this deviation takes place well within the region which influences the dendrite tip temperature. Similarly, measurements of the succinonitrile dendrites used for the tip radius measurements in (1) indicate that branching starts at approximately p 3. Dendrites growing at bath supercooling ~O ranging from 5 X 10 ~ to 6.5 X 10 2 were studied: there was no indication that the value of p at which branching started varied with supercooling. Fig. 3 shows two succinonitrile dendrites with superimposed parabolas and coordinates axes. Thus it is clear that dendrite branching starts well within the region which influences the dendrite tip, at least for P< 10 2 (corresponding to z~O<0.04) and probably for P< 10 1 (corresponding to M <0.2). Steady.state models which include the Gibbs— Thomson effect, which lowers the tip temperature, predict Peclet numbers only one half as large, at a given bath undercooling, as those predicted by the Ivantsov model used here. This fact does not alter the general nature of the observation: in fact it implies
19
_______
0.0068 2
______
a
—
that at a given M the influence on the tip temperature from large p values is even greater than is estimated here. -
.
.
In conclusion, the calculations, together with observations of dendrite tips in pure materials, mdi-
0.066
b Fig. 3. Succinonitrile dendrites with superimposed parabolas and inscribed tip radius circles. Values of p are indicated: (a) ~O = 0.0068, r 0 = 47 X i0~ cm; (b) ~O = 0.066, r0 = 5.4 X 10 ~ cm.
20
R.J. Schaefer / Validity of steady-state dendrite growth models
cate that the use of steady-state models to describe dendrite growth is highly unrealistic except possibly at very large supercoolings. At the low or moderate normalized supercoolings which prevail in most experimental situations, branching starts well within the region which has a thermal influence on the dendrite tip. Therefore a successful theoretical prediction of the dendrite tip velocity must include a consideration of the side branches, either in a time-averaged steady-state way, or in a fully time-dependent model.
Acknowledgement The dendrites in fig. 3 were photographed during an earlier study [1] with J.D. Ayers and M.E. Glicks-
man. Their permission to use the photographs here is gratefully acknowledged.
References [11ME.
Glicksman, R.J. Schaefer and J.D. Ayers, Met. Trans. 7a (1976) 1747. [21G.P. Ivantsov, Dokl. Akad. Nauk USSR 58 (1947) 56.7. [31T. Fujioka, Doctoral Thesis, Carnegie-Mellon University (1976). [41D. Rosenthal, Trans. Am. Soc. Mech. Engrs. 68 (1946) 849. [5] M.E. Glicksman and R.J. Schaefer, Acta Met. 14 (1966) 1126.
THE VALIDITY OF STEADY-STATE DENDRITE GROWTH MODELS R.J. SCHAEFER Material Sciences Division, Naval Research Laboratory, Washington, DC 20375, USA Received 18 May 1977 The thermal interaction between the tip of a dendrite, assumed to be a paraboloid of revolution, and the points along its sides is analyzed. It is found that, except at extreme levels of supercooling, the tip temperature is strongly affected by portions of the paraboloid which experiment reveals to have a well-developed branch structure. It is concluded that a paraboloid or similar steady-state shape is not an adequate model for dendrite heat flow calculations.
Recently, a table has been published [1] listing eleven steady-state theories of dendrite growth in pure supercooled materials and comparing their treatments of the boundary conditions at the solid/liquid interface. None of the theories correctly predicts the velocity supercooling relation for pure materials, leading one to suspect that they all contain some common fault or oversimplification. The theories all assume that the dendrite tip is a paraboloid of revolution or a closely related shape, which advances at steady-state into the supercooled liquid. Thus the theories all carry the implication that the temperature at the tip is not significantly affected by the presence of side branches on the dendrite stem. This note examines the validity of that assumption. During the growth of a crystal, latent heat of fusion is emitted at the solid/liquid interface. The temperature of the interface is thereby elevated above the temperature of the supercooled bath. Ivantsov [2] showed that if the crystal propagates as a paraboloid of revolution growing at steady state along its axis, it has an isothermal surface. Setting this isotherm equal to the equilibrium temperature gives a unique relation between the bath supercooling and the Peclet number defined as
refined Ivantsov’s approach by adding the effects of curvature (the Gibbs Thomson effect) and interfacial kinetics, and by attempting to identify the specific velocity and tip radius of the growing dendrites. It has generally been assumed in these theories, without rigorous justification, that the dendrite grows with a tip radius which gives the maximum possible value of the velocity V. Dendrite growth experiments have generally been designed to measure the rate of growth of the dendrite tip as a function of melt supercooling. Two recent sets of measurements [1,3] gave velocity supercooling relationships which did not agree with any of the theories considered in ref. [1]. Therefore it is reasonable to question how realistically a steadystate model can describe the tip temperature of dendrites, which in reality have branches. For a paraboloid, we can readily calculate the fraction of the temperature rise at the tip which is attributable to latent heat emitted within a given distance of the tip. The paraboloidal model should be expected to work well only if this fraction approaches unity for distances within which branching is imperceptible. The temperature rise at the dendrite tip can be calculated by the theory of moving heat sources [4].
P = yr /2~
When a heat source of strength dq moves at velocity u through a medium of diffusivity a and conductivity
(1)
where v is the velocity of axial growth, r0 is the radius of curvature at the tip of the paraboloid, and a is the thermal diffusivity. Most recent theories of dendritic growth have
k, the temperature rise at any point is given by dT = dq exp [—(v/2a)(E+ R)] 4irk R 17
(2)
18
R.J. Schaefer
/ Validity of steady-state dendrite growth models
where R is the distance from the heat source to the point in question and ~ is the component of R in the direction of motion. The temperature rise at a dendrite tip can be calculated by summing the contribu-
the latent heat times the rate at which material is transforming from liquid to solid: thus dq = Xy2irrdr. Making these substitutions into eq. (3) and using a normalized coordinate system in which y = Y/r0 and
tions from the moving latent heat sources on the paraboloidal surface.to express temperature differences It is convenient dT on a normalized scale by writing dO = CdT/X, where C is the heat capacity and X is the latent heat of fusion. The moving heat source formula then becomes dq exp [ (y/2a)(~+ R)] dO—~~ R (3)
p = r/r0, we obtain2/2) + [p2 + (p4/4)] 1/2}] exp[ P{(p 2 dO = F— i 4, \1 1/2 p dp, (5) + u i4jj where P is the Peclet number as defined by eq. (1). To find the temperature rise at the tip due to heat emitted from that part of the surface from the tip out to radius p = p, we integrate this expression from
—
‘
using the relation a = k/C. Consider the paraboloidal surface
p=Otop=~: -
p
described by the
f
{exp
[
P {(p2/2) +
[p2 + (p4/4)] 1/2 }] }
p-O
equation Y
X{[p2+(p4/4)]”2}
r 2 /2r 0
(4)
,
where r0 is the radius of curvature of the tip. If we wish calculate the temperature rise at due to theto band of moving heat sources in the the tip region between r and r + dr (see fig. 1), we can note that ~ in eq. (3) corresponds to Yin eq. (4), andR in eq. (3) is equal to (Y2 + r2)”2 = [(r4/4r~) + r2] 1/2
‘pdp.
(6)
Letting 2 + (p4/4)] 1/2} U = P{(p/2) 1 + [p this expression+ becomes
}
P~I(l,2/2)+1+I~2÷(~4/4)1 1/2
M
PePf
(eu/u) du. (7)
Using the definition of the exponential integral,
The rate of heat production in this strip is equal to E
~
/c\
(e u/u)~
1(P)f P
we obtain M =Pe~E1(P) 2/2) + 1 + [~2 +(~o4/4)]1/2])} (8) E1(P[(1ô When p = cc, this reduces to the Ivantsov expression M Fe~E 1(P). The fraction of the tip temperature rise due to heat emitted from the surface out to p = ~ is thus
dp P~r/r
f
2/2)+ 1 —
I
y
2
3
4
5
p
Fig. 1. Paraboloid model of a dendrite tip, showing the tip radius r0 and the normalized coordinatesy and p.
—
E1(P) E1(P{co
+
[~2 +~4/4)]h/2})
E 1(P)
This fraction is plotted in fig. several values ofF. The values
2 as a function of p for of M corresponding to
R.J. Schaefer / Validity of steady-state dendrite growth models
Fig. 2. The fraction f of the tip temperature rise which is attributable to heat emitted within the region p <~, as given by eq. (9).
the selected values of P, according to the Ivantsov model, are also indicated in the figure. At small supercoolings, the shape of dendrite tips can be observed directly. The two tin dendrites analyzed by Glicksman and Schaefer [5] had paraboloidal tips, and Peclet numbers of 2.1 X i0~ and 1.0 X io~.These dendrites show a distinct deviation from the paraboloidal shape for p ~ 3. Fig. 2 indicates that this deviation takes place well within the region which influences the dendrite tip temperature. Similarly, measurements of the succinonitrile dendrites used for the tip radius measurements in (1) indicate that branching starts at approximately p 3. Dendrites growing at bath supercooling ~O ranging from 5 X 10 ~ to 6.5 X 10 2 were studied: there was no indication that the value of p at which branching started varied with supercooling. Fig. 3 shows two succinonitrile dendrites with superimposed parabolas and coordinates axes. Thus it is clear that dendrite branching starts well within the region which influences the dendrite tip, at least for P< 10 2 (corresponding to z~O<0.04) and probably for P< 10 1 (corresponding to M <0.2). Steady.state models which include the Gibbs— Thomson effect, which lowers the tip temperature, predict Peclet numbers only one half as large, at a given bath undercooling, as those predicted by the Ivantsov model used here. This fact does not alter the general nature of the observation: in fact it implies
19
_______
0.0068 2
______
a
—
that at a given M the influence on the tip temperature from large p values is even greater than is estimated here. -
.
.
In conclusion, the calculations, together with observations of dendrite tips in pure materials, mdi-
0.066
b Fig. 3. Succinonitrile dendrites with superimposed parabolas and inscribed tip radius circles. Values of p are indicated: (a) ~O = 0.0068, r 0 = 47 X i0~ cm; (b) ~O = 0.066, r0 = 5.4 X 10 ~ cm.
20
R.J. Schaefer / Validity of steady-state dendrite growth models
cate that the use of steady-state models to describe dendrite growth is highly unrealistic except possibly at very large supercoolings. At the low or moderate normalized supercoolings which prevail in most experimental situations, branching starts well within the region which has a thermal influence on the dendrite tip. Therefore a successful theoretical prediction of the dendrite tip velocity must include a consideration of the side branches, either in a time-averaged steady-state way, or in a fully time-dependent model.
Acknowledgement The dendrites in fig. 3 were photographed during an earlier study [1] with J.D. Ayers and M.E. Glicks-
man. Their permission to use the photographs here is gratefully acknowledged.
References [11ME.
Glicksman, R.J. Schaefer and J.D. Ayers, Met. Trans. 7a (1976) 1747. [21G.P. Ivantsov, Dokl. Akad. Nauk USSR 58 (1947) 56.7. [31T. Fujioka, Doctoral Thesis, Carnegie-Mellon University (1976). [41D. Rosenthal, Trans. Am. Soc. Mech. Engrs. 68 (1946) 849. [5] M.E. Glicksman and R.J. Schaefer, Acta Met. 14 (1966) 1126.