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Effects of crystal-melt interfacial energy anisotropy on dendritic morphology and growth kinetics
277
Journal of Crystal Growth 98 (1989) 277—284 North-Holland, Amsterdam
EFFECTS OF CRYSTAL-MELT INTERFACIAL ENERGY ANISOTROPY ON DENDRITIC MORPHOLOGY AND GROWTh KINETICS M.E. GLICKSMAN Materials Engineering Department, Rensselaer Polytechnic Institute, Troy, New York 12181, USA
and N.B. SINGH Westinghouse R&D Center, 1310 Beulah Road, Pittsburgh, Pennsylvania 15235, USA
Received 10 May 1989; manuscript received in final form 29 June 1989
Morphological and kinetic studies of succinonitrile, a BCC crystal with a low (0.5%) anisotropy and pivalic acid, and FCC crystal with relatively large (5%) anisotropy in solid—liquid interfacial energy, show clearly that anisotropy in the solid—liquid interfacial energy does not affect the tip radius-velocity relationship, but has a profound influence on the tip region and the rate of amplification of branching waves. Anisotropy of the solid-liquid interfacial energy may be one of the key factors by which the microstructural characteristics of cast structures reflect individual material behavior, especially crystal symmetry.
1. Infroduction Careful in situ micromorphological observations during dendritic solidification show that the tip configuration and the branched regions of dendrites display prominent tn-axial anisotropy related to the underlying crystallography of the solid [1—3].Indeed, it has always been recognized that dendrites are crystallographically related growth forms which grow by diffusion a generally isotropic transport process! Clearly, some significant link must exist between the isotropic diffusion of heat and/or solute that controls the development of dendrites and the underlying crystal symmetry. In cubic materials the transport properties such as diffusion coefficient, thermal conductivity and thermal diffusivity are all isotropic, leading to strictly isotropic flow processes, although in non-cubic materials, since these quantities are of a tensorial nature, one might observe anisotropic diffusion. Thus, an even more basic attribute of dendrites must be identified that operates in all materials-independent of the specific —
crystallographic symmetry, which may or may not itself induce anisotropy into the kinetic and morphological behavior via the transport process. The basic scaling relationship that controls the kinetics and morphology of dendritic crystals can be shown to be of the form [4—6] (1 yR2 2acF/a*L =
/
‘
Here V is the tip speed; R is the tip radius of curvature; a is the thermal diffusivity; C and L are the specific and latent heats, respectively; a * is the stability parameter; and F y~/~S,where -y is the solid—liquid surface energy, f~is the molar volume, and z~Sis the molar entropy change on melting. This relationship has been shown to be correct for a few well characterized material systems, where all the thermophysical constants are accurately known [2,3,7], and for strictly isotropic systems studied by computer simulation [8]. The results of these studies show that eq. (1) is obeyed for values of a * 0.02, which values appear to be independent of factors not appearing explicitly on the right-hand side of eq. (1). In fact, theoretical
0022-0248/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
=
278
ME. Glicksman, NB. Sing/i
/
Effects of crystal—melt interfacial energy anisotropy
Stability model
Ref.
precise value of a * can explain neither the crystallographic nature of dendrite systems, nor their individual characteristics. The only quantity ap-
0.02 0.0253 0.025 0.0192 * (exp)
Oldfield’s “force balance” Planar front Parabolic eigenstate Spherical harmonic (/ = 6) System
[8] [2,9] [5,10,11] [2—15] Ref.
peaning in eq. (1) which can reflect elements of the underlying crystal symmetry is the solid—liquid surface energy, y. The purpose of this paper is to show the current status of our knowledge of such anisotropy effects and their dependence on the
0.0195 0.022
SCN PVA
[9] This work
crystallographic variation of y.
Table 1 Values of the stability parameter a
*
for dendntes
2. Crystallography of dendrites estimates of the stability parameter, obtained by using a variety of morphological stability models agree quite well with experiment. Table 1 provides an overall comparison of experimental and theoretical a * values, A perusal of table I shows that a* is not a “universal” stability constant, but varies about 30%, depending on the theoretical stability model ro the specific material system. Nonetheless, the
It is well recognized that cubic materials, such as FCC and BCC metals with non-facetting solid—liquid interfaces, grow in the (100) cube directions. Fig. 1 compares the overall dendritic morphologies for two cubic materials: (A) pivalic acid (PVA), an FCC crystal, and (B) succinonitrile (SCN), a BCC crystal. These materials are organic plastic crystals which have freezing characteristics
4~~1J~ Jr
-w~=—
I
_ I300~I
IJ -II_I~
-
__ ___
Fig. 1. Dendrite morphologics for FCC (a) pisalic acid (PVA) and Ili) BCC succinonitrik (SCN.
ME. (j!ickcman. NB. Singh / Effeet3 of crystal—melt interim ml energy
279
anivotropi
b
Ut
to mm
r
1’
_
_
_
I
____
C ~
b
—
_ ______
Fig. 2. (a)—(c) Tip shape anisotropy for PVA (top row) and SCN (bottom row) dendrite tips. (d) Schematic of four-fold distortions around primar~ <100> growth axis. showing the corresponding viewing orientations (a) < 100>. (hI 22.5*, and (C) 110>. for each crsstal.
280
M.E. Glicksman, NB. Singh
/ Effects
[PVA (c), SCN (c)] show that PVA dendrites distort faster and more severely from axisymmetry
Table 2 Solid—liquid interfacial energies and anisotropy factors 2)
System
Symmetry
‘y
PVA SCN
FCC BCC
2.8 8.9
of crystal—melt interfacial energy anisotropy
Y4/Y
0 (erg/cm
0
0.05
0.005
than do SCN dendrites.
L/C (K) 11.1 23.1
_________________________________________
akin to metals with comparable crystal structures. As shown in figs. la and ib, a four-fold distortion spreads out from the nearly axisymmetric tip of both dendrites, although, as shown later, the PVA crystal exhibits a much more exaggerated four-fold shape than does the SCN crystal. Also, it is noted that the side branches grow along the four (100) branching “sheets”, initially developing as regularly spaced oscillations along the edges. The branches amplify much more quickly in the SCN dendrite than they do in the PVA, and tend to grow initially in directions tilted slightly toward the primary growth direction. By contrast, PVA dendrites have relatively long, smooth tips with the side branches growing almost immediately in the <100> direction orthogonal to the primary growth axis (cf. Figs. la and ib). Fig. 2d provides a schematic diagram that shows the tip micromorphologies in each of three viewing orientations: PVA (a) and SCN (a), <100>; PVA (b) and SCN (b), 22.50; PVA (c) and SCN (c), <110>. The <100> viewing orientations [PVA (a), SCN (a)], clearly show that side branching starts much closer to the tip in SCN dendrites than in PVA dendnites, whereas the <110> viewing orientations
3. Anisotropy As discussed in the Introduction, the rationalization of the anisotropic micromorphologies shown in figs. 1 and 2 logically resides in the behavior of -y. Experiments were performed in which droplets of molten SCN and PVA were trapped within their respective solids and allowed to equilibrate. By virtue of the fact that the captive droplets had slightly higher impurity contents than did the sunrounding crystals, they could be held almost indefinitely at temperatures a few millikelvins below the respective solidus temperatures. At such relatively high temperatures, the droplets rapidly reach thermodynamic equilibrium. Thermodynamic equilibrium occurs when the droplet shape at fixed volume adjusts to permit the chemical potential at the solid—liquid interface to become uniform at all points. For a general, i.e., anisotropy system, the chemical potential .t y + y”, where y” is the second angular derivative of the solid—liquid interfacial energy [16]. Procedures for calculating the equilibrium droplet shape for a given angular vanation of y, or conversely, determining the y-plot from the equilibrium shape of the droplet are described elsewhere [17—19]. Figs. 3a and 3b show droplets of PVA and SCN in their respective equilibrium forms on-
‘I,
/
S
~
X -.
I~
C
--.
—
Fig. 3. Equilibrium droplet shapes showing small anisotropies in the solid—liquid surface tension: (a) SCN with weak 0.5% anisotropy; (b) PVA with moderate 5% anisotropy; (c) schematic of principal droplet profile showing four-fold distortions in <100>
directions.
M.E. Glicksman, NB. Singh
/ Effects
ented relative to the crystal axes. Fig. 3c is a schematic diagram of a principal cross-section showing the four-fold distortions in the droplet radius, r(0), compared to an (isotropic) circle of radius Application of the methods outlined in ref. [19] shows that the polar -y-plot, y(0), mimics the polar variation of r(0) except for a scale factor. If the droplet radius r(0) varies around a four-fold <100> axis as ,~.
of crystal—melt interfacial energy anisotropy
281
lack an underlying tn-axial anisotropy, cannot form dendrites. Presumably, all crystalline solids will have some degree of anisotropy in the interface energy, although as shown in table 2, the amount may be almost undetectably small.
4. Scaling laws The basic relationships among the kinetic and
r(0)
=
r
0
+
r4 cos 40
+
higher order terms,
(2)
where r4 is the amplitude of the four-fold distortion, then the minimum ratio of radius the maximum r() to the r() is radius r()/r(<1I0>)
=
morphological properties of dendrites and the process and material parameters devolve from the 2 constant. If the operating states of operating statemeasured equation at(cf. eq. (1)), which predendrites various supercoolings, dicts yR are =
1 + 2r
4/r~.
(3)
then pains of V—R data are obtained at various
The corresponding ratios for Y4/YO can be obtained by solving eq. (3) for r4/t~ for an observed
____________ IC-’
droplet. Here Y4/YO may be considered to be the anisotropy in the principal plane, and Yo is the modulus or mean interfacial energy. Table 2 shows two systems for which ‘y0 and y4 are known accurately. Although y4/y0 is itself a small quantity, it varied by a factor of 10 in SCN and PVA. Also, the magnitude of y~seems to have no relationship to y4, since the high-anisotropy system, PVA, has only about one-third of the mean interfacial energy of the more isotropic SCN. Anisotropies of only 0.005 are apparently enough to select the preferred growth directions, viz. <100>, in cubic materials under realistic crystal growth conditions. The mechanism for selecting the growth dinection relies on localizing the orientations of maximum y at the tip. Any tendency for the dendrite to veer off the direction of maximum y results in a “force” which tends to steer the tip back toward the preferred direction. The steering “force” results from the extra interfacial energy which is produced whenever growth occurs in directions other than the preferred <100>. Thus a type of feedback occurs in which weak anisotropies of the interfacial energy exert the controlling forces during growth. In this manner, we believe, dendrites can “lock” onto directions indefinitely large crystallographic distances. A corollary of over this mechanism, is that non-crystalline solids, which
0-a
~ ~ ~
...••.~.
Io~
Io_’
Tip Radius (cm) 2 constant for pivalic acid (PVA). Slope of Fig. 4. Velocity versus tip radius (log-scales) testing the scaling --2 shows expected correlations among dendritic operating relationship VR states.
~IJf
282
M.E. Glicksman, N.B. Singh
___
,.~T 3~53
L~T~ 0 276
/
Effects of crystal—melt interfacial energy anisotropy
1
~Tz(J1~i~
~T~0 325
~T~O142
~T~O16
L~T~0414
..~T 0532
Fig. 5. Influence of supercooling, ~1T (in K) on dendrite size scale for PVA. All micrographs are displayed at constant magnification to show reciprocal relationship between supercooling and morphological size scale.
levels of supercooling. A log-log plot of the operating state equation should yield a straight line with a slope of —2. Fig. 4 shows a log-log plot of tip speed data, V. versus tip radius data, R, for PVA. The requires scaling behavior is nicely demonstrated here over almost two decades in the velocity parameter. Fig. 5 shows how the dendrite tip configuration changes in response to increasing levels of supercooling in PVA. The micrographs
supercooling, and as indicated there is a comparable reduction in size scale which is in proportion to ~T The theoretically predicted scaling law for the steady-state dendrite tip radius is d 1 R (4) a where D0 FC/L is referred to as the capillary
arranged in fig. 5 cover a 10-fold increase in
length [4], and P
=
—~-
~,
VR/2a is the Péclet number,
ME. Glicksman, N.B. Singh
—
io’
/ Effects
Eq. (8) shows that vd0/2a is independent of material properties and depends only on supercooling. Fig. 7 is a log-log plot of vd0/2a versus iA9 for PVA and SCN, and confirms this scaling relationship as independent of the surface tension anisotropy operating in the crystal growth system. It is clear that the morphological and kinetic data merge with their respective theory line [5], espe-
*
‘N.
‘N.
-
283
of crystal—melt interfacial energy anisotropy
cially at the higher supercoolings where convection processes are generally of less importance and therefore do not appreciably alter the (diffusive)
£
transport fields that surround the dendrites.
I
________
02
ic
DIMENSIONLESS SUPER000LING (~T/(L/C))
1
i0~
Fig. 6. Tip radius (normalized to capillary length, d0) versus dimensionless supercooling (log scales) for PVA and SCN. Note that data tend to merge with theory line particularly at larger supercoolings where thermal convection is relatively unimportant compared with diffusive transport of latent heat.
~/
J L
which is a known function of the dimensionless supercooling, ~A9. Specifically, for paraboloidal dendrite tips Ivantsov has shown [20] that
~
L~I=Pe”Ei(P),
I>
(5)
where Ei(P) is the first exponential integral, which is a tabulated function [22]. If eq. (5) is inverted to find P(z~e)and substituted into eq. (4), we see that
~i
R/do=[P(~O)]~/a*,
~
(6)
so that R/d0 should be a function essentially independent of material properties and dependent only on the supercooling. Fig. 6 shows a log-log plot of R/2d0 versus ~ for PVA and SCN. The velocity of a steady-state dendrite can be predicted from eqs. (1) and (4) when rearranged as 2/d v = 2aa*Pe
=
a
*
~
A
A
A
~ ~ ~
ii5-s
0.
(7)
If we substitute the value of Pe into eq. (7) as determined from the transport equation (5), we find vd0/2a
/
4
~..
P2
.
(8)
io~
io~
Io’
DIMENSIONLESS SUPERCOOLING (AT/IL/C))
Fig. 7. Growth velocity (normalized to the ratio of thermal diffusivity and capillary length, 2a/d0) versus dimensionless supercooling (log scales). Data for both materials (PVA and SCN) lie onratio the of same predicted eq. (8) despite large the line surface tensionfrom anisotropies.
284
ME. Glicksman, N.B. Singh
/
Effects of crystal—melt interfacial energy anisotropy
It is particularly interesting to note here that the anisotropy of the solid—liquid interfacial energies, which differ so markedly for PVA and SCN, plays little if any role in the scaling behavior of the tip radii and axial tip speeds.
Acknowledgements The authors are grateful to Ms. D. Todd for preparing the subject matter. The support provided by Microgravity Science and Application Division, NASA Code EN, under contract NAS 3-25 368 is sincerely acknowledged. Authors thank —
5. Conclusions
Dr. R. Mazelsky and R. Hopkins for encouragements.
The first detailed studies have been performed to compare the influences on dendritic crystal growth of the absolute values of the solid—liquid interfacial energies and their angular variations within a BCC material (SCN) and an FCC material (PVA). The anisotropies in PVA and SCN differ by an order of magnitude, and appear to cause marked differences in the crystallographic details of the dendrites and their ultimate cast microstructures The operating states of dendrites are only slightly affected by the anisotropy in the solid—liquid interfacial energies. The stability parameter, a seems to change from material to material, perhaps in response to the anisotropy, but by amounts that are less than 10—15%. The theoretical models themselves used for estimating a * by linear perturbation techniques lead to vanations of about 30% in a*, so it is difficult at this time to decide what determines the precise value of the stability parameter. More work is needed here to achieve a deeper understanding of the operating states of dendnitic crystals, even in simple crystal growth systems. Scaling laws derived from morphological stability and transport theory work well in those few systems for which the thermophysical properties are well known and for which quantitative kinetics and morphological measurements are available. These properties include thermal and chemical diffusivities, specific and latent heats, entropy of melting, and, most difficult of all to measure, the solid—liquid interfacial energy, which is only roughly estimated for most crystal—melt systems. More materials must be investigated to confirm and delimit the applicability of the known scaling laws to a wider variety of pure materials and their binary alloys, as well as to develop new ones with yet broader utility in crystal growth sciences and practice. .
~,
References [1] M E. Glicksman, R.J. Schaefer and Di. Ayers, Met. Trans. .47 (1976) 1747. [2] S.C. Huang and M.E. Glicksman, Acta Met. 29 (1981) 701. [3] U. Lappe, KFA Report, Kernforschungsanlage Julich, FRG (1980). [4] iS. Langer and H. Miiller-Krumbhaar, Acta Met 26 (1978) 26. [5] J.S. Langer and H. Muller-Krumbhaar, Acta Met. 26 (1978) 1689. [6] J.S. Langer and H. MUller-Krumbhaar, Acta Met. 26 1697. PhD Thesis, Carnegie-Mellon University [7] (1978) T. Fujioka, (1978). [8] W. Oldfield, Mater. Sci. Eng. 11(1973) 211. [9] W.W. Mullins and R.F. Sekerka, J. Appl. Phys. 34 (1963) 323. [10] J.S. Langer and H. MUller-Krumbhaar, J. Crystal Growth 42 (1977) 11. [11] J.S. Langer, Rev. Mod. Phys. 52 (1980) 1. [12] S.R. Coriell and R.L. Parker, J. App1. Phys. 36 (1965) 632. [13] S.R. Coriell and R.L. Parker, in: Crystal Growth, Ed. H.S. Peiser (Pergamon, Oxford, 1967) p. 703. [14] R. Trivedi, H. Fronke and R. Lacmann. J. Crystal Growth 47 (1979) 389. [15] RD. Doherty, B. Cantor and S. Fairs, Met. Trans. A9 (1978) 621. [16] W.W. Mullins, in: Metal Surfaces (American Society for Metals, Metals Park. OH, 1963) ch. 2, p. 17. [17] Conyers Herring, Phys. Rev. 82 (1951) 87. [18] W.W. Mullins, J. Math. Phys. 3 (1962) 3. [19] J.W. Calm and D.W. Hoffman, Acta Met. 22 (1974) 1205. [20] G.P. Ivantsov, Doki. Akad. Nauk SSSR 58 (1947) 567. [21] G. Horvay and J.W. Cahn. Acta Met. 9 (1961) 695. [22] M. Abramowitz and l.A.Appi. Stegun. of Mathematical Functions, Math.Eds., Ser. Handbook 55 (NatI. Bur. Std. (US), Washington, DC, 1965) p. 238.
Journal of Crystal Growth 98 (1989) 277—284 North-Holland, Amsterdam
EFFECTS OF CRYSTAL-MELT INTERFACIAL ENERGY ANISOTROPY ON DENDRITIC MORPHOLOGY AND GROWTh KINETICS M.E. GLICKSMAN Materials Engineering Department, Rensselaer Polytechnic Institute, Troy, New York 12181, USA
and N.B. SINGH Westinghouse R&D Center, 1310 Beulah Road, Pittsburgh, Pennsylvania 15235, USA
Received 10 May 1989; manuscript received in final form 29 June 1989
Morphological and kinetic studies of succinonitrile, a BCC crystal with a low (0.5%) anisotropy and pivalic acid, and FCC crystal with relatively large (5%) anisotropy in solid—liquid interfacial energy, show clearly that anisotropy in the solid—liquid interfacial energy does not affect the tip radius-velocity relationship, but has a profound influence on the tip region and the rate of amplification of branching waves. Anisotropy of the solid-liquid interfacial energy may be one of the key factors by which the microstructural characteristics of cast structures reflect individual material behavior, especially crystal symmetry.
1. Infroduction Careful in situ micromorphological observations during dendritic solidification show that the tip configuration and the branched regions of dendrites display prominent tn-axial anisotropy related to the underlying crystallography of the solid [1—3].Indeed, it has always been recognized that dendrites are crystallographically related growth forms which grow by diffusion a generally isotropic transport process! Clearly, some significant link must exist between the isotropic diffusion of heat and/or solute that controls the development of dendrites and the underlying crystal symmetry. In cubic materials the transport properties such as diffusion coefficient, thermal conductivity and thermal diffusivity are all isotropic, leading to strictly isotropic flow processes, although in non-cubic materials, since these quantities are of a tensorial nature, one might observe anisotropic diffusion. Thus, an even more basic attribute of dendrites must be identified that operates in all materials-independent of the specific —
crystallographic symmetry, which may or may not itself induce anisotropy into the kinetic and morphological behavior via the transport process. The basic scaling relationship that controls the kinetics and morphology of dendritic crystals can be shown to be of the form [4—6] (1 yR2 2acF/a*L =
/
‘
Here V is the tip speed; R is the tip radius of curvature; a is the thermal diffusivity; C and L are the specific and latent heats, respectively; a * is the stability parameter; and F y~/~S,where -y is the solid—liquid surface energy, f~is the molar volume, and z~Sis the molar entropy change on melting. This relationship has been shown to be correct for a few well characterized material systems, where all the thermophysical constants are accurately known [2,3,7], and for strictly isotropic systems studied by computer simulation [8]. The results of these studies show that eq. (1) is obeyed for values of a * 0.02, which values appear to be independent of factors not appearing explicitly on the right-hand side of eq. (1). In fact, theoretical
0022-0248/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
=
278
ME. Glicksman, NB. Sing/i
/
Effects of crystal—melt interfacial energy anisotropy
Stability model
Ref.
precise value of a * can explain neither the crystallographic nature of dendrite systems, nor their individual characteristics. The only quantity ap-
0.02 0.0253 0.025 0.0192 * (exp)
Oldfield’s “force balance” Planar front Parabolic eigenstate Spherical harmonic (/ = 6) System
[8] [2,9] [5,10,11] [2—15] Ref.
peaning in eq. (1) which can reflect elements of the underlying crystal symmetry is the solid—liquid surface energy, y. The purpose of this paper is to show the current status of our knowledge of such anisotropy effects and their dependence on the
0.0195 0.022
SCN PVA
[9] This work
crystallographic variation of y.
Table 1 Values of the stability parameter a
*
for dendntes
2. Crystallography of dendrites estimates of the stability parameter, obtained by using a variety of morphological stability models agree quite well with experiment. Table 1 provides an overall comparison of experimental and theoretical a * values, A perusal of table I shows that a* is not a “universal” stability constant, but varies about 30%, depending on the theoretical stability model ro the specific material system. Nonetheless, the
It is well recognized that cubic materials, such as FCC and BCC metals with non-facetting solid—liquid interfaces, grow in the (100) cube directions. Fig. 1 compares the overall dendritic morphologies for two cubic materials: (A) pivalic acid (PVA), an FCC crystal, and (B) succinonitrile (SCN), a BCC crystal. These materials are organic plastic crystals which have freezing characteristics
4~~1J~ Jr
-w~=—
I
_ I300~I
IJ -II_I~
-
__ ___
Fig. 1. Dendrite morphologics for FCC (a) pisalic acid (PVA) and Ili) BCC succinonitrik (SCN.
ME. (j!ickcman. NB. Singh / Effeet3 of crystal—melt interim ml energy
279
anivotropi
b
Ut
to mm
r
1’
_
_
_
I
____
C ~
b
—
_ ______
Fig. 2. (a)—(c) Tip shape anisotropy for PVA (top row) and SCN (bottom row) dendrite tips. (d) Schematic of four-fold distortions around primar~ <100> growth axis. showing the corresponding viewing orientations (a) < 100>. (hI 22.5*, and (C) 110>. for each crsstal.
280
M.E. Glicksman, NB. Singh
/ Effects
[PVA (c), SCN (c)] show that PVA dendrites distort faster and more severely from axisymmetry
Table 2 Solid—liquid interfacial energies and anisotropy factors 2)
System
Symmetry
‘y
PVA SCN
FCC BCC
2.8 8.9
of crystal—melt interfacial energy anisotropy
Y4/Y
0 (erg/cm
0
0.05
0.005
than do SCN dendrites.
L/C (K) 11.1 23.1
_________________________________________
akin to metals with comparable crystal structures. As shown in figs. la and ib, a four-fold distortion spreads out from the nearly axisymmetric tip of both dendrites, although, as shown later, the PVA crystal exhibits a much more exaggerated four-fold shape than does the SCN crystal. Also, it is noted that the side branches grow along the four (100) branching “sheets”, initially developing as regularly spaced oscillations along the edges. The branches amplify much more quickly in the SCN dendrite than they do in the PVA, and tend to grow initially in directions tilted slightly toward the primary growth direction. By contrast, PVA dendrites have relatively long, smooth tips with the side branches growing almost immediately in the <100> direction orthogonal to the primary growth axis (cf. Figs. la and ib). Fig. 2d provides a schematic diagram that shows the tip micromorphologies in each of three viewing orientations: PVA (a) and SCN (a), <100>; PVA (b) and SCN (b), 22.50; PVA (c) and SCN (c), <110>. The <100> viewing orientations [PVA (a), SCN (a)], clearly show that side branching starts much closer to the tip in SCN dendrites than in PVA dendnites, whereas the <110> viewing orientations
3. Anisotropy As discussed in the Introduction, the rationalization of the anisotropic micromorphologies shown in figs. 1 and 2 logically resides in the behavior of -y. Experiments were performed in which droplets of molten SCN and PVA were trapped within their respective solids and allowed to equilibrate. By virtue of the fact that the captive droplets had slightly higher impurity contents than did the sunrounding crystals, they could be held almost indefinitely at temperatures a few millikelvins below the respective solidus temperatures. At such relatively high temperatures, the droplets rapidly reach thermodynamic equilibrium. Thermodynamic equilibrium occurs when the droplet shape at fixed volume adjusts to permit the chemical potential at the solid—liquid interface to become uniform at all points. For a general, i.e., anisotropy system, the chemical potential .t y + y”, where y” is the second angular derivative of the solid—liquid interfacial energy [16]. Procedures for calculating the equilibrium droplet shape for a given angular vanation of y, or conversely, determining the y-plot from the equilibrium shape of the droplet are described elsewhere [17—19]. Figs. 3a and 3b show droplets of PVA and SCN in their respective equilibrium forms on-
‘I,
/
S
~
X -.
I~
C
--.
—
Fig. 3. Equilibrium droplet shapes showing small anisotropies in the solid—liquid surface tension: (a) SCN with weak 0.5% anisotropy; (b) PVA with moderate 5% anisotropy; (c) schematic of principal droplet profile showing four-fold distortions in <100>
directions.
M.E. Glicksman, NB. Singh
/ Effects
ented relative to the crystal axes. Fig. 3c is a schematic diagram of a principal cross-section showing the four-fold distortions in the droplet radius, r(0), compared to an (isotropic) circle of radius Application of the methods outlined in ref. [19] shows that the polar -y-plot, y(0), mimics the polar variation of r(0) except for a scale factor. If the droplet radius r(0) varies around a four-fold <100> axis as ,~.
of crystal—melt interfacial energy anisotropy
281
lack an underlying tn-axial anisotropy, cannot form dendrites. Presumably, all crystalline solids will have some degree of anisotropy in the interface energy, although as shown in table 2, the amount may be almost undetectably small.
4. Scaling laws The basic relationships among the kinetic and
r(0)
=
r
0
+
r4 cos 40
+
higher order terms,
(2)
where r4 is the amplitude of the four-fold distortion, then the minimum ratio of radius the maximum r(
=
morphological properties of dendrites and the process and material parameters devolve from the 2 constant. If the operating states of operating statemeasured equation at(cf. eq. (1)), which predendrites various supercoolings, dicts yR are =
1 + 2r
4/r~.
(3)
then pains of V—R data are obtained at various
The corresponding ratios for Y4/YO can be obtained by solving eq. (3) for r4/t~ for an observed
____________ IC-’
droplet. Here Y4/YO may be considered to be the anisotropy in the principal plane, and Yo is the modulus or mean interfacial energy. Table 2 shows two systems for which ‘y0 and y4 are known accurately. Although y4/y0 is itself a small quantity, it varied by a factor of 10 in SCN and PVA. Also, the magnitude of y~seems to have no relationship to y4, since the high-anisotropy system, PVA, has only about one-third of the mean interfacial energy of the more isotropic SCN. Anisotropies of only 0.005 are apparently enough to select the preferred growth directions, viz. <100>, in cubic materials under realistic crystal growth conditions. The mechanism for selecting the growth dinection relies on localizing the orientations of maximum y at the tip. Any tendency for the dendrite to veer off the direction of maximum y results in a “force” which tends to steer the tip back toward the preferred direction. The steering “force” results from the extra interfacial energy which is produced whenever growth occurs in directions other than the preferred <100>. Thus a type of feedback occurs in which weak anisotropies of the interfacial energy exert the controlling forces during growth. In this manner, we believe, dendrites can “lock” onto directions indefinitely large crystallographic distances. A corollary of over this mechanism, is that non-crystalline solids, which
0-a
~ ~ ~
...••.~.
Io~
Io_’
Tip Radius (cm) 2 constant for pivalic acid (PVA). Slope of Fig. 4. Velocity versus tip radius (log-scales) testing the scaling --2 shows expected correlations among dendritic operating relationship VR states.
~IJf
282
M.E. Glicksman, N.B. Singh
___
,.~T 3~53
L~T~ 0 276
/
Effects of crystal—melt interfacial energy anisotropy
1
~Tz(J1~i~
~T~0 325
~T~O142
~T~O16
L~T~0414
..~T 0532
Fig. 5. Influence of supercooling, ~1T (in K) on dendrite size scale for PVA. All micrographs are displayed at constant magnification to show reciprocal relationship between supercooling and morphological size scale.
levels of supercooling. A log-log plot of the operating state equation should yield a straight line with a slope of —2. Fig. 4 shows a log-log plot of tip speed data, V. versus tip radius data, R, for PVA. The requires scaling behavior is nicely demonstrated here over almost two decades in the velocity parameter. Fig. 5 shows how the dendrite tip configuration changes in response to increasing levels of supercooling in PVA. The micrographs
supercooling, and as indicated there is a comparable reduction in size scale which is in proportion to ~T The theoretically predicted scaling law for the steady-state dendrite tip radius is d 1 R (4) a where D0 FC/L is referred to as the capillary
arranged in fig. 5 cover a 10-fold increase in
length [4], and P
=
—~-
~,
VR/2a is the Péclet number,
ME. Glicksman, N.B. Singh
—
io’
/ Effects
Eq. (8) shows that vd0/2a is independent of material properties and depends only on supercooling. Fig. 7 is a log-log plot of vd0/2a versus iA9 for PVA and SCN, and confirms this scaling relationship as independent of the surface tension anisotropy operating in the crystal growth system. It is clear that the morphological and kinetic data merge with their respective theory line [5], espe-
*
‘N.
‘N.
-
283
of crystal—melt interfacial energy anisotropy
cially at the higher supercoolings where convection processes are generally of less importance and therefore do not appreciably alter the (diffusive)
£
transport fields that surround the dendrites.
I
________
02
ic
DIMENSIONLESS SUPER000LING (~T/(L/C))
1
i0~
Fig. 6. Tip radius (normalized to capillary length, d0) versus dimensionless supercooling (log scales) for PVA and SCN. Note that data tend to merge with theory line particularly at larger supercoolings where thermal convection is relatively unimportant compared with diffusive transport of latent heat.
~/
J L
which is a known function of the dimensionless supercooling, ~A9. Specifically, for paraboloidal dendrite tips Ivantsov has shown [20] that
~
L~I=Pe”Ei(P),
I>
(5)
where Ei(P) is the first exponential integral, which is a tabulated function [22]. If eq. (5) is inverted to find P(z~e)and substituted into eq. (4), we see that
~i
R/do=[P(~O)]~/a*,
~
(6)
so that R/d0 should be a function essentially independent of material properties and dependent only on the supercooling. Fig. 6 shows a log-log plot of R/2d0 versus ~ for PVA and SCN. The velocity of a steady-state dendrite can be predicted from eqs. (1) and (4) when rearranged as 2/d v = 2aa*Pe
=
a
*
~
A
A
A
~ ~ ~
ii5-s
0.
(7)
If we substitute the value of Pe into eq. (7) as determined from the transport equation (5), we find vd0/2a
/
4
~..
P2
.
(8)
io~
io~
Io’
DIMENSIONLESS SUPERCOOLING (AT/IL/C))
Fig. 7. Growth velocity (normalized to the ratio of thermal diffusivity and capillary length, 2a/d0) versus dimensionless supercooling (log scales). Data for both materials (PVA and SCN) lie onratio the of same predicted eq. (8) despite large the line surface tensionfrom anisotropies.
284
ME. Glicksman, N.B. Singh
/
Effects of crystal—melt interfacial energy anisotropy
It is particularly interesting to note here that the anisotropy of the solid—liquid interfacial energies, which differ so markedly for PVA and SCN, plays little if any role in the scaling behavior of the tip radii and axial tip speeds.
Acknowledgements The authors are grateful to Ms. D. Todd for preparing the subject matter. The support provided by Microgravity Science and Application Division, NASA Code EN, under contract NAS 3-25 368 is sincerely acknowledged. Authors thank —
5. Conclusions
Dr. R. Mazelsky and R. Hopkins for encouragements.
The first detailed studies have been performed to compare the influences on dendritic crystal growth of the absolute values of the solid—liquid interfacial energies and their angular variations within a BCC material (SCN) and an FCC material (PVA). The anisotropies in PVA and SCN differ by an order of magnitude, and appear to cause marked differences in the crystallographic details of the dendrites and their ultimate cast microstructures The operating states of dendrites are only slightly affected by the anisotropy in the solid—liquid interfacial energies. The stability parameter, a seems to change from material to material, perhaps in response to the anisotropy, but by amounts that are less than 10—15%. The theoretical models themselves used for estimating a * by linear perturbation techniques lead to vanations of about 30% in a*, so it is difficult at this time to decide what determines the precise value of the stability parameter. More work is needed here to achieve a deeper understanding of the operating states of dendnitic crystals, even in simple crystal growth systems. Scaling laws derived from morphological stability and transport theory work well in those few systems for which the thermophysical properties are well known and for which quantitative kinetics and morphological measurements are available. These properties include thermal and chemical diffusivities, specific and latent heats, entropy of melting, and, most difficult of all to measure, the solid—liquid interfacial energy, which is only roughly estimated for most crystal—melt systems. More materials must be investigated to confirm and delimit the applicability of the known scaling laws to a wider variety of pure materials and their binary alloys, as well as to develop new ones with yet broader utility in crystal growth sciences and practice. .
~,
References [1] M E. Glicksman, R.J. Schaefer and Di. Ayers, Met. Trans. .47 (1976) 1747. [2] S.C. Huang and M.E. Glicksman, Acta Met. 29 (1981) 701. [3] U. Lappe, KFA Report, Kernforschungsanlage Julich, FRG (1980). [4] iS. Langer and H. Miiller-Krumbhaar, Acta Met 26 (1978) 26. [5] J.S. Langer and H. Muller-Krumbhaar, Acta Met. 26 (1978) 1689. [6] J.S. Langer and H. MUller-Krumbhaar, Acta Met. 26 1697. PhD Thesis, Carnegie-Mellon University [7] (1978) T. Fujioka, (1978). [8] W. Oldfield, Mater. Sci. Eng. 11(1973) 211. [9] W.W. Mullins and R.F. Sekerka, J. Appl. Phys. 34 (1963) 323. [10] J.S. Langer and H. MUller-Krumbhaar, J. Crystal Growth 42 (1977) 11. [11] J.S. Langer, Rev. Mod. Phys. 52 (1980) 1. [12] S.R. Coriell and R.L. Parker, J. App1. Phys. 36 (1965) 632. [13] S.R. Coriell and R.L. Parker, in: Crystal Growth, Ed. H.S. Peiser (Pergamon, Oxford, 1967) p. 703. [14] R. Trivedi, H. Fronke and R. Lacmann. J. Crystal Growth 47 (1979) 389. [15] RD. Doherty, B. Cantor and S. Fairs, Met. Trans. A9 (1978) 621. [16] W.W. Mullins, in: Metal Surfaces (American Society for Metals, Metals Park. OH, 1963) ch. 2, p. 17. [17] Conyers Herring, Phys. Rev. 82 (1951) 87. [18] W.W. Mullins, J. Math. Phys. 3 (1962) 3. [19] J.W. Calm and D.W. Hoffman, Acta Met. 22 (1974) 1205. [20] G.P. Ivantsov, Doki. Akad. Nauk SSSR 58 (1947) 567. [21] G. Horvay and J.W. Cahn. Acta Met. 9 (1961) 695. [22] M. Abramowitz and l.A.Appi. Stegun. of Mathematical Functions, Math.Eds., Ser. Handbook 55 (NatI. Bur. Std. (US), Washington, DC, 1965) p. 238.