- Email: [email protected]
On the dendritic growth of pure materials
603
Journal of Crystal Growth 3, 4 (1968) 603—6 10 © North-Holland Publishing Co., Amsterdam
ON THE DENDRITIC GROWTH OF PURE MATERIALS
G. R. KOTLER Ford Scientific Laboratory, Dearborn, Michigan 48121, U.S.A. and L. A. TARSHIS General Electric Research and Development Center, Schenectady, New York 12301, U.S.A.
The following three theoretical treatments of unconstrained dendritic growth are discussed: (i) The Ivantsov analysis, which assumes the existence of an isothermal liquid—solid interface and concludes that this dendrite body shape must be a paraboloid of revolution; (ii) and (iii) the non-isothermal treatments of both Boiling and Tiller, and Temkin. These latter two works take account of theover variation in both curvature interface attachment kinetics the entire surface of theand dendrite. They also
tip by assuming that the tip region can effectively be described as a paraboloid of revolution. It is proposed that Temkin’s analysis is sensitive enough to determine the excess solid—liquid surface free energy, y, and the linear kinetic coefficient for molecular attachment, ~. Using the best available data for the ice-water system, the Temkin treat2 and 16 + 0.5 cm/sec °C for y and ment yields 20 + 2 erg/cm ~z
enable one to calculate the velocity and curvature of the dendrite
0,respectively. These values are in remarkably good agreement with other experiments and theory.
1. Introduction
COORDINATE TRANSFORMATION 2-~2)/2
/(z/p~O, r/p~O) Z/p -
rip
a’4
6
6
3
2. Review*
3 7
2
2 I
In order to fully appreciate the discussion which follows, we introduce the parabolic coordinate system (cc /3, 0). These are related to polar coordinates (r, 0, z), as shown in fig. 1. In plotting the polar coordinates, we have introduced the radius of curvature, ~ of a parabolic tip for reasons which shall become obvious. The 0 coordinate is identical in both systems and can be obtained by rotation about the zip (~x= /3 = 0) axis. Ivantsov analysis In 1947, G. P. Ivantsov’) of the Soviet Union made the first serious attempt at describing dendritic growth. (i)
*
(a~O,~O)
zip (a
In this paper we briefly review the existing theoretical descriptions of one aspect of dendritic growth and attempt to show how one of the theories can effectively be used to predict experimental observations. As a result, we are able to extract numerical values of two elusive quantities: y, the excess solid—liquid surface free energy and /10, the linear kinetic coefficient of solidification, in the limited cases where there exists applicable data.
A more detailed review of the existing theories is available20),
XIV
-
LI
a~O
7
1 -
z/p~/2-I/2(r/p)2
Fig. 1.
Graphical definition of circular-parabolic and polar coordinates.
Using the suggestion of Papapetrou2) that the tip region of a dendrite trunk may be adequately represented by a paraboloid of revolution, Ivantsov was able to determine the steady-state temperature field in the liquid surrounding such a body shape. He assumed that the dendrite was isothermal and advancing at constant velocity, V, in the z direction, into an iso-
604
G. R. KOTLER AND L. A. TARSHIS
thermal, undercooled bath at temperature T~.For a given bath undercooling, AT, a unique value for the product Vp can be determined by this analysis. However, a paradoxical feature arises owing to the result that (for each AT) the product Vp must be constant. Thus, one has an infinite set of pairs, (V, p), which satisfy the Ivantsov model. Since one would like to assume that nature provides a unique set (V*, p*), we seek some additional condition in the problem which would allow the determination of V* and p’1’. However, it is not apparent what this condition might be. Commonly, a value of p which optimizes V would be sought but this is impossible since Vp = constant. This apparent inconsistency can be traced to Ivantsov’s assumption that dendritic growth is controlled uniquely by thermal diffusion. There are at least two other phenomena that must be simultaneously considered in any one-component phase transformation. These are, the kinetics of molecular attachment and the effect of localized curvature on the equilibrium transformation temperature. In 1960 D. E. Temkin 3 ), also of the Soviet Union, recognized these limitations and published an elegant mathematical description of the needle crystal (dendrite trunk) growing into a pure, undercooled melt.
isothermal dendrite. No attempt was made to justify this assumption, but rather it was considered to be a questionable first approximation to the complex total problem. Although by no means conclusive, we do see some justification for making this critical mathematical simplification. To best understand these arguments, we shall reword the assumption in the following manner. Recalling that our purpose is to predict the growth conditions (V and p) which prevail at the tip of a freely growing dendrite, we will assume that only the regions in the immediate proximity of the tip affect the results. This is in good physical agreement with the small thermal diffusion distances which prevail during dendritic growth. In order to be applied to the problem, consider the most general description of a dendritic body: 1 1 2 -~= — — + A~(—) p 2 2 P1 k 2 \P/
(ii) Temkin analysis
a continuation of the tip region, i.e., a paraboloid of revolution:
.
.
Contrary to the assumption made by Ivantsov ) that the dendrite is an isothermal body, Temkin3) recognized that the very existence of the variation in curvature along the surface of the body implies that the equilibrium melting temperature must vary with position in accordance with the Gibbs—Thompson relationship. Secondly, since the model assumes that the body shape is unchanging (shape preserving) during growth, the actual interface temperature (departed from equilibrium by an amount sufficient to drive the operating kinetic process) must vary with position in a prescribed manner. This is due to the fact that the kinetic driving force must always be related to the normal velocity of crystal growth. As a result ofboth these considerations, the interface temperature cannot be isothermal. Before proceeding, it is essential to state and discuss a principal assumption of the Temkin model. Despite the fact that Ivantsov showed that the steady-state shape of an isothermal dendrite is a paraboloid of revolution, Temkin still chose this shape for his nonXIV
(-f)
—
>
.
=
For regions sufficiently close to the tip, one may neglect 2 all of the terms in the series relative to ~(r/p) As a result of the assumption that it is only this region which will affect the predictions, independent of the actual morphology of the rest of the body, it i~assumed, for obvious mathematical reasons, that the body shape is .
.
.
.
.
..
.
.
z/p
=
3-— 3-(r/p)2
(2)
(equivalently cc = 1 from the coordinate transformation given in fig. 1). Therefore, the dendrite geometry and corresponding coordinate system chosen by Temkin is identical to that shown in fig. 1. Since the interface is allowed to be non-isothermal, one would expect to find temperature fields in both the solid and liquid phases. This mathematical problem was greatly simplified by Ternkin by considering only those cases where Vp/ai 1 while maintaining the basic assumption to restrict solution to regions near the dendrite tip, i.e., in the vicinity of cc = 1, /3 = 0 (a~is the thermal diffusivity in either phase). The important difference between the Temkin and Ivantsov results is that the former exhibits a maximum in the function V versus p. Temkin, faced with the decision as to which of the pairs (V, p) actually de‘~
-
1
605
ON THE DENDRITIC GROWTH OF PURE MATERIALS
they also recognized the fact that a dendrite cannot be isothermal for the reasons of the variation of curvature and kinetics over the surface. However, these authors further showed how this non-isothermal effect physically results in the observed velocity decrease (fig. 2) relative to the isothermal case. They pointed to the fact that the variation in temperature in the solid gives rise to a heat flux in addition to the latent heat term at the tip of the dendrite. Thus, the velocity preand Tiller~))Tifier 5) has suggested that one should
r SOTHERMAL
/(vANTsov)
10~
/////
Q2 -
ISOTHERMAL
2 3599976400 235995~ 764
a
0
20
40
60
80
2336400000
23364
23597640000
100
AT(~C)
100
Fig. 2. Comparison of the predictions of the Ivantsov’) (iso3) (non-isothermal) treatments of denthermal) and the Temkin dritic solidification. Ni dendrite, ,~ = 150 cm/sec °C,y = 375
IO~
erg/cm2.
scribes the physical system, chose the only value for V which can give a unique p, i.e., the pair associated with the maximum velocity, Vmax. This choice corresponds to the maximum velocity principle used by many previous investigators when faced with a similar decision. Thus, according to the Temkin model, one can uniquely determine the velocity and tip radius of a freely growing dendrite for a given bath undercooling. To illustrate the difference in magnitude in the velocities predicted by the Temkin and Ivantsov models, we have calculated the values of V and p as a function of bath undercooling, AT, for both models and present the results for Ni in fig. 2 (using the value of p determined by the Temkin analysis in both cases). In order to understand the large differences in predicted dendrite growth velocity from the two models (fig. 2), it is necessary to reconsider the physical effects of nonisothermality.
L
I~e
(iii) Analysis of Boiling and Tiller
Io~
In 1961 Bolling and Tiller4) presented an excellent physical picture of the effects of non-isothermality ~ the growth of a dendrite. Independent of Temkin3)
Fig. 3.
~
I0~
100
IO~
Non-isothermal interfacetemperature distribution based
on the Temkin3) analysis. Growth conditions: zi T = 25 °C, V = 19.069 cm/sec, p = 4.72x l0~ cm, p~ = 150 cm/sec °C = 375 erg/cm2.
XIV-!
606
0. it KOTLER AND L. A. TARSHIS
compare the magnitudes of the flux terms (liquid and solid) to determine if neglect of the flux in the solid is ever justifiable. Such a situation might arise, for example, if the flux to the tip in the solid (introduced by the non-isothermality) is exactly compensated for by an increased flux away from the tip inthe liquid. However, the results show that the consideration of non-isothermal effects not only gives rise to a flux in the solid but also shallows the temperature gradient in the liquid and thus actually decreases the flux away from the tip as well. To illustrate this point, the calculated heat flux in the liquid at the dendrite tip is compared to the flux which would exist if it were isothermal for the case of pure nickel (table 1) which was considered for
TABLE
TABLE
Comparison
2
heat flux in the liquid at the dendrite tip to
(i) the corresponding isothermal dendrite and (ii) the flux in the 2 solid for the solidification of erg water; y = 20 cm— itl] = 16 cm sec’ °C-~. ~
Bath undercooling (AT)
At dendrite tip True flux in -
liquid/flux in liquid for
True flux in solid/true flux
corresponding isothermal dendrite
in liquid
22.5 3
0.78 0.77 0.76
0.33 0.32 0.33
3.5
0.76
0.33
~ 5.5
0.75 0.74
0.33 0.33
6.5
0.73
0.33
1
Comparison of the heat flux in the liquid at the dendrite tip to (i) the corresponding isothermal dendrite and (ii) the flux in the solid for the solidification of pure nickel; t’o = 150cm sec 1 °C 1, y = 375 erg cm2
Bath undercooling (AT)
of the
the tip. Although the temperature difference is small, note that the actual distance over which these changes occur is also extremely small, thereby giving rise to a
At dendrite tip True flux in True flux in liquid/flux in solid/true flux liquid for corresponding isothermal dendrite
gradient of significant magnitude. Bolling and Tiller4) also attempted to derive expressions which would determine both Vmax and the corresponding p. However, we find their calculations to be incorrect for mathematical reasons. By subjecting B & T’s solution to the far field boundary condition as /3 —co on the interface, the right side of their eq. (19) becomes arbitrarily large, whereas the left side ap-
in liquid
__________—
25
0.64
0.32
40 50 75
0.61 0.58 0.54
0.31 0.31 0.31 -~
fig. 2 and for the solidification of pure water (table 2). It can be seen that, in all cases, the ratio of the former to the latter is less than unity as discussed above. Also given in the last column of tables 1 and 2 are comparisons of the heat fluxes in the solid to that in the liquid at the dendrite tip. For these two systems, the flux in the solid is approximately one-third ofthat in the liquid, The above comparisons conclude that, in general, it is in error to neglect the complicating features introduced by non-isothermality. To complete the description of the non-isothermal nature of the dendrite surface, let us consider fig. 3 in which the temperature distribution on the dendrite surface, relative to the temperature at the tip, is plotted as a function of the parabolic coordinate distance, The upper abscissa represents the corresponding physical distance, z, from /3.
XIV
proaches a nonzero constant. Although the B & T work contains numerous useful contributions to the understanding of dendritic growth, results and conclusions which are based on solutions to their eq. (3) are incorrect. Included among these are the values for the kinetic coefficients shown in their table 1. Before proceeding with the application ofthe Temkin analysis, it is of interest to stress the difference between the imposed interface temperatures in the Ivantsov model as opposed to the Temkin model. The only justifiable choice for Ivantsov’s interface temperature (no capillarity or interface kinetics) is the equilibrium melting temperature TM. However, there are examples in the literature where authors have arbitrarily chosen this value to be the true interface temperature of the tip (including kinetic and curvature effects). This choice recognizes the non-isothermal nature of the interface but does not properly include the heat flux -
1
ON THE DENDRITIC GROWTH OF PURE MATERIALS
in the solid which has been shown to be of significance in determining the tip velocity. Some may argue that by using the true tip temperature for the isothermal dendrite, the gradient in the solid is taken into consideration since the interface temperature is lower than TM by an amount determined by capillarity and interface kinetics. This will effectively reduce the temperature gradient in the liquid and have the fortuitous effect of lowering the tip velocity for any given AT. There are rare cases of dendritic growth in pure systems where the use of the Ivantsov analysis with a uniform temperature correction made on the basis of tip conditions is a valid approximation. For Vp/as>> 1, there is poor thermal communication between parts of the system, and thus, the dendrite tip is relatively isolated from the main body. In such cases, the growth velocity and radius of curvature are entirely determined by the tip. However, such growth conditions are extremely rare and will only be approached for enormous undercoolings. In the aforementioned paper by Boiling and Tiller4), the authors performed Ivantsov-type calculations and attempted to fit experimental data to this theory. Although some of the results show reasonably good fit, we believe, from the statements above, that this agreement is purely fortuitous. Glicksman and Schaefer6) also used such an analysis in their treatment of isenthalpic solidification of white phosphorous. Furthermore, GS’s experimental conditions determined Vp/as on the order of 0.1. For both of these reasons, the validity of their analysis is also questionable7).
from the tip. This certainly is consistent with the assumptions stated previously. (ii) The dendrite must be a freely growing body unaffected by extraneous influences from neighboring portions of the system. (iii) The bulk liquid must be maintained at a constant temperature below the equilibrium melting point. (iv) The range of allowable bath undercoolings, AT, must be limited to those values which maintain Vp/ai I = S, L). The most straightforward test of the validity of the Temkin analysis is to attempt to predict the observed dependence of dendrite growth velocity, V, on the measured bath undercooling, AT. In order to accompush this, one must select values for the linear kinetic coefficient, j~, and the solid—liquid interfacial free energy, y, and then simultaneously solve a set of transcendental equations given by Temkin3) and further extended by Kotler8) which will yield values for Vma,~ and p for a given AT. We seek to find unique values for both /2~and y which, from the analysis, will obtain the V(A T) corresponding to the data. If unique values for these parameters can be found over all, or a portion of the experimental range of AT, then one is justified in ascribing a linear kinetic growth mechanism to that region of growth since such a mechanism was explicitly assumed by Temkin. The applicability of the Temkin analysis, in the above sense, depends very strongly on a knowledge of a number of physical constants for the system under study. Generally, these constants are readily available in the literature with the exception of ~
(j
/1o and y. Commonly, for lack of direct experimental evidence, the interfacial energy is calculated from Turnbull’s empirical result9) that y is directly related to the latent heat of fusion (the proportionality constant is a function of the complexity of the solidifying molecule). However, by applying the Temkin analysis to dendrite growth data, it is possible to “curve-fit” the equation to determine not only /2o but also a value for y. This is made feasible by the fact that both of these parameters affect the degree of non-isothermality of the dendrite and thus greatly affect the predicted growth velocity for a given AT. Now let us consider the available experimental data.
3. Applicabifity of the Temkin analysis The validity of any theory rests solely on its ability to correctly describe the experimental observations, In this section, we will attempt to show that the Temkin analysis, when correctly applied to systems closely resembling the mathematical model, can indeed be used to reliably describe some features of dendritic growth. The experimental conditions which must be imposed to conform to Temkin’s mathematical model are: (1) The body shape must be a semi-infinite paraboloid of revolution. Experimental observations of dendrite tip regions, in various systems, indicate that a paraboloid of revolution is an excellent approximation for the shape over a distance on the order of lOp back XIV
607
(i)
-
Ice—water The experiment which comes closest to satisfying 1
608
G. it KOTLER AND L. A. TARSHIS 2.0
I
I
I
I
f
/
.8-
I.6~
-
/ 7 J [
4
.2
/ /
0.8
6 cm/(sec—°C) 2
0.6
r -20
0.4
ergs/cm
0.2
02
3
8
Fig. 4. Experimentally10) observed and theoretically calculated dendrite tip velocity (V) vs. bath undercooling (AT) for the ice— water system. The solid curve represents both the least squares fit of the data and the calculated results. The values for 1u0 and2). y were adjusted to fit the data. (~= 16 cm/sec °C,~‘ = 20 erg/cm
all of the environmental requirements discussed above was that performed by C. S. Lindenmeyer’ 0) for the dendritic solidification of water. His experimental results and the results predicted by the Temkin analysis are shown in fig. 4. After numerous attempts to fit Lindenmeyer’s data, using various combinations of /1~ and y, we arrive at the conclusion that the only possible values are, in fact, 16 +0.5 cm/sec C
=
a
nd =
20+2 erg/cm
2
,
—
for the pure ice—water system. In accordance with the theory of interface attachment kinetics, as outlined by Cahn, Hillig and Sars’ i), the value of J1~is given by the relationship: —
—
p*
p
supercooled water droplets in clouds. More recently, Dufour and Defayi4(having carefully analyzed the data of Jacobi’ 5) and Carte’ 6), determined that y = 20.24 erg/cm2 (see also ref. 17). We also note, in the way of verification of our results, that Fletcher’ 8) has theoretically predicted this value of y = 20 erg/cm2 based on entropy considerations due to bond reorientation during solidification. (a) Tin system Although direct observation of unconstrained den.
—
y
factor introduced by Cahn et a!.11) to account for the different jump frequency at the interface relative to the bulk (J3* was denoted as /3 in the Cahn et a!. work). Substituting the values for the constants in the equation yields /1o = 4.3 /3” and, for the established value of /1o = 16, gives /3* = 3.7. This value is qualitatively consistent with theory but by no means conclusive. We see no further ways to check our results owing to the lack of sufficient data on solidication kinetics. Regarding the unique value of y we obtain, this does not correspond to that of Turnbull9) who determined y = 32 erg/cm2 for the pure ice—water interface. However, it must be noted that his determination represents an average value over all possible crystallographic orientations. Our value of y = 20 erg/cm2 represents the solid—liquid surface tension in the [110] direction. Aside from the differences with the homogeneous nucleation results of Turnbull, a value of 20 ±2 erg/ cm2 is in excellent agreement with other experimentally determined values. Mason, for example, after taking into consideration errors in his earlier estimates12) of y ~ 22 erg/cm2, concludes that 12 < y < 22 erg/cm2 (ref. 13) derived from nucleation rate experiments of
2
DLAH/aORTM,
where DL is the self diffusion coefficient of the water molecule (~2x l0 ~ cm2/sec); a 0 isRthe unitconstant growth step in the growth direction (4.54 A); the gas (1.98 cal/mol °K);and /3* is the frequency correction
.
.
.
.
.
dritic solidification is not possible for metallic systems, Orrok’ 9) has indirectly measured the growth velocity . . . of pure tin dendrites growing into an isothermal, undercooled melt. Orrok’s data is reproduced in fig. 5 where the dashed curve represents the best fit. The solid curves are the predictions of the Temkin analysis using Turnbull’s9) value of y = 54.5 erg/cm2 and the various, indicated, values of ~ We find that any slight varia. tion in y markedly alters the slope of the calculated V versus A Tcurve. Due to the large degree ofscatter inthese 2o < 8 data, is seen that awith range of /L~ between 3 < /closely fits theit experiments a value, j1,~= 4, most approximating the least-squares curve. The different
XIV-l
ON THE DENDRITIC GROWTH OF PURE MATERIALS I 2
PURE TIN
I
/
(V
,/
2
r~54.5ergs/cm
-
~
~
The initial phases of this study were done at Stanford University with partial support from the U.S. Air
_,,/
-
731-65. We also wish to acknowledge the more recent support, in part, Glicksman and R. of the J. Schaefer United States were also Army appreciated. Frankford Arsenal, Contract DAAA 25-68-C-0409. Force Office of Scientific Research, Grant AF-AFOSR-
-
-
Appendix
-
-
a0
20
a~, as
— -
-
- —
-
1j~STSQUARESDATAFI~v/z
2
4
6
8
0
2
4
16
8
~T(°C)and theoretically calculated 9) observed dendrite Fig. 5. tip Experimentally’ velocity (V) vs. bath undercooling (AT) for pure Sn. (- - - -) least-squares data fit, (——) theory. y = 54.5 erg/cm2.
data symbols observed in fig. 5 represent varying growth configurations with respect to the bath. It is
DL
4H R r TL, T
extremely difficult to ascertain which configuration minimizes the effects of convection and of outside influences on the growth process. Conclusions
T~ TM AT
(1) The isothermal, paraboloid of revolution, dendrite described by Ivantsov’) is only applicable for the limited number of systems in which the degree of supersaturation is very large and Vp/(aL) >~ 1. (2)4),The non-isothermal treatment Bolling and while greatly enhancing ourofknowledge of Til the ler physics of dendritic growth, is not applicable in its present form because of a number of mathematical inconsistencies. (3) The non-isothermal analysis of Temkin3) is seen as the best available attempt to describe dendritic growth. The values of y and ji~predicted via this work are in remarkably good agreement with experiment and theory. This indicates, as assumed, that the region near the very tip (i.e., r/p ~ 1) is of greatest importance in dictating the values of V and p. Acknowledgements The authors wish to express their gratitude to Prof. W. A. Tiller and Dr. S. O’Hara of Stanford University and Dr. W. B. Hillig of the G. E. Research and Development Center for many valuable discussions concerning the nature of dendritic growth. The thoughtprovoking comments of Drs. R. F. Sekerka, M. E. XIV
Intermolecular distance in direction of crystal growth Thermal diffusivity of liquid and solid, respectively Self diffusion coefficient of liquid Latent heat of fusion Gas constant Radial polar coordinate
5
4.
609
v
z cc, /3 /3* y 0 /1o p
Temperature in the liquid and solid, respectively Uniform bath temperature Equilibrium melting temperature for planar interface Bath undercooling (AT = TM — T,~) Velocity of growth of dendrite tip Coordinate direction of dendrite tip growth Parabolic coordinates Vibrational Solid—liquid frequency interfacialfactor free energy Azimuthal coordinate in either polar or parabolic coordinates Linear kinetic coefficient of solidification Radius of curvature of dendrite tip
References 1) G. P. Ivantsov, Dokl. Akad. Nauk SSSR 58 (1947) 567. 2) A. Papapetrou, Z. Krist. 92 (1935) 89. 3) D. E. Temkin, Dokl. Akad. Nauk SSSR 132 (1960) 1307. 4) G. F. Boiling and W. A. Tiller, J. Appl. Phys. 32 (1961) 2587. 5) W. A. Tiller, class notes, Stanford University (1965). 6) M. E. Glicksman and R. J. Schaefer, J. Crystal Growth 1 (1968) 297. 7) 8) 9)
L. 222.A. Tarshis and G. R. Kotler, J. Crystal Growth 2 (1968) G. R. Kotler, Ph. D. thesis, Stanford University (1968). D. Turnbull, J. Appi. Phys. 21 (1950) 1022.
10) C.W. S. Lindenmeyer, Ph. D. thesis, Harvard University (1959). 11) J. Cahn, W. B. Hiliig and 0. W. Sears, Acta Met. 12(1964. 1421. -!
610
G. R. KOTLER AND L. A. TARSHIS
12) B. J. Mason, Quart. J. Meteor. Soc. 78 (1952) 22. 13) B. J. Mason, J. Meteor. 11 (1954) 514. 14) L. Dufour and R. Defay, Thermodynamics of Clouds (Academic Press, New York, 1963) p. 226. 15) W. Jacobi, Z. Naturforsch. lOa (1958) 322. 16) A. E. Carte, Proc. Phys. Soc. (London) B 69 (1956) 1028. 17) I. Prigogine, Chem. Phys. Letters 1(1967) 446. 18) N. H. Fletcher, J. Chem. Phys. 30 (1959) 1473. 19) G. T. Orrok, Ph. D. Thesis, Harvard University (1958). 20) G. R. Kotler and L. A. Tarshis, G. E. Report No. TIS 68-C-152 (June 1968).
Discussion M. E. BILLIG, Borough Poly., London, England I cannot see the relevance of calculating in great detail the heat flow from a paraboloid of revolution for dendritic growth, especially of covalent solids as first described by me some 15 years ago. One of the major problems there is to clarify the reason of growth in the form of thin plates rather than ofneedles, to which your thermal model might perhaps apply. The rate-limiting
XIV
process is not so much heat dissipation as rapid nucleation at the growing tip. No account is taken in your treatment of such factors as, e.g., twinning which has in fact a paramount effect. I believe that in the field of dendritic growth as in other fields of technology the physical phenomena ought to be classified first and foremost, before detailed mathematical investigation are started; only then will one be reasonably sure that the assumed model applies at all to the problem under discussion. The same comments apply equally to the following paper (XIV-2). B. LEwIs, Plessey Co., Allen Clark Res. Centre, England I would like to make a comment from the point of view of one who has found the atomistic model of nucleation theory more informative and useful than the classical capillarity model. In the theoretical analysis of dendritic growth energy associated with the curvature of the liquid—solid interface is taken into account. An alternative view is to consider step density at the interface and note the close correspondence with curvature. If steps facilitate crystal growth, more rapid growth is expected where the step density is high, i.e. at the tip, and the concept also introduces crystal anisotropy. Of course, heat flow remains the dominant factor which controls dendritic growth.
-!
Journal of Crystal Growth 3, 4 (1968) 603—6 10 © North-Holland Publishing Co., Amsterdam
ON THE DENDRITIC GROWTH OF PURE MATERIALS
G. R. KOTLER Ford Scientific Laboratory, Dearborn, Michigan 48121, U.S.A. and L. A. TARSHIS General Electric Research and Development Center, Schenectady, New York 12301, U.S.A.
The following three theoretical treatments of unconstrained dendritic growth are discussed: (i) The Ivantsov analysis, which assumes the existence of an isothermal liquid—solid interface and concludes that this dendrite body shape must be a paraboloid of revolution; (ii) and (iii) the non-isothermal treatments of both Boiling and Tiller, and Temkin. These latter two works take account of theover variation in both curvature interface attachment kinetics the entire surface of theand dendrite. They also
tip by assuming that the tip region can effectively be described as a paraboloid of revolution. It is proposed that Temkin’s analysis is sensitive enough to determine the excess solid—liquid surface free energy, y, and the linear kinetic coefficient for molecular attachment, ~. Using the best available data for the ice-water system, the Temkin treat2 and 16 + 0.5 cm/sec °C for y and ment yields 20 + 2 erg/cm ~z
enable one to calculate the velocity and curvature of the dendrite
0,respectively. These values are in remarkably good agreement with other experiments and theory.
1. Introduction
COORDINATE TRANSFORMATION 2-~2)/2
/(z/p~O, r/p~O) Z/p -
rip
a’4
6
6
3
2. Review*
3 7
2
2 I
In order to fully appreciate the discussion which follows, we introduce the parabolic coordinate system (cc /3, 0). These are related to polar coordinates (r, 0, z), as shown in fig. 1. In plotting the polar coordinates, we have introduced the radius of curvature, ~ of a parabolic tip for reasons which shall become obvious. The 0 coordinate is identical in both systems and can be obtained by rotation about the zip (~x= /3 = 0) axis. Ivantsov analysis In 1947, G. P. Ivantsov’) of the Soviet Union made the first serious attempt at describing dendritic growth. (i)
*
(a~O,~O)
zip (a
In this paper we briefly review the existing theoretical descriptions of one aspect of dendritic growth and attempt to show how one of the theories can effectively be used to predict experimental observations. As a result, we are able to extract numerical values of two elusive quantities: y, the excess solid—liquid surface free energy and /10, the linear kinetic coefficient of solidification, in the limited cases where there exists applicable data.
A more detailed review of the existing theories is available20),
XIV
-
LI
a~O
7
1 -
z/p~/2-I/2(r/p)2
Fig. 1.
Graphical definition of circular-parabolic and polar coordinates.
Using the suggestion of Papapetrou2) that the tip region of a dendrite trunk may be adequately represented by a paraboloid of revolution, Ivantsov was able to determine the steady-state temperature field in the liquid surrounding such a body shape. He assumed that the dendrite was isothermal and advancing at constant velocity, V, in the z direction, into an iso-
604
G. R. KOTLER AND L. A. TARSHIS
thermal, undercooled bath at temperature T~.For a given bath undercooling, AT, a unique value for the product Vp can be determined by this analysis. However, a paradoxical feature arises owing to the result that (for each AT) the product Vp must be constant. Thus, one has an infinite set of pairs, (V, p), which satisfy the Ivantsov model. Since one would like to assume that nature provides a unique set (V*, p*), we seek some additional condition in the problem which would allow the determination of V* and p’1’. However, it is not apparent what this condition might be. Commonly, a value of p which optimizes V would be sought but this is impossible since Vp = constant. This apparent inconsistency can be traced to Ivantsov’s assumption that dendritic growth is controlled uniquely by thermal diffusion. There are at least two other phenomena that must be simultaneously considered in any one-component phase transformation. These are, the kinetics of molecular attachment and the effect of localized curvature on the equilibrium transformation temperature. In 1960 D. E. Temkin 3 ), also of the Soviet Union, recognized these limitations and published an elegant mathematical description of the needle crystal (dendrite trunk) growing into a pure, undercooled melt.
isothermal dendrite. No attempt was made to justify this assumption, but rather it was considered to be a questionable first approximation to the complex total problem. Although by no means conclusive, we do see some justification for making this critical mathematical simplification. To best understand these arguments, we shall reword the assumption in the following manner. Recalling that our purpose is to predict the growth conditions (V and p) which prevail at the tip of a freely growing dendrite, we will assume that only the regions in the immediate proximity of the tip affect the results. This is in good physical agreement with the small thermal diffusion distances which prevail during dendritic growth. In order to be applied to the problem, consider the most general description of a dendritic body: 1 1 2 -~= — — + A~(—) p 2 2 P1 k 2 \P/
(ii) Temkin analysis
a continuation of the tip region, i.e., a paraboloid of revolution:
.
.
Contrary to the assumption made by Ivantsov ) that the dendrite is an isothermal body, Temkin3) recognized that the very existence of the variation in curvature along the surface of the body implies that the equilibrium melting temperature must vary with position in accordance with the Gibbs—Thompson relationship. Secondly, since the model assumes that the body shape is unchanging (shape preserving) during growth, the actual interface temperature (departed from equilibrium by an amount sufficient to drive the operating kinetic process) must vary with position in a prescribed manner. This is due to the fact that the kinetic driving force must always be related to the normal velocity of crystal growth. As a result ofboth these considerations, the interface temperature cannot be isothermal. Before proceeding, it is essential to state and discuss a principal assumption of the Temkin model. Despite the fact that Ivantsov showed that the steady-state shape of an isothermal dendrite is a paraboloid of revolution, Temkin still chose this shape for his nonXIV
(-f)
—
>
.
=
For regions sufficiently close to the tip, one may neglect 2 all of the terms in the series relative to ~(r/p) As a result of the assumption that it is only this region which will affect the predictions, independent of the actual morphology of the rest of the body, it i~assumed, for obvious mathematical reasons, that the body shape is .
.
.
.
.
..
.
.
z/p
=
3-— 3-(r/p)2
(2)
(equivalently cc = 1 from the coordinate transformation given in fig. 1). Therefore, the dendrite geometry and corresponding coordinate system chosen by Temkin is identical to that shown in fig. 1. Since the interface is allowed to be non-isothermal, one would expect to find temperature fields in both the solid and liquid phases. This mathematical problem was greatly simplified by Ternkin by considering only those cases where Vp/ai 1 while maintaining the basic assumption to restrict solution to regions near the dendrite tip, i.e., in the vicinity of cc = 1, /3 = 0 (a~is the thermal diffusivity in either phase). The important difference between the Temkin and Ivantsov results is that the former exhibits a maximum in the function V versus p. Temkin, faced with the decision as to which of the pairs (V, p) actually de‘~
-
1
605
ON THE DENDRITIC GROWTH OF PURE MATERIALS
they also recognized the fact that a dendrite cannot be isothermal for the reasons of the variation of curvature and kinetics over the surface. However, these authors further showed how this non-isothermal effect physically results in the observed velocity decrease (fig. 2) relative to the isothermal case. They pointed to the fact that the variation in temperature in the solid gives rise to a heat flux in addition to the latent heat term at the tip of the dendrite. Thus, the velocity preand Tiller~))Tifier 5) has suggested that one should
r SOTHERMAL
/(vANTsov)
10~
/////
Q2 -
ISOTHERMAL
2 3599976400 235995~ 764
a
0
20
40
60
80
2336400000
23364
23597640000
100
AT(~C)
100
Fig. 2. Comparison of the predictions of the Ivantsov’) (iso3) (non-isothermal) treatments of denthermal) and the Temkin dritic solidification. Ni dendrite, ,~ = 150 cm/sec °C,y = 375
IO~
erg/cm2.
scribes the physical system, chose the only value for V which can give a unique p, i.e., the pair associated with the maximum velocity, Vmax. This choice corresponds to the maximum velocity principle used by many previous investigators when faced with a similar decision. Thus, according to the Temkin model, one can uniquely determine the velocity and tip radius of a freely growing dendrite for a given bath undercooling. To illustrate the difference in magnitude in the velocities predicted by the Temkin and Ivantsov models, we have calculated the values of V and p as a function of bath undercooling, AT, for both models and present the results for Ni in fig. 2 (using the value of p determined by the Temkin analysis in both cases). In order to understand the large differences in predicted dendrite growth velocity from the two models (fig. 2), it is necessary to reconsider the physical effects of nonisothermality.
L
I~e
(iii) Analysis of Boiling and Tiller
Io~
In 1961 Bolling and Tiller4) presented an excellent physical picture of the effects of non-isothermality ~ the growth of a dendrite. Independent of Temkin3)
Fig. 3.
~
I0~
100
IO~
Non-isothermal interfacetemperature distribution based
on the Temkin3) analysis. Growth conditions: zi T = 25 °C, V = 19.069 cm/sec, p = 4.72x l0~ cm, p~ = 150 cm/sec °C = 375 erg/cm2.
XIV-!
606
0. it KOTLER AND L. A. TARSHIS
compare the magnitudes of the flux terms (liquid and solid) to determine if neglect of the flux in the solid is ever justifiable. Such a situation might arise, for example, if the flux to the tip in the solid (introduced by the non-isothermality) is exactly compensated for by an increased flux away from the tip inthe liquid. However, the results show that the consideration of non-isothermal effects not only gives rise to a flux in the solid but also shallows the temperature gradient in the liquid and thus actually decreases the flux away from the tip as well. To illustrate this point, the calculated heat flux in the liquid at the dendrite tip is compared to the flux which would exist if it were isothermal for the case of pure nickel (table 1) which was considered for
TABLE
TABLE
Comparison
2
heat flux in the liquid at the dendrite tip to
(i) the corresponding isothermal dendrite and (ii) the flux in the 2 solid for the solidification of erg water; y = 20 cm— itl] = 16 cm sec’ °C-~. ~
Bath undercooling (AT)
At dendrite tip True flux in -
liquid/flux in liquid for
True flux in solid/true flux
corresponding isothermal dendrite
in liquid
22.5 3
0.78 0.77 0.76
0.33 0.32 0.33
3.5
0.76
0.33
~ 5.5
0.75 0.74
0.33 0.33
6.5
0.73
0.33
1
Comparison of the heat flux in the liquid at the dendrite tip to (i) the corresponding isothermal dendrite and (ii) the flux in the solid for the solidification of pure nickel; t’o = 150cm sec 1 °C 1, y = 375 erg cm2
Bath undercooling (AT)
of the
the tip. Although the temperature difference is small, note that the actual distance over which these changes occur is also extremely small, thereby giving rise to a
At dendrite tip True flux in True flux in liquid/flux in solid/true flux liquid for corresponding isothermal dendrite
gradient of significant magnitude. Bolling and Tiller4) also attempted to derive expressions which would determine both Vmax and the corresponding p. However, we find their calculations to be incorrect for mathematical reasons. By subjecting B & T’s solution to the far field boundary condition as /3 —co on the interface, the right side of their eq. (19) becomes arbitrarily large, whereas the left side ap-
in liquid
__________—
25
0.64
0.32
40 50 75
0.61 0.58 0.54
0.31 0.31 0.31 -~
fig. 2 and for the solidification of pure water (table 2). It can be seen that, in all cases, the ratio of the former to the latter is less than unity as discussed above. Also given in the last column of tables 1 and 2 are comparisons of the heat fluxes in the solid to that in the liquid at the dendrite tip. For these two systems, the flux in the solid is approximately one-third ofthat in the liquid, The above comparisons conclude that, in general, it is in error to neglect the complicating features introduced by non-isothermality. To complete the description of the non-isothermal nature of the dendrite surface, let us consider fig. 3 in which the temperature distribution on the dendrite surface, relative to the temperature at the tip, is plotted as a function of the parabolic coordinate distance, The upper abscissa represents the corresponding physical distance, z, from /3.
XIV
proaches a nonzero constant. Although the B & T work contains numerous useful contributions to the understanding of dendritic growth, results and conclusions which are based on solutions to their eq. (3) are incorrect. Included among these are the values for the kinetic coefficients shown in their table 1. Before proceeding with the application ofthe Temkin analysis, it is of interest to stress the difference between the imposed interface temperatures in the Ivantsov model as opposed to the Temkin model. The only justifiable choice for Ivantsov’s interface temperature (no capillarity or interface kinetics) is the equilibrium melting temperature TM. However, there are examples in the literature where authors have arbitrarily chosen this value to be the true interface temperature of the tip (including kinetic and curvature effects). This choice recognizes the non-isothermal nature of the interface but does not properly include the heat flux -
1
ON THE DENDRITIC GROWTH OF PURE MATERIALS
in the solid which has been shown to be of significance in determining the tip velocity. Some may argue that by using the true tip temperature for the isothermal dendrite, the gradient in the solid is taken into consideration since the interface temperature is lower than TM by an amount determined by capillarity and interface kinetics. This will effectively reduce the temperature gradient in the liquid and have the fortuitous effect of lowering the tip velocity for any given AT. There are rare cases of dendritic growth in pure systems where the use of the Ivantsov analysis with a uniform temperature correction made on the basis of tip conditions is a valid approximation. For Vp/as>> 1, there is poor thermal communication between parts of the system, and thus, the dendrite tip is relatively isolated from the main body. In such cases, the growth velocity and radius of curvature are entirely determined by the tip. However, such growth conditions are extremely rare and will only be approached for enormous undercoolings. In the aforementioned paper by Boiling and Tiller4), the authors performed Ivantsov-type calculations and attempted to fit experimental data to this theory. Although some of the results show reasonably good fit, we believe, from the statements above, that this agreement is purely fortuitous. Glicksman and Schaefer6) also used such an analysis in their treatment of isenthalpic solidification of white phosphorous. Furthermore, GS’s experimental conditions determined Vp/as on the order of 0.1. For both of these reasons, the validity of their analysis is also questionable7).
from the tip. This certainly is consistent with the assumptions stated previously. (ii) The dendrite must be a freely growing body unaffected by extraneous influences from neighboring portions of the system. (iii) The bulk liquid must be maintained at a constant temperature below the equilibrium melting point. (iv) The range of allowable bath undercoolings, AT, must be limited to those values which maintain Vp/ai I = S, L). The most straightforward test of the validity of the Temkin analysis is to attempt to predict the observed dependence of dendrite growth velocity, V, on the measured bath undercooling, AT. In order to accompush this, one must select values for the linear kinetic coefficient, j~, and the solid—liquid interfacial free energy, y, and then simultaneously solve a set of transcendental equations given by Temkin3) and further extended by Kotler8) which will yield values for Vma,~ and p for a given AT. We seek to find unique values for both /2~and y which, from the analysis, will obtain the V(A T) corresponding to the data. If unique values for these parameters can be found over all, or a portion of the experimental range of AT, then one is justified in ascribing a linear kinetic growth mechanism to that region of growth since such a mechanism was explicitly assumed by Temkin. The applicability of the Temkin analysis, in the above sense, depends very strongly on a knowledge of a number of physical constants for the system under study. Generally, these constants are readily available in the literature with the exception of ~
(j
/1o and y. Commonly, for lack of direct experimental evidence, the interfacial energy is calculated from Turnbull’s empirical result9) that y is directly related to the latent heat of fusion (the proportionality constant is a function of the complexity of the solidifying molecule). However, by applying the Temkin analysis to dendrite growth data, it is possible to “curve-fit” the equation to determine not only /2o but also a value for y. This is made feasible by the fact that both of these parameters affect the degree of non-isothermality of the dendrite and thus greatly affect the predicted growth velocity for a given AT. Now let us consider the available experimental data.
3. Applicabifity of the Temkin analysis The validity of any theory rests solely on its ability to correctly describe the experimental observations, In this section, we will attempt to show that the Temkin analysis, when correctly applied to systems closely resembling the mathematical model, can indeed be used to reliably describe some features of dendritic growth. The experimental conditions which must be imposed to conform to Temkin’s mathematical model are: (1) The body shape must be a semi-infinite paraboloid of revolution. Experimental observations of dendrite tip regions, in various systems, indicate that a paraboloid of revolution is an excellent approximation for the shape over a distance on the order of lOp back XIV
607
(i)
-
Ice—water The experiment which comes closest to satisfying 1
608
G. it KOTLER AND L. A. TARSHIS 2.0
I
I
I
I
f
/
.8-
I.6~
-
/ 7 J [
4
.2
/ /
0.8
6 cm/(sec—°C) 2
0.6
r -20
0.4
ergs/cm
0.2
02
3
8
Fig. 4. Experimentally10) observed and theoretically calculated dendrite tip velocity (V) vs. bath undercooling (AT) for the ice— water system. The solid curve represents both the least squares fit of the data and the calculated results. The values for 1u0 and2). y were adjusted to fit the data. (~= 16 cm/sec °C,~‘ = 20 erg/cm
all of the environmental requirements discussed above was that performed by C. S. Lindenmeyer’ 0) for the dendritic solidification of water. His experimental results and the results predicted by the Temkin analysis are shown in fig. 4. After numerous attempts to fit Lindenmeyer’s data, using various combinations of /1~ and y, we arrive at the conclusion that the only possible values are, in fact, 16 +0.5 cm/sec C
=
a
nd =
20+2 erg/cm
2
,
—
for the pure ice—water system. In accordance with the theory of interface attachment kinetics, as outlined by Cahn, Hillig and Sars’ i), the value of J1~is given by the relationship: —
—
p*
p
supercooled water droplets in clouds. More recently, Dufour and Defayi4(having carefully analyzed the data of Jacobi’ 5) and Carte’ 6), determined that y = 20.24 erg/cm2 (see also ref. 17). We also note, in the way of verification of our results, that Fletcher’ 8) has theoretically predicted this value of y = 20 erg/cm2 based on entropy considerations due to bond reorientation during solidification. (a) Tin system Although direct observation of unconstrained den.
—
y
factor introduced by Cahn et a!.11) to account for the different jump frequency at the interface relative to the bulk (J3* was denoted as /3 in the Cahn et a!. work). Substituting the values for the constants in the equation yields /1o = 4.3 /3” and, for the established value of /1o = 16, gives /3* = 3.7. This value is qualitatively consistent with theory but by no means conclusive. We see no further ways to check our results owing to the lack of sufficient data on solidication kinetics. Regarding the unique value of y we obtain, this does not correspond to that of Turnbull9) who determined y = 32 erg/cm2 for the pure ice—water interface. However, it must be noted that his determination represents an average value over all possible crystallographic orientations. Our value of y = 20 erg/cm2 represents the solid—liquid surface tension in the [110] direction. Aside from the differences with the homogeneous nucleation results of Turnbull, a value of 20 ±2 erg/ cm2 is in excellent agreement with other experimentally determined values. Mason, for example, after taking into consideration errors in his earlier estimates12) of y ~ 22 erg/cm2, concludes that 12 < y < 22 erg/cm2 (ref. 13) derived from nucleation rate experiments of
2
DLAH/aORTM,
where DL is the self diffusion coefficient of the water molecule (~2x l0 ~ cm2/sec); a 0 isRthe unitconstant growth step in the growth direction (4.54 A); the gas (1.98 cal/mol °K);and /3* is the frequency correction
.
.
.
.
.
dritic solidification is not possible for metallic systems, Orrok’ 9) has indirectly measured the growth velocity . . . of pure tin dendrites growing into an isothermal, undercooled melt. Orrok’s data is reproduced in fig. 5 where the dashed curve represents the best fit. The solid curves are the predictions of the Temkin analysis using Turnbull’s9) value of y = 54.5 erg/cm2 and the various, indicated, values of ~ We find that any slight varia. tion in y markedly alters the slope of the calculated V versus A Tcurve. Due to the large degree ofscatter inthese 2o < 8 data, is seen that awith range of /L~ between 3 < /closely fits theit experiments a value, j1,~= 4, most approximating the least-squares curve. The different
XIV-l
ON THE DENDRITIC GROWTH OF PURE MATERIALS I 2
PURE TIN
I
/
(V
,/
2
r~54.5ergs/cm
-
~
~
The initial phases of this study were done at Stanford University with partial support from the U.S. Air
_,,/
-
731-65. We also wish to acknowledge the more recent support, in part, Glicksman and R. of the J. Schaefer United States were also Army appreciated. Frankford Arsenal, Contract DAAA 25-68-C-0409. Force Office of Scientific Research, Grant AF-AFOSR-
-
-
Appendix
-
-
a0
20
a~, as
— -
-
- —
-
1j~STSQUARESDATAFI~v/z
2
4
6
8
0
2
4
16
8
~T(°C)and theoretically calculated 9) observed dendrite Fig. 5. tip Experimentally’ velocity (V) vs. bath undercooling (AT) for pure Sn. (- - - -) least-squares data fit, (——) theory. y = 54.5 erg/cm2.
data symbols observed in fig. 5 represent varying growth configurations with respect to the bath. It is
DL
4H R r TL, T
extremely difficult to ascertain which configuration minimizes the effects of convection and of outside influences on the growth process. Conclusions
T~ TM AT
(1) The isothermal, paraboloid of revolution, dendrite described by Ivantsov’) is only applicable for the limited number of systems in which the degree of supersaturation is very large and Vp/(aL) >~ 1. (2)4),The non-isothermal treatment Bolling and while greatly enhancing ourofknowledge of Til the ler physics of dendritic growth, is not applicable in its present form because of a number of mathematical inconsistencies. (3) The non-isothermal analysis of Temkin3) is seen as the best available attempt to describe dendritic growth. The values of y and ji~predicted via this work are in remarkably good agreement with experiment and theory. This indicates, as assumed, that the region near the very tip (i.e., r/p ~ 1) is of greatest importance in dictating the values of V and p. Acknowledgements The authors wish to express their gratitude to Prof. W. A. Tiller and Dr. S. O’Hara of Stanford University and Dr. W. B. Hillig of the G. E. Research and Development Center for many valuable discussions concerning the nature of dendritic growth. The thoughtprovoking comments of Drs. R. F. Sekerka, M. E. XIV
Intermolecular distance in direction of crystal growth Thermal diffusivity of liquid and solid, respectively Self diffusion coefficient of liquid Latent heat of fusion Gas constant Radial polar coordinate
5
4.
609
v
z cc, /3 /3* y 0 /1o p
Temperature in the liquid and solid, respectively Uniform bath temperature Equilibrium melting temperature for planar interface Bath undercooling (AT = TM — T,~) Velocity of growth of dendrite tip Coordinate direction of dendrite tip growth Parabolic coordinates Vibrational Solid—liquid frequency interfacialfactor free energy Azimuthal coordinate in either polar or parabolic coordinates Linear kinetic coefficient of solidification Radius of curvature of dendrite tip
References 1) G. P. Ivantsov, Dokl. Akad. Nauk SSSR 58 (1947) 567. 2) A. Papapetrou, Z. Krist. 92 (1935) 89. 3) D. E. Temkin, Dokl. Akad. Nauk SSSR 132 (1960) 1307. 4) G. F. Boiling and W. A. Tiller, J. Appl. Phys. 32 (1961) 2587. 5) W. A. Tiller, class notes, Stanford University (1965). 6) M. E. Glicksman and R. J. Schaefer, J. Crystal Growth 1 (1968) 297. 7) 8) 9)
L. 222.A. Tarshis and G. R. Kotler, J. Crystal Growth 2 (1968) G. R. Kotler, Ph. D. thesis, Stanford University (1968). D. Turnbull, J. Appi. Phys. 21 (1950) 1022.
10) C.W. S. Lindenmeyer, Ph. D. thesis, Harvard University (1959). 11) J. Cahn, W. B. Hiliig and 0. W. Sears, Acta Met. 12(1964. 1421. -!
610
G. R. KOTLER AND L. A. TARSHIS
12) B. J. Mason, Quart. J. Meteor. Soc. 78 (1952) 22. 13) B. J. Mason, J. Meteor. 11 (1954) 514. 14) L. Dufour and R. Defay, Thermodynamics of Clouds (Academic Press, New York, 1963) p. 226. 15) W. Jacobi, Z. Naturforsch. lOa (1958) 322. 16) A. E. Carte, Proc. Phys. Soc. (London) B 69 (1956) 1028. 17) I. Prigogine, Chem. Phys. Letters 1(1967) 446. 18) N. H. Fletcher, J. Chem. Phys. 30 (1959) 1473. 19) G. T. Orrok, Ph. D. Thesis, Harvard University (1958). 20) G. R. Kotler and L. A. Tarshis, G. E. Report No. TIS 68-C-152 (June 1968).
Discussion M. E. BILLIG, Borough Poly., London, England I cannot see the relevance of calculating in great detail the heat flow from a paraboloid of revolution for dendritic growth, especially of covalent solids as first described by me some 15 years ago. One of the major problems there is to clarify the reason of growth in the form of thin plates rather than ofneedles, to which your thermal model might perhaps apply. The rate-limiting
XIV
process is not so much heat dissipation as rapid nucleation at the growing tip. No account is taken in your treatment of such factors as, e.g., twinning which has in fact a paramount effect. I believe that in the field of dendritic growth as in other fields of technology the physical phenomena ought to be classified first and foremost, before detailed mathematical investigation are started; only then will one be reasonably sure that the assumed model applies at all to the problem under discussion. The same comments apply equally to the following paper (XIV-2). B. LEwIs, Plessey Co., Allen Clark Res. Centre, England I would like to make a comment from the point of view of one who has found the atomistic model of nucleation theory more informative and useful than the classical capillarity model. In the theoretical analysis of dendritic growth energy associated with the curvature of the liquid—solid interface is taken into account. An alternative view is to consider step density at the interface and note the close correspondence with curvature. If steps facilitate crystal growth, more rapid growth is expected where the step density is high, i.e. at the tip, and the concept also introduces crystal anisotropy. Of course, heat flow remains the dominant factor which controls dendritic growth.
-!