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Formation of patterns during growth of snow crystals
Journal of Crystal Growth 128 (1993) 251-257 North-Holland
j. . . . . . . .
CRYSTAL GROW
T H
Formation of patterns during growth of snow crystals Etsuro Yokoyama Faculty of Computer Science and Systems Engineering, Kyushu Institute of Technology, Iizuka 820, Japan
We discuss recent progress in the understanding of the formation of patterns of snow crystals. In particular, we concentrate on the transition from hexagonal to dendritic patterns that occurs with an increase in the degree of supersaturation. Our numerical results demonstrate that the anisotropy of surface kinetics plays a very important role in the formation of faceted shapes of snow crystals in the form of hexagonal prisms and of facets seen at the tips of dendrites. Furthermore, it is shown that the conditions for the transition from a hexagonal to a dendritic pattern are given by the supersaturation and the dimensionless crystal size relative to the mean free path of a water molecule in air.
1. Introduction
Starting from a spherical single crystal of ice of the order of several micrometers in radius, the pattern of snow crystals grown from supersaturated vapor in clouds first becomes a hexagonal prism bounded by two basal and six prism faces [1]. At low supersaturation, such a hexagonal prism can grow in a stable manner and retains its form. For higher supersaturation, however, it changes form by means of preferred growth of edges and corners, becoming a needle crystal for temperatures between - 4 and - 1 0 ° C and becoming a dendrite for temperatures between - 10 and - 2 2 ° C [2-5]. This instability should be distinguished from the Mullins-Sekerka instability [6] for a rough crystal-melt interface. In the latter case, dendrites develop from perturbations at the interface in the absence of interface kinetics [7]. Anisotropy of surface kinetics derived from the lateral motion of steps [8], for which the kinetic coefficient has singularities in the form of deep cusps, plays a very important role in the formation of polyhedral features of snow crystals, such as hexagonal prisms and faceted shapes seen at the tips of dendrites. In most of the previous
studies on the formation of patterns of snow crystals, such surface kinetic processes were neglected [9,10]. Several studies of the conditions for transition from polyhedral to dendritic patterns have been carried out [11-13]. Recently, Xiao, Alexander and Rosenberger have produced faceted patterns and dendritic patterns with various growth parameters using a Monte Carlo method [14,15]. They also find that the critical size for the stable growth of polyhedral crystals scales linearly with the mean free path of a molecule in the nutrient phase [16]. However, the numerical study of time evolution of patterns for the transition from polyhedral to dendritic patterns still remains insufficient. In this article, we present a numerical model [5,17] that accounts for the following elementary processes relevant to the growth: (1) a process for diffusing molecules through air toward the crystal surface and (2) a surface kinetic process for incorporating water molecules into a crystal lattice. Furthermore, we discuss the morphological instability of snow crystals that occurs with an increase in the degree of supersaturation. Finally, we show that the dimensionless crystal size, relative to the mean free path of a water molecule, as well as
0022-0248/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
252
E. Yokoyama / Formation of patterns during growth of snow crystals
the supersaturation, plays an important role in the formation of patterns during the growth of snow crystals.
ment of molecules by diffusion. Therefore, the supersaturation field has sufficient time to relax to its steady state value, and cr is governed by the Laplace equation [17]:
= 0.
2. The model
(5)
The boundary conditions are as follows: 2.1. Diffusion process We consider a two-dimensional model in which a vapor phase surrounds a single crystal. The crystal boundary is initially a circle that is perpendicular to the c axis of a hexagonal snow crystal. The temperature of the whole system is assumed to be uniform at -15°C, which is a typical temperature for the development of dendritic patterns [2,3]. In this model, we rescale all lengths by an amount I* = 4i = 4 D / ~ ,
(i)
where i is the mean free path of a water molecule in the vapor, D is the diffusion coefficient and is the mean speed of molecules in the vapor, given by V = ~/8kT/Trm,
(2)
where k is Boltzmann's constant, T is the absolute temperature and m is the mass of a molecule. Time is scaled by t* = 2 7 r m D / v c p e,
(3)
where vc is the volume of a molecule in the crystal and Pe is the equilibrium vapor pressure of ice. The driving force for growth of snow crystals is characterized by the supersaturation, or= ( p - p ~ ) / p e ,
(4)
where p is the partial pressure of water vapor. From eqs. (1)-(4), the quantity l * o ' J t * represents the H e r t z - K n u d s e n formula, i.e., the maximum growth speed of the surface, where o,s is the supersaturation at the surface. For typical growth conditions, the growth speed at the surface of snow crystals at any time is negligibly small in comparison with the move-
~r(R=) =~r=,
(6)
(O¢/On)s = V,
(7)
where R= is the radius of the outer boundary, (ao-/On)s is the normal derivative of the supersaturation at a position on the surface, and V is the local normal growth speed of the surface. 2.2. Surface kinetic processes The prism faces are molecularly smooth, and the surface inclined by an angle 0 from a prism face is composed of a parallel array of steps which are distributed with a mean step separation A, where A = d / t a n 0, where d is the step height. Such surfaces cannot grow without the following surface kinetic processes: (i) adsorption of water molecules onto the crystal surface, (ii) surface diffusion of admolecules toward the steps, which result from two-dimensional nucleation or with the aid of screw dislocations, and (iii) a lateral motion of the steps. The dimensionless growth speed V parallel to the basal plane can be described by [8] V = / 3 ( 0 , ~s) o's,
(8)
where 0 is the angle made by the surface and a prism face, o-s is the local supersaturation at the surface and /3(0, crs) is the kinetic coefficient. More generally, /3 depends also on the local curvature K of the surface. For the growth of snow crystals, however, we can neglect the effect of K on the growth speed, because the average radius of the snow crystals is much larger than the step separation A. Moreover, the local equilibrium vapor pressure of ice is nearly equal to the equilibrium vapor pressure at a flat surface ( K = 0) for actual snow crystals, i.e., p c ( K ) = pc(0).
E. Yokoyama / Formation of patterns during growth of snow crystals
The kinetic coefficient for may be expressed in the form
/3(0, ~rs) = a,( s / s 0 tanh( s l / s ), s = Itan 01 = d / A ,
sI = d/2xs,
-7r/6
< 0 < ~-/6
(9) (10) (11)
where o~1 is the sticking coefficient, i.e., the ratio of molecules that stick on the surface at the instant they impinge on it, and x s is the mean surface diffusion distance of a molecule on the surface. In eq. (9), the value of/3(0, ~r~) at prism faces is zero. However, since the steps are generated on the prism faces by two-dimensional nucleation or with the aid of screw dislocation, it is not zero but positive in the nonequilibrium state. In the case of spiral growth with the aid of screw dislocations, the slope s o of a growth hillock on prism faces is given by [18] s o = (sx/o-l)o-s,
(12)
o"1 = 9.5 foK/kTXs,
(13)
where K is the edge free f0 is the surface area molecule. From eqs. (9) following expression for the supersaturated state:
energy along a step and occupied by a water and (12), we obtain the /3 of the prism faces in
/30 =/3(0, O's) = oq ~r--Ltanh( ~1 ). trl k °'s
253
two-dimensional nucleation growth, /30 is equal to zero at small o~s, then increases exponentially with crs above the critical value ~r~c and approaches a I as crs ~ ~ [18,19]. In our model, we consider only spiral growth to occur with the aid of screw dislocations and assign K = 2.0 X 10 -6 c m / e r g for the edge free energy, since the growth speed obtained by eqs. (8)-(14) agrees well with experimental values [20]. For growth by two-dimensional nucleation, we must assign K = 4.0 × 10 -7 c m / e r g to agree with the experimental values. However, this value is unreasonably smaller than the value of K = 3.7 × 10 - 6 c m / e r g derived from the broken bond model [21]. Furthermore, we assume that the steps are supplied from screw dislocations emerging at the center of six prism faces until a circular crystal becomes a perfect hexagon, and they are generated at its corners where the supersaturation at the surface is largest along the surface, once a perfect hexagon has been developed.
2.3. Numerical treatment By substituting eq. (8) into eq. (7), we obtain the boundary conditions at the surface as follows: (O~r/0n)s = / 3 ( 0 , Ors) o's.
(14)
The kinetic coefficient /3o increases linearly with tr~ in the range of o'~ << ¢rl, since the frequency of step generation increases with increasing o-s. For ~rs >> o'l,/3o approaches oq asymptotically as trs -~ oo because of the s t e p - s t e p interaction through the surface diffusion of admolecules. The anisotropy of/3(0, trs) increases with decreasing ~rs, since /3o decreases with a decrease in o'~. The degree of anisotropy is, therefore, not a constant but becomes larger during the growth of crystals because of the decrease in ¢rs with increasing the crystal size. Furthermore, from eqs. (8) and (14), the growth speed, V, of prism faces is a nonlinear function for cry. This causes a drastic change of the patterns of snow crystals during the growth in response to the growth conditions. On the other hand, in the case of
(15)
Eq. (15) is important for determining self-consistently of the growth speed V as determined by both the diffusion process and the surface kinetic processes [13]. We derive the boundary integral equation from eq. (5) using Green's theorem and the boundary conditions (eqs. (6) and (15)): ~rsi +
+ / 3 ( 0 , crs)G ~rs d F urface
=
-
"outer boundary [
--
O'~ ~n
dF,
(16)
where O,si is the supersaturation at a point i on the surface, G is the G r e e n function which has 6-fold symmetry and reflection symmetry about the line y = x / v ~ , and q= is the normal gradient of supersaturation at the outer boundary which located on a circle of radius Roo centered at the
254
E. Yokoyama / Formation of patterns during growth of snow crystals
center of the crystal. Assuming that the outer b o u n d a r y is far from the surface of crystal, qoo is approximately o-~ qoo -~ R= In( R = / r c ) '
(17)
which is obtained for the diffusion filed surrounding a circular crystal with a perfect sink, i.e., o-s = 0 , where r c is the average radius of the crystal. It is also assumed that Roo/r c = constant, so that the outer b o u n d a r y grows with increasing crystal size. This is because the normal derivative of the supersaturation at the surface determined by the two-dimensional Laplace equation agrees with that determined by the two-dimensional t i m e - d e p e n d e n t diffusion equation [22]. The surface is divided into N equal segments marked by grids points. Eq. (16) is changed to an algebraic equation with respect to the surface supersaturation o"s. W e use an iteration m e t h o d [13] to solve the nonlinear problem for which ~0 in eq. (14) depends on o-s. By solving the N simultaneous equations for all points i, we obtain the distribution of o-s. Consequently, from eqs. (8), (9) and (14), we find the growth speed of V at each point on the surface at a given moment, and then determine the shape after growth for a short time At. During growth, faster orientations are cut off by neighboring slower growing surfaces, so the slowest growing surface occupies the larger area. T h e growth pattern is, therefore, determined by the inner envelope of lines perpendicular to normal vectors of length VAt at each point [171.
Fig. 1. Growth pattern for ~r~= 1%. Lines are at time intervals of At = 107 (= 7X103 s).
10 -16 cm 2, mass of a water molecule m = 3.0 x 10 -23 g, step height or lattice constant d = 4.5 x 10 -8 cm, m e a n surface diffusion distance of an admolecule x s = 400d [24], edge free energy along a step K = 2.0 x 10 -6 e r g / c m , radius of initial crystal rc0 = 0.3 x 10 3 corresponding to 5 × 10 .3 cm for D = 0.2 c m 2 / s , and ratio of a distance Roo to an average radius r e of the crystal, R ~ / r c = 6.5 [17]. Fig. 1 shows an example of hexagonal patterns for small supersaturation ~r~ = 1%. T h e initial crystal starting from a circle develops into a hexagonal at t = 107, and then the hexagonal crystal grows, retaining its flat surface in spite of the nonuniformity in surface supersaturation [25], since the degree of the nonuniformity is not large.
3. Results T h e following values are used in this numerical calculation: t e m p e r a t u r e T = 258.15 K ( = - 15°C), equilibrium vapor pressure Pe = 1.66 × 103 d y n / c m 2 = 1.66 x 102 Pa corresponding to - 1 5 ° C , diffusion coefficient D = 0.2 c m 2 / s corresponding to 1 atm, i.e., 1.01325 × 105 Pa, sticking coefficient % = 0.01 [23], molecular volume of an ice crystal v c = 3.25 × 10 -23 cm 3, surface area occupied by a water molecule f0---8.3 ×
Fig. 2. Growth pattern for ~= = 16%. Lines are at time intervals of At = 5.7X 105 (= 4x 102 s).
E. Yokoyama / Formation of patterns during growth of snow crystals
Fig. 3. Growth pattern for tr~ = 32%. Lines are at time intervals of At = 5.7x 105 ( = 4 × 102 s).
For example, the surface supersaturation ors = 0.47% at the corners and ors = 0.41% at the face centers at t = 107 in fig. 1. Figs. 2 and 3 show examples of dendritic patterns for large supersaturation, or= = 16%, and for larger supersaturation, o-==32%, respectively. The growth speed at corners is larger than that at face centers because of the large nonuniformity in surface supersaturation, so there is preferred growth of corners. In fig. 2, the surface supersaturation ors = 2.4% at the corners and ors = 1.7% at the face centers at t = 11.4 x 105, and in fig. 3, ors = 3.6% at the corners and ors = 2.3% at the face centers at t = 11.4 x 105. Facets are developed and macrosteps are formed at the tips of primary stalks because of bunching of monomolecular steps on the facets. Such macrosteps might act as triggers for the formation of side branches. The width of a primary stalk in fig. 3 is smaller than that in fig. 2, since the position at which the growth speed is equal to that at corners, moves toward the corner from the face centers, with increasing nonuniformity in supersaturation.
4. Discussion
The excess supersaturation at the corners of the hexagonal crystal causes preferred growth of
255
corners. For small supersaturations o-=, however, the nonuniformity in supersaturation along the surface can be compensated for by variation of the local kinetic coefficient /3(0, ors) made possible by adjustment of the step distribution [12,13]. The steps generated at the corners slow down as they approach the face centers because of decreasing ors, and then the step density increases near the centers. Since the local kinetic coefficient/3 increases with the local slope (eq. (9)),/3 increases from the corners toward the face centers. Therefore, the growth speed V(0) at the center of the surface can reach V(1) at the corners, and the growth speed V can be constant over the whole surface. The surface of a hexagonal crystal such a shown in fig. 1, however, appears to be macroscopically fiat, because such small nonuniformity can be compensated for by a slight local slope of the order of 10 -2 at the face centers. On the other hand, for large supersaturations or=, the value o f / 3 at the face centers reaches the upper limit o~1 (eq. (9)). This is because the local slope s(0) at the face centers increases with increasing or=, because of an increase of the nonuniformity in ors with o-=. Once /3 = o/1 at the face centers at the critical supersaturation or*, the growth speed V(0) at the face centers can never reach V(1) at the corners for larger than or*, even though s(0) at the face centers increases with o'=. This is the onset of dendritic growth. From eqs. (8), (9) and (14), we obtain the following critical condition for the limit of stable growth of the hexagonal crystal, i.e., the transition from a hexagonal to a dendritic pattern: ors(O) = ( ors*/Orl ) t a n h ( o-1//ors* ) ors*,
(18)
where ors(0) is the supersaturation at the face centers and ors* is the critical supersaturation at the corners for o-*. The difference between ors(0) and ors(l), where ors(l) is the supersaturation at the corners, is given by [25] Ad~ L I aor t o-s(1) -ors(O) = ~ ~ n ] s,
(19)
where L is the dimensionless length of a side of
256
E. Yokoyama / Formation of patterns during growth of snow crystals
important role in the formation of patterns during growth of snow crystals. Furthermore, the conditions for the transition depend on the ratio of the edge free energy K along a step to the surface diffusion distance xs of an admolecule through 0"1 (eq. (13)). The ratio of K to x s affects the formation of patterns during growth of crystals.
10 °
N
10'
,/:@
Acknowledgments ¢)
.o
dendritic pattern hexagona~ pattern
E
,
I00
10
"--...,
20
Ooo/01 Fig. 4. Diagram showing the relationship between patterns and conditions (@/~1,-£~) for ~ l = 2.5%. The solid line described by eqs. (20) and (21) represents the transition from hexagonal to dendritic patterns.
hexagon and A~b = 0.554. From eqs. (7), (8), (14), (18) and (19), and o-s(1) = o-s*, we obtain 0-s* tanh 0"1
1
(20)
1 +.g~'
where
_2a= ( Aqb/2rr)alL, is a dimensionless crystal size. Substituting 0"s* into the general expression o"s(1), which is given by [25,26], we find 0"= o"s* = 1 + F 2 / ( 1 + S °) '
(21)
where F is a constant that depends on the ratio
R=/rc. The solid line in fig. 4 represents the transition from hexagonal to dendritic pattern determined by eqs. (20) and (21). The diagram in fig. 4 shows clearly that the dimensionless crystals size .~, relative to the mean free path of a water molecule, as well as the supersaturation 0"=, plays a very
The author is very grateful to the late Professor T. Kuroda for a number of stimulating and helpful discussions. He would like to thank Professor R.F. Sekerka for critical reading of the manuscript and helpful discussions. He would also like to thank Mr. T. Irisawa for his kind suggestions regarding computer programming. This work was partially supported by a Grant-inAid for Scientific Research from the Japanese Ministry of Education, Science and Culture, which is gratefully acknowledged.
References [1] T. Gonda and T. Yamazaki, J. Crystal Growth 45 (1978) 66. [2] U. Nakaya, Snow Crystals - Natural and Artificial (Harvard University Press, Cambridge, MA, 1954). [3] T. Kobayashi, Phil. Mag. 6 (1961) 1363. [4] T. Kuroda, J. Crystal Growth 71 (1985) 84. [5] T. Kuroda and E. Yokoyama, in: Formation, Dynamics and Statistics of Patterns, Eds. K. Kawasaki et al. (World Scientific, Singapore, 1990) p. 285. [6] W.W. Mullins and R.F. Sekerka, J. Appl. Phys. 34 (1963) 323. [7] Y. Saito, G. Goldbeck-Wood and H. Miiller-Krumbhaar, Phys. Rev. A38 (1988) 2148. [8] W.K. Burton, N. Cabrera and F.C. Frank, Phil. Trans. Roy. Soc. London A 243 (1951) 299. [9] J. Nittmann and H.E. Stanley, Nature 321 (1986) 663; J. Phys. A 20 (1987) Ll185. [10] F. Family, D.E. Platt and T. Vicsek, J. Phys. A 20 (1987) Ll177. [11] J.W. Cahn, in: Crystal Growth, Ed. H.S. Peiser (Pergamon, Oxford, 1967) p. 681. [12] A.A. Chernov, J. Crystal Growth 24/25 (1974) 11. [13] T. Kuroda, T. Irisawa and A. Ookawa, J. Crystal Growth 42 (1977) 41.
E. Yokoyama / Formation of patterns during growth of snow crystals [14] R.F. Xiao, J.I.D. Alexander and F. Rosenberger, Phys. Rev. A 38 (1988) 2447. [15] R.F. Xiao, LI.D. Alexander and F. Rosenberger, J. Crystal Growth 100 (1990) 313. [16] R.F. Xiao, J.I.D. Alexander and F. Rosenberger, Phys. Rev. A 43 (1991) 2977. [17] E. Yokoyama and T. Kuroda, Phys. Rev. A 41 (1990) 2038. [18] E. Yokoyama and T. Kuroda, J. Meteor. Soc. Japan 66 (1988) 927. [19] Y. Arima and T. Irisawa, J. Crystal Growth 94 (1989) 297.
257
[20] T. Sei and T. Gonda, J. Crystal Growth 94 (1989) 697. [21] T. Kuroda and R. Lacmann, J. Crystal Growth 56 (1982) 189. [22] S.R. Coriell and R.L. Parker, J. Appl. Phys. 36 (1965) 632. [23] T. Kuroda and T. Gonda, J. Meteor. Soc. Japan 62 (1984) 563. [24] J. Kiefer and B.N. Hale, J. Chem. Phys. 7 (1977) 3206. [25] A. Seeger, Phil. Mag. 44 (1953) 1. [26] E. Yokoyama and T. Kuroda, J. de Physique 47 (1987) C1-347.
j. . . . . . . .
CRYSTAL GROW
T H
Formation of patterns during growth of snow crystals Etsuro Yokoyama Faculty of Computer Science and Systems Engineering, Kyushu Institute of Technology, Iizuka 820, Japan
We discuss recent progress in the understanding of the formation of patterns of snow crystals. In particular, we concentrate on the transition from hexagonal to dendritic patterns that occurs with an increase in the degree of supersaturation. Our numerical results demonstrate that the anisotropy of surface kinetics plays a very important role in the formation of faceted shapes of snow crystals in the form of hexagonal prisms and of facets seen at the tips of dendrites. Furthermore, it is shown that the conditions for the transition from a hexagonal to a dendritic pattern are given by the supersaturation and the dimensionless crystal size relative to the mean free path of a water molecule in air.
1. Introduction
Starting from a spherical single crystal of ice of the order of several micrometers in radius, the pattern of snow crystals grown from supersaturated vapor in clouds first becomes a hexagonal prism bounded by two basal and six prism faces [1]. At low supersaturation, such a hexagonal prism can grow in a stable manner and retains its form. For higher supersaturation, however, it changes form by means of preferred growth of edges and corners, becoming a needle crystal for temperatures between - 4 and - 1 0 ° C and becoming a dendrite for temperatures between - 10 and - 2 2 ° C [2-5]. This instability should be distinguished from the Mullins-Sekerka instability [6] for a rough crystal-melt interface. In the latter case, dendrites develop from perturbations at the interface in the absence of interface kinetics [7]. Anisotropy of surface kinetics derived from the lateral motion of steps [8], for which the kinetic coefficient has singularities in the form of deep cusps, plays a very important role in the formation of polyhedral features of snow crystals, such as hexagonal prisms and faceted shapes seen at the tips of dendrites. In most of the previous
studies on the formation of patterns of snow crystals, such surface kinetic processes were neglected [9,10]. Several studies of the conditions for transition from polyhedral to dendritic patterns have been carried out [11-13]. Recently, Xiao, Alexander and Rosenberger have produced faceted patterns and dendritic patterns with various growth parameters using a Monte Carlo method [14,15]. They also find that the critical size for the stable growth of polyhedral crystals scales linearly with the mean free path of a molecule in the nutrient phase [16]. However, the numerical study of time evolution of patterns for the transition from polyhedral to dendritic patterns still remains insufficient. In this article, we present a numerical model [5,17] that accounts for the following elementary processes relevant to the growth: (1) a process for diffusing molecules through air toward the crystal surface and (2) a surface kinetic process for incorporating water molecules into a crystal lattice. Furthermore, we discuss the morphological instability of snow crystals that occurs with an increase in the degree of supersaturation. Finally, we show that the dimensionless crystal size, relative to the mean free path of a water molecule, as well as
0022-0248/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
252
E. Yokoyama / Formation of patterns during growth of snow crystals
the supersaturation, plays an important role in the formation of patterns during the growth of snow crystals.
ment of molecules by diffusion. Therefore, the supersaturation field has sufficient time to relax to its steady state value, and cr is governed by the Laplace equation [17]:
= 0.
2. The model
(5)
The boundary conditions are as follows: 2.1. Diffusion process We consider a two-dimensional model in which a vapor phase surrounds a single crystal. The crystal boundary is initially a circle that is perpendicular to the c axis of a hexagonal snow crystal. The temperature of the whole system is assumed to be uniform at -15°C, which is a typical temperature for the development of dendritic patterns [2,3]. In this model, we rescale all lengths by an amount I* = 4i = 4 D / ~ ,
(i)
where i is the mean free path of a water molecule in the vapor, D is the diffusion coefficient and is the mean speed of molecules in the vapor, given by V = ~/8kT/Trm,
(2)
where k is Boltzmann's constant, T is the absolute temperature and m is the mass of a molecule. Time is scaled by t* = 2 7 r m D / v c p e,
(3)
where vc is the volume of a molecule in the crystal and Pe is the equilibrium vapor pressure of ice. The driving force for growth of snow crystals is characterized by the supersaturation, or= ( p - p ~ ) / p e ,
(4)
where p is the partial pressure of water vapor. From eqs. (1)-(4), the quantity l * o ' J t * represents the H e r t z - K n u d s e n formula, i.e., the maximum growth speed of the surface, where o,s is the supersaturation at the surface. For typical growth conditions, the growth speed at the surface of snow crystals at any time is negligibly small in comparison with the move-
~r(R=) =~r=,
(6)
(O¢/On)s = V,
(7)
where R= is the radius of the outer boundary, (ao-/On)s is the normal derivative of the supersaturation at a position on the surface, and V is the local normal growth speed of the surface. 2.2. Surface kinetic processes The prism faces are molecularly smooth, and the surface inclined by an angle 0 from a prism face is composed of a parallel array of steps which are distributed with a mean step separation A, where A = d / t a n 0, where d is the step height. Such surfaces cannot grow without the following surface kinetic processes: (i) adsorption of water molecules onto the crystal surface, (ii) surface diffusion of admolecules toward the steps, which result from two-dimensional nucleation or with the aid of screw dislocations, and (iii) a lateral motion of the steps. The dimensionless growth speed V parallel to the basal plane can be described by [8] V = / 3 ( 0 , ~s) o's,
(8)
where 0 is the angle made by the surface and a prism face, o-s is the local supersaturation at the surface and /3(0, crs) is the kinetic coefficient. More generally, /3 depends also on the local curvature K of the surface. For the growth of snow crystals, however, we can neglect the effect of K on the growth speed, because the average radius of the snow crystals is much larger than the step separation A. Moreover, the local equilibrium vapor pressure of ice is nearly equal to the equilibrium vapor pressure at a flat surface ( K = 0) for actual snow crystals, i.e., p c ( K ) = pc(0).
E. Yokoyama / Formation of patterns during growth of snow crystals
The kinetic coefficient for may be expressed in the form
/3(0, ~rs) = a,( s / s 0 tanh( s l / s ), s = Itan 01 = d / A ,
sI = d/2xs,
-7r/6
< 0 < ~-/6
(9) (10) (11)
where o~1 is the sticking coefficient, i.e., the ratio of molecules that stick on the surface at the instant they impinge on it, and x s is the mean surface diffusion distance of a molecule on the surface. In eq. (9), the value of/3(0, ~r~) at prism faces is zero. However, since the steps are generated on the prism faces by two-dimensional nucleation or with the aid of screw dislocation, it is not zero but positive in the nonequilibrium state. In the case of spiral growth with the aid of screw dislocations, the slope s o of a growth hillock on prism faces is given by [18] s o = (sx/o-l)o-s,
(12)
o"1 = 9.5 foK/kTXs,
(13)
where K is the edge free f0 is the surface area molecule. From eqs. (9) following expression for the supersaturated state:
energy along a step and occupied by a water and (12), we obtain the /3 of the prism faces in
/30 =/3(0, O's) = oq ~r--Ltanh( ~1 ). trl k °'s
253
two-dimensional nucleation growth, /30 is equal to zero at small o~s, then increases exponentially with crs above the critical value ~r~c and approaches a I as crs ~ ~ [18,19]. In our model, we consider only spiral growth to occur with the aid of screw dislocations and assign K = 2.0 X 10 -6 c m / e r g for the edge free energy, since the growth speed obtained by eqs. (8)-(14) agrees well with experimental values [20]. For growth by two-dimensional nucleation, we must assign K = 4.0 × 10 -7 c m / e r g to agree with the experimental values. However, this value is unreasonably smaller than the value of K = 3.7 × 10 - 6 c m / e r g derived from the broken bond model [21]. Furthermore, we assume that the steps are supplied from screw dislocations emerging at the center of six prism faces until a circular crystal becomes a perfect hexagon, and they are generated at its corners where the supersaturation at the surface is largest along the surface, once a perfect hexagon has been developed.
2.3. Numerical treatment By substituting eq. (8) into eq. (7), we obtain the boundary conditions at the surface as follows: (O~r/0n)s = / 3 ( 0 , Ors) o's.
(14)
The kinetic coefficient /3o increases linearly with tr~ in the range of o'~ << ¢rl, since the frequency of step generation increases with increasing o-s. For ~rs >> o'l,/3o approaches oq asymptotically as trs -~ oo because of the s t e p - s t e p interaction through the surface diffusion of admolecules. The anisotropy of/3(0, trs) increases with decreasing ~rs, since /3o decreases with a decrease in o'~. The degree of anisotropy is, therefore, not a constant but becomes larger during the growth of crystals because of the decrease in ¢rs with increasing the crystal size. Furthermore, from eqs. (8) and (14), the growth speed, V, of prism faces is a nonlinear function for cry. This causes a drastic change of the patterns of snow crystals during the growth in response to the growth conditions. On the other hand, in the case of
(15)
Eq. (15) is important for determining self-consistently of the growth speed V as determined by both the diffusion process and the surface kinetic processes [13]. We derive the boundary integral equation from eq. (5) using Green's theorem and the boundary conditions (eqs. (6) and (15)): ~rsi +
+ / 3 ( 0 , crs)G ~rs d F urface
=
-
"outer boundary [
--
O'~ ~n
dF,
(16)
where O,si is the supersaturation at a point i on the surface, G is the G r e e n function which has 6-fold symmetry and reflection symmetry about the line y = x / v ~ , and q= is the normal gradient of supersaturation at the outer boundary which located on a circle of radius Roo centered at the
254
E. Yokoyama / Formation of patterns during growth of snow crystals
center of the crystal. Assuming that the outer b o u n d a r y is far from the surface of crystal, qoo is approximately o-~ qoo -~ R= In( R = / r c ) '
(17)
which is obtained for the diffusion filed surrounding a circular crystal with a perfect sink, i.e., o-s = 0 , where r c is the average radius of the crystal. It is also assumed that Roo/r c = constant, so that the outer b o u n d a r y grows with increasing crystal size. This is because the normal derivative of the supersaturation at the surface determined by the two-dimensional Laplace equation agrees with that determined by the two-dimensional t i m e - d e p e n d e n t diffusion equation [22]. The surface is divided into N equal segments marked by grids points. Eq. (16) is changed to an algebraic equation with respect to the surface supersaturation o"s. W e use an iteration m e t h o d [13] to solve the nonlinear problem for which ~0 in eq. (14) depends on o-s. By solving the N simultaneous equations for all points i, we obtain the distribution of o-s. Consequently, from eqs. (8), (9) and (14), we find the growth speed of V at each point on the surface at a given moment, and then determine the shape after growth for a short time At. During growth, faster orientations are cut off by neighboring slower growing surfaces, so the slowest growing surface occupies the larger area. T h e growth pattern is, therefore, determined by the inner envelope of lines perpendicular to normal vectors of length VAt at each point [171.
Fig. 1. Growth pattern for ~r~= 1%. Lines are at time intervals of At = 107 (= 7X103 s).
10 -16 cm 2, mass of a water molecule m = 3.0 x 10 -23 g, step height or lattice constant d = 4.5 x 10 -8 cm, m e a n surface diffusion distance of an admolecule x s = 400d [24], edge free energy along a step K = 2.0 x 10 -6 e r g / c m , radius of initial crystal rc0 = 0.3 x 10 3 corresponding to 5 × 10 .3 cm for D = 0.2 c m 2 / s , and ratio of a distance Roo to an average radius r e of the crystal, R ~ / r c = 6.5 [17]. Fig. 1 shows an example of hexagonal patterns for small supersaturation ~r~ = 1%. T h e initial crystal starting from a circle develops into a hexagonal at t = 107, and then the hexagonal crystal grows, retaining its flat surface in spite of the nonuniformity in surface supersaturation [25], since the degree of the nonuniformity is not large.
3. Results T h e following values are used in this numerical calculation: t e m p e r a t u r e T = 258.15 K ( = - 15°C), equilibrium vapor pressure Pe = 1.66 × 103 d y n / c m 2 = 1.66 x 102 Pa corresponding to - 1 5 ° C , diffusion coefficient D = 0.2 c m 2 / s corresponding to 1 atm, i.e., 1.01325 × 105 Pa, sticking coefficient % = 0.01 [23], molecular volume of an ice crystal v c = 3.25 × 10 -23 cm 3, surface area occupied by a water molecule f0---8.3 ×
Fig. 2. Growth pattern for ~= = 16%. Lines are at time intervals of At = 5.7X 105 (= 4x 102 s).
E. Yokoyama / Formation of patterns during growth of snow crystals
Fig. 3. Growth pattern for tr~ = 32%. Lines are at time intervals of At = 5.7x 105 ( = 4 × 102 s).
For example, the surface supersaturation ors = 0.47% at the corners and ors = 0.41% at the face centers at t = 107 in fig. 1. Figs. 2 and 3 show examples of dendritic patterns for large supersaturation, or= = 16%, and for larger supersaturation, o-==32%, respectively. The growth speed at corners is larger than that at face centers because of the large nonuniformity in surface supersaturation, so there is preferred growth of corners. In fig. 2, the surface supersaturation ors = 2.4% at the corners and ors = 1.7% at the face centers at t = 11.4 x 105, and in fig. 3, ors = 3.6% at the corners and ors = 2.3% at the face centers at t = 11.4 x 105. Facets are developed and macrosteps are formed at the tips of primary stalks because of bunching of monomolecular steps on the facets. Such macrosteps might act as triggers for the formation of side branches. The width of a primary stalk in fig. 3 is smaller than that in fig. 2, since the position at which the growth speed is equal to that at corners, moves toward the corner from the face centers, with increasing nonuniformity in supersaturation.
4. Discussion
The excess supersaturation at the corners of the hexagonal crystal causes preferred growth of
255
corners. For small supersaturations o-=, however, the nonuniformity in supersaturation along the surface can be compensated for by variation of the local kinetic coefficient /3(0, ors) made possible by adjustment of the step distribution [12,13]. The steps generated at the corners slow down as they approach the face centers because of decreasing ors, and then the step density increases near the centers. Since the local kinetic coefficient/3 increases with the local slope (eq. (9)),/3 increases from the corners toward the face centers. Therefore, the growth speed V(0) at the center of the surface can reach V(1) at the corners, and the growth speed V can be constant over the whole surface. The surface of a hexagonal crystal such a shown in fig. 1, however, appears to be macroscopically fiat, because such small nonuniformity can be compensated for by a slight local slope of the order of 10 -2 at the face centers. On the other hand, for large supersaturations or=, the value o f / 3 at the face centers reaches the upper limit o~1 (eq. (9)). This is because the local slope s(0) at the face centers increases with increasing or=, because of an increase of the nonuniformity in ors with o-=. Once /3 = o/1 at the face centers at the critical supersaturation or*, the growth speed V(0) at the face centers can never reach V(1) at the corners for larger than or*, even though s(0) at the face centers increases with o'=. This is the onset of dendritic growth. From eqs. (8), (9) and (14), we obtain the following critical condition for the limit of stable growth of the hexagonal crystal, i.e., the transition from a hexagonal to a dendritic pattern: ors(O) = ( ors*/Orl ) t a n h ( o-1//ors* ) ors*,
(18)
where ors(0) is the supersaturation at the face centers and ors* is the critical supersaturation at the corners for o-*. The difference between ors(0) and ors(l), where ors(l) is the supersaturation at the corners, is given by [25] Ad~ L I aor t o-s(1) -ors(O) = ~ ~ n ] s,
(19)
where L is the dimensionless length of a side of
256
E. Yokoyama / Formation of patterns during growth of snow crystals
important role in the formation of patterns during growth of snow crystals. Furthermore, the conditions for the transition depend on the ratio of the edge free energy K along a step to the surface diffusion distance xs of an admolecule through 0"1 (eq. (13)). The ratio of K to x s affects the formation of patterns during growth of crystals.
10 °
N
10'
,/:@
Acknowledgments ¢)
.o
dendritic pattern hexagona~ pattern
E
,
I00
10
"--...,
20
Ooo/01 Fig. 4. Diagram showing the relationship between patterns and conditions (@/~1,-£~) for ~ l = 2.5%. The solid line described by eqs. (20) and (21) represents the transition from hexagonal to dendritic patterns.
hexagon and A~b = 0.554. From eqs. (7), (8), (14), (18) and (19), and o-s(1) = o-s*, we obtain 0-s* tanh 0"1
1
(20)
1 +.g~'
where
_2a= ( Aqb/2rr)alL, is a dimensionless crystal size. Substituting 0"s* into the general expression o"s(1), which is given by [25,26], we find 0"= o"s* = 1 + F 2 / ( 1 + S °) '
(21)
where F is a constant that depends on the ratio
R=/rc. The solid line in fig. 4 represents the transition from hexagonal to dendritic pattern determined by eqs. (20) and (21). The diagram in fig. 4 shows clearly that the dimensionless crystals size .~, relative to the mean free path of a water molecule, as well as the supersaturation 0"=, plays a very
The author is very grateful to the late Professor T. Kuroda for a number of stimulating and helpful discussions. He would like to thank Professor R.F. Sekerka for critical reading of the manuscript and helpful discussions. He would also like to thank Mr. T. Irisawa for his kind suggestions regarding computer programming. This work was partially supported by a Grant-inAid for Scientific Research from the Japanese Ministry of Education, Science and Culture, which is gratefully acknowledged.
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E. Yokoyama / Formation of patterns during growth of snow crystals [14] R.F. Xiao, J.I.D. Alexander and F. Rosenberger, Phys. Rev. A 38 (1988) 2447. [15] R.F. Xiao, LI.D. Alexander and F. Rosenberger, J. Crystal Growth 100 (1990) 313. [16] R.F. Xiao, J.I.D. Alexander and F. Rosenberger, Phys. Rev. A 43 (1991) 2977. [17] E. Yokoyama and T. Kuroda, Phys. Rev. A 41 (1990) 2038. [18] E. Yokoyama and T. Kuroda, J. Meteor. Soc. Japan 66 (1988) 927. [19] Y. Arima and T. Irisawa, J. Crystal Growth 94 (1989) 297.
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[20] T. Sei and T. Gonda, J. Crystal Growth 94 (1989) 697. [21] T. Kuroda and R. Lacmann, J. Crystal Growth 56 (1982) 189. [22] S.R. Coriell and R.L. Parker, J. Appl. Phys. 36 (1965) 632. [23] T. Kuroda and T. Gonda, J. Meteor. Soc. Japan 62 (1984) 563. [24] J. Kiefer and B.N. Hale, J. Chem. Phys. 7 (1977) 3206. [25] A. Seeger, Phil. Mag. 44 (1953) 1. [26] E. Yokoyama and T. Kuroda, J. de Physique 47 (1987) C1-347.