J. Phys. Chem. B 2009, 113, 4733–4738
4733
Growth of an Ice Disk: Dependence of Critical Thickness for Disk Instability on Supercooling of Water Etsuro Yokoyama* Computer Centre, Gakushuin UniVersity, Mejiro 1-5-1, Toshima-ku, Tokyo 171-8588 Japan
Robert F. Sekerka Physics and Mathematics, Carnegie Mellon UniVersity, Pittsburgh, PennsylVania 15213-3890
Yoshinori Furukawa Institute of Low Temperature Science, Hokkaido UniVersity, Sapporo, 060-0819 Japan ReceiVed: NoVember 6, 2008; ReVised Manuscript ReceiVed: January 28, 2009
The appearance of an asymmetrical pattern that occurs when a disk crystal of ice grows from supercooled water was studied by using an analysis of growth rates for radius and thickness. The growth of the radius is controlled by transport of latent heat and is calculated by solving the diffusion equation for the temperature field surrounding the disk. The growth of the thickness is governed by the generation and lateral motion of steps and is expressed as a power function of the supercooling at the center of a basal face. Symmetry breaking with respect to the basal face of an ice disk crystal is observed when the thickness reaches a critical value; then one basal face becomes larger than the other and the disk loses its cylindrical shape. Subsequently, morphological instability occurs at the edge of the larger basal face of the asymmetrical shape (Shimada, W.; Furukawa, Y. J. Phys. Chem. 1997, B101, 6171-6173). We show that the critical thickness is related to the critical condition for the stable growth of a basal face. A difference of growth rates between two basal faces is a possible mechanism for the appearance of the asymmetrical shape. Introduction We present a model of the time evolution of a disk crystal of ice with radius R and thickness h growing from supercooled water and discuss its morphological stability. Disk thickening, that is, growth along the c axis of ice, is governed by slow molecular rearrangements on the basal faces. Growth of the radius, that is, growth parallel to the basal plane, is controlled by transport of latent heat. Our analysis is used to understand the results obtained experimentally by Shimada and Furukawa.1 Shimada and Furukawa measured simultaneously the radius R and the thickness h of a circular disk by using an interferometer. They found that there are two types of disk growth prior to morphological instability of the disk shape, as illustrated in Figure 1: (type I) The radius and the thickness each increase at nearly constant speed; (type II) initially, the disk grows preferentially in the radial direction and later begins to grow both radially and in thickness. They also observed an asymmetric pattern with respect to the basal plane, that is, one basal face becomes larger than the other so that the surface joining basal faces (formerly sides of a cylinder) is no longer cylindrical. When the thickness exceeds a critical value, hc, morphological instability occurs at the edge of the larger basal face of the asymmetrical shape. They conclude that morphological instability is controlled by disk thickness rather than disk radius and show that hc is inversely proportional to the bulk supercooling ∆T. Effects of the growth kinetics of ice for growth perpendicular to the basal plane have been observed in capillary tubes2,3 and * To whom correspondence should be addressed. E-mail: yokoyama@ gakushuin.ac.jp.
Figure 1. There are two types of disk growth prior to morphological instability of the disk shape: (type I) The radius and the thickness each increase at nearly constant speed; (type II) initially, the disk grows preferentially in the radial direction and later begins to grow both radially and in thickness. When the thickness exceeds a critical value, hc, one basal face becomes larger than the other so that the surface joining basal faces is no longer cylindrical.
in connection with facetted grain boundary grooves.4,5 The basal faces cannot grow without the generation and lateral motion of steps. Such growth kinetics can play an important role in the morphology of ice crystallization. By using phase plane analysis of an ordinary differential equation6 for h with respect to R, we have analyzed the experimental results of ref 1. We have examined quantitatively how interface kinetics on the basal faces affect the time evolution of disk crystals and have shown that the difference between type I and type II corresponds to a
10.1021/jp809808r CCC: $40.75 2009 American Chemical Society Published on Web 03/10/2009
4734 J. Phys. Chem. B, Vol. 113, No. 14, 2009
Yokoyama et al.
difference of kinetics on the basal faces between spiral growth, with the aid of a screw dislocation, and growth by twodimensional nucleation.7 However, the mechanism for appearance of basal faces of different size near hc still remains an unsolved problem. The purpose of this study is to show that step kinetics on the basal faces plays an important role in the appearance of the asymmetrical pattern. We show that the experimental fact that hc is inversely proportional to ∆T can be related to the critical condition for stable growth of a basal face.
ψf0
as
∇2ψ(F, ζ) ) 0
(3)
The temperature field ψ(F, ζ) that satisfies eqs 1, 2, and 3 is given by9
ψ(F, ζ) )
hL dR 2πk dt
∫-11
dζ′
× + 2(ζ - ζ′)2 4F (4) 2 (1 + F) + 2(ζ - ζ)2
√(1 + F)
(
K
The Model Our model is based on the study of ice disk growth by Fujioka and Sekerka.8,9 For a disk crystal of ice with radius R and thickness h, we use a cylindrical coordinate system (F, ζ), in which the xy plane bisects the disk. The radial coordinate F is scaled by R and the ζ coordinate, scaled by h/2, is coaxial with the disk. We assume that the growth rate parallel to the basal plane, dR/dt, is determined by transport of latent heat and that the interfacial temperature is equal to the equilibrium melting temperature; whereas, the growth rate, dh/dt, along the c-axis is governed by slow interfacial kinetics that depends on the supercooling at the center of a disk face. Temperature changes due to latent heat produced at the basal faces is ignored because their growth is so slow. Since the interface moves sufficiently slowly that the thermal field has time to relax practically to its steady state value under the growth conditions of stable disk growth, for example, small crystal thickness and low supercooling, we use the quasi-steady state approximation. The fact that growth on the basal faces is limited by kinetics rather than transport of latent heat (which is released very slowly on those faces) makes the quasi-steady state approximation even better than it would be, for example, for a body whose growth in all directions contributes release of latent heat at about the same rate. After instability occurs and especially after dendritic growth begins, the quasi-steady state approximation would no longer be valid. Furthermore, the thermal properties of ice and water are taken to be equal for tractability. This should not cause significant error because most of the heat loss is through the liquid, which totally surrounds the disk. The relative temperature ψ(F, ζ) ) T - T∞, where T∞ is the temperature of the bulk supercooled water, is governed by
F2 + ζ2 f ∞
2
)
where ) h/2R and K is the complete elliptic integral of the first kind:
K(x) )
∫0π/2
dt
√1 - x2 sin2 t
The temperature ψ(0, 1) at the center of the basal face, ψ(1, 1) at the periphery of the basal face, and ψ(1, 0) at the center of the edge face are expressed by
ψ(0, 1) )
RL dR ln[2 + √1 + (2)2] 2k dt
(5) ψ(1, 1) )
ψ(1, 0) )
hL dR 2πk dt
∫01
hL dR 2πk dt
∫01
dx
√1 + (x)
(√
1
K
1 + (x)2
2
dx
√1 + (x/2)
(√
K
)
1
1 + (x/2)2
2
(6)
)
(7)
For , 1, these temperatures are given approximately by
ψ(0, 1) =
hL dR π 2πk dt
(8)
ψ(1, 1) =
hL dR (L - ln 2) 2πk dt
(9)
ψ(1, 0) =
hL dR L 2πk dt
(10)
(1) where the quantity L is
The boundary conditions are as follows: the heat balance at the edge can be written
( ∂ψ ∂F )
F)1-
-
( ∂ψ ∂F )
F)1+
)
LR dR H(1 - |ζ|) k dt
(2)
where k is the thermal conductivity of water, L is the latent heat of fusion per unit volume, and H(x) is the Heaviside step function defined by
H(x) ) Also
{
1 when 0 when
x>0 x dh-/dt, |pj+(R)| reaches pc at a shorter time than |pj-(R)| because of the monotonic increase function of |pj((R)| with respect to time. As a result, the disk would develop an asymmetrical pattern. Two possible reasons for the difference in growth rate of the basal faces are following: First, the local supercooling ∆Ts(0, 1) at the center of a basal face can be different from ∆Ts(0, -1) in an actual experiment due to some asymmetrical convection in the growth cell and/or the asymmetrical connection of the center of the disk with a glass capillary tube. Then
4738 J. Phys. Chem. B, Vol. 113, No. 14, 2009 |V+(0)| ) µ[∆Ts(0, 1)]2
and
|V-(0)| ) µ[(1 - δ1)∆Ts(0, -1)]
2
where δ1 is a reduction factor at the face center of the basal face ζ ) -1. Second, the point where a screw dislocation intersects the basal plane could be different on opposite basal planes. Then, for example, one could have
|V+(0)| ) µ[∆Ts(0, 1)]2
and
|V-(δ2)| ) µ[∆Ts(δ2, -1)]
2
where δ2 is a deviation of the face center of basal face ζ ) -1. Both cases produce dh+/dt > dh-/dt. In either case, the instability first occurs for the face with the larger growth rate, for which the local slope |pj+(R)| at the periphery first reaches the critical value pc. When |pj+(R)| becomes larger than pc, one can observe a high index plane near the periphery. Thus, the asymmetrical pattern might be due to unstable growth of one basal face due to preferential step bunching on one of the faces that results from a slightly different growth rate of the basal faces. An alternative explanation was put forward by Xu and Shimizu17,18 who analyzed morphological instability with respect to a perturbation of the form r ) R + δs sin ksz + δc cos kcz in the side faces of the disk. With their assumptions about the boundary conditions at the corners where z ) (h/2, they concluded that the first instability to occur was symmetry breaking, namely r ) R + δs sin ksz with ks ) π/h. Further experiments and analysis are needed to resolve which, if any, of these explanations, is correct. Acknowledgment. Support by JAXA in relation to an experiment in the Japanese Experiment Module “Kibo” aboard the International Space Station (Yokoyama and Furukawa) and former support by NASA (Sekerka) is gratefully acknowledged. References and Notes (1) Shimada, W.; Furukawa, Y. Pattern formation of ice crystals during free growth in supercooled water. J. Phys. Chem. 1997, B101, 6171–6173.
Yokoyama et al. (2) Hillig, W. B. The kinetics of freezing of ice in the direction perpendicular to the basal plane. In Growth and Perfection of Crystals; Doremus, R. H., Roberts, B. W., Turnbull, D., Eds.; Wiley: New York, 1958; pp 350-360. (3) Michaels, A. S.; Brian, P. L. T.; Sperry, P. R. Impurity effects on the basal plane solidification kinetic of supercooled water. J. Appl. Phys. 1966, 37, 4649–4661. (4) Wilen, L. A.; Dash, J. G. Giant facets at ice grain boundary grooves. Science 1995, 270, 1184–1186. (5) Dash, J. G.; Hodgkin, V. A.; Wettlaufer, J. S. Dynamics of faceted grain boundary grooves. J. Stat. Phys. 1999, 95, 1311–1322. (6) Bender, C. M.; Orszag, S. A. AdVanced Mathematical Methods for Scientists and Engineers; McGraw-Hill: New York, 1978; pp 171-195. (7) Yokoyama, E.; Sekerka, R. F.; Furukawa, Y. Growth trajectories of disk crystals of ice growing from supercooled water. J. Phys. Chem. 2000, B104, 65–67. (8) Fujioka, T.; Sekerka, R. F. Morphological stability of disc crystals. J. Cryst. Growth 1974, 24/25, 84–93. (9) Fujioka, T. Study of Ice Growth in Slightly Undercooled Water. Doctoral Thesis. Department of Metallurgy and Materials Science, Carnegie Mellon University, Pittsburgh, PA, 1978. (10) Shimada, W. Experimental Studies on the Pattern Formation in Growth of Ice Crystals. Doctoral Dissertation. Institute of Low Temperature Science, Hokkaido University, Sapporo, Japan, 1995. (11) Furukawa, Y.; Yokoyama, E. Ice crystal pattern formation under gravity and microgravity conditions. ESA SP-454 2001, 465–471. (12) Chernov, A. A. Stability of faceted shapes. J. Cryst. Growth 1974, 24/25, 11–31. (13) Kuroda, T.; Irisawa, T.; Ookawa, A. Growth of a polyhedral crystal from solution and its morphological stability. J. Cryst. Growth 1977, 42, 41–46. (14) Yokoyama, E.; Kuroda, T. Pattern formation in growth of snow crystals occurring in the surface kinetic process and the diffusion process. Phys. ReV. A 1990, 41, 2038–2049. (15) Yokoyama, E. Formation of patterns during growth of snow crystals. J. Cryst. Growth 1993, 128, 251–257. (16) Yokoyama, E.; Giga, Y.; Rybka, P. A microscopic time scale approximation to the behavior of the local slope on the faceted surface under a nonuniformity in supersaturation. Phys. D 2008, 237, 2845–2855. (17) Xu, J. J.; Shimizu, J. Asymptotic theory of disc-like crystal growth: (I). Basic sate solution. Discrete Continuous Dyn. Syst. 2004, B4, 1091– 1116. (18) Xu, J. J.; Shimizu, J. Asymptotic theory of disc-like crystal growth: (II). Interfacial instability and patter formation at early stage of growth. Commun. Pure Appl. Anal. 2004, 3, 527–543.
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