J. Phys. Chem. B 2000, 104, 65-67
65
Growth Trajectories of Disk Crystals of Ice Growing from Supercooled Water Etsuro Yokoyama* Department of Perceptual Science and Design Engineering, Yamaguchi UniVersity, Tokiwadai 2-16-1, Ube, 755-8611 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: April 20, 1999; In Final Form: October 12, 1999
The time evolution of a disk crystal of ice with radius R and thickness h growing from supercooled water is studied using a solution of an ordinary differential equation (ODE) for h with respect to R. The growth of thickness, i.e., growth along the c axis of ice, is governed by slow molecular rearrangements on the basal plane and is expressed as a power function of the supercooling at the center of a disk face. Growth of the radius, i.e., growth in the basal plane, is controlled by transport of latent heat and is calculated by solving the diffusion equation for the temperature field surrounding the disk. We combine both functional expressions for growth rate to obtain an ODE for h as a function of R. Our analysis in the (R, h) plane is used to understand the recent results obtained experimentally by Shimada and Furukawa (J. Phys. Chem. 1997, B101, 61716173), in which they found two types of growth trajectories for ice disks prior to morphological instability.
1. Introduction
2. The Model
We present a model of the growth of a disk crystal of ice from pure supercooled water. In particular, we concentrate on the time evolution of the disk radius, R, and the disk thickness, h, under the assumption that the disk shape remains bounded by two basal planes, perpendicular to the c axis, connected by a thin circular cylinder. Our model is based on an expression for the growth rate in the radial direction of an ice disk by Fujioka and Sekerka.1 Recently, Shimada and Furukawa2 observed the growth of ice crystals from supercooled water. They measured simultaneously the radius R and the thickness h of a circular disk and found that morphological instability of the disk shape occurs when the thickness reaches a critical value, hc. They also found that there are two types of trajectories of disks prior to the instability: (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 in both radius and thickness. We conjecture that the difference between these two growth types is due to the fact that different kinetic mechanisms are operative on the basal plane. The purpose of this study is to examine quantitatively how interface kinetics on the basal plane affects the time evolution of disk crystals. Effects of the growth kinetics of ice for growth perpendicular to the basal plane have been observed in capillary tubes3,4 and in connection with faceted grain boundary grooves.5,6 The measured growth rate can be represented as a power function of the supercooling at the center of the nearly planar interface that separates the ice from pure water. We adopt such a power law to represent the growth kinetics of the basal planes of an ice disk.
For a growing ice disk with radius R and thickness h, we assume that the growth rate in the basal plane, dR/dt, is determined by transport of latent heat, whereas the growth rate, dh/dt, along the c axis is governed by slow interfacial kinetics as a function of the supercooling at the center of a disk face. The growth rate, dR/dt, in the radial direction of a growing ice disk with thickness h is obtained by solving the diffusion equation for the temperature field surrounding a disk crystal,1 resulting in
dR k∆T 2π ) dt Lh L
(1)
where k is the thermal conductivity of water, L is the latent heat of fusion per unit volume, and ∆T ) TM - T∞ is the bulk supercooling, where TM is the melting point of ice and T∞ is the temperature of the bulk supercooled water. The quantity L is
L ) 1 + ln(16R/h)
(2)
The growth rate of the thickness h is given by a general interfacial kinetic law, which has the form of a function of the interfacial supercooling ∆Ts ) TM - Ts, where Ts is the temperature at the center of a disk face, i.e., a bounding basal plane of the disk. Several functional forms related to the molecular arrangement of a solid surface have been developed. To explain the difference between types I and II in Figure 2 of ref 2, however, it is important that the functional expression corresponds with the actual measured data of the growth rate of the basal face. Michaels et al.4 have measured the growth of ice on the basal plane from pure water in capillary tubes and
10.1021/jp991280b CCC: $19.00 © 2000 American Chemical Society Published on Web 12/09/1999
66 J. Phys. Chem. B, Vol. 104, No. 1, 2000
Yokoyama et al.
Figure 1. Dependence on the supercooling exponent R of the trajectories of dimensionless thickness h˜ with respect to dimensionless radius R ˜ . R ) 2 corresponds to spiral growth on the basal plane with the aid of a screw dislocation and R ) 10 to layer growth by twodimensional nucleation.
have shown that the growth rate varies in proportion to (∆Ts)R, with measured exponents of approximately R ) 2 and 10. The power R ) 2 corresponds to spiral growth that can occur with the aid of a screw dislocation intersecting the surface. The power R ) 10 implies that growth on the basal plane is governed by two-dimensional nucleation and layer growth in the range of ∆Ts < 0.07 K. Thus, we adopt a growth rate for the thickness h of the form
dh/dt ) 2µ(∆Ts)R
(3)
where µ is a kinetic coefficient and the factor of 2 arises because both faces of the disk are growing in opposite directions. The supercooling ∆Ts at the center of a basal plane is obtained by solving the diffusion equation for the temperature field surrounding a disk crystal,7 resulting in
∆Ts ) ∆T(1 - π/L)
(4)
In the derivation of eqs 1 and 4, the thermal properties of ice and water are taken to be equal for tractability. Furthermore, because the interface moves sufficiently slowly that the thermal field has time to relax practically to its steady-state value, the quasi-steady-state approximation is assumed. We shall discuss the effect of these simplifications later. We can obtain an ordinary differential equation (ODE) for the thickness h with respect of R by combining eqs 1 and 3 to obtain
π dh h ) L 1dR h0 L
(
R
)
(5)
where
h0 )
πk 1 µL ∆TR-1
The solution of eq 5 is a trajectory in the (R, h) plane. We analyze the time evolution of an ice disk in this plane by means of phase plane analysis.8 3. Results and Discussion We first discuss the growth trajectories described by eq 5 in ˜ ) R/h0. This terms of dimensionless variables, h˜ ) h/h0 and R
Figure 2. Trajectories of the dimensional thickness h with respect to the dimensional radius R computed from eq 5, along with experimental data A, B, C, D, and E from Figure 2 of Shimada and Furukawa.2 Curves S0 and S1 correspond to R ) 2 (screw dislocation), whereas T0, T1, and T2 correspond to R ) 10 (two-dimensional nucleation). For S0 and T0, the thermal conductivities of ice and water are assumed to be equal to that of water, whereas for S1 and T1, the conductivity of ice is 4 times that of water. The curve T2, which fits the data for D and E, is for an 18% reduction in ∆Ts relative to that for T0, instead of the 35% reduction used for T1.
analysis is carried out for R ) 2 and 10. Figure 1 shows the dependence on R of the trajectories of dimensionless thickness h˜ with respect to the dimensionless radius R ˜ for the initial conditions R ˜ (0) ) 1 and h˜ (0) ) 0.01. The growth trajectory is strongly affected by R, i.e., the mechanism of interface kinetics on the basal plane. The increase in thickness h˜ with increasing R ˜ for R ) 2 is rapid, while for R ) 10 the thickness h˜ increases gradually with increasing R ˜ . We also calculated such trajectories starting from various initial conditions, R ˜ (0) and h˜ (0), and confirmed that the general shapes of these R ˜ versus h˜ curves are little affected. To calculate the dimensional values of R and h from eq 5, the following values, which correspond to Figure 2 of Shimada and Furukawa,2 are used: supercooling ∆T ) 0.09 K, thermal conductivity of water k ) 0.001 44 cal cm-1 s-1 K-1, latent heat of fusion per unit volume L ) 80 cal cm-3, initial radius of disk R(0) ) 2 × 10-3 cm, and initial thickness of disk h(0) ) 2 × 10-3 cm. Values of the kinetic coefficient µ are estimated from Figure 6 of Michaels et al.:4
µ)
{
2.5 × 10-2 cm s-1 K-2 1.0 × 108 cm s-1 K-10
for R ) 2 for R ) 10
We compare the growth trajectories described by eq 5 with the growth behaviors of types I and II disks measured by Shimada and Furukawa2 with a Mach-Zender interferometer. Figure 2 shows both of our calculated results and the experimental data from Figure 2 of Shimada and Furukawa.2 According to the calculated trajectories, S0, S1, T0, and T2, a thickness of h ) 0.01 cm is reached at the times 113, 214, 573, and 1604 s, respectively. Crude estimates of the corresponding times from nearby experimental curves in Figure 2 of ref 2 are 318, 609, 583, and 1119 s, showing agreement in order of magnitude, which is reasonable considering that the initial conditions are different and the curves do not really coincide.
Growth Trajectories of Disk Crystals of Ice As suggested by Shimada and Furukawa, we conjecture that type I corresponds to R ) 2 and type II to R ) 10. The trajectory S0 for R ) 2 is in general agreement with the experimental data of type I, denoted by A and B. On the other hand, the trajectory T0 for R ) 10 disagrees with the data of type II, denoted by D and E. It is not clear if the experimental data denoted by C is type I, because the trajectory changes rapidly at the point of R ) 1.13 × 10-1 cm and h ) 0.025 × 10-1 cm. For some unknown reason, the basal planes of the disk C seem to grow as (∆Ts)10 in its initial state, but when the radius reaches 1.13 × 10-1 cm, the mechanism of growth appears to agree with (∆Ts)2. This abrupt change in slope might result simply from lack of enough data points, or if real, it might be due to the nucleation of a dislocation during growth. We explore the disagreement between the model for R ) 10 and the experimental results for type II. As noted previously, eq 5 is based on the assumption of identical thermal conductivities of ice and water. Considering that the conductivity of ice is about 4 times larger than that of water, the temperature Ts at the center of the disk face more nearly approaches the melting point, TM. Thus, the correct supercooling ∆Ts should be smaller than that in eq 4. We are also aware that eq 2 should be modified for the case when the thermal conductivities are different, but this has a much smaller effect on the trajectories than the change in ∆Ts. Because this change in dR/dt is difficult to estimate quantitatively, we ignore it as an approximation. Fujioka7 has estimated numerically the reduction in the supercooling ∆Ts ) TM - Ts given by eq 4 to be about 35%, using a distinct conductivity model and the quasi-steady-state approximation. We introduce this 35% reduction in ∆Ts by means of a multiplicative factor of 0.35 on the right-hand side of eq 4 and calculate curves T1 and S1. Because the growth for R ) 2 is not sensitive to ∆Ts, the modified trajectory S1 still agrees with the experimental data of type I. The modified trajectory T1 for R ) 10, however, changes considerably from T0, and the calculated thickness is now smaller than the experimental values, D and E. Possibly this 35% reduction is an overestimate, for the following reasons: First, there is some convection in the growth cell, which would bring colder temperatures closer to the disk. This would tend to cause the actual supercooling to be larger in Shimada and Furukawa’s experiment. In particular, just before dendritic growth begins to occur, the disk is observed to have an asymmetrical shape, presumably because of convection. Second, in the experiment, a glass capillary tube connects with the center of the disk. Although the thermal conductivity of the glass is about the same as that of water, there is ice of higher thermal conductivity inside the tube, which connects the centers of the basal planes to the bulk supercooled water. Thus, the presence of the capillary would tend to increase the actual supercooling. If we compute the trajectory using only an 18% reduction in ∆Ts instead of 35%, we obtain trajectory T2, which agrees with the data D and E for disks of type II. 4. Conclusions We have studied the time evolution of a disk crystal of an ice crystal with radius R and thickness h growing from supercooled water. Using phase plane analysis of the ordinary differential equation for h with respect to R, we have analyzed the recent results obtained experimentally by Shimada and Furukawa,2 in which they found two types of growth of an ice
J. Phys. Chem. B, Vol. 104, No. 1, 2000 67 disk, type I and II. We examined whether the difference between types I and II corresponds to a difference of kinetics on the basal plane between spiral growth, with the aid of a screw dislocation, and growth by two-dimensional nucleation. The type I trajectories measured by Shimada and Furukawa2 with a Mach-Zender interferometer are consistent with the growth rates governed by a screw dislocation mechanism (R ) 2) measured by optical microscopy in capillary tubes by Michaels et al.4 Thus, we conclude that the basal planes of type I disks probably grow with the aid of screw dislocations. On the other hand, the measured growth trajectories of type II disks are consistent with growth rates (R ) 10) measured in capillary tubes for two-dimensional nucleation.4 Because the growth rate for two-dimensional nucleation is very sensitive to the supercooling ∆Ts at the center of a disk face, we can only get quantitative agreement if ∆Ts is somewhere between that estimated by Fujioka for a distinct conductivity model and a model with the same conductivities. Thus, for an 18% reduction of ∆Ts, relative to the same conductivity model, rather than a 35% reduction, which would correspond to the distinct conductivity model, we obtain reasonable agreement (curve T2 of Figure 2). The justification for only an 18% reduction is ascribed to the following: (a) Relative to a model in which the thermal conductivities of ice and water are assumed to be equal to that for water, allowance for a conductivity of ice that is about 4 times larger than that of water reduces ∆Ts by 35%.7 (b) There is some convection in the growth cell, which would tend to increase the actual ∆Ts. (c) A glass capillary tube occupied by ice connects with the center of a disk in the experiment, which would also tend to increase the actual ∆Ts. Thus, we conclude that the growth of the basal planes of type II disks is consistent with a two-dimensional nucleation mechanism. Acknowledgment. The authors are pleased to acknowledge a grant of Monbusho International Scientific Research Program: Joint Research 10044049 from the Ministry of Education, Science, Sports and Culture, Japan. R.F.S. acknowledges the support of the Invitation Fellowship Programs for Research in Japan from the Japan Society for the Promotion of Science (JSPS). References and Notes (1) Fujioka, T.; Sekerka, R. F. Morphological Stability of Disk Crystals. J. Cryst. Growth 1974, 24/25, 84-93. (2) Shimada, W.; Furukawa, Y. Pattern Formation of Ice Crystals during Free Growth in Supercooled Water. J. Phys. Chem. 1997, B101, 61716173. (3) 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. (4) Michaels, A. S.; Brian, P. L. T.; Sperry, P. R. Impurity Effects on the Basal Plane Solidification Kinetic of Supercooled Water. J. Appl. Phys. 1996, 37, 4649-4661. (5) Wilen, L. A.; Dash, J. G. Giant facets at ice grain boundry grooves. Science 1995, 270, 1184-1186. (6) Dash, J. G.; Hodgkin, V. A.; Wettlaufer, J. S. Dynamics of faceted grain boundary grooves. J. Stat. Phys. 1999, 95, 1311-1322. (7) Fujioka, T. Study of ice growth in slightly undercooled water. Doctoral Thesis, Department of Metallurgy and Materials Science, CarnegieMellon University, Pittsburgh, PA, 1978. (8) Bender, C. M.; Orszag, S. A. AdVanced mathematical methods for scientists and engineers; McGraw-Hill: New York, 1978, pp 171-195.
