Measurements of Cylindrical Ice Crystal Growth Limited by
Combined Particle and Heat Diffusion
Kenneth G. Libbrecht
Department of Physics, California Institute of Technology
Pasadena, California 91125
Abstract. We present measurements of the growth of long columnar ice crystals from water vapor over a broad range of temperatures and supersaturation levels in air. Starting with thin, c-axis ice needle crystals, we observed their subsequent growth behavior in a vapor diffusion chamber, extracting the initial radial growth velocities of the needles under controlled conditions. Approximating the hexagonal needle crystals as infinitely long cylinders, we created an analytical growth model that includes effects from particle diffusion of water molecules through the surrounding air along with the diffusion of heat generated by solidification. With only minimal adjustment of model parameters, we obtained excellent agreement with our experimental data. To our knowledge, this is the first time that the combined effects from particle and heat diffusion have been measured in ice growth from water vapor. This analysis further provides an accurate method for calibration of the water-vapor supersaturation levels in experimental growth chambers.
We recently developed a novel dual diffusion chamber for observing the growth of ice crystals from water vapor in air, which allows us to create slender needle crystals and measure their subsequent growth behavior under carefully controlled conditions. The experimental apparatus is described in some detail in . Figure 1 shows an example of a thin, plate-like ice crystal growing on the end of a long ice needle at a temperature of -15 C. Our overarching goal with these observations is to develop a comprehensive model of ice crystal growth from water vapor that can reproduce quantitative growth rates as well as growth morphologies over a broad range of circumstances. Although ice crystal formation has been studied extensively for many decades, our understanding of the physical effects governing growth behaviors at different temperatures and supersaturations is still rather poor [2, 3, 4, 5, 6, 7, 8].
Determining the water vapor supersaturation in ice growth experiments done in air has long been a challenge, and it remains a significance hindrance to making accurate, quantitative ice growth measurements. While a small thermistor probe can easily determine air temperatures with excellent absolute accuracy and little perturbation of the surrounding environment, water vapor probes (hygrometers) are typically bulky and quite limited in absolute accuracy. Moreover, in a supersaturated environment, water vapor condenses on solid surfaces, and the presence of unwanted ice surfaces can greatly affect the supersaturation field in their vicinity. As a result, one often resorts to modeling of the experimental chamber to determine the supersaturation within.
The second diffusion chamber in our dual-chamber apparatus was carefully designed to facilitate accurate modeling of the water vapor supersaturation . The top and bottom surfaces have constant, well controlled temperatures, and the side walls were constructed to maintain a simple, linear vertical temperature gradient throughout the chamber. All walls of the chamber are coated with ice crystals during operation, thus providing well-defined boundary conditions for constructing a heat and water vapor diffusion model of the interior of the chamber.
1.1 Supersaturation in the Diffusion Chamber
The temperatures of the top and bottom of the second diffusion chamber were defined by so (see  for the chamber dimensions), and the temperatures of the four walls were maintained at Solving the heat diffusion equation within the chamber then yields the air temperature and in particular we have at the center of the chamber. Moreover, the temperature gradient inhibits convective air currents within the chamber. A shutter on one wall of the chamber is opened briefly to allow the transport of crystals into the chamber, but it is otherwise kept closed to maintain a stable temperature profile within the chamber.
If we first imagine moving the side walls of the chamber out to infinity, then we can use a plane-parallel approximation to estimate the water-vapor supersaturation at the chamber center, where test crystals are positioned. Solving the diffusion equation for water vapor density yields a linear function with a constant gradient and , yielding the supersaturation at the center of the chamber
where is the saturated vapor pressure above an ice surface. This expression gives the exact value for in the plane-parallel approximation (ignoring small changes in the diffusion constant with temperature).
For small , we expand the above expression to obtain the simpler expression
The function can be calculated using to good accuracy, where is the temperature in Kelvin. In practice, we have found that the quadratic expansion is usually accurate enough for our purposes, as it differs from the exact expression for by less than a percent when C, and it is only a few percent high when C.
To see how the chamber walls and the crystal support structure affected the supersaturation field, we examined a range of computational models of the diffusion chamber under different conditions, with one example shown in Figure 2. In these models we solved the dual-diffusion problem (temperature and water-vapor density) numerically in three dimensions. We found that, over a broad range of conditions near C, the side walls reduced by a factor of approximately 0.8 compared to the plane-parallel approximation, and an ice-covered central stem further reduced by a factor of approximately 0.9. Combined, our models indicated that these effects lowered the supersaturation at the center by a constant geometrical factor of compared with the plane-parallel approximation, so that Equation 1.1 becomes
Our models also indicated, however, that there remained significant uncertainty in our ability to calculate , arising mostly from difficulties in accurately modeling effects from the crystal support structure and from a microscope objective placed inside the chamber about 90 mm away (horizontally) from the growing crystals.
1.2 Modeling Cylindrical Crystal Growth
As we began measuring crystal growth rates with this apparatus, we soon realized that the radial growth of the ice needles could be used to calibrate the supersaturation as a function of and In essence, the prism surfaces of the columnar needles serve as “witness” surfaces, providing a fairly accurate measure of the surrounding water vapor supersaturation. As we will see below, this calibration works because the radial growth of the needles is limited primarily by particle and heat diffusion, and is nearly independent of the attachment coefficient at the needle surface. Therefore we do not need to know with great accuracy to calibrate
To see this, consider the growth of an infinitely long cylindrical ice crystal. Ignoring latent heat generation for the moment, we can solve the particle diffusion equation in cylindrical coordinates to yield the general solution where and are constants to be determined by the boundary conditions in the equation. Here we have approximated the diffusion equation by Laplace’s equation, which is quite accurate in this situation, as the dimensionless Peclet number is much less than unity .
At the outer boundary of this model we assume a constant supersaturation where is the outer boundary of the cylindrically symmetric diffusion field, and is essentially equal to described above. (Note that cannot be set to infinity in cylindrical coordinates, as is commonly done in spherical coordinates.) Equating with ignores the particle density gradient in the diffusion chamber, which is justified by the observation that vertical asymmetries in crystal growth rates are generally quite small.
At the inner boundary , equal to the surface of the cylindrical crystal, we write the radial growth velocity
where m/sec is the particle diffusion constant, is the number density of ice, and is the kinetic velocity defined in . Including these boundary conditions in the solution for then gives
with and m. This cylindrical solution is similar to the spherical case presented in .
Using typical numbers (as we will see below) of m and cm, we obtain the rather small value
Comparing this with the measurements presented in , we find that in most circumstances, allowing us to write
This equation gives us a good prediction for as all the parameters are rather tightly constrained except for (which is determined roughly by our modeling of the experimental chamber, as described above) and However, Equation 8 only applies in the absence of crystal heating from solidification, which produces a significant perturbation of , so we next examine thermal considerations in our cylindrical crystal model.
Heating occurs because the growth of the cylindrical crystal releases a latent heat per unit length of
where J/kg is the latent heat for the vapor/solid transition and kg/m is the ice density. This generated heat must then be removed via conduction through the air surrounding the crystal (ignoring convective air currents). Solving the heat diffusion equation in cylindrical coordinates is similar to solving the particle diffusion equation described above, and doing so yields a temperature rise of the crystal (relative to the air temperature at ) given by
where W m K is the thermal conductivity of air. The temperature rise increases the equilibrium vapor pressure of the ice to
where , and a bit of algebra reveals that this reduces the growth rate to (see )
2 Comparison with Crystal Growth Measurements
For all the crystal growth data presented here, we first measured the needle radius as a function of time at a location m below the needle tip, starting from images similar to that shown in Figure 1. Figure 3 shows one example crystal grown at a temperature of C and C. This choice of was something of a compromise, being close enough to the needle tip to be relevant for subsequent observations of ice structures at the tip, while far enough below the tip that these same structures did not greatly influence for small
The optical microscope used to photograph the crystal had a resolving power of 2.5 m, and the image pixels measured 0.85 m. Our diameter resolution was therefore about m, giving radial measurements that were accurate to about m. We then fit the data to a smooth curve to determine at a time when m, as shown in Figure 3. We chose the smallest practical for which we could accurately measure because the tip structures more greatly perturbed the cylinder growth at later times, when the tip structures (especially plates) were larger in size. We did observe some variation in the measured with changing , with different tip structures, and between different crystals grown in ostensibly the same conditions. But these variations were at roughly the percent level, so they did not alter our analysis greatly.
Figure 4 shows the resulting radial growth velocity m) measured at two representative growth temperatures as a function of These data, along with similar data at other temperatures, were well fit with simple quadratic functions The measured fit coefficients were then compared with calculated from Equation 10, and the results are summarized in Figure 5. The theory curves used , and was adjusted to fit the particle+heat diffusion curve to the data, yielding a best fit cm. As can be seen in Figure 5, our data are in excellent agreement with the expected particle+heat diffusion prediction over the entire temperature range tested, with a physically reasonable choice for
From this comparison between theory and experiment, we can extract a prediction for essentially equal to the supersaturation at the center of the diffusion chamber in the absence of any test crystals. Parameterizing this as the lower theory curve in Figure 5 becomes the curve shown in Figure 6. The calculated using cm extracted from the data, then replaces Equation 3 as our best estimate of the supersaturation at the center of our diffusion chamber, now calibrated using experimental data.
In summary, we have modeled the early growth of ice needle crystals in a vapor diffusion chamber using a cylindrically symmetric approach that approximates the needles as infinitely long cylinders. The largely analytical model (with the geometrical correction factor provided by numerical simulations) then yielded Equation 10, which gives the radial growth velocity as a function of and other experimental parameters. Comparing this predicted with measurements at m, we found excellent agreement using a sensible value of cm for the outer boundary in the model.
The data clearly indicate that both heat diffusion and particle diffusion limit the crystal growth rates, as theory predicts. To our knowledge, this is the first time that ice growth experiments have achieved sufficient absolute accuracy to verify this basic theoretical prediction. Having a reliable understanding of the supersaturation and resulting crystal growth behavior for this simple cylindrical geometry is a major step forward in producing accurate, quantitative measurements and models of more complex ice crystal growth behaviors.
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