Introduction
In 1948Reference Workman and Reynolds Workman and Reynolds (1950) investigated an electrical effect which accompanies the phase change of dilute aqueous solutions from the liquid to the solid phase They observed large electrical potentials (200 V or more in some cases) and discharge currents on the order of 1 μA measured across the interface between the solid and liquid phases of the solutions during the freezing process. This effect has been termed the “Workman-Reynolds effect, the “Costa Ribeiro effect”, and the “Thermodielectric effect” (Reference GrossGross 1954) In this paper we shall use the terminology ofReference Gross Gross (1958) in a recent review article and cal! the above effect the Workman Reynolds (W-R) effect. The electrical potentials will be referred to as the freezing potentials” and the electrical discharge currents as “the freezing currents”.
A qualitative explanation of the W-R effect was offered byReference Workman and Reynolds Workman and Reynolds (1950). Their explanation was based on pi I differences between the unfrozen solutions and the melt water from the solid phase. These authors considered the effect to be due to selective incorporation of one solute ion into the solid phase during the freezing process. The ideas of Workman and Reynolds have been extended and refined, particularly byReference Gross Gross (1965),Reference LeFebre LeFebre (1967), and Reference Drost-HansenDrost-Hansen (1967).
Although the W-R effect has been studied extensively, it appears that the effect of crystal orientation has not been studied in detail. The importance of crystal orientation is suggested by experience in crystal growth from the melt. (Muffin and Hylme, 1960) and by some experiments with dielectrics in which a phenomenon similar to the W-R effect has been observed.Reference Mascarenhas and Freitas Mascarenhas and Freitas (1960) found, for single crystals of napthalene, that the charge separation for growth normal to the cleavage plane is about ten times greater than for growth in any other direction.
In the present investigation, the freezing potentials and the freezing currents have been measured during the phase change of dilute KC1 solutions into single crystals of ice. The measurements were made for two growth directions, parallel and perpendicular to the e-axis. In addition, the distribution of solute in the liquid phase adjacent to the interface was determined.
Theory
Reference LeFebreLeFebre (1967) has derived an equation for the freezing potential
where I is the charging current due to the difference between the rates of incorporation of the positive and negative ions, α is a growth-rate constant, Ci, is the interface capacitance, R i, is the interface resistance, R is the crystal growth rate and t is the time measured from the beginning of crystal growth. This equation for Vs does not apply to constant crystal growth rates which were used in the present experiments. LeFebre's equation reduces to
for the condition of constant crystal growth rates.
If the resistance of the ice can be neglected in comparison to Ri then
where Vm is the maximum freezing potential and I m is the maximum freezing current. The separate ionic distribution coefficients are defined to be
where C+ s and C- s are the positive and negative ion densities in the ice at the ice-liquid interface and C+ L and C- L are the positive and negative ion densities in the liquid at the ice-liquid interface. These unequal ionic distribution coefficients and the separate ionic diffusion coefficients D+ and D- are related (Reference LeFebreLeFebre, 1967) to the freezing potential Vs through the charging current I. In the case of identical ionic diffusion coefficients the freezing potential can be attributed to the difference between the ionic distribution coefficients
where I'is the charge parameter defined by Reference LeFebreLeFebre (1967). The equilibrium solute distribution coefficient is approximately
The data for the solute distribution in the liquid phase have been analyzed using the equation of Reference Tiller, Tiller, Jackson, Rutter and ChalmersTiller and others (1953):
Where CL(x) is the solute concentration in the liquid at a distance x ahead of the ice-liquid interface, C o is the solute concentration in the bulk liquid, K o is the equilibrium solute distribution coefficient, R is the crystal growth rate and D o is the solute diffustion coefficient in the liquid phase. Equation (7) is valid provided (1) a steady-state solute distribution has been established in the liquid; (2) diffusion in the solid is negligible; (3) the value of K o is constant and K o < 1; (4) convection in the liquid is negligible (i.e. no mixing of the solute in the liquid phase).
Experimentally, a steady-state solute distribution was established in the liquid phase after a 40 mm thickness of solid had formed at a growth rate of 5.5 μm/s (Osterkamp, unpublished). It should be emphasized that this result is valid only for the growth conditions of the present investigation. Somewhat different results (Reference GrossGross, 1968) have been obtained under different growth conditions. According to Gross (1968), diffusion of solute in the solid phase is extremely low. Reference Jaccard and LeviJaccard and Levi (1961) have found a concentration dependence of the distribution coefficients of NH3 and NH4F in ice, but it does not appear that an investigation has been made for KCl in ice. Since the present investigation was primarily concerned with a small concentration range (2.2–2.4 X 10 -4 N), condition (3) was satisfied.
The experimental conditions maintained in the present investigation indicate the presence of convection in the liquid phase. Thus, convection in addition to diffusion caused the transfer of solute away from the interface and Equation (7) was not strictly valid for the system. However, the data have been analyzed under the assumption that diffusion was the primary means of so lute transport away from the interface. The distribution and diffusion coefficients were obtained from a semi-logarithmic plot of Equation (7). These were called “apparent” distribution and diffusion coefficients because of the presence of convection in the liquid phase.
The solidification of dilute aqueous solutions during small c rystal growth rates results in the formation of clear, g lass -like ice with a smooth or planar interface (Reference Harrison, Tiller and KingeryHarrison and Tiller, 1963; Reference Weeks, Lofgren and OuraWeeks and Lofgren, 1967) . At larger crystal growth rates the ice changes to a cloudy or milky appearance and the planar interface becomes unstable and breaks d own into a cellular interface . A necessary condition for the formation of a cellular interface is the presence of a constitutionally supercooled layer ahead of the advancing solid-liquid interface. Reference Tiller, Tiller, Jackson, Rutter and ChalmersTiller and others (1953) have derived the expression
where G is the temperature gradient in the liquid and m is the s lop e of the liquidus line. Equation (8) is a criterion for the presence of constitutional supercooling in the liquid phase and was used to obtain estimates of K o since the change from clear to cloudy ice was observed experimentally.
At a cellular interface, the concentration of solute in the ice cells is relatively small and the solute tends to accumulate between the ice cells. The length of the cells is given (Reference Harrison, Tiller and KingeryHarrison and Tiller, 1963) as
where G s is the average temperature gradient in the solid , $$ is a geometrical factor and δ/a is the ratio of the distance between cells to the cell diameter. The value of the term $$δ/a ≈ 0.5 for ice.
Experimental details
Pure water (conductivity ≈ 10-5 Ω-1 m-1) was obtained by processing ordinary tap water The tap water was doubly distilled and then passed through a mixed-bed.ion exchange resin and μm filter. The conductivity measurements were made in a Jones-type conductivity cell with an A C Wheatstone bridge consisting of a General Radio, 1605 AH comparator and General Radio 510 resistors. This pure water was used for the solutions and also as a rinse and soak water.
Large single crystals of ice (26 mm O.D.) were grown in the “Lucite” (polymethyl methacrylate) tubes shown in Figure 1 and were examined optically between crossed polaroids to determine their crystalline perfection. The crystal orientation was determined with a universal stage of the same design as employed byReference Langway Langway (1958) The electrodes were platinum wires (0.36 mm O.D.) placed perpendicular to the axis of the tube. The two lower electrodes were 2–3 mm apart and 18 mm from the single upper electrode. A forepump vacuum connection to the “Lucite” tube allowed the dissolved gases in the solution to be pumped off prior to freezing the solution.
Fig. 1. Sketch of the “Lucite” tubes and the dialed electrical apparatus used for growing single crystals of ice. A, thermistor probe; B, electrical connections to the platinum wire electrodes.
The “Lucite” tubes containing the seed crystal in the desired orientation and a suitable solution were lowered into a cold bath (-50°C ± 1°C) with a constant-speed motor. The cold bath consisted of an ethanol and water mixture in a metal container which was placed inside a deep-freeze unit. The freezing potentials were measured with a Keithley 610A electrometer and recorded with a Bausch Lomb VOM-5 chart recorder.
The solute distribution in the liquid phase was obtained by measuring the resistance between the two lower electrodes in the freezing tube as a function of time as the ice-liquid interface advanced toward these electrodes from below. When the two lower electrodes were frozen into the ice, a switching arrangement allowed measurement of the freezing potential between the single upper and two lower electrodes while the ice-liquid interface was in that region. Thus, both the solute distribution in the liquid phase and the freezing potentials were measured for each crystal specimen.
The freezing currents were obtained by shunting the interface with a high impedance resistor (1.00 × 108 Ω) and measuring the potential drop across the resistor The specific charge separation Q. was calculated from the curve of freezing current against time
The solute concentrations in the present experiments were determined to within ±1% for constant concentrations. However, the solute concentration gradient near the ice-1iquid interface resulted in an estimated error of ±8% because of changes in the cell constant of the wire-type conductivity cell. The position of the ice-liquid interface was determined to wit! ±0.2 mm from the lowering rate of the sample and the resistance change between the lower two electrodes as these were grown into the ice phase.
Results and discussion
Single crystals of ice were grown from dilute (2.2 X 10-4 N) KCl solutions with their c c-axes oriented either parallel or perpendicular to the growth direction. The results are summarized in Table I and typical so lute distribution curves in se mi- logarithmic form are shown in Figures 2 and 3. Equation (7) was used to obtain the apparent distribution and diffusion coefficients in Table 1. The values for K are larger for growth parallel to the c c-axes of the ice crystals and K increases as R increases although the specific dependence of K on R is not clear. The values for D are approximately the same for both crystal orientations and D increases linearly with R as shown in Figure 4. Extrapolation of the curve to R = 0 indicates a value of D ≈ 1 X 10-3 mm2/s. This is in excellent agreement with the known value (Gross, 1968) of D o = 1.00 X 10-3 mm2/s.
Table I. Averacif. values of the apparent distribution and diffusion coefficients and maximum freezing potentials for the phase change of 2.2 x 10-4 n kc1 solutions
Fig. 2. Typical semi-logarithmic plots of the solute distribution in the liquid phase (if. Equation (7)) obtained during the phase change of 2.2 X 10-4 N KCl solutions. The c-axes of the ice crystals were oriented perpendicular to the growth direction.
Fig. 3. Typical semi-logarithmic plots of the solute distribution in the liquid phase (cf. Equation (7)) obtained during the phase change of 2.2 X 10-4 N KCl solutions. The c-axes of the ice crystals were oriented parallel to the growth direction.
Fig. 4. Apparent diffusion coefficient versus growth rate (see Table I) for growth parallel ([Inline Equation]) and perpendicular (○) to the c-axes.of Crystals formed from 2.2 X 10-4 N KCl solutions.
Larger freezing potential, were associated with phase change when the c-axes of the ice crystals were parallel to the growth direction (Table I). The average maximum freezing potentials are shown plotted against growth rate in. The Figure 5 represent the spread in the data. The largest spread was 4.0 V for a crystal growth rate of 8 8 μm/s with the c-axes of the crystals parallel to the growth direction.
Fig. 5. Average maximum freezing potentials versus growth rate (see Table I) for growth parallel (▭) and perpendicular (○)to the c-axes of crystals formed from 2.2 X 10-4 N KCl solutions . The vertical bars represent the spread in the data.
The freezing potentials were small (+ 1 V or less) when R $$ 5.5 μm/s and large (+ 6 V or more) when R $$ 8.8 μm/s. These growth rates bracket the transition from clear ice (planar interface) to cloudy ice Cellular interface) as shown by Figure 5. The transition region was not always as sharply defined as shown in Figure 5. However, a large number of experiments indicated that it was usually within the region shown for the solute concentration and growth conditions used in the present experiments . The length of the cells d can be calculated from Equation (9) . If K o ≈ 10-3, then d ≈ 2 mm for a 2 X 10-4 N KCl solution . This d is of the same order as the depth of the charged layer in the ice (1 mm) as determined experimentally by Reference LeFebreLeFebre (1967) . Thus, the possibility exists that the larger freezing potentials may be associated with the formation of a cellular interface.
The transition from a planar to a cellular interface (as indicated by the change horn clear to cloudy ice) was visually observed and Equation (8) was used to calculate the equilibrium distribution coefficient K0 The value of K0 for KC1 solutions was calculated to be 4 x 10-3 for both crystal orientations; however, the method of visual observation is not sufficiently sensitive to determine small differences (20% or less) due to crystal orientation. Some experiments were also performed with dilute (about 2 X -4 N) NaCl and LiCl solutions (Osterkamp, unpublished), and the values of K0 were calculated to be 9 X 10-3 and 3 X 10-3, respectively or these solutions. The value of. K0 for NaCl solutions has been estimated byReference Weeks, Lofgren and Oura Weeks and Lofgren (1967) at about 10-3, while Reference Harrison, Tiller and KingeryHarrison and Tiller (1963) state that K0 10-4. There does not appear to be any data available for the KCl and LiCl solutions.
The Freezing currents were measured during the phase change of 2.4X10-4 N KCl solu tons. Single crystals of ice were grown with the, c-axes oriented either parallel or perpendicular to the growth direction at a growth rate of 8.8 μm/s. The specific charge separation Q was calculated to be 1. 3 ± 0.1. X 10-6 C/g of ice when the c-axes were oriented parallel and perpendicular, respectively, to the growth direction. These values of the specific charge separation are of the same order of magnitude as found by other investigators (Reference Pruppacher, Pruppacher, Steinbcrger and WangPruppacher and others, 1968).
The interface parameters, R i, C i and I’ of LeFebre's model have been calculated from Equations (2)-(6), and the results are shown in Table II. It appears that data are not available in the literature for the ice-liquid interface parameters for KC1, NaCl and LiC1 solutions. However LeFebre (1967) has found that C i = 2 X 103 pF, R i = 2.8 X 1010 Ω and I’ = 10 X X 10-1 C/m3 for a 2 X 10-5·N KF solution which may be compared with values for the same parameters determined in the present investigation (Table II).
Table II. Ice-liquid interface parameters for the solutions shown as a function of crystal orientation
The difference between the separate ionic distribution coefficients for the KCl solutions may be calculated by Equation (5), Since the negative ion density in the solution CL- ≈ 1.4 X X 10-1 C/m3, then K +—K - ≈ 2 X 10-5. The sum of the separate ionic distribution coefficient the KC1 solutions may be calculated by Equation (6). Since the equilibrium distribution coefficient K0 = 4X10-3, then K+ + K- ≈ 8 X 10-3.
Conclusions
A wire-type conductivity cell may be used to determine the solute distribution in the liquid phase near an advancing ice-liquid interface. The theory of Tiller and others (1953) describes the solute distribution in the liquid phase near the interface (within 4 mm) provided apparent distribution and diffusion coefficients are used in place of the equilibrium distribution coefficient and the usual diffusion coefficient. The apparent distribution coefficients and the maximum freezing potentials are larger for growth parallel to the c c-axes of the ice crystals. It appears that significant freezing potentials may be associated with a cellular interface structure; however, additional experimental work is needed to test this hypothesis.
Acknowledgements
The authors wish to thank Dr G. W. Gross for his review and comments on an earlier draft of this manuscript. One of us (T.E.O.) acknowledges the support of a National Aeronautics and Space Administration Predoctoral Traineeship.
