2.1. The electrical equipment

The equipment used for measurement of capacitance and conductance of ice in the frequency range o.5 Hz to 0.2 MHz has been described elsewhere ( Johari and Jones, 1976[a]). The calibration of the instruments was regularly checked to ensure the accuracy of the data which is the same in the present results as in the earlier ( Johari and Jones, 1976[a]).

2.2. The dielectric cells

All electrical measurements were carried out on disc-shaped ice samples contained in three-terminal dielectric cells with guarded parallel-plate electrodes. One set of electrodes was made of brass and two of stainless steel. For measurements on three samples of ice, two parallel to the c-axis and one perpendicular to the c-axis, the stainless steel electrodes E and K of the cell shown in Figure 1 and described below, were covered with gold foil c. ı μm thick.

Three dielectric cells were used in this study. Two of the dielectric cells, one made of brass and the other of stainless steel, were made to a design similar to McCammon and Work’s (1965). This is shown in Figure 1. The 12.7 mm thick, 152 mm diameter, plates A and j, and the four 12.7 mm thick rods c threaded at both ends form a yoke, the top plate of which had a concentric bore of 12.7 mm diameter. The high electrode E, 65 mm in diameter and 7 mm thick, was embedded inside a groove cut out from an inverted mushroom-shaped ground electrode n, whose stem was 12.7 mm in diameter and 82 mm long. The lubricated stem slid through the bore in plate A with a clearance of 12.7 μm. The low electrode k, 25.4 mm in diameter and 7 mm thick, was embedded in a concentric groove cut out from the disc G, which also received another concentric low electrode H shaped in the form of a ring, 62 mm in outer and 44 mm in inner diameter and 7 mm thick. K and H were 1 o mm apart. G was permanently fixed to the plate j by means of four screws L as shown in Figure t. A 2 mm diameter hole L was drilled in the plate j from the circumference to the centre of the plate to receive a thermocouple. B is a stainless steel spring which maintained an axial load of 15-2o kg on a plate 5 mm thick inserted between E and K and thus ensured contact between the ice and electrodes at the various temperatures. The electrodes E, H, and x were electrically insulated from the ground electrodes n and G by means of mica and water-repellant epoxy cement, and the assemblies DE and GHK were honed to a smooth finish and their faces lapped flat and parallel. The gaps between the guard rings and the electrodes E, H, and K were between 0.5-0.8 mm. The width of the guard planes were two to three times the thickness of the ice sample F to provide negligible fringing of the electric field. Measurement of ϵ₀ of n-hexane with this cell showed a stray capacitance of less than 0.1%.

Fig. 1.

Diagram of the dielectric cell used for the study of single crystals of ice.

The cell has the advantage of measuring the dielectric properties of a sample without explicit knowledge of the sample thickness. When n and G are parallel, the effective air capacitance between E and H and E and K is given by

where C _K and C_H are the effective air capacitance and A _K and A _H are the area of the electrodes K and H, respectively. The factor (A _K/A _H) was determined by placing stainless steel rings between D and G which provided an electrode separation in the range 3–7 mm. The value was 2.893 6 for brass electrodes and 2.907 9 for stainless steel electrodes. The values increased by c. 2 × 10^–4 when the temperature was decreased from 295 K to 223 K due to a combined effect of a decrease in the area of the electrodes, K and H and in the thickness of the steel ring. The air capacitance of an assembly containing a sample of unknown thickness placed on K can thus be obtained by measuring C _H Since the top electrode E is supported by the specimen F, the cell has the further advantage of measuring the isobaric expansivity of materials from a knowledge of the change in C _H with temperature (McCammon and Work, 1965).

A third parallel-plate cell made of stainless steel, which was the model Type KMT Guard-ring Capacitor, was commercially available from Rohde and Schwarz, W. Germany. It consisted of an 80.3 mm diameter low electrode protected by a guard ring. The spacing between the high and low electrodes could be varied up to 9 mm by means of a micrometer. The gap between the low and the guard electrode was 0.4 mm. The cell has been designed such that an axial load of 8 kg can be applied to the electrodes in order to maintain an appropriate electrical contact with the sample. For a sample thickness of 6 mm, the cell has an air capacitance of 7.62 pF and a stray capacitance of o.14 pF. The permittivity and loss measured with the cell were corrected for the stray capacitance.

The temperature was measured by means of a calibrated copper-constantan thermocouple which was kept inside the hole L in plate j shown in Figure 1. The thermocouple was placed under the low electrode in the Type KMT Capacitor. In order to ascertain that the temperature between the plates E and K was the same as inside plate J, a thermocouple embedded in felt material 8 mm thick was kept between E and K. No difference between the temperature of the two thermocouples was detected in the range 200–295 K.

2.3. The ice samples and their preparation

The single crystals of ice were prepared from distilled water contained in polyethylene tubes of 10 cm internal diameter and 1.1 m long, by zone-refining in a cold room maintained at 263?1 K. The treatment of water, the number of passes made during the zone refining, the age of single crystals prior to the study, and the number of samples studied for each orientation are given in Table 1.

Table I.

Preparation and descriptlqn of the ice crystals obtained by zone-refining method, and details of the samples and elegtrodes used in the study

In nearly all crystals the c-axis lay perpendicular to the cylindrical axis of the single crystals. The cylindrical sample was then cut with a band saw to machinable dimensions with a suitably oriented c-axis. Finally, the discs of ice from different stocks, 3–6 mm in thickness and 28–30 mm in diameter, were cut on a lathe. Disc samples of the same diameter but c. 2 mm thickness were also cut from the same stock, and at least one sample from each stock was used for accurately determining the direction of the c-axis by means of X-ray diffraction, prior to making the electrical measurements. The X-ray measurements showed that the c-axis of the disc sample lay within 0.5-2° of the desired direction. Two samples were also examined by X-ray diffraction after they had been taken out of the dielectric cell to ensure that the c-axis orientation had not altered, and nearly all samples were examined after the completion of electrical measurements with crossed polarizing filters to ensure the single crystallinity of the ice. No change in the c-axis orientation or in crystallinity was found in samples which were stored in the dielectric cell for as long as two months at 258?2 K under a uniaxial stress.

The ice samples used with the Type KMT Capacitor were 85–90 mm in diameter and c. 6 mm in thickness. Two samples with the c-axis perpendicular to the electric field and one sample with the c-axis parallel to the electric field were studied with this capacitor and the rest of the samples were studied with the cell shown in Figure 1.

The dielectric cell containing the ice sample was placed in a polyethylene bag, the leads were taken out and the bag closed. The entire assembly was kept inside a thermostatically air-cooled refrigeration unit capable of cooling to a temperature of 199 K. In several experiments, two cells containing the ice samples, having either the same or different c-axis orientations, were kept in the cold chamber. Measurements could thus be made on several samples at identical temperatures after thermal equilibrium was reached. The temperature of the cells remained constant to within ?0.05 K for 2–3 h. Therefore, there is little uncertainty in the values derived from our measurements of capacitance and conductance, which might otherwise exist from any variation in the temperature of the ice sample during the period of the frequency scan.

4. 1. The frequency dependence of ϵ^′ and ϵ″

The relative permittivity ϵ′ and loss ϵ″ of ice, measured within 2 h of sample preparation, exceeded 250 and 200, respectively at o.1 kHz and 268 K. The values decreased rapidly with time and a typical complex-plane plot of ϵ″ against ϵ′ ^’ followed the pattern indicated in Figure 2. In nearly all cases, ϵ″ decreased to a value of 4 or lower at 268 K in 60–80 h. Measurements at low temperatures were made only after the value of ϵ″ at o. 1 kHz was less than 4 at 268 K. In most cases, after the completion of the low-temperature measurement, which took 4–8 d, the sample when measured again at 0.1 kHz had ϵ″< 3 at 268 K. The annealing of the ice was found to be most effective in reducing its ϵ″ to a limiting value when done at temperatures close to 270 K. Therefore, all ice samples were stored in a cold room maintained at 268?1 K, when not being measured. A similar observation was reported by Taubenberger (1973).

Fig. 2.

The change in the shape of the complex-plane plots of the relative permittivity and loss of an ice sample measured paralle to the c-axis. Left half: measured at 268.4 K, (a) after 48 h, (b) after 70 h. Right half: measured at 265.8 K, (a) after 49 h, (b) after the application of 2.1 kV cm^–1 electric field along the c-axis for e.8 h. The numbers besides the filled data points are frequencies in hertz.

The ϵ″ of ice at o. t kHz and 266 K was also found to decrease when an electric field of c. 2 kV cm^–, was applied for 1–2 h. After removal of the field the value of ϵ″ remained constant for nearly to min and then began to increase with time. The behaviour, which seems typical of ice (Gränicher and others, 1955; Von Hippel and others, 1971) and of clathrate hydrates (Wilson and Davidson, 1963; Davidson and Wilson, 1963), is shown in Figure 2. While the application of a d.c. voltage for a period of 2 h sufficiently reduced the ϵ″ of ice to obtain reliable values of its ϵ ₀ , the increase in ϵ″ with time after removal of the field made the measurements somewhat uncertain. The reduction in ϵ″at low frequencies on the application of a d.c. voltage suggests that slow build-up of space charge, arising either from the irreversibility of the electrode material to protons, or from the presence of ions (Davidson and Wilson, 1963) in ice, is responsible for higher values of ϵ″ than anticipated from the orientation polarization alone.

The ϵ′ and ϵ″ of ice are plotted in a complex plane at several temperatures in Figure 3. In the temperature range of study both the low- and high-frequency part of the arc could be well defined, and, as evident in Figure 3, the plots intersect the ϵ′ axis very nearly vertically. With decreasing temperature, the plots depart progressively from a semicircle, although for temperatures above 250 K the plots are semicircular and are indistinguishable from a Debyetype process with a single relaxation time. This feature is clearly seen in Figure 4 in which the normalized dielectric loss ϵ″/ ϵ″ of one sample of ice of each orientation is plotted against f / f_m on a logarithmic scale. Here ϵ _m” is the maximum loss and f_m the frequency of maximum loss. The solid line represents a Debye-type absorption with a single relaxation time.

Fig. 3.

Complex-plane plots of the relative permittivity and loss of ice at several temperatures. The left half is for measurement with the c-axis oriented parallel to the electric field and the right half for that oriented perpendicular to the field. The numbers besides the filled data points are frequencies in hertz. The plots have been progressively shifted upwards as indicated by the axis.

Fig. 4.

Normalized plots of the dielectric loss of ice against frequency at several temperatures. The solid line is the theoretical curve for a Debye-type relaxation.○, parallel to the c-axis at 238.1 K and •, at 201.5 K. △, perpendicular to the c-axis at 235.9 K and ?, at 202.6K.

The half-width of the normalized plots at temperatures <230 K varied from one sample to another. Therefore, no further analysis of the frequency dependence of ϵ′ and ϵ″ into several dispersion regions was attempted. The ambiguity of such an analysis has already been discussed in the literature (Gough and others, 1973; Johari and Jones, 1976) and it is not certain whether the increase in the half-width of the plots with decreasing temperatures is due to an increase in the relative amplitude of several Debye-type relaxations or to an increase in the relative magnitude of the limits of relaxation time representing a distribution. At temperatures near 205 K, the approach of ϵ″ towards the axis at high frequencies indicated the presence of a small relaxation region, similar to the one seen in D₂0 ice (Johan and Jones, 1976). A similar high-frequency relaxation region has also been observed in H₂O ice single crystals by Von Hippel and others (1971), and Ruepp (1973) at kilohertz frequencies but a temperatures below 240 K and 200 K, respectively, and by Johari (1976) at megahertz frequencies at temperatures above 250 K.

4.2. The equilibrium relative permittivity

The equilibrium relative permittivity ϵ ₀ was obtained from a graphical extrapolation of the low-frequency end of 86 plots for samples with the c-axis oriented parallel to the electric field and 61 plots for samples with the c-axis oriented perpendicular to the electric fields. Strictly, ϵ′ = ϵ ₀, when ϵ″ = 0. For ice, and for many other solids, this condition is not observed because space-charge and other non-linear polarizations begin to contribute to ϵ″ at frequencies far above the one at which ϵ″ due to orientation polarization would become zero. Within the reproducibility of ±2%, our values of ϵ ₀ are not likely to be different from the true ϵ ₀, although the condition ϵ″= 0 was not satisfied by any set of our measurements. The ϵ ₀ of ice for both the orientations is plotted in Figure 5, and values near 266 K for five different samples are given in Table II.

Fig. 5.

The variation of the equilibrium relative permittivity of ice with temperature;○, parallel to the c-axis; △, perpendicular to the c-axis.

Table II.

Equilibrium relative permittivity and relaxation time ofhexagonal ice near 266 k for the two orientations

The c_o measured parallel to the c-axis at 248 K and 268 K are 103 and 96 respectively. The corresponding values interpolated from Humbel and others (1953) are 220 and 113, from Von Hippel and others (1971) are 118 and 106, and from Taubenberger (1973) for his 1114 sample are [13 and 106. Thus there is a considerable difference between our ϵ ₀ measured parallel to the c-axis and those given in the literature. While the scatter in ϵ ₀ values, already mentioned in Section, in Von Hippel and others’ (1971) data, may account for some degree of difference, the reasons for a difference between our data and those of Humbel and others (1953) and Taubenberger (1973) are not obvious. Our values agree with Wörz and Cole’s (1969) values in which the errors in ϵ ₀, from space-charge polarization and “d.c. conductivity” were largely eliminated.

The value of ϵ ₀ = 96, measured perpendicular to the c-axis at 268 K, agrees with the values reported by Humbel and others (1953), by Wörz and Cole (1969), by Von Hippel and others (1970 and by Taubenberger (1973), within their respective uncertainty, which Humbel and others (1953) quoted for their ϵ′ values, “… ist deshalb von der Grössenordnung 3,5^–4,5%”.

For a given sample of ice, our ϵ ₀ values showed a scatter from a mean line of less than ?0.7%. The scatter increased when ϵ ₀ of several different samples was plotted on the same figure. In all cases of 14 samples measured parallel to the c-axis and 8 measured perpendicular to the c-axis, the departure of the data from a mean line did not exceed 2%, as indicated in Figure 5. We attribute the apparent uncertainty in ϵ ₀ to the possible non-parallel faces of the sample, as discussed in Section 3, or the nature of ice itself. Within this uncertainty, the ϵ ₀ measured parallel to the c-axis is the same as that measured perpendicular to the c-axis as the data in Table II and Figure 5 indicate.

The ϵ ₀ values were fitted to the Curie-Weiss equation,

where ϵ _? is the limiting high-frequency permittivity of the orientation polarization, T is the absolute temperature, and A and T₀ are empirical constants. The values of A and T₀ obtained, assuming ϵ _?, = 3.2, for ice for the two orientations and for all samples and orientations, are given in Table III, and a plot of (ϵ _0- ϵ _?)^–1 against temperature is shown in Figure 6. The standard deviation of A and T₀ and of the data points obtained from a least-square treatment are also included. The agreement between A and T₀ for the two orientations indicates the consistency of our measured values for different samples. Our A and T₀ values in Table III differ from those given by Wörz and Cole (1969), who found A = 20 715 K and T ₀ = 38 K, and from those obtained from Von Hippel and others’ (1971) plots, where the values of A for the two orientations differ considerably, but the T ₀s are close to o K. The values of A and T ₀ are very sensitive to changes in the input data, and probably have little significance except for the purpose of interpolation.

Fig. 6.

The variation of (ϵ₀ – ϵ_?)^–1 with temperature for ice; ?, parallel to the c-axis; Δ, perpendicular to the c-axis.

Table III.

The empirical constants of the curie-weiss equation, and the pre-exponential factor and activation energy calculated from eyring’s equation for hexagonal ice for the two orientations

From the recent discussion (Hollins, 1964; Rahman and Stillinger, 1972; Gobush and Hoeve, 1972; Stillinger and Cotter, 1973; Nagle, 1974) of the theoretical treatment of the orientation polarization in ice, there is little agreement between the various workers on the choice of dipolar interactions that affect the ϵ ₀ of ice. Consequently, it is not certain whether the short-range interactions, within the constraints of Bernal–Fowler rules (Bernal and Fowler, 1933) and Pauling’s (1935) calculation of residual entropy, used by Onsager and Dupuis (1962) and later by Nagle (1974), should be used in calculating the ϵ ₀, or whether the long-range interactions as in Kirkwood’s theory (1940) for solids, or only short-range interactions, as in Kirkwood’s theory (1939) for liquids, would be more appropriate.

Fig. 7.

The limiting high-frequency relative permittivity plotted against temperature; ?, parallel to the c-axis, and Δ perpendicular to the c-axis. The data points above 240 K are from Johari (1976).

Nagle (1974) has proposed that ϵ ₀ of ice is likely to be isotropic, on the theoretical grounds mentioned above, and the equation of Onsager and Dupuis (1962) can be written in the form:

where G is defined as the polarization factor, N is Avogadro’s number, μ is the dipole moment of a water molecule in ice, V is the molar volume, k is the Boltzmann factor, and T is the temperature. Nagle (1974) calculated G = 3. The Kirkwood–Fröhlich (Fröhlich, 1949) equation, which takes account of short-range interactions, is

where g is the orientational correlation factor, which is a measure of the extent to which hindered rotation of neighbouring molecules tends to align their dipolar orientations, m is a measure of the enhancement of the dipole moment by its own reaction field, and μ ₀ is the dipole moment of a water molecule in the vapour state, according to Clough and others (1973), μ ₀ = 1.855.

From a comparison of Equations (2) and (3), G _\ and g are formally related by,

Since ϵ ₀ gt; ϵ _? for ice, Equation (4) becomes

In contrast to μ ₀, μ is not accessible to direct measurement. It is noteworthy that if g = 2.07 (Rahman and Stillinger, 1972) and G = 3 (Nagle, 1974) are substituted in Equation (4), μ becomes approximately equal to mμ ₀ From Equation (2) the dipole moment of a water molecule μ = 2.97 ? 10^–18 e.s.u. at 243.1 K, which gives the charge associated with a Bjerrum defect (Bjerrum, 1951), q = 1.87 × 10^–10 e.s.u. (6.25 × 1 0^–20 C). This value is close to that estimated by Onsager and Dupuis (1962).

In Equation (3), m is given by either (ϵ ₀+2)/3 or (n ₂+2)/3, where n is the refractive index at optical frequencies. There has been a considerable discussion (Hollins, 1964; Chan and others, 1965; Johari and Jones, 1976) as to which of the two quantities ϵ _? or n ₂ is appropriate in the calculation of m and some ambiguity still remains as to the frequency at which the orientation polarization separates from the distortion polarization. From Figures 5 and 7 ϵ ₀ 105.2 and ϵ _? = 3.15 for both the orientations in ice at 243 K. Using Equation (3), one obtains g = 1.78, if m = (ϵ ₀+2)/3, and g = 3.4o, if m = (nD₂+2)/3 using the refractive index n _D for sodium D light. Evidently the g calculated assuming m = (ϵ _?+2)/3 is close to the theoretical value of 2.07 calculated by Rahman and Stillinger (1972) for ice containing 2 048 molecules using a Monte-Carlo procedure. We have arrived at a similar conclusion from a somewhat different consideration in the case of D₂O ice (Johari and Jones, 1976[b]).

Another consideration concerning the choice between ϵ _? and n ² in calculating m would be useful here. If the (ϵ ₀ – ϵ _?) of ice obeyed Curie’s law, i.e. if T ₀ = 0 K in Equation (1), as Ruepp and Kass (1969), Von Hippel and others (1970, Ruepp (1973), and Johari and Whalley (1973) have found, a comparison of Equation (1) with Equation (2) gives

Since ϵ ₀ gt; ϵ _? for ice, the first term on the right-hand side of Equation (3) becomes 1.5, and then from a comparison of Equations (1) and (3),

The molar volume of ice, V, decreases from 19.66 cm³ mol^–1 at 273 K to 19.45 cm³ mol^–1 at 200 K (Lonsdale, 1958) and G and g remain constant. It follows, then, that A can be constant only if μ ² and m ², in Equations (6) and (7) respectively, decrease with temperature in the same proportion as the molar volume. Such a decrease in μ ² is implicit in Equations (2) and (6), via the temperature dependence of the ?—? distance d, which determines the charge q associated with a Bjerrum defect by q = 3^½ μ/d, if q is assumed to remain constant with temperature. The decrease in ?–? distance corresponding to the above-mentioned decrease in V is from 2.76 Å at 273 K to 2.75 Å at 200 K (Lonsdale, 1958).

The term m in Equations (3) and (7) is temperature dependent via the temperature dependence of ϵ _? or n _D ². As the n _D ² of ice calculated from Clausius and Mossotti’s relation increases ( Johari, 1976) from 1.715 at 273 K to 1.724 at 200 K, but ϵ _? decreases from 3.20 to 3.14 (Fig. 7) in the same temperature range, it is apparent that m should be given by either (ϵ _?+2)/3, or else a quantity that decreases with temperature, if A in Equation (7) were to remain constant.

4.3. The limiting high-frequency relative permittivity

The values of ϵ _? were obtained by a graphical extrapolation of the high-frequency end of the complex-plane plots of ϵ? and ϵ?. Reliable values could be obtained only at temperatures where ϵ? at 100 kHz was less than 0.1. This condition was satisfied at temperatures below 235 K. The ϵ _? values for eight samples measured parallel to the c-axis and six perpendicular to the c-axis are given in Figure 7. The values at temperatures above 240 K, which are taken from the literature (Johari, 1976), extrapolate to those below 253 K in a manner similar to that seen in polycrystalline ice (Gough, 1972). This indicates that our present measurements of ϵ _? perpendicular to the c-axis are consistent with the previous ones (Johari, 1976), although the equipment and the frequencies used in obtaining the ϵ _? values in the two measurements differ. It is evident in Figure 7 that the ϵ _? of ice parallel and perpendicular to the c-axis are the same within ±0.5%.

The reason for a decrease in ϵ _? with temperature of polycrystalline ice has been a subject of discussion earlier (Gough, 1972; Johari, 1976; Johari and Jones, 1976[b]). In view of the observed isotropic behaviour of ϵ _? of ice, the discussion for single crystals of ice should be similar.

4.4. The average relaxation time

The average relaxation time τ _av was calculated from the frequency of maximum loss using the equation τ _av = (2π f _m)^–1 The f _m values were obtained by plotting ϵ? against the logarithm of frequency on a large scale and interpolating. The τ _av for both the orientations of ice agreed, to within +5%, with Wörz and Cole^’s (1969) data and, at temperatures above 248 K, with Von Hippel and others’ (1971), Ruepp’s (1973), and Taubenberger’s (1973) data.

According to Eyring’s rate-process theory, the relaxation time for a molecular reorientation is given by,

where the pre-exponential factor τ ₀ is given by,

h is Planck’s constant, k is the Boltzmann factor, e is the transmission coefficient, S* and E* are the entropy and energy of activation respectively, and R is the gas constant. The product of τ _av and temperature is logarithmically plotted against the reciprocal temperature in Figure 8. The plot is linear in the temperature range 210–270 K. The values of τ ₀ and E* with standard deviation obtained from the least-square treatment of data for each of the two orientations and of the data for all samples and orientations are given in Table III.

Fig. 8.

The variation of the product of the average relaxation time and temperature of ice plotted logarithmically against reciprocal temperature. ?, parallel to the c-axis; Δ, perpendicular to the c-axis.

The combined effect of the difference of τ _av, ϵ ₀ and ϵ _? as a function of temperature between the various samples measured for the two orientations can be seen in Figure 9, where ϵ? and ϵ? at 0.1 kHz are plotted against the temperature. At high temperatures the deviation from the average plot indicates the uncertainty in the measurement of Δϵ for different samples and at low temperatures the uncertainty in r values which contribute by a term (1 +ω ² τ ²) in ϵ? and ωτ /(1+ω ² τ ²) in ϵ?. It is evident from Figures 8 and 9 that the relaxation time of ice is also isotropic between 2 to and 271 K.

Fig. 9.

The relative permittivity and loss of ice at 0.1 kHz plotted against temperature to show the combined effect of variation in τ_av, ϵ₀ and ϵ_? on the orientation of the sample. ?, parallel to the c-axis, and Δ, perpendicular to the c-axis. Open data points are for ϵ? and filled data points for ϵ?.

The τ _av of ice for both the orientations follows the Eyring equation from 271 to 210 K with E* = 51±2 kJ mol^–1 and S* = 18±2 J mol^–1 K^–1, below which temperature the data show a large scatter and a trend towards a slope giving lower values of E*. The change towards a decreasing slope appears in Wörz and Cole’s (1969) plot at c. 240 K, in Von Hippel and other’s (1971) plot at c. 250 K and in Ruepp’s (1973) plot at c. 230 K, although these workers have plotted τ _av, rather than τ _av T, against T ^–1. It is well recognized that the concentration of lattice defects has no effect on the steady-state polarization ultimately established, which is an equilibrium quantity : they provide only the mechanism by which it is established and their concentration determines the rate of establishment. A small amount of substitutionally or interstitially present impurity in zone-refined ice may produce orientational defects which at T < 210 K are more numerous than those produced by the thermal motion of water molecules in an ideal lattice and thus may cause E* and S* to become appreciably smaller. Gough and Davidson ( 1970) found that τ _av of polycrystalline ice, obtained from depressurizing ice II, was much higher than that of any sample of ice obtained by freezing water. The τ _av however, decreased by several orders of magnitude when the ice was annealed. They proposed that the decrease in τ _av on annealing was caused by the dissolution of impurities, which had separated from hexagonal ice during its transformation to ice II and back into the hexagonal ice. It is noteworthy that impurities are thought to be much less soluble in the high-pressure phases of ice than in hexagonal ice, and indeed ice VI (Johan and Whalley, 1976) and ice VII (Johan and others, 1974) show no decrease in E* with temperature. According to this interpretation, our zone-refined ice contains less impurity-produced orientational defects than any other sample of ice reported in the literature.

Gränicher (1969) and Bilgram and Granicher (1974), after considering the τ _av and the d.c. conductivity together, have proposed that a decrease in E* with temperature is caused by a change in the relaxation mechanism from the one dominated by diffusion of orientational defects to the one dominated by diffusion of intrinsic ionic defects. In this view the decrease in E* is intrinsic to the nature of ice, although no high-pressure phase of ice has shown this behaviour. Furthermore, the change in the relaxation mechanism occurs at a temperature when σ_± /p = σ_DL/q (Bilgram and Gränicher, 1974), where σ_± and σ_DL are the conductivity due to ionic and Bjerrum defects and p and q are the effective charge associated with the defects, respectively. At this temperature the ϵ ₀ should reach a minimum value given by, ϵ ₀ ≈ ϵ _? ( Jaccard, 1965). This has not been observed for the zone-refined ice by us, or by other workers. It seems, therefore, more plausible that a decrease in E* with temperature is caused by the impurity-generated orientational defects rather than the intrinsic ionic defects in ice.

4.5. The effect of ageing

As shown in Table I, the stock from which the ice samples were obtained was 1–8 years old, yet their ϵ? and ϵ? decreased on annealing when contained in the dielectric cell. A number of samples of ice, of both the orientations, were stored at 253±2 K for periods ranging from one to five weeks. Measurements made after this period showed a decrease by as much as 50% in ϵ? at the frequency closest to the low-frequency intercept of the complex-plane plot, an increase of 0.5% in τ _av, but no change in Δϵ. One sample of ice oriented with the c-axis perpendicular to the electric field, contained in the Type KMT Capacitor, was kept at 253±2 K for it weeks after the first measurement. Its Δϵ and τ _av measured after this period showed a similar effect.

Taubenberger (1973) has reported that the Δϵ of ice at 258 K measured perpendicular to the c-axis decreased by 1 %, the ϵ _? increased by 10% and the τ decreased by 8%, when the ice contained between the electrodes was annealed at 258 K for 1 795 h. Another sample measured parallel to the c-axis at 248 K, showed no change in Δϵ and τ but an increase in ϵ _? of 2%, when annealed for 110 h at that temperature. In our experience, the values of ϵ _? at 258 K extrapolated from measurements made only at kilohertz frequencies are somewhat uncertain, and little information is given by Taubenberger regarding the contribution to ϵ? and ϵ? from space charge effects at low-frequencies. The change in τ in our measurement is apparently due to a reduction in the space charge polarization, as discussed in Section 4.1. Such changes are not likely to be intrinsic to ice.