Introduction

For many years, research on dielectric properties of ice has mainly dealt with mono-crystals in order to limit experiments to well-defined samples. One may yet wonder whether the microtoming of the samples does not alter the bulk structure of ice. On the other hand, experiments on doped ice have been chiefly conducted with a view to checking the theories on the electrical properties of ice.

A different orientation has lately been tried. With a view to giving glaciologists a better understanding of polar and temperate glacier ice, polycrystalline ice has been systematically studied. Furthermore, the fact that snow crystals are formed by the freezing of supercooled water down to as low as —41 °C, in itself accounts for the interest shown in the study of frozen emulsions of water in oil.

Lafargue (unpublished) has presented evidence for the existence of discrete temperatures at which supercooled water freezes given by the formula $$T_n = (164°C)/n (n an integer 4 < n ;ζ ι?). These temperatures, which do not seem to be dependent on environment and impurities (except ice isomorphs), seem to be related with the structure of the liquid. Whereas the probability of a droplet freezing is dependent on volume, impurities, and, in a complex way, on environment.

Remarkable changes of the dielectric behaviour occur during the ageing of micron-sized droplets formed when emulsions of supercooled water freeze at -41°C. During the ageing, the activation energy of the relaxation frequency falls from 0.58 eV immediately after freezing to as low as 0.16 eV in a way that maintains stable the relaxation frequency at o°C. Certain activation energy values are more likely to occur depending on the age and storage temperature. These values form a discrete sequence U_a = (—0.042+0.069η) eV where n is an integer (Evrard, 1973); intermediate values always occur before reaching a plateau, sometimes after $$10¹ h or even more. Very similar results have been obtained with pure poly-cristalline ice (Reference Boned, Barbier, Whalley, Whalley, Jones and GoldBoned and Barbier, 1973). Some of the values of this sequence seem to have been seen also by Reference Fitzgerald, Fitzgerald, Glen and ParenFitzgerald and others (1977) on polar ice and on finely grained ice micro-crystals. These more likely values have been shown to be related to the more likely temperatures of supercooling breakdown (Reference Evrard, Whalley, Whalley, Jones and GnldEvrard, 1973). As compared to polycrystalline ice, disperse microcrystals present a twofold interest: on the one hand, they make it possible to study crystals formed, much like snow, by the freezing of highly supercooled water, on the other hand emulsions present no d.c. conductivity. For this reason, they can be studied in a very wide frequency range, e.g. 10¹⁰ Hz to io^-3 Hz. In the case of polycrystalline ice, d.c. conductivity can conceal possible very low frequency (V.L.F.) dispersions.

This paper presents results of the study of dielectric properties of disperse ice microcrystals in the frequency range 10⁵ Hz to io~² Hz obtained with two différent devices. A type 716 G General Radio Capacitance Bridge was used in the frequency range 3 x10^s to io³ Hz. Capacitance and dissipation factor are measured by the substitution method, both for the empty cell and for the cell with the sample, E' and e" are deduced with an accuracy S 0.01. In the frequency range 10⁴ to io^zHz, a type 1172 Schlumberger frequency-response analyser was used. The cell is excited by a high-stability sinusoidal output V_x. The resulting amplitude and phase output of the cell V_y is analysed in the form $$ VyjV_x = a+jb with an accuracy of io~⁴ for Aaja and for Abjb. e' and e" are deduced from the values of a and b b both for the empty cell and for the cell with the sample, with an accuracy Ae'je' JK Ae"je" x o.oi. Values obtained by the two devices in the common frequency range are consistent. The sample was stored in a CS 601 type "Ferisol" Dielectric Cell with the "35 pF" electrode.

Ice was obtained by supercooling breakdown at — 41°C ^CC of droplets of water a few microns in diameter in an emulsion of water in oil. The oil phase was a lanolin-paraffin oil mixture with 0.05 lanolin by weight.

Experimental Results

We have studied one sample of ice of weight fraction P = 0.45 (sample a), and two different samples of ice of weight fraction P = 0.40 (samples b and c) ; for all samples, the results are identical. In this paper, the curves relate to sample a, although some numerical values are reported for the other samples. The complex dielectric constant of disperse ice microcrystals is shown to traverse two separate dispersions (Fig. 1). The first one, at audio frequencies, is due to the Debye dispersion of ice; the second one occurs at V.L.F.

An ageing effect is observed, which is greater at the highest storage temperatures. When a sample has been stored at a given temperature (— I°C Θ — 41 °C) for about 20-40 h, so that the evolution at this temperature becomes extremely slow, it is first measured at this temperature and then measured again at other temperatures T < Θ, no further evolution being observed whenever T Θ. This set of measurements is therefore characteristic of the stage of the evolution at the storage temperature Θ. The sample is then heated and maintained at a storage temperature Θ' Θ. Λ further evolution is observed and measured at some temperatures T < Θ', and so on for increasing storage temperatures.

Fig. 1. Complex dielectric constant of disperse ice microcrystals in the frequency range 10⁵ to 10^–2 Hz. Temperature -24°C P = 0.45.The meeting point R of the two dispersions is the point where ∈″ is minimum.

The strength of both dispersions depends on the weight fraction P of the ice in the emulsion, and on the storage temperature. For a given temperature θ, the strength of both dispersions is not very temperature dependent (Figs 2 and 3).

Fig. 2. Audio-frequency dispersion as a function of temperature for a given preserving temperature. Θ = –19.3°C, P = 0.45

I. T = –24.0°C. II. T = –29.6°C. III. T = –34.0°C.

IV. T = –34.0°C. V. T = –45.7°C. VI. T = –51.1°C.

Fig. 3. V.L.F, dispersion as a function of temperature. P = 0.45. Θ = –19.3°C

To get the temperature dependence of the frequency for both dispersions, we have plotted three points on each dispersion curve (Fig. 1) A, B, and c for the first dispersion, refer to fixed values of e' /fea—e»)- In this way, B is shown to refer to the maximum of e”. D, E, and F for the second dispersion, refer to fixed e'— e_R values, because of the existence of the master curve (Fig. 3). R is the meeting point, where e" is at a minimum. For all samples, the frequency associated with each point depends on temperature according to an Arrhenius-type law. A common value of activation energy is observed for A, B, and c (first dispersion), and another common value for $$ D, E, and F (second dispersion). These values depend on the storage temperature Θ, i.e. the stage of the evolution (Fig. 4, Table I). Figure 5 refers to the meeting-point activation energy.

Fig. 4. Temperature dependence of the frequencies related to the points A, B, C, D, E, F (see Fig, 1). Frequency is obtained by; log _v K+Δ log _v (see Table I). P = 0.45.

I. Θ = –39.5°C. II. Θ = –29.4°C. III. Θ = –19.3°C.

IV. Θ = –11.8°C. V. Θ = –6.6°C. VI. Θ = –1.1°C.

Table I. Values of K Related to the Frequencies in Figure 4

Fig. 5. Temperature dependence of the frequency vR, related to the meeting point R. P=0.45

1. Θ = –39.5°C. 4. Θ = –11.8°C.

2. Θ = –29.4°C. 5. Θ = –6.6°C.

3. Θ = –19.3°C. 6. Θ = –1.1°C.

Let L'i be the audio-frequency dispersion activation energy, ?'π the V.L.F. dispersion activation energy, and $$l^T _R the meeting-point activation energy; it can be noticed that $$XJ\ decreases and in increases with increasing preserving temperatures. Conversely $$i/_R remains constant. Table II gives the values of activation energies for samples a and b. It can be noticed that the values of $$Uj are not necessarily close to one of the values of the sequence$$ U» ⁼ (~θ-°4² +0.069?) eV. This can naturally occur because the sample has not been stored long enough to reach a plateau value.

It is shown in Table II that the values of$$ Lh are correlated with those of $$U_u. The sum $$Ui+Uti retains a constant value which is the same for both samples, about 1.05 eV. It must be noticed that sample c, which has been stored for four months at — I°C, presents the following values: $$U_} = 0.17 eV, U_a = 0.91 eV, Ui+Un = 1.08 eV, and aU_R = 1.05 eV. The fact that $$UI+UJI keeps a constant value during the ageing, seems to point to the existence of more likely values of$$ U_u corresponding to plateaux. A study of the evolution will be necessary to confirm this result.

One can also see that $$t/_R retains a constant value during the ageing and that this value is precisely $$f/_R = i(Ui+[f_u).$$ This fact is easy to explain. If we write for the melting point

and if the approximation is made that the dispersions are of the simple Debye type of strengths $$ Ae, and Δ<=₂ which are temperature-independent, then the angular frequency corresponding to the minimum, $$CUR, satisfies

>Thus $$COR should have an activation energy of$$ %{Ui+Un).

Table II. Activation Energies

This accounts for the agreement of 2U _R with UI+UII (This result remains valid even if the dispersions are not of the simple Debye type provided the relaxation times of each dispersion have the same activation energy). If the two dispersions are such that $$ Δ?,/τ, > Δ₂/τ_2!$$ the product σ₀σ_∞ for the dispersion I has magnitude t^Ae^eJr^z. With the same approximations as above, the activation energy for σ₀σ_∞ is UI+UII which would again be a constant. The observations of Reference Camp, Camp, W., Arnold, Rtehl, Riehl, Bullemer and EngelhardtCamp and others (1969) and Reference Camplin, Glen, Whalley, Whalley, Jones and GeldCamplin and Glen (1973) of a nearly constant value for the activation energy of σ₀σ_∞ is therefore of great relevance. Reference Camplin, Glen, Whalley, Whalley, Jones and GeldCamplin and Glen (1973) found in single crystals of HF-doped and pure ice, activation energies between 0.49 eV and 0.61 eV for σ₀σ_∞, the mean value being 0.5356V. Our values in microcrystals for UI+UII lie between 1.02 and 1.09 eV, and the mean value is 1.05 eV. It is most interesting that the mean values differ by just a factor of two.

Conclusion

It is noteworthy that the two dispersions that occur in ice microcrystals present, during the ageing, activation energies whose sum retains a constant value. The more likely values of activation energy for the high-frequency dispersion had already been shown to be related to the more likely temperatures of breakdown of water supercooling. Activation energies for the low-frequency dispersion should therefore also have more likely values corresponding to plateaux. In spite of the difficulties involved, this evolution would be worth studying.

Work up to date does not enable us to understand the complete mechanism of the V.L.F. dispersion. A numerical analysis with a computer, and additional experiments on pure polycrystalline ice are now in progress.

We would like to acknowledge the very valuable remarks on the points raised in this paper by Dr J. W. Glen.