Introduction
Many studies have been devoted to the dielectric and conduction properties of ice in order to obtain information about its internal structure. Most of them concern either monocrystalline or polycrystalline ice samples a few cm3 in volume whose behaviour was investigated under various conditions of temperature, pressure, and ion incorporation.
Other studies deal with dispersions of ice particles a few μm3 in volume obtained from the breakdown of supercooling of waler-in-oil type emulsions. Results in this field have been reported by Evrard (unpublished [b]) and Evrard and others (Reference Evrard1974[a], [Reference Evrardb]), who showed that dispersions of ice microcrystals can exhibit two distinct dielectric relaxations. A detailed study of this phenomenon has been published recently by Lagourette (Reference Lagourette1976, unpublished), who investigated the complex permittivity of dispersions of ice particles versus the frequency of the applied electrical field, at discrete sub-zero temperatures. According to this author, the dielectric relaxation located in the higher frequency range can be related to the Debye dielectric absorption of ice. Between 0°C and —20°C an additional dielectric relaxation, located in the lower frequency range, is noticeable. When the temperature decreases, the intensity of this extra relaxation decreases and eventually vanishes at about —20°C. Moreover, as is shown in Figure 1, it is not possible to define an energy of activation for this relaxation, as the plot of log v C2 , (v C2 being the critical frequency), versus T -1 is not a straight line. It is worth mentioning that, correspondingly, it is not possible between 0°C and — 20°C to define an energy of activation for the relaxation located in the higher frequency range.
Fig. 1. Plots of log vc versus T-1. Mass fraction of disperse water: p = 0.28.
A tentative explanation of these phenomena has been given by Reference LagouretteLagourette and others (1976) from computer calculations based on a theoretical model that has proved suitable for emulsion systems (Reference Hanaï and ShermanHanaï, 1968; Reference ClausseM. Clausse and Royer, 1976). Lagourette and others propose that, as the temperature approaches the melting point of ice microcrystals, the ice crystal lattice undergoes local structural changes leading to the formation of non-crystalline conductive clusters within it.
In an attempt to give a further contribution to the understanding of the phenomena briefly reported above, a study has been undertaken which concerns the variations of the complex permittivity of dispersions of ice microcrystals versus their temperature.
Experimental procedure
(1) Sample preparation
Water-in-oil type emulsions were made by dispersing in a mixture of 25% (wt/wt) lanolin (Prolabo R.P.) and 75% (wt/wt) vaseline oil (Prolabo R.P.) resin-exchanged water or aqueous solutions of analysis-grade ammonium chloride. The emulsification process was carried out at room temperature by means of a propeller-type homogenizer working at 45 000 r.p.m. Each emulsion was characterized by a mass fraction p of disperse aqueous phase and, in the case of saline solutions, by a molar fraction x of solubilized ammonium chloride. For values of p less than 0.6, the emulsions obtained through this method consist of dispersions of droplets a few micrometres in diameter, the distribution of droplet sizes being very narrow, as proved by microscope observation (Reference LagouretteLagourette, 1976, unpublished).
Owing to the small droplet sizes, large supercoolings of the disperse water can be obtained. For instance, Evrard (unpublished [a]) found that the freezing of water occurs in such systems around — 39°C. Later on, this author claimed a freezing temperature of — 41 °C (Reference Evrard, Whatley, Whatley, Jones and GoldEvrard, 1973, unpublished [b]). A statistical analysis of the freezing temperature of water dispersed within emulsions was made by Reference BonedBoned ([1968], unpublished [a]), from DTA measurements on 600 different samples. He obtained a mean value of —38.5°C and observed that freezing temperatures lower than — 40°C occur very seldom. The value of — 38.5°C was also reported by Lafargue and Reference Lafargue, Babin, Servant and CharruBabin (1964) and Babin (unpublished), who investigated the breakdown of supercooling of water dispersed within emulsions through dielectric measurements. It has been recently confirmed by Reference ClausseD. Clausse (1976) from DEA measurements made by means of a Perkin-Elmer DSC 2 calorimeter.
A freshly made emulsion was poured into the dielectric measuring cell which was placed immediately into a temperature-controlled chamber. The cell was cooled down to about — 150°C and then heated up to near the melting point of the disperse phase. The rate of cooling was —1.5 deg/min, the rate of heating being 0.5 deg/min. Some samples were frozen very rapidly by immersing the cell in liquid nitrogen. Their dielectric behaviour was the same as that of steadily cooled samples.
The temperature T of the samples was recorded to within ±0.5 deg by means of a calibrated thermocouple fitted into a cavity drilled within the volume of the outer electrode of the cell.
(2) Dielectric measurements
The dielectric behaviour of the samples was investigated during the initial cooling, then during the following heating and, sometimes, during a subsequent cooling. The dielectric cell used consists mainly of a cylindrico-conical capacitor made of stainless steel and especially designed for the study of liquids (Ferisol CS 601). The active capacity of the cell C a is 10 pF at 20°C, the gap between the electrodes being 3 mm. The variations of Ca with temperature (∆C a/∆T = 0.5 × 10-3 pF/deg), were taken into account for the determination of the real permittivity and loss factor of the samples.
The complex relative permittivity ε⋆ = ε´–jε˝ of the samples was determined at either 400 Hz or 1 kHz by means of a General Radio 1680 A automatically-balancing admittance bridge connected to a two-channel Esterline-Angus E 1102 E data recorder, the maximum value of the voltage applied between the cell electrodes being equal to 1 V. The uncertainty upon both ε´ and ε˝ was estimated to be less than 1%.
Results
(a) Dielectric behaviour of dispersions of ice microcrystals obtained from emulsions of water
A freshly-made water-in-oil sample having been poured into the measuring cell, its complex permittivity ε⋆ = ε´–jε˝ was recorded during continuous cooling down to about —150°C. Owing to the discrepancy existing between the dielectric properties of water and ice, the breakdown of supercooling of the disperse droplets can be easily detected from the sudden decrease of ε´ and increase of ε˝. As the temperature is lowered, the frozen emulsion exhibits a dielectric relaxation arising from the Debye dipolar absorption of the disperse ice particles.
During the following heating from — 150°C up to near 0°C, the study of the complex permittivity reveals the existence of two dielectric relaxations located either side of a temperature Tm, as shown in Figure 2 which represents the plots of ε˝ and ε´ versus T obtained in the case of a dispersion characterized by a mass fraction of water (or ice) equal to 0.40.
Fig. 2. Variations of (a) ε" and (b) ε' versus T. Mass fraction of disperse water: p = 0.40. Frequency: v = 400 Hz.
The relaxation located in the lower temperature range will he referred to as R1 and the relaxation located in the higher temperature range as R2, T c1 and T C2 being the temperatures corresponding to the respective maximal values of ε˝.
From Figure 3, which gives the variations of ε˝(T) versus ε´(T), some characteristic features of the phenomenon can be seen. Experiments show that R1 is similar to the relaxation observed during the initial cooling, after the breakdown of supercooling has occurred. Up to about —50°C, the part of the diagram representative of R1 is a circular arc, while at higher temperatures the points ε˝(ε´) lie below this circular arc. The connection with that part of the diagram representative of R2 is formed, in most of the cases, by a loop.
Fig. 3. Variation of ε" versus ε'. Mass fraction of disperse water: p = 0.40. Frequency: v = 400 Hz.
The features of both R1 and R2 depend upon the value of p, the mass fraction of disperse ice. For values of p up to 0.43, the intensity of R1 is greater than that of R2 while it is the reverse for values of p greater than 0.43. As regards the critical temperatures T C1 and T C2 both of them are higher the greater the value of p (Table I), Tm remaining equal to — 20°C to within ±0.5 deg.
Table I. Variations Of T c1 and T c2 with þ (V = 1kHz)
For v = 400 Hz, the values found for T C1 and Tc2 are smaller. For instance, with p = 0.40, T C1 = 213 K and T C2 = 270 K. This is consistent with the observations made by Lagourette (unpublished) during his study of ice dispersions versus frequency at discrete sub-zero temperatures.
(b) Dielectric behaviour of dispersions of ice microcrystals obtained from emulsions of aqueous solutions of NH 4 Cl
A similar study has been carried out in the case of ice dispersions obtained from the breakdown of supercooling of emulsions of aqueous solutions of NH4Cl, in order to complement the investigations made in this field by Boned (unpublished [b]) and Reference BonedBoned and others (1976). Moreover, the metastable phase behaviour of aqueous solutions of NH4Cl has been thoroughly studied (D. Reference ClausseClausse, 1976), and is well known.
As is shown on Figure 4, the general behaviour of these systems is qualitatively similar to that exhibited by dispersions of ice obtained from emulsions of water, in particular as concerns the lower-tempcrature relaxation R1, which is shifted towards lower temperatures. The salt concentration x has a marked influence upon the features of R2 whose intensity increases with x while T C2 and T m decrease (Table II).
Table II. Variations of T c2 and T m with x (p = 0.40, v = 1kHz)
Fig. 4. Variations of (a)ε' and (b)ε" versus T. Concentration of NH4Cl: x = 7.4 × 10–2 = xE (eutectic concentration). Mass fraction of disperse phase: p = 0.40. Frequency: v = 1000 Hz.
Some experiments have shown that a similar behaviour can be observed in the case of ice dispersions obtained from emulsions of aqueous solutions of NH4NO3.
Discussion
The results that have just been reported concerning the dielectric behaviour of dispersions of ice microcrystals versus temperature are consistent with those obtained by Lagourette (Reference Lagourette1976, unpublished) who investigated versus frequency, at discrete sub-zero temperatures, the dielectric properties of that kind of system.
Figure 5 is a Cole—Cole plot of a dispersion of ice microcrystals (p = 0.40) studied versus frequency at —8°C, one hour or so after the breakdown of supercooling had occurred. A comparison of Figures 3 and 5 shows that R1 is similar to the higher frequency relaxation observed by Lagourette and R2 to the lower frequency one.
Fig. 5. Cole–Cole plot of a dispersion of ice microcrystals studied one hour after the breakdown of supercooling. T = —8°C, p = 0.40. The frequency v is given in kHz.
In the ε"(T) versus ε'(T) plot, the part of the diagram representing R1 is distorted at higher temperatures and, in most of the cases, is connected to that representing R2 by a loop that can be ascribed to a decrease of the static permittivity εs of ice upon increasing the temperature. This kind of phenomenon has been studied both experimentally and theoretically by different authors (Reference Le MontagnerLe Montagner, 1960; Reference SixouSixou and others, 1967; Reference Demau and VicqDemau and others, 1971). For a substance whose static permittivity is a decreasing function of the temperature T, these authors proved that, when T increases, the real part ε' of the complex permittivity recorded at a given frequency increases, reaches a maximum value and then decreases, which leads to a distortion of the ε" versus ε' diagram and to the formation of a loop if a further relaxation exists in the higher temperature range. In the present case, the existence of a loop can be put into evidence from the data of the study made by Lagourette, by plotting, at a given frequency, 1 kHz for instance, the values of ε' obtained at several temperatures ranging from —36°C up to — 1°C against the corresponding values of ε´.
As concerns the activation energy, its values cannot be determined with great accuracy since, for substances whose static permittivity is strongly temperature dependent (like ice), it is difficult to evaluate correctly the activation energy from the ε"(T) versus ε´(T) plot, as Reference Le MontagnerLe Montagner (1960) and Reference SixouSixou and others (1967) have stressed. For instance, the value found for R1 (Fig. 3) is equal to 0.5 eV at most. This high value indicates that the state of evolution in time of the dielectric properties of this system, which was studied during a continuous heating just after the breakdown of supercooling, is not as advanced as that of the system corresponding to Figure 1, which was first heated up to near 0°C and then studied during a subsequent step-by-step cooling. This is in agreement with the observation made by Evrard (unpublished [b]) that the evolution in time of the dielectric properties of ice microcrystal dispersions is the faster the higher the peak reheating temperature. A similar conclusion was arrived at by Lagourette (unpublished) who found values of the activation energy ranging from about 0.40 eV to about 0.16 eV, whatever the mass fraction of disperse ice.
Similarly, the discrepancy existing between the [—8°C, 400 Hz] data points of Figures 3 and 5 can be ascribed to the slightly different age-temperature history of the samples. However, comparison of the ε" versus ε' plots of Figures 3 and 5 shows clearly that the double relaxation phenomena found by Lagourette during fixed-temperature/variable-frequency experiments can be put into evidence as well as from fixed-frequency/variable-temparature investigations.
A freeze-etching study (Broto, unpublished) has shown that dispersions of ice microcrystals consist of individual spherical particles imbedded in the continuous phase of solidified oil (Fig. 6). This result justifies the application to ice dispersions of the formula proposed by Reference Hanaï and ShermanHanaï (1968) to represent the complex permittivity of spherical dispersion systems:
ε*, ε1*, and ε2* represent the complex permittivity of the dispersion, of the disperse phase and of the continuous phase, respectively, Φ being the volume fraction of the disperse phase. By comparing experimental data and computer calculations, M. Clausse (Reference Clausse1972, unpublished) and Reference Clausse, Royer and KerkerM. Clausse and Royer (1976), have established that the dielectric features of emulsions, particularly of the water-in-oil type, can be correctly represented by using Hanaï’s law and its derivative formulas.
Fig. 6. Freeze-etching replica of a dispersion of ice microcrystals, p = 0.20; magnification : 3260.
A first attempt made by Reference ClausseM. Clausse (unpublished) in 1971 to apply Equation (1) to the study of dispersions of ice microcrystals proved that the dielectric behaviour of these systems cannot be explained by assuming that the ice microcrystals exhibit merely a bulk conduction in addition to a Debye dipolar absorption. While the experimental values of either the higher-frequency or the lower-temperature limiting permittivity are in agreement with the computed theoretical ones, discrepancies appear when dealing with the lower-frequency (or higher temperature) limiting permittivity ε1. According to the theoretical model, ε1 should be related to the volume fraction of disperse ice Φ by
where ε2 represents the static permittivity of the continuous phase. Since Φ and the static permittivity ε2 of non-polar oil phases are almost constant in the temperature range concerned, ε1 should be temperature independent as well. It was shown by Reference LagouretteLagourette (1976) that ε1 depends upon temperature within the range 0°C to — 20°C and does not fit Equation (2). This result is consistent with the observations made during the present study. For instance, the experimental value of ε1 corresponding to Figure 3 was found to be equal to 14.8 while the theoretical value is 10. 1, the continuous phase permittivity ε2, almost temperature independent, being equal to 2.4, and the volume fraction of disperse ice Φ to 0.38.
A more sophisticated model has been designed by Reference LagouretteLagourette and others (1976) to interpret the dielectric behaviour of dispersions of ice microcrystals, particularly as concerns the dependence of the lower-frequency relaxation upon temperature. The main hypothesis underlying the model is that, at temperatures higher than — 20°C or so, clusters of disordered water molecules appear in the vicinity of the defects and dislocations of the ice lattice, the number of molecules involved increasing as the temperature approaches the melting point of ice. On this basis, by following a procedure proposed by Reference Clausse and LachaiseM. Clausse and Lachaise (1972) to study the dielectric properties of polydisperse systems, Reference LagouretteLagourette and others (1976) have computed, through an application of Hanaï’s formula, a theoretical model that fits the experimental data satisfactorily and allowed them to derive the following conclusions:
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(i) the higher-frequency (or lower-temperature) dielectric relaxation exhibited by dispersions of ice microcrystals is connected to the Debye dipolar absorption of the ice lattice;
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(ii) the lower-frequency (or higher-temparature) dielectric relaxation could be ascribed to a so-called “pre-melting” phenomenon arising from the formation, in the vicinity of defects or dislocations in the ice lattice, of essentially conductive clusters of disordered molecules, the number and/or size of the clusters increasing as the temperature approaches the melting point of ice.
The so-called “pre-melting” of ice microcrystals is to be considered in connection with the various observations reported by several authors who have suggested the existence of nearliquid disordered clusters or films in ice, at temperatures close to the melting point (Reference JellinekJellinek, 1967; Reference Bell, Bell, Myatt and RichardsBell and others, 1971; Reference Fletcher, Whalley, Whalley, Jones and GoldFletcher, 1973; Reference Kvlividze, Kvlividze, Kiselev, Kursa[y]ev and UshakovaKvlividze and others, 1974; Reference Nason and FletcherNason and Fletcher, 1975; Reference Golecki and JaccardGolecki and Jaccard, 1977; Reference Perez, Perez, Maï and VassoillePerez and others, 1978). In a recent paper, Reference BartisBartis (1977) has shown that the anomalous components acquired by the specific heat of a solid as its temperature approaches its melting point could be ascribed to the growth of “liquid” domains around point defects and along edge dislocations.
In the case of ice microcrystal dispersions obtained from the breakdown of supercooling of water-in-oil emulsions, it is most likely that, owing to the high degree of metastability reached (∆T = 38.5 deg), many defects are generated within the lattice of the ice particles during freezing. That would explain the striking effects of temperature upon the lower-frequency dielectric relaxation of ice microcrystal dispersions and justifies the claim by Reference LagouretteLagourette and others (1976), that, just below the melting temperature, 30% of the ice molecules could belong to disordered clusters. Anyway, it has been proved from DEA measurements (Reference Dumas, Dumas, Clausse and BrotoDumas and others, 1975), that the dependence upon temperature of the lower-frequency dielectric relaxation features cannot be ascribed to a partial melting of the ice particles induced by impurities at sub-zero temperatures.
The results reported in the preceding section concerning the dielectric behaviour of ice microcrystal dispersions obtained from the breakdown of supercooling of emulsions of NH4Cl aqueous solutions in oil add some support to the above conclusions.
Numerous studies have been devoted to the influence of ion incorporation upon the properties of ice (see for instance Reference Jaccard and LeviJaccard and Levi, 1961; Reference Noll, Riehl, Riehl, Bullemer and EngelhardlNoll, 1969; Reference Haltenorth, Klinger, Riehl, Riehl, Bullemer and EngelhardtHaltenorth and Klinger, 1969, Reference Haltenorth and Klinger1977; Reference Sakabe, Sakabe, Ida and KawadaSakabe and others, 1970; Reference BonedLafargue and Boned, 1973; Reference Klinger, Whalley, Whalley, Jones and GoldKlinger, 1973; Reference Camplin, Glen, Whalley, Whalley, Jones and GoldCamplin and Glen, 1973; Reference Gross, Gross, McKee, Wu, Gross, Wu, Bryant and McKeeGross and others, 1975). Most of them consider that impurities induce strong perturbations of the ice lattice. For instance, Sakabe and others consider that one NaOH molecule could facilitate the activation of several hundred water molecules.
From comparative dielectric measurements performed on macroscopic samples of ice doped with NH4Cl and on dispersions of microcrystals obtained from emulsions of NH4Cl aqueous solutions in oil, Reference BonedBoned and others (1976) have concluded that the doping effect is enhanced in the second case. During the present study, it has been observed that, for a same value of the mass fraction of disperse microcrystals p, the intensity of the higher-temperature dielectric relaxation increases with x, the molar fraction of NH4Cl This result could indicate that, as the NH4Cl incorporation increases, the number and/or size of disordered clusters increase. Moreover, the fact that the shift of Tm towards lower temperatures increases with x up to a limit which is close to the eutectic depression of the system H2O-NH4Cl (∆T e = 15.7 deg), could suggest that a correlation exists between the higher temperature dielectric relaxation and the ice microcrystal melting temperature.
