Introduction

The structural features of hexagonal ice as revealed by various experimental techniques and theoretical consideration are that, while the oxygen atoms form a regular hexagonal lattice, the hydrogen atoms are arranged randomly in that structure. Since such a disordered structure contradicts the Third Law of Thermodynamics, search for a possible ordering of the proton system at low temperatures has been one of the main subjects of recent investigation of hexagonal ice. Reference HaidaHaida and others (1974) reported new facts concerning this subject. In that study, the heat capacities were measured for H₂O ice quenched and annealed at several temperatures below 100 K and at around 100 K the heat-capacity values were found to be clearly dependent upon thermal history of the crystal. The spontaneous temperature rise during the annealing experiment was followed to obtain the characteristic time for enthalpy relaxation. They interpreted these phenomena as follows : As the temperature is lowered, change in the proton configuration becomes suppressed on its way to achieving a more ordered state because of the prolonged relaxation time for realizing the process.

If this is the case, we naturally expect a similar behaviour in D₂O ice. Already, there have been some attempts to find a similar anomaly in D₂O ice corresponding to that in H₂O ice around 100 K. Many years ago, Reference Long and KempLong and Kemp (1936) measured the heat capacity of D₂O ice by use of the same calorimeter employed for H₂O ice and showed the existence of finite residual entropy (3.22 J K^–1 mol^–1) just as in the case of H₂O ice. Nevertheless, they were unable to detect any phenomena of slow attainment to thermal equilibrium in spite of their intention inspired by the report of Reference Giauque and StoutGiauque and Stout (1936). On the other hand, recent investigations of elastic moduli for D₂O ice by Reference Helmreich, Whalley, Whalley, Jones and GoldHelmreich (1973) showed anomalous behaviour at temperatures between 100 and 150 K. Thus the present study on D₂O ice was aimed at finding the expected anomaly analogous to that in H₂O ice and also to detect some isotope effects on the relaxational ordering process if any. A brief description of the experimental results was reported by Reference Haida, Haida, Suga and SekiHaida and others (1973).

Experimental

The heat capacities of D₂O ice were measured by the same calorimeter and auxiliary apparatus used in the investigation of H₂O ice. Briefly, it is an adiabatic calorimeter with an intermittent heating method. The performance of the adiabatic control is such that natural heat leakage to the calorimeter cell including self-heating due to thermometer current is about 8 μ J s^–1 at around 100 K. The precision of the measurements was, in general, better than 0.1%.

Heavy water obtained from E. Merck (99.75% deuteration level) was degassed carefully by freeze-pump-thaw cycles in vacuo. The sample was then distilled and the middle fraction (32.381 g) was introduced into the calorimeter cell by vacuum distillation. The D₂0 ice crystal made in the calorimeter cell in situ was annealed at 273.6 K for about 3 d and then cooled down to 77K at a rate of about — 1 K min^–1 (“quenched” sample).

Experimental results

Heat-capacity measurement

The heat capacities of D₂O ice were measured in the temperature range between 14 and 300 K along the same lines as the study on H₂O ice. The first series of measurement is for a “quenched” sample, while the data for Series 2 and 3 correspond to the ones annealed at 102–106 K for 264 h and at 110 K for 166 h, respectively. The heat-capacity data are listed in Table I. For all of these series of measurements, there appeared a lag in equilibrium in the temperature region around 115 K. This observation stands in contrast to Long and Kemp’s result. These thermal anomalies arc illustrated in Figure 1. “Temperature drift”, the ordinate of the figure, is the value {T (30 min) — T (10 min)}, where T(t) is the temperature at time t; the time origin is the end-point of energy input. For Series 1, an exothermic temperature drift was observed, while an endothermic one was observed for Series 2 and 3 in the temperature range between 100 and 130 K. The reversal of the temperature drift from exothermic to endothermic by annealing and also the increase of temperature drift by annealing at lower temperature for longer periods found in these measurements are characteristic of relaxation phenomena which appear in a glass-transition region of the frozen-in disordered system. Thus, some degrees of freedom in D₂O ice crystals must be frozen around 115 K.

Table I. Molar heat capacities of deuterium oxide

Fig. 1. Anomalous temperature drift observed in D₂O ice. ○Series 1, ● Series 2, Series 3.

The heat-capacity data for the quenched and the annealed samples are shown in Figure 2 in the form of (C _ρ /T) versus T. The data for annealed samples (Series 2 and 3) exceed those for the quenched one (Series 1), the excess part being dependent on the annealing temperature. These observations confirm the existence of enthalpy relaxation around 115 K in D₂O ice. Since deuterons in D₂O ice have the disordered configuration, and since all the high-pressure polymorphs of ice reach some degree of order at low temperature, it seems natural to ascribe these anomalies to the relaxational deuteron ordering as described below.

Fig. 2. C_ρ/T of D₂O ice plotted against temperature. ○Series 1, ●Series 2, Series 3.

Residual entropy

The residual entropy calculated by Long and Kemp for D₂O ice had a discrepancy with that for H₂O ice. If the sophisticated theoretical work by Reference NagleNagle (1966) describes well the actual status of the disordered proton system, this value corresponds to 6% of deuteron ordering. This seems to be too much in view of the fact that merely 2% of the residual entropy of H₂O ice was removed by annealing the crystal over a period longer than 30 d as revealed by Reference HaidaHaida and others (1974). It seems worthwhile, therefore, to retest the value using recent more reliable data since the attainment of a more ordered state is eagerly desired for the conclusive interpretation of the anomaly at around 100 K in H₂O ice.

The calculation of the residual entropy was carried out by using our data along with the following values. The values of the molar enthalpy of vapourization, saturated vapour pressure, and the spectroscopic entropy of standard ideal gaseous state at 25°C were obtained from the works by Reference Rossini, Rossini, Knowlton and JohnstonRossini and others (1940) and by Reference Besley and BottomleyBesley and Bottomley (1973). The residual entropies were found to be 3.47 (Series 1), 3.44 (Series 2), and 3.45 (Series 3), ±0.41 J K^–1 mol^–1 respectively, as shown in Table II. These values are comparable in magnitude with those for H₂O ice. The isotope effect on the residual entropy is much less than that considered hitherto.

Table II. Residual Entropy Of Hexagonal Heavy Ice

The theoretical value of the residual entropy for a hypothetical completely disordered crystal based on the Bernal-Fowler-Pauling statistical model is (3.408±0.0008) J K^–1 mol^–1. The experimental value of the residual entropy for a quenched sample coincides with this value within the estimated experimental error. Therefore, the order attained can only be at most 1% even for the most annealed sample in this work (Series 2). Thus we can only investigate the very initial part of the ordering process in D₂O ice just as in H₂O ice, because the relaxation time of the ordering process increases rapidly with decreasing temperatures, at which significant deuteron ordering might proceed.

Activation enthalpy of the relaxation process

A spontaneous temperature rise due to enthalpy release in the sample during the annealing experiment was followed intermittently. Three typical examples of the temperature drift are shown in Figure 3, where the straight lines refer to the temperature drift caused by heat leakage due to incompleteness of the adiabatic control. The asymptotic approach of the temperature to the straight line was analysed by an exponential decay function with a single characteristic time τ, in the same manner as for H₂O ice. The relaxation times determined in this way are summarized in Table II along with other data. From the Arrhenius plot of the relaxation time, the average activation enthalpy of the relaxation process was determined to be (26±5) kJ mol^–1, which is very similar to the value, (22±4) kJ mol^–1, for H₂O ice in spite of large mass ratio of deuteron to proton. These values are within the range of 15 to 25 kj mol^–1 reported by Reference Sakabe, Sakabe, Ida and KawadaSakabe and others (1970), Helmreich (1969), and Reference Bishop, Glen, Riehl, Riehl, Bullemer and EngelhardtBishop and Glen (1969) for the activation enthalpy of the anomaly in pure and doped H₂O ice at around 100 K. These values for the activation enthalpy are very similar to those ascribed to the mobility of the Bjerrum defect, e.g. Reference JaccardJaccard (1959), and Reference Engelhardt and RiehlEngelhardt and Riehl (1966) gave 22 and 17 kJ mol^–1 for this defect, respectively. This seems to suggest, as has been pointed out by Helmreich (1969) and Reference Bishop, Glen, Riehl, Riehl, Bullemer and EngelhardtBishop and Glen (1969), that the Bjerrum fault participates in the ordering process of the deuteron configuration.

Fig. 3. Three typical examples of spontaneous temperature rise due to enthalpy relaxation of D₂O ice.

Table III. Data On Annealing Experiments

Discussion

If we plot the heat capacity data reported by Reference Long and KempLong and Kemp (1936) in the form of (Cp/T) versus T, we can find clearly an anomalous increase in heat capacity at around 120 K. Howeve , they did not mention this anomalous behaviour. Their failure to observe the anomalous temperature drift seems to be due to a difference in calorimetric principles. Since their calorimeter is of the isoperibol type, a correction for heat exchange between calorimeter cell and “quasi-isothermal” shield must be applied. After this correction, their calorimeter is reported to be capable of detecting a heat evolution (or absorption) exceeding an amount of 8 X 10^-4 cal mol^–1 min^–1 (1.9 × 10^-4 J mol^–1 min^–1). The corresponding temperature drift rate is approximately 4 mK/20 min. Therefore, if they had annealed the crystal for a long time, they might have observed the corresponding endothermic temperature drift.

The present calorimetrie observation is of course a kinetic effect, but reflects an underlying equilibrium property of the crystal. Unless the kinetic effect were associated with equilibrium ‘‘excess” heat capacity or enthalpy, we can not observe any spontaneous temperature variation under adiabatic conditions.

Reference Chamberlain and FletcherChamberlain and Fletcher (1971) and Reference Johari and JonesJohari and Jones (1975) have reported that the current peak due to thermally stimulated depolarization (TSD) observed at 110 K and 124 K in H₂O and D₂O ices is due to relaxation of frozen-in orientation polarization of water molecules in both crystals. The relaxation times for reorientation as determined by the TSD current and by the step-response method studied by Reference Johari and JonesJohari and Jones (1975) are plotted logarithmically against reciprocal temperature in Figure 4, along with the characteristic times determined by an enthalpy-relaxation experiment. The three sets of relaxation times for H₂O ice can be connected by a single smoothed curve. The calorimetric data for D₂O ice are higher by about one order of magnitude than the dielectric data and the difference is likely to be due to the different amounts of physical and chemical defects involved in the respective specimens. The deuteration level of the calorimetric specimen is 99.75%, while that of the dielectric specimen is reported to be 98.75%, In spite of some differences in their behaviour, however, this figure indicates essentially two facts. First, the degrees of freedom associated with the enthalpy relaxation are the same as in the dielectric relaxation. This explanation was also proposed by Reference Giauque and StoutStout and Giauque (1936) and by Reference JohariJohari (1976), but this figure provides convincing evidence for that. Secondly, some configurations of water molecule realized by that degree of freedom must differ energetically from others. This is the necessary condition for the relaxation to be observable calorimetrically.

Fig. 4. Relaxation-time data of ice crystals determined by dielectric arul calorimetric methods. ○calorimetric method × TSD method, ● Step-response (or bridge) method.

Reference Campbell, Campbell, Gelernte, Heinen and MoortiCampbell and others (1967) have calculated the lattice energies of an ice crystal in terms of multipole interaction as a function of orientation of the water molecules and found significant differences between the energy of the various configurations which must affect the thermodynamic properties of the crystal at temperatures around 100 K. All of these facts lead to a picture concerning the ordering process of protons (or deuterons) in ice crystals. The water molecules are in an orientationally disordered state at high temperatures; all the six orientations of a water molecule might be realized essentially equally, as the Reference PaulingPauling (1935) model and the subsequent statistical model by Reference LiebLieb (1967) assume ordinarily. As the crystal is cooled down, some of the proton configurations arc favoured by intermolecular interactions among the water molecules, thus realizing a locally ordered state. The corresponding configurational heat capacity increases gradually as the temperature is lowered.

The configurational heat capacity would increase further and reach a maximum value at a transition point where an ordered state might be realized, if we could cool the crystal at an infinitely slow rale. The prolonged relaxation time, however, hinders the crystal from reaching the equilibrium state in an actual fast-cooling experiment and the proton configurational disorder is frozen at a certain temperature. The corresponding decrease in the configurational heat capacity occurs at 100–110 K for H₂O ice and at 120–130 K for D₂O ice, respectively. Consideration of the values of residual entropies for the quenched and the annealed samples of H₂O and D₂O ice crystals shows that we are concerned with very initial stage of the ordering process and the hypothetical transition point seems to be located much lower than too Reference Pitzer and PolissarK. Pitzer and Polissar (1956) predicted the existence of an order-disorder transition in H₂O ice at about 60 K with a polar structure of the resulting phase.

In passing, we may mention the deviation of the calorimetric relaxation time of ice crystals from the high-temperature behaviour of the dielectric relaxation time,

where τ₀, B, and T₀ are empirical constants. This equation was observed by Reference JohariJohari and Jones (1976) to hold for the dielectric relaxation time of D₂O ice in the temperature range 120 < T/K< 170, with the values τ₀ = 3x10^-16s, B = 828 K and T ₀ = 60 Reference Adachi, Adachi, Suga and SekiK. Adachi and others (1968) observed a similar deviation of calorimetric relaxation time in the orientationally disordered crystal cyclohexanol below its “glass-transition” temperature and explained this from the viewpoint of plural relaxation mechanisms with different time constants. The multiplicity of dielectric relaxation times of D₂O ice at low temperature was pointed out by Reference JohariJohari and Jones (1976). Probably, the calorimetric observation corresponds to the average relaxation time at a relatively high temperature but will correspond to the fastest process when the distribution of the relaxation times becomes wider as the temperature is lowered.

The excess enthalpy released during an annealing experiment in a frozen state is regained as an increased heat capacity when the crystal is heated. If we compare curves of C_p/T versus T for H₂O and D₂O ice crystals, we notice that the excess heat capacity of D₂O ice appears about 20 K higher than that of H₂O ice. This temperature difference is likely to reflect the difference in equilibrium properties of both crystals. A Curie-Weiss representation of the dielectric constants of both crystals by Reference JohariJohari and Jones (1976) indicated that the extrapolated Curie point was 27 K higher for D₂O than that for H₂O ice. Therefore, the higher peak temperature of excess heat capacity for D₂O ice seems to reflect the correspondingly higher Curie temperature. An increase in the Curie point of a hydrogen-bonded crystal on deutera- tion is exemplified by many systems. Reference Chan, Whalley, Whalley, Jones and GoldChan (1973) gave a possible interpretation of the increase in the Curie temperature from the viewpoint of isotopic effect on lattice vibrational frequencies and zero-point energy.

Nevertheless, the relaxation times of both crystals reach a calorimetric time scale (≈10⁴ s) at 104 K and 121 K, respectively, and therefore we can observe only a high temperature part (very initial part) of the ordering process for both crystals. The amount of excess heat capacity is nearly the same for both crystals when comparing the same time-scale experiments, in spite of the difference in their respective Curie temperatures. This fact seems to indicate that the relaxation time for the reorientational degree of freedom is governed by a static property. The important role configurational entropy plays in the relaxational properties of frozen system was first proposed by Reference Adam and GibbsAdam and Gibbs (1965). If this mechanism is operative in the case of ice crystals, we must abandon the assumption of single relaxation time employed in the analysis of temperature-time curves from the calorimeter. Configurational order and hence the relaxation time must change as the time elapses in one experiment. The experimental verification of this mechanism requires a much better performance of the adiabatic control because of the smallness of the excess enthalpy involved and because of the ambiguity in determining the “base line” due to heat leakage. In this respect, it may be interesting to study a solid-solution system, H₂O-D₂O. The Curie point of the system may be situated between those of the parent crystals. The complexity of the system is increased by the isotopic substitution, which is expected to shorten the relaxation time of the system. Alternatively the impurity effect might cause a shortening of the relaxation time. Thus, we can hope to observe an enhanced heat-capacity anomaly.

In any event, we could detect the relaxational heat-capacity anomaly in D₂O ice in an analogous way to that in H₂O ice. This means that the rate of rearrangement of protons (deuterons) is on the calorimetric time-scale (10³-10⁶ s) at a temperature range where energy differences among the possible orientations of the disordered water molecules promote ordering even slightly. The recent observation of a heat-capacity anomaly in another orientationally disordered crystal, CO, around 18K by Reference AtakeAtake and others (1976) was interpreted along the same lines. The relaxational nature of the anomalies in CO and ice crystals is seemingly not independent of the existence of residual entropy in both crystals.

Up to this point, we have assumed implicitly that the observed heat-capacity anomaly was an intrinsic property of the ice crystal that was brought about by the reorienting motion of defects which lost their mobility as the temperature was lowered. There might be another possibility, that the anomaly is associated with some local order induced by lattice defects. However, we can scarcely expect to detect calorimetrically the thermal anomaly due to such defects existing in minor concentration. The amount of orientational defects in pure ice is considered to be of the order of 10^-7 mol % even at — 10°C, and that of ionic defects is much less. An experiment to see how the entropy of an ice crystal can be removed by doping with a particular impurity might be helpful to clarify the nature of the relaxational anomaly. If the corresponding defect were to induce some short-range local order in water molecules in the neighbourhood of the defect, we would expect an increased amount of excess heat capacity in a doping experiment, because the increase in defect concentration will increase the number of regions in which some local order is induced. The excess heat capacity will be observed at essentially the same temperature as in a pure crystal. If the defect were to shorten the reorganization rate of the whole system, we could expect an increased heat-capacity anomaly that starts from a lower temperature. Thus the doping experiment will throw some light on the “intensive” and the “extensive” roles of the lattice defects.