Thermal conductivity of ice and ice clathrate

Solid State Communications, Printed in Great Britain.

Vol. 63, No. 2, pp. 167-171, 1987.

THERMAL

CONDUCTIVITY

0038-1098/87 $3.00 + .OO 0 1987 Pergamon Journals Ltd.

OF ICE AND ICE CLATHRATE

N. Ahmad and W.A. Phillips Cavendish Laboratory, (Received

Madingley Road, Cambridge CB3 ODS, UK 20 November

1986 by R.A. Cowley)

We present the results of an investigation of the thermal conductivity, rc, of ice and an ice clathrate (1,3-dioxolane). It is observed that ICfor the clathrate depends upon the cooling rate during solidification and the temperature dependence of JC differs from that of amorphous materials unless the cooling rate is rapid. A consistent picture of thermal conduction in ice and clathrates is given on the basis of tunnelling states arising from proton disorder. 1. INTRODUCTION THE TERM CLATHRATE was introduced [I] for a structural combination of two substances which remain associated with each other not through strong attractive forces, but because the molecules of one substance form a strongly bound cage-like structure, which firmly enclose molecules of the other. The clathrate hydrates thus constitute a class of solids in which molecules of many types of gas or liquid occupy almost spherical holes in icelike lattices made up of hydrogen-bonded water molecules [2]. As expected from the similarities in lattice structure properties such as the sound velocity [3,4] and the electrical resistivity [5] of clathrates are not greatly different from the corresponding properties of ice. However, the reported thermal conductivity, rc, of clathrates hydrates is markedly different from that of ice [6-81 between room temperature and 40K. All these measurements gave roughly the same rc (about 0.5 Wm-i K-l) which is approximately a fifth of the value of ice-Ih near the melting point. For ice-Ih [9], IC is roughly proportional to l/T between 260 and 100 K but in clathrates rc decreases by 15% as T decreases in this range [7]. This behaviour is typically of amorphous solids, although clathrate hydrates are crystalline. The work described in this Communication was undertaken to extend the available data on the thermal conductivity of the clathrate hydrates of type II to liquid helium temperatures and to increase understanding of the nature of heat transport in these substances. As a test of the method of measurement of IC, polycrystalline samples of ice were also measured. The results agreed with measurements on single crystals of ice-Ih [lo, 111, but analysis of existing results on ice [10-131 lead us to reinterpret the microscopic inter-

pretation in terms of scattering from tunnelling states [141. 2. EXPERIMENTAL

TECHNIQUE

The clathrate hydrates are liquids at room temperature. A sample cell is therefore required into which the liquid can be poured and then solidified by cooling. This cell was a nylon cylinder of inner diameter of 5 mm and length 20 mm with wall thickness 0.1 mm [15]. Using a linear heat flow method the temperature difference along the sample was measured with RhFe vs chrome1 thermocouples, used in a differential mode and wrapped around the cell. The conductance of the sample cell, measured separately, was less than 0.55% of ice at 80 K and only 0.007% at 2.5K. The measured thermal conductivity of spectrosil-B using these thermocouples was within f. 5% of the reported measurements [ 161,serving as a test of the temperature calibration. Absolute temperatures from 300 to 4.5 K were measured using a four terminal RhFe standard accurate to & 60 mK above 30 K and + 30 mK below 30 K. For more accurate measurement of the absolute temperatures below 20K, a Ge standard with a calibration of f: 8 mK was used. 3. SAMPLE PREPARATION

AND RESULTS

To prepare ice samples doubly ionized water was poured into the sample cell and cooled to liquid nitrogen temperature at a cooling rate of 0.8 K min-’ in a nitrogen gas atmosphere. X-ray diffraction studies show that samples prepared in this manner are polycrystalline. Nitrogen gas was pumped out before making thermal conductivity measurements. The thermal conductivity of ice corrected for the contributions from the sample cell is shown in curve 1 in Fig.

167

THERMAL

I

LCE-011 I

CONDUCTIVITY

\

I IO

I loo

T/K

Fig. 1. The thermal conductivity of ice and DO clathrate hydrate. 1 (present work) & 7 (Klinger ef al. [lo]) for ice, 3, 5 and 6 for clathrate with rapid cooling during solidification and 2 and 4 for clathrate with slow cooling. The curves are fit to the data assuming tunnelling states arising from proton disorder and scattering from point defects. For ice and the slowly cooled clathrate Umklapp processes are also used. 1. The reproducibility from run to run is 5%. Earlier results [lo] are also shown (curve 7) for comparison. 1,3-dioxolane (DO) clathrate hydrate was selected for the investigation because of all the clathrate hydrates it has the highest K in the liquid nitrogen temperature range [7]. The method of preparation of the sample has been described in [2] and has been used by all investigators. In earlier measurements [7] a mixture of MS 16.46H20, where M is the molecular weight of DO, was cooled under pressure (0.05-0.1 GPa) to obtain clathrate hydrate. The composition of the sample was kept on the H,O-poor side of the nominal clathrate hydrate composition to avoid the presence of ice in the samples. In our system, we could not apply pressure during solidification; in fact, a small hole in the top cap of the cell prevented any rise in pressure due to change in the volume of the liquid on cooling. A mixture MS 16.46H20 was poured into the nylon cell and was cooled at about 8 K min-’ in a continuous flow cryostat in nitrogen atmosphere. After thermal conductivity measurements between 200 and 10 K, the sample cell was moved to a second cryostat where the measurements can be extended to

OF ICE AND ICE CLATHRATE

Vol. 63, No. 2

2K. Although the sample melted completely during transfer, the cooling rate during solidification was roughly the same in both cryostats. These two sets of measurements are in good agreement in their region of overlap as shown in Fig. 1 (curves 5 and 6). The behaviour of the curve in the range 80 to 200K is similar to measurements made by other investigators [3,6,7, 171and typical of glasses. However our results at 100 K are about 50% larger than reported by others [71* It has been claimed [18] on the basis of NMR studies that samples prepared in this way are not perfect crystals and that better samples can be prepared using a procedure in which the clathrate is solidified and remelted until a very small solid piece is left. The sample is then kept at the solidification tempertaure for over two weeks to grow a perfect crystal from the seed. It was not possible to apply this method in our system and so we have attempted to study the effect of sample perfection by varying the cooling rate during solidification. The lowest available cooling rate (0.8 Kmin’) gave results shown in Fig. 1 (curve 2), markedly different from those obtained on rapid cooling. Subsequently the sample was remelted by bringing the system to room temperature and was rapidly cooled by placing the insert into liquid nitrogen. The measurements as shown by curve 3 are in agreement with those obtained previously using fast cooling rates. The sample was remelted and was again slowly cooled (0.8 K min’). The measurements, shown by curve 4 in Fig. 1, are about 20% larger than those previously obtained with slow cooling rate. This implies that a given sample could be made to alternate between the behaviours typified by curves 4 and 5 by cooling quickly or slowly. It is clear from this figure that the temperature dependence of ic depends upon the cooling rate during solidification. At slow cooling rates it is similar to that of crystalline substances, whereas at fast cooling rates it is similar to that of glasses although the X-ray diffraction studies show a similar crystalline structure. 4. DISCUSSION To study the behaviour of IC as a function of temperature T, K can be expressed in the following integral form K(T) = ; 7 C(o, T)l(w, T)v(o,

T)do,

(1)

0

where C(o, T) is the specific heat at temperature T for a mode of angular frequency o, I(o, T) is the mean free path and o(w, T) is the velocity of the phonons. The phonon heat capacity is given by

THERMAL

Vol. 63, No. 2

Ii2 c(0y

T,

=

@

g(0)

CONDUCTIVITY

o2 exp @o/kT) [exp (ho/kT) _ 132’

(2)

where g(o) is the density of phonon states and k is the Boltzmann constant. Published heat capacity data for T > 2K for ice shows a broad peak in C/T3 against temperature at 15 K [ 191and can be fitted to determine g(w) using the procedure adopted elsewhere [15]. However, it was not felt necessary to perform such a calculation, since dispersion, (average zone boundary frequency for transverse acoustic phonons [9] is 53crn’) which increases g(o) over the Debye-like density-of-states, g,(o), also leads to a compensating decrease in velocity with increasing o. Therefore a Debye-like density of vibrational states with a constant v can be a good approximation to the product of g(o) and v(o). In the previously reported measurements [lO-121 g&o) has been used with an upper limit of integration in equation (1) equal to the Debye temperature (226 K or og = 2.97 x 1013s-‘). In contrast we have used an upper limit which gives 3 acoustic modes per unit cell (0” = 1.93 x 1013s-l). Alternatively, if it is assumed that the phonons which contribute to thermal conductivity are only those which have non-zero group velocities then the upper limit of the integral in equation (1) should be 53cm-’ or wz = 1.0 x 1013s-‘. The limiting value of C/T3 at T = 0 obtained by Flubacher et al. [19] is 9.2$g-’ Kp4. Assuming that the mass density of ice is 917 kgrn3, the estimated C/T3 gives an average sound velocity of v of 2.44 x 103ms-‘. This value lies between that determined from elastic constants [9] at - 16°C 2.13 x 103ms-‘, and the sound velocity estimated at OK, 2.55 x 103ms-‘, using a constant value of - 1.45ms-’ K-’ for the temperature coefficient of velocity [19]. Since Table I. Parameters

to$t

thermal conductivity

Parameter

Units

A

s2m-1

B” D L s 8

upper limit of the integral

the temperature coefficient of velocity must tend to zero as T approaches zero, the results of thermal and elastic measurements appear to be in agreement. The total mean free path l(w) can be written as

V(w) = F Vi(W), where I,(o) is the mean free path for a particular scattering mechanism. In a high purity single crystal, Umklapp processes are the dominant source of phonon scattering at high temperature, with an inverse mean free path [20] I”-’ proportional to T”w~~-“~. At low temperatures boundary scattering is important and the mean free path L can be obtained from geometrical arguments. It is frequency independent and in general is approximately equal to the smallest linear dimensions of the crystal. These two processes are however usually insufficient to predict experimentally determined thermal conductivities and other processes [20] must be assumed. The total mean free path therefore can be written as [I(o)]-’

mK K-2 s3m-’ m K Hz

=

(boR + ATSco2e-“‘)

+ d,

(4)

where R can have values from zero to four depending upon the scattering mechanism [20], 8, A, b, d and s are constants with d = l/L. Equation (1) can be solved using numerical techniques to determine the nature of the scattering mechanism. Previous experimental results have been interpreted in terms of Umklapp processes, boundary scattering and a scattering proportional to w3. This last term was originally associated with cylindrical inclusions observed in electron microscopy [13] but has more recently been reinterpreted as arising from dislocation cores [ 111. However, the required dislocation density of 1014m-2 is greater than normally found in

of ice and clathrate hydrates

Ice (Present work) K-1

169

OF ICE AND ICE CLATHRATE

4.17 x 1o-21 0.08 0.02 5.83 x 1O-46 0.005 1 33 1.93 x 1oL3

Clathrate hydrates Curve 4 8.33 x 1O-21

1.1 x 1o-4 0.0125 1.25 x 1O-45 0.005 0.8 10 1.00 x lOI

Curves 3, 5, 6 1.39 x 1o-s 0.001 4.9 x 1o-45 0.005 1.00 x 1o13

D coefficient of w4; B and /3 constants in the tunnelling model; L smallest size of the sample; d = l/L; 8, s? and A constants used in Umklapp processes. The condition ql % 1, where q is the phonon wave vector, was satisfied

in the temperature

range of the fit.

170

THERMAL

CONDUCTIVITY

OF ICE AND ICE CLATHRATE

non-metals (and it is independent of the upper limit (w, or wz) of the integral in equation (1)). A further problem with this analysis is the fact that rc does not vary as T3 at the lowest temperatures, as it should if boundary scattering from sample boundaries is dominant. We have reinterpreted the results on ice by including scattering from tunnelling states associated with proton disorder. For tunnelling states, the mean free path I, is [21] 1;’

=

for

@o/kB] tanh @w/kT) ho/k

>

+ jIT3/4B

PT3,

(5)

and I,-’ = for

[hw/kB] tanh @w/kT) fiiw/k

<

pT3,

+ ho/4Bk (6)

where B and B are constants. An additional term Dw4 for the scattering from point defect is also used to fit the data. With scattering from tunnelling states and point defects together with Umklapp processes the thermal conductivity of ice can be fitted with the parameters given in Table 1, as shown in Fig. 1. This explanation looks more promising as proton disorder is a common phenomenon in materials containing hydrogen bonds, and tunnelling predicts K proportional to T2 at low T, close to that measured by Klinger et al. [l 11.However, more experimental work on the structure of ice is needed to determine the nature and number density of the defects. It is clear from Fig. 1 that the ICfor clathrate hydrates showing crystalline behaviour cannot be obtained by simply scaling the measurements on ice. Equation (l), therefore, must be solved to fit the data to obtain any useful information. The upper limit of the integration once again can be fixed to give 3 acoustic modes per unit cell and because of the greatly increased unit cell size leads to a much smaller value of cut-off frequency. However, the problem is now more complicated because of the presence of the guest molecules although because they do not interact [18, 221 with the host lattice their effect is small. It was shown [3, 41 that the longitudinal sound velocities in the clathrate and ice structure are similar and it is assumed that the transverse velocities are also similar and the density of the empty clathrate structure is 770 kgmm3. As in ice it was found that K cannot be explained by considering Umklapp processes and boundary scattering alone and another parameter is necessary. Assuming the additional scattering arises from dislocation cores it is possible to fit the data shown in curves 2 and 4 but not the data shown by curves 3, 5 and 6. As in ice, therefore, an analysis based on tunnelling states resulting from proton dis-

Vol. 63, No. 2

order looks more promising. Using the tunnelling model, scattering proportional to w4 and Umklapp processes the curve 4 can be fitted with parameters given in Table 1. The quality of the fit is shown in Fig. 1 by a solid curve. For curves 3, 5 and 6, scattering from tunnelling states and point defects alone are considered. The fitting parameters are given in Table 1 and the fit is shown in Fig. 1. The value of B for curves 3, 5 and 6 is comparable with that for vitreous silica. The parameter D given in Table 1 for ice and clathrate hydrates clearly shows that the disorder in ice is lower than in clathrate hydrates and the disorder in clathrate hydrate is lower if the cooling rate during the solidification process is reduced. Notice that B decreases as D increases, a trend consistent with increasing disorder. More work on the structure of these species is needed to determine the exact nature of these defects. 5. CONCLUSION The conclusion to be drawn from these measurements is that the thermal conductivity of clathrate hydrates does not resemble that of amorphous materials, as claimed by many investigators, unless the cooling rate is rapid. A consistent picture of the thermal conduction in ice and clathrates has been given by assuming the presence of tunnelling states arising from proton disorder. - We are thankful to Dr. E.A. Marseglia for performing X-ray diffraction studies on the samples and Prof. F. Franks from this University, and Prof. M.D. Zeidler of the Institute Fur Physikalische Chemie, 5100 Aachen, W. Germany for useful discussion on the preparation of clathrate hydrates. Acknowledgements

REFERENCES 1.

3. 4.

H.M. Powell, J. Chem. Sot. (London) 61, (1948). D.W. Davidson, Water, Vol. 2, (Edited by F. Franks), Plenum Press, N.Y. (1973). E. Whalley, J. Geophys. Res. 85, 2539 (1980). B.L. Whiffen, H. Kiefte & M.J. Clouter,

5.

Geophys. Res Lett. 9, 645 (1982). D.W. Davidson, Natural Gas Hydrates

2.

7.

(Edited by J.L. Cox), Ann Arbor Science Publishers Inc. N.Y. (1982). T. Ashworth, Report Phys. 84-1, (1984) PressPhysics Department South Dakota School of Mines & Technology, Rapid City, South Dakota, 57701. P. Andersson & R.G. Ross, J. Phys. C16, 1423

8.

J.G. Cook & M.J. Laubitz, Thermal Conductiv-

6.

(1983).

Vol. 63, No. 2

9. 10. 11. 12. 13. 14. 15.

THERMAL

CONDUCTIVITY

ity 17, p. 745 (Edited by J.G. Hust), Plenum Press N.Y. (1983). G.A. Slack, Phys. Rev. B22, 3056 (1980). J. Klinger & K. Neumier, C.R. Acad. Paris R269, 945 (1969). J. Klinger & G. Rochas, J. Phys. C15, 4503 (1982). J. Klinger, Solid State Commun. 16, 961 (1975). G.J. Davy & D. Branton, Science 168, 1216 (1970). W.A. Phillips, J. Low Tern. Phys. 7, 351 (1972); P.W. Anderson, B.I. Halperin & C. Varma, Philos. Msg. 25, 1 (1972). N. Ahmad, Ph.D. Thesis 1986, Cambridge University (Unpublished).

OF ICE AND ICE CLATHRATE 16. 17. 18. 19. 20. 21. 22.

171

R.C. Zeller & R.O. Pohl, Phys. Rev. B4, 2029 (1972). J.G. Cook & D.O. Leaist, Geophys. Res. L&t. 10, 397 (1983). C. Albayrak, A. Geiger & M.D. Zeidler (To be Published). P. Flubacher, A.J. Leadbetter & J.A. Morrison, J. Chem. Phys. 33, 1751 (1960). P.G. Klemens, Solid State Physics 7, p.4, (Edited by F. Seitz and D. Turnbull), Academic Press N.Y. (1958). D.P. Jones, Ph.D. Thesis (1982), Cambridge University (Unpublished). M.W.C. Dharma-Wardana, J. Phys. Chem. 87, 4185 (1983).