4155
J. Phys. Chem. 1983, 87, 4155-4156
Influence of Aging on the Heat Conduction Coefficient of Hexagonal Ice J. Kllnger’ Laboratolre de Glacloiogle et de &physique
de I’Environnement, 8. P. 68, F38402 St. Martin dH6res Cedex, France
and G. Rochas Centre d’Etudes Nucl&ires de Grenobie, Servlce des Basses Temp&atures, Grenoble Cedex F 38000, France (Received:August 23, 1982: I n Final Form: January 4, 1983)
We measured the heat conduction coefficient of two hexagonal ice samples cut from the same single crystal with the c axis oriented perpendicular to the heat flux. One sample has been measured immediately after growth whereas the other has been measured after 15 days of aging at 272 K. The aging produces a lowering of the heat conduction coefficient K of 530% for T I10 K. Further aging for 1 week lowers the K by 5 5 % . The role of defect migration in this aging process is discussed.
Introduction The heat conduction coefficient of hexagonal ice has been studied in a rather detailed manner between 1.5 K and the melting point.1*2 In order to improve the understanding of the interactions of phonons with crystal imperfections the temperature range of the measurements has been extended to 0.5 K.3 A more recent paper4 is devoted to the problem of anisotropy. In this work we examine the influence of aging on the heat conduction coefficient of ice. As in earlier ~ o r k f iwe ~ ,analyze ~ our results using a simplified form of Callaway’s modeL5 The heat conduction coefficient K is given in the following form:
‘IT m4eX(ex - 1)-2 dx
(1)
with x = h w / k T ; k is the Boltzmann constant, 2ah is Planck’s constant, is the Debye temperature (225 K), and u = 2.5 X lo5cm/s is the mean velocity of sound. T is the total relaxation time of phonon interactions. The expression used for the relaxation rate (7-l) is the following:
The first term in eq 2 is due to “umklapp” processes.6 The third term is due to interactions with the boundaries of the macroscopic ample.^ The second and third terms are due to phonon interactions with defects. As has been shown earlier’~~ in a great number of cases it is possible to describe nearly all the defect scattering by taking R = 3 and D = 0. But as in all real crystals point defects are present and we used in some cases4v8D # 0 and P = 4. (P = 4 is generally considered to represent the scattering on point defects.6) Experimental Procedure The measurements have been done by the classical stationary heat flow method described in ref 1 and 3. Two different heat sinks are used: In the temperature range (1) Klinger, J. J. Glaciol. 1975, 14, 517. (2) Slack, G.Phys.Reu. 1980,22, 3065. ( 3 ) Varrot, M.; Rochas, G.; Klinger, J. J. Glaciol. 1978, 21, 241. (4) Klinger, J.; Rochas, G.J. Phys. C 1982, 15,4503. (5) Callaway, J. Phys.Reu. 1959, 113, 1046. (6) Klemens, P. Solid State Phys. 1958, 7 , 1. (7) Casimir, H. Physica 1938, 5, 465. (8) Klinger, J. In “Physics and Chemistry of Ice”; Whalley, E.; Jones, S. J.; and Gold, L. W., Ed.;Royal Society of Canada: Ottawa, 1972; p 114. 0022-365418312087-4 155$01.50/0
TABLE I: Parameters Used for Best Fits Sam- lO”A, d e (s K)-’ 1
4A 4B
1.75 1.75 1.75
4
6.5 6.5 6.5
IOBB, 105c, 1 0 ~ ~ 0 , sz s-’ s3 P 1.46 1.46 1.46
1.5 3 3
4.1 4.1 4.1
4
4 4
R 3 3 3
between 1.5 and 20 K a 4He bath was used. Between 0.5 and 4 K we used a 3He bath. The temperature overlap of those two techniques allowed us to check the reproductibility of the measurements. Thermal contact between the sample and the sample holder was established by means of 170 gold-plated copper wires each 0.15 mm thick and 14 mm long. One end of these wires was soldered to the sample holder whereas the other was frozen to the sample. As pointed out in ref 4 this method gave a thermal impedance proportional to 1 / T whereas the method used in ref 1 gave an impedance proportional to 1 / P . For this reason our method was usable at temperatures lower than 1 K. Chaillou and Duval from Laboratoire de Glaciologie provided the ice single crystals. The method of crystal growth has been described earlier.4 The samples referenced 1 and 4 used in this work were cut from the same single crystal. The length of the samples was 40 mm and the cross section 4 X 4 mm; the longest dimension (the direction of the heat flow) was perpendicular to the c axis of the crystal. As in earlier work the cooling rate between -15 “C and liquid nitrogen temperature was 0.6 “C/min. The theoretical maximum error on the absolute value of the measured heat conduction coefficient was about 14%. 6% is due to errors in the measurement of the form factor. But as shown in ref 4, the fact that the reproductibility between independent measurements of identical samples is very good shows that the theoretical maximum errors are very much higher than the real errors. Results Figure 1 shows the heat conduction coefficient of a fresh sample (sample 1) compared to that of an aged one (sample 4). We see that for temperatures of the order of 15 K the two plots are identical as has been stated A sample aged for 15 days at a temperature of -1 “C (sample 4A) has a lower heat conduction coefficient than the fresh sample for temperatures lower than 15 K. The difference reaches about 30% at 2 K and does not increase significantly for lower temperatures. After further aging for 7 days at -1 “C (sample 4B), we see a further lowering of the heat conduction coefficient that does not exceed 5 %. We have to state that this change is not explainable by a @ 1983 American Chemical Society
4156
The Journal of Physical Chemistry, Vol. 87, No. 21, 1983
[W/cm K ]
Klinger and Rochas K exp
4
2L
4-
3-
1
,
I
1
1
,
,
1
1
1
I
0
0.4 0
0.3
0,Ct
I 05
I
I
/
2
3
4
I I 1 1 1 1 5 10
I
,
L
20T [ K y
Figure 1. Heat conduction of a sample of fresh hexagonal ice with the c axis oriented perpendicular to the heat flux (0, sample 1) compared to a sample with the same orientation aged for 15 days at -1 OC (A,sample 4A) and the same sample aged at the same temperature for 22 days (X, sample 48).
variation of the form factor as the sample mounting during the aging experiment was the same. We fitted our experimental results using eq 1and 2. The parameters used for the best fits are given in Table I. The ratio of the heat conduction coefficient calculated with these parameters and the measured heat conduction coefficient as a function of temperature has been plotted in Figure 2. We see that over the whole temperature range the difference between the measured and the calculated heat conduction value is always much lower than the theoretical experimental maximum error.
Discussion The significance of the relaxation rates proportional to w3 and w4 has been discussed in earlier p a p e r ~ . l *These ~ relaxation rates are not affected by aging. The exponential term is due to “umklapp” processes that are intrinsic processes. The only parameter that is changed during aging is the constant relaxation rate C that is due to phonon scattering of the boundaries of the macroscopic sample. The increase of C by a factor of two means that phonons “see” a smaller sample after aging. As the geometry of the crystal evidently remained unchanged during the experiments we have to conclude that a rearrangement
took place within the sample leading to an “apparent” size smaller than the initial one. Clustering of dislocations may be such a rearrangement. In fact it has been observed by X-ray topography of hexagonal ice single crystals comparable to ours that the original dislocation network disappeared during aging. The result is that one part of the crystal does not contain any dislocations after aging whereas the other part of the sample contains so much perturbations that individual dislocation are indisting ~ i s h a b l e . ~This type of aging effect is speeded up in crystals that have a size comparable to our samples.l0 The boundaries between the disturbed and the undisturbed part of the crystal may interact with phonons in the same manner as the boundaries of the sample do. The fact that the result for sample 4B is only slightly different from the result for sample 4A indicates that after 22 days at -1 “C the crystal has nearly reached a stable state. A more physical interpretation of the rearrangement of the crystal in terms of a minimum energy configuration cannot be given at this moment and needs more effort in theoretical and, perhaps, experimental work.
Acknowledgment. The experimental part of this work has been done at Service des Basses Temperatures du Centre d’Etudes Nucleaires at Grenoble. We thank its director Dr. Doulat for allowing us to use the low-temperature facilities. The study has been sponsored by INAG-ATP Planetologie Grant No. 37-86. Dicussions with Dr. A. M. de G&r and Dr. B. Salce from Centre d’Etudes Nucleaires at Grenoble are gratefully acknowledged. We thank Monsieur A. Chaillou and Dr. P. Duval from Laboratoire de Glaciologie for providing the ice samples. Registry No. Water, 7732-18-5. (9)Vassoille, R.; Tatibouet, J.; Mai, C.; Perez, J. IAHS Publication No. 118,1975,p 38. (IO) Tatibouet, J.; Mai, C.; Perez, J.; Vassoille, R. J.Physiq. 1981, 42, 1473.
