Internal Friction in Ice Crystals

Aug 23, 1982 - The attenuation of sound at frequencies between 15 and 85 MHz has been measured by the pulse reflection method in single crystals of ic...
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J. Phys. Chem. 1983, 87, 4054-4059

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laxation for freshly grown and HF-doped ices and for aged ice, implying in the latter case defects different from the classical rotational defects. (ii) Over a large frequency range, the movement of dislocations responsible for the high-temperature internal friction cannot be described by a simple manner implying

a u-l dependence. Supplementary assumptions have to be introduced; in particular a distribution of the values of the restoring force has to be taken into consideration in order to interpret the data in a self-consistent way. Registry No. Water, 7732-18-5.

Internal Friction in Ice Crystals Yoslo Hlkl‘ and Junlchl Tamura Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152, Japan (Received: August 23, 1982; I n Final Form: November 30, 1982)

The attenuation of sound at frequencies between 15 and 85 MHz has been measured by the pulse reflection method in single crystals of ice between -8.0 and -0.1 ‘C. Specimen crystals were grown by the Bridgman method from purified water, and crystals doped with NaCl, KCl, and NaOH were also used for the experiments. A resonance-type peak always appeared in the frequency dependence of the internal friction, and the peak frequency was shifted by the temperature change and by the Na-ion doping. It was also found that the attenuation value increased very rapidly with large fluctuation when the temperature approached the melting point. Experimental data were analyzed consistentlyon the basis of the vibrating-stringmodel of crystal dislocations, with the following conclusions: Dislocations are pinned by Na ions and the elastic interaction between the two kinds of defects is the origin of the pinning; the damping constant for the dislocation vibration is not large and is of the order of cgs; the density of dislocations in the crystals at lower temperatures is 106-107cm-2; the dislocation density fluctuatesand fiially increases to a large value near the melting temperature, due possibly to the spontaneous generation of dislocations.

Introduction A great deal of information can be obtained concerning the structure and character of crystal lattices when their responses to external mechanical forces are observed. When the forces are vibrational, various mechanical relaxations and resonances may appear as the vibrational frequency is varied. By analyzing such a spectrum one can study the dynamical behavior of lattice defects, lattice anharmonicity, excitations in the crystal, and so on.ln2 These studies are usually made by measuring the mechanical vibrational loss in the crystal, or internal friction, with changing external parameters such as the frequency and the amplitude of the vibration, and the temperature. With regard to the mechanical loss in ice crystals, internal friction at low frequencie~”~ and medium frequencies5p6has been studied by several authors, and origins of the losses were explained with reasonable success. However, results obtained at higher frequncies are still c o n t r o v e r ~ i a land ,~~ (1) A. S. Nowick and B. S. Berry, “Anelastic Relaxation in Crystalline Solids”, Academic Press, New York, 1972. (2) R. Truell, C. Elbaum, and B. B. Chick, ‘Ultrasonic Methods in Solid State Physics”, Academic Press, New York, 1969. (3) T. Nakamura and 0. Abe in “Proceedings of the 6th International Conference on Internal Friction and Ultrasonic Attenuation in Solids”, University of Tokyo Press, Tokyo, 1977, p 285. (4) R. Vassoille, C. Mai, and J. Perez, J. Glaciol., 21, 375 (1978). (5) P. Schiller, Z. Phys., 153, l(1958). (6) D. Kuroiwa, Contrib. Inst. Low Temp. Sei., Hokkaido Uniu., Ser. A , 18, 1 (1964). (7) J. P. VanDevender and K. Itagaki, Cold Regions Research and Engineering Laboratory Research Report No. 243, 1973. (8) J. Tatibouet, R. Vassoille, and J. Perez, J.Glacial., 15, 161 (1975). (9) D. M. Joncich, Ph.D. Thesis, Department of Physics, University of Illinois at Urbana-Champaign, 1976.

0022-3654/83/2087-4054$0 1.50/0

further experiments seem necessary. Measurements of ultrasonic attenuation in crystals are usually used in this frequency range to determine the internal friction, and our experiment also adopts this method. Preliminary results of our study were presented earlier,loJ1 and the present paper is a more extended one including our recent investigations.

Experimental Method Measurements of internal friction were made on ice single crystals prepared by the Bridgman method. The sample cell for growing an ice crystal and for measuring the ultrasonic attenuation is shown in Figure 1. The main body is a fused-quartz optical cell (A). The cell is filled with boiled commercially purified water and frozen in a refrigerator. The frozen water is remelted leaving a small seed crystal a t the bottom of B. The ice crystal is grown from the seed upward through the contracted part (C) at a growth rate of about 1.4 mm/h. The orientation of the ice crystal cannot be controlled in the present method of crystal growth. In our specimen crystals, the orientation angle 0, which is the angle between the crystallographic c axis and the direction of sound propagation, was found to be in the range between 10 and 30’. The preferred orientation of the crystal growth seems to be around 20° in the present case. A gold-plated X-cut 5-MHz quartz transducer, 1/4-in.in diameter (E), is bonded to the outer wall of the cell with phenyl salicylate. Pulsed ultrasound produced by the transducer is sent into the crystal, and (10) Y. Hiki and J. Tamura, J . Physiq., 42, C5-547 (1981). (11)J. Tamura and Y. Hiki, Jpn. J.Appl. Phys., 21, Supplement 21-3, 95 (1982).

0 1983 American Chemical Society

Internal Friction in Ice Crystals

The Journal of Physical Chemistry, Vol. 87, No. 21, 1983 4055

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the opposite wall of the cell acts as a reflector for the sound. Multiply reflected sound was used for measuring the sound attenuation. The measurements were made in the temperature range between -8.0 and -0.1 OC, and the temperature of the specimen was determined by a calibrated chromel-alumel thermocouple 0.3 mm in diameter attached to the inner wall of the cell just above position

E. In order to determine the purity of ice crystals used in our experiments, we performed chemical analyses on water sampled from the center of the crystals. The impurity ions contained were as follows: Na+; 0.40, K+; 0.05, C1-; 0.15, F-; 0.04, 0.01 in atomic ppm. No other impurities were detected. Atomic absorption analysis was used to determine the cation impurity contents, and analysis by the method of chromatography was applied to the anion impurities. Typical value of errors in the chemical analyses was, for example, fO.10 ppm for Na+ ions. Ultrasonic attenuation experiments by the pulse reflection method2 were made at sound frequencies between 15 and 85 MHz. The electronic apparatus used was an ultrasonic generator and receiver (Matec Model 6000 760) and an exponential generator (Matec Model 1204A). The usual measuring procedure was to fit the calibrated exponential curve to the envelope of decaying pulse echoes on an oscilloscope. Frequency dependence of attenuation in a specimen at a fixed temperature was determined by this method. Another apparatus used was an automatic attenuation recorder (Matec Model 2470A), by which the height of a pulse echo can be recorded on a chart. Variation of attenuation with temperature at a definite frequency was sometimes observed by this apparatus. The velocity of sound in the specimen was also measured from the flight time with a calibrated delay time marker. The velocity value can be used to determine the crystallographic orientation of the specimen.12 In ice crystals of good quality, the number of observed sound pulse echoes is 30 at a sound frequency of 15 MHz and 5 at 85 MHz. The apparent accuracy of the attenuation measurements by the exponential curve fitting method is 0.005-0.01 dB/Ms, and the errors in the sound velocity measurement are f l O O m/s. The most important source of errors in the attenuation measurement is due to the diffraction of sound waves in the crystaL2 The ap-

+

(12) F. Tsuruoka and Y.Hiki, Phys. Rev. B , 20, 2702 (1979).

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parent attenuation due to the diffraction effect is calculated by CYd = 1.7(u/a2f) [dB/cm], where u is the sound velocity in cm/s, f is the sound frequency in Hz, and a is the radius of ultrasonic transducer in cm. The diffraction loss is fairly large in the present experiment at low frequencies, and the correction for the loss is always made in measured values of the attenuation. There is little effect of nonparallelism between the transducer and the reflector owing to the good quality of the specimen cell, which has a wall parallelism of better than 10 pm/45 mm. The attenuation caused from this effect2 is 10-3-10-2 of the measured value of attenuation at 15-85 MHz. The attenuation arising from the acoustic loss in the cell wall is also very small. It is estimated to be less than of the measured attenuation value by using the existing data of sound attenuation in fused quartz. The obtained experimental attenuation values can be converted into the decrement or the internal friction A (= a v / f ,where CY is the attenuation in Np/cm). The internal friction is a more meaningful quantity, since it represents the energy loss in a cycle of vibration divided by twice of the vibrational energy. Experimental Results and Analysis Preliminary measurements of ultrasonic attenuation were made for an ice crystal, and the results were as shown in Figure 2. When the temperature of the specimen is raised and lowered, a hysteresis can be seen in the attenuation values. The hysteresis becomes less noticeable when the temperature cycle is repeated. This seems to be a kind of annealing effect, showing that lattice defects are probably responsible for the attenuation of sound. It is also interesting to see that the attenuation always increases very rapidly at temperatures near the melting point. The frequency dependence of the sound attenuation was measured at various temperatures for a large number of specimens. Illustrative data are shown in Figure 3. A broad decrement peak always appears and the height and location of the peak are different in different specimens. However, in all specimens, the peak height increases and the peak position moves to lower frequencies as the temperature is raised. These characteristics show that the

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The Journal of Physical Chemistry, Val. 87,No. 21, 1983

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internal friction of ice is structure sensitive and that the peak is not of a thermally activated type, since the peak shift with temperature should then be in the reverse direction. On the basis of the present knowledge, the internal friction spectrum in ice as a function of frequency is shown in Figure 4. The large relaxation peak5 found in the low-frequency range has been attributed to the energy loss arising from reorientation of H20 molecule^.'^ The curve in the figure was calculated by using the experimental values of the relaxation strength and the relaxation time5 for the case of a crystal with the orientation angle 8 = 90". The height of the peak is largest for this orientation, and the height is reduced to 10-1of the shown value for a crystal with 8 = 0". The thermoelastic damping peak, which is originated from macroscopic heat flow in the specimen, should appear a t higher frequenciesS2The expected relaxation is drawn by using values of physical constants of ice crystal for an orientation angle of 8 = 4 5 O . The height of the peak may not be sensitive to the crystallographic orientation. The peak observed in the present experiment is just between these two relaxation peaks, and should have a different origin, which is believed to involve dislocations. (13)R. Bass, 2. Phys., 153, 16 (1958).

Hiki and Tamura

A considerable number of dislocations are usually contained in crystals of common materials. The dislocations are usually pinned or anchored a t many pinning points, the origins of which are different in different crystals. Dislocation segment between the pinning points can vibrate under the influence of alternating stress produced by sound. Generally, the vibration of dislocations produces mechanical loss and a decrement peak of either a relaxation type or a resonance type. We anticipate for the moment that the decrement peak observed in the present experiment is due to the resonance type, and we will proceed to analyze our experimental data on this basis. For the overdamped resonance, where the damping force against the dislocation motion in the crystal in relatively large, the decrement A can be represented as14

where A, = 8nQGb2hL2/a3C,and 7 = nBL2/n2C. Here o is the angular frequency of the sound, A the dislocation density in the crystal, L the pinning length or the distance between the pinning points, B the damping constant which is proportional to the damping force against the dislocation vibration, fl the orientation factor taking into account the fact that a dislocation can vibrate in a definite direction on a definite slip plane, G the shear modulus of the crystal, and b the magnitude of Burgers vector of the dislocation. In addition, C [= 2Gb2/ap; p = 1 - v for an edge dislocation, where v is the Poisson ratio, and p = 1 for a screw dislocation] is the dislocation line tension, and n and n' are constants. As eq 1 shows, the decrement has a maximum value of Am = (1/2)& at the frequency f, = 1/2a7. The above formulas are available even when the pinning points are not uniformly distributed on the dislocation line.14 For example, when the distribution is of an exponential type, the constant values of n and n'are n = 4.4 and n' = 11.9, and L in the formulas represents the average pinning length. When the type of distribution is the same while the pinning length is changed, the shape of the resonance peak is not changed and only the height and position of the peak are altered. The above formulas were used to analyze the frequency-dependencedata as shown in Figure 3. The distribution of the pinning points is always assumed to be of an exponential type. This assumption is reasonable, because such a distribution is realized in the case of point-defect pinning of dis10cations.l~The values of two parameters, the peak height &,and the peak position f,, were adjusted by a least-squares fit of the data to the theoretical formula. The curves in the figure represent the fitted ones, and the result of the fit is extremely good. The height and the position of the decrement peak depend on various quantities contained in & and 7 in eq 1. One can obtain much knowledge about the dislocations in the crystal from such a kind of analysis of the frequency dependence of decrement. It is usual that in various sorts of crystals disloctions are pinned by impurity atoms or ions, since binding forces usually exist between the two kinds of lattice defects. According to the result of chemical analyses, the impurities which may possibly pin dislocations in our ice crystals are Na+ and C1- ions. In order to determine which ions are effective for the dislocation pinning, ultrasonic attenuation experiments were carried out on doped ice crystals. An adequate amount of high-purity reagent NaC1, KC1, or NaOH was added to the purified water, and specimen (14)A. V. Granato and K.Lucke, J.Appl. Phys., 27,583,789(1956).

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The Journal of Physical Chemistty, Voi. 87, No. 21, 1983 4057

Internal Friction in Ice Crystals I 'I'

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accuracy of the atomic absorption analysis. The concentration of Na+ ions is rather high even in the undoped crystals. This may be due to some contamination in the course of the growing crystals. Chemical analysis showed that the Na+ ion concentration in the purified water we used was 0.02 atomic ppm. The frequency dependence of the decrement was measured for four (undoped) ice crystals at various temperatures, and the peak frequency f, was determined as a function of temperature. The results were as shown in Figure 7. The equilibrium concentration of pinning points on dislocations is15

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crystals were grown from the water in the same manner as described before. The final doping of the ions in the ice crystals was 0.0484055 atomic ppm. Frequency dependence data for the decrement in undoped and doped crystals are shown in Figure 5. In comparison with the undoped specimen, the positions of the overdamped resonance peak are at the higher frequencies in the NaC1doped and NaOH-doped specimens, while no appreciable change of the peak position can be seen in the KC1-doped specimen. The results are more clearly shown in Figure 6, where the values of peak frequency f, determined from the fitting of the data to eq 1 are plotted. Measurements were performed on four undoped specimens and two specimens each with the three types of doping. Equation 1indicates that f, is larger for smaller pinning length L , which is equivalent to higher concentration of the pinning points for dislocations. From the result of Figure 6, we can definitely conclude that Na+ ions are effective for the dislocation pinning. From the amount of the shift of resonance peak for Na+-doped specimens, we can readily evaluate the concentration of Na+ ions in the bulk of undoped specimens co, which is found to be 0.33 f 0.03 atomic ppm. This value seems to be in good agreement with the result of the chemical analysis after considering the

(2)

where co is the concentration of the pinning agencies in the bulk crystal, EBis the binding energy between the dislocation and the pinning agency, k B is the Boltzmann constant, and Tis the absolute temperature. If we consider the (10x0) edge dislocations with the (113)(1210) Burgers vector, then the concentration can be written as c = d3b/L. Therefore, from eq 1 and 2, the peak frequency is represented as f, = (ac,2C/6n'b2B)exp(BE~/k,T)

(3)

Figure 7 shows the plots of logarithm off, against the inverse of temperature T'. Except for the data close to the melting temperature, the above two quantities seem to bear a linear relationship, in agreement with eq 3. In principle, one can separately determine the slope and the intercept of the In f, vs. T1line from the data shown in Figure 7. The intercept cannot, however, be determined very accurately because of the scatter of the data points. Therefore, a calculated value will be used for the intercept. All quantities in the preexponential factor in eq 3 are definite except the damping constant B , which will be considered below. A number of theories have been proposed for the origins of the dislocation damping, Le., the energy dissipation due to a frictional force opposing dislocation motion.I5 The most common mechanisms are the phonon-scattering effect16and the thermoelastic effect,17 (15) F. R. N.Naborro, "Theory of Crystal Dislocations", Clarendon Press, Oxford, 1967. (16) A. D. Brailsford, J. Appl. Phys., 43, 1380 (1972). (17) J. D. Eshelby, h o c . R . Soc. London, Ser. A , 197, 396 (1949).

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The Journal of Physical Chemistry, Vol. 87, No. 21, 1983

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Figure 8. Recorded chart of attenuation vs. temperature for a specimen in two different temperature ranges.

x 10-3:l.2x lo-' for Na, K, and C1 ions, even though their bulk concentrations are the same. Thus the predominant pinning by Na ions is most likely, and also the consideration of the elastic interaction of the dislocation and the ion seems to be very reasonable. To conclude this section, we describe the attenuation behavior in the vicinity of the melting point. Sound attenuation values at a definite frequency were recorded by increasing the specimen temperature slowly to the melting point and again decreasing the temperature just before the specimen melts. An example of the recorded chart is shown as Figure 8. Fluctuation in the attenuation can be seen to increase significantly at temperatures near the melting point. We choose the mean values of the fluctuating attenuation and calculated the dislocation density A by using eq 1 and 2 and by adopting the calculated B and experimentally determined co and EB values. In this experiment, we use the highest sound frequency (85 MHz) possible. As can be seen from eq 1, the decrement is not sensitive to the value of L when the sound frequency is sufficiently high, and a more reliable values of A can be obtained. The results for two specimens are as shown in Figure 9. Density of dislocations is almost constant at lower temperatures, and increases rapidly when the temperature approaches the melting point. Hysteresis near the melting point is also clearly apparent.

and these two damping mechanisms are considered to coexist in common crystals. We have calculated values of the damping constant by using formulas given in the articles cited above, with the result B = B(phonon scattering) B(thermoe1astic) = 1.60 X cgs + 0.97 X cgs = 2.57 X cgs a t -5 O C . The temperature change of B is negligible in the range of temperature in our experiments. With this B, together with constant values b = 4.52 X cm, G = 3.48 X 1O1O dyn/cm2, and v = 0.303, the preexponential factor in eq 3 was determined. Then the slope of the In f , vs. T1line was determined by the least-squares fit of the data, and the binding energy EB was obtained. The result was EB = (0.18 f 0.01) eV. The solid line in Figure 7 shows the result of the fit. The data points are within the region limited by the two dashed lines, and the probable value of B is in the range (1.8 - 3.7) X cgs for the above value of binding energy. When the elastic interaction between an edge dislocation and an impurity is considered, the binding energy between the two defects is15 4 l + v GberO3sin 0 (4) E B = s K r where (r, 0) are the polar coordinates of the impurity when the center of the dislocation core is chosen as the origin, t = (ro - r