Low-temperature elastic coefficients of polycrystalline indium

Materials Science and Engineering A252 (1998) 139 – 143

Low-temperature elastic coefficients of polycrystalline indium Sudook Kim *, Hassel Ledbetter Materials Science and Engineering Laboratory, US Department of Commerce, National Institute of Standards and Technology, 325 Broadway, Boulder, CO 80303, USA Received 29 November 1997

Abstract Using an ultrasonic pulse-echo method, we measured the elastic coefficients of polycrystalline indium from 300 to 5 K. All elastic coefficients showed regular temperature behavior, as predicted by an Einstein-oscillator model. The shear and Young moduli showed the largest change, increasing : 55% during cooling. The Poisson ratio was unusually high at 0.45, just below the theoretical upper bound of 0.5. Using a Marx composite oscillator, we measured the internal friction at room temperature. We calculated the acoustic Debye temperature, 108.4 K, that agreed well with a monocrystal acoustic value, 111.3 K and the specific-heat value, 108.8 K. Also, we calculated the Gru¨neisen parameters, gL = 2.04, gH = 2.68, that agreed well with the specific-heat value, g = 2.48 and the shock-wave value, g = 2.24. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Debye characteristic temperature; Elastic constants; Gru¨neisen parameter; Indium; Internal friction; Low temperatures; Marx oscillator; Pulse-echo method

1. Introduction Indium shows good wetting characteristics and high malleability. Indium wets metals, glasses and ceramics and when evaporated onto glass or metal, it forms a mirror that resists atmospheric corrosion more than silver. Indium’s low-temperature ductility is well known [1]. Its low melting point (430 K) is the main reason for its softness. The ratio of 300 K and the melting temperature is 0.698, equivalent to studying copper at 947 K, thus in the usual high-temperature region. Indium shows a body-centered-tetragonal crystal structure. However, in the alternative face-centered-tetragonal ba˚. sis, the unit-cell dimensions are 4.947 and 4.598 A Therefore, with an aspect ratio of 1.08, indium is not far from face-centered cubic. The physical properties of indium, such as elastic stiffness and thermal expansivity, are moderately anisotropic. In the present study, we measured the elastic-coefficient behavior of polycrystalline indium from room temperature to liquid-helium temperature. At room temperature, we measured indium’s internal friction, the energy dissipation per stress cycle. For 0 K and

* Corresponding author. 0921-5093/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0921-5093(98)00490-0

from the temperature-dependent elastic coefficients, we calculated the Debye temperatures and Gru¨neisen parameters.

2. Measurements

2.1. Material We obtained 3-kg ingots of polycrystalline indium with a purity of 99.99% by mass from a commercial source. Principal impurities are Cu, Fe and Mn. We obtained specimens by casting the indium into aluminum molds in a nitrogen atmosphere (99.95% dry). A microstructure in Fig. 1 shows relatively large grains averaging 200 mm, with a broad grain-size distribution. Using Archimedes’s method, we found a mass density of 7.283 g cm − 3. Samples consisted of two cylinders: 1.27 cm diameter and 0.56 cm height; 0.47 cm diameter and 2.27 cm height.

2.2. Pulse-echo method We used a megahertz-frequency (2–9 MHz) pulseecho method, described previously [2], to determine the elastic stiffness C using the general relationship

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C =r6 2.

(1)

Here r and 6 denote mass density and sound velocity. A quartz transducer was bonded to the specimen with phenyl salicylate for room-temperature measurements and with stopcock grease for low-temperature measurements. We determined sound velocities using 6 =2L/t.

(2)

Here L denotes the specimen thickness and t the roundtrip transit time for an ultrasonic wave. Fig. 2 shows a typical pulse-echo pattern. The longitudinal-mode elastic modulus Cl and the shear-mode elastic modulus G were calculated using Eq. (1) and the remaining principal elastic constants by the following relationships: Young modulus E = (3Cl −4G)G/(Cl −G),

(3)

Bulk modulus B =Cl −4G/3,

(4)

Poisson ratio 6= 0.5(Cl −2G)/(Cl −G).

(5)

For low-temperature measurements, the specimen holder was inserted into a cryostat. Liquid-helium and a type-E thermocouple varied and monitored temperatures. We completely filled and continuously supplied a small amount of helium gas into the specimen chamber to prevent air condensation on the specimen. Based on an Einstein solid, Varshni [3] described the temperature-dependent elastic stiffness with the following relationship: C(T)/C(295)=C0 −s/[exp(uE/T) −1].

(6)

Here C(T) denotes a temperature-dependent elastic stiffness and C0 denotes C(0)/C(295), s denotes a parameter discussed in Eq. (10) and uE denotes the

Fig. 2. Oscilloscope composite display of echo patterns obtained using a 4 MHz x-cut quartz transducer. Lower portion of the display shows expanded-superimposed first and second echoes. Transit time was measured for the fourth half-cycle. Full screen is 2 ms for the bottom two traces.

Einstein temperature. We corrected C(T) for thermal expansion using handbook values [4].

2.3. Bar-resonance method We used a Marx-composite-oscillator resonance method, described previously [5], to determine the Young modulus and the associated internal friction. The system consisted of two quartz-rod oscillators and a long cylindrical specimen. From the extensional mode near 30 kHz, we calculated internal friction using Q − 1 = Dfs/fs.

(7a)

Here, f, denotes specimen resonance frequency and Df, the resonance-peak width at 1/ 2 of maximum amplitude. We calculated the Young modulus using E=4rf 2sL 2s.

(7b)

Here, r and Ls denote mass density and specimen length.

3. Results

Fig. 1. Microstructure of polycrystalline indium.

Table 1 gives the measured elastic constants at 295 K and the parameters in Eq. (6) obtained from a leastsquares fit. The results have been corrected for thermal expansion. Fig. 3 shows the C and G normalized to 295 K. The small waviness in the shear measurements represents a temperature-measurement artifact, not material behavior. Fig. 4 shows the other normalized elasticconstants B, E and n versus temperature. Fig. 5 shows the resonance results for both the Young modulus and

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Table 1 Elastic constants at 295 K and the parameters in Eq. (6)

C (295) C0 s uE(K)

Cl (GPa)

G (GPa)

B (GPa)

E (GPa)

n

48.19 1.167 0.08297 120.6

4.394 1.556 0.1806 82.87

42.33 1.110 0.07059 146.4

12.74 1.536 0.1814 86.02

0.4498 0.9574 −0.01518 90.83

the internal friction, Q − 1 =0.0323. The ratio of resonance (kHz) modulus to pulse-echo (MHz) modulus was 12.08/12.74.

Our results compare favorably with previous monocrystal measurements [6]. For room temperature, we calculate the elastic constants of a random polycrystalline aggregate (texture free) as Cl =48.7, E= 13.7, G=4.73, B= 42.4 GPa and n = 0.446 from the known monocrystal elastic constants — C11 = 45.4, C33 = 45.2, C44 = 6.51, C66 =12.1, C12 =40.1, C13 = 41.5 GPa—using a Voigt – Reuss – Hill averaging method. The ratio of our measured values to these calculated values are O.99 (Cl), 0.92 (E), 0.92 (G), 1.00 (B) and 1.01 (n). Thus, the worst-case disagreement is 8% which is attributed to a small texture in the measured polycrystalline aggregate. The monocrystal elas-

tic constants show that texture could produce enormous elastic-constant changes. The elastic-anisotropy ratio C66/(1/2(C11 − C12))=12.1/2.6 =4.7, which means that within a crystallite, the shear and Young moduli vary with direction by a factor of five. The monocrystal–polycrystal discrepancy increases during cooling to 5 K. Indium’s large Poisson ratio, 0.45, near the theoretical limit of 0.5, deserves comment. As described by Slater [7], as temperature increases, so should n because it approaches the liquid state (more vacancies, higher atomic vibrational amplitudes, etc.). However, this explanation is incomplete. At zero temperature, indium’s Poisson ratio remains unusually high at 0.43. A high value of n means that the material is softer against shear deformation than against dilatational deformation. Several other malleable metals—gold, gallium, thallium and lead—show n above 0.43. Thus, we see that a correlation between Poisson-ratio and softness deserves further focus.

Fig. 3. Temperature variation of shear modulus (G) and longitudinal modulus (Cl). Waviness in G caused by poor temperature control.

Fig. 4. Temperature variation of Young modulus (E), bulk modulus (B) and Poisson ratio (n).

4. Discussion

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Fig. 5. Lorentzian-shaped resonance peak at frequency fr and the width at 1/ 2 of maximum amplitude (half-power level), which gives the internal friction as Q − 1 = Dfr/fr.

Between 5 and 300 K, all elastic constants and Poisson ratio behave regularly with the following ratios: 1.17 (Cl), 1.56 (G), 1.11 (B), 1.54 (E) and 0.96 (n). The increase in G and E is huge, :55%. For iron and copper the increase is 5 and 9%, respectively. This large increase arises, in part, because we are measuring a large temperature interval relative to the melting temperature. This means a large thermal expansion and a large elastic-stiffness change with temperature [8]. At high temperatures, internal friction in a polycrystal is sensitive to relative grain motion that dissipates energy at the grain boundaries. Because of indium’s low melting point, we expect a large internal friction caused by displacement of adjacent grains. Dislocations distributed over various slip systems within the grains also contribute to internal friction. We observed that in indium alloying elements or impurities sharply reduce Q − 1. In an indium alloy, 80ln – 15Pb – 5Ag, Q − 1 is lower by a factor of 21 than the value for pure indium. In accordance with the Koehler – Granato – Lucke vibrating-string model [9], internal friction decreases as impurity atoms pin the dislocations. Further studies on this topic should include monocrystals, impurity-varying alloys and measurement of frequency and strainamplitude effects on Q − 1. The latter provides a signature of the dislocation contribution to Q − 1. The Einstein temperature uE in Eq. (6) relates to Debye temperature UD by 4uE/3. The UD associated with elastic moduli and Poisson ratio are 110 K (G), 114 K (E), 193 K (B) and 120 K (n). In addition, from the mean sound velocity, the Debye temperature can be expressed [10] as UD(K)=2933.22 6m/V 1/3.

(8)

˚ 3 and 6m, the Here V denotes the atomic volume in A mean sound velocity in cm/ms, given by 3/6 3m = 1/6 3l + 2/6 3t.

(9)

Here 61 = (Cl/r)0.5 and 6t = (G/r)0.5, where r denotes mass density. At 0 K, we calculate UD =108.4 K, agreeing well with the reported monocrystal value of UD = 111.3 K [6] and the specific-heat value of 108.8 K. The Debye temperature also measures softness, reflecting especially the shear-deformation modes [10]. Finally, we discuss the Gru¨neisen parameter g, which we can obtain from the elastic-constant temperature dependence. Ledbetter [8] recently showed for the bulk modulus that s in Eq. (6) is given by s= 3kuEg(g+ 1).

(10)

Here k denotes Boltzmann’s constant. One can imagine similar relationships for the other elastic constants such as Cl and G, the two that most interest us because a solid transmits two kinds of waves: longitudinal and transverse waves. The Gru¨neisen parameter of ith mode was given by Slater [11]: gi = − d ln UEi /d ln V.

(11)

As shown by Sheard [12], the weighted-mean Gru¨neisen parameter of the thermodynamic low- and high-temperature values are 3 3 −3 gL = [gl6 − + 2(gt6 − l t )]/(36 m ),

(12)

and gH = (gl + 2gt)/3.

(13)

S. Kim, H. Ledbetter / Materials Science and Engineering A252 (1998) 139–143

Using the information from Table 1, these relations yield gL = 2.04 and gH =2.68. These values compare favorably with the specific-heat value g=2.48 and the shock-wave value g = 2.24 [13]. In our opinion, this approach to estimating g from the temperature dependence of the elastic-constant is new and useful. It is new because Eq. (10) appeared only recently. It deserves application to other materials.

143

ture, agree well with reported specific-heat and shockwave values. This suggests a new way to estimate gamma: from the temperature dependence of the second-order elastic coefficients.

Acknowledgements C. McCowan at NIST prepared the indium-sample microstructure.

5. Conclusions From this conclusions:

study,

we

reached

five

principal

1. Although the ratio of specimen- to grain-size exceeded 50, we observed some texture effects in our specimens. 2. Between 300 and 5 K, all the usual elastic constants show a regular temperature dependence, well described by the simple Einstein oscillator model. 3. Indium’s unusually high Poisson ratio, 0.45 at ambient temperature, deserves further attention, especially concerning mechanical deformation properties. 4. The acoustic Debye temperature derived from our elastic- constant measurements agrees well with a previous monocrystal- elastic-constant value and in particular with the specific-heat value. 5. The acoustic Gru¨neisen parameters, derived from our measurements of elastic-constants versus tempera-

.

References [1] R. Reed, C. McCowan, R. Walsh, L. Delgado, J. McColskey, Mater. Sci. Engr. A102 (1988) 227 – 236. [2] H. Ledbetter, N. Frederick, M. Austin, J. Appl. Phys. 51 (1980) 305 – 309. [3] Y. Varshni, Phys. Rev. B2 (1970) 3952 – 3958. [4] R. Corruccini, J. Gniewak, Thermal Expansion of Technical Solids at Low Temperatures, National Bureau of Standards Monograph 29 (1961). [5] H. Ledbetter, Cryogenics 20 (1980) 637 – 640. [6] B. Chandrasekhar, J. Rayne, Phys. Rev. 124 (1961) 1011–1014. [7] J. Slater, Introduction to Chemical Physics, McGraw-Hill, New York, 1939, p. 240. [8] H. Ledbetter, Phys. Stat. Solidi B181 (1994) 81 – 85. [9] A. Granato, K. Lucke, Phys. Acoust. IVA (1966) 225. [10] H. Ledbetter, Z. Metalik. 82 (1991) 820 – 822. [11] J. Slater, Introduction to Chemical Physics, McGraw-Hill, New York, 1939, p. 219. [12] F. Sheard, Phil. Mag. 3 (1958) 1381. [13] K. Gschneidner, Solid State Phys. 16 (1975) 275.