Computations of Ultrasonic Parameters in Zr_{𝟏𝟎𝟎−𝑋}Sn_𝑋 Alloys

Abstract

The ultrasonic properties like ultrasonic attenuation, sound velocity in the hexagonal Zr_100−𝑋Sn_𝑋 alloys have been studied along unique axis at room temperature. The second- and third-order elastic constants (SOEC & TOEC) have been calculated for these alloys using Lennard-Jones potential. The velocities 𝑉_𝐿 and 𝑉_𝑆1 have minima and maxima, respectively, at 45° with unique axis of the crystal, while 𝑉_𝑆2 increases with the angle from unique axis. The inconsistent behaviour of angle-dependent velocities is associated to the action of second-order elastic constants. Debye average sound velocities of these alloys are increasing with the angle and has maximum at 55° with unique axis at room temperature. Hence, when a sound wave travels at 55° with unique axis of these alloys, then the average sound velocity is found to be maximum. The mechanical and ultrasonic properties of these alloys will be better than pure Zr and Sn due to their high SOEC and ultrasonic velocity and low ultrasonic attenuation. The comparison of calculated ultrasonic parameters with available theoretical/experimental physical parameters gives information about classification of these alloys.

1. Introduction

Group IV transition metal zirconium (Zr) and its alloys are very important materials both from scientific and technological points of view. Scientifically, the electronic transfer between the broad sp band and the narrow d band is the driving force behind many structural and electronic transitions in these materials [1–3]. Technologically, these alloys have applications in the aerospace industry due to their light weight, static strength, and stiffness; they do not degrade rapidly as the temperature increases, and they also show oxidation resistance [4]. The mechanical properties of these alloys can be greatly improved by controlling the crystallographic phases present.

Static high-pressure experimental works indicate that, at room temperature and ambient pressure, Zr is a hexagonal close-packed (hcp) structure (𝛼 phase). At high temperature and zero pressure, it transforms martensitically into the body-centered cubic (bcc) structure (𝛽 phase) before reaching the melting temperature [5] while at room temperature and under pressure, the hcp phase transforms into another hexagonal structure.

The first form of tin is called grey tin (𝛼-Sn) and is stable at temperature below 286.2 K. This form crystallizes in the diamond structure [6], in common with the lighter elements in column IV A of the periodic table. At temperature above 286.2 K, grey tin slowly turns into tin’s second form, white tin. This second form (𝛽-Sn) is a tetragonal distortion of diamond with two atoms per unit cell [7]. Tin is used to form many useful alloys.

Up to now, only a few theoretical methods have been applied successfully to calculate the elastic constants of 𝛼-Zr, such as the full potential linear muffin-tin orbital (FP-LMTO) method [8], the tight-binding (TB) approach [9], the embedded-atom (EAM) method [10], and the modified embedded-atom (MEAM) method [11, 12]. However, elastic constants of ZrSn alloys have not yet been reported.

Ultrasonic properties offer the possibility to detect and characterize microstructural properties, as well as flaws in materials, controlling materials behaviour based on physical mechanism to predict future performance of the materials. Various investigators have shown considerable interest on ultrasonic properties of metals and alloys. Wave propagation velocity is key parameter in ultrasonic characterization and can provide information about crystallographic texture. The ultrasonic velocity is directly related to the elastic constants by the relationship 𝑉=𝐶/𝜌, where 𝐶 is the relevant elastic constants, and 𝜌 is the density of that particular material. Also ultrasonic attenuation is very important physical parameter to characterize the material, which is well related to several physical quantities like thermal conductivity, specific heat, thermal energy density and higher-order elastic constants [13, 14]. The elastic constants provide valuable information about the bonding characteristic between adjacent atomic planes and the anisotropic character of the bonding and structural stability [15, 16].

Therefore, in this work I predict the ultrasonic properties of hexagonal-structured Zr_100−𝑋Sn_𝑋 alloys (Zr₈₂Sn₁₈: alloy 1; Zr₈₀Sn₂₀: alloy 2; Zr₇₈Sn₂₂: alloy 3; Zr₇₆Sn₂₄: alloy 4) at room temperature. The ultrasonic attenuation coefficient, acoustic coupling constants, higher-order elastic constants, thermal relaxation time, and ultrasonic wave velocities for these alloys for each direction of propagation of wave are calculated at 300 K. The calculated ultrasonic parameters are discussed with related thermophysical properties for the characterization of the chosen alloys. The obtained results are analyzed in comparison to other hexagonal-structured materials and alloys.

2. Theory

In the present investigation, the theory is divided into three parts.

2.1. Second- and Third-Order Elastic Constants

The elastic energy density (𝑈) is function of the strain components.𝑈=𝐹𝑒_xx;𝑒_yy;𝑒_zz;𝑒_yz;𝑒_zx;𝑒_xy=𝐹𝑒₁;𝑒₂;𝑒₃;𝑒₄;𝑒₅;𝑒₆, (1) where 𝑒_𝑖𝑗(𝑖or𝑗=𝑥,𝑦,𝑧) is component of strain tensor.

The second- (𝐶_𝐼𝐽) and third- (𝐶_𝐼𝐽𝐾) order elastic constants of material are defined by following expressions. 𝐶_𝐼𝐽=𝜕²𝑈𝜕𝑒_𝐼𝜕𝑒_𝐽,𝐼or𝐶𝐽=1,…,6,_𝐼𝐽𝐾=𝜕³𝑈𝜕𝑒_𝐼𝜕𝑒_𝐽𝜕𝑒_𝐾,𝐼or𝐽or𝐾=1,…,6. (2)

The elastic energy density is well related to interaction potential Φ(𝑟) between atoms. Let the interaction potential be the Lennard-Jones Potential or many-body interaction potential, which is formulated as:Φ(𝑟)𝑎=−⁰𝑟^𝑚+𝑏₀𝑟^𝑛, (3) where 𝑎₀, 𝑏₀ are constants, and 𝑚, 𝑛 are integers. The definition of higher order elastic constants (2) with this potential (3) under equilibrium and symmetric condition six second- and ten third-order elastic constants (SOEC and TOEC) for the hexagonal closed packed structured materials [13, 17] 𝐶₁₁=24.1𝑝⁴𝐶^′,𝐶₁₂=5.918𝑝⁴𝐶^′,𝐶₁₃=1.925𝑝⁶𝐶^′,𝐶₃₃=3.464𝑝⁸𝐶^′,𝐶₄₄=2.309𝑝⁴𝐶^′,𝐶₆₆=9.851𝑝⁴𝐶^′,𝐶₁₁₁=126.9𝑝²𝐵+8.853𝑝⁴𝐶^′,𝐶₁₁₂=19.168𝑝²𝐵−1.61𝑝⁴𝐶^′,𝐶₁₁₃=1.924𝑝⁴𝐵+1.155𝑝⁶𝐶^′,𝐶₁₂₃=1.617𝑝⁴𝐵−1.155𝑝⁶𝐶^′,𝐶₁₃₃=3.695𝑝⁶𝐵,𝐶₁₅₅=1.539𝑝⁴𝐶𝐵,₁₄₄=2.309𝑝⁴𝐵,𝐶₃₄₄=3.464𝑝⁶𝐶𝐵,₂₂₂=101.039𝑝²𝐵+9.007𝑝⁴𝐶^′,𝐶₃₃₃=5.196𝑝⁸𝐵, (4) where 𝑝=𝑐/𝑎; axial ratio; 𝐶′=𝜒𝑎/𝑝⁵; 𝐵=𝜓𝑎³/𝑝³. The rest second-order elastic constants have zero value because under 180° rotation, they have equal and opposite value for the same stress. The harmonic and anharmonic parameters (𝜒 and 𝜓) can be calculated using one experimental SOEC [13, 17]. In the present study, I have expanded the theory for theoretical evaluation of parameters 𝜒 and 𝜓. The potential energy can be expanded in the powers of changes in the squares of distances. The expansion up to cubic term can be written as:Φ=Φ₀+𝜒²_𝑖=1Δ𝑟²_𝑖²+𝜓²_𝑖=1Δ𝑟²_𝑖³. (5)

According to (5), 𝜒 and 𝜓 can be written as:1𝜒=𝑑2!²Φ(𝑟)𝑑𝑟²²,1𝜓=𝑑3!³Φ(𝑟)𝑑𝑟²³. (6) In solving (6) for hexagonal closed packed structured materials:1𝜒=8𝑛𝑏₀(𝑛−𝑚)𝑎^𝑛+4,𝜓=−𝜒6𝑎²(.𝑚+𝑛+6) (7)

The parameters 𝜒 and 𝜓 can be calculated using (7) with appropriate values of 𝑚, 𝑛, and 𝑏₀ so that the calculated values of elastic constants justify the experimental data.

2.2. Ultrasonic Velocity

The anisotropic behaviour of the material can be understood with the knowledge of ultrasonic velocity because the velocity is related to the second order elastic constants [17]. On the basis of mode of atomic vibration, there are three types of velocities (longitudinal, quasishear, and shear) in acoustical region [18]. These velocities vary with the direction of propagation of wave from the unique axis of hexagonal-structured crystal [19, 20]. The ultrasonic velocities as a function of angle between direction of propagation and unique axis for hexagonal-structured materials are [21] 𝑉²_𝐿=𝐶₃₃cos²𝜃+𝐶₁₁sin²𝜃+𝐶₄₄+𝐶₁₁sin²𝜃−𝐶₃₃cos²𝜃+𝐶₄₄cos²𝜃−sin²𝜃²+4cos²𝜃sin²𝜃C₁₃+𝐶₄₄^21/2𝑉/2𝜌,_2𝑆1=𝐶₃₃cos²𝜃+𝐶₁₁sin²𝜃+𝐶₄₄−𝐶₁₁sin²𝜃−𝐶₃₃cos²𝜃+𝐶₄₄cos²𝜃−sin²𝜃²+4cos²𝜃sin²𝜃C₁₃+𝐶₄₄^21/2𝑉/2𝜌_2𝑆2=𝐶₄₄cos²𝜃+𝐶₆₆sin²𝜃𝜌, (8) where 𝑉_𝐿, 𝑉_𝑆1 and 𝑉_𝑆2 are longitudinal, quasi shear and pure shear wave ultrasonic velocities. Variables 𝜌 and 𝜃 represent the density of the material and angle with the unique axis of the crystal, respectively. The Debye temperature (𝑇_𝐷) is an important physical parameter for the characterization of materials, which is well related to the Debye average velocity (𝑉_𝐷): 𝑇_𝐷=ℏ𝑉_𝐷6𝜋²𝑛_𝑎^1/3𝑘_𝐵, (9) here 𝑉_𝐷=131𝑉³₁+1𝑉³₂+1𝑉³₃^−1/3, (10) where ℏ is quantum of action and is equal to Planck’s constant divided by 2𝜋;𝑘_𝐵 is Boltzmann Constant; 𝑛_𝑎 is atom concentration.

The above formulae have been used for the evaluation of ultrasonic velocity and related parameters for the selected materials.

2.3. Ultrasonic Attenuation and Allied Parameters

The predominant causes for the ultrasonic attenuation in a solid at room temperature are phonon-phonon interaction (Akhieser loss) and thermoelastic relaxation mechanisms. The ultrasonic attenuation coefficient (𝒜_Akh) due to phonon-phonon interaction and thermoelastic relaxation mechanisms is given by the following expression [17, 22]:𝒜𝑓²_Akh=4𝜋²3𝐸₀𝛾^𝑗_𝑖²−𝛾^𝑗_𝑖²𝐶_𝑉𝑇𝜏2𝜌𝑉³, (11)𝒜𝑓_2Th=4𝜋²𝛾^𝑗_𝑖²𝑘𝑇2𝜌𝑉⁵_𝐿, (12) where, 𝑓: frequency of the ultrasonic wave; 𝑉: ultrasonic velocity for longitudinal and shear wave; 𝑉_𝐿: longitudinal ultrasonic velocity; 𝐸₀: thermal energy density; 𝛾^𝑗_𝑖: Grüneisen number (𝑖,𝑗 are the mode and direction of propagation).

The Grüneisen number for hexagonal-structured crystal along ⟨001⟩ orientation or 𝜃=0^∘ is direct consequence of second- and third-order elastic constants. 𝐷=3(3𝐸₀⟨𝛾^𝑗_𝑖⟩²−⟨𝛾^𝑗_𝑖⟩²𝐶_𝑉𝑇)/𝐸₀ is known as acoustic coupling constant, which is the measure of acoustic energy converted to thermal energy. When the ultrasonic wave propagates through crystalline material, the equilibrium of phonon distribution is disturbed. The time for reestablishment of equilibrium of the thermal phonon distribution is called thermal relaxation time (𝜏) and is given by following expression:𝜏=𝜏_𝑆=𝜏_𝐿2=3𝑘𝐶_𝑉𝑉²_𝐷. (13)

Here 𝜏_𝐿 and 𝜏_𝑆 are the thermal relaxation time for longitudinal and shear wave. 𝑘 and 𝐶_𝑉 are the thermal conductivity and specific heat per unit volume of the material, respectively.

3. Results and Discussion

3.1. Higher-Order Elastic Constants

The unit cell parameters “a” (basal plane parameter) and “p” (axial ratio) for Zr_100−𝑋Sn_𝑋 alloys are 3.17 Å, 3.23 Å, 3.21 Å, 3.21 Å, and 1.622, 1.599, 1.603, 1.600, respectively, [23]. The values of 𝑚 and 𝑛 for chosen materials are 6 and 7. The value of 𝑏₀ is 1.79×10⁻⁶⁴ erg cm⁷ for all alloys. The SOEC and TOEC have been calculated for these alloys using (4) and are presented in Table 1.

Table 1 

SOEC, TOEC, and Bulk Modulus (𝐵) in the unit of 10¹⁰ Nm⁻² of ZrSn alloys at room temperature.

Published theoretical/experimental data on the elastic constants for the chosen alloys are not available for comparison. Comparison for the present values to theoretical/experimental studies of pure metals, as shown in Table 1, demonstrates that results obtained from the present investigation are higher than those of the pure metals as reported by other investigators [24, 25]. The obtained values of the SOEC and TOEC are of the same order as previous experimental and theoretical studies of other metallic alloys and metals [19, 21, 26–28]. Therefore, it is concluded that the theoretical approach to evaluate the SOEC and TOEC is valid for the chosen alloys. The bulk modulus (𝐵) for these alloys can be calculated with the formula 𝐵=2(𝐶₁₁+𝐶₁₂+2𝐶₁₃+𝐶₃₃/2)/9. The evaluated 𝐵 for these alloys is presented in Table 1.

3.2. Ultrasonic Velocity and Allied Parameters

The density (𝜌) and thermal conductivity (𝑘) at room temperature have been taken from the literature [23]. The value of 𝐶_𝑉 and 𝐸₀ are evaluated using tables of physical constants and Debye temperature. The quantities 𝜌,𝐶_𝑉,𝐸₀,𝑘, and calculated acoustic coupling constants (𝐷_𝐿 & 𝐷_𝑆) are presented in Table 2.

Table 2 

Density (𝜌: in 10³ kg m⁻³), specific heat per unit volume (𝐶_𝑉: in 10⁶ Jm⁻³K⁻¹), thermal energy density (𝐸₀: in 10⁸Jm⁻³), thermal conductivity (𝑘: in Wm⁻¹K⁻¹), and acoustic coupling constant (𝐷_𝐿, 𝐷_𝑆) of ZrSn alloys.

The calculated orientation-dependent ultrasonic wave velocities and Debye average velocities at 300 K are shown in Figures 1, 2, 3 and 4. Figures 1–3 show that the velocities 𝑉_𝐿 and 𝑉_𝑆1 have minima and maxima, respectively, at 45° with the unique axis of the crystal while 𝑉_𝑆2 increases with the angle from the unique axis. The combined effect of SOEC and density is reason for abnormal behaviour of angle-dependent velocities.

Figure 1 

𝑉𝐿 versus angle with unique axis of crystal.

Figure 2 

𝑉𝑆1 versus angle with unique axis of crystal.

Figure 3 

𝑉𝑆2 versus angle with unique axis of crystal.

Figure 4 

𝑉𝐷 versus angle with unique axis of crystal.

The nature of the angle-dependent velocity curves in the present work is found similar as that for heavy rare earth metals, Laves-phase compounds, and other hexagonal wurtzite structured materials (GaN, AlN, InN) [13, 17, 19, 21, 22, 28]. Thus, the computed velocities for these alloys are justified.

Debye average velocities (𝑉_𝐷) of these alloys are increasing with the angle and have maxima at 55° at 300 K (Figure 4). Since 𝑉_𝐷 is calculated using 𝑉_𝐿, 𝑉_𝑆1, and 𝑉_𝑆2 [13, 22]; therefore, the angle variation of 𝑉_𝐷 is influenced by the constituent ultrasonic velocities. The maximum 𝑉_𝐷 at 55° is due to a significant increase in longitudinal and pure shear (𝑉_𝑆2) wave velocities and a decrease in quasishear (𝑉_𝑆1) wave velocity. Thus, it can be concluded that when a sound wave travels at 55° with the unique axis of these alloys then the average sound wave velocity is maximum.

Thus, the present average sound velocity directly correlates with the Debye temperature, specific heat, and thermal energy density of theses materials.

The calculated thermal relaxation time is visualised in Figure 5. The angle-dependent thermal relaxation time curves follow the reciprocal nature of 𝑉_𝐷 as 𝜏∝3K/𝐶_𝑉𝑉²_𝐷. This implies that 𝜏 for these are mainly affected by the thermal conductivity. The 𝜏 for hexagonal-structured materials is the order of picoseconds [19]. Hence, the calculated 𝜏 justifies the hcp structure of chosen alloys at room temperature. The minimum 𝜏 for wave propagation along 𝜃=55^∘ implies that the reestablishment time for the equilibrium distribution of thermal phonons will be minimum for propagation of wave along this direction. The value of thermal conductivity of alloy: 3 is greater than alloy: 4. So, in Figures 1–4, the ultrasonic velocities of alloy 3 and alloy 4 seem quasi-identical, but in Figure 5, their relaxation times differ.

Figure 5 

Relaxation time versus angle with unique axis of crystal.

3.3. Ultrasonic Attenuation

In the evaluation of ultrasonic attenuation, it is supposed that wave is propagating along the unique axis (⟨001⟩ direction) of these alloys. The attenuation coefficient over frequency square (𝒜/𝑓²)_Akh for longitudinal (𝒜/𝑓²)_𝐿 and shear wave (𝒜/𝑓²)_S are calculated using (11) under the condition 𝜔𝜏≪1 at room temperature. The thermoelastic loss over frequency square (𝒜/𝑓²)_th is calculated with (12). The values of (𝒜/𝑓²)_𝐿, (𝐴/𝑓²)_𝑆, (𝒜/𝑓²)_th, and total attenuation (𝒜/𝑓²)_Total, are presented in Table 3.

Table 3 

Ultrasonic attenuation coefficient (in 10⁻¹⁷ Nps²m⁻¹) of ZrSn alloys.

Table 3 indicates that the thermoelastic loss is very small in comparison to Akhieser loss, and ultrasonic attenuation for longitudinal wave (𝒜/𝑓²)_𝐿 is greater than that of shear wave (𝒜/𝑓²)_𝑆. This reveals that ultrasonic attenuation due to phonon-phonon interaction along longitudinal wave is governing factor for total attenuation ((𝒜/𝑓²)_Total=(𝒜/𝑓²)_Th+(𝒜/𝑓²)_𝐿+(𝒜/𝑓²)_𝑆). The total attenuation is mainly affected by thermal energy density and thermal conductivity. Thus, it may predict that at 300 K the alloy 1 behaves as its purest form and is more ductile as evinced by minimum attenuation while alloy 2 is least ductile. Therefore impurity will be least in the alloy 1 at room temperature.

Since 𝒜∝𝑉⁻³ and velocity is the largest for alloy 1 among alloy 2; thus, the attenuation 𝒜 should be smallest and material should be most ductile. The minimum ultrasonic attenuation for alloy 1 justifies its quite stable hcp structure state. The total attenuation of these alloys are much larger than third-group nitrides (AlN: 4.441×10⁻¹⁷ Nps²m⁻¹, GaN: 14.930×10⁻¹⁷ Nps²m⁻¹, and InN: 20.539×10⁻¹⁷ Nps²m⁻¹) due to their large thermal conductivity and acoustic coupling constants [19, 29]. This implies that the interaction between acoustical phonon and quanta of lattice vibration for these alloys is large in comparison to third-group nitrides.

4. Conclusions

Based on the above discussion it is worthwhile to state the following:(i)The present method to evaluate second- and third-order elastic constants involving many body interaction potential for hexagonal wurtzite crystal structured materials (alloys) is correct.(ii)All elastic constants and density are mainly the affecting factor for anomalous behaviour of ultrasonic velocity in these alloys.(iii)When a sound wave travels at 55° with the unique axis of these crystals, then the average sound wave velocity is maximum. Since the Debye average velocity is calculated using the constituent velocities 𝑉_𝐿, 𝑉_𝑆1, and 𝑉_𝑆2, hence, a good resemblance in 𝑉_𝐷 implies that our calculated velocities are correct. (iv)The ⟨001⟩ direction is the direction of symmetry for these alloys as they have the same quasishear and pure shear wave velocities.(v)The order of thermal relaxation time for these alloys is found in picoseconds, which justifies their hcp structure at 300 K. The reestablishment time for the equilibrium distribution of thermal phonons will be minimum for the wave propagation along 𝜃=55^∘ due to being the smallest value of 𝜏 along this direction.(vi)The acoustic coupling constant of these alloys for longitudinal wave are found five times larger than third-group nitrides. Hence, the conversion of acoustic energy into thermal energy will be large for these alloys. This shows general suitability of chosen alloys.(vii)The ultrasonic attenuation due to phonon-phonon interaction mechanism is predominant over total attenuation as a governing factor of thermal conductivity.(viii)Zr₈₂Sn₁₈ alloy is more suitable for industrial and other uses, as it has the highest elastic constants as well as wave velocity in comparison to other chosen alloys.(ix)The mechanical and ultrasonic properties of these alloys will be better than pure Zr and Sn due to their high SOEC and ultrasonic velocity and low ultrasonic attenuation.

Thus, obtained results in the present work can be used for further investigations, general and industrial applications. Our theoretical approach is valid for ultrasonic characterization of these materials at room temperature. The ultrasonic behavior in these alloys as discussed above shows important microstructural characteristic feature, which is well connected to thermoelectric properties of the materials. These results, together with other well-known physical properties, may expand future prospects for the application and study of these materials.

Acknowledgments

The author wishes to express sincere thanks to Dr. Ashok K. Chauhan, President, AMITY School of Engineering and Technology (ASET), New Delhi, Prof. B. P. Singh, Senior Director, ASET New Delhi, and Prof. S. K. Kor and Prof. Raja Ram Yadav, Department of Physics, Allahabad University, for encouraging and providing necessary facilities in carrying out the work.