Elastic-Stiffness Coefficients of Titanium Diboride

Using resonance ultrasound spectroscopy, we measured the monocrystal elastic-stiffness coefficients, the Voigt Cij, of TiB2. With hexagonal symmetry, TiB2 exhibits five independent Cij: C11, C33, C44, C12, C13. Using Voigt-Reuss-Hill averaging, we converted these monocrystal values to quasiisotropic (polycrystal) elastic stiffnesses. Briefly, we comment on effects of voids. From the Cij, we calculated the Debye characteristic temperature, the Grüneisen parameter, and various sound velocities. Our study resolves the enormous differences between two previous reports of TiB2’s Cij.


Introduction
Currently, we see a renascence of research on transition-metal diborides. For example, well-known are many current studies on ReB 2 and OsB 2 as superhard materials, rivaling even diamond.
Titanium diboride's physical and mechanical properties received review elsewhere [1,2]. This compound is known well for low mass density, high hardness, high melting point, low electrical resistivity, good thermal conductivity, and good chemical inertness. However, the problems in preparing full-dense, high-quality monocrystals preclude their extensive study. Akimitsu and colleagues [3] reported superconductivity at 40 K in a same-crystal-structure companion compound: The importance of elastic-stiffness coefficients for both science and technology also received review [4].
Titanium diboride's monocrystal elastic stiffnesses appeared in two reports. The values of Gilman and Roberts [5] depart strongly from the more recent report of Spoor and colleagues [6], mainly in the off-diagonal C 12 and C 13 . The first measurements were made by a pulse-echo method, the second by resonance ultrasonic spectroscopy, but by a very different mechanical setup than used in the present study.
Here, we report a third set of measurements, which agree closely with the Spoor et al. results. From our C i j measurements, we estimate the Debye characteristic temperature, the quintessential harmonic-lattice property. Also, we estimate the Grüneisen parameter, the quintessential anharmonic property. Figure 1 shows the TiB 2 crystal structure, obviously a layered structure suggesting strong elastic anisotropy, not observed as shown below, creating a conundrum.

Crystal
An oriented parallelepiped specimen 1.8 mm × 2.3 mm × 3.9 mm was prepared from a larger crystal grown by a floating zone method [7,8]. Fluorescent x-ray analysis revealed no significant impurities. The exact stoichiometry was TiB 1.97 . The crystal faces with respect to the above dimensions were [2 -1 -10], [ -1100], [0001]. These directions represent an a-axis, the c-axis, and the direction orthogonal to both. Laue x-ray diffraction confirmed these orientations within 1°.

Method
To measure the C ij , we used resonance ultrasound spectroscopy, summarized in Fig. 2 [9][10][11]. Briefly, one clamps lightly a regular-shape (cube, cylinder, cube, parallelepiped) specimen between two piezoelectric transducers. One transducer is swept through frequency and the second transducer detects the specimen's macroscopic vibration frequencies (Fig. 3). Frequencies of a specimen are determined by five factors: (1) shape, (2) size, (3) mass or mass density, (4) elastic-stiffness coefficients C ij , and (5) crystal-axis orientation relative to macroscopic shape. Thus, by Small spheres represent boron atoms, large spheres titanium atoms. The hexagonal boron net resembles strongly that of graphitic carbon; thus we expect strong interatomic bonding within the boron net. The titanium atoms nest in interstices provided by the boron net. The axial ratio equals 3.228/3.028 = 1.066, relatively high for the AlB 2 -compound group. Each Ti atom is surrounded by twelve equidistant boron atoms. Each boron atom has three boron atoms at a short distance, and six titanium atoms at a much longer distance.

Fig. 2.
Schematic measurement setup. Specimen (parallelepiped) is clamped loosely between two piezoelectric transducers. One transducer is swept through frequency. The second transducer detects macroscopic resonance frequencies, which depend on specimen shape, size, mass, and elastic-stiffness coefficients, the C ij . Courtesy of A. Migliori (Los Alamos National Laboratory).
measuring the resonance frequencies f n , one can determine by an inverse calculation the stiffnesses C ij . The problem is strongly overdetermined: about one hundred f n to determine five C ij . Not reported here, one can also determine the complete internal-friction tensor Q ij -1 = Δ f n /f n = C ij * / C ij , where Δ f n denotes resonancepeak width, C ij * the imaginary part, and C ij the real part of the total C~i j tensor. Well-described elsewhere [12], the inverse problem involves Lagrangean minimization, the Rayleigh-Ritz method, and a least-squares procedure for measured and deduced f n values.
We determined mass density by careful mass and size measurements: ρ = 4.502 ± 0.016 g/cm 3 . This compares with reported x-ray mass densities of 4.504 to 4.53. We ascribe the differences to heavier impurities on the Ti sites. From this mass density, we concluded our specimen contains no significant void content. From the sharp resonance peaks (Fig. 3), we concluded our specimen contains few cracks.

Errors
Errors arise from many sources: crystal orientation, crystal dimensions, nonparallelism, the inverse problem (measured-frequency to C i j k l conversion), specimen-transducer interactions. Several authors described these errors elsewhere, especially the first reference [13][14][15]. The effect of the slight (1 %) departure from stoichiometry is hard to estimate. Because of the strong covalent bonding within the boron planes, we conjecture a very small error arising from an occasional missing boron atom. Table 1  Highly overdetermined, the problem uses about one hundred resonance frequencies f n to determine the five independent C ij . Vertical bars at bottom show predicted resonance frequencies. The inset shows a resonance-peak profile, with a Lorentzian shape, the half-power width giving the internal friction Q i j -1 , the imaginary part of the total C ij .

Frequency (kHz)
C 66 = (C 11 -C 12 )/2. As usual (they fail to correspond directly to a phonon), the off-diagonal C ij (C 12 and C 13 ) show the largest uncertainties. Table 1 also gives the Spoor et al. results [6], which differ from ours by an average of 2.2 %. The average uncertainty in our six C ij is 0.7 %. The close agreement between our results and the Spoor et al. results suggests strongly that the earlier Gilman-Roberts results [5] are wrong, the largest discrepancies occurring in the off-diagonal C ij : C 12 and C 13 . Further support for the correctness of the Ledbetter-Tanaka / Spoor et al. results arises from ab initio calculations that yielded B = 251 GPa and ν = 0.12 [16], versus 417 and 0.32 for Gilman-Roberts.  (1) Here S ij denotes the elastic-compliance tensor, the tensor inverse of the C ij . As we expect from C 11 > C 33 , E 11 exceeds E 33 ; that is, TiB 2 is much stiffer within the basal plane than along the c-axis. Obviously, this relates to the crystal structure where the covalentbonded boron atoms lie in the plane perpendicular to x 3 . Table 1 shows also the three principal Poisson ratios ν ij computed by (2) Within the boron plane, the Poisson ratio ν 12 is extremely low, reflecting strong covalent bonding and the strong resistance of boron atoms to change their bond angles. The ν 13 Poisson ratio is only slightly below normal, indicating weaker bonds out of the boron-atom planes than those within the planes.
The shear elastic anisotropy of hexagonal crystals can be expressed in various ways. The simplest is C 66 / C 44 , 1.15 for TiB 2 , thus weak elastic anisotropy (the isotropic case corresponding to 1.00). Because the Young modulus depends so strongly on the shear modulus, a Young-modulus variation with direction also serves as an effective shear-anisotropy indicator. Spoor and colleagues showed a Young-modulus representation surface; it is nearly spherical [6]. From the alternating-layer boron-titanium crystal structure (Fig. 1), one expects higher elastic anisotropy than one finds.
The lower part of Table 1 gives the averagedover-direction quasiisotropic elastic constants obtained from the C ij by a Voigt-Reuss-Hill average [17]. These constants include longitudinal modulus C L , bulk modulus B, shear modulus G, Young modulus E, and Poisson ratio ν. These are the elastic constants appropriate to a full-density nontextured polycrystalline aggregate in which the grain boundaries cause no softening. Grain size produces no effect on elastic constants if it is small relative to specimen size. ( A reviewer pointed out that nanosize grains may soften the elastic constants.) The most unusual feature of our results is the high G / B ratio, thus low Poisson ratio. Our nonpublished results on a YB 66 monocrystal gave an averaged-over-direction (Kröner method) Poisson ratio of 0.13. For boron, a handbook value is ν = 0.089 [18]. Both the bulk and shear moduli of TiB 2 showing the strong interatomic bonds in TiB 2 , both for extension-compression and for shear-bending. Beside implying a low Poisson ratio, the very high G / B ratio holds implications for several other crystal-bonding properties: covalency, Cauchy-relation departure, many-body forces, and others. We plan to discuss all these elsewhere.
From the C ij and the mass density, we can compute the acoustic Debye characterisitic temperature Θ D . At zero temperature, the acoustic Θ D becomes identical with the calorimetric Θ D [19]. Θ D is proportional to the mean sound velocity: (6) where ν m denotes mean sound velocity and K is given by (7) Here h denotes Planck's constant, k Boltzmann's constant, and V a atomic volume. The velocity ν m comes from the integration over all directions: (8) Here ν 1 denotes the quasilongitudinal wave velocity, ν 2 and ν 3 the quasitransverse wave velocities, and d Ω an increment of solid angle. Equation (8) can not be integrated analytically, and numerous numerical and approximate methods have been used for its solution. Phase velocities ν a are roots of the Christoffel equations: (9) This expression follows from equations of motion for plane, monochromatic waves, where ρ denotes mass density, C ijkl fourth-rank elastic-stiffness tensor, x i components of the unit wave vector relative to the cubic axes, and δ il the Kronecker operator. Equation (9) usually yields three distinct real roots ρν 2 .
We computed an exact Θ D by using a distribution of 489 vectors proposed by Overton and Schuch [20], which we distributed over the usual 48 [100]-[110]-[111] stereographic unit triangles, thus a total of 23 472 directions. We obtained Θ D = 1217 K. This result differs enormously from the handbook calorimetric value, 1576 K. Often, calorimetric values contain large errors because of large extrapolations to zero temperature and/or large uncertainties in the lowest-temperature lattice specific heat.
From B and from handbook values of heat capacity C, volume thermal expansivity β , and volume V, we computed the effective Grüneisen parameter: (10) We obtained γ = 1.71, the handbook values for boron and titanium being 1.85 and 1.33. Alternative gammas can be computed when one knows the third-order elastic-stiffness coefficients, the C ijklmn [21]. The few gammas known for compounds similar to TiB 2 preclude any comparisons. Following Pearson's reasoning [22], the Ti-B bonds would lead to γ = 2.
Finally, because some authors suspect voids/cracks in TiB 2 , we want to describe briefly how the above fulldense quasiisotropic elastic constants would change with voids. Focusing on Al 2 O 3 , Ledbetter, Lei, and Datta [23] gave a theory for void effects on elastic constants. Principal results include the following: Voids soften the bulk modulus more than the shear modulus. In the dilute limit, for spherical voids, Volume 114, Number 6, November-December 2009 Journal of Research of the National Institute of Standards and Technology (sum over repeated indices) our results agree with the classical results of Mackenzie [24]: Here, c denotes void volume fraction and ν 0 the void-free-state Poisson ratio. However, we emphasize that the often-used rule-of-thumb that elastic stiffness varies as mass density is true only in the dilute limit. Void shape plays a key role, especially if the voids possess an oblate-spheroid (disc) shape. Dunn and Ledbetter [25] focused on the interesting, unexpected effects of voids and cracks on the Poisson ratio. Other authors addressed this problem using various approximations [26].

Conclusions
1. Our elastic-stiffness-coefficient measurements on titanium diboride support the report of Spoor and colleagues rather than the older (now handbook) values of Gilman and Roberts. The principal differences in the two previous C ij sets lie in C 12 and C 13 .