Low-temperature acoustic properties and quasiharmonic analysis for Cu-based bulk metallic glasses

Low-temperature elastic constant $C_{ij}\left(T\right)$ measurements for ${\mathrm{Cu}}_{60}{\mathrm{Zr}}_{30}{\mathrm{Ti}}_{10}$ and ${\mathrm{Cu}}_{60}{\mathrm{Hf}}_{25}{\mathrm{Ti}}_{15}$ bulk metallic glasses (BMGs) have been carried out to clarify their acoustic properties including a nonlinear interaction term. The BMGs show remarkably low shear to bulk modulus ratio $C_{44}∕B$ or equivalently high Poisson’s ratio $\nu$ compared with the corresponding crystalline states. The $C_{ij}\left(T\right)$ behaviors are well described by Varshni’s formula [Phys. Rev. B 2, 3952 (1970)] and estimated Einstein temperatures suggested significant softening in the transverse-mode acoustic phonon. Mean Grüneisen parameters $\gamma$ derived from $dB∕dT$ slopes become almost identical with those of crystalline Cu. Our theoretical analysis based on inhomogeneous microstructure model successively explains the elastic constants and characteristic features of the BMGs within a framework of quasiharmonic thermodynamics, suggesting that microstructural inhomogeneity plays a dominant role in the BMGs.

Figure 1

$A_g$-group free vibration resonance spectra of CuZrTi BMG obtained at (a) $298\mathrm{K}$ and (b) $5\mathrm{K}$. Indices in the figure represent the free resonance vibration modes identified by measuring displacement distributions using a laser Doppler interferometer.

Figure 2

Temperature dependence of elastic constants of (a) CuZrTi and (b) CuHfTi BMGs obtained by EMAR. Open circles represent measurement points normalized by those of extrapolated $0\mathrm{K}$ values. The solid lines represent least-squares fit by Eq. (1).

Figure 3

(a) Temperature dependence of $d^2{C}_{ij}∕d{T}^2$ for CuHfTi BMG calculated from Varshni’s equation. Note that the data are normalized by $-s$. (b) A relationship between peak temperatures $T_P$ and Einstein temperatures ${\Theta }_E$ for the BMGs and crystalline Cu.

Figure 4

(a) A relationship between the volume fraction of WBR and normalized molar volume of the region $V_W∕V_0$. The limit $V_W∕V_0\to 1.0278$ represents the uniform volume dilatation of the model. (b)–(d) plot macroscopic elastic constants of the inhomogeneous microstructure model for three cases of shear moduli as a function of $V_W∕V_0$. Elastic constants of BMGs obtained at $298\mathrm{K}$ show reasonable agreement with the case 1 results at $V_W∕V_0\approx 1.1$.