Nanoscale elastic inhomogeneity of a Pd-based metallic glass: Sound velocity from ultrasonic and inelastic x-ray scattering experiments

In this paper, we show experimentally that the sound velocity of nanometer wavelength exceeds that of millimeter wavelength for a Pd-based metallic glass completely frozen far below the glass transition temperature. This indicates that nanoscale elastically harder regions exist in the glass matrix and, hence, elastically softer regions are also present so as to realize the macroscopic elasticity.

Figure 1

(Color online) Schematic illustrations showing (a) inhomogeneous structural model of fragile metallic glasses (Ref. 12), (b) phonon dispersion relation in a very small $Q$ region predicted from the model, (c) inverse temperature dependence of the characteristic frequencies of $\alpha$ and $\beta$ relaxations (referring to Ref. 33), and (d) typical frequency dependence of the modulus or rigidity in a viscoelastic substance. The set of characteristic temperatures in (c) is an example for the case of ${\mathrm{Pd}}_{42.5}{\mathrm{Ni}}_{7.5}{\mathrm{Cu}}_{30}{\mathrm{P}}_{20}$; point $A$ is just the definition of the glass transition, $B$ and $C$ are based on the experiments (Refs. 12, 22, 23).

Figure 2

(Color online) RUS spectrum measured at room temperature for a ${\mathrm{Pd}}_{42.5}{\mathrm{Ni}}_{7.5}{\mathrm{Cu}}_{30}{\mathrm{P}}_{20}$ bulk metallic glass. The inset gives the sample information used in the present RUS measurement. The labeling of the vibrational modes (e.g., EV1, EX1, OD1, etc.) follows Ohno’s prescription (Ref. 26). About 40 resonance peaks were used in the inverse analysis, and the root-mean-square error indicating the degree of convergence was about 0.2%. The inverse calculation yields $c_{11}=209\mathrm{GPa}$ and $c_{44}=34.2\mathrm{GPa}$.

Figure 3

(Color online) (a) X-ray diffraction profiles of a fresh sample and the sample after the IXS measurement. (b) IXS spectra obtained for ${\mathrm{Pd}}_{42.5}{\mathrm{Ni}}_{7.5}{\mathrm{Cu}}_{30}{\mathrm{P}}_{20}$ at fixed $Q$ values. The high-energy peaks indicated by white squares at low $Q$ values are due to spurious scattering from the silicon windows of the vessel. The blue (solid) and green (dashed) curves indicate the fitted Eq. (1) and the damped terms in Eq. (1), respectively. (c) The refined IXS spectra at very low $Q$ values obtained by the precise measurements performed for a different sample cut from the same ingot, in order to enhance the signal-to-noise ratio by narrowing the slit of the $2\theta$ arm and optimizing the sample thickness.

Figure 4

(Color online) Phonon-dispersion curve obtained from the IXS data for a ${\mathrm{Pd}}_{42.5}{\mathrm{Ni}}_{7.5}{\mathrm{Cu}}_{30}{\mathrm{P}}_{20}$ bulk metallic glass (lower), and comparison of the sound velocity $({\Omega }_Q∕Q)$ of longitudinal sound obtained from the IXS data and the US velocities (upper). The line “Ultrasound linear function” in the lower part denotes $\hslash {\Omega }_Q=v_{\mathrm{L}}^{\mathrm{US}}Q$. The additional two IXS data are also plotted in the figure, and the previous US data (Refs. 23, 27) are also shown for reference. The sound-velocity plot in the low $Q$ region is considered to be connected to the long-wavelength limit by the curve $A$ depicted in the figure.