Thermodynamic stability and contributions to the Gibbs free energy of nanocrystalline ${\mathrm{Ni}}_3\mathrm{Fe}$

The heat capacities of nanocrystalline ${\mathrm{Ni}}_3\mathrm{Fe}$ and control materials with larger crystallites were measured from $0.4–300\mathrm{K}$. The heat capacities were integrated to obtain the enthalpy, entropy, and Gibbs free energy and to quantify how these thermodynamic functions are altered by nanocrystallinity. From the phonon density of states (DOS) measured by inelastic neutron scattering, we find that the Gibbs free energy is dominated by phonons and that the larger heat capacity of the nanomaterial below 100 K is attributable to its enhanced phonon DOS at low energies. Besides electronic and magnetic contributions, the nanocrystalline material has an additional contribution at higher temperatures, consistent with phonon anharmonicity. The nanocrystalline material shows a stronger increase with temperature of both the enthalpy and entropy compared to the bulk sample. Its entropy exceeds that of the bulk material by 0.4 $k_{\mathrm{B}}$/atom at 300 K. This is insufficient to overcome the enthalpy of grain boundaries and defects in the nanocrystalline material, making it thermodynamically unstable with respect to the bulk control material.

Figure 1

(a) Phonon DOS from a polycrystalline powder of an fcc Ni-Fe alloy [3]. Compared to the DOS from the material with the larger 20 nm crystallite size, the 6 nm DOS shows three features: (1) enhancement of the intensity below 15 meV, (2) broadening of all features, (3) a tail of spectral weight above 36 meV. (b) Representative librational or vibrational modes of the nanocrystalline microstructure, with energies of order 1 meV and wavelengths of 10 nm. (c) Continuum behavior at long wavelengths.

Figure 2

Transmission electron microscopy images of the as-milled ${\mathrm{Ni}}_3\mathrm{Fe}$. The dark-field (DF) and bright-field (BF) images were obtained with the (111) fcc diffraction. (a) DF and (b) BF images of the as-milled sample, compared to (c) DF image from previous work, with crystallites of 6 nm [3]. (d) Electron diffraction pattern.

Figure 3

X-ray diffraction patterns of the as-milled material and of the bulk control sample after annealing at 873 K for 1 h.

Figure 4

Heat capacity of ${\mathrm{Ni}}_3\mathrm{Fe}$ measured by calorimetry for the as-milled materials ($+$) and large-grained control samples ($•$), normalized by the number of moles of atoms. Different colors represent results from different samples. (a) Full data sets. The solid black curve is from the annealed control material (see text). (b) Low-temperature data.

Figure 5

Mean heat capacities of the as-milled nanomaterial and control bulk sample compared to literature. The heat capacity of bulk disordered ${\mathrm{Ni}}_3\mathrm{Fe}$ reported by Kollie and Brooks [44] is shown at 300 K ($⧫$). Data compiled by Desai for fcc Ni are shown as small markers and the recommended curve as a thin solid line [73]. The inset compares our low temperature data ($<5$ K) with the fit reported by Kollie et al. [74].

Figure 6

Heat flow measured by DSC while heating the nanocrystalline sample at 20 K/min. Positive values correspond to endothermic heat flows. The release of internal energy from defects and grain growth is highlighted (green) and its enthalpy of $H_0^{\mathrm{n}-\mathrm{c}}=2.3$ kJ/mol was calculated using a linear baseline (dashed line). The labeled grain sizes were determined from measured XRD patterns presented in the Supplemental Material [43].

Figure 7

(a) Phonon DOS curves measured by inelastic neutron scattering at 300 K. Solid curves correspond to the new measurements performed at ARCS, while the markers are measurements from previous work [3]. The solid light curve (Calc), also from previous work, was calculated with a Born-von Kármán model using force constants from [66] and broadened by instrument resolution. (b) Phonon entropies calculated with the curves of panel a, using Eq. (7).

Figure 8

Low temperature heat capacities of the nanocrystalline and control materials plotted as $C_p/T$ vs $T^2$. Solid curves are fits to the data using Eq. (9).

Figure 9

Heat capacities of nanomaterial ($+$) and control samples ($•$) from calorimetry shown in Fig. 5, superposed with phonon heat capacities calculated from the phonon DOS using Eq. (8) (solid lines) and with vibrational plus electronic heat capacities (dashed lines).

Figure 10

Individual contributions to the heat capacity of (a) the control and (b) nanocrystalline samples. The residual is the difference between the sum of the labeled contributions and the total heat capacity measured by calorimetry.

Figure 11

(a) Total enthalpy and (b) entropy of nanocrystalline material and control sample computed from calorimetry measurements of the heat capacity using Eqs. (2) and (3). The grain boundary enthalpy $H_0$ is not included.

Figure 12

Gibbs free energies of as-milled nanocrystalline ${\mathrm{Ni}}_3\mathrm{Fe}$ and control sample, using enthalpies and entropies of Fig. 11, and $\mathrm{\Delta }{H}_0^{\mathrm{n}-\mathrm{c}}=2.3\mathrm{kJ}/\mathrm{mol}$.