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, 51 (Pt 2), 470-480
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The Thermal Expansion of Gold: Point Defect Concentrations and Pre-Melting in a Face-Centred Cubic Metal

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The Thermal Expansion of Gold: Point Defect Concentrations and Pre-Melting in a Face-Centred Cubic Metal

Martha G Pamato et al. J Appl Crystallogr.

Abstract

On the basis of ab initio computer simulations, pre-melting phenomena have been suggested to occur in the elastic properties of hexagonal close-packed iron under the conditions of the Earth's inner core just before melting. The extent to which these pre-melting effects might also occur in the physical properties of face-centred cubic metals has been investigated here under more experimentally accessible conditions for gold, allowing for comparison with future computer simulations of this material. The thermal expansion of gold has been determined by X-ray powder diffraction from 40 K up to the melting point (1337 K). For the entire temperature range investigated, the unit-cell volume can be represented in the following way: a second-order Grüneisen approximation to the zero-pressure volumetric equation of state, with the internal energy calculated via a Debye model, is used to represent the thermal expansion of the 'perfect crystal'. Gold shows a nonlinear increase in thermal expansion that departs from this Grüneisen-Debye model prior to melting, which is probably a result of the generation of point defects over a large range of temperatures, beginning at T/Tm > 0.75 (a similar homologous T to where softening has been observed in the elastic moduli of Au). Therefore, the thermodynamic theory of point defects was used to include the additional volume of the vacancies at high temperatures ('real crystal'), resulting in the following fitted parameters: Q = (V0K0)/γ = 4.04 (1) × 10-18 J, V0 = 67.1671 (3) Å3, b = (K0' - 1)/2 = 3.84 (9), θD = 182 (2) K, (vf/Ω)exp(sf/kB) = 1.8 (23) and hf = 0.9 (2) eV, where V0 is the unit-cell volume at 0 K, K0 and K0' are the isothermal incompressibility and its first derivative with respect to pressure (evaluated at zero pressure), γ is a Grüneisen parameter, θD is the Debye temperature, vf, hf and sf are the vacancy formation volume, enthalpy and entropy, respectively, Ω is the average volume per atom, and kB is Boltzmann's constant.

Keywords: gold; pre-melting phenomena; thermal expansion; vacancies.

Figures

Figure 1
Figure 1
X-ray powder diffraction pattern of gold (+ MgO) at 298 K, collected with the sample in the hot stage. Observed (red points) and calculated patterns (green line) and their differences (purple lower trace) are also shown. The tick markers show the position of the Bragg reflections from top down: MgO (red) and Au (black).
Figure 2
Figure 2
X-ray powder diffraction patterns of gold (black markers) and MgO (red markers) at high 2θ angles, at different temperatures approaching melting. Note that the gold peaks disappear at 1339 K, indicating that gold melted between 1337 and 1339 K, in perfect agreement with the melting temperature reported in the literature.
Figure 3
Figure 3
Unit-cell volume of gold as a function of temperature showing the entire temperature range. The error bars are smaller than the symbols. The solid line represents the fit of the data to a second-order Grüneisen approximation to the zero-pressure equation of state [equation (3)].
Figure 4
Figure 4
Unit-cell volumes of gold in the high-temperature region expanded to show the possible pre-melting zone. The error bars are smaller than the symbols. The solid and dashed lines represent the fits of the data to second-order and third-order Grüneisen approximations to the zero-pressure equation of state [equations (3) and (6)].
Figure 5
Figure 5
Differences between measured and calculated volumes as a function of temperature, when employing a second-order (filled symbols) or a third-order (open symbols) Grüneisen approximation [equations (3) and (6)].
Figure 6
Figure 6
Unit-cell volume of gold expanded in the high-temperature region to show the possible pre-melting region. The error bars are within the symbols. The solid red line represents the fit of the data to equation (10) (real crystal) and the dashed line is the perfect crystal component [i.e. V p(T)]. Differences between measured and calculated volumes including the defect contribution to the volume of the real crystal as a function of temperature are shown in the insets.
Figure 7
Figure 7
Difference in volume (ΔV) between a real and a perfect Au crystal [equation (9)] as a function of temperature (bottom x axis) and homologous temperature, T/T m (top x axis). The difference in volume is also reported as a percentage on the right.
Figure 8
Figure 8
Volumetric thermal expansion coefficient of gold as a function of temperature. The open data points were obtained by numerical differentiation of the data reported in Table S1 and Fig. 2 ▸ [equation (11)], selecting a 20 K window (see text). The red solid line represents the calculated real crystal model [equation (10)] and the dashed black line is the perfect crystal component thereof [i.e. V p(T), see text for details]. Measured values reported in the literature (filled symbols) are also plotted for comparison.
Figure 9
Figure 9
Temperature dependence of vacancy concentrations in gold, determined by a variety of methods. The letters indicate that the vacancy concentration was determined from differential dilatometry (DD; Simmons & Balluffi, 1962 ▸); calorimetry of quenched samples (QC; DeSorbo, 1960; Pervakov & Khotkevich, 1960 ▸); linear extrapolation of thermal expansivity (L; Gertsriken & Slyusar, 1958 ▸); specific heat (C; Kraftmakher & Strelkov, 1966 ▸); dilatometry of quenched samples (QD; Fraikor & Hirth, 1967 ▸); electron microscopy of quenched samples (QM; Cotterill, 1961; Siegel, 1966a ,b ▸).
Figure 10
Figure 10
Calculated isobaric heat capacity of gold at ambient pressure. Measured values reported in the literature are also plotted for comparison: open circles from Anderson et al. (1989 ▸); crosses from Barin & Knacke (1973 ▸) as reported by Yokoo et al. (2009 ▸); open triangles from Hultgren et al. (1973 ▸) as reported by Shim et al. (2002 ▸); and open squares from Touloukian et al. (1975 ▸) as reported by Tsuchiya (2003 ▸). The red solid line represents our calculated real crystal model and the dashed black line indicates our perfect crystal model; for temperatures below about 800 K the heat capacity curves calculated from the two models are effectively identical.

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Grant support

This work was funded by Natural Environment Research Council grants NE/M015181/1, NE/K002902/1 , and NE/H003975/1. Science and Technology Facilities Council grant ST/K000934/1.
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