⇦Chinese Physics Letters, 2022, Vol. 39, No. 3, Article code 036401 A Free-Volume Model for Thermal Expansion of Metallic Glass Tong Lu (卢通)1,2, Song Ling Liu (刘松灵)1,2, Yong Hao Sun (孙永昊)1,2,3*, Wei-Hua Wang (汪卫华)1,2,3, and Ming-Xiang Pan (潘明祥)1,2,3* Affiliations 1Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 3Songshan Lake Materials Laboratory, Dongguan 523808, China Received 7 January 2022; accepted 29 January 2022; published online 1 March 2022 *Corresponding authors. Email: ysun58@iphy.ac.cn; panmx@iphy.ac.cn Citation Text: Lu T, Liu S L, Sun Y H et al. 2022 Chin. Phys. Lett. 39 036401    Abstract Many mechanical, thermal and transport behaviors of polymers and metallic glasses are interpreted by the free-volume model, whereas their applications on thermal expansion behaviors of glasses is rarely seen. Metallic glass has a range of glassy states depending on cooling rate, making their coefficients of thermal expansion vary with the glassy states. Anharmonicity in the interatomic potential is often used to explain different coefficients of thermal expansion in crystalline metals or in different metallic-glass compositions. However, it is unclear how to quantify the change of anharmonicity in the various states of metallic glass of the same composition and to connect it with coefficient of thermal expansion. In the present work, isothermal annealing is applied, and the dimensional changes are measured for La$_{62}$Al$_{14}$Cu$_{11.7}$Ag$_{2.3}$Ni$_{5}$Co$_{5}$ and Zr$_{52.5}$Cu$_{17.9}$Ni$_{14.6}$Al$_{10}$Ti$_{5}$ metallic glasses, from which changes in density and the coefficients of thermal expansion of the specimens are both recorded. The coefficients of thermal expansion linearly decrease with densification reflecting the role of free volume in thermal expansion. Free volume is found to have not only volume but also entity with an effective coefficient of thermal expansion similar to that of gases. Therefore, the local regions containing free volume inside the metallic glass are gas-like instead of liquid-like in terms of thermal expansion behaviors.
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 DOI:10.1088/0256-307X/39/3/036401 © 2022 Chinese Physics Society Article Text Free volume, defined as the volume surrounding an atom that can be exchanged with neighboring atoms without any expenditure of energy,[1] is a conceptual variable that is frequently used to explain the thermal and mechanical behaviors of metallic glasses (MGs), including shear transformation,[2,3] structural relaxation[2] and glass transition.[4,5] Free volume is often quantified by density: the relative change in density after thermo-mechanical treatment reflects the amount of free volume.[6] Linear relationship has been found between free volume and enthalpy change of an MG at its glassy state,[6] and rejuvenation, indicating increases in the free volume and enthalpy, leads to improved room-temperature plasticity of the MG.[7] Wang et al. pointed out an ubiquitous form of $M= \frac{M_{\infty }}{1+X}$, which connects flow units, i.e., the atomic clusters containing free volume, with many properties or behaviors of MGs, including elastic moduli and stored enthalpy at the glassy state.[8] Here, $M$ stands for a material property, $M_{\infty}$ the material's property after infinitely long-time annealing, and $X$ the concentration of flow units. Up to data, many physical properties have been evaluated by the free-volume model. However, no one has ever studied the coefficient of thermal expansion (CTE) using the free-volume model. It is known that CTE measures the relative dimensional change of a material per unit temperature. In crystalline metals, thermal expansion can be explained by anharmonic vibration of atoms.[9] The asymmetric shape of the interatomic potential well reflects the anharmonicity,[10] whose variation explains why different elements have different CTEs. The same argument can be applied to MGs with different compositions, too, but fails to answer why MGs of the same composition can have varied CTEs. Unlike crystalline metals, MGs have a range of glassy states depending on the cooling rate even if their compositions are the same.[11] Although researchers argued that the theory of anharmonicity can explain thermal expansion of MGs,[10,12–14] one would expect the anharmonicity and CTE to be unchanged in the same composition because the interatomic potentials are fixed. However, CTE varies at different glassy states of an MG in reality. For example, CTEs of the Pd$_{40}$Cu$_{30}$Ni$_{10}$P$_{20}$ and Cu$_{55}$Hf$_{25}$Ti$_{15}$Pd$_{5}$ MGs decrease[15,16] whereas CTE of a Pr-based MG increases after thermal annealing.[17] Gangopadhyay et al. proposed that cooperativity of atomic clusters apart from anharmonicity of the interatomic potential must be considered to interpret thermal expansion of MGs.[14] Therefore, thermal expansion of MGs requires a new theory beyond anharmonicity of interatomic potentials. The CTEs of MGs have been measured through two approaches. The first method is dilatometer that characterizes the macroscopic change of sample dimensions in in situ heating or cooling.[17,18] In this case, specimens as long as a few millimeters or centimeters are required to enhance the accuracy of the measurement. For example, linear CTE can be obtained using the expression $\frac{dl}{l_{0}dT}$ from experimental measurement, here $l_{0}$ is the original length of the sample at temperature $T_{0}$, $l$ is the length of the sample at temperature $T$, $dl=l-l_{0}$ and $dT=T - T_{0}$. The second method is synchrotron x-ray scattering that records the microscopic-structure change of the sample on in situ heating or cooling.[9,14,16] Positions of the peak maxima in the structural factor $S(q)$ and the radial distribution function $g(r)$ are analyzed to calculate CTEs. Here $q$ is the scattering vector, and $r$ is the distance from the origin. According to the Ehrenfest relation,[19] $q_{1}=\frac{1.23\times 2\pi }{r_{1}}$, where $q_{1}$ stands for the position of the first-peak maxima of $S(q)$, and $r$ the equilibrium interatomic distance. Thus, $-\frac{dq_{1}}{q_{1}dT}$ is considered as the microscopic CTE of MG because it equals $\frac{dr}{rdT}$. Experimental evidence shows that this type of microscopic CTE agrees well with the macroscopic CTE of many MGs,[15,16] except for a Zr-based MG.[9] On the other hand, discrepancies are found when using $r_{1}$, the position of the first-peak maxima of $g(r)$, to calculate the microscopic CTE. The $\frac{dr_{1}}{r_{1}dT}$ is always found to be much smaller than the macroscopic CTE, e.g., $\frac{dr_{1}}{r_{1}dT}$ is 9% of the macroscopic CTE for the Cu$_{50}$Zr$_{50}$ MG[20] or negative for the Pd$_{62.1}$Al$_{27.3}$Y$_{4.2}$Ni$_{6.4}$ and La$_{62}$Al$_{14}$Cu$_{11.7}$Ag$_{2.3}$Ni$_{5}$Co$_{5}$ MGs[21,22] when the specimens are thermally expanded. The position shift of the peak maximum agrees with the macroscopic CTE only when the interatomic spacing is beyond the 4$^{\rm th}$ nearest neighboring atomic shell.[20,23] The discrepancies existing in the microscopic and macroscopic CTEs suggest that the thermal expansion of MGs varies at different length scales. For crystalline metals, the macroscopic CTE is closely connected to heat capacity.[24] However, the difference between the microscopic CTE (measured by shifts in the diffraction peaks) and the macroscopic CTE (measured by dilatometer) indicates the role of vacancies.[25] This method works because x-ray is only scattered by atoms but not by vacancies. Yavari et al., adopted the idea, conducted the same experiment on MGs, and wished to measure free-volume from such discrepancy.[16] However, they failed with this approach as they observed that $-\frac{dq_{1}}{q_{1}dT}$ is the same as the macroscopic CTE, meaning that unlike vacancies in crystal, free volume contributes to the diffraction of the MG. This argument is further confirmed by the decrease of $q_{1}$ as a result of structural relaxation in the subsequent cooling and re-heating runs.[16] It then opens two fundamental questions: Is free volume really a volume with no mass (or in vacuum)? Or alternatively, does the free volume has an actual entity that can divert x-ray beam? In this Letter, we investigate effects of isothermal annealing on the CTEs of La$_{62}$Al$_{14}$Cu$_{11.7}$Ag$_{2.3}$Ni$_{5}$Co$_{5}$ and Zr$_{52.5}$Cu$_{17.9}$Ni$_{14.6}$Al$_{10}$Ti$_{5}$ MGs, and develop a simple semi-empirical model based on free volume, which can answer what entity the free volume is in response to the thermal expansion of MGs. It is found that the free volume has an entity with an effective CTE that is close to half the CTE of the Ar gas. Decrease of CTE in the annealed MG is attributed to the annihilation of free volume. The relative change of sample volume $v$, i.e., $\frac{dv}{V_{0}}$, can be calculated by the relative change of sample length ($l$) with a first-order approximation, $\frac{dl}{l_{0}}$, from $$ \frac{dv}{V_{0}}=3\times \frac{dl}{l_{0}},~~ \tag {1} $$ where $V_{0}$ and $l_{0}$ are the volume and length, respectively. The volumetric thermal expansion coefficient $\alpha$ and the linear thermal expansion coefficient $\beta$ are defined as $\alpha =\frac{dv}{V_{0}dT}$, $\beta =\frac{dl}{l_{0}dT}$, respectively, where $T$ stands for temperature. Based on Eq. (1), the following relation exists: $$ \alpha =3\beta .~~ \tag {2} $$ We now consider the MG sample as a composite of matrix and free volume, it then results in $$ dv=dv_{\rm m}+dv_{\rm f}=\alpha_{\rm m}v_{\rm m}dT+\alpha_{\rm f}v_{\rm f}dT, $$ $$ \alpha = \frac{dv}{vdT}=\alpha_{\rm m}\frac{v_{\rm m}}{v}+\alpha_{\rm f}\frac{v_{\rm f}}{v} =\alpha _{\rm m}(1-x)+\alpha_{\rm f}x ,~~ \tag {3} $$ where $\alpha_{\rm m}$ and $\alpha_{\rm f}$ are the CTEs of MG matrix and free volume, respectively, $\alpha$ is the complex CTE of metallic glass, $v_{\rm f}$ is free volume in the MG, and $x= \frac{v_{\rm f}}{v}$ for the volume fraction of free volume inside the MG. One should notice that if $x\ll 1$, $x\to 0$ and $1-x \to \frac{1}{1+x}$, Eq. (3) has the exact form of the ubiquitous relationship proposed by Wang et al.[8] When $t=0$, we have $\alpha =\alpha_{0}$, $v=V_{0}$, and $x=x_{0}$; when $t\to \infty$, we have $\alpha \to \alpha_{\rm m}$, $v_{\rm f}\to 0$, $x\to 0$, and $v$ reaches its minimum as $V_{0}-\Delta V$. Here, we define $\Delta V$ as the maximum volumetric change of the MG by annealing at $t\to \infty$. From Eq. (3), the difference between the two states is $$ \Delta \alpha =(\alpha_{\rm f}-\alpha_{\rm m})\Delta x ,~~ \tag {4} $$ where $\Delta \alpha =\alpha_{0}-\alpha_{\rm m}$, $\Delta x=x_{0}=\frac{\Delta V}{V_{0}}$. Then, based on Eq. (4), $$ \alpha_{\rm f}=\alpha_{\rm m}+(\alpha_{0}-\alpha_{\rm m})\frac{V_{0}}{\Delta V} .~~ \tag {5} $$ To resolve $\alpha_{\rm f}$, four variables, i.e., $\alpha_{\rm m}$, $\alpha_{0}$, $V_{0}$, $\Delta V$, need to be determined. On the other hand, $x$ at any given $t$ is $$ x=\frac{\Delta V-dv}{V_{0}},~~ \tag {6} $$ where $dv=V_{0}-v$. Two MGs, La$_{62}$Al$_{14}$Cu$_{11.7}$Ag$_{2.3}$Ni$_{5}$Co$_{5}$ and Zr$_{52.5}$Cu$_{17.9}$Ni$_{14.6}$Al$_{10}$Ti$_{5}$ (Vit105) were selected in this work. Elements with a purity of (at least 99.5%) were melted by an electric-arc furnace in an argon atmosphere to make ingots. Suction casting was then performed to cast the ingots into a copper mold in an Ar atmosphere. The final rods were 3.5 mm in diameter ($\phi$) and 70 mm in length ($l$). Cylinder samples with $\phi$ of 3.0 mm and $l$ of 6.0 mm were machined by Luling precision company in Suzhou. X-ray diffraction tests (XRD D8 Advance, Bruker) and differential scanning calorimetry (DSC 8000, PerkinElmer) were performed. All the specimens were amorphous, and the glass transition temperature ($T_{\rm g}$) is 409 K for the La-based MG and 665 K for the Zr-based MG. Helium pycnometer (AccuPyc II 1345, Micrometritics) and electronic weight-balance (XS105, Mettler, with an accuracy of 0.01 mg) were applied to measure the sample density ($\rho$). Five as-cast cylinders of the same MG composition were inserted in one gas chamber that was 12.2 mm in diameter, 11.3 mm in depth and 0.719 cm$^{3}$ in volume, and the accuracy of volume determination reaches 0.001 cm$^{3}$; $\rho$ are $6.225\pm 0.001$ g/cm$^{3}$ and $6.630\pm 0.001$ g/cm$^{3}$ for the La-based MG and the Zr-based MG at the as-cast state, respectively. Thermal mechanical analysis (TMA 402 F3, Netzsch) was made for thermal expansion measurements. A uniaxial compressive stress of 4.25 kPa was applied on the cylindrical specimens ($\phi = 3.0$ mm and $l = 6.0$ mm) before thermal treatment. The applied stress is less than one in ten thousands of the yield stress of the La-based MG or Zr-based MG, respectively, guaranteeing elastic deformation of the samples below $T_{\rm g}$. The Zr-based and La-based specimens were kept at 303 K for 3 min at the beginning of thermal treatment, which stabilized the thermal environment of the machine. Next, they were heated to 0.90$T_{\rm g}$ at a constant rate of 3 K/min, held at 0.90$T_{\rm g}$ for 3 min, and then cooled back to 350 K at a constant rate of $-$3 K/min and kept for 3 min at this temperature. Similar thermal protocol was conducted by ten more times in the temperature range between 350 K and $0.90T_{\rm g}$ except that the isothermal annealing interval was extended to 2, 3, 5, 5, 15, 30, 60, 120, 240, 480, 960 min in sequence. Thermal mechanical analyzer (TMA) records $l$ during in situ heating and cooling experiments, from which $\alpha$ can be derived. Take Fig. 1 as an example, $\frac{dl}{l_{0}}$ increases on heating (red lines) and decreases on cooling (blue lines), implying expansion on heating and contraction on cooling, respectively. In the 1$^{\rm st}$ heating run, $\frac{dl}{l_{0}}$ linearly increases with the increasing $T$ until 480 K ($= 0.72T_{\rm g}$) as indicated by the dashed line. The slope of the curve, $1.02 \times 10^{-5}$ K$^{-1}$, is taken as $\beta_{0}$, so $\alpha_{0}$ is $3.06 \times 10^{-5}$ K$^{-1}$ according to Eq. (1). Increase of $\frac{dl}{l_{0}}$ continues at $T >480$ K, but the curve is inclined below the dashed line, demonstrating a nonlinear thermal expansion. The 480 K coincides with the onset of the pre-$T_{\rm g}$ exothermic peak in calorimetry and correlated with the structural relaxation of the MG. The 1$^{\rm st}$ heating run stops at 598 K ($= 0.90T_{\rm g}$), and an isothermal annealing (dotted lines) of 3 min started. In the 1$^{\rm st}$ cooling run, a linear decrease of $\frac{dl}{l_{0}}$ with the decreasing $T$ was seen in the range of 350–480 K. A linear fit of the data from this range is performed, and the slope $(\frac{dl}{l_{0}dT})$ is taken as $\beta$, that is, $1.02 \times 10^{-5}$ K$^{-1}$. Here, $\alpha$ of $3.06 \times 10^{-5}$ K$^{-1}$ is directly calculated from $\beta$ based on Eq. (2). At 532 K ($=0.80T_{\rm g}$), a reduction in $\frac{dl}{l_{0}}$ of $2.7 \times 10^{-4}$ is detected between the 1$^{\rm st}$ heating and the 2$^{\rm nd}$ heating runs, implying a decrease in $\frac{dv}{V_{0}}$ of $8.1 \times 10^{-4}$ based on Eq. (1). The same heating and cooling runs between 350 K and 598 K are repeated 11 times and $\alpha$ is determined from the cooling curves, $\frac{dv}{V_{0}}$ is determined from the heating curves to $0.80T_{\rm g}$. Thus, TMA measurements provide information of $\alpha$, $\alpha_{0}$ and $\frac{dv}{V_{0}}$.
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Fig. 1. DSC (a) and TMA (b) curves of a Zr-based MG. Relaxation starts from nearly 480 K (0.70$T_{\rm g}$). Sample length is $l_{0}$, its relative change is $\frac{dl}{l_{0}}$ and temperature is $T$. The heating curves are colored in red, and the cooling curves in blue. The dashed line marks the linear response of $\frac{dl}{l_{0}}$ to $T$ in the 1$^{\rm st}$ heating run to 480 K from 350 K, and the slope of the dashed line is $\beta_{0}$. Between 350 K and 480 K, the slope of the cooling curve is taken as $\beta_{n}$ ($n = 1$–12, and $n$ stands for the $n$th cooling). At 432 K ($=0.80T_{\rm g}$), difference in $\frac{dl}{l_{0}}$ is recorded as the length change after the $n$th cooling ($n = 1$–12). The double-side arrow denotes the length change after the 1$^{\rm st}$ cooling.
In our experiment, long-time isothermal annealing at 0.90$T_{\rm g}$ was performed to obtain $\alpha_{\rm m}$ and $\Delta v$ with the applied $t$ up to $1.2 \times 10^{5}$ s. As shown in Fig. 2(a), $\frac{dv}{V_{0}}$ remains nearly unchanged after $t>4.0\times {10}^{4}$ s, suggesting that the free volume is depleted or $v_{\rm f}\to 0$ (and $x\to 0$). Thus, the maximum $\frac{dv}{V_{0}}$ represents $\frac{\Delta V}{V_{0}}$. Because $V_{0}=\frac{m}{\rho}$ with $m$ being the sample mass, $\Delta V$ is calculated as the product of $V_{0}$ and maximum $\frac{dv}{V_{0}}$. For the La-based MG, $\Delta V$ is $3.8 \times 10^{-5}$ cm$^{3}$; for the Zr-based MG, $\Delta V$ is $8.1 \times 10^{-5}$ cm$^{3}$. Here, $\frac{dv}{V_{0}}$ can be transferred into $x$ based on Eq. (6). Figure 2(b) exhibits how $x$ changes with $t$. With the increasing $t$, $x$ is diminished. On the other hand, $\alpha$ is saturated when $t>4.0\times {10}^{4}$ s [Fig. 2(c)]. Since $x$ is zero, the saturated $\alpha$ denotes $\alpha_{\rm m}$ representing the CTE of the MG matrix. For the La-based MG, $\alpha_{\rm m}$ is $2.3 \times 10^{-4}$ K$^{-1}$; for the Zr-based MG, $\alpha_{\rm m}$ is $7.8 \times 10^{-5}$ K$^{-1}$.
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Fig. 2. Effects of annealing on the La-based and Zr-based MGs. (a) The relative volumetric change $ \frac{dv}{V_{0}} $, (b) the volume fraction of free volume in the MG ($x$), and (c) the volumetric thermal expansion coefficient $\alpha$ with the annealing time $t$. Saturations with the increasing $t$ are seen for $\frac{dv}{V_{0}}$, $x$ and $\alpha$ in both MGs. The saturated volumetric change $\frac{\Delta V}{V_{0}}$ is marked by the double-side arrow, and the matrix's thermal expansion coefficient $\alpha_{\rm m}$ by the arrow.
The validity of Eqs. (3)-(5) needs the thermal expansion coefficient to be tested to see if $\alpha$ is linearly correlated with $x$. Figure 3 plots $\alpha$ against $x$ where linear correlations are found for both MGs. The root mean square ($R^{2}$) of the linear fit is 0.94 for the La-based MG and 0.96 for the Zr-based MG. Therefore, Eqs. (3)-(5) are valid, and $\alpha_{\rm f}$ can be obtained from $\alpha_{\rm m}$, $\alpha_{0}$, $V_{0}$, $\Delta V$ based on Eq. (5). Table 1 lists the measured $\alpha_{\rm m}$, $\alpha_{0}$, $V_{0}$, $\Delta V$ and the calculated $\alpha_{\rm f}$ of both MGs. In this work, the volumetric CTE of the MG matrix, the free volumes of La-based and Zr-based MGs are obtained from Eq. (5). The nonzero $\alpha_{\rm f}$ of both MGs emphasizes that the free volume cannot be interpreted just as “volume” but should be considered as an entity. Furthermore, $\alpha_{\rm f}$ is larger than $\alpha_{\rm m}$, e.g., about $36 \times$ for the La-based MG and $33 \times$ for the Zr-based MG. This magnitude of change is significantly larger than that of the elastic moduli as estimated by Sun et al.,[26] who found that the flow unit has elastic moduli about one tenth of that of the matrix. Moreover, these flow units containing free volume are often interpreted as liquid-like regions. However, the absolute values of $\alpha_{\rm f}$ (Table 1) are at least one order of magnitude larger than most metallic liquids (in the order magnitude of 10$^{-5}$–$10^{-4}$ K$^{-1}$[27]), and are close to half the value of the inert gas. For example, argon has an $\alpha$ of $2.48 \times 10^{-3}$ K$^{-1}$.[28,29] Thus, in terms of CTE, the free volume is more gas-like rather than liquid-like. It is possible that the flow units are so loose that researchers have paid more attention to its volume and ignored its matter. However, evidences from both x-ray scattering[16] and thermal expansion (the present work) emphasize the “entity” of the free volume. Understanding the entity of the free volume may be useful in interpreting the new phenomena of MG nanoparticles,[30] MG nanowires,[31] MG composite,[32] and the MG-forming liquid.[33]
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Fig. 3. The volumetric thermal expansion coefficient $\alpha$ vs the free-volume fraction $x$. Nice linear fits are obtained for both La-based and Zr-based MGs.
Table 1. Parameters $V_{0}$, $\Delta V$, $\alpha_{0}$, $\alpha_{\rm m}$ and $\alpha_{\rm f}$ of the La-based and Zr-based MGs.
Composition ${V}_{{0}}\,({{10}}^{{-2}}{\mathrm{{cm}}}^{\mathrm{{3}}})$ ${\Delta V}\,({{10}}^{{-5}}{\mathrm{{cm}}}^{\mathrm{{3}}})$ ${\alpha }_{{ 0}}\,({{10}}^{{-5}}{\mathrm{{K}}}^{\mathrm{{-1}}})$ ${\alpha }_{\mathrm{{ m}}}\,({{10}}^{{-5}}{\mathrm{{K}}}^{\mathrm{{-1}}})$ ${\alpha }_{\mathrm{{f}}}\,({{10}}^{{-5}}\mathrm{{K}}^{\mathrm{{-1}}})$
La$_{62}$Al$_{14}$Cu$_{11.7}$Ag$_{2.3}$Ni$_{5}$Co$_{5}$ 4.2 3.8 3.8 3.68 137
Zr$_{52.5}$Cu$_{17.9}$Ni$_{14.6}$Al$_{10}$Ti$_{5}$ 4.2 8.1 3.1 3.01 102
A semi-empirical free-volume model is constructed for metallic glass to explain the change of thermal expansion coefficient in annealing. The volumetric thermal expansion coefficient of metallic glass is found linearly decreased with the increasing free volume inside the metallic-glass matrix. The volumetric thermal expansion coefficient of free volume is derived from the annealing data. It is found that the volumetric thermal expansion coefficient of free volume is about 33–36 times that of the matrix. The present work suggests that the free volume inside the metallic glass has an entity, which is more gas-like than liquid-like in terms of the volumetric thermal expansion coefficient. Acknowledgements. This research was supported by the National Natural Science Foundation of China (Grant Nos. 51671211, 51601215, and 51971239), the National Key Research and Development Program of China (Grant No. 2021YFA0703603), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB30000000), and the Natural Science Foundation of Guangdong Province (Grant No. 2019B030302010). References Embrittlement of a bulk metallic glass due to sub- annealingA microscopic mechanism for steady state inhomogeneous flow in metallic glassesHomogeneous flow of metallic glasses: A free volume perspectiveFree‐Volume Model of the Amorphous Phase: Glass TransitionOn the Free‐Volume Model of the Liquid‐Glass TransitionCorrelation between enthalpy change and free volume reduction during structural relaxation of Zr55Cu30Al10Ni5 metallic glassRejuvenation of metallic glasses by non-affine thermal strainStructural perspectives on the elastic and mechanical properties of metallic glassesThermal behaviour of Pd40Cu30Ni10P20 bulk metallic glassRelationship between thermal expansion coefficient and glass transition temperature in metallic glassesThermomechanical processing of metallic glasses: extending the range of the glassy stateDoes the Interaction Potential Determine Both the Fragility of a Liquid and the Vibrational Properties of Its Glassy State?On the Atomic Anisotropy of Thermal Expansion in Bulk Metallic GlassLink between volume, thermal expansion, and bulk metallic glass formabilitySynchrotron X-ray radiation diffraction studies of thermal expansion, free volume change and glass transition phenomenon in Cu-based glassy and nanocomposite alloys on heatingExcess free volume in metallic glasses measured by X-ray diffractionDilatometric measurements and glass-forming ability in Pr-based bulk metallic glassesGlass transition Tg, thermal expansion, and quenched-in free volume ΔVf in pyrex glass measured by time-resolved X-ray diffractionStrong correlation of atomic thermal motion in the first coordination shell of a Cu-Zr metallic glassThermal expansion of Pd-based metallic glasses by ab initio methods and high energy X-ray diffractionThermal expansion of a La-based bulk metallic glass: insight from in situ high-energy x-ray diffractionDifferent Thermal Responses of Local Structures in Pd 43 Cu 27 Ni 10 P 20 Alloy from Glass to LiquidScaling Behavior between Heat Capacity and Thermal Expansion in SolidsMeasurements of Equilibrium Vacancy Concentrations in AluminumCorrelation between local elastic heterogeneities and overall elastic properties in metallic glassesSurface energies of metals in both liquid and solid statesDynamic Crossover in Metallic Glass Nanoparticles*Rejuvenation in Hot-Drawn Micrometer Metallic Glassy Wires*A New Cu-Based Metallic Glass Composite with Excellent Mechanical Properties*Pressure Effects on the Transport and Structural Properties of Metallic Glass-Forming Liquid
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