# Thermophysical Properties of Undercooled Alloys: An Overview of the Molecular Simulation Approaches

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Molecular Simulations Approaches

_{ij}and ϕ

_{ij}are the distance and the short-range pair potential between atoms i and j. (ρ

_{h}

_{,}

_{i}) F

_{i}ρ is the embedding energy of atom i, and ρ

_{h}

_{,}

_{i}denotes the electron density of the host without atom i, which is approximated to a sum of the atomic electron density of j atom at a distance r

_{ij}from site i,

_{i}is the background electron density, E

_{i}

^{0}is the cohesive energy of the reference structure, Z

_{i}is the number of nearest neighbors of reference structure, and A

_{i}is an adjustable parameter. The form of the background electron density varies with the element type, for example,

_{i}

^{(}

^{k}

^{)}is the partial electron density and

_{ij}

^{α}r

_{ij}

^{α}r

_{ij}= (α = x, y, z), r

_{ij}is the distance between atom i and j. ρ

_{i}

^{a}

^{(}

^{k}

^{)}represents electron density contributed from atom j and shows an exponential decay with the distance. Similar to the EAM, the most parameters in the MEAM are determined by fitting the elastic properties of crystals from experiments or first-principle simulations. In the original MEAM, the interactions confined in the nearest-neighbor atoms are considered. The following modified versions are extended to second nearest-neighbor atom interactions (2NN MEAM) for overcoming the discrepancy in reproducing the surface energy of low-index surface that is deemed to be a result of neglecting the second nearest-neighbor atoms interactions [80,81].

#### 2.1. Melting Temperature

_{3}system versus temperature [41]. The solid and dashed lines represent cooling and heating processes from initial liquid and solid systems respectively, and the circles are the results from the sandwich method. Due to the extremely high cooling and heating rates, the crystallization and melting temperatures obtained from initial pure liquid or solid are far from the equilibrium melting point. Instead, the sandwich method can bypass heating and cooling processes in single-phase MD simulations and accurately produce the melting point. The difficulty of this method is that several trivial simulations are required in order to confirm the coexistence of liquid and solid phases, in particular in the region near the melting temperature.

#### 2.2. Specific Heat

^{2}〉 = 〈E

^{2}〉 − 〈E

^{2}〉, N

_{a}is the particle number of systems. In the micocanonical ensemble (constant particle number, volume and energy), 〈δE

^{2}〉 = 0, and the relevant fluctuations to consider are those of kinetic energy E

_{k}or potential energy E

_{u}. Thus the isovolumetric specific heat is

#### 2.3. Density

#### 2.4. Surface Tension

_{l}and z

_{v}are the arbitrary positions in the bulk liquid and vapor phases respectively, P

_{N}and P

_{T}are the normal and tangential components of pressure tensor. In general, the simulation model is a liquid film that has two symmetry surfaces. Therefore the above equation is converted into

_{z}/N parallel to the X–Y panel, the integral can be transformed to the sum in these slabs. From the statistical mechanical expressions, P

_{N}and P

_{T}in the kth slab are written as

_{x}· L

_{y}· L

_{z}/ N is the volume of a slab, r

_{ij}is the distance between particles i and j, φ (r

_{ij}) is the interaction potential. Therefore, Equation 13 has the following form [90],

_{AA}is defined as the work required to pull the liquid bulk system A apart, thus

#### 2.5. Diffusion

_{j}

^{(}

^{i}

^{)}(t) and u⃗

_{(}

_{i}

_{)}

^{j}(0) denote the velocity vector of particle j of species i at time t and 0 respectively, N

_{i}is the number of i particles, and 〈·〉 is the ensemble average. Correspondingly, in the equilibrium thermodynamics, the self-diffusion coefficient is calculated by means of the Green-Kubo relation that integrates over the autocorrelation function [94],

_{j}

^{(}

^{i}

^{)}(t) and (0) r⃗

_{j}

^{(}

^{i}

^{)}are the positions of particle j of species i at time t and 0 respectively. The two self-diffusion coefficients above are equivalent and are fundamental of computer simulations of self-diffusion coefficient.

_{t}is defined as Fickian diffusion coefficient or transport diffusion coefficient. Compared with the diffusion in a single-component system, the Fickian diffusion in a mixture shows more complicated. The interplay between different species becomes pronounced, and even some abnormal diffusion phenomena are observed. For instance, the experiments have demonstrated that N

_{2}molecules diffuse against the concentration gradient in an ideal gas mixture [95,96]. Several theoretical models are proposed to overcome the difficulty. Based on the irreversible thermodynamics, the diffusion driven by the chemical potential gradients is developed. In this model, the diffusion flux, J is given by a formula related to chemical potential gradients [96,97],

_{ij}is the symmetric matrix of Onsager transport coefficients, n is the number of species, ∇μ

_{j}is the chemical potential gradient of species j. The flux can also be independently given by the Fick’s law in multi-component systems as a function of concentration gradients,

_{ij}is not symmetric and is called the Fickian diffusion coefficient or the binary transport diffusion coefficient. Maxwell and Stefan [97–99] proposed another model based on the kinetics process of ideal gas mixtures for describing the diffusion process above. They assumed that the chemical potential gradient is balanced by the frictional drag and the deviation of the force balance will lead to the diffusion flux. The frictional drag is proportional to the relative velocity of species i and j, u

_{i}− u

_{j}and mole fraction c

_{j}, then we get

_{ij}is the Maxwell-Stefan (M-S) diffusion coefficient and can be regarded as an inverse drag coefficient. The matrix Ð

_{ij}is symmetric on the basis of the Onsager reciprocal relations. For the ideal mixture, Ð

_{ij}is independent of the composition, otherwise it is not. It is noted that the diffusion processes in the multi-component systems described in Equations 22–24 are mathematically equivalent, and there are no approximate relations among them.

_{1}of species 1,

_{1,}

_{self}D

_{2,}

_{self}, and concentrations, c

_{1}, c

_{2}of species 1 and 2. In fact, if the velocity cross correlations 〈u⃗

_{i}(t) · u⃗

_{j}(0) 〉, i ≠ j is negligible in Equation 27, the average M-S diffusion coefficient is accordingly simplified to the Darken relation as,

#### 2.6. Viscosity

#### 2.6.1. Equilibrium MD Simulations

_{αβ}is an off-diagonal element (α ≠ β) of pressure tensor matrix. p

_{αj}and p

_{βj}denote the momenta of the jth particle along α and β directions, r

_{jβ}is the particle position component and F

_{jα}is the α component of force on the jth particle. The equilibrium MD method requires a relatively long simulation time to wait for the dynamic relaxation of the system. Especially under high undercooling conditions, the decay scope of the autocorrelation function rapidly increases and the calculation becomes more difficult. Instead, the non-equilibrium MD calculations are proposed.

#### 2.6.2. Non-Equilibrium MD Simulations

_{x}is exerted to the upper wall in the positive x direction, and an opposite and constant velocity −v

_{x}is applied to the lower wall. As a result, a thin and relatively stationary fluid layer forms close to each wall if the walls are rough enough, and finally a planar Couette flow is developed within the liquid confined between the two walls. In the fluid, the linear velocity profile is constructed and the shear rate is given by

_{xy}to shear rate σ,

_{z}is the channel width, T

_{w}is the wall temperature that is maintained constant in the simulations, g

_{z}is the external force driving the flow, and λ is the thermal conductivity. The periodical boundary conditions are held in the x and y directions. Dividing the simulation region into several slices parallel to the walls and calculating the mean flow velocity and temperature in each slice, the viscosity and thermal conductivity can be obtained by fitting the velocity and temperature profiles based on Equations 34 and 35.

_{x}, the momentum flow, j

_{z}(p

_{x}) is given by

_{x}L

_{y}. Müller-Plathe applied the method to liquid Ar, N

_{2}, water and hexane and yielded good agreement with the experimental results.

## 3. Simulated Results and Comparison with Experiments

#### 3.1. Melting Temperature

_{3}and Au-Cu alloys, agreement with the experimental values shows rather good, but the calculated melting point underestimates the experimental result by about 20% in Al

_{50}Ni

_{50}alloy [106].

#### 3.2. Enthalpy and Specific Heat

#### 3.2.1. Pure Liquid Metal

#### 3.2.2. Binary Liquid Alloy

_{3}Cu, AuCu and AuCu

_{3}as a function of temperature are given as follows [41],

#### 3.3. Density

_{m}is the density of liquid metal or alloy at the melting temperature T

_{m}.

_{A}and M

_{B}are the molar masses of A and B components, V

_{A}and V

_{B}are the molar volumes of A and B components respectively, and c

_{A}= 1− c

_{B}is the molar concentration. However, for most alloy melts, Equation 43 does not provide a reasonable description of the composition dependence of alloys. Figure 11(a) shows the simulated density of Au-Cu alloys as a function of composition at different temperatures [41]. The deviation of simulated density from the prediction of the Neumann-Kopp relation becomes more serious as the composition increases, and reaches a maximum in the moderate composition region. Similar behavior is also observed in experiments. For example, the density of Cu-Ni alloy positively deviates from the prediction of ideal solution model in the middle composition region [20]. The main reason for the deviation is the remarkable volume effect that occurs when two pure elements are melted into an alloy. A concept of excess volume ΔV is introduced to correct the density in the ideal solution model,

_{ideal}is the volume of the ideal mixture. The excess volume is further assumed to depend on the concentrations,

_{X}being a constant parameter, independent of temperature and concentration. Figure 11(b) plots the excess volume of Au-Cu alloy versus composition. Obviously, the excess volume reaches the minimum values at 50%, corresponding to the most pronounced volume effect, and the liquid is characterized by the highly non-ideal solution. In experiments, the influence of the excess volume on density is also verified and Equation 44 can well describe the composition dependence of density of some systems such as Cu-Ni alloys [20]. It is pointed out that the composition dependence of the density usually varies with the systems, and some liquid alloys do not exhibit strong excess volume, for instance the zero excess volume observed in Ag-Cu alloy [118]. In general, for the dilute solution, the Neumann-Kopp relation can be applied to interpolate the density to a certain extent.

#### 3.4. Surface Tension

_{R}corresponding to the spherical droplet splits into a set of five frequencies ω

_{m}belonging to different oscillation modes with m = −2, −1, 0, 1, 2, where m is the mode number of axisymmetric shape oscillation. In the case of the spherical droplet without the external fields, the relationship between the Rayleigh frequency ω

_{R}and the surface tension γ is written as

_{L}is the surface tension at the liquidus temperature T

_{L}, γ ′ = dγ dT is the temperature coefficient. This linear relation is also held in some binary alloys. For most liquids around us, the surface tension shows negative temperature dependence. However, we note a few exceptional cases with positive temperature dependence. From the results of Cu-Si systems reported by Egry et al. [120], the temperature dependence of surface tension transforms from negative to positive values when the composition exceeds 30%. Similar observations are also obtained in Zr-based binary and ternary alloys. A possible explanation is the effect of local ordering structures such as clusters of intermetallic compound on the thermodynamic properties. The simulated evidence about the abnormality is not reported and is an interesting issue for further investigation. Tables 5, 6 summarize some measured and simulated results for pure metals and binary liquid alloys.

_{50}Al

_{50}alloy underestimates the available experimental values by 33–35% values at 1900–2000 K [137]; the calculated values of Ni-Cu alloys are 30% to 40% higher than the experimental data [55]. With the increase of Cu composition, the surface tension of Ni-Cu alloys monotonically decreases, which is identical to the experiments. The Ni-Cu alloy is a simple binary system and the monotonic relationship between the surface tension and the composition is in the expectation. For more complicated alloy systems that contain some intermetallic compounds in phase diagram, the composition dependence of the surface tension, or the influence of the local ordering clusters in liquids on the surface properties is an unnegligible factor.

_{A}and γ

_{B}are the surface tension of pure elements. C

_{A}

^{surf}and C

_{B}

^{surf}are the surface concentrations of each component, respectively, and are given by

_{A}and C

_{B}are the compositions of A and B elements, and A denotes the averaged molar surface area. In the Butler model, the surface tension of alloys is written as functions of surface and bulk partial Gibbs excess free energies, ΔG

^{S}and ΔG

^{B}

_{A B}

_{,}are the surface area occupied by A and B atoms respectively, C

_{A B}

^{Bulk}

_{,}are the bulk composition of species A and B. Comparatively, the prediction of Butler model shows closer to the experiments. For systems consisting of semiconductor elements, the modifications involving the bond energy of potential ordering clusters at surface are proved to effectively improve the predications [120]. The microstructural evolution of melts, in particular in undercooled state, significantly influences the thermodynamic appearance.

#### 3.5. Diffusion

#### 3.5.1. Self-diffusion Coefficient

_{0}is the pre-exponential factor and Q is the diffusion activation energy. The MD simulations reveal that this relation also perfectly operates in the undercooling region. Table 7 summarizes the Arrhenius fits to the simulated self-diffusion coefficients of liquid Cu, Co, Ni and Al using the EAM potential. These data are difficult to achieve in experiments. The available value of liquid copper at the melting temperature is about 0.379 × 10

^{−8}m

^{2}s

^{−1}[138], which is comparable to the simulated values.

_{2}[140] in the undercooled region do not obey the Arrhenius law. Instead, the Vogel-Fulcher-Tamman (VFT) equation, D = D

_{0}exp (− AT/

_{0}(T −T

_{0})), and a power law, D = D

_{0}(T/T

_{0}− 1)

^{α}, are used to describe the temperature dependence of dynamic behaviors of these special liquids [141]. Here A is the parameter controlling how closely the system obeys the Arrhenius law. The liquids whose dynamic properties show the non-Arrhenius law are called “fragile” liquids.

#### 3.5.2. Inter-diffusion Coefficient

_{60}Cu

_{40}and Ni

_{50}Cu

_{50}alloys. It is noted that the Arrhenius law well describes the temperature behavior of diffusion down to the moderate undercooling for most alloys. When the temperature approaches the glass transition, the Arrhenius law shows less effective. Horbach et al. [144] simulated the self-diffusion coefficients of Al and Ni in Al

_{80}Ni

_{20}melt as well as the Fickian diffusion coefficients, and the results indicated that they sharply decreased as the critical temperature of the mode-coupling theory was approached, and the temperature dependence was highly non-Arrhenius. Furthermore, the thermodynamic correction factor was found to strongly depend on temperature, in particular in the high undercooling region.

#### 3.6. Viscosity

_{0}is the pre-exponential factor, ΔE is the activity energy and R is the gas constant. Reference 154 lists the values of η

_{0}and ΔE for some typical pure metals. Alternatively, Cohen and Turnbull proposed the VFT relation based on the free volume model [155],

_{0}is the glass transition temperature, and A is a constant. The third relation for describing the temperature dependence of viscosity is the power law, which is developed from the mode coupling theory (MCT) [156],

_{2}[140], a dynamic transition from the “fragile” to the “strong” is predicted theoretically as a result of the liquid-liquid phase transition in those undercooled covalent liquids, which is first proposed for explaining the anomalous thermodynamic properties of undercooled water near 228 K.

_{A}and η

_{B}are viscosities of elements, x

_{A}and x

_{B}the mole fractions, Ω is the regular solution interaction parameter. The model seems to successfully predict the viscosity of some metal systems, whereas fails to give a prediction of low viscosity behavior at the eutectic compositions. The corrections for the atomic size are proven to be necessary.

## 4. Conclusions

## Acknowledgements

## References

- Yin, Z; Smith, FW. Free-energy model for bonding in amorphous covalent alloys. Phys. Rev B
**1991**, 43, 4507–4510. [Google Scholar] - Zhang, T; Inoue, A; Masumoto, T. Amorphous Zr-Al-TM (TM = Co, Ni, Cu) alloys with significant supercooled liquid region of over 100 K. Mater. Trans JIM
**1991**, 32, 1005–1010. [Google Scholar] - Takeuchi, A; Inoue, A. Calculations of mixing enthalpy and mismatch entropy for ternary amorphous alloys. Mater. Trans JIM
**2000**, 41, 1372–1378. [Google Scholar] - Johnson, WL. Bulk glass-forming metallic alloys: Science and technology. MRS Bulletion
**1999**, 24, 42–56. [Google Scholar] - Bormann, R. Thermodynamics of undercooled liquids and its application to amorphous phase formation. Mater. Sci. Eng A
**1994**, 178, 55–60. [Google Scholar] - Inoue, A. Stabilization of metallic supercooled liquid and bulk amorphous alloys. Acta Mater
**2000**, 48, 279–306. [Google Scholar] - Kurz, W; Fisher, DJ. Fundamentals of Solidification; Trans Tech Publications Ltd: Aedermannsdorf, Switzerland, 1986. [Google Scholar]
- Lipton, J; Kurz, W; Trivedi, R. Rapid dendrite growth in undercooled alloys. Acta Metall
**1987**, 35, 957–964. [Google Scholar] - Trivedi, R; Magnin, P; Kurz, W. Theory of Eutectic Growth Under Rapid Solidification Conditions. Acta Metal
**1987**, 35, 971–980. [Google Scholar] - Herlach, DM. Nonequilibrium solidification of undercooled metallic melts. Mater. Sci. Eng R
**1994**, 12, 177–272. [Google Scholar] - Thermophysical properties of liquids. Available on line: http://apps.isiknowledge.com/ (accessed on 29 November 2010).
- Egry, I; Lohoefer, G; Jacobs, G. Surface tension of liquid alloys: results from measurements on ground and in space. Phys. Rev. Lett
**1995**, 75, 4043–4046. [Google Scholar] - Egry, I; Jacobs, G; Schwartz, E; Szekely, J. Surface tension measurements of metallic melts under microgravity. Int. J. Thermophys
**1996**, 17, 1181–1189. [Google Scholar] - Aune, R; Battezzati, L; Brooks, R; Egry, I; Fecht, H-J; Garandet, J-P; Mills, KC; Passerone, A; Quested, PN; Ricci, E; et al. Measurement of thermophysical properties of liquid Metallic alloys in a ground- and microgravity based research programme—the ThermoLab Project. Microgravity Sci. Eng
**2005**, 16, 7–10. [Google Scholar] - Wunderlich, RK. Surface tension and viscosity of industrial Ti-alloys measured by the oscillating drop method on board parabolic flights. High Temp. Mat Process
**2008**, 27, 401–412. [Google Scholar] - Chathoth, SM; Damaschke, B; Samwer, K; Schneider, S. Thermophysical properties of Si, Ge, and Si-Ge alloy melts measured under microgravity. Appl. Phys. Lett
**2008**, 93, 071902. [Google Scholar] - Egry, I; Lohoefer, G; Sauerland, S. Measurements of thermophysical properties of liquid metals by noncontact techniques. Int. J. Thermophys
**1993**, 14, 573–584. [Google Scholar] - Egry, I; Lohoefer, G; Gorges, E; Jacobs, G. Structure and properties of undercooled liquid metals. J. Phys. Condens. Matter
**1996**, 8, 9363–9368. [Google Scholar] - Brillo, I; Egry, I. Surface tension of nickel, copper, iron and their binary alloys. J. Mater. Sci
**2005**, 40, 2213–2216. [Google Scholar] - Brillo, I; Egry, I. Density determination of liquid copper, nickel, and their Alloys. Int. J. Thermophys
**2003**, 24, 1155–1170. [Google Scholar] - Wang, N; Han, XJ; Wei, B. Specific heat and thermodynamic properties of undercooled liquid cobalt. Appl. Phys. Lett
**2002**, 80, 28–30. [Google Scholar] - Wang, N; Wei, B. Thermodynamic properties of highly undercooled liquid TiAl alloy. Appl. Phys. Lett
**2002**, 80, 3515–3517. [Google Scholar] - Han, XJ; Wang, N; Wei, B. Thermophysical properties of undercooled liquid cobalt. Philos. Mag. Lett
**2002**, 82, 451–459. [Google Scholar] - Han, XJ; Wei, B. Thermophysical properties of undercooled liquid Co-Mo alloys. Philos. Mag
**2003**, 83, 1511–1532. [Google Scholar] - Wang, HP; Cao, CD; Wei, B. Thermophysical properties of a highly superheated and undercooled Ni-Si alloy melt. Appl. Phys. Lett
**2004**, 84, 4062–4064. [Google Scholar] - Mathiak, G; Egry, I; Hennet, L; Thiaudiere, D; Pozdnyakova, I; Price, DL. Aerodynamic levitation and inductive heating—A new concept for structural investigations of undercooled melts. Int. J. Thermophys
**2005**, 26, 1151–1166. [Google Scholar] - Gruner, S; Marczinke, J; Hennet, L; Hoyer, W; Cuello, GJ. Neutron diffraction study on liquid Al-Ni alloys. Int. J. Mater. Res
**2010**, 101, 741–745. [Google Scholar] - Greaves, GN; Wilding, MC; Fearn, S; Langstaff, D; Kargl, F; Cox1, S; Vu Van, Q; Majérus, O; Benmore, CJ; Weber, R; Martin, CM; Hennet, L. Detection of first-order liquid/liquid phase transitions in yttrium oxide-aluminum oxide melts. Science
**2008**, 322, 566–570. [Google Scholar] - Glorieux, B; Millot, F; Rifflet, JC. Surface tension of liquid alumina from contactless techniques. Int. J. Thermophys
**2002**, 23, 1249–1257. [Google Scholar] - Wille, G; Millot, F; Rifflet, JC. Thermophysical properties of containerless liquid iron up to 2500 K. Int. J. Thermophys
**2002**, 23, 1197–1206. [Google Scholar] - Perez, M; Salvo, L; Suery, M; Brechet, Y; Papoular, M. Contactless viscosity measurement by oscillations of gas-levitated drops. Phys. Rev E
**2000**, 61, 2669–2675. [Google Scholar] - Okada, JT; Ishikawa, T; Watanabe, Y; Paradis, PF; Watanabe, Y; Kimura, K. Viscosity of liquid boron. Phys. Rev E
**2010**, 81, 140201. [Google Scholar] - Paradis, PF; Ishikawa, T; Yoda, S. Electrostatic levitation research and development at JAXA: Past and present activities in thermophysics. Int. J. Thermophys
**2005**, 26, 1031–1049. [Google Scholar] - Mukherjee, S; Zhou, ZH; Johnson, WL; Rhim, WK. Thermophysical properties of Ni-Nb and Ni-Nb-Sn bulk metallic glass-forming melts by containerless electrostatic levitation processing. J. Non-crystal Solids
**2004**, 337, 21–28. [Google Scholar] - Weber, JKR; Hampton, DS; Merkley, DR; Rey, CA; Zatarski, MM; Nordine, PC. Aeroacoustic levitation—A method for containerless liquid-phase processing at high temperatures. Rev. Sci. Instrum
**1994**, 65, 456–465. [Google Scholar] - Trinh, EH; Ohsaka, K. Measurement of density, sound-velocity, surface-tension, and viscosity of freely suspended supercooled liquids. Int. J. Thermophys
**1995**, 16, 545–555. [Google Scholar] - Herlach, DM; Cochrane, RF; Egry, I; Fecht, HJ; Greer, AL. Containerless processing in the study of metallic melts and their solidification. Int. Mater. Rev
**1993**, 38, 273–347. [Google Scholar] - Lü, YJ; Xie, WJ; Wei, B. Observation of ice nucleation in acoustically levitated water drops. Appl. Phys. Lett
**2005**, 87, 184107. [Google Scholar] - Han, XJ; Chen, M; Guo, ZY. A molecular dynamics study for the thermophysical properties of liquid Ti-Al alloys. Int. J Thermophysics
**2005**, 26, 869–880. [Google Scholar] - Han, XJ; Wang, JZ; Chen, M; Guo, ZY. Molecular dynamics simulation of thermophysical properties of undercooled liquid cobalt. J. Phys.: Condens Matter
**2004**, 16, 2565–2574. [Google Scholar] - Han, XJ; Chen, M; Guo, ZY. Thermophysical properties of undercooled liquid Au-Cu alloys from molecular dynamics simulations. J. Phys.: Condens Matter
**2004**, 16, 705–713. [Google Scholar] - Wang, HP; Wei, B. Thermophysical property of undercooled liquid binary alloy composed of metallic and semiconductor elements. J. Phys. D: Appl. Phys
**2009**, 42, 035414. [Google Scholar] - Wang, HP; Wei, B. Thermophysical properties and structure of stable and metastable liquid cobalt. Phys. Lett A
**2010**, 374, 1083–1087. [Google Scholar] - Wang, HP; Luo, BC; Wei, B. Molecular dynamics calculation of thermophysical properties for a highly reactive liquid. Phys. Rev E
**2008**, 78, 041204. [Google Scholar] - Wang, HP; Chang, J; Wei, B. Density and related thermophysical properties of metastable liquid Ni-Cu-Fe ternary alloys. Phys. Lett A
**2010**, 374, 2489–2493. [Google Scholar] - Yang, H; Lv, YJ; Chen, M; Guo, ZY. A molecular dynamics study on melting point and specific heat of Ni3Al alloy. Sci. China Ser. G-Phys. Mech. Astron
**2007**, 50, 407–413. [Google Scholar] - Yang, C; Chen, M; Guo, ZY. Molecular dynamics simulation of the specific heat of undercooled Fe-Ni melts. Int. J. Thermophys
**2001**, 22, 1303–1309. [Google Scholar] - Yao, WJ; Wang, N. Monte Carlo simulation of thermophysical properties of Ni-15%Mo alloy melt. Acta Physica Sinica
**2009**, 58, 4053–4058. [Google Scholar] - Yang, C; Chen, M; Guo, ZY. Molecular dynamics simulations on specific heat capacity and glass transition temperature of liquid silver. Chin. Sci. Bull
**2001**, 46, 1051–1053. [Google Scholar] - Chen, L; Wang, HP; Wei, B. Measurement and calculation of specific heat for a liquid Ni-Cu-Fe ternary alloy. Acta Physica Sinica
**2009**, 58, 384–389. [Google Scholar] - Wang, HP; Wei, B. Thermophysical properties of stable and metastable liquid copper and nickel by molecular dynamics simulation. Appl. Phys A
**2009**, 95, 661–665. [Google Scholar] - Wang, HP; Wei, B. Theoretical prediction and experimental evidence for thermodynamic properties of metastable liquid Fe-Cu-Mo ternary alloys. Appl. Phys. Lett
**2008**, 93, 171904. [Google Scholar] - Wang, HP; Wei, B. Experimental determination and molecular dynamics simulation of specific heat for high temperature undercooled liquid. Philos. Mag. Lett
**2008**, 88, 813–819. [Google Scholar] - Chen, M; Yang, C; Guo, ZY. A Monte Carlo simulation on surface tension of liquid nickel. Mater. Sci. Eng A
**2000**, 292, 203–206. [Google Scholar] - Chen, M; Yang, C; Guo, ZY. Surface Tension of Ni-Cu Alloys: A Molecular Simulation Approach. Int. J. Thermophys
**2001**, 22, 1295–1302. [Google Scholar] - Yao, WJ; Han, XJ; Chen, M; Wei, B; Guo, ZY. Surface tension of undercooled liquid cobalt. J. Phys. Condens Matter
**2002**, 14, 7479–7485. [Google Scholar] - Wang, HP; Chang, J; Wei, B. Measurement and calculation of surface tension for undercooled liquid nickel and its alloy. J. Appl. Phys
**2009**, 106, 033506. [Google Scholar] - Cheng, H; Lv, YJ; Chen, M. Interdiffusion in liquid Al-Cu and Ni-Cu alloys. J. Chem. Phys
**2009**, 131, 044502. [Google Scholar] - Han, XJ; Chen, M; Lv, YJ. Transport properties of undercooled liquid copper: A molecular dynamics study. Int. J. Thermophys
**2008**, 29, 1408–1421. [Google Scholar] - Mei, J; Davenport, JW. Molecular-dynamics study of self-diffusion in liquid transition metals. Phys. Rev B
**1990**, 42, 9682–9684. [Google Scholar] - Chen, FF; Zhang, HF; Qin, FX; Hu, ZQ. Molecular dynamics study of atomic transport properties in rapidly cooling liquid copper. J. Chem. Phys
**2004**, 120, 1826–1831. [Google Scholar] - Asta, M; Morgan, D; Hoyt, JJ; Sadigh, B; Althoff, JD; de Fontaine, D. Embedded-atommethod study of structural, thermodynamic, and atomic-transport properties of liquid Ni-Al alloys. Phys. Rev B
**1999**, 59, 14271–14281. [Google Scholar] - Horbach, J; Das, SK; Griesche, A; Macht, M-P; Frohberg, G; Meyer, A. Self-diffusion and interdiffusion in Al80Ni20 melts: Simulation and experiment. Phys. Rev B
**2007**, 75, 174304. [Google Scholar] - Kart, SO; Tomak, M; Uludogan, M; Cagin, T. Structural, thermodynamical, and transport properties of undercooled binary Pd-Ni alloys. Mater. Sci. Eng A
**2006**, 435, 736–744. [Google Scholar] - Mukherjee, A; Bhattacharyya, S; Bagchi, B. Pressure and temperature dependence of viscosity and diffusion coefficients of a glassy binary mixture. J. Chem. Phys
**2002**, 116, 4577–4586. [Google Scholar] - Juan-Coloa, F; Osorio-Gonzalez, D; Rosendo-Francisco, P; Lopez-Lemus, J. Structural and dynamic properties of liquid alkali metals: molecular dynamics. Mol. Simul
**2007**, 33, 1167–1172. [Google Scholar] - Jakse, N; Pasturel, A. Local order and dynamic properties of liquid and undercooled Cu
_{x}Zr_{1−}_{x}alloys by ab initio molecular dynamics. Phys. Rev B**2008**, 78, 214204. [Google Scholar] - Alfè, D; Gillan, MJ. First-Principles calculation of transport coefficients. Phys. Rev. Lett
**1998**, 81, 5161–5164. [Google Scholar] - Raty, JY; Godlevsky, VV; Gaspard, JP; Bichara, C; Bionducci, M; Bellissent, R; Ceolin, R; Chelikowsky, JR; Ghosez, P. Distance correlations and dynamics of liquid GeSe: An ab initio molecular dynamics study. Phys. Rev B
**2001**, 64, 235209. [Google Scholar] - Jakse, N; Wax, JF; Pasturel, A. Transport properties of liquid nickel near the melting point: An ab initio molecular dynamics study. J. Chem. Phys
**2007**, 126, 234508. [Google Scholar] - Daw, MS; Baskes, MI. Semiempirical, quantum mechanical calculation of hydrogen embrittlement in metals. Phys. Rev. Lett
**1983**, 50, 1285–1288. [Google Scholar] - Daw, MS; Baskes, MI. Embedded-atom method: Derivation and application to impurities, surfaces and other defects in metals. Phys. Rev B
**1984**, 29, 6443–6453. [Google Scholar] - Foiles, SM. Application of the embedded-atom method liquid transition-metals. Phys. Rev B
**1985**, 32, 3409–3415. [Google Scholar] - Johnson, RA. Analytic nearest-neighbor model for fcc metals. Phys. Rev B
**1988**, 37, 3924–3931. [Google Scholar] - Mei, J; Davenport, JW; Fernando, GW. Analytic embedded-atom potentials for fcc metals: Application to liquid and solid copper. Phys. Rev B
**1991**, 43, 4653–4658. [Google Scholar] - Cai, J; Ye, YY. Simple analytical embedded-atom-potential model including a long-range force for fcc metals and their alloys. Phys. Rev B
**1996**, 54, 8398–8410. [Google Scholar] - Erhart, P; Albe, K. Analytical potential for atomistic simulations of silicon, carbon, and silicon carbide. Phys. Rev B
**2005**, 71, 035211. [Google Scholar] - Baskes, MI. Modified embedded-atom potentials for cubic materials and impurities. Phys. Rev B
**1992**, 46, 2727–2742. [Google Scholar] - Baskes, MI; Nelson, JS; Wright, AF. Semiempirical modified embedded-atom potentials for silicon and germanium. Phys Rev B
**1989**, 40, 6085–6100. [Google Scholar] - Lee, B; Baskes, MI. Second nearest-neighbor modified embedded-atom-method potential. Phys. Rev B
**2000**, 62, 8564–8567. [Google Scholar] - Lee, B; Baskes, MI; Kim, H; Cho, YK. Second nearest-neighbor modified embedded atom method potentials for bcc transition metals. Phys. Rev B
**2001**, 64, 184102. [Google Scholar] - Ravelo, R; Baskes, M. Equilibrium and Thermodynamic Properties of Grey, White, and Liquid Tin. Phys. Rev. Lett
**1997**, 79, 2482–2485. [Google Scholar] - Cherne, FJ; Baskes, MI; Schwarz, RB; Srinivasan, SG; Klein, W. Non-classical nucleation in supercooled nickel. Model. Simul. Mater. Sci. Eng
**2004**, 12, 1063–1068. [Google Scholar] - Toxvaerd, S; Prastgaard, E. Molecular dynamics calculation of the liquid structure up to a solid surface. J. Chem. Phys
**1977**, 67, 5291–5295. [Google Scholar] - Hiwatari, Y; Stoll, E; Schneider, T. Molecular-dynamics investigation of solid-liquid coexistence. J. Chem. Phys
**1978**, 68, 3401–3404. [Google Scholar] - Mori, A; Manabe, R; Nishioka, K. Construction and investigation of a hard-sphere crystal-melt interface by a molecular dynamics simulation. Phys. Rev E
**1995**, 51, R3831–R3833. [Google Scholar] - Morris, JR; Song, X. The melting lines of model systems calculated from coexistence simulations. J. Chem. Phys
**2002**, 116, 9352–9358. [Google Scholar] - Yoo, S; Zeng, XC; Morris, JR. The melting lines of model silicon calculated from coexisting solid-liquid phases. J. Chem. Phys
**2004**, 120, 1654–1656. [Google Scholar] - Allen, MP; Tildesley, DJ. Computer Simulation of Liquids; Clarendon Press: Oxford, UK, 1989. [Google Scholar]
- Nijmeijer, MJP; Bakker, AF; Bruin, C. A molecular dynamics simulation of the Lennard-Jones liquid-vapor interface. J. Chem. Phys
**1988**, 89, 3789–3792. [Google Scholar] - Miyazaki, J; Barker, JA; Pound, GM. A new Monte Carlo method for calculating surface tension. J. Chem. Phys
**1976**, 64, 3364–3369. [Google Scholar] - Padday, JF; Uffindell, ND. Calculation of cohesive and adhesive energies from intermolecular forces at a surface. J. Phys. Chem
**1968**, 72, 1407–1414. [Google Scholar] - Lu, ZR. Monte Carlo calculation of surface tension of liquids. J. China Textile Univ
**1991**, 3, 88–92. [Google Scholar] - Erpenbeck, JJ; Wood, WW. Self-diffusion coefficient for the hard-sphere fluid. Phys. Rev A
**1991**, 43, 4254–4261. [Google Scholar] - Duncan, JB; Toor, HL. Experimental study of three component gas diffusion. AIChE J
**1962**, 8, 38–41. [Google Scholar] - Krishna, R; Wesselingh, JA. The Maxwell-Stefan approach to mass transfer. Chem. Eng. Sci
**1997**, 52, 861–911. [Google Scholar] - Krishna, R; van den Broeke, LJP. The Maxwell-Stefan description of mass transport across zeolite membranes. Chem. Eng. J
**1995**, 57, 155–162. [Google Scholar] - Taylor, R; Krishna, R. Multicomponent Mass Transfer; John Wiley: NewYork, NY, USA, 1993. [Google Scholar]
- Krishna, R; van Baten, JM. Diffusion of Alkane Mixtures in Zeolites: Validating the Maxwell—Stefan formulation using MD simulations. J. Phys. Chem B
**2005**, 109, 6386–6396. [Google Scholar] - Darken, LS. Diffusion, Mobility and Their interrelation through free energy in binary metallic systems. Trans. Inst. Min. Metall. Eng
**1948**, 175, 184–201. [Google Scholar] - Evans, DJ; Morriss, GP. Statistical Mechanics of Non-Equilibrium Liquids; Academic Press: London, UK, 1990. [Google Scholar]
- Rapaport, DC. The Art of Molecular Dynamics Simulation; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Müller-Plathe, F. A simple nonequilibrium molecular dynamics method for calculating the thermal conductivity. J. Chem. Phys
**1997**, 106, 6082–6085. [Google Scholar] - Müller-Plathe, F. Reversing the perturbation in nonequilibrium molecular dynamics: An easy way to calculate the shear viscosity of fluids. Phys. Rev E
**1999**, 59, 4894–4898. [Google Scholar] - Bordat, P; Müller-Plathe, F. The shear viscosity of molecular fluids: A calculation by reverse nonequilibrium molecular dynamics. J. Chem. Phys
**2002**, 116, 3362–3369. [Google Scholar] - Kerrache, A; Horbach, J; Binder, K. Molecular-dynamics computer simulation of crystal growth and melting in Al50Ni50. Eur. Phys. Lett
**2008**, 81, 58001. [Google Scholar] - Feenken, JWM; van der Veen, JF. Observation of surface melting. Phys. Rev. Lett
**1985**, 54, 134–137. [Google Scholar] - Alfè, D. First-principles simulations of direct coexistence of solid and liquid aluminum. Phys. Rev B
**2003**, 68, 064423. [Google Scholar] - Han, XJ. Molecular dynamics study on thermophysical properties and structure of undercooled metallic melts. In Post-doctoral Report; Tsinghua University: Beijing, China, 2004. [Google Scholar]
- Iida, T; Guthrie, RIL. The Physical Properties of Liquid Metals Oxford; Oxford Science Publications: Oxford; UK, 1993. [Google Scholar]
- Smithell, CJ. Smithells Metals Reference Book; Brandes, EA, Ed.; Butterworths: London, UK, 1983. [Google Scholar]
- Yang, C; Chen, M; Guo, Z. Molecular dynamics simulations on specific heat capacity and glass transition temperature of liquid silver. Chin. Sci Bulletin
**2001**, 46, 1051–1053. [Google Scholar] - Bykov, A; Pastukhov, E. A calorimetric study of alloy formation in Au-Cu and An-In systems. J. Therm. Anal. Calorim
**2000**, 60, 845–850. [Google Scholar] - Luo, BC; Wang, HP; Wei, B. Specific heat, enthalpy, and density of undercooled liquid Fe-Si-Sn alloy. Philos. Mag. Lett
**2009**, 89, 527–533. [Google Scholar] - Brillo, J; Egry, I; Giffard, HS; Patti, A. Density and thermal expansion of liquid Au-Cu alloys. Int. J. Thermophys
**2004**, 25, 1881–1888. [Google Scholar] - Brillo, J; Egry, I; Ho, I. Density and thermal expansion of liquid Ag-Cu and Ag-Au Alloys. Int. J. Thermophys
**2006**, 27, 494–506. [Google Scholar] - Adachi, M; Schick, M; Brillo, J; Egry, I; Watanabe, M. Surface tension and density measurement of liquid Si-Cu binary alloys. J. Mater. Sci
**2010**, 45, 2002–2008. [Google Scholar] - Egry, I; Brillo, J. Surface tension and density of liquid metallic alloys measured by electromagnetic levitation. J. Chem. Eng Data
**2009**, 54, 2347–2352. [Google Scholar] - Yao, WJ; Wang, N. Monte Carlo simulation of thermophysical properties for Al-Ce Liquid Alloys. Mater Sci Forum
**2010**. [Google Scholar] - Schmitz, J; Brillo, J; Egry, I. Surface tension of liquid Cu and anisotropy of its wetting of sapphire. J. Mater. Sci
**2010**, 45, 2144–2149. [Google Scholar] - Egry, I; Schneider, S; Seyhan, I; Volkmann, T. Surface tension measurements of high temperature metallic melts. Trans JWRI
**2001**, 30, 195–200. [Google Scholar] - Sarou-Kanian, V; Millot, F; Rifflet, JC. Surface tension and density of oxygen-free liquid aluminum at high temperature. Int. J. Thermophys
**2003**, 24, 227–286. [Google Scholar] - Egry, I; Sauerland, S; Jacobs, G. Surface tension measurements on levitated liquid noble metals High Temp. High Press
**1994**, 26, 217–223. [Google Scholar] - Keene, BJ. Review of data for the surface-tension of pure metals. Int Mater Rev
**1993**, 38, 157–192. [Google Scholar] - Eichel, RA; Egry, I. Surface tension and surface segregation of liquid cobalt-iron and cobalt-copper alloys. Z. Metallkd
**1999**, 90, 371–375. [Google Scholar] - Przyborowski, M; Hibiya, T; Eguchi, M; Egry, I. Surface tension measurement of molted silicon by the oscillation drop method using electromagnetic levitation. J. Cryst Growth
**1995**, 151, 60–65. [Google Scholar] - Fujii, H; Matsumoto, T; Hata, N; Nakano, T; Kohno, M; Nogi, K. Surface tension of molten silicon measured by the electromagnetic levitation method under microgravity. Metall. Mater. Trans A
**2000**, 31, 1585–1589. [Google Scholar] - Wang, HP; Wei, BB. Surface tension and specific heat of liquid Ni70.2Si29.8 alloy. Chin. Sci Bulletin
**2005**, 50, 945–949. [Google Scholar] - Zhou, K; Wang, HP; Chang, J; Wei, B. Surface tension of substantially undercooled liquid Ti-Al alloy. Philos. Mag. Lett
**2010**, 90, 455–462. [Google Scholar] - Han, XJ; Wei, B. Thermophysical properties of undercooled liquid Co-Mo alloys. Philos. Mag
**2003**, 83, 1511–1532. [Google Scholar] - Wang, HP; Yao, WJ; Cao, CD; Wei, B. Surface tension of superheated and undercooled liquid Co-Si alloy. Appl. Phys. Lett
**2004**, 85, 3414–3416. [Google Scholar] - Lohoefer, G; Schneider, S; Egry, I. Thermophysical properties of undercooled liquid Co80Pd20. Int. J. Thermophys
**2001**, 22, 593–604. [Google Scholar] - Brillo, J; Egry, I. Density and surface tension of electromagnetically levitated Cu-Co-Fe alloys. Int. J. Thermophys
**2007**, 28, 1004–1016. [Google Scholar] - Seyhan, I; Egry, I. The Surface tension of undercooled binary iron and nickel alloys and the effect of oxygen on the surface tension of Fe and Ni. Int. J. Thermophys
**1999**, 20, 1017–1028. [Google Scholar] - Egry, I; Lohoefer, G; Jacobs, G. Surface tension of liquid metals: Results from measurements on ground and in space. Phys. Rev. Lett
**1995**, 75, 4043–4046. [Google Scholar] - Lohoefer, G; Brillo, J; Egry, I. Thermophysical properties of undercooled liquid Cu-Ni alloys. Int. J. Thermophys
**2004**, 25, 1535–1550. [Google Scholar] - Levchenko, EV; Evteev, AV; Beck, DR; Belova, IV; Murch, GE. Molecular dynamics simulation of the thermophysical properties of an undercooled liquid Ni50Al50 alloy. Comput. Mater. Sci
**2010**, 50, 465–473. [Google Scholar] - Cheng, H. Molecular Dynamics Simulations on the Thermodynamic and Transport Properties of Liquid Al-Cu and Cu-Ni Alloys. Master-degree Thesis, Tsinghua University, Beijing, China, 2008. [Google Scholar]
- Sastry, S; Angell, CA. Liquid-liquidphase transition in supercooled silicon. Nature Mater
**2003**, 2, 739–743. [Google Scholar] - Roberts, CJ; Panagiotopoulos, AZ; Debenedetti, PG. Liquid-liquid immiscibility in pure fluids: polyamorphism in simulations of a network-forming fluid. Phys. Rev. Lett
**1996**, 77, 4386–4389. [Google Scholar] - Angell, CA; Nhell, PH; Shao, J. Glass-forming liquids, anomalous liquids, and polyamorphism in liquids and biopolymers. Il Nuovo Cimento D
**1994**, 16, 993–1025. [Google Scholar] - Pasturel, A; Tasci, ES; Sluiter, MHF; Jakse, N. Structural and dynamic evolution in liquid Au-Si eutectic alloy by ab initio molecular dynamics. Phys. Rev B
**2010**, 81, 140202. [Google Scholar] - Asta, M; Morgan, D; Hoyt, JJ; Sadigh, B; Althoff, JD; de Fontaine, D; Foiles, SM. Embedded-atom-method study of structural, thermodynamic, and atomic-transport properties of liquid Ni-Al alloys. Phys. Rev B
**1999**, 59, 14271–14281. [Google Scholar] - Horbach, J; Das, SK; Griesche, A; Macht, M-P; Frohberg, G; Meyer, A. Self-diffusion and interdiffusion in Al80Ni20 melts: Simulation and experiment. Phys. Rev B
**2007**, 75, 174304. [Google Scholar] - Klassen, M; Cahoon, JR. Interdiffusion of Sn and Pb in liquid Pb-Sn alloys. Metall. Mater. Trans A
**2000**, 31, 1343–1352. [Google Scholar] - Cahoon, J; Jiao, Y; Tandon, K; Chaturvedi, M. Interdiffusion in liquid Tin. J. Phase Equilib. Diffus
**2006**, 27, 325–332. [Google Scholar] - Lee, JH; Liu, S; Miyahara, H; Trivedi, R. Diffusion-coefficient measurements in liquid metallic alloys. Metall. Mater. Trans B
**2004**, 35, 909–917. [Google Scholar] - Tanaka, Y; Kajihara, M. Evaluation of interdiffusion in liquid phase during reactive diffusion between Cu and Al. Mater. Trans
**2006**, 47, 2480–2488. [Google Scholar] - Watson, MP; Hunt, JD. The Measurement of liquid diffusion coefficients in the Al-Cu system using temperature gradient zone melting. Metall. Trans A
**1977**, 8, 1793–1798. [Google Scholar] - Holian, BL; Evans, DJ. Shear viscosities away from the melting line—A comparison of equilibrium and non-equilibrium molecular-dynamics. J. Chem. Phys
**1983**, 78, 5147–5150. [Google Scholar] - Cherne, FJ; Deymier, PA. Calculation of the transport properties of liquid aluminum with equilibrium and non-equilibrium molecular dynamics. Scr. Mater
**2001**, 45, 985–991. [Google Scholar] - Cherne, FJ; Deymier, PA. Calculation of viscosity of liquid nickel by molecular dynamics methods. Scr. Mater
**1998**, 39, 1613–1616. [Google Scholar] - Koishi, T; Shirakawa, Y; Tamaki, S. Shear viscosity of liquid metals obtained by non-equilibrium molecular dynamics. J. Non-crystal Solids
**1996**, 205, 383–387. [Google Scholar] - Dinsdale, AT; Quested, PN. The viscosity of aluminium and its alloys—A review of data and models. J. Mater. Sci
**2004**, 39, 7221–7228. [Google Scholar] - Cohen, MH; Turnbull, D. Molecular transport in liquids and glasses. J. Chem. Phys
**1959**, 31, 1164–1169. [Google Scholar] - Bengtzelius, U; Goetze, W; Sjoelander, A. Dynamics of supercooled liquids and the glass transition. J. Phys C
**1984**, 17, 5915–5934. [Google Scholar] - Lazarev, NP; Bakai, AS; Abromeit, C. Molecular dynamics simulation of viscosity in supercooled liquid and glassy AgCu alloy. J. Non-Crystal Solids
**2007**, 353, 3332–3337. [Google Scholar] - Moelwyn-Hughes, EA. Physical Chemistry; Pergamon Press: Oxford, UK, 1961. [Google Scholar]

**Figure 3.**A rectangular simulated cell with a liquid film in the middle and vapor on each side of the cell.

**Figure 10.**Simulated mixing enthalpy H

_{mix}and H

_{mix}/H

_{0}of liquid Ni-Si alloys. H

_{0}is the enthalpy of the ideal mixture [42].

**Figure 12.**M-S diffusion coefficients as a function of concentration. (

**a**) Al-Cu alloy; (

**b**) Ni-Cu alloy.

${T}_{m}^{Calc.}$ (K) | ${T}_{m}^{Exp.}$ (K) | Deviation | |
---|---|---|---|

Cu [41] | 1,320.5 ± 1.5 | 1,356 | −2.62% |

Au [41] | 1,182.5 ± 1.5 | 1,336 | −11.49% |

AuCu_{3} [41] | 1,240.5 ± 1.5 | 1,250 | −0.76% |

AuCu [41] | 1,173.5 ± 0.5 | 1,185 | −0.97% |

Au_{3}Cu [41] | 1,151.5 ± 1.5 | 1,220 | −5.61% |

Ni_{3}Al [46] | 1,705 | 1,663 | 2.5% |

1,725 | 3.8% | ||

Al_{50}Ni_{50} [106] | 1,520 | 1,920 | −20% |

Enthalpy H (J mol^{−1}) | |
---|---|

Co | H = −4.09 ×10^{5} + (32.51 ± 0.19) ·T [40] |

H = −4.21×10^{5} + (38.60 ± 0.08) ·T [40] | |

Cu | H = −3.45 ×10^{5} + (33.68 ± 0.19) ·T [41] |

Au | H = −3.81×10^{5} + (34.30 ± 0.38) ·T [41] |

Ti | H = − 4.661×10^{5} + 39.193 · T − 2.430 × 10^{−3} · T^{2} [39] |

Al | H = −3.197 × 10^{5} + 35.303 × 10^{−4} T − 2.68 × 10^{−3} · T^{2} · [39] |

Simulated Values (J mol^{−1} K^{−1}) | Experimental Values (J mol^{−1} K^{−1}) | |
---|---|---|

Co | 32.509 ± 0.194 [40] | 40.38 T > 1768 K [110] |

38.595 ± 0.084 [40] | 40.6 1541 K < T < 1768 K [21] | |

Cu | 33.68 ± 0.19 [41] | 31.5 1356 K < T < 1873 K [111] |

Au | 34.30 ± 0.38 [41] | 29.3 1336 K < T < 1673 K [111] |

Ti | C_{P} = 39.1991–4.8650 × 10^{−3} · T [39] | 33.53 T = 1933 K [111] |

Al | C_{P} = 35.3091–5.3544 × 10^{−3} · T [39] | 31.8 T = 933 K [111] |

Ag | C_{P} = 30.5357–4.8555 × 10^{−3} · T [112] | 30.5 1233.7 K < T < 1573 K [111] |

Simulated values | Experimental values | |||
---|---|---|---|---|

ρ_{m} (×10^{3} kg/m^{3}) | ∂ρ/∂T (×10^{−1} kg m^{−3}K^{−1}) | ρ_{m} (×10^{3} kg/m^{3}) | ∂ρ/∂T (×10^{−1} kg m^{−3}K^{−1}) | |

Co | 7.74 | −7.70 [40] | 7.76 | −10.9 [111] |

7.49 | −9.17 [40] | |||

Cu | 7.868 | −8.91 [41] | 8.00 | −8.0 [111] |

Au | 16.85 | −24.15 [41] | 17.36 | −15 [111] |

Ti | 4.55 | −2.16 [39] | 4.13 | −2.3 [111] |

Al | 2.55 | −1.18 [39] | 2.385 | −3.5 [111] |

Cu_{75}Au_{25} | 10.98 | −12.39 [41] | 11.39 | −19.5 [115] |

Ag_{80}Cu_{20} | - | - | 9.0 | −6 [116] |

Ni_{11}Cu_{89} | - | - | 7.97 | −7.95 [20] |

Si_{10}Cu_{90} | - | - | 7.591 | −6.308 [117] |

Element | T_{m}(K) | γ_{0} (N·m^{−1}) | dγ/dT (×10^{−4} N·m^{−1}·K^{−1}) | Ref. |
---|---|---|---|---|

Cu | 1357 | 1.30 | −2.64 | Egry et al. [117] |

1.29 | −2.34 | Egry et al. [19] | ||

Ni | 1726 | 1.764 | −3.30 | Wei et al. [57] |

1.770 | −3.30 | Egry et al. [19] | ||

Al | 933 | 0.88 | −2.0 | Egry et al. [121] |

1.024 ± 0.048 | −2.74 ± 0.25 | Sarou-Kanian et al. [122] | ||

Fe | 1811 | 1.92 | −3.97 | Egry et al. [19] |

1.888±0.031 | −2.85 ± 0.15 | Wille et al. [30] | ||

Ag | 1235 | 0.91 | −1.8 | Egry et al. [123] |

Au | 1337 | 1.149 | −1.4 | Egry et al. [13] |

1.875 | −3.48 | Wei et al. [23] | ||

Co | 1773 | 1.881 | −3.4 | Keene [124] |

1.887 | −3.3 | Egry et al. [125] | ||

Si | 1683 | 0.784 | −6.5 | Przyborowski et al. [126] |

0.73 | −6.2 | Fujii et al. [127] |

Alloys (at%) | T_{L} (K) | γ_{0} (N·m^{−1}) | dγ/dT (×10^{−4} N·m^{−1}·K^{−1}) | Ref. |
---|---|---|---|---|

Ni_{90.1}Si_{9.9} | 1623 | 1.697 | −3.97 | Wei et al. [25,57,128] |

Ni_{70.2}Si_{29.8} | 1488 | 1.693 | −4.23 | |

Ti_{49}Al_{51} | 1753 | 1.094 | −1.422 | Wei et al. [129] |

Co_{93}Mo_{7} | 1744 | 1.895 | −3.1 | Wei et al. [130] |

Co_{75}Si_{25} | 1607 | 1.604 | −4 | Wei et al. [131] |

Co_{80}Pd_{20} | 1613 | 1.687 | −1.5 | Egry et al. [132] |

Zr_{64}Ni_{36} | 1283 | 1.54 | 1.07 | Egry et al. [18] |

Cu_{70}Co_{30} | 1638 | 1.22 | −2.9 | Egry et al. [125] |

Co_{50}Fe_{50} | 1752 | 1.8 | −3.72 | Egry et al. [133] |

Cu_{90}Si_{10} | 1246 | 1.332 | −2.686 | Egry et al. [120] |

Fe_{80}Ni_{20} | 1748 | 2.056 | −0.15 | Egry et al. [134] |

Au_{56}Cu_{44} | 1183 | 1.21 | −0.15 | Egry et al. [135] |

Cu_{90}Ni_{10} | 1409 | 1.31 | −2.21 | Egry et al. [136] |

**Table 7.**Arrhenius fits to self-diffusion and inter-diffusion coefficients of liquid metals and alloys.

Simulations | ||
---|---|---|

D_{0} (m^{2}s^{−1}) | Q (kJ·mol^{−1}) | |

Cu | 6.69 × 10^{−8} | 34.2 [134] |

7.38 × 10^{−8} | 42.0 [59] | |

Ni | 9.69 × 10^{−8} | 47.1 [134] |

Co | 1.29 × 10^{−7} | 48.8 [40] |

Al | 1.78 × 10^{−7} | 25.8 [134] |

Ni(Ni_{50}Al_{50}) | 1.01 × 10^{−7} | 45.3 [137] |

Al(Ni_{50}Al_{50}) | 0.94 × 10^{−7} | 46.3 [137] |

Al_{60}Cu_{40} | 9.16 × 10^{−8} | 17.0 [58] |

Ni_{50}Cu_{50} | 8.03 × 10^{−8} | 48.9 [58] |

Methods | η _{0} (mPa·s) | E_{a}/k_{b} (K) |
---|---|---|

NEMD | 4.09301 | 2591.23024 |

RNEMD | 3.94354 | 2624.62262 |

EMD | 3.88996 | 2570.48062 |

Experimental | 4.35932 | 2867.86973 |

© 2011 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Lv, Y.J.; Chen, M.
Thermophysical Properties of Undercooled Alloys: An Overview of the Molecular Simulation Approaches. *Int. J. Mol. Sci.* **2011**, *12*, 278-316.
https://doi.org/10.3390/ijms12010278

**AMA Style**

Lv YJ, Chen M.
Thermophysical Properties of Undercooled Alloys: An Overview of the Molecular Simulation Approaches. *International Journal of Molecular Sciences*. 2011; 12(1):278-316.
https://doi.org/10.3390/ijms12010278

**Chicago/Turabian Style**

Lv, Yong J., and Min Chen.
2011. "Thermophysical Properties of Undercooled Alloys: An Overview of the Molecular Simulation Approaches" *International Journal of Molecular Sciences* 12, no. 1: 278-316.
https://doi.org/10.3390/ijms12010278