The melting temperature is thermodynamically defined as the temperature at which the Gibbs free energies of liquid and solid are identical,

Due to the influence of the interfacial energy, the values calculated by directly heating the simulation system with the periodical boundary conditions are usually larger than the experimental results. Therefore, the liquid/solid interface should first be constructed properly in simulations.

There are three methods to simulate the melting point. The first one is the sandwich method [84–86]. Two solid and one liquid subsystems are first equilibrated at different temperatures. Then the two solid systems sandwich the liquid layer and construct a solid-liquid-solid interlayer structure along the Z direction, as shown in Figure 1. The periodical boundary conditions are applied in the X and Y coordinate directions. In simulations, the whole system is annealed at a certain temperature, and the structural evolution of the interlayer is monitored during this process. If the sandwich structure completely melts into the liquid state, the current temperature is higher than the melting point, and contrariwise lower than the melting point. If the liquid/solid structure can stably exist, the temperature is approximated to the melting point. For example, Figure 2 shows the internal energy of a sandwich AuCu₃ 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.

The second method is the moving interface method [41]. At first, a liquid/solid interface is constructed at a desired temperature. After beginning the simulations, the moving velocity of interface is measured at the current temperature and then the temperature is changed with the interval ΔT.

The solid and dashed lines describe the cooling and heating from initial liquid and solid systems. The circle symbols predict the melting point with the sandwich method to repeat the measurement. Given enough measurements, we can plot a velocity-temperature curve, and extrapolate the moving velocity to zero, so the corresponding temperature is approximately regarded as the melting point. Similar to the sandwich method, this method requires many trials at different temperatures. The third one is the NVE ensemble method [87,88]. Similar to the second method, the liquid/solid double-layer structure is first set up near the melting point. Then, the structural relaxation of the system is performed in a constant volume, particle number, and energy (NVE) ensemble. If the system is initially assigned to a higher temperature than the melting point, the solid phase will partially melt. This process consumes the latent heat, and converts some of the kinetic energy into potential energy, resulting in the decrease in temperature. When the system is finally stabilized, the corresponding temperature is approximated to the melting point. Similarly, if the initial temperature is lower than the melting point, it will evolve toward the melting temperature from below. In contrast to the former two methods, the NVE ensemble method can determine the melting temperature in a single simulation. However, it is noted that the initial temperature should be set as near the melting temperature as possible. In fact, the NVE ensemble method integrated with the moving interface method is more advisable for calculating melting temperature. A rough estimate is performed using the moving interface method, and then a fine calculation is undertaken by the NVE ensemble method.

Specific heat is a fundamental parameter in thermophysics, and plays an important role in understanding the thermodynamic and dynamic behaviors of liquids. It is also widely studied by MD simulations, particularly for undercooled liquids. The simulated systems involve the pure metals, binary and even ternary alloys.

In general, there are two methods used in calculating the specific heat. The first one is the statistical average of the energy fluctuation [89]. The expression of specific heat depends on the ensemble adopted in the simulations. In the canonical ensemble (constant particle number, volume and temperature), the isovolumetric specific heat is defined as

where 〈δE²〉 = 〈E²〉 − 〈E²〉, N_a is the particle number of systems. In the micocanonical ensemble (constant particle number, volume and energy), 〈δE²〉 = 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

This method requires a large-size simulation system and relatively simple interactions, such as gas, so its application is relatively limited.

The second method is the derivative of the thermodynamic function. The isobaric specific heat is written as the differential of enthalpy,

where enthalpy H(T ) = E + PV is readily calculated in MD simulations. This method is more popular in calculating the specific heat, and the estimated results are also in good agreement with experimental measurements.

The density is calculated by the ratio of mass to volume in MD simulations. In the constant particle number, pressure and temperature (NPT) ensemble, the volume of system is sampled directly at a constant temperature. Its agreement with the experimental measurements depends on the potential function. This conclusion is driven for not only density but most thermophysical properties. Density is closely related to many thermophysical properties, and experimental measurements about it are intensively carried out. Therefore, density is considered to be a good candidate for the test of potentials.

There are three methods to produce the surface tension of fluids in MD simulations. The first one is to construct a liquid-vapor surface in the simulations and directly calculate the surface tension from the mechanical expression. The surface tension can be expressed as an integral of the difference between the normal and tangential pressure tensor components over the surface

where z_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

If the film is divided into N slabs with the length of L_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

where n(k ) is the number density of the kth slab, V = L_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],

In the MD simulations, the liquid-vapor surface is constructed as follows: A NPT system with the periodical boundary conditions along the three coordination directions is set up first and equilibrated, then the periodical boundary condition in the Z direction is withdrawn and the simulation box is prolonged in the Z direction, forming two vacuum regions sandwiching the liquid phase. After equilibrating in the NVT ensemble, two surfaces form. Figure 3 shows a schematic diagram of the simulation region.

The second method is derived from the thermodynamic expression of surface tension, namely, the additional free energy required by forming a new surface. One of these operations was carried out by Miyazaki et al. [91], in which the free energy for reversibly creating a surface in the bulk liquid was calculated using the MC simulation. Generally speaking, the first approach suffers from rather high fluctuation and statistical uncertainty, and the second method incurs additional complexity in the performance.

A bulk system is assumed to be split into two parts, and the difference of the free energy during this process is equal to the surface free energy, which is also known as cohesive energy. Therefore, the third method to simulate the surface tension, calculates the cohesion work of the liquid. This method was adopted by Padday and Uffindell [92] to calculate the surface tension of n-alkanes, as well as the interfacial tension between the n-alkanes and water. Lu [93] applied the method to calculate the surface tension of liquid argon, nitrogen and oxygen by summing the interaction energy across an assumed surface in the MC simulation. Chen et al. [54–56] extended the method to the liquid metal and alloy systems. They calculated the surface tension of liquid nickel, cobalt and Ni-Cu alloy using the MC simulations and the results showed reasonable agreement with experimental values.

The simulated model is shown in detail in Figure 4. The bulk liquid phase A is separated into two liquid subsystems with equal volumes. During this process, two surfaces are created and the change of total free energy is equal to the sum of the surface energy of the two surfaces. The work of the cohesion W_AA is defined as the work required to pull the liquid bulk system A apart, thus

Note that the equilibration of the surface region is not taken into account during the process of separation. It is assumed that the formation of surface phase cannot influence the thermodynamic behavior of the internal bulk phase. The quantity of heat exchange for obtaining equilibrium is Q, the entropy increase during the surface formation is S, and Equation 16 is modified as

In the calculation of the surface tension of n-alkanes, it is found to be reasonable to neglect the last two terms in Equation 17 [92]. Chen [54,55] extended the conclusion to the metal systems, and the surface tension is approximated by statistically calculating the free energy difference.

Diffusion is one of the most important research topics of liquid physics. It closely relates to many physical and chemical phenomena associated with mass transfer in liquid systems. During the solidification of alloys, the solute diffusion in front of the liquid/solid interface plays a key role in quantitative prediction of microstructural morphologies. For example, the stability of the solid/liquid interface in the directional solidification depends on the diffusion coefficient and growth velocity, and the former directly determines the microstructural evolution of the interface from planar to cells and final dendrites.

In general, the diffusion refers to all types of mass transporting processes, and thus shows a diversity of definitions. The simplest diffusion mode is self-diffusion. The mass transfer of the pure liquid metals can be described by this diffusion mode. In the more general definition, the self-diffusion is regarded as a kind of random-walk motion of a tagged particle of species i in a multi-component system and its velocity autocorrelation function is expressed by,

where u⃗_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],

Alternatively, the self-diffusion coefficient can also be defined by the long time limit of the mean-squared displacement (MSD), namely the Einstein’s equation [89],

where r⃗_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.

Like the case of the mass transport of absorbed molecules in porous materials, another diffusion process, based on the mass flux driven by concentration gradients, is defined, in which the diffusion flux is expressed by the well-known Fick’s law,

here D_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 N₂ 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],

where L_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,

where the diffusion coefficient D_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

where Ð_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.

For a binary system, the diffusion seems to be much simplified. The Fickian diffusion coefficient is related to the M-S diffusion coefficient by only multiplying a factor,

where Ω is the thermodynamic correction factor that is related to the activity a₁ of species 1,

In MD simulations, the self-diffusion coefficient can be obtained using Equations 19 or 20. The M-S diffusion coefficient is calculated by the autocorrelation function of velocity flux [89],

where the velocity flux, J → ( t ) = c 1 ∑ i ∈ 2 u → i ( t ) - c 2 ∑ i ∈ 1 u → j ( t ), and N is the total number of particles.

M-S diffusion coefficient shows more difficult to measure in experiments due to its independence of concentration. Instead, an approximate relation is proposed for the binary system, namely, the Darken relation [100],

which relates the M-S diffusion coefficient to the self-diffusion coefficients, D_1,self D_2,self, and concentrations, c₁, c₂ 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,

The Darken relation has been proved to operate well in some metals and organic systems, and provides an alternative way to obtain the M-S diffusion coefficients indirectly. Although this is so, the diffusion mechanism and diffusion data in the binary and multi-component systems still largely depend on the computer simulations.

Table 1 summarizes the simulated melting temperatures of Au, Cu, Au-Cu, Ni-Al and Al-Ni alloys at zero pressure. The accuracy of MD simulations varies with systems. For AuCu₃ 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 Al₅₀Ni₅₀ alloy [106].

The accuracy of these methods depends largely on the potential function. The parameter, function forms and the truncation of potential are believed to be the important factors that influence the simulated results, but the system size shows independent. On the other hand, the experimental measurements of melting temperature are often influenced by the surface melting [107]. The melting process of metal initiates from the surface pre-melting, and the temperature corresponding to the surface melting is usually larger than the thermodynamic melting point, which is also a factor responsible for the discrepancy between experiments and simulations. With the development of computation technique, large scale simulations based on the first principle are also applied to the coexistence of liquid and solid metals, such as aluminum [108], and the results show a good agreement with other approaches.

As the elementary thermophysical quantity, density of liquid alloys as well as its temperature and pressure dependences is crucial for describing thermodynamic state of liquids. For the undercooled melts, the difficulty of measuring the density lies in realizing the undercooling state. Therefore, the normal contact measurement of density shows impracticable. At present, the electromagnetic levitation technique, in combination with the image acquisition method, is widely employed in measuring density of undercooled alloys. The principle of measuring density follows the definition of density,

In experiments, the conservation of mass is held and the mass of sample can be approximated by weighing the solidified sample after experiments. The key for the method is the accurate measurement of volume. In the ground-based experiments, the shape of droplets is deformed induced by the gravity and electromagnetic fields. The equilibrium shape is assumed to have rotational symmetry. Therefore, the droplet volume is expressed by,

where r is the radial radius and θ is the polar angle. The radial radius can be written as a sum of Legendre polynomials, which is determined by fitting the captured images. It is experimentally found that the temperature dependence of the density of liquid metals follows a linear relation,

where ρ_m is the density of liquid metal or alloy at the melting temperature T_m.

Compared with the experimental values above the melting temperature, the simulated densities of pure metals generally display a good agreement. Table 4 shows a comparison of the simulated and experimental (above melting temperature) values of some pure liquid metals.

The error between the simulations and experiments is less than 5% near the melting temperatures; particularly for liquid titanium, the error is controlled within 1%. In fact, for some special systems such as aluminum, germanium, the density seems to deviate from the linear relation in a wide temperature range. This deviation may be related to the existence of local ordering clusters in the undercooled region. Moreover, the alloys, such as Ni-Si [42] and Ti-Al [39], also show non-linear temperature dependences in some composition ranges. Instead, the quadratic polynomial as

is used to describe the temperature behavior.

For an ideal binary mixture, the density can be represented by the Neumann-Kopp relation associated with the thermodynamic quantities of the two pure elements,

where 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,

where

V is the real volume, and V_ideal is the volume of the ideal mixture. The excess volume is further assumed to depend on the concentrations,

with V_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.

The thermophysical properties of undercooled melts are experimentally measured mainly by means of electromagnetic and aerodynamic levitation techniques. The advantages of electromagnetic levitation consist of two aspects. Firstly, the levitation state induced by the rf electromagnetic field can effectively exclude the contamination of container wall and enhance undercooling. The undercooling of many alloys can thus reach 200–300 K using the levitation technique, which is difficult under normal experimental conditions. Secondly, the eddy currents in the sample caused by the inhomogeneous electromagnetic field simultaneously heat and eventually melt the sample, even for some high melting-point alloys. In combination with noncontact diagnostic tools, this method is preferred in the study of bulk liquid alloys. For the aerodynamic levitation technique, the levitation and heating are manipulated independently, and usually the laser resource is supplied to control the heating and cooling processes by adjusting the power. The aerodynamic levitation is suitable to the measurements of semiconductors, metal oxides and some non-metal liquids especially.

The measurements of surface tension are carried out by the oscillating drop method based on the levitation techniques. The droplet surface will oscillate around the rest position with respect to the small perturbations. The restoring force driving the oscillation is provided by the surface tension of the droplet. As for the droplet levitated in terrestrial electromagnetic fields, the equilibrium shape deviates from the regular sphere. As a result, the Rayleigh frequency ω_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

where M is the mass of the droplet. Equation 47 can approximately describe the surface oscillation in the microgravity environment. For the levitated droplet in ground-based experiments, the frequency splits and droplet translation induced by the gravity fields should be taken into account.

The reported simulations of the surface tension mostly employ the method described in Figure 4. Although the operation of the approach is simple, the results show a relatively large error compared with the experimental measurements [54,119]. The calculated surface tension of liquid nickel in the temperature range of 1208–2273 K is 20–40% lower than the experimental data, and the value of liquid cobalt at the melting point is 17% larger than the experimental result [54]. Comparatively, the temperature dependence of the surface tension is in reasonable agreement with the experiments, in particular for liquid cobalt [56]. The large difference between predicted and measured values is attributed to the EAM potential used in the simulations. Chen [55] argued that the high-order terms of many body contributions were dropped in constructing the effective pair potential, which significantly affected the accuracy of predictions.

Both simulations and experiments suggest that there is a universal linear relationship for describing the temperature dependence of surface tension of pure metals,

where γ_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.

Like the pure liquid metals, the calculated surface tension of binary alloys shows large deviation from experimental results. For example, the surface tension of Ni₅₀Al₅₀ 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.

The composition dependence of surface tension can be theoretically described by ideal solution model and Butler model approximately. In the ideal solution model, the surface tension of binary alloys is given by,

where γ_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

here C_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

where A_{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.

The validity of the NEMD method for calculating the viscosity always attracts the researchers interests during its development. Holian and Evans [150] examined the shear viscosities of LJ liquids produced by NEMD and Green-Kubo methods and revealed that the disagreement between them near the triple point originates from the abnormally large tail of the shear-stress autocorrelation function, and they are in good agreement far from this region. For the liquid aluminum, the comparisons between NEMD and EMD show an excellent agreement from 935 to 1175 K [151]. However, the systematic difference in liquid nickel using the two methods is up to 20%, and the difference intends to reduce in low viscosity systems. Cherne and Deymier [152] ascribed the difference to the thermostate used in the NEMD, whereas there is no further evidence to support this conclusion. Contrarily, in the calculations of shear viscosity of liquid copper, EMD, NEMD and RNEMD produce good consistence over the temperature range from 900 to 1700 K [59], and moreover well agree with the experimental measurements, as listed in Table 8. Essentially, the accuracy of calculated viscosities depends on the potentials. Koishi et al. [153] employed the pseudopotential method, tight-binding method and empirical potential to calculate the viscosities of Na and Fe, and the results varied from quantitative to qualitative levels.

The temperature dependence of viscosity shows more complicated compared to density and surface tension. For most alloys in moderate undercooling region, the temperature dependent viscosity can be described by Arrhenius law

where η₀ is the pre-exponential factor, ΔE is the activity energy and R is the gas constant. Reference 154 lists the values of η₀ and ΔE for some typical pure metals. Alternatively, Cohen and Turnbull proposed the VFT relation based on the free volume model [155],

where T₀ 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],

In fact, the three relations describe different dynamic behaviors of undercooled liquids. If the macroscopic dynamics properties such as diffusion and shear viscosity or structural relaxation time obey the Arrhenius law, the liquid is generally called “strong”. It reflects a fast structural relaxation process and features the most undercooled alloys. The Arrhenius law cannot accurately describe the dynamic behavior when the undercooled liquid approaches the glass transition temperature, and the temperature dependence in the region intends to follow the VFT relation. In contrast to the “strong” liquid, the liquid ruled by the VFT relation is called “fragile”. For some special melts such as Si [139] and SiO₂ [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.

Lazarev et al. [157] simulated the self-diffusion coefficients and viscosities of Ag-Cu alloy from high temperature above the melting point to the region near the glass transition. The temperature dependence of viscosity shows a high non-Arrhenius behavior in the high undercooling region, suggesting a typical fragile liquid as approaching the glass transition. Due to the significant slow dynamic relaxation of high undercooled liquids, the calculation of dynamic properties shows rather difficult and the error is also inevitably enlarged. The NEMD provides a reliable manner to study the dynamic behavior near the glass transition in detail. For Ag-Cu alloy, the predicted divergence temperature for viscosity is comparable to the Kauzmann temperature from the thermodynamic properties, which implies a possible thermodynamic origin of the dynamic abnormalities. They also argued that the Stokes-Einstein law breaks down near the glass transition. Han et al. discussed the problem in liquid copper [59]. The Stokes-Einstein law describes the correlation between the viscosity and self-diffusion as,

Another similar equation, Sutherland-Einstein law differs in the pre-factor,

Therefore, the universal relation is written as

The constant is equal to 1/6π for Stokes-Einstein law and 1/4π for Sutherland-Einstein law. According to Han’s simulations [59], this constant is about 0.099, which is larger than Stokes-Einstein and Sutherland-Einstein laws’ results and close to 1/3π. This conclusion needs further simulations in more metals and alloys.

Moelwyn-Hughes developed a simple model to set up the composition dependence of viscosity for a binary system [158],

where η_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.