Thermotransport in binary system: case study on

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Thermotransport in binary system: case study on Ni50Al50 melt a

a

c

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Alexander V. Evteev , Elena V. Levchenko , Irina V. Belova , Rafal b

Kozubski , Zi-Kui Liu & Graeme E. Murch a

The University Centre for Mass and Thermal Transport in Engineering Materials, Priority Research Centre for Geotechnical and Materials Modelling, School of Engineering, The University of Newcastle, Callaghan, NSW 2308, Australia b

M. Smoluchowski Institute of Physics, Jagiellonian University, Krakow, Poland c

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Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA Published online: 07 Oct 2014.

To cite this article: Alexander V. Evteev, Elena V. Levchenko, Irina V. Belova, Rafal Kozubski, ZiKui Liu & Graeme E. Murch (2014) Thermotransport in binary system: case study on Ni50Al50 melt, Philosophical Magazine, 94:31, 3574-3602, DOI: 10.1080/14786435.2014.965236 To link to this article: http://dx.doi.org/10.1080/14786435.2014.965236

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Philosophical Magazine, 2014 Vol. 94, No. 31, 3574–3602, http://dx.doi.org/10.1080/14786435.2014.965236

Thermotransport in binary system: case study on Ni50Al50 melt Alexander V. Evteeva*, Elena V. Levchenkoa, Irina V. Belovaa, Rafal Kozubskib, Zi-Kui Liuc and Graeme E. Murcha


a

The University Centre for Mass and Thermal Transport in Engineering Materials, Priority Research Centre for Geotechnical and Materials Modelling, School of Engineering, The University of Newcastle, Callaghan, NSW 2308, Australia; bM. Smoluchowski Institute of Physics, Jagiellonian University, Krakow, Poland; cDepartment of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA (Received 5 May 2014; accepted 8 September 2014) The formalism of thermotransport in a binary system is analysed. Focus is put on a detailed consideration of the heat of transport parameter characterizing diffusion driven by a temperature gradient. We introduce the reduced heat of transport parameter Q0 c , which characterizes part of the interdiffusion flux that is proportional to the temperature gradient. In an isothermal system Q0 c represents the reduced heat flow (pure heat conduction) consequent upon unit interdiffusion flux. It is demonstrated that Q0 c is independent of reference frame and is useful in a practical way for direct comparison of simulation and experimental data from different sources obtained in different reference frames. In the case study of the liquid Ni50Al50 alloy, we use equilibrium molecular dynamics simulations in conjunction with the Green–Kubo formalism to evaluate the heat transport properties of the model within the temperature range of 1500–4000 K. Our results predict that in the presence of a temperature gradient Ni tends to diffuse from the cold end to the hot end whilst Al tends to diffuse from the hot end to the cold end. Keywords: thermotransport; heat of transport; Green–Kubo method; liquid alloy; Ni–Al system

molecular

dynamics;

1. Introduction A phenomenon that leads to the tendency of an alloy of two or more components to segregate as a result of a temperature gradient is often called thermotransport [1–3]. This phenomenon is also known as thermodiffusion and the Ludwig–Soret effect, or simply the Soret effect. Thermotransport has an important role in the fundamental understanding of the solidification and crystal growth processes [4,5]. In addition, it is important to have an accurate measure of the thermotransport phenomenon in alloys when considering them in applications involving large temperature gradients. In fact, it can dramatically change the composition and thus potentially change the properties of engineering materials that are subjected to in-service temperature gradients. In particular, investigation of thermotransport in molten metal alloys is initiated by its potential application in the isotopic enrichment of materials [6], design of electronic devices [7] *Corresponding author. Email: [email protected] © 2014 Taylor & Francis


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and nuclear reactors [8,9]. However, a significant amount of controversy exists in this field due to inaccuracies in the measurements or inconsistency in the interpretation of the results [10,11]. At the same time, theoretical modelling of thermotransport still lacks a clear and convincing fundamental understanding of the process at the atomistic level. Accordingly, computer simulation techniques such as molecular dynamics (MD) which operate at the atomistic level are useful for obtaining valuable insight into the thermotransport phenomenon [12–26]. The underlying phenomenological description of thermotransport is through a formulation based on non-equilibrium thermodynamics (also known as irreversible thermodynamics) [1–3]. In this context, the effect of the temperature gradient on the diffusion of atoms is characterized by a parameter called the heat of transport, Q*. This description has been extensively used in experimental and theoretical studies in applications to both liquids and solids [10–33]. As it follows from the Onsager reciprocal relations [34,35] the heat of transport, besides characterizing part of the flux of a species that is proportional to the temperature gradient, is also the coefficient of proportionality between the heat flux and the flux of the species in an isothermal system. This remarkable property means that there are two possible routes to an atomic-level calculation of the heat of transport [31]. One of the routes is based on an approach that employs non-equilibrium MD simulations of a model system subjected to either a temperature gradient or suitable spatially uniform external forces (homogeneous perturbations) [13–16,20,25]. As a result, assuming linear response of the system and computing either the flux of species in the presence of a temperature gradient or the heat flux and the flux of the species in the presence of an external force can be used to determine the heat of transport somewhat by analogy with the experimental determination of this quantity. However, this approach usually requires the use of thermal gradients or other perturbation fields that are large enough to observe thermotransport. This in turn can significantly affect the atomic dynamics of a model system and can take the system response well outside the linear regime [15,20]. Another route is based on equilibrium MD simulations [15–17,21,25,26]. Statistical mechanics is able to provide a general expression for the heat of transport in terms of an integral of the time correlation function between heat flux and matter flux evaluated for the system in thermodynamic equilibrium. This is widely known as the Green–Kubo formulae [36,37]. The demands on computing time are considerably heavy in this approach due to a slow convergence of the time correlation function of the coupled heat and matter fluxes. Nonetheless, the Green–Kubo formalism provides the more convenient method for the calculation of the heat of transport by MD because the behaviour of a model system in thermal equilibrium is far easier to deal with. Indeed, implementation of the Green–Kubo method has no effect on the atomic dynamics and the timeaveraged system temperature is uniform and constant. In addition, this elegant method allows for calculation of the temperature dependence of the heat of transport [26]. In the past, equilibrium MD simulations making use of the Green–Kubo method have been employed several times to analyse thermotransport in pair-potential models of binary systems where at least one of the components has a high mobility. These include an empirical pair-potential model of hydrogen in f.c.c. palladium [15,25,26] and the Lennard–Jones pair-potential model of argon-krypton binary liquid mixtures [17,21].


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Gillan [15] was the first to demonstrate the practicality of this approach for thermotransport calculations in a system with high diffusion mobility of one of the components. In particular, Gillan found that for hydrogen in f.c.c. palladium the reduced heat of transport Q0 H (the heat flux accompanying a unit flux of hydrogen, excluding the partial enthalpy of hydrogen) has a small positive value of 0.05 ± 0.05 eV at a temperature of about T ≈ 980 K. This is in qualitative agreement with experimental measurements, which give a value of 0.065 eV in the temperature range of 670–830 K [38]. In practical terms, it means that hydrogen tends to diffuse from the hot end to the cold end of a specimen. It should be mentioned that the calculations were performed in a velocity reference frame fixed relative to the Pd lattice (Pd was assumed to be immobile). Non-equilibrium MD simulations on the basis of the external-force method, also reported by Gillan [15], confirmed this result. More recently, Schelling and Le [25] revisited the Green–Kubo calculations of the reduced heat of transport Q0 H for the same model of hydrogen in f.c.c. palladium. For the reduced heat of transport Q0 H they found a value of –0.21 ± 0.05 eV at T ≈ 980 K, which is in disagreement with the previously published calculations by Gillan [15] and experimental measurements [38], suggesting now that hydrogen moves from low to high temperatures. Schelling and Le [25] pointed out that despite using a velocity reference frame fixed relative to the centre of mass of the system (in contrast to the calculations by Gillan [15]) the results obtained in the two different reference frames should not differ in any significant fashion since the mass of hydrogen is much smaller than the mass of palladium. Disagreement between the two above-mentioned calculations was attributed in [25] to different ways of calculating the partial enthalpy per atom of hydrogen in the system. Gillan [15] expressed the partial enthalpy of hydrogen exactly in terms of more easily calculated thermodynamic functions and these were calculated by computer simulations in which the number of hydrogen atoms in a fixed amount of f.c.c. palladium was varied. On the other hand, Schelling and Le [25] assumed that the partial enthalpy of hydrogen could be calculated by using the average energy and the average trace of the stress tensor of hydrogen atoms, neither of which quantities is concerned with a change in the relative amounts of hydrogen and palladium. Disagreement with the experimental measurements [38] was attributed in [25] to inadequacy of the empirical pair-potential model used for palladium–hydrogen system. Schelling and Le [25] also confirmed their Green–Kubo calculations by using non-equilibrium MD simulations in the presence of a large temperature gradient as well as a recently developed constrained-dynamics approach [24]. Furthermore, Schelling and co-authors [26] found that the reduced heat of transport Q0 H for the model of hydrogen in f.c.c. palladium is approximately temperature independent in the temperature range of 580–1280 K. Vogelsang and co-authors [17] used the Green–Kubo method to calculate the thermal diffusion coefficient, DT , for the Lennard–Jones pair-potential model of an equimolar argon-krypton binary liquid mixture in a velocity reference frame fixed relative to the centre of mass of the system. The thermal diffusion coefficient is a closely related alternative to the heat of transport parameter in characterizing the Soret effect [1–3]. For the two state points considered, they found: DT 2:2 1011 m2s−1 K−1 at T ≈ 116.4 K and P ≈ 5.8 MPa, and DT 1:7 1011 m2s−1 K−1 at T ≈ 115 K and P ≈ −2.8 MPa. To obtain the partial enthalpies of the components, Vogelsang and co-authors [17] used the total enthalpy of the Lennard–Jones pair-potential model of argon-krypton binary liquid mixture calculated as a function of composition. Moreover,



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Vogelsang and co-authors [17] concluded that for this particular model of argon-krypton binary liquid mixture the representation of the partial enthalpies by using the average energies and the average traces of the stress tensors of components works well. This was attributed to the fact that the ideal mixture approximation is valid for argon-krypton binary liquid mixtures, as can be expected for systems having a low excess enthalpy [16,17]. Lastly, Perronace and co-authors [21] reported on successful Green–Kubo calculations of the thermal diffusion coefficient, DT , in a velocity reference frame fixed relative to the centre of mass of the system for two state points (DT 2:93 1011 m2s−1 K−1 for Ar67.59Kr32.41 at T ≈ 95.2 K and P ≈ 0.1 MPa, and DT 2:77 1011 m2s−1 K−1 for Ar78.7Kr21.3 at T ≈ 93 K and P ≈ 0.1 MPa) of the Lennard–Jones pair-potential model of the argon–krypton liquid mixture. Taking into account the results of the previously discussed work [17], Perronace and co-authors [21] used the ideal mixture approximation to calculate the partial enthalpies of the components. Comparison of the simulation results with available experimental data [39] showed remarkable agreement. Thus, the above brief literature review reveals a rather limited number of applications of equilibrium MD simulations using the Green–Kubo method to describe the thermotransport phenomenon. Furthermore, in these applications only rather simple pair-potential models of binary systems were considered. In addition, there is a significant amount of contradiction over the simulations and presentation of results. The problem comes from the fact that thermotransport represents a cross-correlation (offdiagonal) effect in a binary system, which is generally difficult to describe with good accuracy since it is usually rather small compared with the direct (diagonal) transport phenomena such as matter flux that is induced by a concentration gradient and heat flux induced by a temperature gradient. In this paper, first we discuss an elaborated formalism of thermotransport in a binary system and then we employ it, as a case study, for interpretation of equilibrium MD (Green–Kubo based) calculations of thermotransport in a binary liquid Ni50Al50 alloy. For the MD simulations, we use state-of-the-art first-principles-based many body potential developed for the Ni-Al intermetallic compound within the framework of the embedded atom method (EAM) [40]. We believe that the calculations of the crosscoupling effect of the heat and matter flows with this EAM potential, besides the fundamental interest to investigate the heat of transport in a highly non-ideal binary system, will also contribute towards a better understanding of such microscopic processes as crystal nucleation and crystal growth, which are essential for efficiently controlling composition, microstructure and properties of this compound [41–44]. This hope is additionally supported by the recent considerable success of the EAM potential model in the description of mass diffusion in the liquid Ni50Al50 alloy [45–47] in accordance with available experimental data [48]. We must also note that in metallic alloys the heat of transport parameter may be divided into two parts, one associated with atomic movements and the other associated with conduction electrons [22,23,27,49,50]. However, estimates of the electronic contribution to heat of transport parameter by the method of Gerl [50] indicated it to be relatively small [22,23,27,49,50]. We are therefore concerned here only with the contribution from atomic movements. The present paper is set out as follows. In Section 2, we specify the formalism we use. We give the essential macroscopic relations on which the work is based. We introduce the heat of transport parameter in a binary system which is invariant to a change

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of reference frame and which appears to be practically useful for direct comparison of simulation and experimental data from different sources obtained in different reference frames. Also, we give a brief description of the Green–Kubo and thermodynamic relations employed in this work. In Section 3, we describe the chosen model and the calculation techniques used. In Section 4, we present and discuss the simulation results for the heat of transport. Finally, conclusions are drawn in Section 5.


2. Formalism of transport in a binary system Let us consider an isotropic three-dimensional binary system of N = N1 + N2 atoms in which N1 and N2 are the number of atoms of species 1 and 2, respectively. We assume that the system is in mechanical equilibrium and no external forces are supposed to be present. 2.1. Material species and heat fluxes The fluxes J 1 and J 2 of species 1 and 2 in a velocity reference frame fixed relative to the centre of mass of the system are given by: J1 ¼

N c 1 v1 ; V

(1a)

J2 ¼

N c 2 v2 ; V

(1b)

where V is the volume of the system, c1 = N1/N and c2 = N2/N (c1 + c2 = 1) are the atomic (mole) fractions of species 1 and 2, respectively, and v1 and v2 are the mean velocities of species 1 and 2, respectively, relative to the centre of mass of the system. Since the velocities of both species are referred to the centre of mass of the system, the fluxes J 1 and J 2 are not independent and a linear relation exists between the fluxes (zero total momentum) [1–3]: m1 J 1 þ m2 J 2 ¼ 0;

(2)

where m1 and m2 are the masses of atoms of species 1 and 2, respectively. Thus, in a binary system there exists only one independent matter flux. One appropriate choice in the binary system, as we will see below, can be made by using the chemical interdiffusion flux, J c , to represent the material species along with the heat flux in the phenomenological transport equations. The interdiffusion flux can be given by: Jc ¼

N c 1 c 2 ð v1 v2 Þ ¼ c 2 J 1 c 1 J 2 ; V

(3)

The interdiffusion flux is invariant with respect to the choice of reference frame in the sense that it describes the fluxes of components J 1 and J 2 relative to each other. For example, if a reference frame fixed relative to the centre of mass of the system moves with a velocity v relative to an alternative reference frame then the fluxes of components J~ 1 and J~ 2 in this alternative reference frame are given by [1,2]:


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N J~ 1 ¼ J 1 þ c1 v; V

(4a)

N J~ 2 ¼ J 2 þ c2 v: V

(4b)

Substitution of the fluxes components J 1 and J 2 expressed from Equations (4a) and (4b) into Equation (3) shows that the interdiffusion flux J c is invariant to a change of reference frame:


J c ¼ c2 J 1 c1 J 2 ¼ c2 J~ 1 c1 J~ 2 :

(5)

From Equations (2) and (3), we can find relations between the interdiffusion flux J c and the fluxes J 1 and J 2 in a reference frame fixed relative to the centre of mass of the system: Jc ¼

m m J 1 ¼ J 2; m2 m1

(6)

¼ c1 m1 þ c2 m2 is the total mass per atom of the system. where m Moreover, we need to show that the heat flux can be transformed to also be invariant with respect to the choice of reference frame, as is the interdiffusion flux. For this reason, it is convenient to consider the so-called reduced heat flux J 0q , which can be obtained by subtracting from the total heat flux J q that part associated with the heat transfer due to the movement of material species: 1J 1 H 2J 2; J 0q ¼ J q H

(7)

1 and H 2 are the partial enthalpies (per atom) of components 1 and 2, respecwhere H tively. Then, if a reference frame fixed relative to the centre of mass of the system moves with a velocity v relative to an alternative reference frame then the heat flux J~ q in this alternative reference frame is given by [1]: N J~ q ¼ J q þ hv; V

(8)

1 þ c2 H 2 is the total enthalpy per atom of the system. Substitution of the where h ¼ c1 H fluxes of components J 1 and J 2 and the heat flux J q expressed from Equations (4a), (4b) and (8), respectively, into Equation (7) shows that the reduced heat flux J 0q , as the interdiffusion flux J c , is invariant to change of reference frame: 1J 1 H 2 J 2 ¼ J~ q H 1 J~ 1 H 2 J~ 2 ; J 0q ¼ J q H

(9)

Although below we analyse the phenomenological transport equations in a reference frame fixed relative to the centre of mass of the system, a similar consideration can also be made in other reference frames. 2.2. Phenomenological transport equations According to the thermodynamics of irreversible processes, at sufficiently small departures from thermodynamic equilibrium the rate of entropy production per unit volume, σ, for a binary system is [1–3,35]:

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T r ¼ J 1X 1 þ J 2X 2 þ J qX q;

(10)

where T is the absolute temperature, and X 1 , X 2 and X q are thermodynamic vector forces which are measures of the imbalances generating the fluxes of material species and heat J 1 , J 2 and J q , respectively. The thermodynamic forces X 1 and X 2 are given by [1–3]: l X 1 ¼ T r 1 ; (11a) T l X 2 ¼ T r 2 ; (11b) T where μ1 and μ2 are the chemical potentials of components 1 and 2, respectively. At a constant temperature T and pressure P, the chemical potential of a species is the partial derivative of the Gibbs free energy of the system G ¼ H TS ¼ E þ PV TS (H is the enthalpy, E is the internal energy and S is the entropy of the system) with respect to the number of atoms of the species, considering that the amounts of other species remain constant. Accordingly, for a binary system we have [1–3]: @G 1 T S1 ¼ E 1 þ PV1 T S1 ; ¼H (12a) l1 ¼ @N1 T;P;N2 l2 ¼

@G @N2

2 T S2 ¼ E 2 þ PV2 T S2 ; ¼H

(12b)

T;P;N1

2 , the partial internal energies (per 1 and H where the partial enthalpies (per atom) H atom) E1 and E2 and the partial entropies (per atom) S1 and S2 of components 1 and 2 derived from the related extensive properties are defined as: @Z Z1 ¼ ; (12c) @N1 T ;P;N2 Z2 ¼

@Z @N2

(12d) T ;P;N1

1; E 1 ; V1 ; S1 ; Z2 ¼ H 2; E 2 ; V2 ; S2 and Z = H, E, V, S). By means of Equations (Z1 ¼ H (12a) and (12b) one can then rewrite Equations (11a) and (11b) in the form: 1 H l1 @l1 l rT X 1 ¼ rl1 þ rT ¼ rðl1 ÞT rT þ 1 rT ¼ rðl1 ÞT þ T @T P;N1 T T (13a) X 2 ¼ rl2 þ

2 H l2 @l2 l rT ¼ rðl2 ÞT rT rT þ 2 rT ¼ rðl2 ÞT þ T @T P;N2 T T (13b)

where rðl1 ÞT and rðl2 ÞT denote the parts of the gradients of chemical potentials of components 1 and 2 that are due to gradients in pressure and concentration but not temperature. Thus, Equations (13a) and (13b) allow for a representation in which the


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contribution to the thermodynamic forces X 1 and X 2 due to the gradient in temperature is separated from the contributions due to the gradients in pressure and concentration [1–3]. The thermodynamic force X q depends only on the temperature gradient [1–3]: 1 X q ¼ rT : T

(14)


The phenomenological transport equations represent generalized laws of transport in the form of linear relations between fluxes J 1 , J 2 and J q and the thermodynamic forces X 1 , X 2 and X q [1–3,35]. For an isotropic binary system, the phenomenological transport equations can be written in the following form: J1 ¼ L11 X1 þ L12 X2 þ L1q Xq ;

(15a)

J2 ¼ L21 X1 þ L22 X2 þ L2q Xq ;

(15b)

Jq ¼ Lq1 X1 þ Lq2 X2 þ Lqq Xq ;

(15c)

where, for simplicity, we drop the vector notation (all Cartesian components of a vector in an isotropic system must be the same). The matrix of phenomenological coefficients (L-coefficients) is symmetric according to the well-known Onsager reciprocal relations [34,35]: L12 ¼ L21 ;

(16a)

L1q ¼ Lq1 ;

(16b)

L2q ¼ Lq2 :

(16c)

Moreover, it should be noted that in an isotropic binary system there are only three independent phenomenological coefficients. In particular, upon substituting Equations (15a) and (15b) into Equation (2) and keeping in mind that the forces X1, X2 and Xq are linearly independent quantities, in general, it is easy to show that in a reference frame fixed relative to the centre of mass of the system the additional relations between the phenomenological coefficients are: m1 L12 ¼ L11 ; (17a) m2 L22 ¼

2 m1 L11 ; m2

L2q ¼

m1 L1q : m2

(17b) (17c)

Thus, in an isotropic binary system one can eliminate, for example, the flux J2 and then employ the formalism of the phenomenological transport equations for the two remaining fluxes J1 and Jq [1–3]. However, as mentioned above, an alternative phenomenological representation of the transport equations in an isotropic binary system can be made by using the interdiffusion flux Jc (which is invariant to change of reference frame, see Equations (3)–(5)) to represent the material species. Taking into account

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Equations (2), (3), (6) and (15)–(17), one can consider the following set of two phenomenological equations in order to describe material species and heat transport in an isotropic binary system in a reference frame fixed relative to the centre of mass of the system: Jc ¼ c2 J1 c1 J2 ¼ Lcc Xc þ Lcq Xq ;

(18a)

Jq ¼ Lqc Xc þ Lqq Xq ;

(18b)

m2 m1 X1 X2 ; m m

(18c)


where Xc ¼

Lcc ¼

m m2

2

Lcq ¼ Lqc ¼

L11 ;

(18d)

m L1q : m2

(18e)

In order to make a representation of the heat flux also invariant to change of reference frame, we can rewrite the phenomenological transport equations, Equations (18a) and (18b), by making use of the reduced heat flux Jq0 (see Equations (7)–(9)): Jc ¼ c2 J1 c1 J2 ¼ Lcc Xc0 þ L0cq Xq ;

(19a)

1 J1 H 2 J2 ¼ L0 X 0 þ L0 Xq ; Jq0 ¼ Jq H qc c qq

(19b)

c Xq ¼ m2 X 0 m1 X 0 ; Xc0 ¼ Xc þ H 1 m 2 m

(19c)

c ¼ m2 H 1 m1 H 2; H m m

(19d)

1 Xq ¼ rðl1 Þ ; X 01 ¼ X 1 þ H T

(19e)

2 Xq ¼ rðl2 Þ ; X 02 ¼ X 2 þ H T

(19f)

c Lcc ; L0cq ¼ L0qc ¼ Lcq H

(19g)

where

c Lcq þ H 2 Lcc ; L0qq ¼ Lqq 2H c

(19h) X10

X20

It is important to note here that the new thermodynamic forces and (and accordingly Xc0 ) do not now depend on the temperature gradient. Thus, the importance of the representation of the phenomenological transport equations in an isotropic binary system by Equations (19a) and (19b) is contained in the two following observations: (i) both the interdiffusion flux Jc in Equation (19a) and the reduced heat flux Jq0 in Equation (19b) are invariant to change of reference frame, and (ii) each phenomenological coefficient in the equations clearly characterizes the origin of the



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contribution to the fluxes. Regarding the second observation, one can see from Equations (19a) and (19b) that the diagonal coefficients Lcc and L0qq characterize part of the interdiffusion flux that is proportional to the concentration gradient (at constant pressure) and part of the reduced heat flux that is proportional to the temperature gradient, respectively; while the off-diagonal coefficient L0cq ð¼ L0qc Þ: characterizes both part of the interdiffusion flux that is proportional to the temperature gradient and part of the reduced heat flux that is proportional to the concentration gradient (at constant pressure). Finally, let us briefly consider a representation of the phenomenological transport equations in an isotropic binary system by using the fluxes J1, J2 and Jq0 along with the thermodynamic forces X10 , X20 and Xq. This will be useful for the following consideration of the heats of transport. By using Equations (7), (19e) and (19f) one can replace Jq, X1 and X2 in Equations (15) by Jq0 , X10 and X20 , so that: J1 ¼ L11 X10 þ L12 X20 þ L01q Xq ;

(20a)

J2 ¼ L21 X10 þ L22 X20 þ L02q Xq ;

(20b)

Jq0 ¼ L0q1 X10 þ L0q2 X20 þ L0qq Xq ;

(20c)

where the coefficients L11, L12, L21 ð¼ L12 Þ and L22 are the same as in Equations (15), while the other coefficients can be straightforwardly defined as: 1 L11 H 2 L12 ; L01q ¼ L0q1 ¼ L1q H

(21a)

1 L12 H 2 L22 ; L02q ¼ L0q2 ¼ L2q H

(21b)

2 L12 : 1 L1q 2H 2 L2q þ H 2 L11 þ H 2 L22 þ 2H 1H L0qq ¼ Lqq 2H 1 2

(21c)

As can be seen, the new phenomenological coefficients L01q ð¼ L0q1 Þ and L02q ð¼ L0q2 Þ in the linear relations given by Equation (20) also obey the Onsager reciprocal relations. In a reference frame fixed relative to the centre of mass of the system, the additional relations between the phenomenological coefficients are given by Equations (17a) and (17b) as well as by the expression (shown below) which is similar to Equation (17c) and which can be easily obtained from Equations (21a) and (21b) by using Equations (17) and (19d): m1 0 m1 m 0 L2q ¼ L1q ¼ Hc L11 : L1q (22) m2 m2 m2 Lastly, it is worth pointing out here that the rate of entropy production per unit volume given by Equation (10) is invariant to the above-described transformations to the new fluxes and thermodynamic forces: T r ¼ Jc Xc þ Jq Xq ¼ Jc Xc0 þ Jq0 Xq ¼ J1 X10 þ J2 X20 þ Jq0 Xq :

(23)

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2.3. Heats of transport To explicitly describe thermotransport and other related phenomena in the system, let us consider first of all the heat of transport Qc . Following the usual way to introduce a heat of transport [1–3], the formal definition of the heat of transport Qc in the present case can be done via the coefficients Lcq and Lcc: Lcq ¼ Lcc Qc ;

(24)

so that


Qc ¼

Lcq m2 L1q ¼ : L11 Lcc m

(25)

By substitution of Equation (24) into Equation (18b) and using Equation (18a) to eliminate Xc one can obtain: (26) Jq ¼ Qc Jc þ Lqq Qc Lcq Xq ; or, for an isothermal system (where Xq = 0): Jq ¼ Qc Jc :

(27)

Qc

Thus, in an isothermal binary system represents the total heat flow (consisting of heat conduction and heat transfer) consequent upon unit interdiffusion flux. We recall again that in Equation (27) the interdiffusion flux Jc is invariant to change of reference frame, while the heat flux Jq depends on the choice of reference frame. This means that the considered heat of transport Qc must always be calculated in the same reference frame as the heat flux. However, in the present consideration we can alternatively introduce in the same 0 manner the so-called reduced heat of transport Q0 c via the coefficients Lcq and Lcc: L0cq ¼ Lcc Q0 c ;

(28)

so that Q0 c ¼

L0cq Lcq c ¼ Q H c: ¼ H c Lcc Lcc

(29)

By substitution of Equation (28) into Equation (19b) and using Equation (19a) to eliminate Xc0 one can obtain: 0 0 0 (30) Jq0 ¼ Q0 c Jc þ Lqq Qc Lcq Xq ; or, for an isothermal system (where Xq = 0) Jq0 ¼ Q0 c Jc :

(31)

Thus, in an isothermal binary system, Q0 c represents the reduced heat flow (pure heat conduction) consequent upon unit interdiffusion flux. Since, both the interdiffusion flux Jc and the reduced heat flux Jq0 in Equation (31) are invariant to change of reference frame, the reduced heat of transport Q0 c must also be invariant to change of reference frame. Also, we note that in an isothermal binary system the part of the heat flux associated with the heat transfer due to the movement of material species can be expressed, using Equations (27), (29) and (31), in the form:



3585

cJ c; Jq Jq0 ¼ H

(32)

c (given by Equation (19d)) appears to have the meaning of the characteristic where H enthalpy (per atom) of species carried by the interdiffusion flux in a reference frame fixed relative to the centre of mass of the system. Thus, substitution of Equations (19d) and (25) into Equation (29) allows for calculation of the reduced heat of transport Q0 c in a reference frame fixed relative to the centre of mass of the system if the phenomenological coefficients L11 and L1q, and the partial enthalpies (per atom) of components 1 and H 2 are known. H Furthermore, we should point out that since only one independent heat of transport can be defined in a binary system from the phenomenological transport equations [1–3], the choice of the reduced heat of transport Q0 c carried by the interdiffusion flux is somewhat unique, owing its important feature of invariance to change of reference frame. Hence, the reduced heat of transport Q0 c can be used as a reference quantity to compare results of different theoretical modelling and experimental studies of the thermotransport in binary systems, where significant inconsistency and ambiguity exists in the formulation and presentation of the data [10,11]. For instance, the formal definition of the heats of transport Q1 and Q2 of components in a binary system can be done via the coefficients L11, L12, L21 ð¼ L12 Þ, L22, L1q and L2q appearing in Equation (15) [1–3]: L1q ¼ L11 Q1 þ L12 Q2 ;

(33a)

L2q ¼ L21 Q1 þ L22 Q2 :

(33b)

Hence by substitution of Equation (33) into Equation (15c) and using Equations (15a) and (15b) to eliminate X1 and X2, one can obtain (keeping in mind the Onsager reciprocal relations Equation (16)): (34) Jq ¼ Q1 J1 þ Q2 J2 þ Lqq Q1 L1q Q2 L2q Xq ; or, for an isothermal system (where Xq = 0): Jq ¼ Q1 J1 þ Q2 J2 : Q1

(35)

Q2

Thus, in an isothermal binary system and represent the total heat flows (consisting of heat conduction and heat transfer) consequent upon unit fluxes of component 1 and 2, respectively. 0 Furthermore, the reduced heats of transport Q0 1 and Q2 of the components in a binary system can be defined in the same manner via the coefficients L11, L12, L21 ð¼ L12 Þ, L22, L01q and L02q appearing in Equation (20): 0 L01q ¼ L11 Q0 1 þ L12 Q2 ;

(36a)

0 L02q ¼ L21 Q0 1 þ L22 Q2 :

(36b)

3586

A.V. Evteev et al.

Hence by substitution of Equation (36) into Equation (20c) and using Equations (20a) and (20b) to eliminate X10 and X20 , one can obtain (keeping in mind the Onsager reciprocal relations Equations (21a) and (21b)): 0 0 0 0 0 0 J þ Q J þ L Q L Q L (37) Jq0 ¼ Q0 1 2 1 2 qq 1 1q 2 2q Xq ; or, for an isothermal system (where Xq = 0):


0 Jq0 ¼ Q0 1 J1 þ Q 2 J2 :

(38)

By Equations (7), (21), (34) and (37) (or by Equations (7), (35) and (38)) one can see that: Q0 1 ¼ Q1 H1 ;

(39a)

Q0 2 ¼ Q2 H2

(39b)

represent the reduced heat flows (pure heat conduction) consequent upon unit fluxes of components 1 and 2, respectively. However, in the case when J1 and J2 are linearly dependent a direct approach to find the heats of transport of components from Equations (33) and (36) cannot be performed. For instance, in a reference frame fixed relative to the centre of mass of the system, the matrix of the phenomenological coefficients L11, L12, L21 ð¼ L12 Þ and L22 in Equations (33) and (36) is singular (see also Equations (17a) and (17b)) and its inverse does not exist. See [51,52] for a detailed discussion on a procedure for inverting the linear laws of irreversible thermodynamics in situations where the fluxes of the components of a system are not independent. The main result of this discussion is that the solution for the heats of transport is arbitrary in this case, so that to complete the definition of the heats of transport one needs to specify a linear dependence among them [52]. As was stated in [52] there is no unique way (which would have physical significance) to define such a linear relation. For this reason, we show here only how the 0 heats of transport Q0 1 and Q2 of the components of a binary system in the reference frame fixed relative to the centre of mass of the system can be related to the heat of transport Q0 c associated with the interdiffusion flux. In particular, by comparing Equations (31) and (38) while keeping Equation (6) in mind, one can find the following 0 0 relation between Q0 c , Q1 and Q2 m2 0 m1 0 Q Q : Q0 (40) c ¼ 1 2 m m Thus, we have demonstrated that the heat of transport Q0 c associated with the interdiffusion flux determines the entire phenomenon of thermotransport in a binary system. Once Q0 c is known, then one can determine the fluxes of the components in any reference frame. There is no physical significance in providing an explicit definition for Q0 1 and Q0 2 and reporting their values. Lastly, we note that from Equations (19d), (29), (39) and (40), it follows that a similar relation holds between Qc , Q1 and Q2 m2 m1 Q Q : Qc ¼ (41) 1 m 2 m



3587

A similar way of presenting results on the heat of transport in a binary system obtained in other reference frames appears to be practically useful for direct comparison of simulation and experimental data from different sources. In particular, in a volumefixed frame of reference all expressions for the heats of transport will be similar to the expressions (discussed above) presented here in a reference frame fixed relative to the centre of mass of the system. It is only necessary to change the atomic masses of components on the partial atomic volumes of components. Lastly, we should note that in the literature, quite often the thermal diffusion coefficient, DT , is used to characterize part of a flux of species that is proportional to the imposed temperature gradient [1–3]. From Equations (14) and (19a), we can see that DT should be proportional to L0cq =T . Following the usual way to define the thermal diffusion coefficient [1,2] wherein the mean velocities of species are used rather than the fluxes of species (see Equation 3), one can easily express DT in terms of L0cq , Q0 c and Lcc, namely: DT ¼

VL0cq Nc1 c2 T

¼

VQ0 c Lcc : Nc1 c2 T

(42)

In addition, from Equations (28) and (30), it can be seen that in a binary system not 0 only Q0 c but also three other quantities DT , Lcq and Lcc, in Equation (42) are invariant to change of reference frame. 2.4. Green–Kubo formalism for phenomenological coefficients Statistical mechanics shows that the phenomenological coefficients can be calculated as time integrals of correlation functions involving the microscopic fluxes of material species and heat. These relations are generally known as the Green–Kubo formulae [36,37]. In the present work, as follows from Equation (25), we are interested in determining the phenomenological coefficients L11 and L1q which can be calculated as: L11

V lim ¼ 3kB T t!1

Zt

C11 ðt 0 Þdt 0 ;

(43)

C1q ðt 0 Þdt 0 ;

(44)

0

L1q

V lim ¼ 3kB T t!1

Zt 0

where kB is the Boltzmann constant, and C11 ðt Þ ¼ hJ 1 ðt ÞJ 1 ð0Þi

(45)

is the autocorrelation function of the flux of component 1, and C1q ðt Þ ¼ Cq1 ðt Þ ¼

1 hJ 1 ðt ÞJ q ð0Þi þ hJ q ðt ÞJ 1 ð0Þi 2

(46)

is the correlation function between flux of component 1 and the heat flux. The time averaging 〈⋯〉 in Equations (45) and (46) should be taken at thermodynamic equilibrium.

3588

A.V. Evteev et al.

The microscopic expression for the total flux of component 1 in a velocity reference frame fixed relative to the centre of mass of the system is simply given by:


J1 ¼

N1 1X v1i ; V i¼1

(47)

where v1i denotes the velocity vectors of atoms of species 1 (i = 1…N1). The total heat flux in the same reference frame can be obtained using the microscopic Irving–Kirkwood expression given by [53]: ! N N N 1 d X 1X 1X dei ; (48) ei r i ¼ ei vi þ ri Jq ¼ V dt i¼1 V i¼1 V i¼1 dt where ri , vi and ei denote the radius vector, velocity vector and energy (kinetic and potential) of atoms without specifying species (i = 1…N), and t is the time. 2.5. Partial enthalpies In order to calculate the reduced heat of transport Q0 c , we need to determine first the 1 and H 2 as it follows from Equations (19d) and (29). We should partial enthalpies H 1 and H 2 are pure thermodynamic quantities (see note that the partial enthalpies H Equation 12) [1–3,15] and cannot be strictly defined in terms of simple microscopic expressions (see the discussion in [14,16,17]). In general, evaluation of these quantities by computer calculations requires additional methods of determination, involving simulations at different compositions [14–17]. Only in some special cases, for example for argon-krypton binary liquid mixtures as was mentioned above, the ideal mixture approximation may be invoked to estimate, in the thermodynamic limit, the partial enthalpies of components using microscopic expressions for the average energies and the average traces of the stress tensors of components [16,17,21]. The main problem in such estimations of the partial enthalpies comes from the fact that it is not clear how to correctly split up the energy and stress contributions between each species in the microscopic expressions [14,16]. Hence, such estimations can be strictly exact only in the case of isotopic mixtures, when the components simply differ in masses but have the same interatomic potentials [16]. In the case of the Lennard–Jones pair-potential model of an equimolar argon-krypton binary liquid mixture a rough estimate showed that an approximation, which attributes to each species 50% of the energy and stress contributions in the microscopic expressions, gives difference from the thermodynamic definition of the partial enthalpies (see Equation (12)) of about 4% for argon and of about 9% for krypton [16,17]. This may indicate that the Lennard–Jones pair-potential model of an equimolar argon-krypton binary liquid mixture can be considered as quasi-ideal because of the similarity between Ar–Ar, Kr–Kr and Ar–Kr interactions [16,17]. In contrast, the model of the Ni50Al50 melt considered in this paper can be characterized as a strongly non-ideal alloy, so that, similar to the case of hydrogen in f.c.c. palladium considered by Gillan [15], we do not have any other choice but to undertake the rather heavy task of calculating the partial enthalpies from their thermodynamic definition given by Equation (12). Nevertheless, we also present in the paper, for comparison reason, the results of an assessment of the partial enthalpies of components using


3589

microscopic expressions for the average energies and the average traces of the stress tensors of components [16,17,21,25]. The relations used for this assessment are presented in Section 3. Thus, in this work, to calculate the partial enthalpies we will make use of a number of supplementary simulations at different concentrations, temperatures and volumes, as was suggested by Gillan [15]. For instance, according to Equation (12c) we have @H @E @V 1 þ PV1 : H1 ¼ ¼ þP ¼E (49) @N1 T ;P;N2 @N1 T ;P;N2 @N1 T;P;N2


Since 1 ¼ E

@E @N1

¼ T;P;N2

@E @N1

@E þ V1 ; @V T ;P;N2 T ;N1 ;N2

(50)

we have that: ! @E @E @E @P H1 ¼ þ V1 þP þ þ V1 T : (51) @N1 T ;V ;N2 @V T ;N1 ;N2 @N1 T;V ;N2 @T T;N1 ;N2 Finally, V1 ¼

@V @N1

@V @P @P ¼ ¼ jT V ; @P T ;N1 ;N2 @N1 T ;V ;N2 @N1 T;V ;N2 T ;P;N2

where jT ¼

1 V

@V @P

(52)

(53) T ;N1 ;N2

is the isothermal compressibility. 1 via supplementary constant volume Thus, in order to obtain the partial enthalpy H simulations, to Equations(51)–(53), one has to calculate four thermodynamic according @P @E @P , @T V ;N1 ;N2 , @N1 and @V quantities @N1 @P T;N1 ;N2 . The partial enthalpy H2 T ;V ;N2

T ;V ;N2

can be found then from the relation: 1 þ N2 H 2; H ¼ E þ PV ¼ N1 H

(54)

since the internal energy E and pressure P are obtained as a matter of course during calculations of the correlation functions C11 ðt Þ and C1q ðt Þ (see Equations (45) and (46)). 3. Model and calculation methods In general, the total energy of an atom i is represented in the EAM model [54] as: 1 1X ei ¼ mi m2i þ Fli ð qi Þ þ Vl l ðrij Þ; (55) 2 2 jð6¼iÞ i j where mi is the mass of the atom, mi is the absolute value of the velocity vector of the atom, Fli ð qi Þ is the embedding energy of the atom as a function of the host electron

3590

A.V. Evteev et al.

i induced at site i by all other atoms in the system and Vli lj ðrij Þ is the pair density q interaction potential as a function of distance rij between atoms i and j (μi and μj indicate whether the functional form for the species of atom i or atom j is used). The host i is given by: electron density q X i ¼ qli ðrij Þ; (56) q


jð6¼iÞ

where qlj rij is the electron density induced by an atom j at the location of atom i. In particular, for a binary system this model involves seven potential functions that can be conveniently divided into three groups: (i) V11 ðrÞ F1 ð qÞ and q1 ðrÞ; (ii) V22 ðrÞ F2 ð qÞ and q2 ðrÞ; and (iii) V12 ðrÞ. These seven functions can be treated as some fitting functions that have to be reasonably parameterized. In this work, the interactions between atoms in the MD model of liquid Ni50Al50 alloy are described by using an EAM potential developed by Mishin et al. [40]. The potential functions were obtained by fitting to: (i) experimental properties of B2-NiAl, the L12-ordered compound Ni3Al and pure Ni and Al; and (ii) a large set of ab initio data that were generated in the form of energy-volume relations for a number of alternative structures of NiAl, Ni3Al and pure Ni and Al. This potential not only accurately reproduces various properties of crystalline NiAl but also gives a realistic description of the principal properties of the liquid Ni50Al50 alloy, even though no liquid data were included in the fitting of the potential. In particular, the results of the MD simulations [45–47] of the liquid Ni50Al50 alloy with this EAM potential were found to be in accordance with recent experimental data [48,55,56] on density, surface energy and Ni selfdiffusion in the liquid alloy, thus demonstrating a good transferability of this potential [40] for an atomistic simulation of the liquid Ni50Al50 alloy. The melting temperature of B2-NiAl predicted by this potential is 1520 K. This is about 20% lower than the experimental value of 1911 K [57]. When Equations (48), (55) and (56) are taken into account, the Cartesian components of the microscopic heat flux in a system described using an EAM potential model can be represented as [58]: ! 1 d X 1X 1X ðpÞ ei xia ¼ ei via Xi riab vib ; (57) Ja ¼ V dt V V i i i where ðpÞ riab Xi

" # X @Flj ð qj Þ @qli ðrij Þ 1 @Vlilj ðrij Þ xija xijb ¼ þ ; 2 @rij @rij rij @ qj jð6¼iÞ

(58)

Xi is the volume of atom i, the symbols α and β enumerate Cartesian components of vectors and tensors: xiα, xijα (or xijβ) and mia (or mib ) are the components of the vectors ðpÞ ri , rij and vi , respectively, while riab denotes the potential energy contribution to the components of the stress tensor of atom i. For the symbols which enumerate Cartesian components of vectors and tensors, the Einstein summation notation is implied. Now, let us consider a quantity which can be denoted as the enthalpy related to atom i: hi ¼ e i þ pi X i ;

(59)


3591

where pi is the local hydrostatic pressure in the direct vicinity of atom i, so that pi Xi ¼

1 2 ðpÞ mi vi dab riab Xi 3

(δαβ is the Kronecker delta). Then, the time averaged quantities * + N1 X 1 1 ¼ h1i ; h N1 i¼1


* h2 ¼

N2 1 X h2i N2 i¼1

(60)

(61a)

+ (62a)

(h1i and h2i denote hi of atoms of species 1 (i = 1…N1) and 2 (i = 1…N2), respectively), calculated at thermodynamic equilibrium, can be considered [16,21] as certain approximations, but unfortunately not necessary good approximations, of the thermody 1 and H 2 (see Equation (49)–(54)). It is easy to namically defined partial enthalpies H 2 (see Equation (54)), the following see that for h1 and h2 , as in the case of H1 and H relation holds: H ¼ N1 h1 þ N2 h2 :

(62) Furthermore, in an analogy with Equation (19d), we can introduce hc for the direct c as: comparison with H c ¼ m2 h 1 m1 h2 : h (63) m m In the present calculations, we considered in detail eight state points (see Table 1) within the temperature range of 4000–1500 K. In all MD simulations reported here, the equations of atomic motion were numerically integrated according to the well-known Verlet algorithm [59] with a time step Δt = 1.5 fs. During the simulation, the total momentum of the model system was conserved at zero value. We started our MD simulations by melting a perfect B2-NiAl crystal in a cubic simulation cell of N = 4394 atoms (NNi = NAl = 2197) with periodic boundary conditions in all three directions using NPT (isothermal–isobaric) ensemble dynamics at the temperature 4000 K and zero pressure for 6 ns. We used a Nosé-Hoover thermostat and a Nosé-Hoover barostat for the NPT ensemble. Next, the MD ensemble was switched to NVE (microcanonical) ensemble and an additional 9 ns long equilibration run was performed. The average equilibrium value of the volume of the cubic simulation cell obtained in the NPT ensemble was used as an input for the MD simulations in the NVE ensemble. During this NVE run, the temperature and pressure levelled out at some average equilibrium values that were close to those given in the NPT ensemble. After this, the velocities of the atoms were reinitialized to the next considered temperature 3000 K and the same sequence of runs (6 ns NPT run followed by a 9 ns NVE run) was employed to bring the system to equilibrium at 3000 K. Such simulations were repeated to obtain the equilibrium state of the liquid Ni50Al50 alloy at each progressively lower temperature using the equilibrium state of the melt at the preceding higher temperature. The calculations were made with a temperature step 1000 K in the temperature range of 4000–2000 K and with a temperature step 100 K in the temperature range of 2000–1500 K.

@T V ;N1 ;N2

@P

V1 V2 1 H 2 H c H Qc Q0 c DT

T

@E @N1 V ;T ;N 2 @P V @N 1 V ;T;N 2 κT

−5.14 4.81 2.04 1.27 6.83 10.9 9.8 18.4 −4.47 −3.13 1.47 0.70 −0.77 −0.70

eV Å3 eV−1 10−2 GPa−1 10−2 eV Å−3 GPa Å3 Å3 eV eV eV eV eV 10−11 m2 s−1 K−1

1510 4.5 72.1 14.1 −3.81 −3.80 −4.83 −2.77 0.76

eV

K 10−4 eV Å−3 MPa Å3 eV eV eV eV eV

T P

X ¼ NV e ¼ NE h ¼ HN h1 h2 hc

Units

Quantities

4.69 2.11 1.32 6.86 11.0 9.9 18.5 −4.45 −3.09 1.43 0.69 −0.74 −0.76

−5.13

1605 2.5 40.0 14.2 −3.77 −3.77 −4.80 −2.74 0.74

4.49 2.16 1.35 6.93 11.1 9.7 18.9 −4.45 −3.05 1.38 0.66 −0.72 −0.77

−5.12

1689 −4.2 −67.3 14.3 −3.74 −3.75 −4.77 −2.72 0.72

4.41 2.23 1.39 6.97 11.2 9.8 19.2 −4.41 −2.99 1.32 0.64 −0.68 −0.81

−5.09

1798 −0.9 −14.4 14.5 −3.70 −3.70 −4.73 −2.68 0.70

Values

4.36 2.26 1.41 7.09 11.4 9.9 19.3 −4.36 −2.96 1.31 0.63 −0.68 −0.86

−5.06

1906 2.2 35.3 14.6 −3.67 −3.66 −4.68 −2.64 0.67

4.22 2.30 1.44 7.19 11.5 9.7 19.7 −4.33 −2.91 1.26 0.59 −0.67 −0.91

−5.03

2012 4.4 70.5 14.7 −3.63 −3.62 −4.64 −2.61 0.65

3.37 3.04 1.90 7.42 11.9 10.2 21.6 −4.02 −2.58 1.00 0.55 −0.45 −0.75

−4.78

3017 4.3 68.9 15.9 −3.31 −3.30 −4.29 −2.32 0.48

2.71 3.82 2.39 7.33 11.7 10.4 23.8 −3.73 −2.29 0.79 0.48 −0.31 −0.52

−4.49

4004 0.8 12.8 17.1 −3.01 −3.01 −3.95 −2.07 0.35

Table 1. Thermodynamic quantities, heats of transport and thermal diffusion coefficient calculated for the eight studied state points of the model of the liquid Ni50Al50 alloy. Subscripts 1 and 2 represent Ni and Al, respectively. All quantities present in this table are introduced in the text.


3592 A.V. Evteev et al.

3593

Then, the equilibration at the considered eight state points (see Table 1) was followed by 3 successive 9 ns long (each) production runs in the NVE ensemble. The reported results were averaged over these 3 production runs. The correlation length 3 ps and the total number of time origins about 1.8 × 107 were used for calculations of the correlation functions C11 ðt Þ and C1q ðt Þ (see Equations (43)–(46) and Figures 1 and 2)). @P @E @P , , Finally, four thermodynamic partial derivatives @N @T V ;N1 ;N2 @N1 T ;V ;N 1 T ;V ;N 2 2 @V and @P T ;N1 ;N2 were calculated in the direct vicinity of the eight state points, using the NVT (canonical) ensemble (see Table 1 and, as example, Figures 3–6 for a state point at 1605 K). Each point in Figures 3–6 (the methodology described here was exactly the same for other state points) was obtained using a combination of 1.5 ns long equilibra@E and tion run and 1.5 ns long production run. During calculations of @N 1 T ;V ;N 2 @P , the initial configurations for each point in Figures 3 and 4 were obtained @N1 T ;V ;N2

10

-4

10

-5

10

-6

10

-7

10

-5

10

-6

10

-7

10

-5

10

-6

10

-7

10

-5

10

-6

10

-7

T=4004 K

T=1798 K

T=3017 K

T=1689 K

T=2012 K

T=1605 K

T=1906 K

T=1510 K

2 -1

VkBTL11 (m s )



0.01

0.1

1

0.01

0.1

1

Time (ps)

Figure 1. The time integral of the correlation function C11 ðtÞ (see Equations (43), (45) and (47)) calculated for the MD model of liquid Ni50Al50 alloy containing N = 4394 atoms in a reference frame fixed relative to the centre of mass of the system.

A.V. Evteev et al. 10

-4

10

-5

10

-6

10

-7

10

-5

10

-6

10

-7

10

-5

10

-6

10

-7

10

-5

10

-6

10

-7

T=4004 K

T=1798 K

T=3017 K

T=1689 K

T=2012 K

T=1605 K

T=1906 K

T=1510 K

2 -1

VkBTL1q (eVm s )


3594

0.01

0.1

1

0.01

0.1

1

Time (ps)

Figure 2. The time integral of the correlation function C1q ðtÞ (see Equations (44), (46), (48) and (57)) calculated for the MD model of liquid Ni50Al50 alloy containing N = 4394 atoms in a reference frame fixed relative to the centre of mass of the system.

by creating or deleting atoms one by one, starting from the models used for calculations of the correlation functions C11 ðt Þ and C1q ðt Þ. To avoid highly overlapped atoms in the initial configurations during the creation of new atoms, we perform an energy minimization to remove the overlapping before performing the combination of equilibration and production runs. 4. Results and discussion First of all, we must match our model system to the notation used in Sections 2 and 3. For this reason, in all our calculations we arbitrarily assume that Ni and Al are the first and the second components in the system (Ni 1 and Al 2, so that m1 = 58.71 amu. and m2 = 26.98 amu.), respectively. Hence, for the MD model of the liquid Ni50Al50 alloy, we can rewrite the interdiffusion flux given by Equation (3) in the following form,


3595

-16530 -16540 T=1605 K

-16550 -16560

E (eV)

-16570 -16580

-16600 -16610 -16620 -16630 2189 2191 2193 2195 2197 2199 2201 2203 2205

N Ni

Figure 3. The internal energy E of the model system containing NAl=2197 aluminium atoms and NNi nickel atoms at constant temperature 1605 K. 60 50 T =1605 K

40 30 20

PV (eV)


-16590

10 0 -10 -20 -30 2189

2191

2193

2195

2197

2199

2201

2203

2205

NNi

Figure 4. The product of pressure and volume PV of the model system containing NAl=2197 aluminium atoms and NNi nickel atoms at constant temperature 1605 K.

3596

A.V. Evteev et al. (64)

In Figures 1 and 2, we show the time integrals of the calculated correlation functions C11 ðt Þ and C1q ðt Þ, respectively, at the considered eight state points for the MD model of the liquid Ni50Al50 alloy. The average system temperature was always close to the target temperature, while the system pressure was close to zero. The time integrals of the both correlation functions C11 ðt Þ and C1q ðt Þ attain their asymptotic values in relatively short times which are within 0.5 ps at all eight state points studied. The asymptotic values of the integrals allow us to estimate, according to Equations (43) and (44), the phenomenological coefficients L11 and L1q in a reference frame fixed relative to the centre of mass of the system. Using these asymptotic values of the phenomenological coefficients L11 and L1q, we estimated, according to Equation (25), the heat of transport Qc at different temperatures in a reference frame fixed relative to the centre of mass of the system. We report the heat of transport Qc as well as h1 , h2 and hc (obtained as a matter of course during these calculations) at the eight considered state points in Table 1. In Figures 3 and 4, we show, as an example, results for the internal energy E and the product of pressure and volume PV of the system, respectively, as functions of the number of Ni atoms at a constant temperature 1605 K. These data fall almost exactly on straight lines at all eight state points From the of the dependencies, studied. slopes @E @P we obtained the numerical values for @N and V @N1 T;V ;N which are reported 1 T ;V ;N 2 2 in Table 1. The temperature dependence of the pressure in the vicinity of 1605 K is shown in Figure 5, as an example. The linear slopes of the plots in the direct vicinity of the eight

0.0020 0.0015 0.0010

T =1605 K

0.0005 3

P (eV/Å )


1 J c ¼ cAl J Ni cNi J Al ¼ ðJ Ni J Al Þ: 2

0.0000 -0.0005 -0.0010 -0.0015 -0.0020 1560

1580

1600

1620

1640

1660

T (K)

Figure 5. The pressure P of the model of liquid Ni50Al50 alloy containing N = 4394 atoms as function of temperature T in the vicinity of 1605 K.


3597

63500

T=1605 K


3

V (Å )

63000

62500

62000

61500 -0.0050

-0.0025

0.0000

0.0025

0.0050

3

P (eV/Å )

Figure 6. The volume V of the model of liquid Ni50Al50 alloy containing N = 4394 atoms as function of pressure P at constant temperature 1605 K.

state @Ppoints allowed us to estimate the numerical values of the thermodynamic quantity T @T which are also reported in Table 1. Finally, in Figure 6 we show, as an V ;N1 ;N2 example, a volume versus pressure plot at a constant temperature 1605 K. In this case, the linear slopes of the plots in the direct vicinity of the eight state points allowed us to estimate the numerical values of the isothermal compressibility given by Equation (53) which are presented in Table 1 too. The thermodynamic quantities reported in Table 1 allow us to obtain numerical val 1 and H 2 by using Equations (51)–(54). The numerical ues for the partial enthalpies H values of H1 and H2 as well as Hc given by Equation (19d) are also shown in Table 1. We estimated the reduced heat of transport Q0 c given by Equation (29) at the eight state points considered (see Table 1). Moreover, for convenience, we report in Table 1 the thermal diffusion coefficient given by Equation (42). 1, H 2 and H c from one side and h1 , h2 Now, let us make a comparison between H and hc from another side. For this reason, it is useful to consider two relative quantities: ec ¼

c hc hc H ¼ 1; Hc Hc

(65)

e1 ¼

1 h1 h1 H ¼ 1: H1 H1

(66)

and, for instance,

The relation between εc and ε1, which can be readily presented as

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A.V. Evteev et al. (67)

(see Equations (19d), (54), (62) and (63), and recall that h = H/N), clearly shows how 1 is amplified in the relative difference between the relative difference between h1 and H hc and Hc . In particular, as can be seen in Table 1, a relatively small percentage differ 1 (ε1 ≈ 0.08) is amplified to about 48% for hc and H c ence of about 8% for h1 and H (εc ≈ −0.48) at 1510 K, while a somewhat smaller percentage difference of about 6% 1 (ε1 ≈ 0.06) is amplified even stronger to about 56% for hc and H c for h1 and H (εc ≈ −0.56) at 4004 K. Thus, using h1 and h2 instead of H1 and H2 would result in an c , roughly speaking, by around 50% in the whole studied temperaunderestimation of H ture range. Moreover, it would lead for Q0 c to: (i) an underestimate of its negative value by one order of magnitude in the temperature range of 1500–2000 K, and (ii) prediction of a sign change from negative to positive in the temperature range of 2000–3000 K, following by an additional increase of its positive value in the temperature range of 3000–4000 K. In this context, it is interesting also to estimate εc (supposing that ε1 is known) for the above-mentioned Lennard–Jones pair-potential model of an equimolar argon-krypton binary liquid mixture which was suggested [16,17] to be considered as quasi-ideal because of the similarity between Ar–Ar, Kr–Kr and Ar–Kr interactions. According to the data reported in [17], we have: Ar 1 and Kr 2; c1 = c2 = 0.5; m1 = 39.95 amu. 2.0

1.5

Heat of transport (eV)


m1 h 1 m1 h 1 ec ¼ e1 1 ¼ e1 1 ð1 þ e1 Þ H1 m h1 m

1.0

0.5

0.0

-0.5

-1.0 1000

1500

2000

2500

3000

3500

4000

4500

Temperature (K)

c Figure 7. Temperature dependencies of the heat of transport parameters Qc (diamonds), H (upward facing triangles) and Q0 (downward facing triangles), representing the total heat flow, c the heat transfer due to movement of materials species and the pure heat conduction, respectively, consequent upon unit of interdiffusion flux in the model of liquid Ni50Al50 alloy. The lines show the linear fits of the data.


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Table 2. Results for the linear fits, a + bT, of the temperature dependencies of the heat of trans c and Q0 , of the model of the liquid Ni50Al50 alloy. port parameters Qc , H c


a (eV) b (10−4 eV K−1)

Qc

c H

Q0 c

0.80 −0.82

1.83 −2.66

−1.03 1.84

0.646); and h= and m2 = 83.8 amu. (m1 =m h1 1.258. Hence, if we assume ε1 ≈ 0.04 1 [16,17]) then according to (a small percent difference of about 4% for h1 and H c is preEquation (67) we have εc ≈ 0.26, i.e. the relative difference between hc and H dicted to be about 26%. Thus, in a problem dealing with thermotransport phenomena great attention should be taken to an accurate evolution of the partial enthalpies even for such simple systems as the above-mentioned argon-krypton binary liquid mixture. Finally, in Figure 7, we show temperature dependencies of the heat of transport c and Q0 , which, it is recalled, represent the total heat flow, the heat parameters Qc , H c transfer due to movement of materials species and the pure heat conduction, respectively, consequent upon unit of interdiffusion flux. It can be seen in Figure 7 that each of the three quantities can be fitted by a linear function with a reasonably good accuracy in the temperature range of 1500–4000 K. The fitting parameters of the linear functions are reported in Table 2. On the basis of the obtained results, the following behaviour of the reduced heat of transport Q0 c for the model of the liquid Ni50Al50 alloy can be predicted. The reduced heat of transport Q0 c of the model exhibits a quite large negative value −0.77 eV near the melting temperature which changes fairly slowly towards zero with increasing temperature. The effective temperature at which Q0 c is expected to reach a zero value can be estimated by extrapolation on the basis of the fitting parameters reported in Table 2. It is about 5600 K. We should also note that near the melting temperature the magnitude of the absolute value of the reduced heat of transport Q0 is notably greater than the activation energies 0.47–0.48 eV and c 0.48–0.49 eV of Ni and Al self-diffusion, respectively, in this model [45,46] (experimental value of the activation energy of the Ni self-diffusion is 0.42 eV [48]). Thus, our results predict the following picture of thermotransport in the MD model of the liquid Ni50Al50 alloy. Over the whole studied temperature range of 1500–4000 K, the interdiffusion flux (see Equation (64)) in the presence of a temperature gradient is expected to be directed from the cold end to the hot end. This suggests that Ni tends to diffuse from the cold end to the hot end while Al tends to diffuse from the hot end to the cold end. 5. Conclusions The formalism of transport in a binary system has been discussed, using, as an example, a velocity reference frame fixed relative to the centre of mass of the system. Special emphasis has been put on a detailed analysis of the heat of transport parameter. The reduced heat of transport parameter Q0 c associated with the interdiffusion flux in a binary system has been introduced. It is invariant to a change of reference frame and appears to be practically useful for direct comparison of simulation and experimental data from different sources obtained in different reference frames. In a binary system


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subjected to a temperature gradient, Q0 c characterizes part of the interdiffusion flux that is proportional to the temperature gradient, while in an isothermal binary system Q0 c represents the reduced heat flow (pure heat conduction) consequent upon unit interdiffusion flux. In the case study, the heat transport properties of the model of the liquid Ni50Al50 alloy have been investigated in detail within the temperature range of 1500–4000 K. The simulations have been performed within the framework of equilibrium MD simulations in conjunction with the Green–Kubo formalism. We have found that the reduced heat of transport Q0 c of the model, associated with the interdiffusion flux given by Equation 64, demonstrates a quite large negative value −0.77 eV near the melting temperature. This value changes fairly slowly with increasing temperature towards zero in a reasonably good agreement with a linear law. The effective temperature at which Q0 c is expected to reach a zero value has been estimated by extrapolation to be about 5600 K. Our results predict that in the liquid Ni50Al50 alloy in the presence of a temperature gradient Ni tends to diffuse from the cold end to the hot end while Al tends to diffuse from the hot end to the cold end. Acknowledgements We would like to thank Emeritus Professor Alan Allnatt (University of Western Ontario) for his very valuable comments on this research and his encouragement. This research was supported by the Australian Research Council through its Discovery Project Grants Scheme.

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