­­Potential energies of characteristic atoms on basis of experimental heats of formation of AuCu and AuCu₃ compounds (Ⅰ)

XIE You-qing(谢佑卿)^{1, 2, 3}, LIU Xin-bi(刘心笔)^{1, 2, 3}, LI Xiao-bo(李晓波)⁴,

PENG Hong-jian(彭红建)^{1, 2, 3}, NIE Yao-zhuang(聂耀庄)^{1, 2, 3}

1. School of Materials Science and Engineering, Central South University, Changsha 410083, China;

2. Powder Metallurgy Research Institute, Central South University, Changsha 410083, China;

3. State Key Laboratory of Powder Metallurgy, Central South University, Changsha 410083, China;

4. College of Mechanical Engineering, Xiangtan University, Xiangtan 411105, China

Received 30 March 2009; accepted 2 June 2009

Abstract: The systematic science of alloys(SSA) is a framework of the total energy and total volume able to be separated. The potential energy sequences of characteristic atoms at the central sites of the basic clusters in the fcc-based lattice Au-Cu system are separated out from smaller experimental heats of formation of L1₀-AuCu and L1₂-AuCu₃ compounds only, by nine potential energy E-functions and through the use of structural unit inversion method. From these potential energy sequences, the potential energies and heats of formation of the disordered Au_1-xCu_x alloys at 0 K are calculated. The potential energies, heats of formation and T_c-temperatures of order-disorder transitions of the L1₀-AuCu, L1₂-Au₃Cu and L1₂-AuCu₃ compounds, as well as the Au₃Cu-, AuCu- and AuCu₃- type ordered alloys with maximal ordering degrees are calculated too. The results show that the 5th E-function may be chosen for developing it into the free energy-, enthalpy-, vibrational energy- and vibrational entropy-functions for describing thermodynamic properties of the compounds, ordered and disordered phases and for establishing the phase diagram of the Au-Cu system in the future.

Key words: systematic science of alloys; Au-Cu system; potential energy; heat of formation; order-disorder transition temperature

1 Introduction

An alloy system contains three structure levels: the phase level of organizations, atomic level of phases and electronic level of atoms. In order to get a entirely understanding of the alloy systems, to establish phase diagrams and to search a method for designing of alloys, the SSA framework[1-4] has been established based on two scientific philosophy propositions. A diversity of structures, properties and features of whether matter or nonmatter systems should be attributed to combination and arrangement of structural units in the structural unit sequence. For examples, the diversity of atoms in the atomic system is attributed to arrangement of electrons in the electronic orbital (state) sequence described by four quantum numbers, which are the main quantum number, angular quantum number, magnetic quantum number and spin quantum, under the influence of nuclei; the diversity of substances in the matter system is attributed to compositions of elements in the periodic sequence of elements; the diversity of quantities in metrology is described by combination and arrangement of basic numbers in the basic number sequence, which are Arabic numerals; the diversity of species in the biological system is attributed to splices of the genes in the gene sequence; and the diversity of individual characters in character system is described by combination and arrangement of stroke elements in the stroke element sequence, which are the dot, horizontal, vertical, left-falling, right-falling, turning and hook strokes. And a systematic theory of any complex system described quantitatively should be constructed by structural unit sequence-, equation- and information-chains. Besides the quantum-mechanics of free atom system, there is few or no one described quantitatively. The SSA framework is constructed indeed by the structural unit sequence-, equation- and information-chains.

1.1 Models and structure unit sequence chain

The atomic level of alloy phases in the SSA framework involves three models for constructing diversity of structures and properties of alloy phases [5-8].

1) The basic cluster overlapping(BCO) model, of which the structural units are a pair of  basic cluster sequences in a based lattice (such as fcc, hcp and bcc) of binary alloy systems, with each cluster consisting of a central atom, the first neighbor configuration, the second neighbor configuration and the third neighbor configuration. The actions of the BCO model are to give information about an atomic arrangement of alloy phases described by overlapping pattern of basic clusters and to determine the splitting order on the potential energy, volume and electronic structure of the central atoms, which may have one, two and three splitting orders respectively corresponding to the basic clusters with one, two and three neighbor configurations. Using one order split the basic consists of a central atom and the first neighboring configuration [(I-i)Au, iCu], which contains i Cu-atoms and (I-i) Au-atoms. Here the symbol α denotes Au or Cu; I is the coordinative number and equals 12 for the fcc based lattice; and i can change from 0 to 12.

2) The characteristic atom arranging(CAA) model, of which the structural units are a pair of characteristic atom sequences, with each characteristic atom being the central atom of a specific basic cluster. The characteristic  atom has own characters at the ground state: potential energy  volume  and electronic structure  The actions of the CAA model are to give information about an atomic arrangement of alloy phases described by potential energy pattern, volume pattern and electronic structure pattern of the characteristic atoms occupied at the various lattice sites in intermetallics[5-8], to derive equations for calculating concentrations of various characteristic atoms and configurational entropy of ordered and disordered alloys as functions of composition(x) and ordering degree(σ) and to establish composition-ordering degree dependent E(x, 0, σ), V(x, 0, σ) and ψ(x, 0, σ) functions of potential energy, volume and electronic structure of alloy phases at 0 K or without considering temperature effect.

Therefore, in a based lattice the variations in potential energy, volume and electronic structure of intermetallics, various type ordered alloys and disordered alloys with composition and ordering degree can be calculated by the same information about a pair of potential energy sequences, a pair of volume sequences and a pair of electronic structure sequences, which belong to a pair of characteristic atom sequences.

3) The characteristic crystal mixing(CCM) model, of which the structure units are a pair of invented characteristic crystal sequences, with each characteristic crystal consisting of the same characteristic atoms of the identical potential energy, identical volume and identical electronic structure. The actions of the CCM model are to establish a set of temperature(T) dependent functions of the energetic and volumetric properties consisting of the general thermal expansion coefficient β^v(T) function, general thermal expansion volume V^v(T) function, general vibrational capacity  function, general vibrational energy U^v(T) function, general vibrational entropy S^v(T) function, enthalpy H(T) function and free energy G(T) function of the characteristic crystals and to derive a set of  composition-temperature-ordering degree dependent functions of the the general thermal expansion coefficient β^v(x, T, σ), general thermal expansion volume V^v(x, T, σ), general vibration heat capacity  general vibrational energy U^v(x, T, σ), general vibration entropy S^v(x, T, σ), enthalpy H(x, T, σ) and characteristic free energy G*(x, T, σ) without containing configuration entropy S^c(x, σ) of alloy phases. And finally, by combining the G*(x, T, σ) function with partition function, the free energy G(x, T, σ) function can be derived, where the G(x, T, σ) function contains configuration entropy S^c(x, σ).

These three models are proposed in order to overcome disappointments of the atomic pair interaction model and central atom model[9] as well as effective cluster interaction model[10-12].

1.2 Equation and information chains

1.2.1 Additive law of q-properties of characteristic crystals

According to CCM model, the extensive properties q(x, T, σ), q_A(x, T, σ), q_B(x, T, σ) of a given alloy phase and its components can be obtained by an additive law of the q-properties of characteristic crystals (in terms of CCA law)[5-8]:

 (1)

where  and  are the concentrations of the characteristic atoms.

1.2.2 Nine E(x, 0, σ)- and nine V(x, 0, σ)-functions of alloy phases at 0 K

In order to make Eq.(1) become a simple, applicable and separable q(x, 0, σ) function (here q denotes E or V), three type relations of  and  with i have been designed[1, 8]:

TypeⅠof straight line relation:

                (2)

Type Ⅱ of concave parabola relation:

               (3)

Type Ⅲ of convex parabola relation:

    (4)

where  and denote, respectively, potential energies of the primary  and  characteristic crystals;  and  denote, respectively, potential energies of the terminal  and  characteristic crystals.

By combining Eqs.(2), (3), (4) and substituting them into Eq.(1), nine E(x, 0, σ) functions can be obtained. In Table 1 the nine general E(x, 0, σ)-functions of alloy phases can be used to compounds, ordered and disordered alloy phases. In Table 2, the nine E(x, 0, 0)- functions can be used to the disordered alloy phase only.

Table 1 Nine general potential energy E(x, 0, σ)-functions of alloy phases at 0 K

Table 2 Nine potential energy E(x, 0, 0)-functions of disordered alloy phases at 0 K

1.3 Methodology

From the first proposition, an inference can be drawn: “The whole can be reproduced from parts”. For example, the whole of a tree can be reproduced from a seed, a leaf or a branch of the tree in biologic systems; The whole information of an alloy system can be reproduced from disordered alloys, ordered alloys or intermetallics. This means that “The total potential energies and total volumes of a few alloys can be separated into the potential energy sequence and volume sequence of the characteristic crystals (atoms), from which the whole information about energetic and volumetric properties, electronic and crystalline structures of all alloy phases can be reproduced”. Therefore, the systematic study for an alloy system in the SSA framework is divided into three step investigations.

1) The aims of the first step investigation are to choose E(x, 0, σ) and V(x, 0, σ) functions and to determine a pair of potential energy sequences and a pair of volume sequences of characteristic crystals (atoms), through resolving nine E(x, 0, σ)- and V(x, 0, σ)- functions on the basis of FP-calculated heats and volumes of formation of several intermetallics only, or of experimental heats and volumes of formation of several intermetallics only and several disordered alloys only by the structural unit inversion method; to determine a pair of electronic structure sequences of characteristic crystals (atoms), according to potential energies and volumes of characteristic crystals (atoms), then to study energetic properties, volumetric properties and electronic structures of alloy phases at ground state (0 K).

2) The aims of the second step investigation are to obtain energetic and volumetric properties of characteristic crystals as function of temperature, to obtain the energetic and volumetric property equations of alloy phases as functions of composition, temperature and ordering degree and then to study energetic and volumetric properties, crystalline parameters and electronic structures of alloy phases in the whole ranges of composition, temperature and ordering degree.

3) The aims of the third step investigation are to study stability, transformation and equilibrium of alloy phases, to establish phase diagram and to provide a systematic knowledge about energetic and volumetric and structural properties, crystalline and electronic structures of characteristic crystal (atom) sequences for designing applied alloys. That is called as design engineering of characteristic crystal (atom) sequences of alloy systems.

The Au-Cu system has a diversity of alloy phases, which include intermetallics with various space groups, ordered alloys with Au₃Cu-, AuCu-, and AuCu₃-types and disordered alloys, a relative simplicity for systematical study, which indicates all solid phases exist in the fcc-based lattice only, and is rich in order-disorder phenomena with phase transitions at Au₃Cu, AuCu and AuCu₃, that leads to the fact that it has become a studying platform for nearly all theories and experimental techniques of alloys. Now, it is also used as the platform systematically to represent the SSA framework through a series of works. The outline of remainder of this work is as follows. In the next section, the results are presented. The potential energy sequences of the  and characteristic atoms are calculated by nine E-functions at 0 K; The potential energies and heats of formation of disordered Au_1-xCu_x alloys are calculated by nine E-functions at 0 K; The potential energies and heats of formation and critical T_c-temperatures of the order-disorder transformation of Au₃Cu, AuCu and AuCu₃ compounds are calculated by nine E-functions; The potential energies and heats of formation and critical T_c-temperatures of the order-disorder transformation of Au₃Cu-, AuCu- and AuCu₃-type ordered Au_1-xCu_x alloys, and some general conclusions are then presented.

2 Results

2.1 Potential energy sequences of characteristic atoms

The experimental data show that from 0 K to 560 K for AuCu compound and from 0 K to 400 K for AuCu₃ compound, their heats of formation have no variations with temperature[13]. Therefore, it can be considered that the ?H_exp(AuCu, 298, 1)=?E(AuCu, 0, 1)=-8 745 J/mol, and ?H_exp(AuCu₃, 298, 1)=?E(AuCu₃, 0, 1)= -7 164 J/mol[13], which are smaller respectively than ?H_exp(AuCu, 298, 1)=-9 337 J/mol and ?H_exp(AuCu₃, 298, 1)=-7 268J /mol[14-15]. From the smaller experimental heats of formation, the potential energy  and  sequences of the  and  characteristic atoms in the Au-Cu system separated out by nine E-functions in Table1 are listed in Table 3 and shown in Fig.1. From these results, it can be known that the potential energy sequences obtained from nine E-functions are different and that the  and  sequences obtained by the 4th, 5th and 7th E-functions are different too, even the average potential energies of the compounds, ordered and disordered alloys calculated by them are, respectively, equivalent. The 2nd and 3rd E-functions have the same characters (see results of 2.2, 2.3 and 2.4 in the present section).

Fig.1 Potential energy  and  sequences of  and  characteristic atoms in Au-Cu system obtained by the 4th (a), 5th (b) and 7th (c) E-functions

Table 3 Potential energy sequences of  and  characteristic atoms in Au-Cu system calculated by nine E-functions at 0 K (J/mol)

2.2 Energetic properties of disordered Au_1-xCu_x alloys and their components

Fig.2 shows the distribution of the  and  of characteristic atoms in the disordered Au_1-xCu_x alloys.

Fig.2 Concentrations  and  in disordered Au_1-xCu_x alloys as functions of composition x(Cu)

According to the potential energy sequences of the characteristic atoms in Table 3, the E(x, 0, 0), E_Au(x, 0, 0) and E_Cu(x, 0, 0) potential energies of disordered Au_1-xCu_x alloys and their components as well as their ?H(x, 0, 0) heats of formation calculated by nine E-functions in Table 2 are listed in Table 4 and shown in Fig.3. From these results, the following knowledge can be obtained.

Fig.3 Average potential energies E(x, 0, 0), E_Au(x, 0, 0) and E_Cu(x, 0, 0) of disordered Au_1-xCu_x alloys and their components calculated from the 2nd, 3rd, 4th, 5th and 7th E-functions (a) and ?H(x, 0, 0) for the disordered Au_1-xCu_x alloys calculated from nine E-functions, together with experimental heats of formation at 320 K[15] (denoted by circles) (b)

Table 4 Potential energies E(x, 0, 0), E_Au(x, 0, 0), E_Cu(x, 0, 0) and heats of formation ?H(x, 0, 0) of disordered Au_1-xCu_x alloys calculated by nine E-functions at 0 K (J/mol)

1) The average E(x, 0, 0) potential energies of the disordered Au_1-xCu_x alloys obtained from nine E-functions are different each other. The average E_Au(x, 0, 0) and E_Cu(x, 0, 0) potential energies of the Au and Cu components obtained by nine E-functions are different each other too. But there is no E-function, which can well describe the experimental heats of formation in the whole compositional range. It has been discovered that the transformation of the disordered Au_0.5Cu_0.5 alloy to ordered AuCu alloy is fast at temperatures below 573 K so that it is rather difficult to fully suppress ordering on quenching[17-20]. Therefore, the experimental heats of formation of so-called disordered alloys in the compositional range (atomic fraction) 25%≤x(Cu)≤75% are referred to partly-ordered alloys with short- range ordering degrees. The details will be reported in the second part of this investigation.

2) The average potential energies and heats of formation of the disordered Au_1-xCu_x alloys obtained by the 5th E-functions are higher than corresponding experimental values. It may be chosen to describe the Au-Cu system, considering that the experimental heats of formation of the so-called disordered alloys are referred to those of partly-ordered alloys with short-range ordering degrees.

2.3 Energetic properties of L1₂-Au₃Cu, L1₀-AuCu and L1₂-AuCu₃ compounds

In the SSA framework, the crystalline structures of compounds are described by the characteristic atom occupation pattern, in which one can distinguish characteristic atoms occupied at the lattice sites (see Fig.4).

Fig.4 Characteristic atom occupation patterns of L1₂-Au₃Cu (a), L1₀-AuCu (b) and L1₂-AuCu₃ (c) crystalline structures

According to the potential energy sequences of the characteristic atoms in the fcc-based Au-Cu system, the potential energies  and  of characteristic atoms occupied at the ith sites in the cell, the average potential energies E(x, 0, 0) and heats of formation ?H(x, 0, 0) of the compounds can be calculated by CCA law. If the vibrational contribution to and the effect of variation in ordering degree on the entropy and enthalpy are neglected, the critical T_c(0)-temperature of the order-disorder phase transformation is calculated by the potential energy difference divided by the congfigurational entropy difference:

        (10)

The results are listed in Table 5 and shown in Fig.5, from which the following knowledges can be obtained:

Table 5 Potential energies of characteristic atoms (ε, in 10^-19 J/atom), total potential energies (E, in J/mol), heats of formation (?H, in J/mol) and order-disorder transformation temperatures (T_c, in K) of L1₂-Au₃Cu, L1₀-AuCu and L1₂-AuCu₃ compounds at 0 K

Fig.5 ?H_ord(0)-?H_dis(0)—T_c interrelated patterns of AuCu (a), AuCu₃ (b) and Au₃Cu (c) compounds relative to corresponding disordered Au_1-xCu_x alloys calculated from nine E-functions together with experimental data

1) The average potential energies of these compounds obtained from the 2nd to 9th E-functions are equal each to each, but their potential energies of characteristic atoms are different each to each.

2) The average potential energies and heats of  formation of the L1₀-AuCu and L1₂-AuCu₃ compound obtained by the 2nd to 9th E-functions are equal respectively to corresponding experimental values, but only the T_c-temperatures obtained by 5th E-function are closer by experiment values than ones obtained by other E-functions.

3) The average potential energies and heats of formation of the L1₂-Au₃Cu compound calculated from the 2nd to 9th E-functions are equal each to each and closer by the experiment values, but only the T_c-temperature obtained by the 5th E-function is closer by experiment value than ones obtained by other E-functions.

4) The ?H_ord(0)-?H_dis(0)—T_c(0) interrelated patterns of the AuCu, AuCu₃ and Au₃Cu compounds relative to corresponding disordered alloys obtained respectively from nine E-functions show that the heats of formation and T_c-temperatures calculated by 5th E-function can be satisfactory for describing experimental values at the same time.

2.4 Energetic properties of Au₃Cu-, AuCu- and AuCu₃-type ordered Au_1-xCu_x alloys

In the SSA framework, the ordering of disordered alloys is attributed to degeneracy in the energy, volume and electronic structure states of characteristic atoms, and the disordering of ordered alloys is attributed to split in the energy, volume and electronic structure states of characteristic atoms. According to the potential energy sequences and concentrations  and  in the ordered Au_1-xCu_x alloys with ordering degree σ, their energetic properties can be calculated by nine E-functions in Table 1. The concentrations  and  of characteristic atoms and some energetic properties of Au₃Cu-, AuCu- and AuCu₃-type ordered Au_1-xCu_x alloys with maximal ordering degrees are shown in Figs.6 and 7,  but only the calculated energetic properties of the Au-Cu-type ordered alloys are listed in Table 6.

Fig.6 Concentrations  and of characteristic atoms in Au₃Cu-, AuCu- and AuCu₃-type ordered Au_1-xCu_x alloys with maximal ordering degrees

Fig.7 Calculated heats of formation  of Au₃Cu-, AuCu- and AuCu₃-type ordered Au_1-xCu_x alloys with maximal ordering degrees by nine E-functions at 0 K (Experimental values[13] denoted by circles are respectively: ?H(Au₃Cu) =-5 736 J/mol, ?H(AuCu)=-8 746 J/mol, ?H(AuCu₃)=-7 164 J/mol

Table 6 Potential energies  and heats of formation  of AuCu-type ordered Au_1-xCu_x alloys with maximal ordering degree calculated by nine E-functions at 0 K (J/mol)

According to Eq.(10), the critical temperatures T_c(x, 0, σ_max) of Au₃Cu-, AuCu- and AuCu₃-type Au_1-xCu_x alloys translated into the disordered alloys described by nine E-functions have been calculated. The T_c(0)-temperatures obtained from 5th E-function are closer respectively by the experimental values than from other E-functions (see Fig.8).

Fig.8 Potential energy differences (a, b, c), configurational entropies  differences (d) and critical temperatures (e, f, g) of Au₃Cu-, AuCu- and AuCu₃-type ordered alloys with maximum ordering degrees relative to disordered Au_1-xCu_x alloys

3 Conclusions

1) From the smaller experimental heats of formation of the L1₀-AuCu and L1₂-AuCu₃ compounds, only the potential energy sequences have been separated out by nine E-functions. Although the potential energies of the L1₂-Au₃Cu, L1₀-AuCu and L1₂-AuCu₃ compounds calculated from the 2nd to 9th E-functions are equal each to each, the potential energies of their characteristic atoms are not equal each to each.

2) According to the potential energy sequences of the nine E-functions, the heats of formation of the disordered Au_1-xCu_x alloys have been calculated. But there is no E-function, which can well describe the experimental heats of formation in the whole compositional range. It has been discovered that the so-called experimental disordered Au_1-xCu_x alloys are the partly ordered alloys with short range ordering degree, which will be reported in the second part of this investigation.

3) The T_c(0) -temperatures of the Au₃Cu, AuCu and AuCu₃ compounds obtained from the 5th E-function are closer by the experimental values than from other E-functions. The T_c(0)-temperatures of these three compounds obtained from the other E-functions are greatly lower than experimental ones.

4) The 5th E-function may be chosen for developing it into the free energy, enthalpy, vibrational energy and vibrational entropy functions for describing the composition- temperature-ordering degree dependence of thermodynamic properties of the compounds, ordered and disordered phases, and for establishing the phase diagram of the Au-Cu system in the future.

5) Although the potential energy sequences of characteristic atoms are separated out from the experimental heats of formation of both L1₀-AuCu and L1₂-AuCu₃ compounds only, which can be used for calculating energetic properties of compounds, ordered and disordered alloys. This demonstrates that the first scientific philosophy proposition about the diversity of structures and properties of the system is correct, and that the whole energetic information can be reproduced from a part energetic information.

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Foundation item: Project(50471058) supported by the National Natural Science Foundation of China; Project(08JJ3099) supported by the Natural Science Foundation of Hunan Province, China

Corresponding author: XIE you-qing; Tel: +86-731-88879287; E-mail: xieyouq2000@yahoo.com.cn

DOI: 10.1016/S1003-6326(08)60436-7

(Edited by YANG Hua)