Critical Temperature for Ordered-Disordered Phase Transformation in Cu3Au under Pressure

The ordered-disordered phase transformation in Cu3Au under pressure has been studied by experimental methods such as electrical-resistance measurements made while the sample is at high temperature and under pressure [1] and the X-ray diffraction and resistance measurement. The transitional process in the Cu3Au alloy from the disordered to the ordered state and the relaxation process in the phase change in Cu3Au have been investigated by measuring the time dependence of the X-ray superstructure line width and the electrical resistivity [2].Theordering kinetics of the orderdisorder phase transition in Cu3Au has been investigated by X-ray diffraction [3]. A coarse-grained model for a Cu3Au system undergoing an order-disorder transition is constructed. The model is characterized by a Ginzburg-Landau Hamiltonian with a three-component order parameter and the symmetry of the Cu3Au system. The ordering dynamics of this model subjected to a temperature quench are then studied by use of Langevin dynamics.Themodel is analyzed with a generalization of the recently developed first-principles theory of unstable thermodynamic systems [4]. The ordered-disordered phase transition in alloy Cu3Au also is investigated theoretically by applying statistical methods for ordered phenomena such as the Kirkwood method, the pseudopotential method, and the pseudochemicalmethod [5, 6]. However, these works only considered the dependence of ordered parameter on temperature and considered the critical temperature at zero pressure. In this paper, the dependence of critical temperature on pressure in alloy Cu3Au is studied by using the model of effective metals and the statistical moment method (SMM). We obtained a rather simple equation describing this dependence. Our numerical calculations are in a good agreement with the experimental data.


Introduction
The ordered-disordered phase transformation in Cu 3 Au under pressure has been studied by experimental methods such as electrical-resistance measurements made while the sample is at high temperature and under pressure [1] and the X-ray diffraction and resistance measurement.The transitional process in the Cu 3 Au alloy from the disordered to the ordered state and the relaxation process in the phase change in Cu 3 Au have been investigated by measuring the time dependence of the X-ray superstructure line width and the electrical resistivity [2].The ordering kinetics of the orderdisorder phase transition in Cu 3 Au has been investigated by X-ray diffraction [3].
A coarse-grained model for a Cu 3 Au system undergoing an order-disorder transition is constructed.The model is characterized by a Ginzburg-Landau Hamiltonian with a three-component order parameter and the symmetry of the Cu 3 Au system.The ordering dynamics of this model subjected to a temperature quench are then studied by use of Langevin dynamics.The model is analyzed with a generalization of the recently developed first-principles theory of unstable thermodynamic systems [4].The ordered-disordered phase transition in alloy Cu 3 Au also is investigated theoretically by applying statistical methods for ordered phenomena such as the Kirkwood method, the pseudopotential method, and the pseudochemical method [5,6].However, these works only considered the dependence of ordered parameter on temperature and considered the critical temperature at zero pressure.
In this paper, the dependence of critical temperature on pressure in alloy Cu 3 Au is studied by using the model of effective metals and the statistical moment method (SMM).We obtained a rather simple equation describing this dependence.Our numerical calculations are in a good agreement with the experimental data.

Hemholtz Free Energy of Cu 3 Au Alloy by Using the SMM
Using the thermodynamic theory of alloy in [5,7], we analyze the ordered alloy Cu 3 Au into a combination of four effective metals Cu * 1, Cu * 2, Au * 1, and Au * 2.Then, the Helmholtz free energy of alloy Cu 3 Au can be calculated through the Helmholtz free energy of these effective metals and has the form ( (1)  Cu  Cu * 1 + 3 (2)  Cu  Cu * 2 +  (1)  Au  Au * 1 where  ()  ( = Cu, Au;  = 1, 2) is the probability so that the atom  occupies the lattice site of -type and these probabilities are determined in [8],  Cu * 1 ,  Cu * 2 ,  Au * 1 , and  Au * 2 are the Helmholtz free energy of effective metals Cu * 1, Cu * 2, Au * 1, and Au * 2, respectively, and   is the configurational entropy.
The Helmholtz free energy of effective metals  *  ( = Cu, Au;  = 1, 2) is calculated by the SMM analogously as for pure metals [9] and is equal to where   ,   are parameters of the pure metal  [9],    is the probability so that the atom of -type and the atom of   -type (,   = Cu, Au;  ̸ =   ) are side by side, and  is the ordered energy and is determined by [6] where  Cu-Cu ,  Au-Au , and  Cu-Au are the interaction potential between atoms Cu-Cu, Au-Au, and Cu-Au, respectively, on same distance and Δ (0)   and Δ (2)    are the difference of interaction potentials and the difference of derivatives of second degree for interaction potential to displacement of atom pairs   −   and  − , respectively, on same distance .

Critical Temperature for Cu 3 Au Alloy under Pressure
The ordered-disordered phase transition in alloy Cu 3 Au is the phase transition of first type [8], where the following relations are satisfy simultaneously: where  is the equilibrium long-range ordered parameter at the temperature  and pressure  and is determined from condition ( 6) and  0 is the equilibrium long-range ordered parameter at the critical temperature   .
The probabilities  ()  and    are represented through the ordered parameter  by the following relations [6,8]: where  AuCu is the correlational parameter.This parameter has small value and is ignored.Substituting (4) into ( 6) and (7), paying attention to (8), and carrying out some calculations, we obtain two equations in order to determine  0 and   as follows: of alloy Cu 3 Au at the critical temperature   in the ordered zone and the disordered zone, respectively.From ( 9) we find the dependence of  on temperature and pressure as follows: Second term in right side of ( 10) depends on temperature and pressure.At phase transition point in (9),  =   and  =  0 .Therefore, from ( 9) and ( 10) we find the equation in order to determine  0 as follows: Because the parameters  and   are somewhat different, Δ(,   ) has very small contribution to (12).Therefore, Δ(,   ) approximately does not depend on temperature and pressure and is determined at the critical point and zero pressure.

The Results and Discussion
At the critical temperature   (∼100 K),  Cu ,  Au are very near unit and we can take  Cu =  Au = 1.On the other hand, from [11] we find Δ (2)  Cu-Au = (1/6)( Au −  Cu ).So, (13) has the following simple form: Applying the potential Lennard-Jones ( − ) [12] to interactions Cu-Cu and Au-Au and the expression of   in [11], we have where  = 0.052, () = (1 − 0.02 3.5 )/(1 − 0.002 6 ),  is measured in unit of 10 −10 m.From ( 14), (15) and the equation of parameter  for alloy Cu 3 Au in [10], we find the dependence of the critical temperature   on pressure.Our numerical calculations of the dependence of   () with the values of pressure from 0 to 30 kbar are given in Table 1 and represented in Figure 1.
From Figure 1 we see that, in the interval of pressure from 7 to 21 kbar, the critical temperature   depends near linearly on pressure with the mean speed of changing Δ/Δ ≈ 1.8 K/kbar.This result agrees with experiments [1], where the rate of change of the critical temperature with pressure is 2.1 K/kbar from 7 to 21 kbar.The kinetics of the order transformation below   are adequately described by the homogeneous reaction rate equation and an activation volume of 6.8 cm 3 /mole of atoms.The magnitude of this activation volume indicates that the formation of vacancies on the Advances in Condensed Matter Physics gold sublattice is the rate-limiting step in the homogeneous ordering process.
If ignoring the second term in right side of (11) (this term depends on pressure and temperature), we obtain the expression of ordered parameter  calculated by other statistical methods [8].
In conclusion, the obtained dependence of critical temperature on pressure (see ( 14)) in alloy Cu 3 Au has simple analytic form and numerical result in a good agreement with the experimental data.

Figure 1 :
Figure 1: The dependence of the critical temperature   for alloy Cu 3 Au on pressure.