We use the statistical moment method to study the dependence of the critical temperature Tc for Cu3Au on pressure in the interval from 0 to 30 kbar. The calculated mean speed of changing critical temperature to pressure is 1.8 K/kbar. This result is in a good agreement with the experimental data.
HNUE1. Introduction
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]. The ordering kinetics of the order-disorder 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. 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 Cu3Au 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 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.
2. Hemholtz Free Energy of Cu3Au Alloy by Using the SMM
Using the thermodynamic theory of alloy in [5, 7], we analyze the ordered alloy Cu3Au into a combination of four effective metals Cu∗1, Cu∗2, Au∗1, and Au∗2. Then, the Helmholtz free energy of alloy Cu3Au can be calculated through the Helmholtz free energy of these effective metals and has the form(1)ψCu3Au=14PCu1ψCu∗1+3PCu2ψCu∗2+PAu1ψAu∗1+3PAu2ψAu∗2-TSc,where Pα(β)α=Cu,Au;β=1,2 is the probability so that the atom α occupies the lattice site of β-type and these probabilities are determined in [8], ψCu∗1,ψCu∗2,ψAu∗1, and ψAu∗2 are the Helmholtz free energy of effective metals Cu∗1, Cu∗2, Au∗1, and Au∗2, respectively, and Sc is the configurational entropy.
The Helmholtz free energy of effective metals α∗βα=Cu,Au;β=1,2 is calculated by the SMM analogously as for pure metals [9] and is equal to(2)ψα∗β=3Ruα∗β6kB+TXα∗β+ln1-exp-2Xα∗β,uα∗β=uα+Pαα′CαΔαα′0-2ω,Xα∗βkα∗βmα∗β,kα∗β=kα+3Pαα′CαΔαα′2,where uα,kα are parameters of the pure metal α [9], Pαα′ is the probability so that the atom of α-type and the atom of α′-type α,α′=Cu,Au;α≠α′ are side by side, and ω is the ordered energy and is determined by [6](3)ω=12φCu-Cu+φAu-Au-φCu-Au,where φCu-Cu,φAu-Au, and φCu-Au are the interaction potential between atoms Cu-Cu, Au-Au, and Cu-Au, respectively, on same distance and Δαα′0 and Δαα′2 are the difference of interaction potentials and the difference of derivatives of second degree for interaction potential to displacement of atom pairs α′-α′ and α-α, respectively, on same distance a.
Substituting (2) into (1), we obtain the expression of the Helmholtz free energy for alloy Cu3Au as follows:(4)ψCu3Au=143ψCu+ψAu+6R3TXCukCu-XAukAuΔCu-Au2-ωkBPCu-Au-TSc,where Xα=xαcothxα,xα=ħ/2kBTkα/mαα=Cu,Au,mα is the mass of atom α,ψCu and ψAu are the Helmholtz free energies of pure metals Cu and Au, respectively, and the configurational entropy of alloy has the form [6](5)Sc=-R4PCu1lnPCu1+3PCu2lnPCu2+PAu1lnPAu1+3PAu2lnPAu2.
3. Critical Temperature for Cu3Au Alloy under Pressure
The ordered-disordered phase transition in alloy Cu3Au is the phase transition of first type [8], where the following relations are satisfy simultaneously:(6)∂ψCu3Au∂ηη=η0=0,(7)ψCu3Auη=η0=ψCu3Auη=0,where η is the equilibrium long-range ordered parameter at the temperature T and pressure p and is determined from condition (6) and η0 is the equilibrium long-range ordered parameter at the critical temperature Tc.
The probabilities Pα(β) and Pαα′ are represented through the ordered parameter η by the following relations [6, 8]:(8)PAu1=14+34η,PAu2=14-14η,PCu1=34-34η,PCu2=34+14η,PAuCu=316+116η2+εAuCu,where εAuCu 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 η0 and Tc as follows:(9)3XCukCu-XAukAuΔCu-Au2-ωkBTη=-14ln1+3η3+η1-η3-3η,(10)3XCukCu-XAukAuΔCu-Au2-ωkBTη02=233ln3-4ln4-34-34η0ln34-34η0-334+14η0ln34+14η0-14+34η0ln14+34η0-314-14η0ln14-14η0-Δa,Tc,where Δa,Tc=2/RTψCua-ψCua′+2/3RTψAua-ψAua′ and a and a′ are the lattice parameters of alloy Cu3Au at the critical temperature Tc in the ordered zone and the disordered zone, respectively.
From (9) we find the dependence of η on temperature and pressure as follows:(11)ωkBT=14ηln1+3η3+η1-η3-3η+3XCukCu-XAukAuΔCu-Au2T,p.
Second term in right side of (10) depends on temperature and pressure. At phase transition point in (9), T=Tc and η=η0. Therefore, from (9) and (10) we find the equation in order to determine η0 as follows:(12)-η0ln1+3η03+η031-η02=-4Δa,Tc+833ln3-4ln4-34-34η0ln34-34η0-334+14η0ln34+14η0-14+34η0ln14+34η0-314-14η0ln14-14η0.
Because the parameters a and a′ are somewhat different, Δa,Tc has very small contribution to (12). Therefore, Δa,Tc approximately does not depend on temperature and pressure and is determined at the critical point and zero pressure.
Using the expressions of ψα and a in [9, 10] at the temperature T=Tc=665K and pressure p=0, we obtain Δa,Tc=0.6526η02.
Substituting this value of Δa,Tc into (12), we find the ordered parameter η0=0.37. Substituting this value of η0 into (11), the dependence of critical temperature Tc on pressure has the form(13)kBTc=1.207+3XCukCu-XAukAuΔCu-Au2Tc,p-1ω.
4. The Results and Discussion
At the critical temperature Tc (~100 K), XCu,XAu are very near unit and we can take XCu=XAu=1. On the other hand, from [11] we find ΔCu-Au(2)=1/6kAu-kCu. So, (13) has the following simple form:(14)kBTcω=1.207+12kAu-kCu2kAukCu-1.
Applying the potential Lennard-Jones (n-m) [12] to interactions Cu-Cu and Au-Au and the expression of kα in [11], we have(15)kAu-kCu2kAukCu=Aa2.5Xa+1Aa2.5Xa-2,where A=0.052,Xa=1-0.02a3.5/1-0.002a6,a is measured in unit of 10-10 m.
From (14), (15) and the equation of parameter a for alloy Cu3Au in [10], we find the dependence of the critical temperature Tc on pressure. Our numerical calculations of the dependence of Tcp with the values of pressure from 0 to 30 kbar are given in Table 1 and represented in Figure 1.
Solutions of (13) at different pressures ω/kB=910.6K.
p (kbar)
0
5
10
15
20
25
30
a10-10m
2.7618
2.7591
2.7563
2.7536
2.7509
2.7480
2.7453
Tc (K)
665
676
686
695
704
711
718
The dependence of the critical temperature Tc for alloy Cu3Au on pressure.
From Figure 1 we see that, in the interval of pressure from 7 to 21 kbar, the critical temperature Tc depends near linearly on pressure with the mean speed of changing ΔT/Δp≈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 Tc are adequately described by the homogeneous reaction rate equation and an activation volume of 6.8 cm3/mole of atoms. The magnitude of this activation volume indicates that the formation of vacancies on the 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 η calculated by other statistical methods [8].
In conclusion, the obtained dependence of critical temperature on pressure (see (14)) in alloy Cu3Au has simple analytic form and numerical result in a good agreement with the experimental data.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was carried out by the financial support from HNUE, the Le Quy Don University of Technology.
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