# Critical Temperature for Ordered-Disordered Phase Transformation in [Cu.sub.3]Au under Pressure.

1. IntroductionThe ordered-disordered phase transformation in [Cu.sub.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.sub.3]Au alloy from the disordered to the ordered state and the relaxation process in the phase change in [Cu.sub.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 order-disorder phase transition in [Cu.sub.3]Au has been investigated by X-ray diffraction [3].

A coarse-grained model for a [Cu.sub.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.sub.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.sub.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.sub.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.

2. Hemholtz Free Energy of [Cu.sub.3]Au Alloy by Using the SMM

Using the thermodynamic theory of alloy in [5,7], we analyze the ordered alloy [Cu.sub.3]Au into a combination of four effective metals [Cu.sup.*]1, [Cu.sup.*]2, [Au.sup.*]1, and [Au.sup.*]2. Then, the Helmholtz free energy of alloy [Cu.sub.3]Au can be calculated through the Helmholtz free energy of these effective metals and has the form

[mathematical expression not reproducible], (1)

where [P.sup.([beta])].sub.[alpha]] ([alpha] = Cu, Au; [beta] = 1, 2) is the probability so that the atom [alpha] occupies the lattice site of [beta]-type and these probabilities are determined in [8], [mathematical expression not reproducible] are the Helmholtz free energy of effective metals [Cu.sup.*]1, [Cu.sup.*]2, [Au.sup.*]1, and [Au.sup.*]2, respectively, and [S.sub.c] is the configurational entropy.

The Helmholtz free energy of effective metals [alpha] * [beta] ([alpha] = Cu, Au; [beta] = 1,2) is calculated by the SMM analogously as for pure metals [9] and is equal to

[mathematical expression not reproducible], (2)

where [u.sub.[alpha]],[k.sub.[alpha]] are parameters of the pure metal [alpha] [9], [P.sub.[alpha][alpha]]' is the probability so that the atom of [alpha]-type and the atom of [alpha]'-type ([alpha], [alpha]' = Cu, Au; [alpha] = [alpha]') are side by side, and [omega] is the ordered energy and is determined by [6]

[omega] = 1/2 ([[phi].sub.Cu-Cu] + [[phi].sub.Au-Au]) - [[phi].sub.Cu-Au], (3)

where [[phi].sub.Cu-Cu], [[phi].sub.Au-Au], and [[phi].sub.Cu-Au] are the interaction potential between atoms Cu-Cu, Au-Au, and Cu-Au, respectively, on same distance and [[DELTA].sup.(0).sub.[alpha][alpha]'], and [[DELTA].sup.(2).sub.[alpha][alpha]'], are the difference of interaction potentials and the difference of derivatives of second degree for interaction potential to displacement of atom pairs [alpha]' - [alpha]' and [alpha] - [alpha], respectively, on same distance [alpha].

Substituting (2) into (1), we obtain the expression of the Helmholtz free energy for alloy [Cu.sub.3] Au as follows:

[mathematical expression not reproducible] (4)

where [X.sub.[alpha]] = [x.sub.[alpha]] coth [x.sub.[alpha]] = (h/2[k.sub.B]T)[square root of [k.sub.[alpha]]/[m.sub.[alpha]] ([alpha] = Cu, Au), [m.sub.[alpha]] is the mass of atom [alpha], [[psi].sub.Cu] and [[psi].sub.Au] are the Helmholtz free energies of pure metals Cu and Au, respectively, and the configurational entropy of alloy has the form [6]

[mathematical expression not reproducible] (5)

3. Critical Temperature for [Cu.sub.3]Au Alloy under Pressure

The ordered-disordered phase transition in alloy [Cu.sub.3] Au is the phase transition of first type [8], where the following relations are satisfy simultaneously:

[mathematical expression not reproducible], (6)

[mathematical expression not reproducible], (7)

where [eta] is the equilibrium long-range ordered parameter at the temperature T and pressure p and is determined from condition (6) and [[eta].sub.0] is the equilibrium long-range ordered parameter at the critical temperature [T.sub.c].

The probabilities [P.sup([beta]).sub.[alpha] and [P.sub.[alpha][alpha]'] are represented through the ordered parameter [eta] by the following relations [6, 8]:

[mathematical expression not reproducible] (8)

where [[epsilon].sub.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 [[eta].sub.0] and [T.sub.c] as follows:

[mathematical expression not reproducible], (9)

[mathematical expression not reproducible], (10)

where [DELTA](a, [T.sub.c]) = (2/RT)[[[psi].sub.Cu](a) - [[psi].sub.Cu](a')] + (2/3RT) [[[psi].sub.Au](a) - [[psi].sub.Au](a')] and a and a' are the lattice parameters of alloy [Cu.sub.3]Au at the critical temperature [T.sub.c] in the ordered zone and the disordered zone, respectively.

From (9) we find the dependence of [eta] on temperature and pressure as follows:

[mathematical expression not reproducible] (11)

Second term in right side of (10) depends on temperature and pressure. At phase transition point in (9), T = [T.sub.c] and [eta] = [[eta].sub.0]. Therefore, from (9) and (10) we find the equation in order to determine [[eta].sub.0] as follows:

[mathematical expression not reproducible]. (12)

Because the parameters a and a' are somewhat different, [DELTA] (a, [T.sub.c]) has very small contribution to (12). Therefore, [DELTA] (a, [T.sub.c]) approximately does not depend on temperature and pressure and is determined at the critical point and zero pressure.

Using the expressions of [[psi].sub.[alpha]] and a in [9, 10] at the temperature T = [T.sub.c] = 665 K and pressure p = 0, we obtain [DELTA](a, [T.sub.c]) = 0.6526[[eta].sup.2.sub.0].

Substituting this value of [DELTA](a, [T.sub.c]) into (12), we find the ordered parameter [[eta].sub.0] = 0.37. Substituting this value of [[eta].sub.0] into (11), the dependence of critical temperature [T.sub.c] on pressure has the form

[mathematical expression not reproducible]. (13)

4. The Results and Discussion

At the critical temperature [T.sub.c] (~100 K), [X.sub.Cu], [X.sub.Au] are very near unit and we can take [X.sub.Cu] = [X.sub.Au] = 1. On the other hand, from [11] we find [[DELTA].sup.(2).sub.Cu-Au] = (1/6)([k.sub.Au] - [k.sub.Cu]). So, (13) has the following simple form:

[k.sub.B][T.sub.c]/[omega] = [[1.207 + 1/2 ([[k.sub.Au] - [k.sub.Cu]).sup.2]/ [[k.sub.Au][k.sub.Cu]].sup.-1] (14)

Applying the potential Lennard-Jones (n - m) [12] to interactions Cu-Cu and Au-Au and the expression of [k.sub.[alpha]] in [11], we have

([[k.sub.Au] - [k.sub.Cu]).sup.2]/[[k.sub.Au][k.sub.Cu] = [Aa.sup.2.5]X(a) + 1/[Aa.sup.2.5]X(a) - 2, (15)

where A = 0.052, X(a) = (1 - 0.02[a.sup.3.5])/(1 - 0.002[a.sup.6]), a is measured in unit of [10.sup.-10] m.

From (14), (15) and the equation of parameter a for alloy [Cu.sub.3]Au in [10], we find the dependence of the critical temperature [T.sub.c] on pressure. Our numerical calculations of the dependence of [T.sub.c](p) 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 [T.sub.c] depends near linearly on pressure with the mean speed of changing [DELTA]T/[DELTA]p = 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 [T.sub.c] are adequately described by the homogeneous reaction rate equation and an activation volume of 6.8 [cm.sup.3] /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 [eta] calculated by other statistical methods [8].

In conclusion, the obtained dependence of critical temperature on pressure (see (14)) in alloy [Cu.sub.3] Au has simple analytic form and numerical result in a good agreement with the experimental data.

https://doi.org/10.1155/2018/5686394

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.

References

[1] M. C. Franzblau and R. B. Gordon, "The Order-Disorder Transformation in Cu," Journal of Applied Physics, vol. 38, no. 1, pp. 103-110, 1967.

[2] T. Hashimoto, K. Nishimura, Y. Takeuchi, and J. Phys, "Dynamics on transitional ordering process in [Cu.sub.3]Au alloy from disordered state to ordered state," Journal of The Physical Society of Japan, vol. 45, 9 pages, 1978.

[3] K. Torli, T. Tamaki, N. Wakabayashi, and J. Phys, "Order-disorder kinetics of [Cu.sub.3]Au studied by X-Ray diffraction," Jounal of The Physical Society of Japan, vol. 59, 9 pages, 1990.

[4] Z. Lai, "Theory of ordering dynamics for [Cu.sub.3]Au," Physical Review B: Covering Condensed Matter and Materials Physics, vol. 41, no. 13, 1990.

[5] P. D. Tam and N. Q. Hoc, "Effective Metal's Model for Alloy," in Proceedings of the 6th National Congress on Physics, vol. 1, pp. 125-128, Hanoi, 2005.

[6] L. V. Panina, K. Mohri, K. Bushida, and M. Noda, "Giant magneto-impedance and magneto-inductive effects in amorphous alloys (invited)," Journal of Applied Physics, vol. 76, no. 10, pp. 6198-6203, 1994.

[7] K. Masuda-Jindo, V.V.Hung, and P. D. Tam, "Application of statistical moment method to thermodynamic quantities of metals and alloys," Calphad, vol 26, no. 1, pp. 15-32, 2002.

[8] A. A. Smirnov, Molecular Dynamic Theory of Metals, Nauka, Moscow, Russia, 1966.

[9] K. Masuda-Jindo, V. V. Hung, and P. D. Tam, "Thermodynamic quantities of metals investigated by an analytic statistical moment method," Physical Review B: Condensed Matter and Materials Physics, vol 67, article 094301, no. 9, 2003.

[10] P. D. Tam, "The lattice spacings for binary alloys AB," Communication in Physics, vol 2, pp. 78-83, 1998.

[11] P. D. Tam, "The melting temperature for binary alloys AB at various pressures," VNU Journal of Science: Natural Sciences and Technology, vol. 2, no. 35, 1999.

[12] S. Zhen and G. J. Davies, "Calculation of the Lennard-Jones n m potential energy parameters for metals," Physica Status Solidi, vol. 78, no. 2, pp. 595-605, 1983.

Pham Dinh Tam, (1) Bui Duc Tinh (iD), (2,3) Nguyen Quang Hoc, (3) and Pham Duy Tan (4)

(1) Le Quy Don University of Technology, 100 Hoang Quoc Viet, Cau Giay District, Hanoi, Vietnam

(2) Institute of Research and Development, Duy Tan University, 03 Quang Trung, Da Nang, Vietnam

(3) Hanoi National University of Education, 136 Xuan Thuy Street, Cau Giay District, Hanoi, Vietnam

(4) College of Armor, Tam Dao, Vinh Phuc, Vietnam

Correspondence should be addressed to Bui Duc Tinh; tinhbd@hnue.edu.vn

Received 18 July 2017; Accepted 19 October 2017; Published 12 April 2018

Academic Editor: Jorg Fink

Caption: Figure 1: The dependence of the critical temperature [T.sub.c] for alloy [Cu.sub.3]Au on pressure.

Table 1: Solutions of (13) at different pressures ([omega]/ [k.sub.B] = 910.6K). p (kbar) 0 5 10 15 a ([10.sup.-10] m) 2.7618 2.7591 2.7563 2.7536 [T.sub.c] (K) 665 676 686 695 p (kbar) 20 25 30 a ([10.sup.-10] m) 2.7509 2.7480 2.7453 [T.sub.c] (K) 704 711 718

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Title Annotation: | Research Article |
---|---|

Author: | Tam, Pham Dinh; Tinh, Bui Duc; Hoc, Nguyen Quang; Tan, Pham Duy |

Publication: | Advances in Condensed Matter Physics |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Jan 1, 2018 |

Words: | 2129 |

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