Studying the effects of vacancies on the melting temperature of agce alloy - Dang Thanh Hai

Communications in Physics, Vol. 28, No. 2 (2018), pp. 155-162 DOI:10.15625/0868-3166/28/2/11367 STUDYING THE EFFECTS OF VACANCIES ON THE MELTING TEMPERATURE OF AGCE ALLOY DANG THANH HAIa,†, VU VAN HUNGb AND GIANG THI HONGc aVietnam Education Publishing House bUniversity of Education, Vietnam National Uiversity Hanoi cFaculty of Physics, Hanoi National University of Education †E-mail: dthai@nxbgd.vn Received 21 February 2018 Accepted for publication 7 May 2018 Published 16 June 2018 Abstract. In order to evaluate the effects of vacancies on the melting temperature of AgCe alloy, the statistical moment method was used to find out the analytical expressions to determine the Gibbs free energy in the AB substitution alloy and the expression to calculate the melting tem- perature of perfect and defect AB substitution alloys. The melting temperature was calculated by the numerical calculation method on the perfect and defect AgCe alloys. The calculating results showed that the melting temperature of alloys increases with increasing pressure. The melting temperatures of defect alloys of AgCe, Ag2Ce and Ag3Ce are always slightly lower than that of perfect alloys. Especially, the melting temperatures value of the perfect and defect alloys were almost the same at high pressure. The melting temperature of alloys increase with an increase in amount of Ag containing in the alloys. The calculated results agreed well with the experimental results at zero pressure. Keywords: statistical moment method, rare earth, rare earth alloys, melting temperature, vacancy. Classification numbers: 61.66.Dk, 64.30.Ef, 64.70.kd, 64.70.qd. c©2018 Vietnam Academy of Science and Technology 156 DANG THANH HAI, VU VAN HUNG AND GIANG THI HONG I. INTRODUCTION Metals and rare earth alloys are a strategic material for high-tech industries such as elec- tronics, nuclear, optics, superconducting materials, super-magnets, metallurgy... Some materials made from rare earth oxides in nanoscale are used for polishing high precision tools such as optical glasses. . . [1]. Studying and understanding the mechanical, thermal, electrical, and optical prop- erties of rare earth materials is an urgent need, attracting much attention of scientists. The study of the melting temperature versus pressure is one of the noticeable research directions, especially the thermodynamic properties. Recent studies of AB substitution alloys have shown that a change in the composition of alloy results in a change in the thermodynamic properties and the melting temperature of the alloy. The theoretical and experimental results have shown that the vacancies concentration also influences the melting temperature of the alloy. There are many methods have been used to study the melting temperature of alloys such as diamond-anvil cell (DAC) [2], ab initio [3], first principle [4], shock-wave experiments [5]. However, these methods have high errors at high pressure and their analytical expressions are complicated. These limitations have made difficulties in numeric calculations. The experimental approaches are costly and difficult to carry out. The statistical moment method used in this study has been expected to solve the current problems. The purpose of this paper is to study the effects of vacancies on the melting temperature of rare earth substitutional alloys at different pressures. The statistical moment methods [6] were used to find out the analytical expressions to determine the Gibbs free energy needed to create a vacancy. The melting temperature of perfect and defect alloys were also calculated. All calculations were carried out on AgCe, Ag2Ce and Ag3Ce alloys. II. THEORY II.1. The free energy of defect AB substitutional alloy We determine the AB substitutional alloy containing N (NA +NB = N) atoms and n vacan- cies (n¡¡ N). The Gibbs free energy is described by following expression: G = G0 +nAg f Av +n Bg f Bv −T Snc , (1) where G0 is the free energy of the AB substitutional alloy; g f A v is the difference of Gibbs free energy of A metal when it creates one vacancy, g f Bv is the difference of Gibbs free energy of B metal when it creates one vacancy and g f Av ≈ U A 0 2 ,g f B v ≈ U B 0 2 [7]. n A and nB are vacancies that are created by moving A atom (or B atom) from their lattice point; Uα0 is the interaction energy, Uα0 = 1 2 ∑ i φαi0 (|~ai|) (α = A,B), and φi0 is a paired interaction potential between 0th and ↔ h¯ atoms. It is assumed that nAis proportional to NA(NA is the total numbers of A atoms in AB alloy); nBis proportional to NB(NB is the total numbers of B atoms in AB alloy). nA NA = nB NB = nA +nB NA +NB = n N , (2) where n = nA +nB is the total of vacancies of AB alloy. The expression (2) should be rewritten as following: STUDYING THE EFFECTS OF VACANCIES ON THE MELTING TEMPERATURE OF AgCe ALLOY 157 nA =n NA N = nCA, nB =n NB N = nCB. (3) Substitute (3) for (1) we get G = G0 +n [ CAg f Av +CBg f B v ]−T Snc . (4) If putting gABf =CAg f A v +CBg f B v , the expression (4) should be rewritten as following: G = G0 +ng f ABv −T Snc = G0 +nvNg f ABv −T Snc , (5) From the minimum condition of the Gibbs function, we obtain the expression to calculate the vacancy concentration: ( ∂G ∂nv ) P,T,N = 0⇒ nv = e− g f ABv θ , (6) where nv = nN+n is the vacancy concentration. From above expression, we get the expression to calculate the difference of the free Gibbs energy when one vacancy is created in AB substitutional alloy: gABf =− 1 2 [ CAUA0 +CBU B 0 ] . (7) II.2. The melting temperature of AB substitutional alloy at difference pressure It is known that the isothermal bulk modulus BT of metals is a function of pressure. There- fore, BT = BT (P) can be written as [7, 8]: BT (P) = B0 +B′0P+ 1 2 B′′0P 2 + ... If we only consider for a first-order approximation, BT will be writen as: BT (P) = B0 +B′0P, (8) where B0 is the isothermal bulk modulus at a pressure P = 0, and B′0, B ′′ 0 are determined by the expressions: B′0 = ( dBT dP ) P=0 ; B′′0 = ( d2BT dP2 ) P=0 . (9) Therefore the ratio of the melting temperature of crystal at pressure that differs from zero and the melting temperature of metal at pressure that equals to zero is written: Tm(P) Tm(0) = G(P) G(0) exp − P∫ 0 dP′ B(P′)  , (10) where G(P) is shear modulus at pressure P 6= 0, G(0) is shear modulus at pressure P = 0. If we only expand to first order, we will obtain: TmAB(P) TmAB(0) = G(P) G(0) exp − P∫ 0 dP′ B0 +B′0P′  . (11) 158 DANG THANH HAI, VU VAN HUNG AND GIANG THI HONG From this, we find out the expression to calculate the melting temperature of AB TmAB(P) = TmAB(0)B 1 B′0 0 G(0) . G(P)( B0 +B′0P ) 1 B′0 . (12) Therefore, if G(P), G(0), B0, B′0 and TmAB(0) are known, we will determine the melting temperature of perfect AB substitutional alloy at non zero pressure. The isothermal compressibility coefficient BT of AB substitutional alloy is determined as following [9]: BT,AB = 1 χT,AB , (13) where χT,AB = 3 ( aAB a0AB )3 2P+ 13NvAB ( ∂ 2ψAB ∂a2AB ) , (14) a0AB = cAa0A B0T,A B¯0,T + cBa0B B0T,B B¯0,T , (15) aAB = cAaA BT,A B¯T + cBaB BT,B B¯T , (16) where cA and cB are the composition of A metal and B metal in AB substitutional alloy, respectively; a0α , aα(α = A, B) is in turn the distance between neighbourα atoms at 0(K) and T (K); B0T,α and BT,α is the isothermal bulk modulus of αmetals at temperature of T ; B¯0T and B¯T,α are the average of isothermal bulk modulus in AB substitutional alloy at 0(K) and T (K). B¯0T should be written as following: B¯0T = cABT,A + cBBT,B. (17) The free energy of AB substitutional alloy is determined by: ψAB = cAψA + cBψB−T Scn, (18) where ψA , ψBis free energy of A metal and B metal, respectively. Shear modulus GAB and G0AB are determined by the Young modulus E as following [10]: G0AB = E0AB 2(1+υAB) ; GAB = EAB 2(1+υAB) ; (19) where, υ is the Poisson coefficient and E is the Young modulus. They are determined by: υAB = cAυA BT,A B¯T + cBυB BT,B B¯T , (20) E0AB = 1 pia0ABAAB ; EAB = 1 piaABAAB , (21) where AAB in the expression (21) is determined as following: AAB = cAAA BT,A B¯T + cBAB BT,B B¯T , (22) with Aα = 1kα [ 1+ 2γ 4 αθ 2 k4α ( 1+ xα cothxα2 ) (1+ xα cothxα) ] , α = A, B. STUDYING THE EFFECTS OF VACANCIES ON THE MELTING TEMPERATURE OF AgCe ALLOY 159 k = 1 2∑i ( ∂ 2φi0 ∂ 2uiβ ) eq ; γ = 1 12∑i (∂ 4φi0 ∂u4iβ ) eq +6 ( ∂ 4φi0 ∂ 2uiβ∂ 2uiγ ) eq  , (23) in which α 6= β 6= γ; x = h¯ω2θ ; θ = kBT , ω = √ k m . II.3. Melting temperature of defect AB substitutional alloy It is known that the temperature of crystals is a function of volume V , pressure P and balance vacancies concentration. This means Tm = Tm(P,V,nv),thus: Tm = Tm(P)+ ( ∂T ∂P ) nv,V P+ ( ∂T ∂nv ) V,P nv. (24) From the expression calculating the balance vacancies concentration (6), we have:( ∂nv ∂T ) P,V =kB ( ∂nv ∂θ ) P,V =kB ∂ ∂θ ( exp { −g f v θ }) P,V =kB exp { −g f v θ } ∂ ∂θ ( −g f v θ ) P,V =kBnv ( ∂g fv θ ) P,V θ −g fv θ 2 ⇒ ( ∂T ∂nv ) P,V = θ 2 kBnv . ( θ ∂g fv ∂θ −g fv )−1 . (25) Substitute (25) to (24), we get the expression to calculate melting temperature of defect AB substitutional alloy [7]: T vacancymAB = TmAB(P)+ ( ∂T ∂P ) nv,V P+ T 2mAB(P) TmAB(0) ∂g fv ∂θ − g f v kB . (26) Based on the melting temperature of perfect substitutional alloy, if we know the energy needed to create a vacancy, we will determine the melting temperature of defect substitutional alloy. 160 DANG THANH HAI, VU VAN HUNG AND GIANG THI HONG III. RESULTS AND DISCUSSIONS In this paper, we used the Lennard-Jones potential that was written as following [6]: φ(r) = D (n−m) [ m (r0 a )n −n (r0 a )m] , (27) where r0 is a distance between two atoms corresponding to minimum potential at D; n and m, the numbers determined by experiment corresponding to A metal and B metal. The potential parameters and the Poisson constant of Ag, Ce are listed in Table 1. Table 1. Parameters D, r0, m, n of Ag, Ce [11] and the Poisson constant υ of Ag, Ce [12]. Metals D/kB(K) r0(A˚) m n υ Ag 5737.19 2.8760 2.82 12.7 0.37 Ce 8510.85 3.6496 2.28 8.22 0.24 In order to carry out numerical calculations, it is necessary to know the factors such as: the isothermal compressibility coefficient B0T ,BT (13); B¯0T (17), the nearest neighbour distances (15), (16), the Poisson coefficient (20), the shear modulus (19) and the Young modulus (21). Based on these the melting temperature of perfect AB substitutional alloy (12), the Gibbs free energy (7) and the melting temperature of defect AB substitutional alloy (26) are calculated. Table 2. The melting temperature of rare earth alloys at zero pressure. Alloys TmAB-SMM(K) TmAB-Exp[13](K) AgCe 1112.2 1151.0 Ag2Ce 1191.3 1138.0 Ag3Ce 1230.4 1263.0 Table 2 is the calculation of the melting temperature results of AgCe, Ag2Ce and Ag3Ce at zero pressure by statistical moment method and the experimental results [13]. The results shown that the calculating results by the statistical moment method agreed well with experimental results. The error is less than 5%. Tables 3, 4 and 5 are the results of melting temperature calculations for perfect and defect alloys. The results show that the melting temperature of alloys increases with the increase in pres- sure. The melting temperatures of defect alloys of AgCe, Ag2Ce and Ag3Ce are always lower than those of perfect alloys. From above tables, when the pressure rises, the melting temperatures of defect alloys approach the melting temperatures value of perfect alloys. The melting temperature of alloys increases with an increase in amount of Ag containing in the alloys because the melting temperature of Ag higher the melting temperature of Ce. STUDYING THE EFFECTS OF VACANCIES ON THE MELTING TEMPERATURE OF AgCe ALLOY 161 Table 3. The pressure dependences of the melting temperature of AgCe alloy. P(GPa) 0 10 20 30 40 50 60 70 TmAB 1112.20 1250.84 1372.06 1480.80 1580.00 1671.64 1757.06 1837.28 TmABvacancy 1110.25 1248.98 1370.36 1479.32 1578.82 1670.80 1756.62 1837.28 Table 4. The pressure dependences of the melting temperature of Ag2Ce alloy. P(GPa) 0 20 40 60 80 90 100 105 TmAB 1191.28 1493.62 1743.21 1960.85 2156.29 2247.58 2335.24 2377.85 TmABvacancy 1188.77 1491.09 1740.95 1959.08 2155.21 2246.90 2335.01 2377.85 Table 5. The pressure dependences of the melting temperature of Ag3Ce alloy. P(GPa) 0 20 40 60 80 100 120 128.5 TmAB 1230.44 1553.45 1823.57 2061.37 2276.67 2475.01 2660.15 2735.53 TmABvacancy 1227.56 1550.41 1820.67 2058.82 2274.67 2473.73 2659.74 2735.53 Fig. 1. The effect of vacancies on the melting temperature of AgCe, Ag2Ce and Ag3Ce alloys under various pressures. Fig.1 presents the ratio of (TmAB−TmABvacancy)/TmAB of AgCe, Ag2Ce and Ag3Ce alloys under various pressures. This ratio is a function of pressure, the value of this ratio decreases when the pressure increases. It is concluded from the Fig.1 that the vacancy contribution is negligible 162 DANG THANH HAI, VU VAN HUNG AND GIANG THI HONG on the melting temperatures. This contribution accounts for some thousand percent of the melting temperatures at low pressures. It is also indicated from the Fig.1 shown that at zero pressure, the impact of vacancy of AgCe alloy is about 1.75%o, Ag2Ce is 2.10%o and Ag3Ce is 2.34%o. When the pressure in- creases, the contribution of vacancy gradually decreases, and the impact of vacancy decreases by zero at a definite pressure. For the alloy AgCe when pressure is 70GPa, Ag2Ce when pressure is 105GPa and Ag3Ce when the pressure is 128.5GPa the impact of vacancy is zero and at that point the melting temperature of defect alloys is equal to the melting temperature of perfect ones. IV. CONCLUSIONS The analytical expressions to calculate the Gibbs free energy needed to create a vacancy and the melting temperature of perfect and defect AB substitutional alloys were found out by the statistical moment method. The numerical calculations showed that the vacancies influence slightly the melting tem- perature of alloys. There were no effects of vacancies on the melting temperature of the alloys at high pressure. REFERENCES [1] S. D. Barrett and S. S. Dhesi, The structure of rare-earth metal surfaces, World Scientific, 2001. [2] S.-N. Luo and D. C. Swift, Physica B: Condensed Matter 388 (2007) 139. [3] A. B. Belonoshko, S. Simak, A. Kochetov, B. Johansson, L. Burakovsky and D. Preston, Phys. Rev. Lett. 92 (2004) 195701. [4] Y. Wang, R. Ahuja and B. Johansson, Phys. Rev. B 65 (2001) 014104. [5] Y. Zhang, T. Sekine, H. He, Y. Yu, F. Liu and M. Zhang, Sci. Rep. 6 (2016) 22473. [6] N. Tang and V. Van Hung, physica status solidi (b) 149 (1988) 511. [7] V. Van Hung, D. T. Hai et al., Comput. Mater. Sci. 79 (2013) 789. [8] L. Burakovsky, D. L. Preston and R. R. Silbar, J. Appl. Phys. 88 (2000) 6294. [9] D. T. Hai, V. V. Hung and H. P. T. Minh, J. Science HUE 60 (2015) 104. [10] V. V. Hung and N. T. Hai, Comput. Mater. Sci. 14 (1999) 261. [11] M. N. Magomedov, High Temperature 44 (2006) 513. [12] Periodictable, Accessed: 2018-01-30. [13] P. Nash and A. Nash, Bull. Alloy Phase Diagr.‘ 6 (1985) 350.