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The melting curve of defective substitutional alloy AB with body-centered cubic (BCC) structure under pressure is derived by the statistical moment method (SMM). The temperature of absolute stability for crystalline stateand the equilibrium vacancy concentration have been used to calculate the melting temperature.
Scientific Journal − No27/2018 63 THE MELTING CURVE OF BCC SUBSTITUTIONAL ALLOY MoNi WITH DEFECTS Nguyen Quang Hoc1, Bui Duc Tinh1, Tran Dinh Cuong1, Le Hong Viet2 Hanoi National University of Education Tran Quoc Tuan University Abstract: The melting curve of defective substitutional alloy AB with body-centered cubic (BCC) structure under pressure is derived by the statistical moment method (SMM) The temperature of absolute stability for crystalline stateand the equilibrium vacancy concentration have been used to calculate the melting temperature In limit case, we obtain the melting theory of main metal A with BCC structure The theoretical results are numerically applied for molybdenum-nickel alloy (MoNi) with using Mie-Lennard-Jones potential These results are in good agreements with experimental data and other calculations Keywords: Substitutional alloy, equilibrium vacancy concentration, absolute stability for crystalline state, statistical moment method Email: hocnq@hnue.edu.vn Received December 2018 Accepted for publication 18 December 2018 INTRODUCTION The melting temperatureis very important physical characteristic of alloys Study on the effect of pressure and impurities on the melting point of crystal pays particular attention to many researchers [1, 2] On the experimental side, we have Simon equation to describe the pressure- temperature relationship at the melting point in the case of low pressure [3] In the case of high pressure, we can find the melting curve of crystal by using Kumari - Dass equation [4] On the theoretical side, the melting happens when the Gibbs energy of solid phase is equal to the one of liquid phase However, we cannot find the explicit expression of melting temperature Tmby solving this condition, so building a theory for defining the melting properties of crystal is one of the most interesting research topics in materials science In aid of the statistical moment method (SMM), Nguyen Tang and Vu Van Hung [5, 6] show that we absolutely only use the solid phase of crystal to determine the melting Ha Noi Metroplolitan University 64 temperature Firstly they determine the absolute stability temperature TSat different pressures by using the SMM and then carry out a regulation in order to find Tm from TS The obtained results from the SMM are better than that from other methods in comparison with experiments [7, 8] Besides, in [9, 10] the authors proved that the point defects including the vacancieshas the significantly contribution on thermodynamic quantities of crystals at high temperature But in most studies the melting theory is only applied for perfect crystal For the reasons above, we will use the SMM to consider the effect of pressure, the substitutional atoms and the equilibrium vacancy concentration on the melting temperature of substitutional alloy AB with body-centered cubic (BCC) structure The melting curve of alloy MoNi has been builded in this paper METHODOF CALCULATION 2.1 The melting of perfect BCC substitutional alloy AB The cohesive energy u0Xand the crystal’s parameters k X , γ X , γ X , γ X for pure metal X (X = A, B) with BCC structure in the approximation of two coordination spheres have the form u0 X = 8ϕ XX (a1 X ) + 6ϕ XX (a2 X ), (1) (2) (1) (1) (2) k X = ϕ XX (a1 X ) + ϕ XX (a1 X ) + ϕ XX (a2 X ) + ϕ AA (a2 X ), 3a1 X a1 X (2) γ 1X = ( 4) (3) ( 2) (1) ϕ XX ( a1 X ) + ϕ XX ( a1 X ) − ϕ XX ( a1 X ) + ϕ XX (a1 X ) + 54 9a1 X 9a1 X 9a1 X + (4) 3 (1) (2) ϕ XX (a2 X ) + ϕ XX (a2 X ) − ϕ XX (a2 X ), 24 16a1 X 32a13X (4) γ X = ϕ XX ( a1 X ) + + (2) (1) ϕ XX (a1 X ) − ϕ XX ( a1 X ) + 3a1 X 3a1 X (3) 9 (1) (2) ϕ XX (a2 X ) − ϕ XX (a2 X ) + ϕ XX (a2 X ), 4a1 X 16a1 X 32a13X γ X = 4(γ X + γ X ), (3) (4) (5) Scientific Journal − No27/2018 65 where ϕ XX is the pair interaction potential between atoms X-X, a1X is the nearest neighbor distance, a2 X ∂ mϕ XX (aiX ) ( m) = a1 X , ϕ XX (aiX ) = (m = 1, 2, 3, 4; i = 1, 2) ∂aiXm By using the equation of state at K and pressure P ∂u0 X ℏω X ∂k X + PvX = −a1 X ∂a1X 4k X ∂a1 X , (6) we can find the nearest neighbor distance a1 X ( P,0) and then we can calculate the displacements of atom X from the following formula 2γ X ( P, 0)θ y X ( P, T ) = 3k X3 ( P, 0) AX ( P, 0) , (7) where θ = k BoT , k Bo is the Boltzmann constant and AX ( P,0) was given in [11] From that, we derive the nearest neighbor distance a1 X ( P, T ) at temperature T and pressure P a1 X ( P, T ) = a1 X ( P,0) + y X ( P, T ) (8) In the model of perfect BCC substitutionalalloy AB, the main atoms A stay in the peaks and the substitutional atoms B stay in the body centers of cubic unit cell (Figure 1) Figure The model of perfect BCC substitutional alloy AB The mean nearest neighbor distance for alloy AB is determined by ∑c B a = ∑c B X a AB TX 1X X X (9) TX X ∑c where c X is the atomic concentration X X = 1 , BTX is the isothermal bulk modulus [11] Ha Noi Metroplolitan University 66 From the condition of absolute stability limit for crystalline state ∂P = 0, ∂a AB T =T (10) S and the equation of state of the substitutional alloy AB P=− ∂u 3γ θ a AB cX X + G , ∑ ∂a1X 6v AB X v AB (11) where γ G is the Grüneisen parameter γG = − a AB c X ∂k X ℏω x X coth x X , x X = X , ∑ X k X ∂a1X 2θ (12) we can derive the absolute stability temperature for crystalline state in the form TS = TS , MS a AB TS = Pv AB + a AB koB MS = c ∑X k X2 X ∂ 2u0 X ℏa AB ∑X cX ∂a − 1X c X ω X ∂k X ∂ k X ∑X k 2k ∂a − ∂a 1X X X 1X , ∂k X ∂a1 X (13) Solving equation (13) will give us the value of TS Note that TSand MSmust be calculated at TS After that, because TSis not far from Tmat the same physical condition, so we can carry out a regulation in order to find Tm from TS a −a Tm ≈ TS + m S kBoγ G Pv ∂ 2u0 X cX ∂u0 X AB + ∑ + aS X 18 ∂a1 X ∂a1 X T =T aS T =T S S , (14) where am = aAB ( P, Tm ) , aS = aAB ( P, TS ) If we know the melting temperature Tm (0) at zero pressure, we have another way to calculate the melting temperature Tm ( P) at pressure P [2]as follows Tm ( P) = Tm (0) B0 G (0) B0′ G( P) ( B0 + B0′ P) B0′ , (15) Scientific Journal − No27/2018 67 where G(P)and G(0) respectively are the rigidity bulk modulus at pressure P and zero dBT , dP P =0 pressure, B0 is the isothermal elastic modulus at zero pressure, B0′ = BT = BT ( P) is the isothermal elastic modulus at pressure P 2.2 The melting of defective BCC substitutional alloy AB From the minimum condition of real Gibbs thermodynamic potential ∂G XR = , we can find the equilibrium vacancy concentration nvX in defective ∂nvX P ,T ,nvX metal X as follows nvX g vf X = exp − θ , (16) where g vf X is the change of Gibbs thermodynamic potential when a vacancy is formed g vf X = − u0 X u + ∆ Xψ X0 + P∆v ≈ − X (17) Figure The model of perfect metal (left) and defective metal (right) with BCC structure The equilibrium vacancy concentration in defective substitutional alloy AB is determined by g vf AB nv = exp − θ ∑ c X g vf X = exp − X θ (18) Ha Noi Metroplolitan University 68 At constant pressure and constant substitutional atom concentration, the melting temperature TmR of defective alloy is the function of the equilibrium vacancy concentration nv In first approximation the melting temperature TmR can be expanded in term of nvas ∂T Tm2 ( P ) TmR ( P ) = Tm ( P ) + m nv = Tm ( P ) + , f f g vAB ∂g vAB ∂nv Tm ( P ) − koB ∂θ f ∂gvAB a ∂u = − AB αT ∑ c X X , ∂θ ∂a X 4koB X (19) where α T is the thermal expansion coefficient of perfect alloy AB NUMERICAL RESULTS AND DISCUSSION For alloy MoNi, we use the Mie-Lennard-Jones pair potential where potential parameters are given in Table D r0 r0 ϕ (r ) = m − n n − m r r n m (20) Table The potential parameters D, m, n, r0for materials Mo [12] and Ni [13] Interaction D (eV) m n r0 (10-10 m) Mo-Mo 1.7042 1.93 7.68 2.72 Ni-Ni 0.3727 8.0 9.0 2.478 Approximately ϕ Mo-Ni ≈ (ϕ Mo-Mo + ϕ Ni-Ni ) (21) We have some comments about the melting temperature of alloy MoNi Firstly, at the same concentration of substitutional atoms when pressure increases, the melting temperature of alloy MoNi also increases For example, at cNi = 1.8 % when pressure P increases from to 80 GPa, the melting temperature Tm of perfect alloy MoNi increases from 1754 to 3183 Kand the melting temperature Tm of defective alloy MoNi increases from 1703 to 3023 K.Secondly, at the same pressure when the concentration of substitutional atoms increases, the melting temperature of alloy MoNi also decreases For Scientific Journal − No27/2018 69 example, at zero pressure when c Ni increases from to 1.8%, the melting temperature Tm of perfect alloy MoNi decreases from 3089 to 1754 K and the melting temperature Tm of defective alloy MoNi decreases from 2948 to 1703 K Thirdly, the melting temperature Tm of defective alloy MoNi is smaller than the melting temperature Tm of perfect alloy MoNi at the same physical condition Maximum melting temperature decreases about 8.6 % The calculated results of melting temperature for alloy MoNi with defect are nearer with experiments and the other theoretical results than that for ideal alloy Table The melti temperature Tmof alloy Mo-1.8%Ni at zero pressure from SMM, CALPHAD [18] and experimental data (EXPT) [19 – 24] SMM (perfect alloy) SMM (defective alloy) CALPHAD [18] [19] [20] [21] [22] [23] [24] 1754K 1703K 1622K 1616K 1619K 1643K 1623K 1635K 1633K EXPT Figure The melting curve of Mo from SMM, EXPT [14] and the other calculations [15-17] Ha Noi Metroplolitan University 70 Figure The melting curve of alloy Mo-1.8%Ni from the SMM CONCLUSION The melting temperature of defective substitutional alloy AB with BCC structure has been studied by using SMM The theoretical results are numerically applied for alloy MoNi with using Mie-Lennard-Jones potential in the interval of pressure from to 80 GPa and in the interval of concentration of substitutionalatoms from to 1.8% Our calculated results are in good agreement with experiments and the other calculations That proved that the concentration of equilibrium vacancies has 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equilibrium in the systems Ni Nb Mo, Ni—Nb and Ni Mo Ecole Nationale Superieure d'Electrochimie et d'Electrometallurgie, Grenoble, France ĐƯỜNG CONG NÓNG CHẢY CỦA HỢP KIM THAY THẾ MONI VỚI CẤU TRÚC LPTK CÓ KHUYẾT TẬT Tóm tắ tắt: Rút đường cong nóng chảy hợp kim thay AB có khuyết tật với cấu trúc lập phương tâm khối (LPTK) tác dụng áp suất phương pháp thống kê mômen Nhiệt độ bền vững tuyệt đối trạng thái tinh thể nồng độ nút khuyết cân dùng để tính nhiệt độ nóng chảy Trong trường hợp giới hạn, chúng tơi thu lý thuyết nóng chảy kim loại A với cấu trúc LPTK Các kết lý thuyết áp dụng tính số cho hợp kim MoNi sử dụng Mie-Lennard-Jones Các kết phù hợp tốt với số liệu thực nghiệm kết tính tốn khác Từ khóa: Hợp kim thay thế, nồng độ nút khuyết cân bằng, bền vững tuyệt đối trạng thái tinh thể, phương pháp thống kê mômen ... substitutional atoms and the equilibrium vacancy concentration on the melting temperature of substitutional alloy AB with body-centered cubic (BCC) structure The melting curve of alloy MoNi has been builded... In the model of perfect BCC substitutionalalloy AB, the main atoms A stay in the peaks and the substitutional atoms B stay in the body centers of cubic unit cell (Figure 1) Figure The model of. .. some comments about the melting temperature of alloy MoNi Firstly, at the same concentration of substitutional atoms when pressure increases, the melting temperature of alloy MoNi also increases