Cent Eur J Phys • 9(1) • 2011 • 222-229 DOI: 10.2478/s11534-010-0065-1 Central European Journal of Physics A thermodynamic lattice theory on melting curve and eutectic point of binary alloys Application to fcc and bcc structure Research Article Nguyen V Hung1∗ , Dung T Tran1† , Nguyen C Toan1 , Barbara Kirchnner2 Department of Physics, University of Science, VNU-Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam Wilhelm-Ostwald-Institute for Physical and Theoretical Chemistry, University of Leipzig, Linnéstr 2, 04103 Leipzig, Germany Received 10 February 2010; accepted July 2010 Abstract: A thermodynamic lattice theory has been developed for determination of the melting curves and eutectic points of binary alloys Analytical expressions for the melting curves of binary alloys composed of constituent elements with the same structure have been derived from expressions for the ratio of root mean square fluctuation in atomic positions on the equilibrium lattice positions and the nearest neighbor distance This melting curve provides information on Lindemann’s melting temperatures of binary alloys with respect to any proportion of constituent elements, as well as on their eutectic points The theory has been applied to fcc and bcc structure Numerical results for some binary alloys provide a good correspondence between the calculated and experimental phase diagrams, where the calculated results for Cu1−x Nix agree well with the measured ones, and those for the other alloys are found to be in a reasonable agreement with experiment PACS (2008): 61.10.Ht Keywords: thermodynamic lattice theory • Lindemann’s melting temperature • eutectic point • binary alloys © Versita Sp z o.o Introduction The melting of materials has great scientific and technological interest The problem is understanding how to determine the temperature at which a solid melts, i.e., its ∗ E-mail: hungnv@vnu.edu.vn Present address: Dept of Metallurgy and Materials, University of Birmingham, Edgbaston, Birmingham, UK B15 2TT † 222 melting temperature The atomic vibrational theory has been successfully applied by Lindemann and others [1– 4] The Lindemann’s criterion [1, 2] is based on the concept that the melting occurs when the ratio of the root mean square fluctuation (RMSF) in atomic positions on the equilibrium lattice positions and the nearest neighbor distance reaches a threshold value The validity of this criterion was tested by experiment [5] This criterion relates melting to a lattice vibrational instability Hence, the thermodynamic lattice theory is one of the most important fundamentals for interpreting the thermodynamic Nguyen V Hung, Dung T Tran, Nguyen C Toan, Barbara Kirchnner properties and melting of materials [1–6, 9, 18–23] Binary alloys having liquidus consisting of two branches in their phase diagram or melting curve are called eutectics [6] and the minimum solidification temperature is called the eutectic temperature [6] The binary alloy phase diagrams have been experimentally studied [7] Phenomenological theory of the phase diagrams of the binary eutectic systems [8] has been developed to show qualitatively the temperature-concentration diagrams of eutectic mixtures using a Landau-type approach, which involves a coupling between the liquid-solid transition order-parameters and a specific nonlinear dependence on concentration of the free-energy coefficients Here the eutectic point is considered more generally as the minimum of the melting curve X-ray absorption fine structure (XAFS) [9] in studying melting is focused mainly on the Fourier transform magnitudes and cumulants of XAFS The melting curve of materials from theory versus experiments [10] has been studied based on quantum mechanics within the framework of density functional theory, with use of the generalized gradient corrections, but this is focused mainly on the dependence of the melting temperature of single elements on pressure Empirical rules [1, 11–13] have been used to characterize the melting transition of solids as useful procedures in computer simulations without performing free energy calculations [14] The mechanism for the solidliquid phase transition based on the Lindemann’s criterion has been studied using Monte-Carlo simulation [15], but a complete “ab initio” theory for the melting transition is not available [11, 15] As such, the calculation of melting temperature curves versus proportions of constituent elements of binary alloys and their eutectic points still can be a useful contribution to the field The purpose of this work is to derive a thermodynamic lattice theory for analytical calculation and analysis of the melting curves or phase diagrams and eutectic points of binary alloys composed of any constituent elements with the same structure Our development in Sec is a derivation of the analytical expressions for the atomic mean square displacement (MSD), mean lattice energy, atomic mean square fluctuation (MSF), and then the ratio of the RMSF in atomic positions on the equilibrium lattice positions and the nearest neighbor distance, as well as the melting curves of binary alloys This melting curve contains the atomic proportion of constituent elements and their melting temperatures in the limiting cases when the whole elementary cell is occupied by the atoms of one of the constituent elements Our theory is based on Lindemann’s idea regarding the melting [1, 2, 11, 18] so that the derived melting curve provides information on the Lindemann’s melting temperatures of the binary alloys with respect to any proportion of the constituent elements and on their eutectic points Further we have shown that the Gibbs energy of the eutectic binary alloy system described by the present theory is always at minimum, satisfying the condition of equilibrium The theory has been applied to fcc and bcc binary alloys by carrying out several numerical calculations (Sec 3) The results are compared to experiment [7, 16, 17, 24] and show a good correspondence between the calculated melting curves and the experimental phase diagrams More specifically, the calculated results for Cu1−x Nix agree well with the measured ones, and those for the other binary alloys are found to be in a reasonable agreement with experiment The conclusions mentioning the main results and possible applications of the derived theory are presented in Sec Formalism The binary alloy lattice is always in a state of atomic thermal vibration so that in the lattice cell n the atomic fluctuation function, denoted by number for the 1st element and by number for the 2nd element composing the binary alloy, is given by U1n = U2n where = u1q eiq.Rn + u∗1q e−iq.Rn , q u2q e iq.Rn + u∗2q e−iq.Rn (1) , q u1q = u1 eiωq t , u2q = u2 eiωq t , (2) with ωq being the lattice vibration frequency and q the wave number The atomic oscillating amplitude is characterized by the MSD or Debye-Waller factor (DWF) [9, 18, 20–23] which has the form [18] W = |K ¯ uq |2 , (3) q where K is the scattering vector equaling a reciprocal lat¯ q is the mean atomic vibration amplitude tice vector, and u If each binary alloy lattice cell contains p atoms, where on average s is the number of atoms of type and (p − s) ¯ q is is the number of atoms of type 2, then the quantity u given by su1q + (p − s)u2q ¯q = (4) u p The potential energy of an oscillator is equal to its kinetic energy, so that the mean energy of atom k vibrating with 223 A thermodynamic lattice theory on melting curve and eutectic point of binary alloys Application to fcc and bcc structure wave number q has the form ˙ ¯ q = Mk u ¯ kq ε (5) The lattice vibrations quantized as phonons obey BoseEinstein statistics Transforming the sum over q into the corresponding integral, and applying this to the high temperature area (T θD ) due to the melting with θD being the Debye temperature we obtain the DWF from Eq (12) Using Eqs (2), (5), the mean energy of the crystal consisting of N lattice cells is given by N sM1 ωq2 |u1q |2 + (p − s)M2 ωq2 |u2q |2 , ¯q = ε ¯= E q q (6) where M1 , M2 are the masses of atoms of types and 2, respectively Using the relationship between u2q and u1q [21], i.e W = [M2 s + (p − s)M1 ] 2p M1 M2 kB θD2 m= M1 , M2 |u1q |2 = ωq , (8) ¯ q + 12 p n NM1 ωq [s + (p − s)m] [s + (p − s) m]2 |u1q |2 p2 ¯= E ˙ kn Mk U 2p K [s + (p − s)m]2 q (10) (11) ¯ q + 12 n NM1 ωq [s + (p − s) m] (12) 224 (14) Mk ωq2 |Uknq |2 = k,n k,n (15) q Using this expression and Eqs (6), (7) we derive the atomic MSF in the form N N To study the MSD Eq (3) we use the Debye model, where all three vibrations have the same velocity [18] Hence, we calculate the contribution of each polarization, taking Eq (11) into account, and then using Eq (11), the MSD or DWF Eq (3) with all three polarizations is given by W = [s + (p − s)m]2 |U2n |2 = m2 n |u1q |2 , (16) q which, by use of Eq (14), is given by From Eq (4) and Eq (7) we get the mean atomic vibration amplitude for qth lattice mode in the form |¯ uq | = 6p2 W K2 (9) so that, using Eq (8) and Eq (9) we obtain |u1q |2 = (13) The mean crystal lattice energy has been calculated as The mean energy for this qth lattice mode with p atoms in a lattice cell calculated using the phonon energy with ¯ q as the mean number of phonons is given by n ¯q + ¯q = p n ε , (7) and Eqs (5), (6) we obtain the mean energy for the atomic vibration of the qth lattice mode as ¯q = Nωq2 |u1q |2 sM1 + M2 (p − s) m2 ε K 2T which is linearly proportional to the temperature T as was mentioned already [18, 22] From Eq (11), and using Eq (3) for W we obtain q u2q = mu1q , |U2n |2 = n 6p2 m2 W K [s + (p − s)m]2 (17) Further, using W from Eq (13) this expression is resulted as 9pm2 T |U2n |2 = (18) N n M1 [s + (p − s)m] kB θD2 Hence, at T θD the MSF in atomic positions about the equilibrium lattice positions is determined by Eq (18) which is linearly proportional to the temperature T Therefore, at a given temperature T the quantity R defined by the ratio of the RMSF in atomic positions on the equilibrium lattice positions and the nearest neighbour distance d is given by R= d 9pm2 T M1 [s + (p − s)m] kB θD2 (19) This expression for R contains the parameters p and s which are different for different binary alloy structures such as fcc and bcc, which will be determined below Nguyen V Hung, Dung T Tran, Nguyen C Toan, Barbara Kirchnner (Sec 3), as well as the parameter m concerning the atomic mass M1 of element and atomic mass M2 of element composing the binary alloys So that it represents the contribution of different binary alloys consisted of different pairs of elements having the same crystal structures Based on the Lindemann’s criterion, the binary alloy will be melted when this ratio R of Eq (19) reaches a threshold value Rm , then the Lindemann’s melting temperature Tm for a binary alloy using Eq (19) is defined as Tm = [sM2 + (p − s)M1 ] χ, 9pm (20) where χ= Rm2 kB θD2 d2 , Rm2 = Nd2 |U2n |2 (21) n This expression for the Lindemann’s melting temperature can be applied to different binary alloys composed of different pairs of elements, with the atomic masses M1 and M2 having the same crystal structures defined by the parameters p and s If we denote x as proportion of the mass of the element in the binary alloy, then we have x= sM1 sM1 + (p − s) M2 px m(1 − x) + x (22) (23) We consider one element to be the host and another the dopant Since the tendency to be the host is equal for both constituent elements we take the average of the parameter m with respect to the atomic mass proportion of the constituent elements in the binary alloy as ¯ = m M1 M2 s + (p − s) p M1 M2 (24) This equation can be solved using the successive approximation Substituting the zero-order term with s from Eq (23), we obtain the 1st order term equation as (1 − x) m ¯ + x − (1 − x) M2 M1 ¯ −x m = 0, M2 M1 ¯ = m − x − (1 − x) (25) M1 M2 √ + ∆ , (1 − x) M1 M2 ∆ = x − (1 − x) + 4x (1 − x) , M2 M1 (26) replacing m in Eq (20) for the calculation of Lindemann’s melting temperature The threshold value Rm of the ratio of RMSF in atomic positions on the equilibrium lattice positions and the nearest neighbor distance at melting is contained in χ which will be obtained by an averaging procedure The average of χ can not be directly based on χ1 and χ2 because it has the form of Eq (21) containing Rm2 , i.e., the second order of Rm , while the other averages have been realized based on the first order of the displacement That is why we have to perform the average for χ and then obtain χ= √ √ s χ1 + (p − s) χ2 , p2 (27) containing χ1 for the 1st element and χ2 for the 2nd element, for which we use the following limiting values 9Tm(2) , M2 9Tm(1) χ1 = , M1 χ2 = From this equation we obtain the mean number of atoms of the element in each binary alloy lattice cell s= which provides the following solution s = 0, (28) s = p, containing Tm(1) and Tm(2) as melting temperatures of the 1st or doping element and of the 2nd or host element, respectively, which compose the binary alloy Therefore, the melting temperature of binary alloys has been obtained from our calculated ratio of RMSF in atomic positions on the equilibrium lattice positions and nearest neighbour distance Eq (19) The eutectic point is calculated using the condition for the minimum of the melting curve, i.e., dTm = dx (29) Focusing on a thermodynamic model for a binary alloy, we will show further that the Gibbs energy of the eutectic binary alloy system described by the present theory is always at minimum, satisfying the condition of thermodynamic equilibrium The total Gibbs energy G of a system can be written formally as G= ni gi + ∆mix , (30) i 225 A thermodynamic lattice theory on melting curve and eutectic point of binary alloys Application to fcc and bcc structure where ni is the number of moles, gi is the Gibbs energy per mole of phase i, and ∆Gmix the change in Gibbs energy due to inter-phase interactions Applying this to an AB binary system, we have possible phases: liquid A (Aliq), solid A (Asol), liquid B (Bliq), and solid B (Bsol), so that G = nAliq gAliq + nAsol gAsol + nBliq gBliq + nBsol gBsol + ∆Gmix (31) According to the definition in thermodynamics, the Gibbs energy G of a binary alloy has the following form G = U + PV − T S, model for a eutectic binary alloy is always at a minimum, and the system is in a state of thermodynamic equilibrium Hence, we can determine the melting curves, from which the Lindemann’s melting temperatures of the binary alloys with respect to any proportions of their constituent elements, using Eq (20) with Eqs (22), (23), (26), (27), (28) and their eutectic points using Eq (29), can also be determined The eutectic isotherm is the one for which T equals the eutectic temperature TE (32) where U, P, V , T , S are internal energy, pressure, volume, temperature, and entropy of the system, respectively Taking the differentiation of the Gibbs energy Eq (32) we obtain dG = dU + PdV + V dP − T dS − SdT (33) Since dU = T dS − PdV the differential of Eq (33) is changed into dG = V dP − SdT (34) In our approach, the time-dependence of pressure is not taken into account so that dP = The direct relation between the DWF and the mean module of atomic vibration described by Eq (3) is time-independent, so that the time-differential of our derived DWF is equal to zero dW = dt q d |K ¯ uq |2 = dt (35) Alternatively, the time-differential of DWF of Eq (13) can be written as dW dT =C , (36) dt dt where C is a symbol denoting all the time constants in Eq (13) Hence, comparing Eq (36) to Eq (35) we obtain dT = 0, dt (37) which shows that in our model if the system is energetically isolated, the temperature does not change with time This is consistent with the meaning of temperature being that the mean atomic vibration amplitude is timeindependent Therefore, in Eq (34) we have dP = and dT = 0, so that dG = This means that the Gibbs energy in our 226 Figure Possible typical phase diagrams of a binary alloy formed by components A and B Nguyen V Hung, Dung T Tran, Nguyen C Toan, Barbara Kirchnner Application to fcc and bcc binary alloys and discussions of numerical results Now we apply the derived theory to the fcc and bcc binary alloys It is apparent that 81 atom on the vertex and 12 atom on the surface of the fcc are localized in the elementary cell Hence, the total number of atoms in a fcc elementary cell is p(fcc)=4 Similarly, 18 atom on the vertex and one atom in the center of the bcc are localized in the elementary cell Therefore, the total number of atoms in a bcc elementary cell is p(bcc)=2 According to the phenomenological theory for the phase diagrams of the binary eutectic systems [8] Figure shows qualitative schemes of typical possible phase diagrams of a binary alloy formed by the components A and B, i.e., the dependence of the temperature T on the proportion x of element B doped in the host element A Below the isotropic liquid mixture L, the liquidus or melting curve beginning from the melting temperature TA of the host element passes through a temperature minimum TE known Table as the eutectic point E and ends at the melting temperature TB of the doping element The phase diagrams contain two solid crystalline phases α and β The eutectic isotherm T = TE passes through the eutectic point The eutectic temperature TE can have a value either lower TA and TB (Figure 1a) or equaling TA (Figure 1b) or TB (Figure 1c) The mass proportion x appears to characterise the proportion of the doping element mixed in the host element to form the binary alloy Our numerical calculations using the derived theory are focused mainly on the melting curves giving the Lindemann’s melting temperatures with respect to any proportion of the constituent elements, and eutectic points of binary alloys composed by fcc or bcc elements The eutectic isotherm is apparently T = TE All input data have been taken from Ref [6] The calculated melting temperatures TE and their respective proportions xE of doping elements for eutectic points of binary alloys Cu1−x Agx , Cu1−x Alx , Cu1−x Nix (fcc) and Cr1−x Rbx , Cs1−x Rbx , Cr1−x Mox (bcc) are presented in Table 1, compared to experiment [7, 16, 17, 24] Alloys Cu1−x Agx Cu1−x Alx Cu1−x Nix Cr1−x Rbx Cs1−x Rbx Cr1−x Mox xE , Present xE , Expt TE , Present TE , Expt 0.7107 0.719 [24] 1170.0 1123.5 [17] 0.7089 0.672 [16] 887.0 870.0 [7] 0.0 1.0 0.0 [7] 1358.0 312.6 1356.0 [7] 0.3212 0.357 [7] 288.0 285.8 [7] 0.1977 0.17 [7] 2127.0 2127.0 [7] Calculated eutectic melting temperatures TE (K ) and their respective proportions xE of doping elements for binary alloys Cu1−x Agx , Cu1−x Alx , Cu1−x Nix , (fcc) and Cr1−x Rbx , Cs1−x Rbx , Cr1−x Mox (bcc) compared to experiment [7, 16, 17, 24] Here the calculated results for Cu1−x Nix agree well with the measured values [7], and those for the other binary alloys are found to be in a reasonable agreement with experiment [7, 16, 17, 24] Figure illustrates the calculated melting curves providing information on the Lindemann’s melting temperatures and eutectic points of binary alloys Cu1−x Agx (fcc) and Cs1−x Rbx (bcc) compared to experiment [7] They correspond to their experimental phase diagrams [7] and belong to the types presented in Figure 1a, i.e., their eutectic temperatures are lower than the melting temperature of the host element Cu and Cs and those of the doping elements Ag and Rb, respectively Figure illustrates the calculated melting curves of Cu1−x Nix and Cr1−x Rbx The results for Cu1−x Nix agree well with experiment [7] and belong to the types presented in Figure 1b, where its eutectic temperature is equal to the melting temperature of the host element Cu The results for Cr1−x Rbx belong to those of Figure 1c, where its eutectic point is equal to the melting temperature of the doping element Rb The calculated melting curves represented in Figures and correspond to their experimental phase diagrams [7], showing that the Lindemann’s melting temperatures of the considered binary alloys vary, with respect to increasing proportions x of the doping elements Ag, Rb, Ni and Rb, between the melting temperatures of the pure host elements, when the whole elementary cell is occupied by the host atoms, and the pure dopant elements, where the whole elementary cell is occupied by the dopant atoms Figure also shows the rate at which the atoms become more weakly bonded after Cu and Cs were mixed by the doping elements Ag and Rb, respectively, because the 227 A thermodynamic lattice theory on melting curve and eutectic point of binary alloys Application to fcc and bcc structure Figure Calculated melting curves providing information on Lindemann’s melting temperature and eutectic points of binary alloys Cu1−x Agx (fcc) and Cs1−x Rbx (bcc), where the results for Cs1−x Rbx are compared to experiment [7] Figure Calculated melting curves of binary alloys Cu1−x Nix (fcc) and Cr1−x Rbx (bcc), where the results for Cu1−x Nix are compared to experiment [7] x melting temperature decreases up to the eutectic point, and more tightly bonded after the eutectic point because the melting temperature increases Figure for Cu1−x Nix shows the rate that the atoms become more tightly bonded after the host element Cu was doped by Ni because the melting temperature increases But for Cr1−x Rbx it shows the rate that the atoms become more weakly bonded after the host element Cr was doped by Rb, because its melting temperature decreases Table shows the good agreement of the Lindemann’s melting temperatures of Cu1−x Nix (fcc) and Cs1−x Rbx (bcc), taken from the calculated melting curves of these binary alloys, with experiment [7] for different proportions x (it is also for any proportion x) of Ni and Rb doped in Cu and Cs, respectively, to form these binary alloys The present calculation has been carried out for the binary alloys composed of the constituent elements with the same fcc or bcc 228 Cu1−x Nix , Cu1−x Nix , Cs1−x Rbx , Cs1−x Rbx , Table Present Expt [7] Present Expt [7] 0.10 0.30 0.50 0.70 0.90 1396 1388 292.6 291.4 1468 1461 287.5 286.0 1538 1531 290.0 287.4 1611 1605 295.0 293.5 1687 1684 305.0 304.0 Calculated Lindemann’s melting temperatures Tm (K ) of Cu1−x Nix (fcc) and Cs1−x Rbx (bcc) with respect to different proportions x of Ni and Rb doped in Cu and Cs, respectively, compared to experiment [7] structure, but it also can be generalized to alternate structures for those composed of constituent elements with the same other structures by calculation of the atomic number in their elementary cells Nguyen V Hung, Dung T Tran, Nguyen C Toan, Barbara Kirchnner Conclusions In this work a thermodynamic lattice theory on the melting curves, eutectic points and eutectic isotherms of binary alloys composed by the constituent elements with the same structure has been derived Our development is derivation of the analytical expressions for the ratio of RMSF in atomic positions on the equilibrium lattice positions and the nearest neighbor distance, which lead to expressions for the melting curve providing information on the Lindemann’s melting temperatures of the binary alloys composed of any proportions of constituent elements, and on their eutectic points Focusing on a thermodynamic model for a binary alloy system, it has resulted that the Gibbs energy of the eutectic binary alloy system described by the present theory is always at minimum, so that the system is in a state of thermodynamic equilibrium The theory has been derived based on averaging several physical quantities with the hope of getting an estimative mean melting curve, yet provides quantitative results for the melting of binary alloys The calculated melting curves of binary alloys correspond to their experimental phase diagrams, where the results for Cu1−x Nix agree well with the measured ones and those for the other fcc and bcc binary alloys are found to be in a reasonable agreement with experiment The calculated melting curve also shows the rate that the atoms of binary alloys become either more tightly or more weakly bonded (the host element becomes either harder or softer) after the host element was mixed by the doping element to be a binary alloy This behavior may be useful for technological applications The present numerical calculations have been carried out for fcc and bcc binary alloys, but it also can be applied to those composed by the constituent elements with the same other structure by calculation of the atomic number in their elementary cells For this reason, the derived theory may prove to be a simple and effective method for prediction of the Lindemann’s melting temperatures, eutectic points and eutectic isotherms of binary alloys composed by any proportions of constituent elements with the same structure ical Chemistry, University of Leipzig for the supports and hospitality during his stay here This work ist supported by the research project No 103.01.09.09 of NAFOSTED 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thermodynamic lattice theory on melting curve and eutectic point of binary alloys Application to fcc and bcc structure wave number q has the... lattice theory on melting curve and eutectic point of binary alloys Application to fcc and bcc structure Figure Calculated melting curves providing information on Lindemann’s melting temperature and. .. typical phase diagrams of a binary alloy formed by components A and B Nguyen V Hung, Dung T Tran, Nguyen C Toan, Barbara Kirchnner Application to fcc and bcc binary alloys and discussions of numerical