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Investigation of segregation for AlxIn1-x liquid binary alloys
Physica B 422 (2013) 56–63
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Investigation of segregation for AlxIn1-x liquid binary alloys Mir Mehedi Faruk a, G.M. Bhuiyan b,n a b
Department of Applied Physics, Electronics and Communication Engineering, University of Dhaka, Bangladesh Department of Theoretical Physics, University of Dhaka, Bangladesh
art ic l e i nf o
a b s t r a c t
Article history: Received 8 December 2012 Received in revised form 21 March 2013 Accepted 11 April 2013 Available online 26 April 2013
Segregation of AlxIn1-x liquid binary alloys is systematically investigated from the energetic point of view using the electronic theory of metals. The free energy of mixing is calculated at different thermodynamic states characterized by temperatures for the full range of concentration by using the perturbation approach. The interionic interaction is described by a local pseudopotential. This study enables us to predict the correct miscibility gap as well as critical temperature (T¼1160 K) and critical concentration (x¼ 0.5) of segregation for the concerned alloys. These results agree well with available experimental data. Most importantly, results of our calculations have precisely identified for the first time that, the volume dependent term of the energy of mixing is mostly responsible for the total energy of mixing to be positive, which is one of the most significant indicators of segregation of liquid metals in binary alloys. & 2013 Elsevier B.V. All rights reserved.
Keywords: Segregation Liquid binary alloys Miscibility gap Free energy of mixing Critical temperature Critical concentration
1. Introduction All liquid binary alloys can be grouped, in a broader sense, into two distinct classes according to their deviations from the Raoult's law. Alloys exhibiting positive and negative deviations [1,2] fall in the group of segregating system and short ranged ordered alloys, respectively. Considerably large deviation may lead to phase separation or compound forming. However, there are liquid alloys which do not belong to any of these classes; these alloys rather exhibit concentration dependent positive and negative thermophysical properties. That is, an alloy forming from the same elemental metals shows positive energy of mixing at some concentration range and negative at others [1–4]. Understanding of segregation of liquid–liquid binary metallic alloys is a very interesting subject to study. Recently, an increasing attraction towards this subject is observed [5–10]. We note here that the study of segregation of metallic alloys is important not only for fundamental microscopic understanding but also for its importance in application for technological advancement [11,12]. Our knowledge about the immiscible binary alloys through their structural and thermophysical properties [1,2,7,13] and computer simulation [14] has been somewhat advanced so far, but pinpointing the microscopic origin of miscibility gap of segregating alloys is yet to be seen. Although the experimental studies of immiscible
n Corresponding author. Tel.: +880 2 9661900/7022, 9672080, +880 2 9671257. fax: +880 2 8615583. E-mail address: [email protected] (G.M. Bhuiyan).
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alloys began in the fifties of the last century, the theoretical attempts in this regard, from the microscopic point of view, have been very few. We note here that some empirical or semiempirical models [15–17] are also attempted to understand miscibility gap, but these are lacking of microscopic view. Microscopic theoretical approaches based on statistical mechanics and electronic theory of metals are also attempted [3,4]. An excellent review article by Singh and Sommer [1] reports, within the framework of quasilattice theory, that, a competition between enthalpy and entropy dictates the degree of segregation in the system. Most importantly, it is shown that the composition-composition structure factors in the long wavelength limit, Scc ð0Þ, acts as a probe to measure the degree of concentration fluctuation and thus immiscibility of liquid binary alloys. In Ref. [1] it is also described that, Scc ð0Þ is connected to the second derivative of the Gibbs free energy of mixing ΔG which in turn can be expressed in terms of interchange or order energy w and the size ratio γ. It is noted that segregation to be happened w must be positive and, the values of γ describe the asymmetry nature of Scc ð0Þ. Beyond the quasilattice model some attempts also have been spent to study the segregation of liquid binary alloys from the hard sphere PY theory of liquids. For the additive hard sphere model there are two contradictory predictions; one shows that no segregation whatever the size ratio of hard spheres be [18] and, another predicts that segregation is possible for large mismatch of hard sphere diameters [19]. But the latter prediction depends on the approximation [19] of the method of computation. That is, segregation comes out only if Roger and Young [20] closure
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relation is used instead of Percus-Yevick, and Ballone et al [21]. Note that the grand canonical Monte Carlo simulation for a mixture of large and small cubes [22] predicts demixing of hard sphere in the frame work of additive hard sphere theory. Another MC simulation study [23] reports that a suitable repulsive interaction can lead to phase separation of liquid binary alloys under high pressure. However, stability and other physical properties of condensed state depend, from the microscopic point of view, on both structure and energetics. The electronic theory convincingly provides a platform where the energetics through band structure, a characteristic feature of metals, and the structure of liquid metals can be combined to understand and describe interesting physical properties more accurately [3]. Singh and Sommer in their review article [1] on phase separation emphasize that electronic theory is needed to understand the underlying mechanism of phase separation, without incurring appreciable error. A study on a number of binary alloys shows that the volume dependent part of the enthalpy favors formation of alloys and the structure dependent part opposes it (see Ref. [1]). Finally, the phase separation is a result of the delicate balance between these two contributions. The work on phase separation using the electronic theory [3] together with variational approach demonstrates how the miscibility gap reduces with increasing temperature. But the fundamental cause involved in the process remains to be understood clearly from the microscopic view of interaction. We have made an attempt to address this question in the present study. In this report, we intend to apply a microscopic approach involving the electronic theory of metals and statistical mechanics to study systematically the immiscibility of AlxIn1-x liquid binary metals. In this approach we describe the segregation from the point of view of free energy of mixing like Stroud [3] but unlike him we use the perturbative approach instead of variational one. The advantage we have thus had is to be able to calculate each and every contributions of different interactions to the energy separately which finally lead us to discover for the first time that the volume dependent (structure independent) term of interaction is mostly responsible to generate miscibility gap. The alloy of our interest, AlxIn1-x, is formed by the elements Al and In. Al based alloys are known to be good candidates for a new advanced antifriction materials, which may be used as plain sliding bearings for automotive applications. However, Al and In belongs to the group of less simple polyvalent metals [24–26]. There are different models to describe the interionic interaction of less simple polyvalent metals [27–29]. Among them the Bretonnet and Silbert (BS) model [27] is a promising candidate. The BS model for interionic interactions, originally proposed for liquid transition metals, combines both, the sp and d band effects within the frame work of the pseudopotential theory [30]. Its form is easily transferable to other systems as less simple and simple metals [25,26,31]. The sp-band is treated by the empty core pseudopotential and the term inside the core derived from the scattering phase shift by employing the inverse scattering approach. The resulting model is thus reduced to a simple form, which appears similar to that of the well known Abarenkov–Heine model [32]. This model has already proved to be successful in describing liquid structure [31,33,34], atomic transport [35–37], thermodynamics [25,26], electronic transport [38,39] properties of elemental and binary alloys. The well established first principles perturbation approach requires such a reference system which can closely resemble the concerned real system. There are many theoretical and experimental evidences [40,41] that the HS model can describe the structure of simple and less simple liquid metals and their binary alloys well [34]. As our alloys are formed from simple and less simple metals we employ the hard sphere (HS) liquid within the
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Percus–Yevick approximation (HSPY) as our reference system without trying others [42,43]. We note that the knowledge of the effective hard sphere diameters (HSD) is required to describe the HSPY liquid system. We have determined HSD from the Linearized Weeks–Chandler–Anderson thermodynamic perturbation theory [44]. It is worth mentioning that the theoretical study on the segregation of liquid binary alloys by Stroud [3] also employed the HSPY liquid in his variational study. This report is organized in the following way. Theories relevant to the present calculation of energy of mixing are briefly described in Section 2. Results are presented and discussed in Section 3. Finally we conclude this paper with some remarks in Section 4.
2. Theory In order to calculate energy of mixing, the required theories such as the perturbation theory for energy of mixing, theory for partial interionic interactions, and for pair distribution functions are described below. 2.1. Energy of mixing for liquid binary alloys The total Helmoltz Free energy per ion for an alloy can be written, in general, as [25] F ¼ F vol þ F eg þ F HS þ F Tail ;
ð1Þ
where subscripts vol, eg, HS, Tail denote volume dependent, electron gas, hard sphere and tail of potential contributions, respectively. Finally, the energy of mixing is expressed as ΔF ¼ ΔF Vol þ ΔF HS þ ΔF eg þ ΔF Tail ;
ð2Þ
The explicit expressions for the above terms are available in Ref. [25]. 2.2. The effective partial pair potential The local electron-ion pseudopotential for i-th component of the metallic alloy systems may be modeled by the superposition of the sp and d-band contributions [27] 8 2 > < ∑ BðiÞ expð−r=mai Þ if r o Rci ; m W i ðrÞ ¼ m ¼ 1 ð3Þ > : −Z =r if r 4 Rci ; i where a, Rc and Z are softness parameter, core radius and the effective s-electron occupancy number, respectively. The term outside the core is just the bare electron-ion Coulomb interaction. The coefficients in the core are determined by setting the continuity condition at the core, which yields BðiÞ 1 ¼ ðZ i =Rci Þð1−2ai =Rci Þ expðRci =ai Þ BðiÞ 2 ¼ ð2Z i =Rci Þðai =Rci −1Þ expðRci =2ai Þ;
ð4Þ
Finally, the partial interionic interaction within the pseudopotential formalism has the form Z ZiZj 2 dq q2 F N 1− ð5Þ V ij ðrÞ ¼ ij sin ðqrÞ=q ; π r where, i and j denote ionic components i and j, respectively. In (Eq. 5) FijN is the normalized energy wave number characteristics qffiffiffiffiffiffiffiffi 1 1 2 2 : ð6Þ FN ¼ ½q =ð4πρ Z Z W ðqÞW ðqÞ 1− i j i j ij ϵðqÞ 1−GðqÞ In the above equation Wi(q) denotes the unscreened form factor of the i-th component. ϵðqÞ and G(q) are dielectric function and the local field factor in momentum space, respectively. These are taken
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from Ichimaru and Utsumi [45] because their theory satisfies the compressibility sum rule and the short range correlation conditions. 2.3. The LWCA theory The starting point of the LWCA thermodynamic perturbation theory is the WCA theory [46]. The blip function in the WCA theory is given as BðrÞ ¼ Y s ðrÞ½exp½−βvij ðrÞ−exp½−βvsðiÞsðjÞ ;
ð7Þ
where vij and vsðiÞsðjÞ denote the soft and HS potentials, respectively, Y s is the cavity function associated with the HS distribution function and is continuous at r ¼ s. Within the linearized WCA approximation, thermodynamic condition that for an effective HSD, B(q ¼0) vanishes leads to the transcendental equation −2βsij v′ij ðsij Þ þ Y þ 2 βvij ¼ ln ; ð8Þ −βsij v′ij ðsij Þ þ Y þ 2
Fig. 1 illustrates the effective partial pair potentials at 1173 K for three different alloys of concentrations x ¼ 0:1; 0:5 and 0.9. Figures show that, V11 (Al–Al) and V22 (In–In) have the deepest and shallowest potential well, respectively, and the well of V12 (Al–In) lies in between. The reason of it is obvious, the electron-ion interaction outside the core is just the Coloumb interaction, which is negative and directly proportional to the valence Z. Since Z is same for both Al and In, the depth of the well is determined by the ionic number density ρ, which is inversely proportional to vij(r). Here, ρ for Al is 0:0517 1030 ð1=m3 Þ and for In is 0:03415 1030 ð1=m3 Þ, where former is larger than the latter one. As a result the In–In interaction in the matrix of AlxIn1-x alloy would be stronger than that of Al–Al, this is what reflected in the figures. A little variation of the depth of the wells with the change of concentrations is also noticed. Actually, the depth and position of the well is a result of delicate balance between the repulsive and
where prime on vij denotes the first derivative of the partial pair potentials at r ¼ sij . Eq. (8) is solved either graphically or numerically to obtain the effective hard sphere diameter. However, it is important to note that s12 calculated from the LWCA theory is very close to the average value, ðs11 þ s22 Þ=2 and the deviation is within 0:1%. So, for the theoretical advantage we follow Ref. [47] and use the additive hard sphere theory (i.e. average value) instead of s12 . 2.4. Pair distribution functions The Ashcroft-Langreth partial pair distribution function gij(r) is related to the static structure factors , Sij(q), in the following way [48] Z ∞ 1 3 g ij ðrÞ ¼ 1 þ ðSij −δij Þ expðiq:rÞd r; ð9Þ 3 pffiffiffiffiffiffiffiffi ð2πÞ ρ xi xj 0 where, ρ is the ionic number density of the alloys, x is the concentration of component one and q denotes the momentum transfer. The effective hard sphere diameter required to evaluate Sij(q) is obtained by using the linearized WCA method [44].
3. Results and discussion In this section, we have presented results for the energy of mixing and miscibility gap of segregation for AlxIn1-x liquid binary alloys at different thermodynamic states characterized by temperatures 1140, 1145, 1150, 1155, 1160 and 1173 K. In order to evaluate free energy we employ the generalized microscopic theory for metals (GMT) along with the perturbative approach. We note that the evaluation of the free energy within the framework of GMT requires the knowledge of partial pair potentials as well as pair distribution functions. The latter is related to the former one through the statistical mechanics [49]. The partial interionic interactions for the concerned polyvalent metals are calculated from the BS-model. The BS model has three parameters to be determined for an effective calculation. These are the core radius Rc, the softness parameter aand the effective s-electron occupancy number Z. The value of Rc is generally determined by fitting to some physical properties of the system of interest [48,50,51], for example, bulk modulus [51], electrical resistivity [48] or structural data [50]. In the present study the values of Rc are taken from Ref. [30] and for Z we have chosen the chemical valence Z¼3 for both Al and In. The softness parameter a is determined by best fitting the VMHNC structure [53] to the experimental ones [41] of liquid elementary systems Al and In near their melting temperatures (see Fig. 8). The values of a thus obtained are 0.3 and 0.29 a.u. for Al and In, respectively.
Fig. 1. Partial pair potentials for AlxIn1-x liquid binary alloys for x¼ 0.1, 0.5 and 0.9 at T ¼ 1173 K.
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adequately. This result in turn reflects the accuracy of our calculations. The partial HSDs sii are derived from LWCA method [44] (See Table 1). Table 1 shows that the values of HSD s11 and s22 vary with the change of concentration, the underlying cause of it is the charge transfer from an atom of one species to an atom of another species in the alloy state [54]. Fig. 3 illustrates breakdown details of different contributions to the total free energy of mixing as a function of concentration x at 1173 K. Here ΔF HS and ΔF Tail values are negative for all concentrations. On the other hand, ΔF eg and ΔF vol are positive but the former is very close to zero and much smaller in magnitudes than the latter one. The combinations of all four contributions, however, yields total energy of mixing, ΔF, which is in good agreement with available experimental data [55]. It is worth noting that the ΔF vol depends on the square of the screened electron-ion pseudopotential in momentum space (see Eq. (2) of Ref. [25]) and hence varies from metal to metal. For example, ΔF vol becomes negative at all concentrations in the case of AgxIn1-x binary alloys [25]. We should mention here that, as far as components of the total free energy F are concerned, the largest contribution comes from the HS reference system, as it should be whenever perturbation treatment is applied. The least contribution to the total free energy arises from the volume term. Despite all, the scenario turns out to be different when we look at different contributions to the energy of mixing; in this case the volume term contribution dominates others. One can attribute this to the fact that, the volume term contains the square of the form factor of the electron ion interaction whereas the other contributions are directly related to interionic pair potentials. Consequently, the change in electron-ion interaction in the alloy states affects the volume term much more
Table 1 Variation of HSD of AlxIn1-x liquid binary alloys with respect to concentration at temperature 1173 K. x
s11
s22
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
– 2.605 2.6495 2.6485 2.6465 2.6452 2.622 2.6 2.551 2.531 2.524
2.914 2.8954 2.8824 2.8794 2.878 2.8748 2.8688 2.8638 2.8548 2.8438 –
Fig. 2. Partial pair correlation function for AlxIn1-x liquid binary alloys for x¼ 0.1, 0.5 and 0.9 at T ¼ 1173 K.
attractive interaction experienced by the ions in the metals. We note here that, experimental densities for different concentrations of AlxIn1-x alloy are not available to us, so we have calculated them from the concept of ideal volume of mixing. Fig. 2(a)–(c) illustrate the LWCA pair correlation functions gij(r) at 1173 K for x¼ 0.1, 0.5 and 0.9, respectively. For x¼ 0.1 the principal peak of g22 is much larger than that of g11 and, peak value of g12 lies in between. That means, the probability of finding atoms of component 2 at the nearest neighbourhood is much larger than that of component 1. The reason is that at x¼0.1 the alloy is rich in In, it is thus natural to be of high probability to find In ions in the neighborhood. Obviously, the situation will be reversed at x¼ 0.9 where the alloy is Al rich, this has been reflected exactly in Fig. 2(c). For equiatomic concentration the probability of finding both Al and In at a distance r from the origin (where one atom of Al/In lies) should be equal, Fig. 2(b) illustrates it
Fig. 3. Energy of mixing for AlxIn1-x liquid binary alloys at 1173 K. Solid lines are theoretical calculation and dots are experimental results.
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Fig. 4. Energy of mixing for AlxIn1-x liquid binary alloys at different temperatures T ¼ 1160, 1155, 1150, 1145, and 1140 K.
(relative to the elemental systems) than the others. This result is corroborated by the work of Stroud [3]. Fig. 4(a)–(e) show the energy of mixing as well as breakdown details of different contributions for AlxIn1-x liquid binary alloys at different temperatures 1160, 1155, 1150, 1145 and 1140 K. The hard sphere contribution to the free energy of mixing is found to be minimum, as in Fig. 3 and, the volume contribution is positive and maximum across the whole range of concentration for all thermodynamic states. The values of different contributions follow the relation ΔF HS o ΔF tail o ΔF eg o ΔF vol . ΔF HS and ΔF tail are negative for all concentrations. The values of ΔF eg at different concentrations are almost constant and very close to zero. It is noticed that the combination of all contributions yielding total energy of mixing, ΔF, increases and moves towards positive values with decreasing temperature. As this feature of positiveness manifests
the miscibility gap, it demands a finer description to understand the segregation of liquid alloys. So, total energy of mixing only at different temperatures are separately illustrated in Fig. 5. Fig. 5 illustrates that the critical temperature of segregation is 1160 K and below this temperature the energy of mixing is partly positive and partly negative across the concentration range. Specifically, at temperatures 1155, 1150 and 1145 K the free energy of mixing are partially positive and partially negative. But ΔF, becomes more positive as temperature is decreased further and, consequently, the miscibility gap increases. The largest miscibility gap span the whole range of concentration at temperature 1140 K. From this trend it is observed that the critical concentration of segregation is at x ¼0.5. The experimental data of Campbell et al. [56] and Campbell and Wagemann [57] show a critical temperature of about 1220 K, whereas the data of Predel [58,61] shows
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Fig. 5. Free energy of mixing for AlxIn1-x liquid binary alloys at different temperatures. Fig. 7. Concentration width of the miscibility gap as a function of temperature. Dots are the calculated results and dashed line represents the mean field theory.
Fig. 8. Static Structure factors, S(q), for liquid Al and In. Lines (dashed and solid) are calculated values and solid circles and triangles are corresponding experimental data [41]. The values for a determined from the fitting procedure are 0.3 and 0.29 a. u. for Al and In, respectively.
Fig. 6. (a) HS part and (b) tail part of energy of mixing for AlxIn1-x liquid binary alloys at different temperatures.
Tc ¼1100 K. On the other hand, a careful differential thermal analysis show a critical temperature of 1112 K [59]. The predicted value for the critical temperature, 1160 K, in the present study, agrees well with the average value of the above three experimental data. The critical concentration we have predicted exactly matches with that of Predel [58,61] (xc ¼0.5) but largely deviates from that of Campbell and Wagemann [57] (xc ¼ 0.34). In order to examine whether there is any other signature of segregation in the present study, we evaluate V ord ¼ V 12 −V 11 ðrÞ þ V 22 ðrÞ=2 and found it to be positive in the concentration range of the miscibility gap. This result provides further support to our
claim. Since the hard sphere structure is related to the repulsive part of the potential, it is unable to describe segregation [18]. So, we have not attempted in this direction to evaluate Scc ð0Þ, for hard spheres. As the experimental data for energy of mixing are not available to us we could not compare our results with experiments at temperatures other than 1173 K. Let us now focus our attention to understand the fundamental mechanism involved, why the concentration width of the miscibility gap of AlxIn1-x liquid binary alloy alters with change of temperature. Fig. 6(a) shows ΔF HS as a function of concentration for different temperatures. Here it is noticed that ΔF HS decreases (i.e. becoming more negative) with increasing temperature. The reason is that the reference part (HS) of the Helmholtz free energy is directly proportional to the temperature. Since, FHS has negative value (see Ref. [25]), it will decrease with increasing temperature. On the other hand, FHS also depends on the effective HSD through (Eq. 6) of Ref. [25], which (HSD) decreases with increasing temperature [24]. As a consequence, the HS part of free energy increases with decreasing value of HSD. But the first contribution dominates in this case and the combined effect thus pushes ΔF HS further down at higher temperature.
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Fig. 6(b) illustrates the tail part component ΔF Tail as a function of concentration for different temperatures. Here ΔF Tail decreases with increasing temperature as in ΔF HS . In the thermodynamic perturbation theory [46] the tail part of the pair potential has V(r) ¼Vmin ¼constant for r o s and V(r) ¼V(r) for r 4 s. As the temperature increases kinetic energy of the HS increases; as a result, two ions approaches closer to each other. This results s to be smaller, and consequently the lower limit of integration becomes smaller relative to the position of Vmin. This turns out the integrand to be more negative. We have not, however, found in the present study any significant change in the ΔF vol and ΔF eg with the increase of temperature. The major cause of it is that these terms are not directly connected to temperature but depends on temperature through the ionic or electronic number density which remains almost constant in the temperature range ðΔT ¼ 34 KÞ of the present study. Now it is transparent clear that the combine effect of four components together conspires to shift gradually the value of total energy of mixing from negative to positive with decreasing temperature and, as a result, the partial miscibility gap as well as its width increase according. Fig. 7 illustrates the variation of the concentration range of the miscibility gap with the change of temperature. The gap is exhibited by AlxIn1-x liquid binary alloys below 1160 K and, above it the gap completely disappears. Figure also shows a nonlinear T vs concentration gap behavior. We have found that the mean field relation [60] jxb −xa j∝ðT c −TÞ1=2 fits the curve reasonably well. This result justifies that our prediction of critical temperature, TC, is at least qualitatively reliable.
4. Conclusion Energy of mixing for AlxIn1-x liquid binary alloys at six different thermodynamic states characterized by T¼ 1140, 1145, 1150, 1155, 1160 and 1173 K are calculated and presented and, from these results the segregating properties in terms of miscibility gap are described in this report. From the results and discussion we are now in a position to draw the following conclusions. (i) The GMT together with the perturbation approach can describe, at least qualitatively, the energy of mixing and the thermodynamic state dependent partial miscibility of AlxIn1-x liquid binary alloys. Our theoretical prediction that AlxIn1-x alloy becomes completely miscible at 1173 K, is confirmed by the experiment. (ii) Below the critical temperature, this alloy starts getting partially miscible. The critical temperature and concentration as predicted by the theory are found to be 1160 K and 0.5, respectively. The critical concentration matches exactly with the observed value of Predel [58]. In the case of critical temperature, the theoretically predicted value lies in between the experimental data of Predel, and Campbell and Wagemann [57]. (iii) Finding positiveness of the energy of mixing would have been impossible if the volume dependent term, ΔF vol were not there as a dominant positive quantity. (iv) Most importantly, the present study has pinpointed for the first time the fundamental interaction responsible for the opening of liquid–liquid miscibility gap for AlxIn1-x liquid binary alloys, and it is attributed to the volume dependent (structure independent) contribution of the interionic interaction. We believe that further application of the present approach to other segregating liquid binary alloys would underpin our findings on stronger basis.
Acknowledgments One of us (MMF) would like to thank Mr. Amitabh Biswas and Mr. Md. Onirban Islam for their cordial help in demonstrating him
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Physica B journal homepage: www.elsevier.com/locate/physb
Investigation of segregation for AlxIn1-x liquid binary alloys Mir Mehedi Faruk a, G.M. Bhuiyan b,n a b
Department of Applied Physics, Electronics and Communication Engineering, University of Dhaka, Bangladesh Department of Theoretical Physics, University of Dhaka, Bangladesh
art ic l e i nf o
a b s t r a c t
Article history: Received 8 December 2012 Received in revised form 21 March 2013 Accepted 11 April 2013 Available online 26 April 2013
Segregation of AlxIn1-x liquid binary alloys is systematically investigated from the energetic point of view using the electronic theory of metals. The free energy of mixing is calculated at different thermodynamic states characterized by temperatures for the full range of concentration by using the perturbation approach. The interionic interaction is described by a local pseudopotential. This study enables us to predict the correct miscibility gap as well as critical temperature (T¼1160 K) and critical concentration (x¼ 0.5) of segregation for the concerned alloys. These results agree well with available experimental data. Most importantly, results of our calculations have precisely identified for the first time that, the volume dependent term of the energy of mixing is mostly responsible for the total energy of mixing to be positive, which is one of the most significant indicators of segregation of liquid metals in binary alloys. & 2013 Elsevier B.V. All rights reserved.
Keywords: Segregation Liquid binary alloys Miscibility gap Free energy of mixing Critical temperature Critical concentration
1. Introduction All liquid binary alloys can be grouped, in a broader sense, into two distinct classes according to their deviations from the Raoult's law. Alloys exhibiting positive and negative deviations [1,2] fall in the group of segregating system and short ranged ordered alloys, respectively. Considerably large deviation may lead to phase separation or compound forming. However, there are liquid alloys which do not belong to any of these classes; these alloys rather exhibit concentration dependent positive and negative thermophysical properties. That is, an alloy forming from the same elemental metals shows positive energy of mixing at some concentration range and negative at others [1–4]. Understanding of segregation of liquid–liquid binary metallic alloys is a very interesting subject to study. Recently, an increasing attraction towards this subject is observed [5–10]. We note here that the study of segregation of metallic alloys is important not only for fundamental microscopic understanding but also for its importance in application for technological advancement [11,12]. Our knowledge about the immiscible binary alloys through their structural and thermophysical properties [1,2,7,13] and computer simulation [14] has been somewhat advanced so far, but pinpointing the microscopic origin of miscibility gap of segregating alloys is yet to be seen. Although the experimental studies of immiscible
n Corresponding author. Tel.: +880 2 9661900/7022, 9672080, +880 2 9671257. fax: +880 2 8615583. E-mail address: [email protected] (G.M. Bhuiyan).
0921-4526/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2013.04.039
alloys began in the fifties of the last century, the theoretical attempts in this regard, from the microscopic point of view, have been very few. We note here that some empirical or semiempirical models [15–17] are also attempted to understand miscibility gap, but these are lacking of microscopic view. Microscopic theoretical approaches based on statistical mechanics and electronic theory of metals are also attempted [3,4]. An excellent review article by Singh and Sommer [1] reports, within the framework of quasilattice theory, that, a competition between enthalpy and entropy dictates the degree of segregation in the system. Most importantly, it is shown that the composition-composition structure factors in the long wavelength limit, Scc ð0Þ, acts as a probe to measure the degree of concentration fluctuation and thus immiscibility of liquid binary alloys. In Ref. [1] it is also described that, Scc ð0Þ is connected to the second derivative of the Gibbs free energy of mixing ΔG which in turn can be expressed in terms of interchange or order energy w and the size ratio γ. It is noted that segregation to be happened w must be positive and, the values of γ describe the asymmetry nature of Scc ð0Þ. Beyond the quasilattice model some attempts also have been spent to study the segregation of liquid binary alloys from the hard sphere PY theory of liquids. For the additive hard sphere model there are two contradictory predictions; one shows that no segregation whatever the size ratio of hard spheres be [18] and, another predicts that segregation is possible for large mismatch of hard sphere diameters [19]. But the latter prediction depends on the approximation [19] of the method of computation. That is, segregation comes out only if Roger and Young [20] closure
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relation is used instead of Percus-Yevick, and Ballone et al [21]. Note that the grand canonical Monte Carlo simulation for a mixture of large and small cubes [22] predicts demixing of hard sphere in the frame work of additive hard sphere theory. Another MC simulation study [23] reports that a suitable repulsive interaction can lead to phase separation of liquid binary alloys under high pressure. However, stability and other physical properties of condensed state depend, from the microscopic point of view, on both structure and energetics. The electronic theory convincingly provides a platform where the energetics through band structure, a characteristic feature of metals, and the structure of liquid metals can be combined to understand and describe interesting physical properties more accurately [3]. Singh and Sommer in their review article [1] on phase separation emphasize that electronic theory is needed to understand the underlying mechanism of phase separation, without incurring appreciable error. A study on a number of binary alloys shows that the volume dependent part of the enthalpy favors formation of alloys and the structure dependent part opposes it (see Ref. [1]). Finally, the phase separation is a result of the delicate balance between these two contributions. The work on phase separation using the electronic theory [3] together with variational approach demonstrates how the miscibility gap reduces with increasing temperature. But the fundamental cause involved in the process remains to be understood clearly from the microscopic view of interaction. We have made an attempt to address this question in the present study. In this report, we intend to apply a microscopic approach involving the electronic theory of metals and statistical mechanics to study systematically the immiscibility of AlxIn1-x liquid binary metals. In this approach we describe the segregation from the point of view of free energy of mixing like Stroud [3] but unlike him we use the perturbative approach instead of variational one. The advantage we have thus had is to be able to calculate each and every contributions of different interactions to the energy separately which finally lead us to discover for the first time that the volume dependent (structure independent) term of interaction is mostly responsible to generate miscibility gap. The alloy of our interest, AlxIn1-x, is formed by the elements Al and In. Al based alloys are known to be good candidates for a new advanced antifriction materials, which may be used as plain sliding bearings for automotive applications. However, Al and In belongs to the group of less simple polyvalent metals [24–26]. There are different models to describe the interionic interaction of less simple polyvalent metals [27–29]. Among them the Bretonnet and Silbert (BS) model [27] is a promising candidate. The BS model for interionic interactions, originally proposed for liquid transition metals, combines both, the sp and d band effects within the frame work of the pseudopotential theory [30]. Its form is easily transferable to other systems as less simple and simple metals [25,26,31]. The sp-band is treated by the empty core pseudopotential and the term inside the core derived from the scattering phase shift by employing the inverse scattering approach. The resulting model is thus reduced to a simple form, which appears similar to that of the well known Abarenkov–Heine model [32]. This model has already proved to be successful in describing liquid structure [31,33,34], atomic transport [35–37], thermodynamics [25,26], electronic transport [38,39] properties of elemental and binary alloys. The well established first principles perturbation approach requires such a reference system which can closely resemble the concerned real system. There are many theoretical and experimental evidences [40,41] that the HS model can describe the structure of simple and less simple liquid metals and their binary alloys well [34]. As our alloys are formed from simple and less simple metals we employ the hard sphere (HS) liquid within the
57
Percus–Yevick approximation (HSPY) as our reference system without trying others [42,43]. We note that the knowledge of the effective hard sphere diameters (HSD) is required to describe the HSPY liquid system. We have determined HSD from the Linearized Weeks–Chandler–Anderson thermodynamic perturbation theory [44]. It is worth mentioning that the theoretical study on the segregation of liquid binary alloys by Stroud [3] also employed the HSPY liquid in his variational study. This report is organized in the following way. Theories relevant to the present calculation of energy of mixing are briefly described in Section 2. Results are presented and discussed in Section 3. Finally we conclude this paper with some remarks in Section 4.
2. Theory In order to calculate energy of mixing, the required theories such as the perturbation theory for energy of mixing, theory for partial interionic interactions, and for pair distribution functions are described below. 2.1. Energy of mixing for liquid binary alloys The total Helmoltz Free energy per ion for an alloy can be written, in general, as [25] F ¼ F vol þ F eg þ F HS þ F Tail ;
ð1Þ
where subscripts vol, eg, HS, Tail denote volume dependent, electron gas, hard sphere and tail of potential contributions, respectively. Finally, the energy of mixing is expressed as ΔF ¼ ΔF Vol þ ΔF HS þ ΔF eg þ ΔF Tail ;
ð2Þ
The explicit expressions for the above terms are available in Ref. [25]. 2.2. The effective partial pair potential The local electron-ion pseudopotential for i-th component of the metallic alloy systems may be modeled by the superposition of the sp and d-band contributions [27] 8 2 > < ∑ BðiÞ expð−r=mai Þ if r o Rci ; m W i ðrÞ ¼ m ¼ 1 ð3Þ > : −Z =r if r 4 Rci ; i where a, Rc and Z are softness parameter, core radius and the effective s-electron occupancy number, respectively. The term outside the core is just the bare electron-ion Coulomb interaction. The coefficients in the core are determined by setting the continuity condition at the core, which yields BðiÞ 1 ¼ ðZ i =Rci Þð1−2ai =Rci Þ expðRci =ai Þ BðiÞ 2 ¼ ð2Z i =Rci Þðai =Rci −1Þ expðRci =2ai Þ;
ð4Þ
Finally, the partial interionic interaction within the pseudopotential formalism has the form Z ZiZj 2 dq q2 F N 1− ð5Þ V ij ðrÞ ¼ ij sin ðqrÞ=q ; π r where, i and j denote ionic components i and j, respectively. In (Eq. 5) FijN is the normalized energy wave number characteristics qffiffiffiffiffiffiffiffi 1 1 2 2 : ð6Þ FN ¼ ½q =ð4πρ Z Z W ðqÞW ðqÞ 1− i j i j ij ϵðqÞ 1−GðqÞ In the above equation Wi(q) denotes the unscreened form factor of the i-th component. ϵðqÞ and G(q) are dielectric function and the local field factor in momentum space, respectively. These are taken
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from Ichimaru and Utsumi [45] because their theory satisfies the compressibility sum rule and the short range correlation conditions. 2.3. The LWCA theory The starting point of the LWCA thermodynamic perturbation theory is the WCA theory [46]. The blip function in the WCA theory is given as BðrÞ ¼ Y s ðrÞ½exp½−βvij ðrÞ−exp½−βvsðiÞsðjÞ ;
ð7Þ
where vij and vsðiÞsðjÞ denote the soft and HS potentials, respectively, Y s is the cavity function associated with the HS distribution function and is continuous at r ¼ s. Within the linearized WCA approximation, thermodynamic condition that for an effective HSD, B(q ¼0) vanishes leads to the transcendental equation −2βsij v′ij ðsij Þ þ Y þ 2 βvij ¼ ln ; ð8Þ −βsij v′ij ðsij Þ þ Y þ 2
Fig. 1 illustrates the effective partial pair potentials at 1173 K for three different alloys of concentrations x ¼ 0:1; 0:5 and 0.9. Figures show that, V11 (Al–Al) and V22 (In–In) have the deepest and shallowest potential well, respectively, and the well of V12 (Al–In) lies in between. The reason of it is obvious, the electron-ion interaction outside the core is just the Coloumb interaction, which is negative and directly proportional to the valence Z. Since Z is same for both Al and In, the depth of the well is determined by the ionic number density ρ, which is inversely proportional to vij(r). Here, ρ for Al is 0:0517 1030 ð1=m3 Þ and for In is 0:03415 1030 ð1=m3 Þ, where former is larger than the latter one. As a result the In–In interaction in the matrix of AlxIn1-x alloy would be stronger than that of Al–Al, this is what reflected in the figures. A little variation of the depth of the wells with the change of concentrations is also noticed. Actually, the depth and position of the well is a result of delicate balance between the repulsive and
where prime on vij denotes the first derivative of the partial pair potentials at r ¼ sij . Eq. (8) is solved either graphically or numerically to obtain the effective hard sphere diameter. However, it is important to note that s12 calculated from the LWCA theory is very close to the average value, ðs11 þ s22 Þ=2 and the deviation is within 0:1%. So, for the theoretical advantage we follow Ref. [47] and use the additive hard sphere theory (i.e. average value) instead of s12 . 2.4. Pair distribution functions The Ashcroft-Langreth partial pair distribution function gij(r) is related to the static structure factors , Sij(q), in the following way [48] Z ∞ 1 3 g ij ðrÞ ¼ 1 þ ðSij −δij Þ expðiq:rÞd r; ð9Þ 3 pffiffiffiffiffiffiffiffi ð2πÞ ρ xi xj 0 where, ρ is the ionic number density of the alloys, x is the concentration of component one and q denotes the momentum transfer. The effective hard sphere diameter required to evaluate Sij(q) is obtained by using the linearized WCA method [44].
3. Results and discussion In this section, we have presented results for the energy of mixing and miscibility gap of segregation for AlxIn1-x liquid binary alloys at different thermodynamic states characterized by temperatures 1140, 1145, 1150, 1155, 1160 and 1173 K. In order to evaluate free energy we employ the generalized microscopic theory for metals (GMT) along with the perturbative approach. We note that the evaluation of the free energy within the framework of GMT requires the knowledge of partial pair potentials as well as pair distribution functions. The latter is related to the former one through the statistical mechanics [49]. The partial interionic interactions for the concerned polyvalent metals are calculated from the BS-model. The BS model has three parameters to be determined for an effective calculation. These are the core radius Rc, the softness parameter aand the effective s-electron occupancy number Z. The value of Rc is generally determined by fitting to some physical properties of the system of interest [48,50,51], for example, bulk modulus [51], electrical resistivity [48] or structural data [50]. In the present study the values of Rc are taken from Ref. [30] and for Z we have chosen the chemical valence Z¼3 for both Al and In. The softness parameter a is determined by best fitting the VMHNC structure [53] to the experimental ones [41] of liquid elementary systems Al and In near their melting temperatures (see Fig. 8). The values of a thus obtained are 0.3 and 0.29 a.u. for Al and In, respectively.
Fig. 1. Partial pair potentials for AlxIn1-x liquid binary alloys for x¼ 0.1, 0.5 and 0.9 at T ¼ 1173 K.
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adequately. This result in turn reflects the accuracy of our calculations. The partial HSDs sii are derived from LWCA method [44] (See Table 1). Table 1 shows that the values of HSD s11 and s22 vary with the change of concentration, the underlying cause of it is the charge transfer from an atom of one species to an atom of another species in the alloy state [54]. Fig. 3 illustrates breakdown details of different contributions to the total free energy of mixing as a function of concentration x at 1173 K. Here ΔF HS and ΔF Tail values are negative for all concentrations. On the other hand, ΔF eg and ΔF vol are positive but the former is very close to zero and much smaller in magnitudes than the latter one. The combinations of all four contributions, however, yields total energy of mixing, ΔF, which is in good agreement with available experimental data [55]. It is worth noting that the ΔF vol depends on the square of the screened electron-ion pseudopotential in momentum space (see Eq. (2) of Ref. [25]) and hence varies from metal to metal. For example, ΔF vol becomes negative at all concentrations in the case of AgxIn1-x binary alloys [25]. We should mention here that, as far as components of the total free energy F are concerned, the largest contribution comes from the HS reference system, as it should be whenever perturbation treatment is applied. The least contribution to the total free energy arises from the volume term. Despite all, the scenario turns out to be different when we look at different contributions to the energy of mixing; in this case the volume term contribution dominates others. One can attribute this to the fact that, the volume term contains the square of the form factor of the electron ion interaction whereas the other contributions are directly related to interionic pair potentials. Consequently, the change in electron-ion interaction in the alloy states affects the volume term much more
Table 1 Variation of HSD of AlxIn1-x liquid binary alloys with respect to concentration at temperature 1173 K. x
s11
s22
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
– 2.605 2.6495 2.6485 2.6465 2.6452 2.622 2.6 2.551 2.531 2.524
2.914 2.8954 2.8824 2.8794 2.878 2.8748 2.8688 2.8638 2.8548 2.8438 –
Fig. 2. Partial pair correlation function for AlxIn1-x liquid binary alloys for x¼ 0.1, 0.5 and 0.9 at T ¼ 1173 K.
attractive interaction experienced by the ions in the metals. We note here that, experimental densities for different concentrations of AlxIn1-x alloy are not available to us, so we have calculated them from the concept of ideal volume of mixing. Fig. 2(a)–(c) illustrate the LWCA pair correlation functions gij(r) at 1173 K for x¼ 0.1, 0.5 and 0.9, respectively. For x¼ 0.1 the principal peak of g22 is much larger than that of g11 and, peak value of g12 lies in between. That means, the probability of finding atoms of component 2 at the nearest neighbourhood is much larger than that of component 1. The reason is that at x¼0.1 the alloy is rich in In, it is thus natural to be of high probability to find In ions in the neighborhood. Obviously, the situation will be reversed at x¼ 0.9 where the alloy is Al rich, this has been reflected exactly in Fig. 2(c). For equiatomic concentration the probability of finding both Al and In at a distance r from the origin (where one atom of Al/In lies) should be equal, Fig. 2(b) illustrates it
Fig. 3. Energy of mixing for AlxIn1-x liquid binary alloys at 1173 K. Solid lines are theoretical calculation and dots are experimental results.
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Fig. 4. Energy of mixing for AlxIn1-x liquid binary alloys at different temperatures T ¼ 1160, 1155, 1150, 1145, and 1140 K.
(relative to the elemental systems) than the others. This result is corroborated by the work of Stroud [3]. Fig. 4(a)–(e) show the energy of mixing as well as breakdown details of different contributions for AlxIn1-x liquid binary alloys at different temperatures 1160, 1155, 1150, 1145 and 1140 K. The hard sphere contribution to the free energy of mixing is found to be minimum, as in Fig. 3 and, the volume contribution is positive and maximum across the whole range of concentration for all thermodynamic states. The values of different contributions follow the relation ΔF HS o ΔF tail o ΔF eg o ΔF vol . ΔF HS and ΔF tail are negative for all concentrations. The values of ΔF eg at different concentrations are almost constant and very close to zero. It is noticed that the combination of all contributions yielding total energy of mixing, ΔF, increases and moves towards positive values with decreasing temperature. As this feature of positiveness manifests
the miscibility gap, it demands a finer description to understand the segregation of liquid alloys. So, total energy of mixing only at different temperatures are separately illustrated in Fig. 5. Fig. 5 illustrates that the critical temperature of segregation is 1160 K and below this temperature the energy of mixing is partly positive and partly negative across the concentration range. Specifically, at temperatures 1155, 1150 and 1145 K the free energy of mixing are partially positive and partially negative. But ΔF, becomes more positive as temperature is decreased further and, consequently, the miscibility gap increases. The largest miscibility gap span the whole range of concentration at temperature 1140 K. From this trend it is observed that the critical concentration of segregation is at x ¼0.5. The experimental data of Campbell et al. [56] and Campbell and Wagemann [57] show a critical temperature of about 1220 K, whereas the data of Predel [58,61] shows
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Fig. 5. Free energy of mixing for AlxIn1-x liquid binary alloys at different temperatures. Fig. 7. Concentration width of the miscibility gap as a function of temperature. Dots are the calculated results and dashed line represents the mean field theory.
Fig. 8. Static Structure factors, S(q), for liquid Al and In. Lines (dashed and solid) are calculated values and solid circles and triangles are corresponding experimental data [41]. The values for a determined from the fitting procedure are 0.3 and 0.29 a. u. for Al and In, respectively.
Fig. 6. (a) HS part and (b) tail part of energy of mixing for AlxIn1-x liquid binary alloys at different temperatures.
Tc ¼1100 K. On the other hand, a careful differential thermal analysis show a critical temperature of 1112 K [59]. The predicted value for the critical temperature, 1160 K, in the present study, agrees well with the average value of the above three experimental data. The critical concentration we have predicted exactly matches with that of Predel [58,61] (xc ¼0.5) but largely deviates from that of Campbell and Wagemann [57] (xc ¼ 0.34). In order to examine whether there is any other signature of segregation in the present study, we evaluate V ord ¼ V 12 −V 11 ðrÞ þ V 22 ðrÞ=2 and found it to be positive in the concentration range of the miscibility gap. This result provides further support to our
claim. Since the hard sphere structure is related to the repulsive part of the potential, it is unable to describe segregation [18]. So, we have not attempted in this direction to evaluate Scc ð0Þ, for hard spheres. As the experimental data for energy of mixing are not available to us we could not compare our results with experiments at temperatures other than 1173 K. Let us now focus our attention to understand the fundamental mechanism involved, why the concentration width of the miscibility gap of AlxIn1-x liquid binary alloy alters with change of temperature. Fig. 6(a) shows ΔF HS as a function of concentration for different temperatures. Here it is noticed that ΔF HS decreases (i.e. becoming more negative) with increasing temperature. The reason is that the reference part (HS) of the Helmholtz free energy is directly proportional to the temperature. Since, FHS has negative value (see Ref. [25]), it will decrease with increasing temperature. On the other hand, FHS also depends on the effective HSD through (Eq. 6) of Ref. [25], which (HSD) decreases with increasing temperature [24]. As a consequence, the HS part of free energy increases with decreasing value of HSD. But the first contribution dominates in this case and the combined effect thus pushes ΔF HS further down at higher temperature.
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Fig. 6(b) illustrates the tail part component ΔF Tail as a function of concentration for different temperatures. Here ΔF Tail decreases with increasing temperature as in ΔF HS . In the thermodynamic perturbation theory [46] the tail part of the pair potential has V(r) ¼Vmin ¼constant for r o s and V(r) ¼V(r) for r 4 s. As the temperature increases kinetic energy of the HS increases; as a result, two ions approaches closer to each other. This results s to be smaller, and consequently the lower limit of integration becomes smaller relative to the position of Vmin. This turns out the integrand to be more negative. We have not, however, found in the present study any significant change in the ΔF vol and ΔF eg with the increase of temperature. The major cause of it is that these terms are not directly connected to temperature but depends on temperature through the ionic or electronic number density which remains almost constant in the temperature range ðΔT ¼ 34 KÞ of the present study. Now it is transparent clear that the combine effect of four components together conspires to shift gradually the value of total energy of mixing from negative to positive with decreasing temperature and, as a result, the partial miscibility gap as well as its width increase according. Fig. 7 illustrates the variation of the concentration range of the miscibility gap with the change of temperature. The gap is exhibited by AlxIn1-x liquid binary alloys below 1160 K and, above it the gap completely disappears. Figure also shows a nonlinear T vs concentration gap behavior. We have found that the mean field relation [60] jxb −xa j∝ðT c −TÞ1=2 fits the curve reasonably well. This result justifies that our prediction of critical temperature, TC, is at least qualitatively reliable.
4. Conclusion Energy of mixing for AlxIn1-x liquid binary alloys at six different thermodynamic states characterized by T¼ 1140, 1145, 1150, 1155, 1160 and 1173 K are calculated and presented and, from these results the segregating properties in terms of miscibility gap are described in this report. From the results and discussion we are now in a position to draw the following conclusions. (i) The GMT together with the perturbation approach can describe, at least qualitatively, the energy of mixing and the thermodynamic state dependent partial miscibility of AlxIn1-x liquid binary alloys. Our theoretical prediction that AlxIn1-x alloy becomes completely miscible at 1173 K, is confirmed by the experiment. (ii) Below the critical temperature, this alloy starts getting partially miscible. The critical temperature and concentration as predicted by the theory are found to be 1160 K and 0.5, respectively. The critical concentration matches exactly with the observed value of Predel [58]. In the case of critical temperature, the theoretically predicted value lies in between the experimental data of Predel, and Campbell and Wagemann [57]. (iii) Finding positiveness of the energy of mixing would have been impossible if the volume dependent term, ΔF vol were not there as a dominant positive quantity. (iv) Most importantly, the present study has pinpointed for the first time the fundamental interaction responsible for the opening of liquid–liquid miscibility gap for AlxIn1-x liquid binary alloys, and it is attributed to the volume dependent (structure independent) contribution of the interionic interaction. We believe that further application of the present approach to other segregating liquid binary alloys would underpin our findings on stronger basis.
Acknowledgments One of us (MMF) would like to thank Mr. Amitabh Biswas and Mr. Md. Onirban Islam for their cordial help in demonstrating him
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