J. Min. Metall. Sect. B-Metall., 56 (2) B (2020) 183 - 191 Journal of Mining and Metallurgy, Section B: Metallurgy An effectIve AlgOrIthM tO IdentIfy the MIScIBIlIty gAp In A BInAry SuBStItutIOnAl SOlutIOn phASe t. fua, y. du b,*, Z.-S. Zheng a,*, y.-B. peng c, B. Jin b, y.-l. liu b, c.-f. du a, S.-h. liu b, c.-y. Shi b, J. Wang b a* School of Mathematics and Statistics, Central South University, Changsha, Hunan, China b State Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan, China c College of Metallurgy and Materials Engineering, Hunan University of Technology, Zhuzhou, Hunan, China (Received 16 September 2019; accepted 18 May 2020) Abstract In the literature, no detailed description is reported about how to detect if a miscibility gap exists in terms of interaction parameters analytically. In this work, a method to determine the likelihood of the presence of a miscibility gap in a binary substitutional solution phase is proposed in terms of interaction parameters. The range of the last interaction parameter along with the former parameters is analyzed for a set of self-consistent parameters associated with the miscibility gap in assessment process. Furthermore, we deduce the first and second derivatives of Gibbs energy with respect to composition for a phase described with a sublattice model in a binary system. The Al-Zn and Al-In phase diagrams are computed by using a home-made code to verify the efficiency of these techniques. The method to detect the miscibility gap in terms of interaction parameters can be generalized to sublattice models. At last, a system of equations is developed to efficiently compute the Gibbs energy curve of a phase described with a sublattice model. Keywords: Computational thermodynamics; Equilibrium calculations; Phase diagram; Miscibility gap; Algorithm 1. Introduction interaction parameters are analyzed mathematically in the solution model. The other objective in the present Miscibility gap occurs when a two-phase work is to provide a new approach to calculate the coexistence line in a phase diagram ends at a critical miscibility gap and guide selection for the interaction point [1]. On one hand, miscibility gap has been well parameters during the thermodynamic assessment in understood and extensively used to develop high- binary systems. performance materials through the spinodal-type In section 2, we present a simple method to decomposition of the microstructure, such as (Ti,Zr)C identify if there is a miscibility gap in terms of the [2, 3], TiAlN [4], etc. On the other hand, for some interaction parameters analytically. Subsequently, in alloys like high-entropy alloys, intermediate phase or order to investigate the existence of the miscibility miscibility gap should be avoided in order to form a gap, we deduce the first and second derivatives of single-phase microstructure thereby obtaining Gibbs energy with respect to composition for a phase excellent performance. As a result, it is of great described with a sublattice model in section 3 and importance to acquire the information about section 4. After that in section 5, the calculations of miscibility gap of materials during the developing the Al-Zn and Al-In phase diagrams are demonstrated process. In the literature, many authors presented to show the unique features of the presently developed mathematic equations for understanding miscibility new algorithm. Finally, a summary is made in section gap in terms of Gibbs energies [5-8], and numerically 6. It could be mentioned that the outcome of the detected them by a discretization of composition axis present work is of both scientific and educational [9-13]. To the best of our knowledge, however, there interests. is no detailed description about how to detect if a miscibility gap exists in terms of interaction 2. On the existence of miscibility gap in view of parameters analytically. One purpose of this paper is interaction parameters to develop a novel method to determine the existence of miscibility gap analytically for given In thermodynamic equilibrium calculations, if the thermodynamic parameters. In addition, the ranges of miscibility gap exists it should be considered. In this *Corresponding author: [email protected]; [email protected] https://doi.org/10.2298/JMMB190916004F 184 T. Fu et al. / J. Min. Metall. Sect. B-Metall., 56 (2) B (2020) 183 - 191 i section, we will investigate the existence of the RT =-xx()1 miscibility gap in a binary phase with a substitutional 2L (3) solution model. 0 x For fixed T and P, let 1 - x and be the molar R It< is obvious that Eq. (3) has two solutions in fractions of components A and B, respectively. Then, RT 1 0 << the- x molar Gibbs energy of a phase described with a interval (0, 1) only if . In other words, there 2L0 4 substitutional solution model can be expressed as LRT> 2 exists a miscibility gap in> this phase when 0 . GxGxGRTxxxx=-1100++È -ln1- + ln ˘ M ()AB Î()() ˚ nn = 1 n 2.2 Case i (1) 1 +-112xx- x LL ()Â()i L ¹ 0 Li=02 i=0 Similarly, for the case 1 and i (), Gx¢¢ = 0 G0 G0 the equation () is equivalent toG ¢ 0 where A (respectively, B ) is the Gibbs energy R Lini ()0 ££ RT=- x x112 L x-- 6 L 2 L of pure A (respectively, B). are the ()()110 (4) interaction energy parametersi between¢ A and B. 0 G h( Generally, a miscibility gap involves phase ( hx Let () denote the right hand side of Eq. (4). It separation within a single phase. The corresponding 3 should1 beh( noted that there exist two real solutions to Gibbs energy curve for fixed T shows two coexisting hx Eq. (4)x if and only if the maximum of () in interval compositions and two inflection points, except for the hx() consolute point [1], cf. Fig. 1. Mathematically, this (0, 1) is greater than RT. The function h ( is a cubic implies that there are two real number solutions to polynomial and has three zero points,h( 0, 1 and Gx¢¢ = 0 3LL+ equation M () in interval (0, 1). Here the second 10 hx¢ = 0 G x . This indicates that equation () has order derivativeG of M with respect to is given by 6L1 xx12, ()xx< hx¢()x RT two solutions 12where is the first GM¢¢ = -+2126Lx0 (() -L1 x hx() h xx()1- order derivative 1of h ¢ and ( (2) h( +-48xxL2 + 48 - 10 +L hx¢ =-36 Lx2 36 L++ 462Lx++ L L ()2 () 1 ()1 010(5) G 1 h( 0 1 We now discuss the existence of the miscibility 0 4L Thanks to the feature of the cubic function, there gap with the increase in the number of the interaction max hx()> 0 parameters. arex only two cases such that xŒ()01, , namely, Ï 3LL+ hx, ifL> 00 and 10> 0, 2.1 Case n = 0 Ô ()11 Ô 6L1 max hx()= Ì m h (6) L ¹ 0 Li=01 xŒ()01, Œ 0 3LL+ For the case 0 and i () , by expression Ô hx(), ifL< 0 and 10<1 Gx¢¢ = 0 Ô 21 (2), equation () canG be¢ expressed0 as: Ó 6L1 R We are now in a position to construct the judgement for the existence >of the miscibility gap L > 0 L > 0 30LL+> according to 1 orL1 < 0 . For1 , if 10 hx> RT and()1 , there is a miscibility gap inh this( phase, L < 0 30LL-< hx()> RT1 whileL for 1 , if 10 and2 0 , the miscibility gap existsh( in2 this phase. L When dealing with the2 miscibility gap in assessment process, to evaluate a set of self-consistent parameters, the last interaction parameter should be reasonably selected on the basis of2 former interaction parameters. LRT> 2 0 G¢¢ 05. = 4RRTL-<20 In fact, if0 , since M () 0 M G¢¢ Æ+• x Æ1 and the factG that M asG ¢x Æ 0 or , equation Gx¢¢ ()= 0 M always hasÆ two solutions0 in interval (0, 1) no L matterL what value the parameter L1 takes, i.e., there is Figure 1. Modified Gibbs energy curves after a linear always a miscibility gap in this phase. GG---00()1 xGx LRT< 2 shifting, MAl Zn , at different On the other hand, if0 , we rewrite Eq. (4) T = . temperatures 0in Al-Zn binary system. The curve as R at T0 = 625.7111 K shows the Gibbs energy RT M related to the consolute point while equation +=2126LxL01() -1 G¢¢ (()x = 0 xx-1 M has only one solution () (7) T. Fu et al. / J. Min. Metall. Sect. B-Metall., 56 (2) B (2020) 183 - 191 185 The functions from the both sides of Eq. (7) are exists. Now consider the case n = 2 and rewrite Eq. plotted against x in Fig. 2, we can observe that when (9) as LL= 0 x x 11, only one value 0 satisfies Eq. (7), where 0 and RT 2 0 ++2484810126LxxLxL0 () -+ 21=-() Lx1 can be computed by x()xx -1 (10) Ï RT 0 1 +=2126LxL001() - x = Ô xx-1 Ô 00() where the left hand side- is a function symmetry Ì 1 RT()12- x0 0 (8) x = Ô =12L with respect to , and for different L2 the Ô 2 2 1 0 2 ÓÓ xx0 ()0 -1 corresponding function curves invariably pass 0 0 Ê 66- 24RT ˆˆ LL> LL<- , 2L - Hence, we can conclude that if 11or11 , Á 0 ˜ through the point Ë 12 5 ¯6, as shown in Eq. (7) has two solutions in intervalL (0, 1),n namely,³ 66- x = there will exist a miscibility gap in this phase. Fig. 3. In Eq. (10), by taking 12 , the right hand 24RT n ³ 2 - 6 L1 26L ->-6 L 2.3 case side becomes .