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 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 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 11 or11 , Á 0 ˜ through the point Ë 12 5 ¯6, as shown in Eq. (7) has two solutions in intervalL (0, 1),n namely, 66- x = there will exist a miscibility gap in this phase. Fig. 3. In Eq. (10), by taking 12 , the right hand 24RT n 2 - 6 L1 26L ->-6 L 2.3 case side becomes . Thus, if 01 5 , The analysis described in sections 2.1 and 2.2 would there is always a miscibility gap- in this phase Ln 02 0 be complex and expensive for the case n () Lini = +01() whatever the parameter L2 takes. and . Therefore, a numericalL = 0 method 24RT for findingG¢ the0 solutions of polynomial will be 2L01-<-6 L G¢¢ x = 0 If 5 , it can be observed, from employed here. Rewrite the equation M ( ) as x = LL= 0 00 Fig. 3 that2 when taking 2 21, orLL222= , , Eq. (10) Ï-2L +-12xL 6 + ¸ Ô 001() Ô xx()-1 Ì ˝ -=RT 0 L , -+-48xxL2 48 10 +L (9) has only one solution x0,1 or x0,2 , respectively. These ÓÔ()2 ˛Ô Lx0 , Lx0 , n £ two sets of values 21 ,, 01 and 22,,02 can be obtained L 0 It is easy to check that the function from the left L2, 2 Lx, side of Eq. (9) is a polynomial of degree n+2. By by solving Eq. (11) of unknowns2 0 : computing the eigenvalues of the corresponding Ï RT R companion matrix [17], if there are at least two real Ô ++2L0 xx()-1 eigenvalues in interval (0,1), a miscibility gap exists; Ô 00 Ô otherwise, it does not exist. 48x2 - 44810126xLxL+ 0 =- Ì()0 02() 01 (11) Generally, in a binary system, a substitutional Ô RT12- x solution model is considered with no more than 4 Ô () 0 0 n £ 3 2 +-48Lx2 () 20 1= 122 L1 interaction parameters, i.e., , and mostly we just Ô 2 Ó xx0 ()0 -1 take n = 2. In thermodynamic assessment process, according to previously introduced interaction LL> 0 LL< 0 parameters, we can compute the range of the last Thus, if 221, or222, , there exists a interaction parameter in which a miscibility gap miscibility gapL in the phase.L
Figure 2. Geometric illustration for Eq. (7) Figure 3. Geometric illustration for Eq. (11)
186 T. Fu et al. / J. Min. Metall. Sect. B-Metall., 56 (2) B (2020) 183 - 191 Ï
It is remarkable that since the analysis for the È yGSnV(,hh ZrSnSn : : ;)00++ yGa (, ZrSnVa : : ;) GM = Í existence of miscibility gap in this section is in terms RT yln y+ y ln y Î ()Sn Sn Va VVa of interaction parameters at different discrete values (13) + y yL(h,::,;) ZrSnSnVa0 + of temperature, the present approach can also be used Sn Va ˘ ˙ ()9 - yVa . yy()(,: y- y Lh ZrSn::,;)Sn Va 0 V to check the, existence of the miscibility gaps for Sn Va Sn Va ˚ 0 different expressions of T-dependence of interaction B parameters [14-16] quickly and efficiently. In this section, we shall study the existence of the In addition, for the linear model for the T- miscibility gap for a phase described with a sublattice dependence of interaction parameters, generally given model under fixed T and P where the Gibbs energy LabT=+ = yy, byiii , Kaptay described that for the simplest expression involves two internal variables BVa or yyAB, casen = of n = 0 an artificial inverted miscibility gap : yA a < 0 bR0 > 2 appears at high temperature when 0 and Gy()BVa, y Gy(()AB, y GMG = G = [15]. In fact, this simple rule can be used for bthe case M ay- or M a (14) bR> 2 Va of n = 1 due to the fact that if0 , there always a G¢¢ ()05. =-< 4RT 2 L 0 holds MM 0 atb high temperature no whereG constant a is the sum of the numbers of sites M matter what value L1 is. for all sublattices, and bivariate function G has the 2L - . sameB form as expression (1). We next focus on the Furthermore, for the case n = 2, according to( the yy, yy, case ofBVa . The case of AB is similar to that discussion about Eq. (10) and Fig. 3, we find that if yyBVa, y 24RT of .A To this end, we startyB by deducing the first 2L ->-6 L y 015 , a miscibility gap appears and secondA derivatives of GM with respect to x. LT() Generally, the mole fraction of element B in that regardless of the value of L . Since the parameter i can 2 phase is defined by bT b B be approximated by i at sufficiently high ay+ ay+ = aa--1 24RT x = BB= BB=-1 B 2bT01->-6 bT ay ay1 aay1 (15) temperatures, we have b 5 . This - Va -+ B -+ B 0 - y + implies that an artificial inverted miscibility gap yyBVa+=1 b0 Here Eq. (15) follows from the relation , 5() 2b001+ 624bR> bR0 > 12 5 arises if , or roughly, . and aB is a constant and can be zero. ThenaB GM is 6 x 1 regarded as a function of x: On the otherb0 hand,1 the function from the right side of Gy(), y Eq. (10) always passes though the point (1/2, 0), and GGyx= ()()= BVa MBay-+1 (16) when taking x = 1/2, the left side in Eq. (10) is equal B 224LLRT-- G = to 02 . Analogous to the analysis above, Consequently, we obtain the first derivative of GM with respect to x as: 2L0 - 224LLRT-- > 0 in view of Fig. 3, when 02 , i.e. Ê G G ˆ b -bR > 2 b - 2 - ayG-+1 - 02 , an artificial 0 inverted miscibility gap Á ˜()B dGM dG dyB Ë yBVa y ¯ dyB = = (17) appears at high temperatures. dx ddy= dx 2 dx B ()ay-+1 B y d 3. finding the miscibility gap for a phase B From= the relation (15), there holds that described with a sublattice model dx aaa--1 = B dy 2 (18) B ()ay-+1 B The method descried in section 2 deals with a B phase described with a substitutional solution model. Next we consider a sublattice model in Thus, we arrive at d which the corresponding Gibbs energy can also be = Ê G G ˆ Á - ˜(ay-+1 B )) - G expressed by a function of two variables, such as dG dG dy y y y M ==B Ë BVa¯ (19) phase η in the Zr-Sn [18] binary system, which is dx dy dx aa-+1 described with the sublattice model B B G ySSnVaSnVa, yy()+= y 1 (Zr)5(Sn)3(Sn,Va)1. Let be the Similarly,d we get the following second derivative: site fractions of the third sublattice for phase η. = B The molar fractions of components Sn and Zr are ÊÊ G G ˆ ˆ 2 - 3 y xS = dG 1 d ÁÁ ˜ ˜ dy + Sn 5 M y y B x = x = 2 = ÁË BVa¯ ˜ Sn and Zr (12) dx a-+1 a dy dx 9 - yVa 9 - yVa ()BBÁ ˜ Ë()ay-+1 B -- G ¯ (20) 3 ThenZ the corresponding Gibbs energy expression 2 22 Ê G G G ˆ ()ay-+1 BB reads: = Á - 2 + ˜ 2 y 2 yy y 2 Ë BBVaVa¯ ()aa-+1 B y = T. Fu et al. / J. Min. Metall. Sect. B-Metall., 56 (2) B (2020) 183 - 191 187 ( yy=-1 By takingVa B , it is easily verified that the Under fixed T and P, G is considered as a function 2G 22G G y¢ y¢¢ y¢ + yM¢ = 1 y¢¢ + y¢¢ =1 - 2 G + of B and B since A B andA B . term 2 2 has the same expression yBBV yyaV y a Together with Eq. (21), GM can also be regarded as a B y¢ as Eq. (2). From the discussion in section 2, we know function of B and x, that is, M ) G y¢ ,y¢ ,y¢¢,,y¢¢ = G y¢ ,y¢¢(y¢ , x) when the miscibility gap exists, there are two real M ( A B A B ) M ( B B B ) (22) number solutions to equation 3GxM¢¢ ()= 00 in interval ()ay-+1 B 0 Thus we can express the partial derivative of GM (0,1). Noting that 2 , we can conclude with respect to x, ()aa-+1 B ) ) that the method described in section 2 works equally GM GM GM y¢¢ Ê G G ˆ y¢¢ = = B =Á M - M ˜ B x x ¢¢ ¢¢ ¢¢ (23) well in the case of ya sublattice model. yB x Ë yB yA ¯ x The phase η in Zr-Sn [18] binary system is used Similarly, we have as an example to verify the correctness of these G Ê G G ˆ y¢ y = y MMM B B Sn = Á - ˜ derivatives. Here we take , and set a x y¢ y¢ x (24) temperature T = 1200 K and a number of axis Ë B A ¯ subdivisions N = 50. In Fig. 4 we find that the first Clearly, there holds that for any x, and second derivatives computed directly by the Ê G G ˆ y¢ Ê G G ˆ y¢¢ Á M - M ˜ BM= Á - M ˜ B expressions (19) and (20) coincide with those y¢ y¢ x y¢¢ y¢¢ x (25) calculated numerically by the finite difference Ë B A ¯ Ë B A ¯ approximation schemes. In fact, Eq. (25) can also be obtained from the partial Gibbs energies of the end-members in [19, 20]. 4. the derivatives in a sublattice model Applying the end-members (A:A) and (A:B) yields
GM GM In this section, we shall expand the derivation GGAA: =+ M + - ()yyyy¢¢¢¢¢¢AB,,, AB . method introduced in section 3 to a general sublattice y¢A y¢¢A¢ model. These derivatives can help to obtain the Gibbs T Ê G G G G ˆ (26) energy curve efficiently by a simple Newton Á M ,,,M M M ˜ algorithm. Ë y¢A yB¢ y¢¢A yBB¢¢ ¯ In order to make it easier to understand the G G derivation for the derivatives of G with respect to x, M M M GGAB: =+ M + - ()yyyy¢¢¢¢¢¢ABAB,,, . a phase described with (A,B) (A,B) is taken as an y¢A y¢¢B y¢ , y¢ y¢¢, y¢¢a b example. Let A B and A B denote the constituent T Ê G G G G ˆ (27) fractions of the first and second sublattices, Á M ,,,M M M ˜ respectively. Then we have Ë y¢¢A yB¢ y¢¢A y¢¢B ¯ ay¢B + bby¢B¢ x = : a + b (21) Note that the partial Gibbs energies of the end- members are related to the chemical potentials of the
(a) (b)
Figure 4. The first (a) and second (b) derivatives of GM of phase η at T = 1200 K in Zr-Sn system computed directly via formulas (19), (20) and numerically by finite difference schemes , y 188 T. Fu et al. / J. Min.A Metall. Sect. B-Metall., 56 (2) B (2020) 183 - 191
Gab elements, AA: =+mm A A and GaAB: =+m A bbmB. multi-component phase. And in the Newton method,
Thus we deduceGA = thatm there are just 10 iterations at each node xi. In Table 1,
GM 1 1 Ê GM GM ˆ for a fixed N = 100, we present the maxima of the =-=mmBA()GG ABAA:: - = Á - ˜ xb b y¢¢ y¢¢ (28) differences of the above two Gibbs energies at all Ë B A ¯ nodes against various N`. As shown in Table 1, the Analogously, by using the end-members (A:A) computation time from the present algorithm is much and (B:A), there consequently holds that shorter than that due to the discretization method.
GM 11Ê GM GM ˆ Ê GM GM ˆ In Fig. 5(b), the first derivatives computed by Eq. = Á - ˜= Á - ˜ (29) (23), equal to those by Eq. (24), nearly coincide with xaË yB¢ yb¢A ¯ Ë y¢¢B y¢¢A ¯ the results calculated by the finite difference method. It is obvious that Eq. (25) and Eq. (29) are And the maximum of the differences of the results by ¢ equivalent. Eq. (24) (denoted by G M ) and the finite difference x µ For a given x, we now have 4 equations of 4 method (denoted by G’ M ) is nearby line x = 1/3. In Table ¢¢¢¢ ¢¢ unknowns yA ,yyyyBAB,, as follows: 2, the maxima of those differences near line x = 1/3 are presented against various N. Ï11Ê G G M ˆ Ê G G ˆ Ô Á M - M ˜= ÁÁ M - M ˜ GGn - d a y¢ a yb¢ y¢¢ y¢¢ Table 1. The maxima of M M and the computation Ô Ë B A ¯ Ë B A ¯ times of the discretization method against different Ô ay¢ + by¢¢ N`. Since the NewtonG method is independent of the Ì x = BB ab+ (30) number of subdivisions N`, the computation time Ô does not change Ô yy¢¢AB+1= Ô Computation Computation Ó yy¢¢AB+ ¢¢ = 1 n d time of time of N` Max GG M - M Eq.G (30) can be solved by the simple Newton discretization Newton method. G method method Now we take the phase Cu Mg described with 2 1000 51.5938 0.0198 s (Cu,Mg)2(Cu,Mg)1 in Cu-Mg [21] binary system as an example to observe the efficiency of Eqs. (23), (24) 2000 23.8303 0.0263 s and (30). At T = 1000K, let N = 100 be the number of 5000 8.5057 0.0407 s 0.0003 s x axis partitions. In Fig. 5(a), the values of Gibbs 10000 3.8669 0.0605 s energy at each node by solving Eqs. (30) with Newton G n 100000 0.259 0.4217 s method (denoted by M ) are close to those calculated G d by traditional discretization method (denoted by M ). µ In the discretization method, the value of Gibbs Table 2. The maxima of G’MM- G’ against different N energy at any discrete node xi is approximately obtained by computing the minimum of Gibbs N 1000 2000 5000 10000 100000 energies at all the gridpoints with the number of Max internal variable axis subdivisions N`= 1000. Such µ G’MM- G’ 162.1459 41.7066 6.724 1.6499 0.0029 calculations are time-consuming, in particular for a
(a) (b)
Figure 5. The Gibbs energies (a) and the first derivatives (b) of the GM of Cu2Mg phase at T = 1000K in Cu-Mg system computed by present methods and general numerical methods T. Fu et al. / J. Min. Metall. Sect. B-Metall., 56 (2) B (2020) 183 - 191 189
5. results and discussion (a) Some numerical results are presented to verify the efficiency of the proposed algorithm in section 2. By using the method described in section 2 to find the existence of the miscibility gap in a phase described with a substitutional solution model, the Al-Zn [22] and Al-In [23] binary phase diagrams are computed with a home-made MATLAB code. The corresponding procedure to calculate binary phase diagrams is described in detail in our recent work [24]. It is stressed that in the calculation of this phase diagram, after finding a miscibility gap in a phase, one still needs to check whether it is the global equilibrium or not. In the calculation of Fig. 6(a), we can easily find the miscibility gap in phase fcc_a1 with the interaction parameters, instead of the discretization of composition axis. And in Fig. 6(b) (b) the miscibility gap exists in the liquid phase. The parameters L0, L1 and L2 for fcc_a1 phase in the Al-Zn and liquid in the Al-In system are given in Table 3. By inserting L0, L1 into Eq. (11), we can calculate the range for L2 where a miscibility gap exists in this phase. At T = 625K, using Eq. (11), it shows that only when L2 < -994.14 or L2 > 1292.17 a miscibility gap can exist in phase fcc_a1 for the Al-Zn system. The value of L2 given in the Al-Zn system is within that range, which coincides with the fact that there is a miscibility gap in phase fcc_a1 at 625 K. On the contrary, at T = 626 K, it can be seen that the value of L2 in the fcc_a1 phase of the Al-Zn system is outside the range resulting from Eq. (11); Figure 6. Al-Zn (a) and Al-In (b) phase diagrams computed also, there is no miscibility gap in phase fcc_a1 at this by the present algorithm via MATLAB temperature. In addition, except for the solid phase, this method can also be applied to the liquid phase in Table 4. Parameters L0 and L1 are given in the the Al-In system. corresponding TDB file at these temperatures. And Analogously, the parameters L0, L1 for η phase in the L1 parameter from Eq. (8) is solved with the given Zr-Sn system are given in Table 4. By inserting L0 into L0 Eq. (9), we calculate the range for L where a miscibility 1 Results gap exists. At T = 1340K, using Eq. (8), we find that only Phase T L L 0 from Eq.(9) 2 when L1 < -28772 or L1 > 28772 a miscibility gap can < - 28772 exist in phase η. The value of L1 given in the Zr-Sn 1340 -17520 -28780 > 28772 system is within that range. At T = 1341 K, the value of η (Zr-Sn) L is outside the range computed from Eq. (8). And there < -28801 1 1341 -17548 -28747 is no miscibility gap in phase η at this temperature. > 28801
Table 3. Parameters L0, L1 and L2 are given in the corresponding TDB file at these temperatures. And L2 parameter from Eq. (11) is solved with the given L0, L1
Phase TL0 L1 Results from Eq.(11) L2
625 7594.45 3743.46 < - 994.14 >1292.17 -1030.73 fcc_a1 (Al-Zn) 626 594.92 3738.87 < -1042.02 >1344.52 -1027.42 1208 20670.85 2393.88 < 1675.25 >5021.26 1652.19 liquid (Al-In) 1209 20670.36 2392.68 < 1641.15 >5063.49 1649.02 -
190 T. Fu et al. / J. Min. Metall. Sect. B-Metall., 56 (2) B (2020) 183 - 191
Table 5. Implication and guidance to thermodynamic calculations and optimization from the analytical results to identify if a miscibility gap exists for a binary phase a
n An analytical equation Implication and guidance ( 0 RT =-xx()1 LRT> 2 0 2L 0 : there is a miscibility gap. 0 L L > 30LL+> hx> RT hx() = RT (1) L 1 > 0 , 1 0 and () 1 , there is a miscibility gap. 1 h( h hhx():=- xx()112() Lx1 -- 6 L10 2 L L < 0 hx> RT (2) 1 , 3 LL 1 -< 0 0 and () 2 , there is a miscibility gap. L < 0 1 È ( xx()-1- 2LLxL01+-()12 6 + + Î 0Solved by Eigenvalue method. If there are at least two real roots ≥ 2 x -+-48xxL2 48 10+L˘ -= RT 0 in interval (0,1), a miscibility gap exists. ()2 ˚ L
exin a G = 1 Gx=-()112 x ( - x )LL Excess Gibbs energy is given as follows: Âi =0 i The outcome of the present work is of interest for compound function, we deduce the first and second the thermodynamic optimization in which a derivatives of Gibbs energy with respect to miscibility gap is involved. During the composition for a phase described with a sublattice thermodynamic optimization of a binary system in model in a binary system. The method to detect the which a miscibility gap exists, Eq.(8) and Eq.(11) can miscibility gap in terms of interaction parameters can be used to detect numerical regions of the interaction be generalized to a sublattice model in which the parameters very easily, which will reduce the amount Gibbs energy has two internal variables. Moreover, in of work for assessments. As a summary, Table 5 view of the derivatives, we have developed a system presents the implication and guidance to of equations to efficiently compute the Gibbs energy thermodynamic calculations and optimizations from curve of a phase described with a sublattice model. the analytical results for each of the cases n = 0, 1, 2 and > 2. Acknowledgements To the best of our knowledge, no detailed description is presented in textbooks on how to This work was supported by the National Key identify if a miscibility gap exists in terms of the Research and Development Program of China (grant given thermodynamic parameters analytically in a numbers 2017YFB0701700, 2017YFB0305601). binary system. The present work shows all of the Thank is also due to Prof. Bo Sundman for helpful details in order to detect the existence of the discussion. miscibility gap in a binary system, being of interest to undergraduates and graduates. references
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efIkASAn AlgOrItAM ZA OdređIvAnJe prekIdA u rAStvOrlJIvOStI u BInArnOM SupStItucIOnOM rAStvOru 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* Fakultet za matematiku i statistiku, Centralno južni univerzitet, Čangša, Hjunan, Kina b Glavna državna laboratorija metalurgije praha, Centralno južni univerzitet, Čangša, Hjunan, Kina c Fakultet za metalurgiju i inženjerstvo materijala, Univerzitet tehnologije u Hjunanu, Kina Apstrakt
U postojećoj literaturi nema detaljnih opisa o tome kako analitički detektovati postojanje prekida u rastvorljivosti na osnovu interakcionih parametara. U ovom radu je predložen metod za određivanje verovatnoće postojanja prekida u rastvorljivosti u binarnom supstitucionom rastvoru, a uz pomoć interakcionih parametara. Raspon između poslednjeg interakcionog parametra i prethodnih parametara se analizira da bi se dobio skup doslednih parametara povezanih sa prekidom u rastvorljivosti. Osim toga, izračunati su prvi i drugi izvod Gibsove energije u odnosu na sastav faze opisane pomoću modela podrešetke u binarnom sistemu. Fazni dijagrami Al-Zn i Al-In su proračunati uz pomoć sopstvenog koda da bi se verifikovala efikasnost ovih tehnika. Metod za otkrivanje prekida u rastvorljivosti na bazi interakcionih parametara može se generalizovati i na modele podrešetki. Na kraju, razvijen je sistem jednačina da bi se efikasno izračunala kriva Gibsove energije kod faze opisane pomoću modela podrešetke.
Ključne reči: Termodinamički proračuni; Proračuni ravnoteže; Fazni dijagram; Prekid u rastvorljivosti; Algoritam.