An effective algorithm to identify the miscibility gap in a binary substitutional solution phase

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.


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Miscibility gap occurs when a two-phase coexistence line in a phase diagram ends at a critical point [1]. On one hand, miscibility gap has been well understood and extensively used to develop high-performance materials through the spinodal-type decomposition of the microstructure, such as (Ti,Zr)C [2,3], TiAlN [4], etc. On the other hand, for some alloys like high-entropy alloys, intermediate phase or miscibility gap should be avoided in order to form a singe-phase microstructure thereby obtaining excellent performance. As a result, it is of great importance to acquire the information about miscibility gap of materials during the developing process. In the literature, many authors presented mathematic equations for understanding miscibility gap in terms of Gibbs energies [5][6][7][8], and numerically detected them by a discretization of composition axis [9][10][11][12][13]. To the best of our knowledge, however, there is no detailed description about how to detect if a miscibility gap exists in terms of interaction parameters analytically. One purpose of this paper is to develop a novel method to determine the existence of miscibility gap analytically for given thermodynamic parameters. In addition, the ranges of interaction parameters are analyzed mathematically in the solution model. The other objective in the present work is to provide a new approach to calculate the miscibility gap and guide selection for the interaction parameters during the thermodynamic assessment in binary systems.
In section 2, we present a simple method to identify if there is a miscibility gap in terms of the interaction parameters analytically. Subsequently, in order to investigate the existence of the miscibility gap, we deduce the first and second derivatives of Gibbs energy with respect to composition for a phase described with a sublattice model in section 3 and section 4. After that in section 5, the calculations of the Al-Zn and Al-In phase diagrams are demonstrated to show the unique features of the presently developed new algorithm. Finally, a summary is made in section 6. It could be mentioned that the outcome of the present work is of both scientific and educational interests.

On the existence of miscibility gap in view of interaction parameters
In thermodynamic equilibrium calculations, if the miscibility gap exists should be considered. In this section, we will investigate the existence of the miscibility gap in a binary phase with a substitutional solution model.
For fixed T and P, let x be the molar fractions of components A and B, respectively. Then, the molar Gibbs energy of a phase described with a substitutional solution model can be expressed as   Generally, a miscibility gap involves phase separation within a single phase. The corresponding Gibbs energy curve shows two coexisting compositions, or two inflection points [1]. Mathematically, this implies that there are two real number solutions to equation Here the second order derivative of M G with respect to x is given by We now discuss the existence of the miscibility gap with the increase in the number of the interaction parameters.

Case
can be expressed as: It is obvious that Eq. In other words, there exists a miscibility gap in this phase when RT L 2 0  .

 n
Similarly, for the case Thanks to the feature of the cubic function, there are only two cases such that

Case
It is easy to check that the function from the left side of Eq. (9) is a polynomial of degree 2  n . By computing the eigenvalues of the corresponding companion matrix [17], if 6 there are at least two real eigenvalues in interval   1 , 0 , a miscibility gap exists; otherwise, it does not exist.
Generally, in a binary system, a substitutional solution model is considered with no more than 4 interaction parameters, i.e., 3  n , and mostly we just take 2  n . In thermodynamic assessment process, according to previously introduced interaction parameters, we can compute the range of the last interaction parameter in which a miscibility gap exists. Now consider the case where the left hand side is a function symmetry with respect to 1 2 x  , and for different 2 L the corresponding function curves invariably pass through the point , there exists a miscibility gap in the phase. It is remarkable that since the analysis for the existence of miscibility gap in this section is in terms of interaction parameters at different discrete values of temperature, the present approach can also be used to check the existence of the miscibility gaps for different expressions of T-dependence of interaction parameters [14][15][16] quickly and efficiently.
In addition, for the linear model for the T-dependence of interaction parameters, generally given by Analogous to the analysis above, in view of Fig. 2, when 0 , an artificial inverted miscibility gap appears at high temperatures.

Finding the miscibility gap for a phase described with a sublattice model
The method descried in section 2 deals with a phase described with a substitutional solution model. Next we consider a sublattice model in which the corresponding Gibbs energy can also be expressed by a function of two variables, such as phase  in the Zr-Sn [18] binary system, which is described with the sublattice model In this section, we shall study the existence of the miscibility gap for a phase described with a sublattice model under fixed T and P where the Gibbs energy expression involves two internal variables ,  Fig. 3 we find that the first and second derivatives computed directly by the expressions (19) and (20) coincide with those calculated numerically by the finite difference approximation schemes. in Zr-Sn system computed directly via formulas (19), (20) and numerically by finite difference schemes.

The derivatives in a sublattice model
In this section, we shall expand the derivation method introduced in section 3 to a general sublattice model. These derivatives can help to obtain the Gibbs energy curve efficiently by a simple Newton algorithm.
In order to make it easier to understand the derivation for the derivatives of Clearly, there holds that for any In fact, Eq. (25) can also be obtained from the partial Gibbs energies of the endmembers in [19,20].
Note that the partial Gibbs energies of the end-members are related to the chemical potentials of the elements, Eq. (30) can be solved by the simple Newton method. Now we take the phase Cu2Mg described with (Cu,Mg)2(Cu,Mg)1 in Cu-Mg [21] binary system as an example to observe the efficiency of Eqs. (23), (24) Table 1, for a fixed 100 N  , we present the maxima of the differences of the above two Gibbs energies at all nodes against various N . As shown in Table 1, the computation time from the present algorithm is much shorter than that due to the discretization method.

Results and discussion
Some numerical results are presented to verify the efficiency of the proposed algorithm in sections 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 need to check whether it is the global equilibrium or not. In the calculation of Fig. 5(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. 5(b) the miscibility gap exists in the liquid phase.  L 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 K T 626  , we can see that the value of 2 L in the fcc_a1 phase of the Al-Zn system is outside the range resulting from Eq. (11). And there is no miscibility gap in phase fcc_a1 at this temperature. In addition, except for the solid phase, this method also can be applied to liquid phase in the Al-In system. Analogously, the parameters 0 L , 1 L for  phase in the Zr-Sn system are given in  The outcome of the present work is of interest for the thermodynamic optimization in which a miscibility gap is involved. During the thermodynamic optimization of a binary system in which a miscibility gap exists, Eq.(8) and Eq. (11) can be used to detect numerical regions of the interaction parameters very easily, which will reduce the amount work of assessment. As a summary, Table 5 presents the implication and guidance to thermodynamic calculations and optimizations from the analytical results for each of the cases n = 0, 1, 2 and > 2. 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 . a Excess Gibbs energy is given as follows: To the best of our knowledge, no detailed description is presented in textbooks on how to identify if a miscibility gap exists in terms of the given thermodynamic parameters analytically in a binary system. The present work shows all of the details in order to detect the existence of the miscibility gap in a binary system, being of interest to undergraduates and graduates.

Summary
In this work, the existence of the miscibility gap in a phase described with a substitutional solution model is analyzed for a binary system. When a miscibility gap exists in the phase, the second derivative of Gibbs energy   Based on the chain rule of the derivative of compound function, 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 method to detect the miscibility gap in terms of interaction parameters can be generalized to a sublattice model in which the Gibbs energy has two internal variables. Moreover, in view of the derivatives, we have developed a system of equations to efficiently compute the Gibbs energy curve of a phase described with a sublattice model.     Table 3. Parameters L0, L1 and L2 are given in corresponding TDB file at these temperatures. And L2 parameter from Eq. (12) is solved with the given L0, L1 Table 4. Parameters L0 and L1 are given in corresponding TDB file at these temperatures. And L1 parameter from Eq. (9) is solved with the given L0 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