Third Generation of Unary Calphad Descriptions and the Avoidance of Re-Stabilized Solid Phases and Unexpected Large Heat Capacity

Abstract

The problem of solid phase re-stabilization at high temperature in the latest version of 3rd generation data for pure solid elements is studied and the remedy to apply new Calphad software with the Equal-Entropy Criterion (EEC) implemented is critically examined. The concept of Equal-Entropy Temperature (EET) as very top limit for crystal existence is used to develop amendments by a second temperature range without introducing any adjustable parameter to an existing set of 3rd generation unary functions of solid and liquid. This is shown to safely prevent by definition of the mathematical Gibbs energy function the violation of the EEC axiom, namely that no solid phase should be more stable than the liquid at high temperature. The comprehensive amendment also solves the problem of unexpected large C_p contributions at high temperature. Detailed analyses are provided for the examples of pure Al and for binary Al-Ir, Al-Cr and Al-C systems. Serious risks of unforeseen pitfalls and artifacts are exemplified for some cases where an automatic exclusion of an stable phase by new “EEC-software” may occur. It is argued that this is not acceptable at least for applied Calphad simulations.

1 Introduction

In the third-generation data for pure elements^[1,2,3,4,5] the molar Gibbs energy function of an element i in any crystalline phase φ, is generally described by an equation of the following form

$$G_{i}^{0,\varphi } \left( T \right) = G_{i}^{\varphi } \left( T \right) - H_{i}^{SER} = E_{0} + \frac{3}{2}R\theta_{E} + 3RT\ln \left[ {1 - \exp \left( { - \frac{{\theta_{E} }}{T}} \right)} \right] - \frac{a}{2}T^{2} - \frac{b}{n(n + 1)}T^{n + 1} + G^{mag}$$

(1)

where G^mag is the magnetic contribution^{[1, 6]} Which is not in the focus here. R is the gas constant and T is the absolute temperature. E₀ is a total energy at 0 K and the parameters a, b and n are determined from the molar heat capacity, C_p, where n usually takes the values 2, 3 or 4. θ_E is the Einstein temperature for the phase φ of this element. The focus of this work is on the proposal made for the latest version of the 3rd generation data to use a single set of these parameters in Eq 1 to cover the temperature range from 0 to 6000 K^[3,4,5] as opposed to a second high temperature range.^[2]

Note that $H_{i}^{SER}$ is the molar enthalpy of the element i at 298.15 K and 1 bar in its standard element reference (SER) state. In the present notation the superscript “0” in $G_{i}^{0,\varphi } \left( T \right)$ already defines the SER reference, which avoids the distracting replication of subtracting the terms $H_{i}^{SER}$ in any other phase, for example in stoichiometric compounds. The parameters in Eq 1 must be set so that the enthalpy calculated from $H_{i}^{0,\varphi } = G_{i}^{0,\varphi } - T\partial G_{i}^{0,\varphi } /\partial T$ at 298.15 K and 1 bar is zero only for φ = SER, e.g.,$H_{Al}^{0,Fcc} (298.15\;K) = 0$ but not for other phases, e.g. $H_{Al}^{0,Liquid} (298.15\;K) = + 1105$ J/mol. The background is the missing absolute enthalpy scale in thermodynamics, as opposed to entropy. Thus, an arbitrary choice must be made for the enthalpy derived from the general $G_{i}^{\varphi } \left( T \right)$ for one phase of this element. This choice is made by formally subtracting the same quantity $H_{i}^{SER}$ in Eq 1 for all phases φ of this element to define $G_{i}^{0,\varphi } \left( T \right)$. The result of this arbitrary choice is that the enthalpy derived from $G_{i}^{0,\varphi } \left( T \right)$ results in $H_{i}^{0,SER} (298.15\;K) = 0$, but only for the SER phase. In practice this is done by adjusting the value of E₀ in Eq 1. Even though this is not stated explicitly in the equations in Ref 1, 3,4,5 it is exactly this procedure providing the same SER as in the SGTE unary database^[7].

The basic principle in the SGTE unary database is that the C_p functions coincide for all condensed phases at high (and low) temperatures because it was well known that the direct extrapolation into the metastable ranges may lead to absurd results^[8]. Thus, a second temperature range for $G_{i}^{0,\varphi }$ above the melting point ensures that C_p of solid phases φ approach the values of C_p(Liquid) by using a term T⁻⁹ in the high-temperature range.^[7] This safely prevents the incorrect re-stabilization of solid at high temperatures at the–relatively small–cost of a kink in C_p at the melting point, which may be distracting in solid solution or intermetallic phases. That was already recognized and stated earlier “… SGTE has adopted an interim solution for many elements…” in this seminal contribution.^[7] These so-called “2nd generation” unaries served as basis for many multicomponent Calphad databases in the past three decades. This kink is avoided by using Eq 1 with a single temperature range in the 3rd generation unaries. The author emphasizes and acknowledges that significant progress was made in the 3rd generation of unaries to describe the stable and metastable solid and the liquid/amorphous phases with $G_{i}^{0,\varphi } \left( T \right)$ functions down to 0 K on a more physical basis. The supplementary zip file to this manuscript includes 9 commented TDB files for all the calculations performed in this work. It also includes 5 batch files (*.pbfx) to reproduce Fig. 1, 2, 3, and 4 with Pandat software.^[9]

Fig. 1

Calculated thermodynamic functions of the stable Al phases FCC and Liquid; (a) the Gibbs energy data for G(Liquid) and G(FCC_HE) are calculated from the description by He et al.,³ the data for G(FCC_RSF) are calculated using Eq 4 with EET = 2366.5 K and shown by the thick green line, the data for G(FCC_BIG) are calculated from the description by Bigdeli et al.²; (b) the corresponding Gibbs energy data relative to G(FCC_HE); (c) the corresponding entropy data; (d) the corresponding enthalpy data; (e) the corresponding heat capacity data. (f) same as (b) with additional isobars of G(GAS) relative to G(FCC_HE), p = 1-40 bar

Fig. 2

Calculated Gibbs energy functions of all Al phases Liquid, FCC, BCC and HCP relative to the SER phase FCC; (a) calculated from the original description by He et al.³ with the single temperature range; (b) calculated using Eq 4, 5, and 6 for the solid phases denoted as FCC_RSF, BCC_RSF, and HCP_RSF and the original description of liquid by He et al.³

Fig. 3

Calculated thermodynamic functions of the stable Al phases FCC and Liquid; (a) the Gibbs energy data, relative to G(FCC_HE) from Fig. 1(b) with the additional G(FCC_RSF2) from Eq 12 shown by the thick line in magenta; (b) the corresponding heat capacity data

Fig. 4

Calculations in the binary Al-Ir system using data by Zhang et al.^[13] which are based on the SGTE unaries; (a) Al-Ir phase diagram in T-x view; (b) heat capacities of the FCC phase at 90 at.% Ir and of the B2 phase at 50 at.% Ir, stable single-phase ranges are highlighted by thick curves; (c) Al-Ir phase diagram in H–x view, SER-reference with zero enthalpy at 298 K and pure components; (d) enthalpy development of FCC phase at 90 at.% Ir, the SGTE-no-break-unaries curve shows the significantly exaggerated H(FCC) at higher temperature. (e) heat capacity of the FCC phase at 930 and 2150 K compared to the values at 2150 K (pointed red curve) if the SGTE data are used without the break at T_m, the “SGTE-no-break-unaries”

2 Problems Created by Using a Single Temperature Range for Unary Gibbs Energy

2.1 Problem of Solid Phase Re-Stabilization at High Temperature

This work is prompted by the publication of the paper “A third generation Calphad description of Fe: revisions of fcc, hcp and liquid” by He et al. in this issue.^[4] However, the problem of solid phase re-stabilization at high temperature noted in that work is also observed in the latest version of 3rd generation data for pure Al^[3] and pure W.^[5] It is obviously imminent to the method applied for these unary Calphad descriptions because the parameters a and b in Eq 1 are generally positive, resulting in an unlimited growth of C_p of solid phases at high T, far above C_p(Liquid).

The first serious consequence of this extrapolation method in Ref 4 for Fe is that not only the solid phase BCC becomes stable beyond 3630 K, G^BCC(T) < G^Liquid(T) for T > 3630 K. In addition, liquid becomes less stable compared to all solid phases at high T: G^FCC(T) < G^Liquid(T) for T > 6320 K, and G^HCP(T) < G^Liquid(T) for T > 7380 K. The authors suggest that, as remedy for this artifact, new software should be used: “Thus, the Equal-Entropy Criterion (EEC) suggested by Sundman et al.^[10] must be applied to exclude any solid phase with higher entropy than the liquid phase. If the calculation reaches the Equal-Entropy Temperature (EET) of the solid phase, the software will automatically suspend this solid phase.” Similar statements are made for pure Al,^[3] and pure W.^[5] Currently (as of March 2022) the EEC is not implemented in the software Thermo-Calc,^[11] but only in the latest software version of OpenCalphad (www.opencalphad.com). The re-stabilization problem is addressed by the simple amendment in section 3.1.

2.2 Problem of Unexpected Large Heat Capacity at High Temperature

The second serious consequence of this extrapolation method with single temperature range is the unrestricted growth of heat capacity of all solid phases. That results in unexpected large C_p contribution of low-melting elements in solid solutions at high temperatures, far above the melting or instability temperature of this contributing solid phase. The solution for this problem, which at the same time solves the re-stabilization issue detailed above, is presented in the comprehensive amendment in section 3.2 and discussed in detail for the case of pure Al and the Al-Ir binary system.

3 Proposed Amendment of 3rd Generation data by Addition of Second T-Range

3.1 Simple Amendment to Avoid Re-Stabilization

The author entirely agrees that the physical basis (or axiom) of the Equal-Entropy Criterion (EEC) as given in,^[10] is truly valid: No solid phase should be more stable than the liquid at high temperature. The Equal-Entropy Temperature (EET) of any solid phase φ is defined by

$$S^{\varphi } \left( {{\text{EET}}} \right) \, = S^{{{\text{Liquid}}}} \left( {{\text{EET}}} \right)$$

(2)

where the (extrapolated) entropy of solid, S^φ(T) has reached the entropy of the liquid, S^Liquid(T) and even exceeds it for T > EET. It is also entirely agreed that certainly at T = EET the thermodynamic description of the solid phase φ becomes physically unrealistic.^[10] The Calphad method requires the handling of phase φ above EET but there is no physical basis for any extrapolation of $G_{i}^{0,\varphi } \left( T \right)$.

In the present work a simple alternative extrapolation is considered:

$$G_{i}^{0,\varphi } \left( T \right) = G_{i}^{0,\varphi } \left( T \right)\;{\text{of}}\;{\text{ Eq}}.\left( 1 \right){\text{ for}}\;T < {\text{ EET}}$$

(3a)

$$G_{i}^{0,\varphi } \left( T \right) = G_{i}^{{0,{\text{Liquid}}}} \left( T \right) + c^{\varphi } {\text{ for}}\;T > {\text{ EET}}$$

(3b)

where c^φ is a constant to be determined by the continuity condition of $G_{i}^{0,\varphi } \left( T \right)$ at the chosen break point, T = EET. By defining $G_{i}^{0,\varphi } \left( T \right)$ of the solid phase φ to be equal to the parallel shifted function of liquid $G_{i}^{0,Liquid} \left( T \right)$ for T > 2366.5 K, as seen by the green curve for FCC_RSF in Fig. 1(b), any further intersection is excluded. This safely prevents by definition of the mathematical Gibbs energy function the violation of the EEC axiom and the incorrect re-stabilization of solid at high temperatures at the cost of a step in C_p at EET.

This is exemplified below for pure aluminum because it is a simple element without magnetic contribution and two different 3rd generation data,^[2,3] can be shown for comparison. Figure 1 shows the thermodynamic functions of the stable Al phases FCC and Liquid. The gas phase, which would become stable at 2792 K and 1 bar, is considered later. The data for liquid and FCC are first calculated using the description proposed in 2021 by He et al.^[3] and denoted by G(Liquid) and G(FCC_HE) in Fig. 1(a), respectively. The corresponding EET of FCC_HE equals 2366.5 K and is marked in the diagram. The data for G(FCC_RSF) are calculated using Eq 3, with

$$G\left( {{\text{FCC}}\_{\text{RSF}}} \right) \, = G\left( {{\text{FCC}}\_{\text{HE}}} \right)\;{\text{for}}\;T < \, 2366.5 \, {\text{K}}$$

(4a)

$$G\left( {{\text{FCC\_RSF}}} \right) \, = G\left( {{\text{Liquid}}} \right) \, + \, 10086.62 \, {\text{J/mol for}}\;T > \, 2366.5 \, {\text{K}}.$$

(4b)

In addition, the data proposed in 2019 by Bigdeli et al.^[2] are also calculated and shown for comparison, denoted by G(FCC_BIG) in Fig. 1(a). It is seen that FCC_HE becomes more stable than Liquid for T > 3538 K whereas FCC_RSF does not re-stabilize at any high temperature. This is more clearly seen in Fig. 1(b) which is plotted not with G referring to SER but shown relative to G(FCC_HE), denoted by the label G(*:FCC_HE[Al]).

Figure 1(c), (d), and (e) shows the corresponding thermodynamic functions entropy, enthalpy and heat capacity derived from G with analog notations as in Fig. 1(a). The Equal-Entropy Temperature is most clearly recognized in Fig. 1(c). The continuity of S(FCC_RSF) and H(FCC_RSF), also ensured by the construction in Eq 3, is demonstrated in Fig. 1(c) and (d). Consequently, the heat capacity C_p(FCC_RSF) exhibits a step in Fig. 1(d) at EET = 2366.5 K whereas the values of C_p(FCC_HE) grow unlimited with temperature. These results are discussed later.

The equilibria with gaseous Al are given for completion. Values for G(GAS), comprising the species Al and Al₂, are shown in Fig. 1(f), also relative to G(FCC_HE). This is essentially the superposition of the additional isobars of G(GAS) for the pressure range p = 1-40 bar to Fig. 1(b), showing the Liquid + Gas equilibria at the intersection of the G-curves. The intersection at 1 bar occurs at 2757 K; that is below the accepted standard evaporation point of Al, 2792 K, because the G(LIQUID) curve of He et al.^[3] already bends towards G(GAS) at much lower temperature, requiring an adjustment. Despite that minor numerical difference it gives an impression of the equilibria with gas. At 20 bar the re-stabilization temperature of FCC_HE at 3538 K and the (metastable) evaporation at 3579 K are almost at the same temperature. At 40 bar this leaves even room for the predicted artifact of stable solid FCC_HE between 3538 K and sublimation (!) at 3982 K.

The same procedure of Eq 3 is applied to provide the second temperature range also for the metastable solid phases of Al, BCC and HCP, as given in Eq 5 and 6, respectively:

$$G\left( {{\text{BCC\_RSF}}} \right) \, = G\left( {{\text{BCC\_HE}}} \right){\text{ for}}\;T < \, 1949.0 \, {\text{K}}$$

(5a)

$$G\left( {{\text{BCC\_RSF}}} \right) \, = G\left( {{\text{Liquid}}} \right) \, + \, 9660.47 \, {\text{J/mol for}}\;T > \, 1949.0 \, {\text{K}}$$

(5b)

$$G\left( {{\text{HCP\_RSF}}} \right) \, = G\left( {{\text{HCP\_HE}}} \right){\text{ for}}\;T < \, 2218.5 \, {\text{K}}$$

(6a)

$$G\left( {{\text{HCP\_RSF}}} \right) \, = G\left( {{\text{Liquid}}} \right) \, + \, 11370.06 \, {\text{J/mol for}}\;T > \, 2218.5 \, {\text{K}}.$$

(6b)

The values of each EET are calculated from the original description by He et al.^[3] Figure 2(a) shows the Gibbs energy functions for Liquid, FCC, BCC and HCP relative to the SER phase (FCC_HE) calculated from the original description by He et al.^[3] with the single temperature range. The re-stabilization temperatures of each solid phases compared to Liquid are marked. Figure 2(b) shows these Gibbs energy functions calculated using Eq 4, 5, and 6 based on the original description of Liquid by He et al.^[3] relative to FCC_RSF, where the EET-values are marked. It is emphasized that below each EET the values for all phases in both diagrams are exactly identical. Towards higher temperature also the curve for Liquid bends, reflecting the parallel shift given in Eq 4b. Re-stabilization of a solid phase cannot occur based on the Gibbs energy functions in Fig. 2(b). These results are discussed in section 4.

3.2 Comprehensive Amendment to Avoid Re-Stabilization and Unexpected Large Heat Capacity Contributions

A most important point of the simple amendment is that no additional parameters are introduced to an existing set of 3rd generation unary functions of solid and liquid. That is also the case for the proposed comprehensive amendment and, again, the concept of EET as very top limit for crystal existence is used. A simple protocol is given to generate the amended unary functions, demonstrated below for FCC Al.

The basic idea is that EET is a pragmatic but very crude approximation for the extrapolated temperature at which the crystalline phase loses any physical reality. The actual instability temperature, T_inst, where the superheated solid collapses into a liquid-like structure, is much lower for FCC Al, T_inst = 1200 K^[2] as discussed later. We have the relation for FCC Al

$$\left( {T_{{\text{m}}} = \, 933.47 \, {\text{ K}}} \right) \, < \, \left( {T_{{{\text{inst}}}} = \, 1200 \, {\text{ K}}} \right) \, < \, \left( {{\text{EET }} = \, 2366.5 \, {\text{ K}}} \right).$$

(7)

The elaborate determination of the actual T_inst^[2] involves too much effort for an application to each phase of the element, not to speak of a systematic development for all unaries. Therefore, it is considered to evaluate the estimated instability temperature, T_inst,est, based only on the

data in an existing 3rd generation set, for example as the mean value of T_m and EET:

$$T_{{\text{inst,est}}} = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \, \left( {T_{{\text{m}}} + {\text{ EET}}} \right).$$

(8a)

As a reasonable choice for C_p(solid) at very high temperature the value of C_p(Liquid) at EET may be considered. Using only the 3rd generation set of He et al.^[3] we have for FCC Al with EET(FCC, Al) = 2366.5 K:

$$T_{{\text{inst,est}}} \left( {\text{FCC, Al}} \right) = 1650 \, {\text{ K}}$$

(9a)

$$C_{{\text{p}}} ({\text{FCC}},{\text{ Al}},T \to \infty ) \, = C_{{\text{p}}} ({\text{Liquid}},{\text{ Al}},T = {\text{EET}}) \, = 28.1{\text{ Jmol}}^{ - 1} {\text{K}}^{ - 1} .$$

(9b)

Note that a simpler alternative to the key temperatures in Eq 8a and 9b is given in Eq 8b and 9c at the end of this section as the final recommendation. The second temperature range in the comprehensive amendment is proposed to start at T_inst,est using the following equations

$$G\left( {{\text{FCC}}_{\text{RSF2}}} \right) \, = G\left( {{\text{FCC\_HE}}} \right){\text{ for}}\;T < \, 1650 \, {\text{ K}}$$

(10a)

$$G\left( {{\text{FCC}}_{\text{RSF2}}} \right) \, = c_{0} + c_{{1}} T + c_{{2}} T\,{\text{ln}}\left( T \right) \, + c_{{3}} T^{ - p} + c_{{4}} T^{ - q}\;{\text{ for }}T > { 165}0{\text{ K}}$$

(10b)

Suitable exponents are chosen as p = 5 and q = 11, following the example used for a second temperature range by Bigdeli et al.^[2] All of the five constants c_i are determined by the four continuity constraints of G, S, C_p, and dC_p/dT at T = 1650 K and the one value of C_p from Eq 9b. The explicit equations for the second temperature range are

$$G = c_{0} + c_{1} T + c_{2} T\ln \left( T \right) \, + c_{3} T^{ - 5} + c_{4} T^{ - 11}$$

(11a)

$$S = \, {-} \, c_{1} {-} \, c_{2} {-} \, c_{2} \ln \left( T \right) \, + \, 5 \, c_{3} T^{ - 6} + \, 11 \, c_{4} T^{ - 12}$$

(11b)

$$C_{p} = \, {-} \, c_{2} {-} \, 30 \, c_{3} T^{ - 6} {-} \, 132 \, c_{4} T^{ - 12}$$

(11c)

$${\text{dC}}_{p} /{\text{dT}} = + \, 180 \, c_{3} T^{ - 7} + \, 1584 \, c_{4} T^{ - 13} .$$

(11d)

Using the values provided by G(FCC_HE) and its derivatives at 1650 K and solving the set of four linear Eq 11 with c₂ = –28.1 Jmol⁻¹ K⁻¹ one obtains

$$\begin{gathered} G\left( {{\text{FCC\_RSF2}}} \right) \, = \, 6085.70781 \, + \, 152.0127767T - 28.1T\ln \left( T \right) \hfill \\ - 2.287787267 \times 10^{19} T^{ - 5} + \, 6.032531948 \, \times 10^{37} T^{ - 11} {\text{ in Jmol}}^{ - 1} {\text{ for}}\;T > \, 1650 \, {\text{K}}. \hfill \\ \end{gathered}$$

(12)

These data, relative to G(FCC_HE), are shown in Fig. 3(a), which is an extension of Fig. 1(b). The corresponding heat capacity data are shown in Fig. 3(b), where also the key temperatures of the comprehensive amendment (T_m, T_inst,est, EET) of FCC Al are marked, together with T_inst.^[2]

The same procedure given in Eq 8, 9, 10 and 11 should be followed to generate the second temperature range in the comprehensive amendment for the metastable Al phases, BCC and HCP, which have their own values of EET calculated from the same 3rd generation set,^[3] as shown in Fig. 2(b). Using the melting point of the stable phase FCC, T_m = 933.47 K, the estimated instability temperatures are obtained as T_inst,est(BCC, Al) = 1441 K and T_inst,est(HCP, Al) = 1576 K, similar to the FCC value. The same protocol can be followed for any element with existing 3rd generation set without introducing any adjustable parameter. Using always the stable melting point as fixed value for T_m is the only generally valid viable method. From Fig. 2 we see that the metastable melting points cannot be used, HCP shows a strangely low value and BCC shows none at all. This observation is found valid for many elements by reviewing the SGTE data.^[7]

As the more practical alternative to the estimation of instability temperature, T_inst,est in Eq 8a, it is proposed to use the simpler relation

$$T_{{\text{inst,est}}} = 1.3 \cdot T_{{\text{m}}} .$$

(8b)

The estimation factor 1.3 in Eq 8b is based on the value for FCC Al. We get T_inst,est(FCC, Al) = T_inst,est(BCC, Al) = T_inst,est(HCP, Al) = 1214 K. In addition, the reasonable choice for C_p(solid) at very high temperature is proposed to be taken from the value of C_p(Liquid) at 2.5·T_m instead of EET:

$$C_{p} ({\text{FCC, Al}},T \to \infty ) \, = C_{p} ({\text{Liquid}},{\text{ Al}},2.5 \cdot T_{m} ) \, = 28.1{\text{ Jmol}}^{ - 1} {\text{K}}^{ - 1} .$$

(9c)

The estimation factor 2.5 in Eq 9c is also based on the value for FCC Al, and 2.5·T_m = 2334 K. This does not change the terminal C_p value at all. Therefore, the second temperature range, Eq 10b, shall start at 1214 K and, together with the value of c₂ = –28.1 Jmol⁻¹ K⁻¹, the solution of the Eq 11 provides the proper equation of G(FCC) for T > 1214 K.

Setting the second temperature according to Eq 8b, 9c, and 10, 11 is the final recommendation for the comprehensive amendment. The advantages are (i) the simplicity and equal start point of the second range for all stable and metastable phases (also found in Ref 7), (ii) no need to evaluate EET for each phase, (iii) the flattening of the unexpected large extrapolated values of C_p starts right at T_inst,est  ≈ T_inst,, shortening the extrapolation into the unphysical range. One may argue that a meticulous study of instability temperatures of other elements might give somewhat different factors compared to Al. However, for a general recommendation the present heuristic approach appears more reasonable. For the present study the data calculated from Eq 8a are maintained to put the results into perspective to the value of EET which is the key point for the re-stabilization issue.

4 Discussion

4.1 Elaboration of Amendments

While the simple amendment solves the first problem and safely prevents the incorrect re-stabilization of solid at high temperatures it does not address the second problem. The extrapolated values of C_p(solid) far beyond T_inst are unexpected high. In addition, C_p(solid) is discontinuous with a step at EET. However, it is emphasized that even the simple amendment of Eq 3 provides continuous function for G, H, and S, so that equilibrium calculation based on the minimization of G and its continuous first derivatives are not impaired. The following discussion is focused only on the recommended comprehensive amendment.

Chen and Sundman^[1] proposed another selection of functions for the second temperature range starting already at the melting point of solid, T_m, which is significantly lower compared to Eq 13. It is certainly important to start a second temperature range not at the melting point but at a higher temperature, also for physical reasons. Bigdeli et al.^[2] have studied that aspect in their careful 3rd generation Calphad modeling of FCC Al. Finite temperature density-functional theory (DFT) calculations and molecular dynamics simulations of solid Al in the superheated temperature region revealed that the superheated solid collapses into the liquid at a certain instability temperature, T_inst. Combining these data with experimental superheating limits a value of T_inst ≈ 1200 K is assessed,^[2] clearly above T_m = 933.47 K for FCC Al. In the range of T_m < T < T_inst the extrapolated DFT data are still considered to be physically reasonable, but not beyond T_inst.

Consequently, a second temperature range for 1200 K (T_inst) < T < 6000 K is provided for G(FCC_BIG).^[2] As seen in Fig. 3(b) C_p(FCC_BIG) does not decay down to C_p(Liquid) at high T, as proposed in the equations of Ref 1 or in Eq. 9(c), but remains almost constant at approximately the chosen value of 36-37 Jmol⁻¹ K⁻¹ in the range of 1200-6000 K. Thus, the C_p(FCC_BIG) function with a very small bump and continuous C_p, and dC_p/dT at T = T_inst is achieved.^[2] Note that, due to this relatively large value of C_p^FCC and the difference to C_p^Liquid, the entropy of FCC_BIG intersects S(Liquid) at 4120 K and for T > 9130 K re-stabilization occurs, G(FCC_BIG) < G(Liquid).

To obtain the complete set of unary data for Al the determination of T_inst for all metastable structures of the element is required. The value of T_inst(FCC) = 1200 K was finally estimated,^[2] based on the experimental data and only partly on the DFT and other simulations. However, experimental superheating data are obviously not available for pure Al in BCC and HCP structure. The key message from that elaborate work is that an instability limit above the melting point exists, suggesting that the solid “explodes”^[2] at high temperatures, turning into a more liquid-like structure at T_inst. One may think about a two-state model for FCC for this feature, as accepted for the liquid phase,^[3] but that effort appears also not justified. It is concluded,^[2] that the strong increase of C_p between T_m and T_inst is realistic, but that any extrapolation to higher temperature cannot be derived from the physics of the FCC structure. The unlimited extrapolation of C_p beyond T_inst^[3] avoids any bump or kink, however, extremely large values of C_p are obtained in the region T > T_inst where any physical basis for the FCC crystal is gone. The adverse consequences are discussed in section 4.2.

The thermal vacancy contribution was also discussed by Bigdeli et al.^[2] and is found to be significant above some 600 K and may be a main driver for the strong increase of C_p between T_m and T_inst. They considered the Calphad modeling, also using Eq 1 up to T_inst, to picture a perfect superheated solid without an explicit vacancy term. Consequently, they subtracted the ab initio vacancy contribution from the experimental C_p data and subsequently determined the parameters of Eq 1 to fit. Therefore, the orange curve of C_p(FCC_BIG) in Fig. 1(e) and 3(b) is increasingly below the actual experimental data in that range, while fitting previous DFT data.^[2] Note that C_p(FCC_HE) is even below C_p(FCC_BIG) between T_m and T_inst, and a correction may be considered. More elaborate and general Calphad modeling of the thermal vacancy contribution is still pending.^[2] However, it is certainly most important that the G-functions for the pure elements in Eq 1 are parameterized so as to fit the best actual experimental C_p data as close as possible. That is crucial because the function assessed in Eq 1 is then used for the extrapolation from T_m to T_inst in the 3rd generation data as well as in the proposed amendments, see Fig. 3(b). Note that the final recommendation of Eq. 8(b) limits the extrapolation to T_inst,est = 1214 K.

In summary, the comprehensive amendment, Eq 8b, 9c, and 10, 11, is preferred over the simple amendment. Both safely prevent the artificial re-stabilization of solid and include the physics-supported strong increase of C_p between T_m and T_inst. However, the smooth and earlier reversal of the strongly growing C_p(solid) towards a value close to C_p(liquid) at very high temperature is also beneficial and important to avoid exaggerated predicted C_p values as discussed in the next section.

4.2 Why are the Unary Data Important for T > T_inst ?

Even far above the melting or instability temperature, especially of low-melting elements, the extrapolated C_p data of the solid phase become very important for solid solutions at high temperatures based on the same crystal structure. Using the extrapolation method with single temperature range results in unrestricted growth of heat capacity of all solid phases and the consequence of unexpected large C_p contribution will be discussed next. Another problem is the completely unphysical prediction of thermodynamic driving force for nucleation/formation of the solid phase from liquid in a wide temperature range around and above EET, which is discussed at the end of this section.

For multicomponent systems the Calphad method requires the Gibbs energy data for each phase, generally based on the unaries, in the entire temperature range. For stoichiometric compounds it is disapproved to use the floating reference state (i.e. Neumann–Kopp’s rule) and instead the fixed reference state at 298.15 K with a stand-alone modeling of C_p of the compound is recommended.^[12] That also avoids the reflection of any artifacts in the pure element reference functions in the properties of the compound. However, for solid solution phases the end-members generally refer to the unaries, both for terminal solid solutions, emerging from the stable unary phase, and for intermetallic solutions. To study the impact for the example of Al it is most relevant to consider very high melting second components.

The highest melting FCC elements are Ir (2719 K), Rh (2237 K), Pt (2041 K), Pd (1828 K), Co (1768 K), Ni (1728 K), and so on, with the melting points given in decreasing order in brackets.^[7] Relevant is only iridium, all other candidates show T_m < EET(FCC, Al) = 2366.5 K. Again, Al proves to be a perfect example because an experimentally validated Calphad assessment is available for the binary Al-Ir system by Zhang et al.^[13] It shows both an extended solution of up to 11 at.% Al in FCC-Ir at 2152 K and a congruent melting point of the ordered B2 intermetallic solution at 2403 K as given in Fig. 4(a), the Al-Ir phase diagram with gas phase suspended. Figure 4(b) shows the calculated heat capacity of the FCC phase at 90 at.% Ir (red curve). Note that this alloy is stable as single-phase FCC only from solvus to solidus, 2009-2213 K, as also seen along the dashed composition line in Fig. 4(a). Also the calculated heat capacity of the B2 phase at 50 at.% Ir is given in Fig. 4(b), which is stable below 2403 K.

The C_p(FCC)-data in Fig. 4(b) exhibit a kink at the melting point of Al because the modelling,^[13] is based on the SGTE unaries.^[7] That kink at 933.47 K is only minor for the 10Al-90Ir (at%) FCC composition because the unary data for Al enter only with the factor 0.1 and the kink from pure Ir would occur only at 2719 K. A complete re-modeling of the Al-Ir system, based on the 3rd generation unary functions, would be necessary, however, the data for Ir are still missing. Simply replacing the Al data by 3rd generation or, as an approximation, by the SGTE-no-break-unaries, falsifies the phase diagram completely. Therefore, the readers are asked to still refer to the phase diagrams in Fig. 4(a) and (c) in the following discussion.

Using the Al data from He et al.^[3] the kink at 933.47 K is removed and a smooth and apparently “artifact-free” curve results. That may give the deceptive impression that even the C_p of Ir-rich FCC can be calculated/predicted without problems at any temperature. However, C_p(FCC_HE, Al) contributes a large value of 58 Jmol⁻¹ K⁻¹ at 2366.5 K,^[3] see Fig. 1(e), much higher than the 37 Jmol⁻¹ K⁻¹ evaluated at T_inst for the collapsing FCC structure^[2] or the 28.1 Jmol⁻¹ K⁻¹ for the Liquid.^[3] The latter possibly reflects a more reasonable estimate for C_p(FCC, Al), which has already collapsed into a liquid-like structure at about 1160 K lower temperature. It may be argued that the liquid-like and, with T > 1200 K, ever more open and disordered structure should contribute a decreasing instead of an increasing value from C_p(FCC, Al), based on the elaborate study of Bigdeli et al.^[2] but this is an open question. But it is clear that any the physical basis for the FCC Al is long gone, since T_inst = 1200 K. Using the comprehensive amendment of Eq 8, 9, 10, and 11 avoids any kink, moreover the predicted values above T_inst,est are considered less exaggerated as shown in Fig. 3(b) and discussed below in more detail.

Exaggerated C_p data have a significant impact on heat increment and may impair heat balance calculations which are very important for industrial application. It is instructive to show the Al-Ir phase diagram in enthalpy (H)—composition (x) view in Fig. 4(c). Note that this phase diagram is produced without an additional calculation from the data stored by Pandat software,^[9] during the calculation of the T–x diagram in Fig. 4(a). The difference is only that the molar state variable H instead of the potential state variable T is plotted as ordinate. Consequently, the three-phase regions, degenerated to a line in in Fig. 4(a), become true phase “regions” (triangles) because each phase has its own enthalpy. For example, the eutectic at 2152 K corresponds to the triangle Liquid + B2 + FCC and this region is also isothermal. Each of the green tie lines also indicates a constant temperature and the one at 1000 K in the B2 + FCC region is highlighted by a thick green line. More instructions for applications of this H–x and other phase diagram views are given in Ref 14. Note that the vertical red dashed line at alloy composition 10Al-90Ir starts at 1000 K and extends to the liquidus point at 2647 K, exactly matching the same range of the dashed line in Fig. 4(a). One can read the enthalpy required to heat this alloy from 1000 K to the solidus point at 2213 K directly along the line in Fig. 4(c), starting at 1500 J and ending at 45,610 J, thus, 44.1 kJ/mol-atoms alloy are required.

For the numerical calculation let us focus on the major contribution from the FCC phase. The enthalphy evolution of FCC at 90 at.% Ir is shown in Fig. 4(d) as solid line, starting at 4755 J/mol FCC, the end value at the solidus point, marked by the arrow at 2213 K, is also 45,610 J/mol. To demonstrate the impact of unrestricted growth of C_p(FCC) the same calculation is performed using SGTE-no-break-unaries for the Al-Ir data set. These are obtained by removing the break point at T_m with unrestricted extrapolation of the solid unary functions to T > T_m. The impact on C_p(FCC) in Fig. 4(b) is the removal of the kink at the Al melting point and a value of the extrapolated C_p(FCC, 2213 K) = 48.7 Jmol⁻¹ K⁻¹, much larger than the value of 39.3 Jmol⁻¹ K⁻¹ at the same point in the accepted C_p(FCC) curve. The impact on H(FCC) in Fig. 4(d) is shown by the red dashed curve, reaching 50,520 J/mol at the solidus marked by the arrow. The starting point at 1000 K is almost the same (4765 J/mol), and subtracting it from each end point we obtain 45.8 kJ (no-break) compared to 40.9 kJ (real) for the heat energy required to heat 1 mol FCC from 1000 to 2213 K. The exaggerated value is 12% higher, which is a significant deviation and only due to the unexpected large C_p contribution from pure Al.

This is also evident in Fig. 4(e), the isothermal values of C_p(FCC), in the pointed red curve at 2150 K with the unrealistic steep C_p-slope in the range of Ir-rich solution for “SGTE-no-break-unaries”, ending at C_p(FCC, Al) = 119 Jmol⁻¹ K⁻¹. The 3rd generation data,^[3] 53 Jmol⁻¹ K⁻¹, are lower but they also suggest a steeply rising C_p(FCC, Al) at 2150 in the stable range up to 11 at.% Al in the Ir-rich solution. That is considered unrealistic, knowing that FCC Al collapsed to liquid-like structure at 1200 K with a probably much smaller C_p. A lower value of C_p(FCC, Al) = 34 Jmol⁻¹ K⁻¹ at 2150 K is obtained using the comprehensive amendment, similar to the standard SGTE data in the green curve at 2150 K in Fig. 4(e). Using the comprehensive amendment with the lower value of T_inst,est from Eq. 8(b), 1214 K, reduces C_p(FCC, Al) at 2150 K even further. The curve at 930 K calculated/predicted from the data by Zang et al.^[13] using the standard SGTE unaries is shown for comparison.

Figure 4(b) also shows the calculated heat capacity of the B2 phase at 50 at.% Ir where the kink at 933.47 K is more pronounced because the unary data for Al enter with the factor 0.5. It is obvious that by using the extrapolation of Al data from He et al.^[3] the kink is removed but the C_p(B2) curve would increase even more drastically than discussed for FCC above. There are more material systems, such as Ga-Ni or Ag-Hg, where the difference in melting point of the pure elements as well as the solubility in the high-melting element are even larger. The problem of exaggerated C_p extrapolations cannot be neglected.

The smooth flattening of C_p(FCC, Al) shown in Fig. 3(b), is considered to be the second significant advantage of the comprehensive amendment, in addition to avoiding the re-stabilization of the solid phase. The avoidance of unexpected large C_p contribution from the low-melting is most important for realistic Calphad based enthalpy and heat balance predictions of technological relevance.

As an alternative solution to the problem of unexpected large heat capacity contribution in solid solutions the inclusion of interaction terms affecting heat capacity may be considered. That can be discussed for the example of the Al-Ir FCC solution where the Redlich–Kister interaction parameter is L_Al,Ir = −158,872 + 13*T, in J/mol.^[13] This typical linear T-dependence with zero value of C_p-excess results in the linear composition dependence of C_p(x_Ir) shown by the straight line in Fig. 4(e) at 2150 K. The most simple term to be added to L_Al,Ir for a variation of of C_p(x_Ir) is c·T·ln(T). Positive values of c are required to bend the straight line of of C_p(x_Ir) downwards into a bell shape. For c =  + 10 Jmol⁻¹ K⁻¹ the maximum reduction at C_p(0.5) is only −2.5 Jmol⁻¹ K⁻¹, hardly visible on the graph and the steep slope at pure Ir is hardly reduced. For c =  + 12 Jmol⁻¹ K⁻¹ already the demixing of FCC below 97.2 at% Ir (2.8 At.% Al dissolved) occurs, which is unacceptable. Therefore, this is not a viable approach. The only option is to make sure that the end member values of C_p are realistic (or not too unrealistic), as in the comprehensive amendment. This also leaves room for a fine tuning of C_p(x) based on experimental data. However, a term such as c·T·ln(T) provides the same C_p-excess values at all temperatures, e.g. also at 930 K, where it may not be justified.

Another problem inherent in the latest version of 3rd generation data is that the predicted thermodynamic driving force (DF) for nucleation/formation of the solid phase from liquid starts increasing from negative values around EET with increasing T and even becomes positive at re-stabilization temperature as seen in Fig. 3(a). That may be most distractive in kinetic software for phase transformation using such a 3rd generation Calphad thermodynamic database. If a search region for DF > 0 is initiated in a range of evaporation of Al-based alloys at modest pressure of 1-30 bar, see Fig. 1 (f), this may lead into the wrong direction due to the completely unphysical decreasing absolute values of DF. This problem is also avoided using the comprehensive amendment, FCC_RSF2, even better than compared to the simple amendment, FCC_RSF, seen from Fig. 3(a).

4.3 Why is the use of “EEC-Software” Precarious ?

The remedy suggested for the artifact of solid phase re-stabilization^[3,4,5] is the use of modified Calphad software, outlined in section 2.4 of the detailed work on the EEC by Sundman et al.^[10] The author entirely agrees not only that the axiom of the EEC is truly valid but also that the EET must occur well before that solid phase would become more stable than the liquid,^[10] thus, it indicates a suitable maximum temperature to cease the unphysical extrapolation of solid data. The statement by Sundman et al.^[10] in section “3. Unary database and models: the EEC is useful to detect extrapolations that are nonphysical “ is also entirely agreed with. That was clearly demonstrated by the violation of EEC if the database COST507 is combined with modified “SGTE-no-break-unaries”^[10] as an example of an artifact-infested thermodynamic database.

The criticism of the present work is focused on two adverse facts if the latest version of 3rd generation unary data is used without amendment: (i) the automatic suspension of solid phases above EET by new “EEC-software” is no longer optional, and (ii) unexpected large C_p contributions occur. This is not a criticism of the EEC-paper by Sundman et al.^[10]

One future step would be to implement the EEC into software such as Thermo-Calc^[11] or Pandat^[9] whereby the detection of metastable phases at their own composition or the calculation speed, mentioned in Ref 10 is not a problem at all. The major problem is the possible inadvertent and wrong (automatic) exclusion of a solid phase from the equilibrium calculation. It was already recognized that “There is a small risk that a metastable solid phase will be removed due to the EEC when in fact it could become stable with another composition and with lower entropy than that of the liquid. This should mean that the solid has a miscibility gap and it is the responsibility of the software to detect this and adjust the composition of the phase. If the liquid has a miscibility gap the liquid with the highest entropy should be used for the EEC”, cited from Ref 10

One further possible pitfall is revealed by the high melting B2 phase of Al-Ir in Fig. 4(a). The B2 phase itself, described by the stand-alone (Al,Ir)(Ir,Va) two- sublattice model^[13] is safe because the EEC is not met, the calculated entropies show S(B2) << S(Liquid) at 50 at.% Ir and any temperature. However, more generally, it would be modeled as BCC_B2 with the ordered CsCl structure, (Al,Ir)_0.5(Al,Ir)_0.5, disordering on heating to BCC, (Al,Ir), with the disordered part of the BCC function included into the BCC_B2 description, following the works of Ansara, Dupin and Sundman.^[15,16] Now the composition dependent EET of the BCC phase comes into play, starting at 1949 K for pure Al if the latest 3rd generation data,^[3] are used. Who knows if the developer of the revised Al-Ir binary would take care that the EET(BCC, x_Ir) would not intersect the stable B2 range around 50 at.% Ir? Since the disordered BCC phase itself does not appear in the stable phase diagram it might be assessed in a way to make the intersection of EET(BCC, x_Ir) with the stable B2 happen. In this case the BCC basis for the BCC_B2 will probably be excluded, and serious artifacts may be expected.

Another possible pitfall may occur if certain binary interaction parameters are tried out as is always the case during a parameter optimization run. As a simple example the Al-Cr system shown in Fig. 5 of Ref 10 is used, where only the three phases Liquid, FCC and BCC are considered, all taken as ideal solutions and with SGTE unaries. Using SGTE-no-break-unaries the BCC phase is re-stabilized above the contiguous liquid region where a second BCC region occurs, separated from the stable terminal solution of (Cr). The composition dependent EET curve of the BCC phase is located inside the contiguous liquid region, well below the artificially re-stabilized BCC region which ranges from 2287 K at pure Al to 3822 K at pure Cr. Note that in Fig. 2(a) the BCC Al re-stabilizes at 3184 K, which is higher because these extrapolated 3rd generation functions^[3] differ from the SGTE-no-break-unaries. It was shown that this BCC artifact can be removed by “EEC-software”.^[10]

Fig. 5

Calculations of the binary Al-Cr phase diagram, only Liquid, FCC and BCC phases considered with ideal Liquid and FCC and L_Al,Cr = −20 kJmol⁻¹ for BCC. SGTE unary data with and without break and the simple amendment are compared. The author is grateful to Jean-Marc Joubert for the hint to this example

However, even with a small negative interaction coefficient of regular solution for the BCC phase, L_Al,Cr = − 5 kJmol⁻¹, the two BCC phase regions merge into one and the liquid phase exists in the isolated terminal regions only. That is shown by the dotted green line in the calculated Al-Cr phase diagram in Fig. 5 for the more drastic case of L_Al,Cr = − 20 kJmol⁻¹. It is unclear how “EEC-software” would handle this situation, it may or may not exclude the BCC or it has to stop with an error report. That would impede any parameter optimization run. Also shown in Fig. 5 by the red solid line is the phase diagram calculated with just the simple amendment of Eq. 3(b) used for the SGTE-no-break data, where EET(BCC, Al) = 1531.7 K, EET(FCC, Al) = 1735.7 K, EET(BCC, Cr) = 3083.7 K and EET(FCC, Cr) = 3093.9 K. This demonstrates that the BCC re-stabilization artifact is safely prevented even by the simple amendment. The comprehensive amendment provides the even safer distance of G(FCC) to G(Liquid) at high temperature as seen in Fig. 3(a). For comparison the dashed blue line in Fig. 5 shows the diagram using the standard SGTE data where the break occurs at the melting points, also preventing the BCC re-stabilization.

An expert may see that something went wrong by carefully checking the calculated binary phase diagram if an unforeseen artifact occurs due to the already recognized possibilities, e.g., solid state miscibility gap, the Al-Ir B2 example or an optimization run in the Al-Cr example. Even for experts the detection is not possible/easy in multicomponent applications. Such artifacts will never ever be recognized in simple Calphad simulations of solidification of a, say six-component alloy, not to speak of the use as embedded equilibrium calculation in higher-level software, e.g., phase-field simulation.

It is the nature of unforeseen artifacts that more of them may pop up unexpectedly. A major progress of the Calphad method,^[17,18] was the development of robust Calphad software that always finds the true thermodynamic equilibrium based on the global mathematical minimum of Gibbs energy functions of the phase assembly, starting more than 20 years ago.^[19,20] Numerous examples of overlooked solid phase re-stabilization at high temperature, taken from published Calphad assessment, were revealed in that work.^[19] The latest revision of the 3rd generation data, ^[3,4,5] breaks this principle because with these new unaries there will be no binary or multicomponent system in which the thermodynamic equilibrium can be found simply based on the minimum of Gibbs energy functions without the incorrect re-stabilization of solid at high temperatures.

As an example, the Al-C phase diagram calculated with two different datasets is given in Fig. 6. The open symbols are calculated with the data from He et al.,^[3] see Fig. 2(a), and the solid lines with these data including the simple amendment of Eq 3, see Fig. 2(b). Note that the physically correct part of the phase diagram in Fig. 6 is exactly the same in both calculations, while the unacceptable re-stabilization of the BCC phase and the two-phase region with liquid occurs only if the latest version of 3rd generation data,^[3] are used. That occurrence was already discussed in,^[3] and suggested to be removed by a new “EEC-software”. Figure 6 demonstrates that just by including the simple amendment of Eq 3 for unary Al the correct and artifact-free phase diagram is calculated by standard Calphad software, such as Pandat.^[9] Binary interaction terms are not touched by the amendment. Of course, the recommended comprehensive amendment, where the safe distance between liquid and solid Gibbs energy, G(Solid)–G(Liquid) >> 0, is even larger at high T, see Fig. 3(a), produces the same correct Al-C phase diagram. Note that the amendment must be made for all solid phases, stable or metastable, of an element. Otherwise, the artificial HCP + Liquid or FCC + Liquid phase regions would pop up in Fig. 6 at the re-stabilization temperatures given in Fig. 2(a).

Fig. 6

Calculations of the binary Al-C phase diagram, gas phase suspended. The open symbols are calculated with the data from He et al. ³ and the solid lines with these data including the simple amendment proposed in section 3 of this work

It is obvious that certainly for multicomponent and applied Calphad simulations such software leaves no room for an option to “switch off” the EEC-mode and to control the results by an expert. In such calculations the “EEC-software” must automatically exclude the phases above their calculated EET, maybe without the user not even knowing what’s going on, to avoid re-stabilization of solid phases at high temperature. The risk of unforeseen artifacts–some of them discussed above–renders such use of “EEC-software” precarious.

5 Conclusions

• The latest version of 3rd generation data for pure solid elements generate Gibbs energy functions which all intersect and then fall below the value of G(Liquid) at high temperatures, suggesting an incorrect re-stabilization of solid.

• The remedy to apply new Calphad software with the EEC implemented is critically examined in this work. Instead, a simple and a comprehensive amendment of such 3rd generation data with a second temperature range are proposed without introducing adjustable parameters to an existing set of 3rd generation unary functions of solid and liquid. This safely prevents by definition of the mathematical Gibbs energy function the violation of the EEC axiom, namely that no solid phase should be more stable than the liquid at high temperature.

• It is argued that for applied Calphad simulations the remedy of “EEC-software” bears serious risks of unforeseen pitfalls and artifacts, for example with high melting B2 phase of Al-Ir or parameter variation in Al-Cr.

• The second serious disadvantage of this latest version of 3rd generation data is the unrestricted growth of heat capacity of all solid phases. The resulting unexpected large C_p contribution of low-melting elements in solid solutions at high temperatures is discussed in detail for the case of pure Al and the Al-Ir binary system. It is removed by the comprehensive amendment, which at the same time eliminates the artifact of re-stabilization.

• The comprehensive amendment, Eq 8b, 9c, and 10, 11 in section 3.2, is recommended and preferred over the simple amendment in Eq 3. It includes the physics-supported strong increase of C_p between T_m and T_inst, estimated at T_inst,est = 1.3·T_m, and the smooth reversal of the strongly growing C_p(solid) towards a value close to C_p(liquid) at 2.5·T_m.

• The best practice to check new Calphad assessments of condensed phases is the calculation of the phase diagram up to extremely high temperatures, based solely on the minimum of Gibbs energy. If incorrect re-stabilization of solid at high temperatures occur a revision of the multicomponent data/models is imperative. This requires that any new unary data are amended to not breaking this principle.