Prediction of Activity of Au-Sn-Based Lead-Free Solder Using Modified Molecular Interaction Volume Model

Abstract

Controlling thermodynamic properties is critical for the rational design and development of advanced lead-free solders, especially in high-temperature applications. Au–Sn-based alloys have emerged as promising candidates for high-performance electronic packaging, yet reliable thermodynamic descriptions of their multicomponent systems remain limited. The Modified Molecular Interaction Volume Model (M-MIVM) provides a effective approach for characterizing strongly asymmetric liquid alloys that are typical in Au–Sn-based systems. This work focuses on the thermodynamic modeling of Au–Sn-containing ternary and quaternary solder systems within a physically consistent and computationally efficient framework. The study aims to support the database development, composition design, and optimization of next-generation high-temperature lead-free solders.

1. Introduction

Over the past few decades, Sn-Pb solder has been the dominant interconnect material in electronic packaging, owing to its suitable solidification range, low melting point, and low cost [1,2]. However, growing awareness of lead toxicity and strict environmental legislation have driven a global transition toward lead-free solder systems [3]. This shift has motivated extensive research into environmentally friendly alternatives, establishing the development of high-performance lead-free solders as a major direction in advanced materials engineering.

Despite extensive investigations into various lead-free solder compositions, no fully universal replacement for conventional Sn-Pb solders has been established. The most promising candidates are Sn-based alloys, including Sn-Ag, Sn-Zn, and Sn-Cu systems, which satisfy general packaging requirements but often struggle in high-temperature operating conditions. With the rapid development of high-power and optoelectronic devices, the demand for reliable high-temperature lead-free solders has increased significantly. In this scenario, Au-Sn-based alloys have attracted considerable attention as promising high-temperature lead-free solders [4].

Au-Sn solders possess numerous attractive properties, such as high mechanical strength, excellent oxidation and thermal fatigue resistance, suitable melting characteristics, and good wettability, making them well-suited for high-temperature packaging applications [5,6]. Nevertheless, their practical application is restricted by inherent brittleness, challenges in processing and solder preform fabrication, and high material cost. Alloying with additional elements such as Ag, Cu, Zn, and In has become a common strategy to improve manufacturability, mechanical performance, and economic efficiency.

Thermodynamic properties, especially component activities, play a critical role in the rational design and optimization of lead-free solders. Current experimental databases for Au-Sn-based multicomponent systems remain limited, which hinders efficient alloy development. Therefore, the use of theoretical models to predict thermodynamic behavior from restricted experimental data has become both necessary and effective.

In recent years, the CALPHAD (Computer Coupling of Phase Diagrams and Thermochemistry) method has become the standard framework for modeling phase equilibria and thermodynamic properties of multicomponent alloys [7], supported by increasingly mature thermodynamic databases for solder systems. Contemporary liquid-phase descriptions include substitutional solution models, Redlich–Kister formalisms, and physically motivated models that account for atomic size difference and short-range ordering. Meanwhile, semi-empirical models such as the Wilson equation, NRTL, and the Molecular Interaction Volume Model (MIVM) have also been widely adopted for predicting thermodynamic properties [8]. However, conventional models often show limited accuracy for highly asymmetric liquid alloys like Au-Sn and its derivatives. To address this limitation, the Modified Molecular Interaction Volume Model (M-MIVM) was proposed, offering improved performance for strongly asymmetric systems [9].

In this study, the M-MIVM is employed to predict the activities of components in Au-Sn-based lead-free solder systems. First, the binary interaction parameters of the M-MIVM are determined by fitting experimental activity data from asymmetric binary systems such as Au-Zn and Ag-Sn. These parameters are then used to predict the component activities in the ternary Au-Sn-Ag, Au-Sn-Cu, and Au-Sn-Zn solder alloys. The predictions are compared with available experimental data, and the associated errors are analyzed. Iso-activity contours for all components in the ternary systems are presented graphically. Furthermore, the activities of all components in the quaternary Au-Cu-In-Sn system are predicted. The influence of indium addition is illustrated by comparing the activities in Au-Sn-Cu-In alloys at fixed indium mole fractions of x_In = 0.1 (800 K and 1000 K) and x_In = 0.2 (800 K). This work aims to provide a simple and reliable thermodynamic model to support the development of new high-temperature lead-free solders and the performance optimization of existing tin-based alloys.

2. Calculation Method

As the Wilson equation, NRTL and MIVM are widely used, there have been a large number of studies describing them in detail, so we will not describe them here [10].

The M-MIVM adopted in this work differs from these methods. The M-MIVM is rigorously derived based on the physical principles of random mixing and strict conservation of pairwise molecular interactions. Unlike the traditional local composition models, it does not introduce empirical local composition assumptions that break the interaction balance. Instead, it employs physically meaningful coordination numbers and interaction parameters that inherently satisfy the conservation law of intermolecular interactions throughout the entire derivation. Therefore, the M-MIVM maintains strict thermodynamic consistency and physical validity without violating the conservation balance of intermolecular interactions.

M-MIVM continues to follow the configuration partition function of MIVM liquid mixture and achieves parameter separation by introducing Scatchard–Hildebrand theory assumption.

The general formula of the molar excess Gibbs energy ($G_m^E$) of liquid mixture of the M-MIVM can be expressed as [11,12,13]:

$${\displaystyle \frac{G^E}{RT}}=-{\displaystyle \sum _{\mathrm{i}=1}^C{x}_i\ln \left({\displaystyle \sum _{j=1}^C{x}_j\frac{V_{mj}}{V_{mi}}{B}_{ji}}\right)+{\displaystyle \sum _{j>i}^C{\displaystyle \sum _{i=1}^{C-1}{x}_i{x}_j}}}\left[{\displaystyle \frac{A_{ji}}{{\displaystyle \sum _{l=1}^C{x}_l\frac{V_{ml}}{V_{mi}}{B}_{li}}}}+{\displaystyle \frac{A_{ij}}{{\displaystyle \sum _{l=1}^C{x}_l\frac{V_{ml}}{V_{mj}}{B}_{lj}}}}\right]$$

(1)

where V_mt (t = i, j and l) represents the molar volume of component t, x_t (t = i, j and l) is the mole fraction of component t, A_ij and A_ji are energy parameters, and B_ij and B_ji are volume parameters of molecular pair i-j and j-i, respectively. A_ij is mainly related to the atomic interaction energy between component i and j, characterizing the strength of interatomic attraction or repulsion. B_ij is associated with the coordination number and atomic size difference, reflecting the geometric matching and short-range order in the liquid structure.

Here, we define P_ji as the probability that the molecule j appears in the first coordination layer of the central molecule i.

$$B_{ji}={\displaystyle \frac{P_{ji}}{P_{ii}}} \mathrm{and} B_{ij}={\displaystyle \frac{P_{ij}}{P_{jj}}}$$

(2)

In addition, we assume that the volume parameter is temperature dependent, and its relationship to temperature is shown as follows:

$$T_1 \ln {{\displaystyle B}}_{ ji}^{T_1}=T_2 \ln {{\displaystyle B}}_{ ji}^{T_2}$$

(3)

${{\displaystyle B}}_{ji}^{T_1}$ and ${{\displaystyle B}}_{ji}^{T_2}$ are the parameters of B at temperature T₁ and T₂ respectively [14].

A_ij and A_ji is temperature independent in the systems with weak intermolecular interaction, but they will be temperature dependent in strong negative deviation system because of the strong intermolecular interaction in this system. Heterogeneous molecules have strong attraction, which limits the degree of freedom of molecules and requires conversion of parameters at different temperatures.

$${{\displaystyle A}}_{\mathrm{ij}}^{\prime }={\displaystyle \frac{T_1}{T_2}} {{\displaystyle A}}_{ij}$$

(4)

A_ij and ${{\displaystyle A}}_{\mathrm{ij}}^{\prime }$ are the parameters of at temperature T₁ and T₂ respectively [15].

In a strongly negative deviation system, we generally consider A_ij < −1 and A_ji < −1.

The natural logarithm value of the activity coefficient of any component i in the multicomponent system can be expressed as:

$$\begin{array}{c}\ln {{\displaystyle \gamma }}_i=-\ln \left({\displaystyle \sum _{j=1}^C{x}_j{D}_{ji}}\right)-{\displaystyle \sum _{j=1}^C\frac{x_j}{{\displaystyle \sum _{l=1}^C{x}_l{D}_{lj}}}}\left[D_{ij}-{\displaystyle \sum _{t=1}^C{x}_t{D}_{tj}}\right]\\ +{\displaystyle \sum _{j\ne i}^C{x}_j\left[\frac{A_{ji}}{{\displaystyle \sum _{l=1}^C{x}_l{D}_{li}}}+\frac{A_{ij}}{{\displaystyle \sum _{l=1}^C{x}_l{D}_{lj}}}\right]}-{\displaystyle \sum _{k>1}^C{\displaystyle \sum _{j=1}^{C-1}{x}_j{x}_k}}\left[{\displaystyle \frac{A_{kj}}{{\displaystyle \sum _{l=1}^C{x}_l{D}_{lj}}}}+{\displaystyle \frac{A_{jk}}{{\displaystyle \sum _{l=1}^C{x}_l{D}_{lk}}}}\right]\\ -{\displaystyle \sum _{k>j}^C{\displaystyle \sum _{j=1}^{C-1}{x}_j{x}_k}}\left[{\displaystyle \frac{A_{kj}\left(D_{ij}-{\displaystyle \sum _{t=1}^C{x}_t{D}_{tj}}\right)}{{\left({\displaystyle \sum _{l=1}^C{x}_l{D}_{lj}}\right)}^2}}+{\displaystyle \frac{A_{jk}\left(D_{ik}-{\displaystyle \sum _{t=1}^C{x}_t{D}_{tk}}\right)}{{\left({\displaystyle \sum _{l=1}^C{x}_l{D}_{lk}}\right)}^2}}\right]\end{array}$$

(5)

In the expression here, D_ji = B_jiV_mj/V_mi.

The parameter acquisition in this paper relies on the least square method through MATLAB 2014 programming to solve the optimal parameters. The selected expression is as follows:

$$S_{\mathrm{i}}={\displaystyle \sum }_{\mathrm{n}}^{\mathrm{i}=1}\left|a_{\mathrm{i},\exp }-a_{\mathrm{i},\mathrm{pre}}\right|$$

(6)

where a_{i exp} and a_{i pre} are the respective experimental and predicted activity values of component i. The accuracy of the predictions is assessed using the mean relative error and standard deviation of the experimental and predicted values.

$$S_i^{\ast }=\pm {\left[{\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle \sum }_{\mathrm{n}}^{\mathrm{i}=1}{\left(a_{\mathrm{i} \exp }-a_{\mathrm{i} \mathrm{pre}}\right)}^2\right]}^{{\displaystyle \frac{1}{2}}},S_{\mathrm{i}}=\pm {\displaystyle \frac{100}{\mathrm{n}}}{\displaystyle \sum }_{\mathrm{n}}^{\mathrm{i}=1}\left|{\displaystyle \frac{a_{\mathrm{i} \exp }-a_{\mathrm{i} \mathrm{pre}}}{a_{\mathrm{i} \exp }}}\right|$$

(7)

where a_{i exp} and a_{i pre} are the respective experimental and predicted activity values of component i, while n is the number of compared values [16].

The pure metal parameters required for M-MIVM are listed in

Table 1. The pure metal parameters required for M-MIVM [17,18].

Element	Vmi(cm3/mol)
Ag	11.64 [1 + 0.98 × 10−4 (T − 1235)]
Au	11.37 [1 + 0.69 × 10−4 (T − 1337)]
Sn	17.03 [1 + 0.87 × 10−4 (T − 505)]
Cu	7.99 [1 + 1 × 10−4 (T − 1311)]
Zn	9.99 [1 + 1.5 × 10−4 (T − 693)]
In	16.30 [1 + 0.97 × 10−4 (T − 430)]

.

3. Result and Discussion

3.1. Fitting Effect of Binary Alloy Systems

The parameters of the M-MIVM were obtained by fitting the model to the experimental activity data using Equation (5). The resulting binary interaction parameters and the corresponding fitting results are summarized in

Table 2. The fitting parameters of different models.

Alloy	T (K)	M-MIVM				MIVM		Wilson		NRTL
		B12	B21	A12	A21	B12	B21	Λ12	Λ21	τ12	τ21	α12
Au-Sn [19]	823	0.810	1.209	−3.523	−3.548	1.091	2.012	13.581	4.136	−0.823	−4.550	0.171
Au-Zn [19]	1080	0.821	1.172	−6.170	−2.717	1.513	1.672	8.971	7.757	−2.867	−3.157	0.170
Sn-Zn [19]	750	0.696	0.474	−0.061	0.182	1.137	0.599	0.133	1.197	−0.143	1.692	0.450
Ag-Au [19]	1350	1.024	1.117	−0.714	−0.515	0.438	1.820	1.781	1.781	−0.628	−0.628	0.170
Ag-Sn [19]	1250	0.539	0.938	−2.077	2.040	0.763	1.380	5.273	0.190	1.367	−1.782	0.448
Au-Cu [19]	1400	2.365	0.558	11.933	−4.878	0.212	2.148	2.049	3.503	−2.039	−0.158	0.170
Cu-Sn [19]	1373	0.790	0.885	−3.181	4.978	0.739	1.372	6.986	0.143	1.368	−2.009	0.445
Au-In [19]	1400	1.591	0.774	−1.550	−2.951	1.347	1.375	5.022	4.163	−1.579	−2.199	0.171
Cu-In [19]	1373	0.834	1.062	−8.669	20.000	0.800	1.282	5.001	0.221	1.154	−1.641	0.444
In-Sn [19]	600	0.918	0.838	−1.580	1.172	1.572	0.547	1.852	0.540	0.524	−0.586	0.390

and

Table 3. The values of S* and S( % ) of the binary liquid alloys in different models.

Alloy	M-MIVM		MIVM		Wilson		NRTL
	S*	S(%)	S*	S(%)	S*	S(%)	S*	S(%)
Au-Sn	0.0068	9.42	0.0067	12.13	0.0335	86.69	0.0075	13.35
Au-Zn	0.0333	16.56	0.0234	106.88	0.0605	268.18	0.0272	81.16
Sn-Zn	0.0059	10.06	0.0083	10.38	0.0060	10.05	0.0077	10.29
Ag-Au	0.0004	9.13	0.0006	9.17	0.0042	9.70	0.0007	9.18
Ag-Sn	0.0076	11.62	0.0570	23.47	0.0318	17.15	0.0380	19.31
Au-Cu	0.0155	14.64	0.0205	16.56	0.0313	17.50	0.0265	15.42
Cu-Sn	0.0103	12.41	0.0569	25.17	0.0244	16.43	0.0329	19.05
Au-In	0.0007	10.7	0.0032	11.6	0.0193	21.88	0.0029	11.54
Cu-In	0.0114	11.61	0.057	22.91	0.0392	19.06	0.0473	21.05
In-Sn	0.0004	9.17	0.0059	10.34	0.0114	11.41	0.0122	11.59
overall	0.0092	11.53	0.0240	24.86	0.0262	47.81	0.0203	21.19

, respectively. The mean standard deviation and mean relative deviation for each component were evaluated according to Equations (6) and (7), and are also provided in Table 3.

As shown in Table 3, the standard deviation of M-MIVM across the ten studied systems (six of which are asymmetric) is only 0.0092, significantly lower than those obtained with other models. For the strongly asymmetric systems Au-Zn, Cu-Sn, and Cu-In, the standard deviation given by M-MIVM is notably smaller than that of all other models. This indicates that M-MIVM possesses a clear advantage in describing intermolecular interactions in binary systems compared to conventional local composition models. In contrast to the original MIVM, M-MIVM implements parameter separation, employing two distinct parameter sets to separately account for enthalpic and entropic contributions. Moreover, while models such as Wilson and NRTL primarily focus on enthalpic effects, M-MIVM explicitly incorporates entropic contributions, leading to superior fitting performance in highly asymmetric systems.

Overall, the M-MIVM significantly improves prediction accuracy over MIVM, Wilson, and NRTL. It performs excellently in systems with weak to moderate interactions (Ag-Au, In-Sn, Au-In, Au-Sn, Sn-Zn), where predictions are reliable for materials design. Larger deviations occur in systems with strong intermetallic interactions (Au-Zn, Cu-In, Au-Cu, Cu-Sn), mainly due to systematic physical effects rather than random error. Building on these results, the following section will employ M-MIVM to predict the activities of ternary asymmetric alloy systems based on Au-Sn with additions of Ag, Cu, and Zn.

3.2. Research on Prediction Model of Ternary Alloy System

The primary strength of the model lies in its capacity to predict the thermodynamic properties of multicomponent systems solely from the binary thermodynamic data available in the literature. Based on Equation (5), the activity of Sn in the ternary Au-Sn-Ag, Au-Sn-Cu, and Au-Sn-Zn systems was calculated. The predicted results, along with the corresponding experimental data for each system, are summarized in

Table 4. Comparison of the experimental data [20] with the predicted values of activity obtained from different models in ternary Ag-Au-Sn at 973 K.

xAg/xAu	xAg	xAu	xSn	aSn,exp	M-MIVM	MIVM	Wilson	NRTL
2	0.5333	0.2667	0.2000	0.0410	0.0433	0.0271	0.0439	0.0282
2	0.4667	0.2333	0.3000	0.1394	0.1300	0.1070	0.1151	0.0878
2	0.4000	0.2000	0.4000	0.2512	0.2564	0.2369	0.2192	0.1879
2	0.3333	0.1667	0.5000	0.4308	0.3992	0.3876	0.3468	0.3201
2	0.2667	0.1333	0.6000	0.5098	0.5402	0.5354	0.4869	0.4692
2	0.2000	0.1000	0.7000	0.6593	0.6707	0.6697	0.6294	0.6206
2	0.1333	0.0667	0.8000	0.8040	0.7893	0.7896	0.7662	0.7632
2	0.0667	0.0333	0.9000	0.9026	0.8979	0.8982	0.8911	0.8907
1	0.4000	0.4000	0.2000	0.0217	0.0252	0.0157	0.0339	0.0199
1	0.3500	0.3500	0.3000	0.0922	0.0872	0.0738	0.0935	0.0673
1	0.3000	0.3000	0.4000	0.2097	0.1948	0.1883	0.1867	0.1557
1	0.2500	0.2500	0.5000	0.3242	0.3346	0.3403	0.3082	0.2827
1	0.2000	0.2000	0.6000	0.4655	0.4868	0.5011	0.4493	0.4353
1	0.1500	0.1500	0.7000	0.6546	0.6355	0.6508	0.5998	0.5965
1	0.1000	0.1000	0.8000	0.7770	0.7721	0.7823	0.7488	0.7507
1	0.0500	0.0500	0.9000	0.8991	0.8934	0.8968	0.8856	0.8872
0.5	0.2667	0.5333	0.2000	0.0177	0.0136	0.0089	0.0250	0.0130
0.5	0.2333	0.4667	0.3000	0.0631	0.0557	0.0472	0.0750	0.0492
0.5	0.2000	0.4000	0.4000	0.1636	0.1429	0.1373	0.1581	0.1249
0.5	0.1667	0.3333	0.5000	0.3068	0.2739	0.2774	0.2726	0.2446
0.5	0.1333	0.2667	0.6000	0.4654	0.4322	0.4445	0.4123	0.3987
0.5	0.1000	0.2000	0.7000	0.6452	0.5974	0.6123	0.5683	0.5692
0.5	0.0667	0.1333	0.8000	0.8283	0.7526	0.7634	0.7287	0.7361
0.5	0.0333	0.0667	0.9000	0.9435	0.8881	0.8919	0.8787	0.8831
		S*			0.0269	0.0266	0.0297	0.0266
		S(%)			6.67	11.56	9.72	11.56

,

Table 5. Comparison of the experimental data [21] with the predicted values of activity obtained from different models in ternary Au-Cu-Sn at 900 K.

xAu/xCu	xAu	xCu	xSn	aSn,exp	M-MIVM	MIVM	Wilson	NRTL
3	0.6000	0.2000	0.2000	0.0184	0.0122	0.0226	0.0224	0.0088
3	0.5984	0.1995	0.2021	0.0193	0.0126	0.0235	0.0231	0.0092
3	0.5240	0.1747	0.3014	0.0611	0.0524	0.0938	0.0678	0.0389
3	0.4488	0.1496	0.4016	0.1591	0.1388	0.2104	0.1436	0.1081
3	0.3728	0.1243	0.5029	0.3100	0.2738	0.3539	0.2520	0.2262
3	0.2995	0.0998	0.6007	0.4676	0.4331	0.4988	0.3842	0.3788
3	0.2263	0.0754	0.6983	0.6464	0.5984	0.6387	0.5371	0.5507
3	0.1485	0.0495	0.8020	0.7370	0.7604	0.7763	0.7116	0.7312
3	0.0749	0.0250	0.9001	0.8953	0.8907	0.8936	0.8709	0.8809
1	0.3998	0.3998	0.2005	0.0385	0.0530	0.0316	0.0404	0.0198
1	0.3993	0.3993	0.2014	0.0472	0.0535	0.0321	0.0408	0.0201
1	0.3498	0.3498	0.3005	0.1297	0.1357	0.1173	0.1005	0.0684
1	0.3495	0.3495	0.3011	0.1321	0.1364	0.1180	0.1009	0.0688
1	0.3002	0.3002	0.3996	0.2772	0.2561	0.2525	0.1900	0.1579
1	0.2997	0.2997	0.4007	0.2912	0.2576	0.2542	0.1912	0.1591
1	0.2509	0.2509	0.4982	0.4542	0.3974	0.4062	0.3055	0.2847
1	0.2488	0.2488	0.5025	0.4588	0.4037	0.4129	0.3111	0.2908
1	0.2001	0.2001	0.5998	0.5942	0.5455	0.5576	0.4453	0.4402
1	0.1501	0.1501	0.6999	0.7221	0.6801	0.6882	0.5946	0.6006
1	0.1498	0.1498	0.7005	0.7278	0.6809	0.6889	0.5955	0.6016
1	0.0995	0.0995	0.8011	0.8408	0.7992	0.8020	0.7464	0.7548
1	0.0499	0.0499	0.9003	0.9294	0.9020	0.9021	0.8844	0.8883
1	0.0496	0.0496	0.9008	0.9419	0.9025	0.9026	0.8851	0.8890
1/3	0.1999	0.5997	0.2004	0.0955	0.1119	0.0314	0.0538	0.0284
1/3	0.1754	0.5262	0.2984	0.1833	0.2173	0.1022	0.1300	0.0930
1/3	0.1499	0.4496	0.4006	0.3607	0.3555	0.2246	0.2406	0.2043
1/3	0.1255	0.3764	0.4982	0.5066	0.4911	0.3693	0.3655	0.3383
1/3	0.0998	0.2995	0.6007	0.6643	0.6197	0.5262	0.5059	0.4903
1/3	0.0748	0.2244	0.7008	0.7546	0.7268	0.6678	0.6431	0.6364
1/3	0.0499	0.1498	0.8003	0.8849	0.8193	0.7911	0.7732	0.7713
1/3	0.0249	0.0746	0.9006	0.9330	0.9073	0.9000	0.8936	0.8935
		S*			0.0331	0.0589	0.0894	0.0589
		S(%)			10.37	16.8	19.02	16.8

, and

Table 6. Comparison of the experimental data [22] with the predicted values of activity obtained from different models in ternary Au-Sn-Zn at 973 K.

xAu/xSn	xAu	xSn	xZn	aZn,exp	M-MIVM	MIVM	Wilson	NRTL
3	0.7126	0.2375	0.0499	0.0003	0.0003	0.0009	0.0020	0.0008
3	0.6749	0.2250	0.1001	0.0009	0.0009	0.0029	0.0057	0.0024
3	0.5998	0.1999	0.2003	0.0048	0.0052	0.0123	0.0204	0.0103
3	0.5249	0.1750	0.3001	0.0176	0.0194	0.0354	0.0475	0.0299
3	0.4499	0.1500	0.4001	0.0508	0.0561	0.0821	0.0909	0.0707
3	0.4124	0.1375	0.4501	0.0840	0.0882	0.1178	0.1201	0.1027
3	0.3750	0.1250	0.5000	0.1110	0.1324	0.1633	0.1552	0.1444
3	0.3375	0.1125	0.5500	0.1956	0.1906	0.2199	0.1971	0.1973
3	0.3000	0.1000	0.6000	0.2683	0.2635	0.2881	0.2468	0.2624
3	0.2250	0.0750	0.7000	0.4819	0.4489	0.4573	0.3749	0.4299
3	0.1500	0.0500	0.8000	0.6968	0.6634	0.6572	0.5513	0.6363
3	0.0751	0.0250	0.8999	0.8722	0.8613	0.8547	0.7828	0.8466
1	0.4749	0.4749	0.0502	0.0046	0.0039	0.0087	0.0096	0.0069
1	0.4511	0.4511	0.0978	0.0132	0.0098	0.0199	0.0212	0.0160
1	0.4001	0.4001	0.1998	0.0376	0.0337	0.0562	0.0547	0.0466
1	0.3501	0.3501	0.2998	0.0833	0.0790	0.1118	0.1011	0.0958
1	0.3000	0.3000	0.4000	0.1596	0.1550	0.1914	0.1638	0.1696
1	0.2500	0.2501	0.4999	0.2706	0.2667	0.2966	0.2459	0.2714
1	0.2251	0.2250	0.5499	0.3383	0.3356	0.3585	0.2954	0.3331
1	0.2001	0.2000	0.5999	0.4172	0.4120	0.4262	0.3515	0.4017
1	0.1500	0.1499	0.7001	0.5864	0.5789	0.5751	0.4856	0.5560
1	0.1000	0.1000	0.8000	0.7350	0.7449	0.7315	0.6507	0.7206
1	0.0500	0.0501	0.8999	0.8846	0.8882	0.8795	0.8396	0.8764
1/3	0.2250	0.6746	0.1004	0.0543	0.0517	0.0705	0.0553	0.0602
1/3	0.2000	0.6002	0.1998	0.1307	0.1218	0.1534	0.1207	0.1353
1/3	0.1750	0.5249	0.3001	0.2141	0.2122	0.2486	0.1980	0.2262
1/3	0.1500	0.4500	0.4000	0.3165	0.3206	0.3528	0.2873	0.3302
1/3	0.1250	0.3751	0.4999	0.4342	0.4428	0.4632	0.3894	0.4439
1/3	0.1001	0.3000	0.5999	0.5558	0.5713	0.5762	0.5040	0.5627
1/3	0.0750	0.2251	0.6999	0.6827	0.6972	0.6885	0.6300	0.6814
1/3	0.0500	0.1500	0.8000	0.7839	0.8108	0.7969	0.7624	0.7948
1/3	0.0250	0.0750	0.9000	0.9031	0.9082	0.9001	0.8907	0.9001
			S*		0.016	0.0247	0.0576	0.0239
			S(%)		7.55	48.02	97.54	33.17

, respectively.

The mean standard deviation and mean relative deviation for each component in the ternary Au-Sn-Ag, Au-Sn-Cu, and Au-Sn-Zn systems were calculated using Equations (6) and (7), and the results are compiled in

Table 7. Comparison of predicting effect among different models.

Alloy	M-MIVM		MIVM		Wilson		NRTL
	S*	S(%)	S*	S(%)	S*	S(%)	S*	S(%)
Au-Sn-Zn	0.016	7.55	0.0247	48.02	0.0576	97.54	0.0239	33.17
Ag-Au-Sn	0.0269	6.67	0.0266	11.56	0.0414	11.65	0.051	15.38
Au-Cu-Sn	0.0331	10.37	0.0589	16.80	0.0894	19.02	0.1009	30.16
overall	0.0253	8.20	0.0367	25.46	0.0628	42.74	0.0586	26.24

. Table 7 also compares the deviations obtained from the four considered models when predicting the component activities in these three ternary systems.

To comprehensively and intuitively assess the reliability and stability of M-MIVM in predicting the thermodynamic properties of liquid alloys, the predicted activities are plotted against the corresponding experimental values in Figure 1, Figure 2 and Figure 3.

Ag-Au-Sn (973 K): The maximum deviation occurs at x_Ag/x_Au = 2, x_Sn = 0.5 (a_Sn,exp = 0.4308, M-MIVM = 0.3992) and x_Ag/x_Au = 0.5, x_Sn = 0.8 (a_Sn,exp = 0.8283, M-MIVM = 0.7526); deviations are mainly attributed to strong interatomic association effects. The model yields S* = 0.0269 and S(%) = 6.67, showing good overall agreement.

Au-Cu-Sn (900 K): The largest deviation appears at x_Au/x_Cu = 3, x_Sn = 0.8 (a_Sn,exp = 0.7370, M-MIVM = 0.7604) and x_Au/x_Cu = 1/3, x_Sn = 0.8 (a_Sn,exp = 0.8849, M-MIVM = 0.8193); the main reason is limited binary data quality and strong compound formation. The model has S* = 0.0331 and S(%) = 10.37.

Au-Sn-Zn (973 K): Maximum deviation is observed at x_Au/x_Sn = 3, x_Zn = 0.5 (a_Zn,exp = 0.1956, M-MIVM = 0.1906) and x_Au/x_Sn = 1/3, x_Zn = 0.8 (a_Zn,exp = 0.7839, M-MIVM = 0.8108); this is due to incomplete capture of intermetallic compound formation effects. The model achieves S* = 0.016 and S(%) = 7.55, with the best overall performance.

Based on the results, M-MIVM demonstrates a clear advantage over traditional local composition models in predicting the thermodynamic properties of Au-Sn-based (with Ag, Cu, and Zn) systems. Both the average relative error and the standard deviation are significantly reduced when using M-MIVM, whereas the calculations from the MIVM, Wilson, and NRTL models show considerably larger deviations. This discrepancy stems primarily from the substantial errors in the binary parameter fitting for these systems using the latter models. Compared to the Wilson equation, NRTL equation, and original MIVM, M-MIVM exhibits superior performance in fitting the activity coefficients of binary asymmetric systems, which in turn enhances the accuracy of activity predictions for multicomponent systems that contain such asymmetric binaries.

Relative to MIVM, M-MIVM incorporates the coordination number within its parameters, thereby minimizing prediction errors associated with the explicit selection of this number. A key improvement of M-MIVM lies in its parameter separation, which allows the entropy and enthalpy contributions of the system to be adjusted independently. Consequently, the parameters in M-MIVM possess a clearer physical meaning compared to those in MIVM. Moreover, while models such as Wilson and NRTL primarily account for enthalpic effects, M-MIVM explicitly incorporates entropic contributions. Due to its more accurate description of elemental interactions, M-MIVM provides more reliable predictions for complex ternary systems.

For binary and ternary alloys, the predicted activities from the M-MIVM show an average relative deviation less than 0.04 and an average standard deviation less than 17% compared with experimental data, demonstrating the reliability of the model.

3.3. Prediction of Activity of All Components

To further analyze the influence of each component on the properties of the alloy, this study aims to predict the activity of the entire component system. The results are shown in Figure 4, Figure 5, Figure 6 and Figure 7.

In the Au-Sn-Ag system, both Au-Sn and Sn-Ag exhibit asymmetric, strongly negative deviations from ideality, while Au-Ag behaves as a symmetric, strongly negative deviation system. From the iso-activity contours of Ag (Figure 4a), it can be observed that in the low-concentration region of Ag, the activity remains very small near a_Ag ≈ 0.3 and in the most dilute region, where the variation in γ_Ag is minimal. The distribution of the Ag activity coefficient is relatively symmetric. When a_Ag is held constant, the required amount of Ag is lowest at x_Au/x_Sn ≈ 0.5. Figure 7a further indicates that Ag exerts a considerable influence on γ_Au and γ_Sn, especially at x_Au/x_Sn = 2, where the behavior distinctly differs from that at x_Au/x_Sn = 1 and 0.75—a point worthy of attention. Unlike the trend observed with increasing concentrations of other components, γ_Au decreases when x_Au/x_Sn = 2.

For the Au-Sn-Cu system (Figure 5), Au-Sn, Sn-Cu, and Au-Cu all display asymmetric, strongly negative deviations. As seen in Table 5, when x_Au/x_Sn is fixed, γ_Sn gradually increases with rising x_Sn. Figure 7b shows that at x_Au/x_Sn = 1 and x_Au ≈ 0.43, γ_Sn reaches its maximum value while still remaining below unity. With increasing x_Cu, γ_Au gradually decreases, and the decline becomes more pronounced at higher x_Au/x_Sn ratios. This suggests that the addition of Zn in the Au-Sn-Zn system tends to promote compound formation with Au and Sn, simultaneously lowering the activity coefficients of all components. When Zn enters a high-concentration region, γ_Au and γ_Sn approach zero. Figure 7b also reveals that for any given x_Au/x_Sn, γ_Sn exhibits a maximum, which we attribute to the combined interaction of Au and Cu with Sn across different concentration ranges.

In the Au-Sn-Zn system, Sn-Zn shows an asymmetric positive deviation, whereas Au-Sn and Au-Zn are asymmetric strong negative and symmetric strong negative deviation systems, respectively. According to Table 6, when x_Au/x_Sn is constant, a_Zn gradually increases with x_Zn but decreases as x_Au/x_Zn rises. From Figure 7c, at x_Au/x_Sn values of 2, 1, and 0.75, γ_Au declines gradually with increasing x_Zn, while γ_Sn increases linearly with x_Au. This indicates that Zn addition in Au-Sn-Zn favors compound formation with Au, thereby reducing the activity coefficients of both Zn and Au. In the high-Zn concentration region, γ_Zn approaches unity, γ_Au tends toward zero, and γ_Sn rises linearly. Figure 6a demonstrates that Au exhibits a strong negative deviation relative to an ideal solution across the entire concentration range. In Figure 6b, γ_Sn attains a relatively high value near x_Au/x_Zn ≈ 4. Figure 6c shows that Zn displays a positive deviation when x_Au < 0.03 and a negative deviation when x_Au > 0.03. It is also evident that the minimum amount of Sn required to achieve a specified activity coefficient occurs when the molar ratio x_Au/x_Zn is approximately 0.17.

3.4. Predicting Effect of Quadratic Alloy Systems

In this section, M-MIVM is used to predict the activity of all components of the Au-Cu-In-Sn quaternary system. The activity predictions of each component when x_In = 0.1 and x_In = 0.2 at 800 K are listed in

Table 8. Effect of In addition on the activities of all components in Au-Cu-In-Sn at 800 K when x_In = 0.1.

xAu/xCu	xAu	xCu	xSn	aAu	aCu	aIn	aSn
3.0000	0.4500	0.1500	0.3000	0.0658	0.0443	0.0883	0.0550
3.0000	0.3750	0.1250	0.4000	0.0283	0.0428	0.0950	0.1488
3.0000	0.3000	0.1000	0.5000	0.0119	0.0388	0.0949	0.2929
3.0000	0.2250	0.0750	0.6000	0.0049	0.0330	0.0914	0.4653
3.0000	0.1500	0.0500	0.7000	0.0018	0.0254	0.0871	0.6366
2.0000	0.4000	0.2000	0.3000	0.0461	0.0654	0.0887	0.0724
2.0000	0.3333	0.1667	0.4000	0.0199	0.0618	0.0938	0.1781
2.0000	0.2667	0.1333	0.5000	0.0086	0.0550	0.0932	0.3270
2.0000	0.2000	0.1000	0.6000	0.0036	0.0461	0.0899	0.4941
2.0000	0.1333	0.0667	0.7000	0.0014	0.0351	0.0861	0.6535
1.0000	0.3000	0.3000	0.3000	0.0191	0.1209	0.0866	0.1222
1.0000	0.2500	0.2500	0.4000	0.0085	0.1091	0.0897	0.2490
1.0000	0.2000	0.2000	0.5000	0.0039	0.0938	0.0889	0.3994
1.0000	0.1500	0.1500	0.6000	0.0018	0.0764	0.0866	0.5492
1.0000	0.1000	0.1000	0.7000	0.0008	0.0566	0.0840	0.6834
0.5000	0.2000	0.4000	0.3000	0.0055	0.1987	0.0805	0.1949
0.5000	0.1667	0.3333	0.4000	0.0027	0.1708	0.0835	0.3315
0.5000	0.1333	0.2667	0.5000	0.0014	0.1419	0.0838	0.4702
0.5000	0.1000	0.2000	0.6000	0.0007	0.1125	0.0830	0.5960
0.5000	0.0667	0.1333	0.7000	0.0004	0.0813	0.0819	0.7058
0.3333	0.1500	0.4500	0.3000	0.0024	0.2468	0.0763	0.2373
0.3333	0.1250	0.3750	0.4000	0.0013	0.2075	0.0800	0.3722
0.3333	0.1000	0.3000	0.5000	0.0007	0.1697	0.0812	0.5004
0.3333	0.0750	0.2250	0.6000	0.0004	0.1329	0.0813	0.6135
0.3333	0.0500	0.1500	0.7000	0.0002	0.0950	0.0809	0.7132

and

Table 9. Effect of In addition on the activities of all components in Au-Cu-In-Sn at 800 K when x_In = 0.2.

xAu/xCu	xAu	xCu	xSn	aAu	aCu	aIn	aSn
3.0000	0.3750	0.1250	0.3000	0.0407	0.0380	0.1880	0.0791
3.0000	0.3000	0.1000	0.4000	0.0164	0.0354	0.1968	0.1908
3.0000	0.2250	0.0750	0.5000	0.0063	0.0307	0.1928	0.3463
3.0000	0.1500	0.0500	0.6000	0.0022	0.0240	0.1827	0.5171
3.0000	0.0750	0.0250	0.7000	0.0006	0.0147	0.1711	0.6742
2.0000	0.3333	0.1667	0.3000	0.0289	0.0552	0.1896	0.0985
2.0000	0.2667	0.1333	0.4000	0.0118	0.0505	0.1954	0.2186
2.0000	0.2000	0.1000	0.5000	0.0047	0.0431	0.1903	0.3735
2.0000	0.1333	0.0667	0.6000	0.0017	0.0333	0.1805	0.5353
2.0000	0.0667	0.0333	0.7000	0.0005	0.0200	0.1698	0.6806
1.0000	0.2500	0.2500	0.3000	0.0124	0.0988	0.1890	0.1496
1.0000	0.2000	0.2000	0.4000	0.0053	0.0869	0.1903	0.2817
1.0000	0.1500	0.1500	0.5000	0.0023	0.0719	0.1843	0.4286
1.0000	0.1000	0.1000	0.6000	0.0009	0.0539	0.1757	0.5686
1.0000	0.0500	0.0500	0.7000	0.0003	0.0315	0.1673	0.6918
0.5000	0.1667	0.3333	0.3000	0.0038	0.1576	0.1830	0.2176
0.5000	0.1333	0.2667	0.4000	0.0018	0.1332	0.1820	0.3502
0.5000	0.1000	0.2000	0.5000	0.0009	0.1068	0.1767	0.4794
0.5000	0.0667	0.1333	0.6000	0.0004	0.0779	0.1702	0.5958
0.5000	0.0333	0.0667	0.7000	0.0002	0.0441	0.1645	0.7003
0.3333	0.1250	0.3750	0.3000	0.0017	0.1936	0.1778	0.2555
0.3333	0.1000	0.3000	0.4000	0.0009	0.1604	0.1767	0.3826
0.3333	0.0750	0.2250	0.5000	0.0005	0.1269	0.1723	0.5005
0.3333	0.0500	0.1500	0.6000	0.0003	0.0914	0.1672	0.6060
0.3333	0.0250	0.0750	0.7000	0.0001	0.0508	0.1631	0.7033

, respectively. The activity prediction of each component when x_In = 0.1 at 1000 K is listed in

Table 10. Effect of In addition on the activities of all components in Au-Cu-In-Sn at 1000 K when x_In = 0.1.

xAu/xCu	xAu	xCu	xSn	aAu	aCu	aIn	aSn
3.0000	0.4500	0.1500	0.3000	0.0459	0.0680	0.0929	0.0334
3.0000	0.3750	0.1250	0.4000	0.0172	0.0634	0.1035	0.1072
3.0000	0.3000	0.1000	0.5000	0.0063	0.0543	0.1032	0.2386
3.0000	0.2250	0.0750	0.6000	0.0022	0.0426	0.0966	0.4130
3.0000	0.1500	0.0500	0.7000	0.0007	0.0296	0.0875	0.5994
2.0000	0.4000	0.2000	0.3000	0.0318	0.0978	0.0923	0.0457
2.0000	0.3333	0.1667	0.4000	0.0120	0.0889	0.1008	0.1313
2.0000	0.2667	0.1333	0.5000	0.0045	0.0746	0.0999	0.2696
2.0000	0.2000	0.1000	0.6000	0.0017	0.0579	0.0938	0.4410
2.0000	0.1333	0.0667	0.7000	0.0006	0.0399	0.0856	0.6167
1.0000	0.3000	0.3000	0.3000	0.0131	0.1687	0.0883	0.0825
1.0000	0.2500	0.2500	0.4000	0.0052	0.1456	0.0937	0.1912
1.0000	0.2000	0.2000	0.5000	0.0021	0.1181	0.0926	0.3366
1.0000	0.1500	0.1500	0.6000	0.0008	0.0895	0.0879	0.4957
1.0000	0.1000	0.1000	0.7000	0.0003	0.0608	0.0818	0.6481
0.5000	0.2000	0.4000	0.3000	0.0041	0.2514	0.0809	0.1396
0.5000	0.1667	0.3333	0.4000	0.0018	0.2070	0.0854	0.2641
0.5000	0.1333	0.2667	0.5000	0.0008	0.1633	0.0852	0.4052
0.5000	0.1000	0.2000	0.6000	0.0004	0.1218	0.0824	0.5449
0.5000	0.0667	0.1333	0.7000	0.0002	0.0820	0.0783	0.6739
0.3333	0.1500	0.4500	0.3000	0.0020	0.2938	0.0767	0.1751
0.3333	0.1250	0.3750	0.4000	0.0009	0.2373	0.0813	0.3026
0.3333	0.1000	0.3000	0.5000	0.0004	0.1854	0.0818	0.4372
0.3333	0.0750	0.2250	0.6000	0.0002	0.1378	0.0798	0.5658
0.3333	0.0500	0.1500	0.7000	0.0001	0.0927	0.0766	0.6842

.

Analysis of Table 8 and Table 9 reveals that all components exhibit negative deviations from ideality. Specifically, γ_In shows only slight negative deviations, whereas the remaining components display strongly negative deviations. As the indium content increases from x_In = 0.1 to x_In = 0.2, γ_In remains essentially unchanged, while γ_Sn and γ_Cu increase slightly. In contrast, γ_Au decreases more markedly, which can be attributed to the greater tendency of In to form compounds with Au. The addition of In changes the component activities and the Gibbs free energy of formation of intermetallic phases. By reducing the activity of Sn and modifying the chemical affinity between Au and Sn, In can suppress excessive growth of brittle Au-Sn intermetallics and improve phase stability and mechanical reliability. The resulting In-Au intermetallic phases are associated with a significant lowering of the melting point, consistent with experimental observations. Moreover, as the ratio x_Au/x_Cu increases, γ_Sn drops sharply, indicating that Au promotes compound formation more effectively than Cu does.

Conversely, a comparison between Table 8 and Table 10 shows that with increasing temperature, γ_Au shifts noticeably, γ_Cu displays an upward trend, γ_In increases slightly, and γ_Sn undergoes only a minor shift. These trends suggest that at higher temperatures, Sn exhibits a stronger tendency to form compounds with Au, thereby reducing its propensity to combine with Cu.

Although the M-MIVM provides good predictions for component activities in Au-Sn-based multicomponent liquid alloys, it still has certain limitations: The model is calibrated for liquid phases and does not directly describe solid–liquid equilibria or solid-state phase transformations. It does not explicitly consider temperature dependence of interaction parameters or the formation of long-range ordered intermetallic phases in the solid state.

4. Conclusions

(1) M-MIVM retains the theoretical framework of the radial distribution function from the original MIVM and incorporates the Scatchard–Hildebrand theory to achieve separation between the energy and volume parameters. This modification endows M-MIVM with a clear advantage in describing asymmetric systems. The improved accuracy in fitting binary systems directly translates into enhanced predictive performance for multicomponent alloys. Consequently, M-MIVM provides a more reliable means of predicting the activities of Au-Sn-based multicomponent systems that contain binary asymmetric subsystems.

(2) For binary and ternary alloys, the predicted activities from the M-MIVM show an average relative deviation less than 0.04 and an average standard deviation less than 17% compared with experimental data, demonstrating the reliability of the model.

(3) In terms of both accuracy and applicability, M-MIVM demonstrates strong potential for supplementing thermodynamic databases of high-temperature lead-free solders that include asymmetric systems. It can be applied with greater confidence to phase equilibrium calculations and the prediction of various alloy properties.