Preprint and STEM images from article "Effect of Ag on the precipitation stability in Al-Mg-Si-Ag alloy: First-principles calculations, Calphad modeling and experimental validation"

Effect of Ag on the precipitation stability in Al-Mg-Si-Ag alloy: First-principles calculations, Calphad modeling and experimental validation Wei shao1,*, Witold Chrominski1, Mark Fedorov1, Chenying Shi2, Javier LLorca2,3,* and Jan S. Wróbel1,* 1Faculty of Materials Science and Engineering, Warsaw University of Technology, ul. Wołoska 141, 02-507, Warsaw, Poland 2IMDEA Materials Institute, C/Eric Kandel 2, 28906 Madrid, Spain 3Department of Materials Science. Polytechnic University of Madrid/Universidad Politécnica de Madrid. E. T. S. de Ingenieros de Caminos, 28040 Madrid, Spain. Abstract: The Gibbs free energy of different phases in the Al-Mg-Si-Ag alloy system was determined by combining first-principles calculations, cluster expansion method and Monte Carlo simulations. This information was used to develop a thermodynamic database of the AlMg-Si-Ag system, enabling accurate determination of phase stability and phase distribution. The analysis of short-range order parameters showed that the solute cluster was dominated by Mg-Si co-clusters in Al-Mg-Si alloys, while Mg-Ag co-clusters formed prior to the Mg-Si-Ag clusters when the Al-Mg-Si-Ag alloy is quenched from high temperature. Experimental observations by means of transmission electron microsocopy confirmed the existence of such clusters. The addition of Ag also modified the crystal structure of Guinier-Preston zones, whose composition was Al1-zAgz, Al1-x-zMgxAgz and MgxAgz. In contrast, the crystal structure of metastable β″ and stable β phases was not influenced by the presence of Ag. Phase diagram calculations of the Al-Mg-Si-Ag system revealed the existence of a fcc single-phase in Al-Ag, Al-Mg, Ag-Mg binary and Al-Ag-Mg ternary systems. In the Al-rich region of the quaternary system, two-phase regions fcc + β and fcc + β′/β′Ag were observed. As temperature increases, the fcc+β'/β'Ag two-phase region gradually transforms into the fcc+β two-phase region. These results reveal the thermodynamic stability of metastable precipitate and, in certain cases, elucidate compositional changes in precipitates during the aging process. Keywords: Al-Mg-Si-Ag alloys; First-principles calculations; Monte Carlo simulations; Calphad; Transmission electron microscopy This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5311375 Preprint not peer reviewed 1. Introduction Al-Mg-Si alloys are widely used in construction, automotive engineering, and aviation industries due to their good corrosion resistance, excellent formability and weldability [1,2]. The mechanical properties of Al-Mg-Si alloys are controlled by the type, dimension, and distribution of nano-sized precipitates that form during aging at different temperatures and times [3,4]. Therefore, it is useful to study the precipitate evolution of Al-Mg-Si alloys to gain insight into their age-hardening response. The precipitation sequence in the Al-Mg-Si alloys has been reported as [5,6]: supersaturated solid solution (SSSS) → Mg-Si co-clusters → GP zones →β″ → β' → β(Mg2Si). The main hardening phase in the peak hardness condition is the needle-like β", which has a monoclinic unit cell with a=15.16 Å, b=4.05 Å, c=6.74 Å and β=105.3º [7]. The chemical composition of β″ was initially determined to be Mg5Si6 using high-resolution transmission electron microscopy (HRTEM) and electron diffraction (ED) [8]. Subsequently, Ninive provided the atomistic insight into the β″ phase by scanning transmission electron microscopy with highresolution high-angle annular dark field (HAADF-STEM), and proposed that the most likely composition of β″ was Al3Mg4Si4 [9]. Another possible composition of β″ reported was Al2Mg5Si4 [10]. Hence, the composition of β" is still a subject of controversy. β′ precipitates form after β″ precipitates in the ageing sequence. The β(Mg2Si) phase with anti-fluorite (CaF2) structure is formed at equilibrium [5]. Early investigations have demonstrated that the strength of Al-Mg-Si alloys was enhanced after aging [11], and that considerable precipitation strengthening can be achieved by adding microalloying elements [12,13]. Ding et al. [14] indicated that the addition of Cu suppresses the precipitation of β″ and results in the formation of quaternary phases (QP) during aging. Therefore, the precipitation sequence of Al-Mg-Si alloys with Cu is: SSSS → atomic clusters → GP zones → β″, QP1, QP2, C (Mg4Al1Si3+xCu1-x, x∼0.3) → Q′ (Al3Cu2Mg9Si7), QP2, C→ Q (Al4Cu2Mg8Si7), Si. Ding et al. [15] found that relatively high concentrations (3 wt.%) of Zn could enhance the age hardening response of Al-0.99Mg-0.54Si (wt. %) alloy by forming GP (II)-zones of η-MgZn2 and its precursor. Compared with Zn and Cu, Ag not only significantly improves the age-hardening response and hardening kinetics of Al-Mg-Si alloys during artificial aging, but also ensures relatively low strength in the natural aging, thereby providing a good balance between formability and bakehardening potential [16]. Ag atoms could enter Mg-Si clusters and refine the distribution of This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5311375 Preprint not peer reviewed clusters during the early aging, which resulted in a finer and denser distribution of β″ to improve the peak hardness for these alloys [17]. Marioara et al. [18] observed through HAADF-STEM that the addition of Ag transforms the structure of β ′ from Mg9Si5 to Al3Mg3Si2Ag. More recently, Weng et al. [19] found that another quaternaty phase QAg′ Al3Mg3SiAg2 structure may also be formed, and the precipitate sequence was: SSSS → atomic clusters → GP zones → β″ → β′, β′Ag1, β′Ag2, QAg′→ β, Ag. Although many studies of the precipitate phases have been reported [18-21], complete crystal structure information of these phases is not always available or unambiguous. Specifically, the solute clustering behavior was not examined, which is essential for understanding the early stages of precipitation and plays a vital role in determining the thermodynamic stability and strengthening mechanisms of Al-Mg-Si(-Ag) alloys. The GP zones are the precursors and nucleation core of the main strengthening phase β″, while the effect of Ag on the atomic structures of GP zones and β″ phase formed at the early and peak aging stage is still unknown. Additionally, the some studies indicated that structure of β was not influenced by Ag atoms [19,22], while Zhang et al. [23]revealed that Ag atoms entered the crystal structure of β by replacing Si atoms in the corners of the unit cell. The remarkable solubility of Ag in β (Mg2Si) has been confirmed by Udono et al. [24] and Prytuliak et al. [25]. However, it is very difficult to answer these questions in the Al-Mg-Si-Ag alloys only from experimental observations. Currently, the phase stability for a system is typically evaluated using thermodynamic databases (TDBs) obtained through the Calphad (calculation of phase diagrams) method [26]. However, due to the limitation of slow kinetics at low temperature and incomplete thermodynamic information in some phases [25-27], the TDBs availble may not be accurate for alloys within the complex Al-Mg-Si-Ag system. Within this framework, the objective of this investigation is to develop a strategy that integrates first-principles calculations and the cluster expansion (CE) method coupled with statistical mechanics principles to construct the Al-MgSi-Ag TDB without relying on experiments [30,31]. To this end, the Gibbs free energies of each phase in the system are obtained as a function of temperature and composition from the formation enthalpies at 0 K calculated from density functional theory (DFT) and thermodynamic properties - formation enthalpies and entropy - at finite temperatures obtained from Monte Carlo (MC) simulations [32,33]. These TDBs for the Al-Mg-Si-Ag system are implemented into the OpenCalphad software [32-34]. They allow to determine the stability region of key precipitate phases β″, β′, β′Ag and β, as well as phase coexistence and their fractions. This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5311375 Preprint not peer reviewed The atomic structure and composition of GP zones are also determined based on this methodology. The simulation results were validated through STEM-HAADF. Moreover, the accurate TDB for this quaternary alloy will facilitate the prediction of thermodynamic properties in binary and ternary systems involving Al, Mg, Si and Ag atoms. It also provides a robust foundation for extrapolating to more complex Al-Mg-Si-based alloy systems, overcoming the limitations of experimental approaches due to the extensive number of measurements required to explore the entire compositional space. 2. Methodology 2.1 DFT calculations DFT calculations were carried out in the Vienna ab initio simulation package (VASP) [37], using the projector augmented wave (PAW) method [38]. Exchange-correlation is treated in the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) [39]. The electronic configurations considered are 3s23p1 for Al, 2p63s2 for Mg, 3s23p2 for Si, and 4d105s1 for Ag, respectively. Convergence tests indicated that a cutoff of 400 eV is sufficient to ensure total energy differences are less than 1 meV/atom. The Γ-centered Monkhorst-Pack mesh [40] of 𝑘-points in the Brillouin zone, with the k-mesh spacing of 0.02 Å-1, corresponding to a 12×12×12 𝑘-point mesh for a fcc conventional unit cell. The total energy convergence criterion is set to 10-5 eV/cell, and force components are relaxed to 10-3 eV/Å. All structures are fully relaxed concerning volume as well as cell-internal and -external coordinates. 2.2 Cluster expansion The formation enthalpies, Hform, of a set of AlxMgySizAg1-x-y-z configurations that comprise the whole composition range of the system are calculated as [41] 𝐻 Al x Mg y Si z Ag 1 ― x ― y ― z 𝑓𝑜𝑟𝑚 = 𝐸 Al x Mg y Si z Ag 1 ― x ― y ― z 𝑡𝑜𝑡 ― 𝑥 𝐸 𝐴𝑙 𝑓𝑐𝑐 ― 𝑦 𝐸 𝑆𝑖 𝑑𝑖𝑎𝑚𝑜𝑛𝑑 ― 𝑧 𝐸 𝑀𝑔 ℎ𝑐𝑝 ― (1 ― 𝑥 ― 𝑦 ― 𝑧) 𝐸 𝐴𝑔 𝑓𝑐𝑐 (1) where x, y and z are atomic fractions of Al, Si and Mg in a given configuration, respectively. 𝐸 Al x Mg y Si z Ag 1 ― x ― y ― z 𝑡𝑜𝑡 is the total energy per atom of AlxMgySizAg1-x-y-z after relaxation, 𝐸 𝐴𝑙 𝑓𝑐𝑐 , 𝐸 𝐴𝑔 𝑓𝑐𝑐 , 𝐸 𝑆𝑖 𝑑𝑖𝑎𝑚𝑜𝑛𝑑 , and 𝐸 𝑀𝑔 ℎ𝑐𝑝 are the energies of the stable phases for Al and Ag with fcc lattice, Si with diamond lattice, and Mg with hcp lattice, respectively. This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5311375 Preprint not peer reviewed In the CE formalism [40-42], the formation enthalpy can be parametrized as a polynomial in the occupational variables σ: 𝐻 𝑓𝑜𝑟𝑚 ( 𝜎 ) = ∑ 𝜔,𝑛,𝑠 𝐽 (𝑠) 𝜔,𝑛 𝑚 (𝑠) 𝜔,𝑛 〈 𝜑 ( 𝜎 ) 〉 (𝑠) 𝜔,𝑛 (2) where 𝜎 = { 𝜎 1 , 𝜎 2 , …, 𝜎 𝑁 } is the vector of occupation variables that define the atomic species in lattice site i in a quaternary system ( 𝜎 𝑖 = 0, 1, 2, 3 ) with 0 for Al, 1 for Mg, 2 for Si and 3 for Ag, following the emthodology previously applied to the quaternary Fe-Cr-Mn-Ni system [45]. The summation is performed over all the clusters, distinct under symmetry operations in the studied lattice, represented by the following parameters: 𝜔 and 𝑛 are the cluster size (the number of lattice points in the cluster) and its label (related to the maximal distance between two atoms in the cluster), respectively. ( 𝑠 ) is the decoration of cluster by a point function 𝛾 𝑗,𝐾 ( 𝜎 𝑖 ) . 𝐽 (𝑠) 𝜔,𝑛 represents the effective cluster interaction (ECI) coefficient corresponding to the same ( 𝑠 ) decorated cluster. 𝑚 (𝑠) 𝜔,𝑛 denotes the site multiplicity of the decorated clusters. 〈 𝜑 ( 𝜎 ) 〉 𝜔,𝑛,𝑠 is the cluster basis function, which is expressed as a product of orthonormal point functions 𝛾 𝑗,𝐾 ( 𝜎 𝑖 ) over all sites included in the specific cluster described by 𝜔 and 𝑛 [42]: 〈 𝜑 ( 𝜎 ) 〉 (𝑠) 𝜔,𝑛 = 𝛾 𝑗 1 𝐾 ( 𝜎 1 ) 𝛾 𝑗 2 𝐾 ( 𝜎 2 ) ⋯ 𝛾 𝑗 𝜔 𝐾 ( 𝜎 𝜔 ) (3) where 𝑗 𝑖 = (0, 1, 2, 3) has a similar meaning to 𝜎 𝑖 , indicating which type of atom is located on the lattice site i that belongs to the cluster. K is the number of alloy components, which is equal to 4 in our system. The cluster basis functions of each pair of clusters, α and β, should satisfy 〈 𝜑 ( 𝜎 ) (𝑠) α , 𝜑 ( 𝜎 ) (𝑠) β 〉 = 0 if they are different and 〈 𝜑 ( 𝜎 ) (𝑠) α , 𝜑 ( 𝜎 ) (𝑠) β 〉 = 1 if they are identical. This is achieved by 𝛾 𝑗,𝐾 ( 𝜎 𝑖 ) as [34,44]: 𝛾 𝑗,𝐾 ( 𝜎 𝑖 ) = { 1 if 𝑗 = 0, ― cos ( 2𝜋 [ 𝑗 𝑖 2 ] 𝜎 𝑖 𝐾 ) if 𝑗 > 0 and odd, ― sin ( 2𝜋 [ 𝑗 𝑖 2 ] 𝜎 𝑖 𝐾 ) if 𝑗 > 0 and even (4) where [ 𝑗 𝑖 2 ] denotes the ceiling function - rounding up to the closest integer. The optimal values of ECIs are calculated to minimize the Cross-Validation (CV) Score between 𝐻 DFT 𝑓𝑜𝑟𝑚 and 𝐻 CE 𝑓𝑜𝑟𝑚 through the structure inversion method, as implemented in the ATAT package [43]: 𝐶𝑉 = ∑ 𝑁 𝑖 = 1 ( 𝐸 DFT 𝑖 ― 𝐸 CE ′ 𝑖 ) 2 𝑁 (5) This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5311375 Preprint not peer reviewed Where 𝐸 CE ′ 𝑖 is the predicted energy of ith structure by fitting CE energies from DFT, and excluding the ith structure. 2.3 Monte Carlo simulations At finite temperatures, the stability of the different phases may change significantly due to entropic contributions [46,47]. In this work, we mainly focus on configurational entropy as the strongest contribution to the Gibbs free energy of formation, 𝐺 𝑓𝑜𝑟𝑚 . 𝐺 𝑓𝑜𝑟𝑚 for a given composition at finite temperature can be calculated by performing a canonical MC simulation - the chemical composition and number of atoms are fixed - within the CE formalism for each lattice: 𝐺 𝑀𝐶 𝑓𝑜𝑟𝑚 ( 𝜎 ,𝑇) = 𝐻 𝑓𝑜𝑟𝑚 ( 𝜎 ,𝑇) ― 𝑇 𝑆 𝑐𝑜𝑛𝑓 ( 𝜎 ,𝑇) (6) where formation enthalpy 𝐻 𝑓𝑜𝑟𝑚 ( 𝜎 ,𝑇) is calculated from eq. (2). The configurational entropy 𝑆 𝑐𝑜𝑛𝑓 ( 𝜎 ,𝑇) per site is calculated from the specific heat 𝐶( 𝜎 ,𝑇′) by thermodynamic integration as [48] 𝑆 𝑐𝑜𝑛𝑓 ( 𝜎 ,𝑇) = ∫ 𝑇 0 𝐶( 𝜎 ,𝑇′) 𝑇′ 𝑑𝑇′ (7) where 𝐶( 𝜎 ,𝑇′) can be obtained from the fluctuations of the formation enthalpy in MC simulations at temperature 𝑇 ′, by using the expression [48]: 𝐶( 𝜎 ,𝑇′) = 〈 𝐻 2 𝑓𝑜𝑟𝑚 ( 𝜎 ,𝑇′ ) 〉 ― 〈 𝐻 𝑓𝑜𝑟𝑚 ( 𝜎 ,𝑇′ ) 〉 2 𝑇′ 2 (8) where 〈 𝐻 𝑓𝑜𝑟𝑚 ( 𝜎 ,𝑇′ ) 〉 and 〈 𝐻 2 𝑓𝑜𝑟𝑚 ( 𝜎 ,𝑇′ ) 〉 are the mean and mean-squared average formation enthalpies, respectively, which describe the variance of 𝐻 𝑓𝑜𝑟𝑚 ( 𝜎 ,𝑇′ ) . They are computed by averaging all the MC steps at the accumulation stage for a given temperature, as in [48]. Canonical MC simulations were performed using the ATAT package separately for fcc, CaF2 (β), β′ and 𝛽 ' 𝐴𝑔 lattices, because these phases were found in the Al-Mg-Si-Ag alloys [19]. Supercells were generated at 1 at. % intervals over the whole composition range using the special quasi-random structure (SQS) method [49], comprising 10 × 10 × 10 supercells with 4000 atoms for fcc, 5 × 5 × 5 supercells with 1500 atoms for β, 5 × 3 × 2 supercells with 840 atoms for β′, and 5 × 3 × 6 supercells with 810 atoms for β ' Ag . For each composition, MC simulations are performed starting from a high-temperature disordered state at 𝑇 = 1000 K. The alloy is then cooled down to 10 K within the temperature step of 𝛥𝑇 = 20 K. At each This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5311375 Preprint not peer reviewed temperature, a MC simulation includes 3000 passes for equilibrium and thermodynamic averages. The phase boundary in binary systems can be determined by the common tangent of Gibbs free energy of formation 𝐺 𝑓𝑜𝑟𝑚 , and then the fraction of the considered phases as functions of concentration can be calculated by using the lever rule [50,51]. In the case of multicomponent Al-Mg-Si-Ag alloys, the calculation of fractions of both phases in a four-dimensional composition space cannot be achieved by the application of one common tangent [34]. Instead, the well-established Calphad method provides robust algorithms for calculating multicomponent equilibria, which are essential for evaluating the corresponding phase fractions [35,52]. 2.4 Development of thermodynamic databases The Gibbs free energy of formation for a solid solution phase within the framework of Calphad can be expressed as [53] 𝐺 𝑓𝑜𝑟𝑚 = 𝐺 𝑟𝑒𝑓 + 𝐺 𝑖𝑑𝑒𝑎𝑙 + 𝐺 𝑥𝑠 (9) where 𝐺 𝑟𝑒𝑓 = ∑ 4 𝑖 = 1 𝑥 𝑖 0 𝐺 𝑖 is the reference energy of pure elements (Al, Mg, Si and Ag) for the stable phase. 𝐺 𝑖𝑑𝑒𝑎𝑙 = ― 𝑇 𝑆 𝑖𝑑𝑒𝑎𝑙 describes the contribution to the Gibbs free energy from ideal random mixing of the constituents on the crystal lattice. 𝑆 𝑖𝑑𝑒𝑎𝑙 = ― 𝑅 ∑ 4 𝑖 = 1 𝑥 𝑖 ln 𝑥 𝑖 , where R is the gas constant, and 𝑥 𝑖 the mole fraction of component i. 𝐺 𝑥𝑠 is the excess Gibbs free energy of formation, describing the influence of non-ideal mixing behaviour on the thermodynamic properties of a solution phase. 𝐺 𝑥𝑠 is usually described by the semi-empirical Redlich-Kister (RK) polynomial [54,55]: 𝐺 𝑥𝑠 = 𝑥𝑠2 𝐺 + 𝑥𝑠3 𝐺 + ⋯ = 𝑛 ― 1 𝑖 = 1 𝑛 𝑗 = 𝑖 + 1 𝑥 𝑖 𝑥 𝑗 𝑣 𝐿 𝑖,𝑗 ( 𝑥 𝑖 ― 𝑥 𝑗 ) 𝑣 + ∑ 𝑛 ― 2 𝑖 = 1 ∑ 𝑛 ― 1 𝑗 = 𝑖 + 1 ∑ 𝑛 𝑘 = 𝑗 + 1 𝑥 𝑖 𝑥 𝑗 𝑥 𝑘 ( 𝑢 𝑖 𝐿 𝑖 + 𝑢 𝑗 𝐿 𝑗 + 𝑢 𝑘 𝐿 𝑘 ) + ⋯ (10) 𝑢 𝑖 = 𝑥 𝑖 + 1 ― 𝑥 𝑖 ― 𝑥 𝑗 ― 𝑥 𝑘 3 (11) 𝑢 𝑗 = 𝑥 𝑗 + 1 ― 𝑥 𝑖 ― 𝑥 𝑗 ― 𝑥 𝑘 3 (12) This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5311375 Preprint not peer reviewed 𝑢 𝑘 = 𝑥 𝑘 + 1 ― 𝑥 𝑖 ― 𝑥 𝑗 ― 𝑥 𝑘 3 (13) where 𝑥𝑠2 𝐺 and 𝑥𝑠3 𝐺 are binary and ternary excess terms, respectively. L is the interaction parameter between components and 𝑣 is the maximum power used in the optimization. It should be noted that complex phases are usually modelled using the compound energy formalism (CEF) in the Calphad methodology. Thus, β′ and β′Ag phases in Al-Mg-Si-Ag alloys are described by (Al,Mg)18(Si,Ag)10 and (Al,Mg)6(Si,Ag)3, respectively. Therefore, 𝐺 𝑟𝑒𝑓 is expressed as 𝐺 𝑟𝑒𝑓 = ∑ 𝑠 , 𝑡 = 1,2 𝑦 𝑠 𝑖 𝑦 𝑡 𝑗 0 𝐺 𝑠 , 𝑡 𝑖 , 𝑗 (i=Al, Mg; j=Si, Ag) (14) where y is denoted as site fraction and defined as the composition of each constituent in the sublattice s. The 𝐺 𝑠:𝑡 𝑖,𝑗 represents the Gibbs free energy of formation of compounds where each sublattice is occupied by the same constituent, such as Al18Si10 and Al6Si3. In this work, TDBs based only on the results of MC simulations obtained using the DFT-based CE models were generated for fcc, β, β′ and β ' Ag in the Al-Mg-Si-Ag alloys using sqs2tdb tool in ATAT package [56]. Therefore, two important details have to be pointed out in the fitting: (1) temperature dependence of 𝐺 𝑓𝑜𝑟𝑚 is contained in the 𝐺 𝑟𝑒𝑓 supplied from SGTE databases in the traditional Calphad model [57], which contradicts our objective to create TDBs just from DFT simulations, so the SGTE database has not been used. (2) Excess Gibbs free energy of formation 𝐺 𝑥𝑠 does not include quaternary interactions ( 𝑥𝑠4 𝐺 ) in the sqs2tdb fitting because they would not contribute to the description of the Gibbs free energies of the phases [58]. Finally, the TDBs obtained from the MC simulations were used to calculate the phase equilibrium diagram using OpenCalphad at each temperature using the corresponding TDB [34]. 2.5 Experiments and microstructural observation In order to verify the actual phase tranformations, an ingot with chemical composition Al 0.89Mg 0.96Si 0.15Ag (at. %) was prepared by mould casting from pure elements (> 99.9% purity). After homogenization and recrystallization at 520 °C, billets were heated up to 520°C during one hour followed by water quenching. Afterwards, they were artificially aged at 160ºC in afurnace. The results of heat treatment were evaluated on the basis of hardness measurements. Three samples, aged during 0.5h, 2h and 12h were selected as representative of the precipitation evolution to support calculations in this study. They stand for the early stages (clustering), the beginning of hardness stabilization and the peak hardness, respectively. This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5311375 Preprint not peer reviewed Thin foils for microscopic evaluation were prepared with convention electrochemical polishing of 3 mm discs. A solution of nitric acid in methanol has been used for this purpose. STEMHAADF observation (collection angle range of 45-200 mrda) was carried out in Cs-corrected ThermoFisherScientific Spectra 200 microscope. Beam current below 25 pA has been kept during the observations and energy dispersive spectroscopic analysis in order to maintain the stability of clusters and precipitates. 3. Results 3.1 Phases of interest The stable and metastable phases in the Al-Mg-Si-Ag alloys - according to the information in the literature[5,8,19,59] - are listed in Table. 1. Their crystal structures are depicted in Fig. 1. Based on our previous experience [60,61], some phases with complex structure can be treated as a distorted configuration on a simpler lattice. Al has a fcc lattice (Fig.1(a)). The β″ (Mg5Si6) (Fig. 1(b)) also presents a fcc lattice, where the lattice vector bβ″ of β″ is parallel to the [001] axis of α-Al, and the other two lattice vectors of β″ are defined by aβ″=2aAl+3bAl (the [230] direction in α-Al) and cβ″=- 3 2 aAl+ 1 2 Al (the [3 1 0] direction in α-Al). However, the crystal structure of β″ is not uniquely determined from Refs. [8-10], which may contain a certain fraction of Al. β′ (Mg9Si5) (Fig.1(c)) has a hexagonal unit cell, and there are multiple orientation relationships between α-Al and β' [62]. β'Ag1 ((Fig.1d)) and β'Ag2 ((Fig.1(f)) have a hexagonal unit cell that is similar to that of β' phase, while their lattice parameter along the a-axis is slightly different from the corresponding direction in β'. β (Mg2Si) has a CaF2 structure. It is worth noting that all phases, except for the equilibrium phases β-Mg2Si and Si, are metastable. Table 1. Structural information of the different phases in the Al-Mg-Si-Ag alloys. Composition Symmetry Lattice parameters (Å) phase Al Fm 3 m a = b = c = 4.05 α Mg5Si6 C2/m a=15.16, b=4.05, c=6.74 β=105.3º β″ Mg9Si5 P6₃/m a = b =7.15, c=12.15 α=β=90º, γ=120° β' Al3Mg3 Si2Ag P62m a = b =6.9, c=4.05 α=β=90º, γ=120° β'Ag1 Al3Mg3SiAg2 P62m a = b =7.2, c=4.05 β'Ag2 This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5311375 Preprint not peer reviewed Fig. 6 (a) Formation enthalpy per solute atom, ∆ 𝐻 𝑠𝑜𝑙𝑢𝑡𝑒 𝑓𝑜𝑟𝑚 , calculated from eq. (15) as a function of as a function of 𝑥 𝑀𝑔 + 𝑦 𝑆𝑖 + 𝑧 𝐴𝑔 . (b) ∆ 𝐻 𝑠𝑜𝑙𝑢𝑡𝑒 𝑓𝑜𝑟𝑚 as a function of as a function of 𝑧 𝐴𝑔 for configurations with 𝑥 𝑀𝑔 + 𝑦 𝑆𝑖 + 𝑧 𝐴𝑔 = 1 . 3.5 Phase stability at elevated temperatures 3.5.1 Comparison with MC and MC-fitted TDB The CE of each lattice structure can be used to predict the thermodynamic properties of the corresponding structures with this lattice, following the methodology presented in Section 2.3. The compositional grid of the Al-Mg-Si-Ag quaternary system for fcc and β lattice was systematically explored using an interval of 10 at.% for each one of the constituents. The β′ and β′Ag only exist in a small compositional range, so the interval of 3.5 at.% for each component is used. Gibbs free energies of formation Gform for each composition were calculated as indicated in Section 2.3. In the end, Gform as a function of composition at elevated temperatures was fitted as indicated in Section 2.4 to develop TDBs of Al-Mg-Si-Ag alloys. The Gform at 500 K obtained using MC-fitted TDB and those calculated directly from MC simulations were first compared in the six binary systems underlying fcc and β lattice to verify the accuracy of TDBs, as shown in Fig. 7. The Gform obtained using MC-fitted TDB is in good agreement with those calculated from MC simulations in the six binary systems. The fcc phase is more stable than β in the whole composition range for Ag-Al, Ag-Mg and Al-Mg binary systems. The stability of the phase changed between fcc and β phase in the whole composition range for Ag-Si, Al-Si and Mg-Si binary systems. Such as the fcc phase is more stable than the β phase in the Ag-rich Ag-Si and Al-rich Al-Si. The β phase is more stable than the fcc phase in the Si composition larger than 0.2 for the Mg-Si binary system. Additionally, the Gform at 500 K obtained from MC-fitted TDB and MC simulations in the pseudo-binary systems, such as the composition of Al and Mg are fixed, are also compared and This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5311375 Preprint not peer reviewed shown in Fig. 8. It should be noted that only Gform obtained using MC-fitted TDB for fcc and β lattice are depicted in Fig. 8. The β′ and β′Ag phases can co-exist with the fcc phase in this composition range. It should be noted that there are some differences between MC and MCfitted TDB for β′Ag lattice, which may be improved by performing MC simulations on a denser compositional grid. In fact, the MC-fitted TDB remains suitable for calculating the Gform of this lattice. This is supported by the low CV score of 0.0061 eV/atom, calculated based on the Gform from MC and MC-fitted TDB. The Gform at 300 K and 800 K obtained from the two approaches can be found in Figs. S1-S4 in the Supplementary Material. Except for a small difference in the Gform of β′Ag lattice, the Gform for fcc, β and β′ lattices obtained using MC-fitted TDB are in good agreement with those calculated directly from MC simulations. The TDBs for the temperature range from 100 K to 1000 K were constructed using the Calphad method based on the Gform obtained from the MC simulation. These databases are provided in the Supplementary Material as a compressed file named TDB.zip, which includes individual TDB files for each temperature interval: Al-Mg-Si-Ag-100K.TDB, Al-Mg-Si-Ag-200K.TDB, ... , up to Al-Mg-Si-Ag1000K.TDB. This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5311375 Preprint not peer reviewed Fig. 7. The comparison of Gibbs free energies of formation Gform calculated from MC simulations and the OpenCalphad calculations using TDB based on MC simulations at 500 K for (a) Ag-Al. (b) Ag-Mg. (c) Ag-Si. (d) Al-Mg. (e) Al-Si and (f) Mg-Si binary systems underlying fcc and β lattice. Green and orange circles represent the Gform with fcc and β lattice calculated from MC simulations, respectively. Green and orange lines represent the Gform with fcc and β lattice calculated using TDB, respectively. This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5311375 Preprint not peer reviewed Fig. 8. The comparison of Gibbs free energies of formation Gform calculated from MC simulations and the OpenCalphad calculations using TDB based on MC simulations at 500 K for (a) Ag0.339-xAl0.0357Mg0.307Six and (b) Ag0.296-xAl0.037Mg0.63Six pseudo-binary systems underlying fcc, β, β' and β'Ag lattice. Green and orange lines represent the Gform with fcc and β lattice calculated using TDB, respectively. Purple circles and lines represent the Gform with β' or β'Ag lattice calculated from MC simulations and the TDB, respectively. 3.5.2 fcc, β, β' and β'Ag phase distribution The methodology based on the development of TDBs and their implementations in the OpenCalphad calculations, as described in Section 2.4, was used to investigate the distribution of different phases and their fractions in Al-Mg-Si-Ag alloys. As mentioned in Section 2.3, the phase composition is determined using the common tangent of Gibbs free energy of formation Gform, and the phase fractions are subsequently calculated using the lever rule. The percentage of fcc phase across the whole composition range at 300 K is plotted in Fig. 9(a), while the corresponding phase distribution is depicted in Fig. 9(b). The phase composition is plotted in different colors, each indicating the stability region of the four different phases. It is well known that the stable structure of Si and Mg is diamond and hcp lattice, thus they are only located on the vertices of the tetrahedron and plotted in grey and pink circles in Fig. 9(b), (d) and (f). Furthermore, the β' and β'Ag phases are not analyzed separately in this work because they always co-exist at the over-aging stage [19]. It can also be seen that the Gibbs free energy of formation Gform for them with similar composition is very close. In the end, only the fcc solid solution (the fcc phase percentage is 100%) appeared in the Al-Mg, Ag-Mg, Al-Ag binary systems, and one Al-Mg-Ag ternary system. The same results are also observed at 500 K (Fig. 9(c)) and 800 K (Fig. 9(c)), indicating that the ordering states are easily formed among Al, Mg This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5311375 Preprint not peer reviewed and Ag atoms. The percentage of fcc phase decreased with temperature from 800 K (Fig. 9(e)) to 300 K (Fig. 9(e)), indicating some other phases precipitated from the Al matrix. It can be seen from the phase distribution that the single β phase region is observed in the Si-rich AgMg-Si and Al-Mg-Si ternary systems at 500 K (Fig. 9(d)) and 800 K (Fig. 9(f)). The β phase coexists with fcc and β' or β'Ag phase in the Al-Mg-Si ternary system at 300 K (Fig. 9(b)). However, it is not easy to analyse the phase change in the interior from a four-dimensional phase diagram, so a more detailed analysis will be described in Section 3.5.3. Fig. 9 Percentage of fcc phase and different phase distribution obtained from OpenCalphad calculations at 300 K (a), (b), 500 K (c), (b) and 800 K (e), (f). 3.5.3 Phase stability in the Al-Mg-Si-Ag alloys The fcc fractions and phase distribution for Al-Mg-Si ternary alloys -computed using the OpenCalphad calculations based on MC-fitted TDB as described in Section 2.4are shown in Fig. 10. There is a high fcc fraction (>70 %) in the Al-rich part of the Al-Mg-Si phase diagram, whether at 300 K (Fig. 10(a)), 500 K (Fig. 10(c)) or 800 K (Fig. 10(e)). In parallel, a low fcc fraction (< 20%) is also observed in the Si-rich part of the Al-Mg-Si phase diagram from three different temperatures (Fig. 10(a), (c) and (e)). The phase compositions of the stability region at 300 K, 500K and 800 K are plotted in Fig. 10(b), (d) and (f), respectively. A fcc+β two-phase This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5311375 Preprint not peer reviewed region plotted in green color is found in the Mg-rich part at 300 K (Fig. 10(b)). It can be seen that there is a high fcc fraction (>60%) in this two-phase region, and the fcc fraction has a slight change as temperature increases to 800 K (Fig. 10(e)), indicating that the fcc phase remains thermodynamically stable over this temperature range. The stable β+β' two-phase region is plotted in purple color, and the fcc+β' two-phase region is plotted in red color. There is a fcc+β+β' three-phase region that is plotted in brown color. As the temperature increases, these regions gradually transform into a two-phase region where fcc and β coexist. To figure out the effect of Ag on the phase stability, the fcc fractions and phase distribution for (AlMgSi)90Ag10 composition are analyzed and also compared with Al-Mg-Si in Fig. 11. The stable two-phase regions changed from fcc+β' (or β+β') to fcc+β'/β'Ag (or β+β'/β'Ag) when Ag is added into Al-Mg-Si alloys, indicating that the addition of Ag improves the stability of β' phase. As temperature increases from 300 K (Fig. 11(a), (c)) to 500 K (Fig. 11(b), (d)), the fcc+β'/β'Ag two-phase regions in the Mg-rich part gradually transform into the fcc+β two-phase region, while the stable β+β'/β'Ag two-phase regions in the Si-rich part completely transform into the fcc+β two-phase region and a small single β phase region. As temperature increases to 800 K (Fig. 11(e), (f)), except for the fcc+β'/β'Ag two-phase region in the Al-rich gradually transforms into the fcc+β two-phase region, the fcc+β+β'/β'Ag three-phase region also partly converts into the fcc+β two-phase region. Additionally, only the fcc single-phase region (blue color) appeared in the Al-Mg-Ag tenary system at three different temperatures (Fig. 11(b), (d) and (f)), indicating that Al-Mg-Ag is more likely to form atomic clusters in the Al-Mg-Si-Ag alloys. A significant difference in the fcc fraction was observed between Fig.10(a) and Fig. 11(a), particularly in the Mg-rich part. It can be seen that there is a fcc+β two-phase region in the Mg-rich part in Fig.11(b), while the stable two-phase region is fcc+β'/β'Ag in Fig. 11(b). This indicates that β' or β'Ag is more stable than β in this composition range. The fcc fractions and phase distribution for (AlMgSi)70Ag30 can be found in Fig. S5 in the supplementary Materials. The stable phase region in this composition range at 300 K (Fig. S5(a), (b)) is composed of fcc+β+β'/β'Ag three-phase region and fcc+β'/β'Ag two-phase region. As temperature increases to 500 K (Fig. S5(b), (c)), they are gradually transformed into the fcc+β two-phase region. As temperature increases to 800 K (Fig. S5(e), (f)), the stable phase region is dominated by the fcc+β two-phase region, and there is a single fcc phase region observed in the Al-rich part. This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5311375 Preprint not peer reviewed Fig. 10. Fcc fraction and phase distribution in the Al-Mg-Si ternary phase diagram at 300 K (a), (b), 500 K (c), (d) and 800 K (e), (f). This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5311375 Preprint not peer reviewed Fig. 11. Fcc fraction and phase distribution in the (AlMgSi)90Ag10 pseudo-ternary phase diagram at 300 K (a), (b), 500 K (c), (d) and 800 K (e), (f). 4. Discussion 4.1 Effect of Ag on the solute clusters It is well known from the precipitation sequence of Al-Mg-Si-based alloys that solute clusters were first precipitated from SSSS. These clusters serve as precursors to influence the artificial ageing response and govern the transformation pathways of the precipitation sequence. However, in contrast to the detailed knowledge of metastable precipitates - structures, evolution, This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5311375 Preprint not peer reviewed and their influence on material properties -, the knowledge about the atomic clusters that preceded them is very limited. Therefore, to figure out the effect of Ag on the solute clusters in the Al-Mg-Si alloys, a fcc solid solution with the composition Al98Mg1Si1 (at.%) - same as experiments - was generated using the SQS method and then used to perform MC simulations, as described in Section 2.3. In parallel, a series of fcc solid solution models with varying Ag composition, ranging from 0.1 at. % to 1 at. % in increments of 0.1 at. %, were also generated to perform MC simulations. These models were used to compare the clustering behaviors with those of the Ag-free fcc solid solution. Formation enthalpies 𝐻 𝑀𝐶 𝑓𝑜𝑟𝑚 of each solid solution model as a function of temperature were calculated using eq. (2), as shown in Fig. 12 (a). 𝐻 𝑀𝐶 𝑓𝑜𝑟𝑚 is a constant for Ag-free fcc solid solution above 180 K, indicating that it is a completely disordered state. 𝐻 𝑀𝐶 𝑓𝑜𝑟𝑚 decreases sharply below 180 K, and it might form local chemical clustering and short-range ordering because there are differences in the atomic bonding between elements under the enthalpy effect. As Ag concentration increases, the temperature at which the 𝐻 𝑀𝐶 𝑓𝑜𝑟𝑚 starts to significantly decrease shifts to a higher temperatures. For instance, 𝐻 𝑀𝐶 𝑓𝑜𝑟𝑚 of Al97.9Mg1Si1Ag0.1 starts to decrease significantly at 230 K, Al97Mg1Si1Ag0.5 at 420 K and Al97Mg1Si1Ag1 at 480 K. The atomic model where 𝐻 𝑀𝐶 𝑓𝑜𝑟𝑚 drops sharply is drawn in the Fig. 12(b)-(e). Al atoms are not visualized in these figures to shown more clearly the clustering of solute atoms. A Mg-Si cocluster is formed in the Al98Mg1Si1 model (Fig. 12(b)). Mg-Si co-clusters have been observed in the early stage in the Al-Mg-Si alloy by three-dimensional atom probe (3DAP) and TEM [6]. The cluster type changed by adding Ag to Al-Mg-Si alloys, even if the amount of Ag is very small, and the Mg-Si-Ag co-clusters are formed in Al97.9Mg1Si1Ag0.1 (Fig. 12(c)). As the Ag concentration increases to 0.5 at. % (Fig. 12(d)), the Mg-Ag co-clusters are dominant at 420 K in Al97.9Mg1Si1Ag0.5, while Si atoms exist in a disordered state. Si atoms entered Mg-Ag co-clusters and Mg-Si-Ag co-clusters are formed when the temperature decreased to 200 K, as shown in Fig. S6(a) in the supplementary material. Similar clustering behavior is also observed in Al97.9Mg1Si1Ag1 (Fig. 12(e) and Fig. S6(b)). This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5311375 Preprint not peer reviewed Fig. 12 (a) Formation enthalpies 𝐻 𝑀𝐶 𝑓𝑜𝑟𝑚 with varying Ag concentration as a function of temperature calculated by eq. (2). Solute clusters in the fcc solid solution with the Ag concentration of 0 at.% (b), 0.1 at.% (c), 0.5 at.% (d) and 1.0 at.% (e), respectively. The effect of Cu on the clustering behaviour in the Al-Mg-Si alloys has been widely studied [74-76]. They found that although the solute cluster was dominated by Mg-Si clusters although This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5311375 Preprint not peer reviewed [8] S.J. Andersen, H.W. Zandbergen, J. Jansen, C. TrÆholt, U. Tundal, O. Reiso, The crystal structure of the β″ phase in Al-Mg-Si alloys, Acta Materialia 46 (1998) 3283-3298. https://doi.org/10.1016/S1359-6454(97)00493-X. [9] P.H. Ninive, A. Strandlie, S. Gulbrandsen-Dahl, W. Lefebvre, C.D. Marioara, S.J. 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