E ec o Ag on he p ecipi a ion s abili y in Al-Mg-Si-Ag alloy: Fi s -p inciples
calcula ions, Calphad modeling and expe imen al alida ion
Wei shao1,*, Wi old Ch ominski1, Ma k Fedo o 1, Chenying Shi2, Ja ie LLo ca2,3,* and Jan
S. W óbel1,*
1Facul y o Ma e ials Science and Enginee ing, Wa saw Uni e si y o Technology, ul. Wołoska
141, 02-507, Wa saw, Poland
2IMDEA Ma e ials Ins i u e, C/E ic Kandel 2, 28906 Mad id, Spain
3Depa men o Ma e ials Science. Poly echnic Uni e si y o Mad id/Uni e sidad Poli écnica
de Mad id. E. T. S. de Ingenie os de Caminos, 28040 Mad id, Spain.
Abs ac : The Gibbs ee ene gy o di e en phases in he Al-Mg-Si-Ag alloy sys em was
de e mined by combining i s -p inciples calcula ions, clus e expansion me hod and Mon e
Ca lo simula ions. This in o ma ion was used o de elop a he modynamic da abase o he Al-
Mg-Si-Ag sys em, enabling accu a e de e mina ion o phase s abili y and phase dis ibu ion.
The analysis o sho - ange o de pa ame e s showed ha he solu e clus e was domina ed by
Mg-Si co-clus e s in Al-Mg-Si alloys, while Mg-Ag co-clus e s o med p io o he Mg-Si-Ag
clus e s when he Al-Mg-Si-Ag alloy is quenched om high empe a u e. Expe imen al
obse a ions by means o ansmission elec on mic osocopy con i med he exis ence o such
clus e s. The addi ion o Ag also modi ied he c ys al s uc u e o Guinie -P es on zones, whose
composi ion was Al1-zAgz, Al1-x-zMgxAgz and MgxAgz. In con as , he c ys al s uc u e o
me as able β″ and s able β phases was no in luenced by he p esence o Ag. Phase diag am
calcula ions o he Al-Mg-Si-Ag sys em e ealed he exis ence o a cc single-phase in Al-Ag,
Al-Mg, Ag-Mg bina y and Al-Ag-Mg e na y sys ems. In he Al- ich egion o he qua e na y
sys em, wo-phase egions cc + β and cc + β′/β′Ag we e obse ed. As empe a u e inc eases,
he cc+β'/β'Ag wo-phase egion g adually ans o ms in o he cc+β wo-phase egion. These
esul s e eal he he modynamic s abili y o me as able p ecipi a e and, in ce ain cases,
elucida e composi ional changes in p ecipi a es du ing he aging p ocess.
Keywo ds: Al-Mg-Si-Ag alloys; Fi s -p inciples calcula ions; Mon e Ca lo simula ions;
Calphad; T ansmission elec on mic oscopy
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1. In oduc ion
Al-Mg-Si alloys a e widely used in cons uc ion, au omo i e enginee ing, and a ia ion
indus ies due o hei good co osion esis ance, excellen o mabili y and weldabili y [1,2].
The mechanical p ope ies o Al-Mg-Si alloys a e con olled by he ype, dimension, and
dis ibu ion o nano-sized p ecipi a es ha o m du ing aging a di e en empe a u es and
imes [3,4]. The e o e, i is use ul o s udy he p ecipi a e e olu ion o Al-Mg-Si alloys o gain
insigh in o hei age-ha dening esponse.
The p ecipi a ion sequence in he Al-Mg-Si alloys has been epo ed as [5,6]: supe sa u a ed
solid solu ion (SSSS) → Mg-Si co-clus e s → GP zones →β″ → β' → β(Mg2Si). The main
ha dening phase in he peak ha dness condi ion is he needle-like β", which has a monoclinic
uni cell wi h a=15.16 Å, b=4.05 Å, c=6.74 Å and β=105.3º [7]. The chemical composi ion o
β″ was ini ially de e mined o be Mg5Si6 using high- esolu ion ansmission elec on
mic oscopy (HRTEM) and elec on di ac ion (ED) [8]. Subsequen ly, Nini e p o ided he
a omis ic insigh in o he β″ phase by scanning ansmission elec on mic oscopy wi h high-
esolu ion high-angle annula da k ield (HAADF-STEM), and p oposed ha he mos likely
composi ion o β″ was Al3Mg4Si4 [9]. Ano he possible composi ion o β″ epo ed was
Al2Mg5Si4 [10]. Hence, he composi ion o β" is s ill a subjec o con o e sy. β′ p ecipi a es
o m a e β″ p ecipi a es in he ageing sequence. The β(Mg2Si) phase wi h an i- luo i e (CaF2)
s uc u e is o med a equilib ium [5].
Ea ly in es iga ions ha e demons a ed ha he s eng h o Al-Mg-Si alloys was enhanced a e
aging [11], and ha conside able p ecipi a ion s eng hening can be achie ed by adding
mic oalloying elemen s [12,13]. Ding e al. [14] indica ed ha he addi ion o Cu supp esses
he p ecipi a ion o β″ and esul s in he o ma ion o qua e na y phases (QP) du ing aging.
The e o e, he p ecipi a ion sequence o Al-Mg-Si alloys wi h Cu is: SSSS → a omic clus e s
→ GP zones → β″, QP1, QP2, C (Mg4Al1Si3+xCu1-x, x∼0.3) → Q′ (Al3Cu2Mg9Si7), QP2, C→
Q (Al4Cu2Mg8Si7), Si. Ding e al. [15] ound ha ela i ely high concen a ions (3 w .%) o
Zn could enhance he age ha dening esponse o Al-0.99Mg-0.54Si (w . %) alloy by o ming
GP (II)-zones o η-MgZn2 and i s p ecu so .
Compa ed wi h Zn and Cu, Ag no only signi ican ly imp o es he age-ha dening esponse and
ha dening kine ics o Al-Mg-Si alloys du ing a i icial aging, bu also ensu es ela i ely low
s eng h in he na u al aging, he eby p o iding a good balance be ween o mabili y and bake-
ha dening po en ial [16]. Ag a oms could en e Mg-Si clus e s and e ine he dis ibu ion o
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clus e s du ing he ea ly aging, which esul ed in a ine and dense dis ibu ion o β″ o imp o e
he peak ha dness o hese alloys [17]. Ma ioa a e al. [18] obse ed h ough HAADF-STEM
ha he addi ion o Ag ans o ms he s uc u e o β
′
om Mg9Si5 o Al3Mg3Si2Ag. Mo e
ecen ly, Weng e al. [19] ound ha ano he qua e na y phase QAg′ Al3Mg3SiAg2 s uc u e may
also be o med, and he p ecipi a e sequence was: SSSS → a omic clus e s → GP zones → β″
→ β′, β′Ag1, β′Ag2, QAg′→ β, Ag.
Al hough many s udies o he p ecipi a e phases ha e been epo ed [18-21], comple e c ys al
s uc u e in o ma ion o hese phases is no always a ailable o unambiguous. Speci ically, he
solu e clus e ing beha io was no examined, which is essen ial o unde s anding he ea ly
s ages o p ecipi a ion and plays a i al ole in de e mining he he modynamic s abili y and
s eng hening mechanisms o Al-Mg-Si(-Ag) alloys. The GP zones a e he p ecu so s and
nuclea ion co e o he main s eng hening phase β″, while he e ec o Ag on he a omic
s uc u es o GP zones and β″ phase o med a he ea ly and peak aging s age is s ill unknown.
Addi ionally, he some s udies indica ed ha s uc u e o β was no in luenced by Ag a oms
[19,22], while Zhang e al. [23] e ealed ha Ag a oms en e ed he c ys al s uc u e o β by
eplacing Si a oms in he co ne s o he uni cell. The ema kable solubili y o Ag in β (Mg2Si)
has been con i med by Udono e al. [24] and P y uliak e al. [25]. Howe e , i is e y di icul
o answe hese ques ions in he Al-Mg-Si-Ag alloys only om expe imen al obse a ions.
Cu en ly, he phase s abili y o a sys em is ypically e alua ed using he modynamic
da abases (TDBs) ob ained h ough he Calphad (calcula ion o phase diag ams) me hod [26].
Howe e , due o he limi a ion o slow kine ics a low empe a u e and incomple e
he modynamic in o ma ion in some phases [25-27], he TDBs a ailble may no be accu a e o
alloys wi hin he complex Al-Mg-Si-Ag sys em. Wi hin his amewo k, he objec i e o his
in es iga ion is o de elop a s a egy ha in eg a es i s -p inciples calcula ions and he clus e
expansion (CE) me hod coupled wi h s a is ical mechanics p inciples o cons uc he Al-Mg-
Si-Ag TDB wi hou elying on expe imen s [30,31]. To his end, he Gibbs ee ene gies o
each phase in he sys em a e ob ained as a unc ion o empe a u e and composi ion om he
o ma ion en halpies a 0 K calcula ed om densi y unc ional heo y (DFT) and
he modynamic p ope ies - o ma ion en halpies and en opy - a ini e empe a u es ob ained
om Mon e Ca lo (MC) simula ions [32,33]. These TDBs o he Al-Mg-Si-Ag sys em a e
implemen ed in o he OpenCalphad so wa e [32-34]. They allow o de e mine he s abili y
egion o key p ecipi a e phases β″, β′, β′Ag and β, as well as phase coexis ence and hei ac ions.
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The a omic s uc u e and composi ion o GP zones a e also de e mined based on his
me hodology. The simula ion esul s we e alida ed h ough STEM-HAADF. Mo eo e , he
accu a e TDB o his qua e na y alloy will acili a e he p edic ion o he modynamic
p ope ies in bina y and e na y sys ems in ol ing Al, Mg, Si and Ag a oms. I also p o ides a
obus ounda ion o ex apola ing o mo e complex Al-Mg-Si-based alloy sys ems,
o e coming he limi a ions o expe imen al app oaches due o he ex ensi e numbe o
measu emen s equi ed o explo e he en i e composi ional space.
2. Me hodology
2.1 DFT calcula ions
DFT calcula ions we e ca ied ou in he Vienna ab ini io simula ion package (VASP) [37],
using he p ojec o augmen ed wa e (PAW) me hod [38]. Exchange-co ela ion is ea ed in he
gene alized g adien app oxima ion (GGA) o Pe dew-Bu ke-E nze ho (PBE) [39]. The
elec onic con igu a ions conside ed a e 3s23p1 o Al, 2p63s2 o Mg, 3s23p2 o Si, and 4d105s1
o Ag, espec i ely. Con e gence es s indica ed ha a cu o o 400 eV is su icien o ensu e
o al ene gy di e ences a e less han 1 meV/a om. The Γ-cen e ed Monkho s -Pack mesh [40]
o 𝑘-poin s in he B illouin zone, wi h he k-mesh spacing o 0.02 Å-1, co esponding o a
12×12×12 𝑘-poin mesh o a cc con en ional uni cell. The o al ene gy con e gence c i e ion
is se o 10-5 eV/cell, and o ce componen s a e elaxed o 10-3 eV/Å. All s uc u es a e ully
elaxed conce ning olume as well as cell-in e nal and -ex e nal coo dina es.
2.2 Clus e expansion
The o ma ion en halpies, H o m, o a se o AlxMgySizAg1-x-y-z con igu a ions ha comp ise he
whole composi ion ange o he sys em a e calcula ed 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)
whe e x, y and z a e a omic ac ions o Al, Si and Mg in a gi en con igu a ion, espec i ely.
𝐸
Al
x
Mg
y
Si
z
Ag
1
―
x
―
y
―
z
𝑡𝑜𝑡
is he o al ene gy pe a om o AlxMgySizAg1-x-y-z a e elaxa ion,
𝐸
𝐴𝑙
𝑓𝑐𝑐
,
𝐸
𝐴𝑔
𝑓𝑐𝑐
,
𝐸
𝑆𝑖
𝑑𝑖𝑎𝑚𝑜𝑛𝑑
, and
𝐸
𝑀𝑔
ℎ𝑐𝑝
a e he ene gies o he s able phases o Al and Ag wi h cc la ice, Si wi h
diamond la ice, and Mg wi h hcp la ice, espec i ely.
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In he CE o malism [40-42], he o ma ion en halpy can be pa ame ized as a polynomial in
he occupa ional a iables σ:
𝐻
𝑓𝑜𝑟𝑚
(
𝜎
)
=
∑
𝜔,𝑛,𝑠
𝐽
(𝑠)
𝜔,𝑛
𝑚
(𝑠)
𝜔,𝑛
〈
𝜑
(
𝜎
)
〉
(𝑠)
𝜔,𝑛
(2)
whe e
𝜎
=
{
𝜎
1
,
𝜎
2
, …,
𝜎
𝑁
}
is he ec o o occupa ion a iables ha de ine he a omic species in
la ice si e i in a qua e na y sys em (
𝜎
𝑖
=
0, 1, 2, 3
) wi h 0 o Al, 1 o Mg, 2 o Si and 3 o
Ag, ollowing he em hodology p e iously applied o he qua e na y Fe-C -Mn-Ni sys em [45].
The summa ion is pe o med o e all he clus e s, dis inc unde symme y ope a ions in he
s udied la ice, ep esen ed by he ollowing pa ame e s:
𝜔
and
𝑛
a e he clus e size ( he numbe
o la ice poin s in he clus e ) and i s label ( ela ed o he maximal dis ance be ween wo a oms
in he clus e ), espec i ely. (
𝑠
) is he deco a ion o clus e by a poin unc ion
𝛾
𝑗,𝐾
(
𝜎
𝑖
)
.
𝐽
(𝑠)
𝜔,𝑛
ep esen s he e ec i e clus e in e ac ion (ECI) coe icien co esponding o he same (
𝑠
)
deco a ed clus e .
𝑚
(𝑠)
𝜔,𝑛
deno es he si e mul iplici y o he deco a ed clus e s.
〈
𝜑
(
𝜎
)
〉
𝜔,𝑛,𝑠
is he
clus e basis unc ion, which is exp essed as a p oduc o o hono mal poin unc ions
𝛾
𝑗,𝐾
(
𝜎
𝑖
)
o e all si es included in he speci ic clus e desc ibed 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)
whe e
𝑗
𝑖
=
(0, 1, 2, 3)
has a simila meaning o
𝜎
𝑖
, indica ing which ype o a om is loca ed on
he la ice si e i ha belongs o he clus e .
K
is he numbe o alloy componen s, which is equal
o 4 in ou sys em. The clus e basis unc ions o each pai o clus e s, α and β, should sa is y
〈
𝜑
(
𝜎
)
(𝑠)
α
,
𝜑
(
𝜎
)
(𝑠)
β
〉
=
0
i hey a e di e en and
〈
𝜑
(
𝜎
)
(𝑠)
α
,
𝜑
(
𝜎
)
(𝑠)
β
〉
=
1
i hey a e iden ical. This
is achie ed by
𝛾
𝑗,𝐾
(
𝜎
𝑖
)
as [34,44]:
𝛾
𝑗,𝐾
(
𝜎
𝑖
)
=
{
1
i
𝑗
=
0,
―
cos
(
2𝜋
[
𝑗
𝑖
2
]
𝜎
𝑖
𝐾
)
i
𝑗
>
0
and odd,
―
sin
(
2𝜋
[
𝑗
𝑖
2
]
𝜎
𝑖
𝐾
)
i
𝑗
>
0
and e en
(4)
whe e
[
𝑗
𝑖
2
]
deno es he ceiling unc ion - ounding up o he closes in ege .
The op imal alues o ECIs a e calcula ed o minimize he C oss-Valida ion (CV) Sco e
be ween
𝐻
DFT
𝑓𝑜𝑟𝑚
and
𝐻
CE
𝑓𝑜𝑟𝑚
h ough he s uc u e in e sion me hod, as implemen ed in he
ATAT package [43]:
𝐶𝑉
=
∑
𝑁
𝑖
=
1
(
𝐸
DFT
𝑖
―
𝐸
CE
′
𝑖
)
2
𝑁
(5)
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Whe e
𝐸
CE
′
𝑖
is he p edic ed ene gy o i h s uc u e by i ing CE ene gies om DFT, and
excluding he i h s uc u e.
2.3 Mon e Ca lo simula ions
A ini e empe a u es, he s abili y o he di e en phases may change signi ican ly due o
en opic con ibu ions [46,47]. In his wo k, we mainly ocus on con igu a ional en opy as he
s onges con ibu ion o he Gibbs ee ene gy o o ma ion,
𝐺
𝑓𝑜𝑟𝑚
.
𝐺
𝑓𝑜𝑟𝑚
o a gi en
composi ion a ini e empe a u e can be calcula ed by pe o ming a canonical MC simula ion
- he chemical composi ion and numbe o a oms a e ixed - wi hin he CE o malism o each
la ice:
𝐺
𝑀𝐶
𝑓𝑜𝑟𝑚
(
𝜎
,𝑇)
=
𝐻
𝑓𝑜𝑟𝑚
(
𝜎
,𝑇)
―
𝑇
𝑆
𝑐𝑜𝑛𝑓
(
𝜎
,𝑇)
(6)
whe e o ma ion en halpy
𝐻
𝑓𝑜𝑟𝑚
(
𝜎
,𝑇)
is calcula ed om eq. (2). The con igu a ional en opy
𝑆
𝑐𝑜𝑛𝑓
(
𝜎
,𝑇)
pe si e is calcula ed om he speci ic hea
𝐶(
𝜎
,𝑇′)
by he modynamic in eg a ion
as [48]
𝑆
𝑐𝑜𝑛𝑓
(
𝜎
,𝑇)
=
∫
𝑇
0
𝐶(
𝜎
,𝑇′)
𝑇′
𝑑𝑇′
(7)
whe e
𝐶(
𝜎
,𝑇′)
can be ob ained om he luc ua ions o he o ma ion en halpy in MC
simula ions a empe a u e
𝑇
′, by using he exp ession [48]:
𝐶(
𝜎
,𝑇′)
=
〈
𝐻
2
𝑓𝑜𝑟𝑚
(
𝜎
,𝑇′
)
〉
―
〈
𝐻
𝑓𝑜𝑟𝑚
(
𝜎
,𝑇′
)
〉
2
𝑇′
2
(8)
whe e
〈
𝐻
𝑓𝑜𝑟𝑚
(
𝜎
,𝑇′
)
〉
and
〈
𝐻
2
𝑓𝑜𝑟𝑚
(
𝜎
,𝑇′
)
〉
a e he mean and mean-squa ed a e age o ma ion
en halpies, espec i ely, which desc ibe he a iance o
𝐻
𝑓𝑜𝑟𝑚
(
𝜎
,𝑇′
)
. They a e compu ed by
a e aging all he MC s eps a he accumula ion s age o a gi en empe a u e, as in [48].
Canonical MC simula ions we e pe o med using he ATAT package sepa a ely o cc, CaF2
(β), β′ and
𝛽
'
𝐴𝑔
la ices, because hese phases we e ound in he Al-Mg-Si-Ag alloys [19].
Supe cells we e gene a ed a 1 a . % in e als o e he whole composi ion ange using he
special quasi- andom s uc u e (SQS) me hod [49], comp ising 10
×
10
×
10 supe cells wi h
4000 a oms o cc, 5
×
5
×
5 supe cells wi h 1500 a oms o β, 5
×
3
×
2 supe cells wi h
840 a oms o β′, and 5
×
3
×
6 supe cells wi h 810 a oms o
β
'
Ag
. Fo each composi ion,
MC simula ions a e pe o med s a ing om a high- empe a u e diso de ed s a e a
𝑇
= 1000
K. The alloy is hen cooled down o 10 K wi hin he empe a u e s ep o 𝛥𝑇 = 20 K. A each
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empe a u e, a MC simula ion includes 3000 passes o equilib ium and he modynamic
a e ages.
The phase bounda y in bina y sys ems can be de e mined by he common angen o Gibbs ee
ene gy o o ma ion
𝐺
𝑓𝑜𝑟𝑚
, and hen he ac ion o he conside ed phases as unc ions o
concen a ion can be calcula ed by using he le e ule [50,51]. In he case o mul icomponen
Al-Mg-Si-Ag alloys, he calcula ion o ac ions o bo h phases in a ou -dimensional
composi ion space canno be achie ed by he applica ion o one common angen [34]. Ins ead,
he well-es ablished Calphad me hod p o ides obus algo i hms o calcula ing mul i-
componen equilib ia, which a e essen ial o e alua ing he co esponding phase ac ions
[35,52].
2.4 De elopmen o he modynamic da abases
The Gibbs ee ene gy o o ma ion o a solid solu ion phase wi hin he amewo k o Calphad
can be exp essed as [53]
𝐺
𝑓𝑜𝑟𝑚
=
𝐺
𝑟𝑒𝑓
+
𝐺
𝑖𝑑𝑒𝑎𝑙
+
𝐺
𝑥𝑠
(9)
whe e
𝐺
𝑟𝑒𝑓
=
∑
4
𝑖
=
1
𝑥
𝑖
0
𝐺
𝑖
is he e e ence ene gy o pu e elemen s (Al, Mg, Si and Ag) o he
s able phase.
𝐺
𝑖𝑑𝑒𝑎𝑙
=
―
𝑇
𝑆
𝑖𝑑𝑒𝑎𝑙
desc ibes he con ibu ion o he Gibbs ee ene gy om ideal
andom mixing o he cons i uen s on he c ys al la ice.
𝑆
𝑖𝑑𝑒𝑎𝑙
=
―
𝑅
∑
4
𝑖
=
1
𝑥
𝑖
ln
𝑥
𝑖
, whe e R is he
gas cons an , and
𝑥
𝑖
he mole ac ion o componen i.
𝐺
𝑥𝑠
is he excess Gibbs ee ene gy o o ma ion, desc ibing he in luence o non-ideal mixing
beha iou on he he modynamic p ope ies o a solu ion phase.
𝐺
𝑥𝑠
is usually desc ibed by he
semi-empi ical Redlich-Kis e (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)
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𝑢
𝑘
=
𝑥
𝑘
+
1
―
𝑥
𝑖
―
𝑥
𝑗
―
𝑥
𝑘
3
(13)
whe e
𝑥𝑠2
𝐺
and
𝑥𝑠3
𝐺
a e bina y and e na y excess e ms, espec i ely. L is he in e ac ion
pa ame e be ween componen s and
𝑣
is he maximum powe used in he op imiza ion.
I should be no ed ha complex phases a e usually modelled using he compound ene gy
o malism (CEF) in he Calphad me hodology. Thus, β′ and β′Ag phases in Al-Mg-Si-Ag alloys
a e desc ibed by (Al,Mg)18(Si,Ag)10 and (Al,Mg)6(Si,Ag)3, espec i ely. The e o e,
𝐺
𝑟𝑒𝑓
is
exp essed 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)
whe e y is deno ed as si e ac ion and de ined as he composi ion o each cons i uen in he
subla ice s. The
𝐺
𝑠:𝑡
𝑖,𝑗
ep esen s he Gibbs ee ene gy o o ma ion o compounds whe e each
subla ice is occupied by he same cons i uen , such as Al18Si10 and Al6Si3.
In his wo k, TDBs based only on he esul s o MC simula ions ob ained using he DFT-based
CE models we e gene a ed o cc, β, β′ and
β
'
Ag
in he Al-Mg-Si-Ag alloys using sqs2 db ool
in ATAT package [56]. The e o e, wo impo an de ails ha e o be poin ed ou in he i ing:
(1) empe a u e dependence o
𝐺
𝑓𝑜𝑟𝑚
is con ained in he
𝐺
𝑟𝑒𝑓
supplied om SGTE da abases in
he adi ional Calphad model [57], which con adic s ou objec i e o c ea e TDBs jus om
DFT simula ions, so he SGTE da abase has no been used. (2) Excess Gibbs ee ene gy o
o ma ion
𝐺
𝑥𝑠
does no include qua e na y in e ac ions (
𝑥𝑠4
𝐺
) in he sqs2 db i ing because
hey would no con ibu e o he desc ip ion o he Gibbs ee ene gies o he phases [58]. Finally,
he TDBs ob ained om he MC simula ions we e used o calcula e he phase equilib ium
diag am using OpenCalphad a each empe a u e using he co esponding TDB [34].
2.5 Expe imen s and mic os uc u al obse a ion
In o de o e i y he ac ual phase an o ma ions, an ingo wi h chemical composi ion Al
0.89Mg 0.96Si 0.15Ag (a . %) was p epa ed by mould cas ing om pu e elemen s (> 99.9%
pu i y). A e homogeniza ion and ec ys alliza ion a 520 °C, bille s we e hea ed up o 520°C
du ing one hou ollowed by wa e quenching. A e wa ds, hey we e a i icially aged a 160ºC
in a u nace. The esul s o hea ea men we e e alua ed on he basis o ha dness measu emen s.
Th ee samples, aged du ing 0.5h, 2h and 12h we e selec ed as ep esen a i e o he p ecipi a ion
e olu ion o suppo calcula ions in his s udy. They s and o he ea ly s ages (clus e ing), he
beginning o ha dness s abiliza ion and he peak ha dness, espec i ely.
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Thin oils o mic oscopic e alua ion we e p epa ed wi h con en ion elec ochemical polishing
o 3 mm discs. A solu ion o ni ic acid in me hanol has been used o his pu pose. STEM-
HAADF obse a ion (collec ion angle ange o 45-200 m da) was ca ied ou in Cs-co ec ed
The moFishe Scien i ic Spec a 200 mic oscope. Beam cu en below 25 pA has been kep
du ing he obse a ions and ene gy dispe si e spec oscopic analysis in o de o main ain he
s abili y o clus e s and p ecipi a es.
3. Resul s
3.1 Phases o in e es
The s able and me as able phases in he Al-Mg-Si-Ag alloys - acco ding o he in o ma ion in
he li e a u e[5,8,19,59] - a e lis ed in Table. 1. Thei c ys al s uc u es a e depic ed in Fig. 1.
Based on ou p e ious expe ience [60,61], some phases wi h complex s uc u e can be ea ed
as a dis o ed con igu a ion on a simple la ice. Al has a cc la ice (Fig.1(a)). The β″ (Mg5Si6)
(Fig. 1(b)) also p esen s a cc la ice, whe e he la ice ec o bβ″ o β″ is pa allel o he [001]
axis o α-Al, and he o he wo la ice ec o s o β″ a e de ined by aβ″=2aAl+3bAl ( he [230]
di ec ion in α-Al) and cβ″=-
3
2
aAl+
1
2
Al ( he [3
1
0] di ec ion in α-Al). Howe e , he c ys al
s uc u e o β″ is no uniquely de e mined om Re s. [8-10], which may con ain a ce ain
ac ion o Al. β′ (Mg9Si5) (Fig.1(c)) has a hexagonal uni cell, and he e a e mul iple o ien a ion
ela ionships be ween α-Al and β' [62]. β'Ag1 ((Fig.1d)) and β'Ag2 ((Fig.1( )) ha e a hexagonal
uni cell ha is simila o ha o β' phase, while hei la ice pa ame e along he a-axis is sligh ly
di e en om he co esponding di ec ion in β'. β (Mg2Si) has a CaF2 s uc u e. I is wo h
no ing ha all phases, excep o he equilib ium phases β-Mg2Si and Si, a e me as able.
Table 1. S uc u al in o ma ion o he di e en phases in he Al-Mg-Si-Ag alloys.
Composi ion
Symme y
La ice pa ame e s (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
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Fig. 6 (a) Fo ma ion en halpy pe solu e a om,
∆
𝐻
𝑠𝑜𝑙𝑢𝑡𝑒
𝑓𝑜𝑟𝑚
, calcula ed om eq. (15) as a unc ion
o as a unc ion o
𝑥
𝑀𝑔
+
𝑦
𝑆𝑖
+
𝑧
𝐴𝑔
. (b)
∆
𝐻
𝑠𝑜𝑙𝑢𝑡𝑒
𝑓𝑜𝑟𝑚
as a unc ion o as a unc ion o
𝑧
𝐴𝑔
o
con igu a ions wi h
𝑥
𝑀𝑔
+
𝑦
𝑆𝑖
+
𝑧
𝐴𝑔
=
1
.
3.5 Phase s abili y a ele a ed empe a u es
3.5.1 Compa ison wi h MC and MC- i ed TDB
The CE o each la ice s uc u e can be used o p edic he he modynamic p ope ies o he
co esponding s uc u es wi h his la ice, ollowing he me hodology p esen ed in Sec ion 2.3.
The composi ional g id o he Al-Mg-Si-Ag qua e na y sys em o cc and β la ice was
sys ema ically explo ed using an in e al o 10 a .% o each one o he cons i uen s. The β′ and
β′Ag only exis in a small composi ional ange, so he in e al o 3.5 a .% o each componen is
used. Gibbs ee ene gies o o ma ion G o m o each composi ion we e calcula ed as indica ed
in Sec ion 2.3. In he end, G o m as a unc ion o composi ion a ele a ed empe a u es was i ed
as indica ed in Sec ion 2.4 o de elop TDBs o Al-Mg-Si-Ag alloys. The G o m a 500 K ob ained
using MC- i ed TDB and hose calcula ed di ec ly om MC simula ions we e i s compa ed
in he six bina y sys ems unde lying cc and β la ice o e i y he accu acy o TDBs, as shown
in Fig. 7. The G o m ob ained using MC- i ed TDB is in good ag eemen wi h hose calcula ed
om MC simula ions in he six bina y sys ems. The cc phase is mo e s able han β in he whole
composi ion ange o Ag-Al, Ag-Mg and Al-Mg bina y sys ems. The s abili y o he phase
changed be ween cc and β phase in he whole composi ion ange o Ag-Si, Al-Si and Mg-Si
bina y sys ems. Such as he cc phase is mo e s able han he β phase in he Ag- ich Ag-Si and
Al- ich Al-Si. The β phase is mo e s able han he cc phase in he Si composi ion la ge han
0.2 o he Mg-Si bina y sys em.
Addi ionally, he G o m a 500 K ob ained om MC- i ed TDB and MC simula ions in he
pseudo-bina y sys ems, such as he composi ion o Al and Mg a e ixed, a e also compa ed and
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shown in Fig. 8. I should be no ed ha only G o m ob ained using MC- i ed TDB o cc and β
la ice a e depic ed in Fig. 8. The β′ and β′Ag phases can co-exis wi h he cc phase in his
composi ion ange. I should be no ed ha he e a e some di e ences be ween MC and MC-
i ed TDB o β′Ag la ice, which may be imp o ed by pe o ming MC simula ions on a dense
composi ional g id. In ac , he MC- i ed TDB emains sui able o calcula ing he G o m o his
la ice. This is suppo ed by he low CV sco e o 0.0061 eV/a om, calcula ed based on he G o m
om MC and MC- i ed TDB. The G o m a 300 K and 800 K ob ained om he wo app oaches
can be ound in Figs. S1-S4 in he Supplemen a y Ma e ial. Excep o a small di e ence in he
G o m o β′Ag la ice, he G o m o cc, β and β′ la ices ob ained using MC- i ed TDB a e in good
ag eemen wi h hose calcula ed di ec ly om MC simula ions. The TDBs o he empe a u e
ange om 100 K o 1000 K we e cons uc ed using he Calphad me hod based on he G o m
ob ained om he MC simula ion. These da abases a e p o ided in he Supplemen a y Ma e ial
as a comp essed ile named TDB.zip, which includes indi idual TDB iles o each empe a u e
in e al: Al-Mg-Si-Ag-100K.TDB, Al-Mg-Si-Ag-200K.TDB, ... , up o Al-Mg-Si-Ag-
1000K.TDB.
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Fig. 7. The compa ison o Gibbs ee ene gies o o ma ion G o m calcula ed om MC
simula ions and he OpenCalphad calcula ions using TDB based on MC simula ions a 500 K
o (a) Ag-Al. (b) Ag-Mg. (c) Ag-Si. (d) Al-Mg. (e) Al-Si and ( ) Mg-Si bina y sys ems
unde lying cc and β la ice. G een and o ange ci cles ep esen he G o m wi h cc and β la ice
calcula ed om MC simula ions, espec i ely. G een and o ange lines ep esen he G o m wi h
cc and β la ice calcula ed using TDB, espec i ely.
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Fig. 8. The compa ison o Gibbs ee ene gies o o ma ion G o m calcula ed om MC
simula ions and he OpenCalphad calcula ions using TDB based on MC simula ions a 500 K
o (a) Ag0.339-xAl0.0357Mg0.307Six and (b) Ag0.296-xAl0.037Mg0.63Six pseudo-bina y sys ems
unde lying cc, β, β' and β'Ag la ice. G een and o ange lines ep esen he G o m wi h cc and β
la ice calcula ed using TDB, espec i ely. Pu ple ci cles and lines ep esen he G o m wi h β' o
β'Ag la ice calcula ed om MC simula ions and he TDB, espec i ely.
3.5.2 cc, β, β' and β'Ag phase dis ibu ion
The me hodology based on he de elopmen o TDBs and hei implemen a ions in he
OpenCalphad calcula ions, as desc ibed in Sec ion 2.4, was used o in es iga e he dis ibu ion
o di e en phases and hei ac ions in Al-Mg-Si-Ag alloys. As men ioned in Sec ion 2.3, he
phase composi ion is de e mined using he common angen o Gibbs ee ene gy o o ma ion
G o m, and he phase ac ions a e subsequen ly calcula ed using he le e ule.
The pe cen age o cc phase ac oss he whole composi ion ange a 300 K is plo ed in Fig. 9(a),
while he co esponding phase dis ibu ion is depic ed in Fig. 9(b). The phase composi ion is
plo ed in di e en colo s, each indica ing he s abili y egion o he ou di e en phases. I is
well known ha he s able s uc u e o Si and Mg is diamond and hcp la ice, hus hey a e only
loca ed on he e ices o he e ahed on and plo ed in g ey and pink ci cles in Fig. 9(b), (d)
and ( ). Fu he mo e, he β' and β'Ag phases a e no analyzed sepa a ely in his wo k because
hey always co-exis a he o e -aging s age [19]. I can also be seen ha he Gibbs ee ene gy
o o ma ion G o m o hem wi h simila composi ion is e y close. In he end, only he cc solid
solu ion ( he cc phase pe cen age is 100%) appea ed in he Al-Mg, Ag-Mg, Al-Ag bina y
sys ems, and one Al-Mg-Ag e na y sys em. The same esul s a e also obse ed a 500 K (Fig.
9(c)) and 800 K (Fig. 9(c)), indica ing ha he o de ing s a es a e easily o med among Al, Mg
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and Ag a oms. The pe cen age o cc phase dec eased wi h empe a u e om 800 K (Fig. 9(e))
o 300 K (Fig. 9(e)), indica ing some o he phases p ecipi a ed om he Al ma ix. I can be
seen om he phase dis ibu ion ha he single β phase egion is obse ed in he Si- ich Ag-
Mg-Si and Al-Mg-Si e na y sys ems a 500 K (Fig. 9(d)) and 800 K (Fig. 9( )). The β phase
coexis s wi h cc and β' o β'Ag phase in he Al-Mg-Si e na y sys em a 300 K (Fig. 9(b)).
Howe e , i is no easy o analyse he phase change in he in e io om a ou -dimensional
phase diag am, so a mo e de ailed analysis will be desc ibed in Sec ion 3.5.3.
Fig. 9 Pe cen age o cc phase and di e en phase dis ibu ion ob ained om OpenCalphad
calcula ions a 300 K (a), (b), 500 K (c), (b) and 800 K (e), ( ).
3.5.3 Phase s abili y in he Al-Mg-Si-Ag alloys
The cc ac ions and phase dis ibu ion o Al-Mg-Si e na y alloys -compu ed using he
OpenCalphad calcula ions based on MC- i ed TDB as desc ibed in Sec ion 2.4- a e shown in
Fig. 10. The e is a high cc ac ion (>70 %) in he Al- ich pa o he Al-Mg-Si phase diag am,
whe he a 300 K (Fig. 10(a)), 500 K (Fig. 10(c)) o 800 K (Fig. 10(e)). In pa allel, a low cc
ac ion (< 20%) is also obse ed in he Si- ich pa o he Al-Mg-Si phase diag am om h ee
di e en empe a u es (Fig. 10(a), (c) and (e)). The phase composi ions o he s abili y egion
a 300 K, 500K and 800 K a e plo ed in Fig. 10(b), (d) and ( ), espec i ely. A cc+β wo-phase
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egion plo ed in g een colo is ound in he Mg- ich pa a 300 K (Fig. 10(b)). I can be seen
ha he e is a high cc ac ion (>60%) in his wo-phase egion, and he cc ac ion has a sligh
change as empe a u e inc eases o 800 K (Fig. 10(e)), indica ing ha he cc phase emains
he modynamically s able o e his empe a u e ange. The s able β+β' wo-phase egion is
plo ed in pu ple colo , and he cc+β' wo-phase egion is plo ed in ed colo . The e is a
cc+β+β' h ee-phase egion ha is plo ed in b own colo . As he empe a u e inc eases, hese
egions g adually ans o m in o a wo-phase egion whe e cc and β coexis .
To igu e ou he e ec o Ag on he phase s abili y, he cc ac ions and phase dis ibu ion o
(AlMgSi)90Ag10 composi ion a e analyzed and also compa ed wi h Al-Mg-Si in Fig. 11. The
s able wo-phase egions changed om cc+β' (o β+β') o cc+β'/β'Ag (o β+β'/β'Ag) when Ag
is added in o Al-Mg-Si alloys, indica ing ha he addi ion o Ag imp o es he s abili y o β'
phase. As empe a u e inc eases om 300 K (Fig. 11(a), (c)) o 500 K (Fig. 11(b), (d)), he
cc+β'/β'Ag wo-phase egions in he Mg- ich pa g adually ans o m in o he cc+β wo-phase
egion, while he s able β+β'/β'Ag wo-phase egions in he Si- ich pa comple ely ans o m
in o he cc+β wo-phase egion and a small single β phase egion. As empe a u e inc eases o
800 K (Fig. 11(e), ( )), excep o he cc+β'/β'Ag wo-phase egion in he Al- ich g adually
ans o ms in o he cc+β wo-phase egion, he cc+β+β'/β'Ag h ee-phase egion also pa ly
con e s in o he cc+β wo-phase egion. Addi ionally, only he cc single-phase egion (blue
colo ) appea ed in he Al-Mg-Ag ena y sys em a h ee di e en empe a u es (Fig. 11(b), (d)
and ( )), indica ing ha Al-Mg-Ag is mo e likely o o m a omic clus e s in he Al-Mg-Si-Ag
alloys. A signi ican di e ence in he cc ac ion was obse ed be ween Fig.10(a) and Fig.
11(a), pa icula ly in he Mg- ich pa . I can be seen ha he e is a cc+β wo-phase egion in
he Mg- ich pa in Fig.11(b), while he s able wo-phase egion is cc+β'/β'Ag in Fig. 11(b). This
indica es ha β' o β'Ag is mo e s able han β in his composi ion ange. The cc ac ions and
phase dis ibu ion o (AlMgSi)70Ag30 can be ound in Fig. S5 in he supplemen a y Ma e ials.
The s able phase egion in his composi ion ange a 300 K (Fig. S5(a), (b)) is composed o
cc+β+β'/β'Ag h ee-phase egion and cc+β'/β'Ag wo-phase egion. As empe a u e inc eases o
500 K (Fig. S5(b), (c)), hey a e g adually ans o med in o he cc+β wo-phase egion. As
empe a u e inc eases o 800 K (Fig. S5(e), ( )), he s able phase egion is domina ed by he
cc+β wo-phase egion, and he e is a single cc phase egion obse ed in he Al- ich pa .
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Fig. 10. Fcc ac ion and phase dis ibu ion in he Al-Mg-Si e na y phase diag am a 300 K (a),
(b), 500 K (c), (d) and 800 K (e), ( ).
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Fig. 11. Fcc ac ion and phase dis ibu ion in he (AlMgSi)90Ag10 pseudo- e na y phase
diag am a 300 K (a), (b), 500 K (c), (d) and 800 K (e), ( ).
4. Discussion
4.1 E ec o Ag on he solu e clus e s
I is well known om he p ecipi a ion sequence o Al-Mg-Si-based alloys ha solu e clus e s
we e i s p ecipi a ed om SSSS. These clus e s se e as p ecu so s o in luence he a i icial
ageing esponse and go e n he ans o ma ion pa hways o he p ecipi a ion sequence.
Howe e , in con as o he de ailed knowledge o me as able p ecipi a es - s uc u es, e olu ion,
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and hei in luence on ma e ial p ope ies -, he knowledge abou he a omic clus e s ha
p eceded hem is e y limi ed. The e o e, o igu e ou he e ec o Ag on he solu e clus e s in
he Al-Mg-Si alloys, a cc solid solu ion wi h he composi ion Al98Mg1Si1 (a .%) - same as
expe imen s - was gene a ed using he SQS me hod and hen used o pe o m MC simula ions,
as desc ibed in Sec ion 2.3. In pa allel, a se ies o cc solid solu ion models wi h a ying Ag
composi ion, anging om 0.1 a . % o 1 a . % in inc emen s o 0.1 a . %, we e also gene a ed
o pe o m MC simula ions. These models we e used o compa e he clus e ing beha io s wi h
hose o he Ag- ee cc solid solu ion.
Fo ma ion en halpies
𝐻
𝑀𝐶
𝑓𝑜𝑟𝑚
o each solid solu ion model as a unc ion o empe a u e we e
calcula ed using eq. (2), as shown in Fig. 12 (a).
𝐻
𝑀𝐶
𝑓𝑜𝑟𝑚
is a cons an o Ag- ee cc solid
solu ion abo e 180 K, indica ing ha i is a comple ely diso de ed s a e.
𝐻
𝑀𝐶
𝑓𝑜𝑟𝑚
dec eases
sha ply below 180 K, and i migh o m local chemical clus e ing and sho - ange o de ing
because he e a e di e ences in he a omic bonding be ween elemen s unde he en halpy e ec .
As Ag concen a ion inc eases, he empe a u e a which he
𝐻
𝑀𝐶
𝑓𝑜𝑟𝑚
s a s o signi ican ly
dec ease shi s o a highe empe a u es. Fo ins ance,
𝐻
𝑀𝐶
𝑓𝑜𝑟𝑚
o Al97.9Mg1Si1Ag0.1 s a s o
dec ease signi ican ly a 230 K, Al97Mg1Si1Ag0.5 a 420 K and Al97Mg1Si1Ag1 a 480 K. The
a omic model whe e
𝐻
𝑀𝐶
𝑓𝑜𝑟𝑚
d ops sha ply is d awn in he Fig. 12(b)-(e). Al a oms a e no
isualized in hese igu es o shown mo e clea ly he clus e ing o solu e a oms. A Mg-Si co-
clus e is o med in he Al98Mg1Si1 model (Fig. 12(b)). Mg-Si co-clus e s ha e been
obse ed in he ea ly s age in he Al-Mg-Si alloy by h ee-dimensional a om p obe (3DAP) and
TEM [6]. The clus e ype changed by adding Ag o Al-Mg-Si alloys, e en i he amoun o Ag
is e y small, and he Mg-Si-Ag co-clus e s a e o med in Al97.9Mg1Si1Ag0.1 (Fig. 12(c)). As
he Ag concen a ion inc eases o 0.5 a . % (Fig. 12(d)), he Mg-Ag co-clus e s a e dominan a
420 K in Al97.9Mg1Si1Ag0.5, while Si a oms exis in a diso de ed s a e. Si a oms en e ed Mg-Ag
co-clus e s and Mg-Si-Ag co-clus e s a e o med when he empe a u e dec eased o 200 K, as
shown in Fig. S6(a) in he supplemen a y ma e ial. Simila clus e ing beha io is also obse ed
in Al97.9Mg1Si1Ag1 (Fig. 12(e) and Fig. S6(b)).
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Fig. 12 (a) Fo ma ion en halpies
𝐻
𝑀𝐶
𝑓𝑜𝑟𝑚
wi h a ying Ag concen a ion as a unc ion o
empe a u e calcula ed by eq. (2). Solu e clus e s in he cc solid solu ion wi h he Ag
concen a ion o 0 a .% (b), 0.1 a .% (c), 0.5 a .% (d) and 1.0 a .% (e), espec i ely.
The e ec o Cu on he clus e ing beha iou in he Al-Mg-Si alloys has been widely s udied
[74-76]. They ound ha al hough he solu e clus e was domina ed by Mg-Si clus e s al hough
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