Ci a ion: Jimenez, A.; Sanchez, J.M.;
Gi o , F.; Rende os, M.; Egizabal, P.
Nume ical Modeling o he In luence
o Nanome ic Ce amic Pa icles on
he Nuclea ion o AlSi10MnMg Alloy.
Me als 2022,12, 855. h ps://doi.o g/
10.3390/me 12050855
Academic Edi o : F ank Cze winski
Recei ed: 6 Ap il 2022
Accep ed: 13 May 2022
Published: 17 May 2022
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me als
A icle
Nume ical Modeling o he In luence o Nanome ic Ce amic
Pa icles on he Nuclea ion o AlSi10MnMg Alloy
Ane Jimenez 1,*, Jon Mikel Sanchez 1, F anck Gi o 2,3 , Ma io Rende os 2and Ped o Egizabal 1
1TECNALIA, Basque Resea ch and Technology Alliance (BRTA), Mikele egi Pasealekua, 2,
20009 Donos ia-San Sebas ián, Spain; [email p o ec ed] (J.M.S.);
[email p o ec ed] (P.E.)
2Depa men o Mechanical Enginee ing, Enginee ing School o Bizkaia, Uni e si y o he Basque
Coun y (UPV/EHU), Alameda de U quijo s/n, 48013 Bilbao, Spain; ank.gi [email p o ec ed] (F.G.);
ma ioal edo. [email p o ec ed] (M.R.)
3IKERBASQUE, Basque Founda ion o Science, 48013 Bilbao, Spain
*Co espondence: [email p o ec ed]; Tel.: +34-667-10-25-43
Abs ac :
In ecen yea s, many esea che s ha e a emp ed o model he solidi ica ion p ocess o
nano- ein o ced ma e ials. In he p esen documen , he e ec on he he e ogeneous solidi ica ion
egime o he di e en sizes, shapes, and chemical composi ions o nanome ic ce amic pa icles in an
AlSi10MnMg alloy is s udied. This a icle de elops a ma hema ical model o p edic he solidi ica ion
beha io o a gene al nano- ein o ced alloy, hen alida es he esul s using expe imen al echniques.
The main objec i e o he model is o minimize he cos ly and ime-consuming expe imen al p ocess
o ab ica ing nano- ein o ced alloys. The p oposed model p edic s he c i ical Gibbs ene gy and he
c i ical adius equi ed o nuclea ion in he he e ogeneous solidi ica ion egime. Con e sely, he
expe imen al pa ocuses on unde s anding he solidi ica ion p ocess om he di e en ial he mal
analysis (DTA) o he solidi ica ion cu es. I was concluded ha i subcooling is in ol ed, cubic and
py amidal pa icles wo k be e as nuclea ing pa icles in he s udied alloy.
Keywo ds:
he e ogeneous solidi ica ion; modeling; shaped pa icles; mechanical p ope ies; i anium
ca bide; alumina
1. In oduc ion
In he las ew yea s, he au omo i e indus y has p omo ed he de elopmen o
ligh e ma e ials wi h be e mechanical p ope ies, seeking economic, en i onmen al, and
poli ical bene i s. The e o e, esea ch is needed on he imp o emen o ligh e ma e ials as
a way o lowe uel consump ion, which will enable a d op in pollu an gas emissions [
1
].
One way o mee his challenge is o in oduce ce amic nanopa icles in o a ligh weigh
alloy o modi y he inal mic os uc u e o he ma e ial, wi h he pu pose o imp o ing
i s mechanical p ope ies [
2
,
3
]. To achie e imp o emen in he mechanical p ope ies o
me al ma e ials i.e., s eng h and duc ili y, nanome ic pa icles ha e p o en o be a easible
app oach o ob aining he equi ed esul [4].
The p ocess used o imp o e he mechanical p ope ies o he aluminum alloy consis s
o in oducing nanome ic pa icles in o he ma e ial be o e cas ing so ha he pa icles ac
as solidi ica ion ge ms o he
α
-Al phase. By loca ing he nanopa icles in his phase,
wo o
he ele an mechanical p ope ies, i.e., he s eng h and he duc ili y, can be enhanced due
o he g ain e inemen o he mic os uc u e. The e o e, i is necessa y o quan i y how
much he inco po a ed nanopa icles can ac as he e ogeneous solidi ica ion ge ms. Among
he ac o s ha mos in luence he e ec i eness o he pa icles a e hei size, shape, and
su ace p ope ies [5,6].
In me allic alloys, he nuclea ion p ocess is unde s ood as he beginning o a phase
ans o ma ion in a small egion [
7
,
8
]. In his p ocess, a oms come oge he and o ganize
Me als 2022,12, 855. h ps://doi.o g/10.3390/me 12050855 h ps://www.mdpi.com/jou nal/me als
Me als 2022,12, 855 2 o 12
hemsel es o mo e om he liquid phase o a solid phase. Fo he nuclea ion p ocess o oc-
cu , small emb yos mus i s o m. As hese emb yos each he c i ical adius, hey will end
o o m a s able nucleus, which g ows o c ea e a c ys al. O he wise, he a oms o molecules
sepa a e, dissol ing back in o he liquid. Based on he model ha o he esea che s ha e
de eloped o he s udy o nuclea ion om sphe ical pa icles, i was conside ed desi able
o s udy he e ec o in oducing di e en ly shaped pa icles,
i.e., cubic
and py amidal
o ms [
9
,
10
]. Cu en ly, wi h he boom in he de elopmen o new nanopa icle- ein o ced
alloys using pa icles o di e en geome ies, i is in e es ing o ex end he sphe ical model
o o he shapes, such as cubes and py amids. The in luence o geome y on he e ogeneous
ge mina ion has ye o be p ope ly unde s ood. In ea lie s udies, a amewo k o mod-
eling coupled nuclea ion, g ow h, and coa sening in dilu ed alloys, based on di e en
me hods, has been de eloped [11–16].
In he p esen wo k, wi h he aim o unde s anding he in luence o in oducing
di e en ce amic pa icles in o a liquid aluminum alloy, a model o di e en pa icle
shapes and sizes has been de eloped. Wi h he model, he c i ical Gibbs ene gies o each
ein o ced alloy can be ob ained so ha sphe ical, cubic, and py amidal pa icles can be
compa ed. In cases in which he needed ene gy o nuclea ion is less han ha o he
homogeneous ma e ial, he pa icles a e going o beha e as a solidi ica ion ge m. Mo eo e ,
he in luence o in oducing nanopa icles on he solidi ica ion cu e has been s udied,
o which DTA has been ca ied ou . In he case o sphe ical and cubic nanopa icles, he
ma hema ical model is alida ed using expe imen al esul s. In he case o py amidal
pa icles, only he ma hema ical model is shown. The expe imen al de elopmen was
pe o med on wo di e en samples o AlSi10MnMg alloy, one o hem alloyed wi h cubic
TiC and he o he wi h sphe ical Al
2
O
3
ce amic nanopa icles. The de eloped ma hema ical
models p edic he c i ical Gibbs ene gy and he c i ical adius ha he nuclei mus achie e
o ini ia e s able nuclea ion. The model allows p edic ing nuclea ion beha io as a unc ion
o he shape, size, and chemical composi ion o he in oduced pa icles.
2. Ma e ials and Me hods
The The mo-Calc so wa e ( . 2020, The mo-Calc So wa e AB, S ockholm, Sweden)
was used in conjunc ion wi h he TCAL7 he modynamic da abase o calcula e he non-
equilib ium solidi ica ion p ope ies o he alloys [
17
]. The AlSi10MnMg ingo s we e mel ed
in a silicon ca bide c ucible, hea ed in an elec ic esis ance u nace ha can hold up o
3 kg
o aluminum alloy. The s i -cas ing p ocess [
18
] was used o in oduce he nanome ic
ce amic pa icles o he AlSi10MnMg alloy. The p ocess consis ed o mel ing he aluminum,
aising he u nace empe a u e o 710–740
◦
C; once he ma e ial was mel ed, he s i e
was in oduced o gene a e a o ex. To ensu e he p ope agi a ion o he alloy, he s i e
emained cen e ed. In his wo k, pa icles o wo di e en shapes and composi ions (Al
2
O
3
and TiC) we e compa ed. In bo h cases, he size o he nanopa icles (equi alen diame e )
was 80 nm. The nanopa icles we e in oduced o he mol en alloy and, o ensu e hei
homogeneous dis ibu ion, he s i ing was main ained o 15 min. Finally, he mel ed
mix u e was cooled down o 700–720
◦
C and pou ed in o a s eel mold p ehea ed o 100
◦
C.
Be o e cas ing, samples we e pou ed in o s anda d QuiK-Cup
®
sand cups. The da ase o
DTA was collec ed using a high-speed Na ional Ins umen s®da a acquisi ion sys em.
3. Resul s and Discussion
3.1. Composi ional Analysis
The elemen al composi ion o he mol en ba h was analyzed using op ical emission
spec ome y (OES); he esul s a e shown in Table 1. I should be no ed ha in he case o
he alloy ein o ced wi h Al
2
O
3
(Base + Al
2
O
3
), he chemical composi ion is qui e simila o
ha o he base alloy. This is due o he ac ha he spec ome e is no capable o de ec ing
aluminum oxides in an aluminum base alloy.
Me als 2022,12, 855 3 o 12
Table 1. Elemen al composi ion (in w %) o he mol en alloys ob ained by OES.
Re . Al Fe Si Mn C Ni Zn Mg Ti S V
Base Bal. 0.16 10.92 0.55 0.01 0.01 0.01 0.24 0.05 0.01 0.01
Base + TiC Bal. 0.19 11.31 0.54 0.01 0.01 0.01 0.25 0.10 0.01 0.01
Base + Al2O3Bal. 0.17 11.12 0.54 0.01 0.01 0.01 0.24 0.05 0.01 0.01
3.2. He e ogeneous Solidi ica ion Model o Nano-Me ic Pa icles
In his wo k, he de eloped ma hema ical model analyses he phase ans o ma ion
o he aluminum by s udying he Gibbs ene gy, which allows he p edic ion o he c i ical
adius needed o s a he phase ans o ma ion om liquid o solid [7].
The model ha has been de eloped is based on he known equa ion o he homo-
geneous scena io, in which he ee ene gy o he homogenous su ace (
∆Ghom
) can be
exp essed as:
∆Ghom =V·∆G +As·γ(1)
whe e Vis he olume o he nuclei,
∆G
is he olume ic Gibbs ene gy di e ence,
As
is
he supe icial a ea o he nuclei, and
γ
is he su ace ension. Mo eo e , acco ding o he
hi d law o he modynamics, he olume ic Gibbs ene gy can be exp essed as [19]:
∆G =∆H −T·∆S(2)
whe e
∆G
is he olume ic Gibbs ene gy, Tis he phase ans o ma ion empe a u e,
∆
H
is he inc ease in en halpy o he sys em, and ∆Sis he inc ease in en opy o he sys em.
Conside ing ha in he icini y o he mel ing empe a u e, he liquid and solid s a es
coexis , he olume ic Gibbs ene gy di e ence can be exp essed as:
∆G =∆T·∆H
Tm(3)
whe e
∆T
is he di e ence be ween he mel ing and phase ans o ma ion empe a u es.
Conside ing he ma hema ical deduc ion, he c i ical adius * o he homogeneous solidi-
ica ion case can be calcula ed as ollows:
∗
hom =−2γ
∆G (4)
Fo his speci ic adius, Gibbs ene gy can be exp essed as:
∆G∗
hom =16π·γ3
∆G 2Tm
∆T·∆H (5)
Once he o mula ion and he me hodology used o he homogeneous case a e known,
in he he e ogeneous nuclea ion model, i is assumed ha nuclea ion occu s when he
nucleus o ms on impu i ies o on he walls o he essel con aining he liquid. When he
mel con ains solid pa icles, o hey o m om simple con ac wi h he c ucible wall o an
oxide laye , he p obabili y o he nuclea ion o small nuclei inc eases, e en wi h a small
numbe o a oms, i.e., he ac i a ion ene gy equi ed o he o ma ion o a s able nucleus
dec eases. In his wo k, based on he shape o he in oduced nanopa icles, he e a e h ee
case s udies o he e ogeneous nuclea ion (sphe ical, cubic, and py amidal), in which he
c i ical adius equa ion is ob ained om he de i a i e o he Gibbs ene gy equa ion since
he c i ical adius coincides wi h he poin a which he maximum ene gy is eached.
3.2.1. Sphe ical Pa icles
Figu e 1shows he schema ic illus a ion o a scena io in which he ein o cing pa icles
ha e a sphe ical shape.
Me als 2022,12, 855 4 o 12
Me als 2022, 12, x FOR PEER REVIEW 4 o 12
3.2.1. Sphe ical Pa icles
Figu e 1 shows he schema ic illus a ion o a scena io in which he ein o cing
pa icles ha e a sphe ical shape.
Whe e he su aces S a e:
=2π(1−β),
=2π(1−α)
and
=
3(1−β)(2+β)
−(1−α)(2+α)
Figu e 1. Schema ic illus a ion o he sphe ical pa icle nuclea ion.
Conside ing he ma hema ical ela ionships ob ained om Figu e 1, in which θ is he
we abili y angle [20], in he case o sphe ical pa icles, he Gibbs ene gy o he sys em
() is de ined as:
= ·+
γ
·+ (
γ
−
γ
) (6)
whe e γ
12
is he con ac angle be ween he nuclei and he liquid me al, γ
23
is he con ac
angle be ween he nuclei and he sphe ical pa icle, is he su ace o he nucleus in
con ac wi h he liquid, and is he su ace o he nucleus in con ac wi h he sphe ical
nanopa icle.
3.2.2. Cubic Pa icles wi h a Nucleus on One Face
Figu e 2a shows he schema ic illus a ion o he scena io in which he ein o cing
pa icles ha e a cubic shape. In he case o cubic pa icles, he nuclea ion occu s on one
ace o he cube, as shown in Figu e 2b. The c i ical adius and Gibbs ene gy a e ob ained
by imposing he condi ion o a cons an we ing angle.
Figu e 2. (a) Schema ic illus a ion o cubic pa icles wi h he nucleus on one ace, and (b) he
de ailed sec ion o he o ma ion o he nucleus om he ace o a cubic pa icle.
In he ma hema ical modeling o he cubic pa icles, he leng h o he cubic pa icle
was de ined as “2c”, he dis ance be ween he cen e o he cubic pa icle and he cen e
o he nuclei was de ined as “a”, he coo dina e o he ci cum e ence insc ibing he cap o
Figu e 1. Schema ic illus a ion o he sphe ical pa icle nuclea ion.
Conside ing he ma hema ical ela ionships ob ained om Figu e 1, in which
θ
is he
we abili y angle [
20
], in he case o sphe ical pa icles, he Gibbs ene gy o he sys em
(∆Gsph) is de ined as:
∆Gsph =∆GV·V+γ12 ·S12 +(γ23 −γ13)S23 (6)
whe e
γ12
is he con ac angle be ween he nuclei and he liquid me al,
γ23
is he con ac
angle be ween he nuclei and he sphe ical pa icle,
S12
is he su ace o he nucleus in
con ac wi h he liquid, and
S23
is he su ace o he nucleus in con ac wi h he sphe ical
nanopa icle.
3.2.2. Cubic Pa icles wi h a Nucleus on One Face
Figu e 2a shows he schema ic illus a ion o he scena io in which he ein o cing
pa icles ha e a cubic shape. In he case o cubic pa icles, he nuclea ion occu s on one
ace o he cube, as shown in Figu e 2b. The c i ical adius and Gibbs ene gy a e ob ained
by imposing he condi ion o a cons an we ing angle.
Me als 2022, 12, x FOR PEER REVIEW 4 o 12
3.2.1. Sphe ical Pa icles
Figu e 1 shows he schema ic illus a ion o a scena io in which he ein o cing
pa icles ha e a sphe ical shape.
Whe e he su aces S a e:
=2π(1−β),
=2π(1−α)
and
=
3(1−β)(2+β)
−(1−α)(2+α)
Figu e 1. Schema ic illus a ion o he sphe ical pa icle nuclea ion.
Conside ing he ma hema ical ela ionships ob ained om Figu e 1, in which θ is he
we abili y angle [20], in he case o sphe ical pa icles, he Gibbs ene gy o he sys em
() is de ined as:
= ·+
γ
·+ (
γ
−
γ
) (6)
whe e γ
12
is he con ac angle be ween he nuclei and he liquid me al, γ
23
is he con ac
angle be ween he nuclei and he sphe ical pa icle, is he su ace o he nucleus in
con ac wi h he liquid, and is he su ace o he nucleus in con ac wi h he sphe ical
nanopa icle.
3.2.2. Cubic Pa icles wi h a Nucleus on One Face
Figu e 2a shows he schema ic illus a ion o he scena io in which he ein o cing
pa icles ha e a cubic shape. In he case o cubic pa icles, he nuclea ion occu s on one
ace o he cube, as shown in Figu e 2b. The c i ical adius and Gibbs ene gy a e ob ained
by imposing he condi ion o a cons an we ing angle.
Figu e 2. (a) Schema ic illus a ion o cubic pa icles wi h he nucleus on one ace, and (b) he
de ailed sec ion o he o ma ion o he nucleus om he ace o a cubic pa icle.
In he ma hema ical modeling o he cubic pa icles, he leng h o he cubic pa icle
was de ined as “2c”, he dis ance be ween he cen e o he cubic pa icle and he cen e
o he nuclei was de ined as “a”, he coo dina e o he ci cum e ence insc ibing he cap o
Figu e 2.
(
a
) Schema ic illus a ion o cubic pa icles wi h he nucleus on one ace, and (
b
) he de ailed
sec ion o he o ma ion o he nucleus om he ace o a cubic pa icle.
In he ma hema ical modeling o he cubic pa icles, he leng h o he cubic pa icle
was de ined as “2c”, he dis ance be ween he cen e o he cubic pa icle and he cen e o
he nuclei was de ined as “a”, he coo dina e o he ci cum e ence insc ibing he cap o he
g owing nucleus was de ined as
Ω
, and he adius o he cap on he ace o he nuclea ing
pa icle was de ined as X.
3.2.3. Cubic Pa icles wi h a Nucleus on he Ve ex
Figu es 3and 4show he schema ic illus a ion o di e en possible cases in which he
ein o cing pa icles ha e a cubic shape wi h a nucleus on he e ex, so ha , he nucleus
has de eloped on a co ne o he pa icle. To keep he we abili y angle cons an , i assumed
Me als 2022,12, 855 5 o 12
ha he cen e o he pa icle mo es owa ds he inside o ou side o he cube. Due o
geome ical easons, he minimum we ing angle is 45
◦
(Figu e 3b). The e o e, o lowe
angle alues (Figu e 3a), he p e e en ial nuclea ion will be on he ace o he cube.
Me als 2022, 12, x FOR PEER REVIEW 5 o 12
he g owing nucleus was de ined as Ω, and he adius o he cap on he ace o he
nuclea ing pa icle was de ined as X.
3.2.3. Cubic Pa icles wi h a Nucleus on he Ve ex
Figu es 3 and 4 show he schema ic illus a ion o di e en possible cases in which
he ein o cing pa icles ha e a cubic shape wi h a nucleus on he e ex, so ha , he
nucleus has de eloped on a co ne o he pa icle. To keep he we abili y angle cons an ,
i assumed ha he cen e o he pa icle mo es owa ds he inside o ou side o he cube.
Due o geome ical easons, he minimum we ing angle is 45° (Figu e 3b). The e o e, o
lowe angle alues (Figu e 3a), he p e e en ial nuclea ion will be on he ace o he cube.
Figu e 3. Scheme o he c i ical we abili y angles, wi h a nucleus on he e ex o a cubic pa icle.
(a) Case o θ < 45
o
. (b) Case o θ = 45
o
. (c) Case o θ > 45
o
.
In Figu e 3c, he cen e o he nucleus and he co ne o he cube (D) a e he same and
he g ains g ew cen e ed a he co ne . As done in he p e ious cases, he Gibbs ene gy
will be ob ained om he a io o angles, con ac su aces, and olumes.
Figu e 4. Schema ic illus a ion o cubic pa icles wi h a nucleus on he e ex. (a) Pe spec i e iew
o nucleus g ow h in he co ne o he cubic pa icle; (b) Sec ion iew (A-A) o he geome ic
de i a ion o model olumes and su aces o his sec ion.
3.2.4. Py amidal Pa icles
Figu e 5 shows he schema ic illus a ion o he py amidal pa icles; he leng h o he
py amidal pa icle was de ined as “a”. Conside ing he o med sphe ical cap, he Gi a d
heo em was applied [21]. Following his heo em, i α, β, and γ a e he measu es in
Figu e 3.
Scheme o he c i ical we abili y angles, wi h a nucleus on he e ex o a cubic pa icle.
(a) Case o θ< 45◦. (b) Case o θ= 45◦. (c) Case o θ> 45◦.
Me als 2022, 12, x FOR PEER REVIEW 5 o 12
he g owing nucleus was de ined as Ω, and he adius o he cap on he ace o he
nuclea ing pa icle was de ined as X.
3.2.3. Cubic Pa icles wi h a Nucleus on he Ve ex
Figu es 3 and 4 show he schema ic illus a ion o di e en possible cases in which
he ein o cing pa icles ha e a cubic shape wi h a nucleus on he e ex, so ha , he
nucleus has de eloped on a co ne o he pa icle. To keep he we abili y angle cons an ,
i assumed ha he cen e o he pa icle mo es owa ds he inside o ou side o he cube.
Due o geome ical easons, he minimum we ing angle is 45° (Figu e 3b). The e o e, o
lowe angle alues (Figu e 3a), he p e e en ial nuclea ion will be on he ace o he cube.
Figu e 3. Scheme o he c i ical we abili y angles, wi h a nucleus on he e ex o a cubic pa icle.
(a) Case o θ < 45
o
. (b) Case o θ = 45
o
. (c) Case o θ > 45
o
.
In Figu e 3c, he cen e o he nucleus and he co ne o he cube (D) a e he same and
he g ains g ew cen e ed a he co ne . As done in he p e ious cases, he Gibbs ene gy
will be ob ained om he a io o angles, con ac su aces, and olumes.
Figu e 4. Schema ic illus a ion o cubic pa icles wi h a nucleus on he e ex. (a) Pe spec i e iew
o nucleus g ow h in he co ne o he cubic pa icle; (b) Sec ion iew (A-A) o he geome ic
de i a ion o model olumes and su aces o his sec ion.
3.2.4. Py amidal Pa icles
Figu e 5 shows he schema ic illus a ion o he py amidal pa icles; he leng h o he
py amidal pa icle was de ined as “a”. Conside ing he o med sphe ical cap, he Gi a d
heo em was applied [21]. Following his heo em, i α, β, and γ a e he measu es in
Figu e 4.
Schema ic illus a ion o cubic pa icles wi h a nucleus on he e ex. (
a
) Pe spec i e iew o
nucleus g ow h in he co ne o he cubic pa icle; (
b
) Sec ion iew (A-A) o he geome ic de i a ion
o model olumes and su aces o his sec ion.
In Figu e 3c, he cen e o he nucleus and he co ne o he cube (D) a e he same and
he g ains g ew cen e ed a he co ne . As done in he p e ious cases, he Gibbs ene gy will
be ob ained om he a io o angles, con ac su aces, and olumes.
3.2.4. Py amidal Pa icles
Figu e 5shows he schema ic illus a ion o he py amidal pa icles; he leng h o he
py amidal pa icle was de ined as “a”. Conside ing he o med sphe ical cap, he Gi a d
heo em was applied [
21
]. Following his heo em, i
α
,
β
, and
γ
a e he measu es in adians
o he angles o he sphe ical iangle ABC, on a sphe e o adius , he a ea o his sphe ical
iangle is equal o (
α
+
β
+
γ−π
)
2
. This assump ion was aken in o conside a ion o
calcula e he Gibbs ene gy equa ion o he sys em.
Me als 2022,12, 855 6 o 12
Me als 2022, 12, x FOR PEER REVIEW 6 o 12
adians o he angles o he sphe ical iangle ABC, on a sphe e o adius , he a ea o his
sphe ical iangle is equal o (α + β + γ − π)
2
. This assump ion was aken in o
conside a ion o calcula e he Gibbs ene gy equa ion o he sys em.
Figu e 5. Schema ic illus a ion o he nuclei g ows in a py amidal pa icle. (a) The nucleus g owing,
(b) he shape o he pa icle, and (c) sphe ical cap shape de ail.
3.2.5. Nume ical Modeling
To ob ain he equa ions o p edic he c i ical Gibbs ene gy o each scena io, he
olumes and con ac su aces we e ob ained and subs i u ed in o Equa ion (1). In Table 2,
he alues o he s udied cubic and py amidal cases a e summa ized. These alues we e
ob ained in a simila way o he p edic ion made o he sphe ical pa icles.
Table 2. Calcula ed olume and con ac su ace in he unc ion o he pa icle shape.
Pa icle Shape Volume Con ac Su ace
Cubic, wi h a nucleus
on one ace =
3
·(2+cos()·(1−cos
())
=2
·
(1−cos())
=
·
cos ()
Cubic, wi h a nucleus
on he e ex
θ ≥ 45° = 4π
3
·
()
=4
()
=3
2
()
Py amidal =4
3
−
√
2
12
=5−3·1
3
·
=3
·
√
3
4
·
He e, o a cubic pa icle wi h a nucleus on he e ex, (Ɵ)=+ +
, as ollows:
n a b c
1 0.8815967897 −0.654080314 −0.5033064535
2 0.8665609048 −0.276518036 −0.4422067834
3 0.3570758524 0.7369844695 1.179334371
The de eloped ma hema ical ela ionships o he c i ical Gibbs ene gy and o m
ac o in he unc ion o he pa icle shape a e summa ized in Table 3. These equa ions
we e ob ained om he geome ies shown in Figu es 1–5 and ollow he me hodology
p e iously de ailed in he case o sphe ical pa icles.
Figu e 5.
Schema ic illus a ion o he nuclei g ows in a py amidal pa icle. (
a
) The nucleus g owing,
(b) he shape o he pa icle, and (c) sphe ical cap shape de ail.
3.2.5. Nume ical Modeling
To ob ain he equa ions o p edic he c i ical Gibbs ene gy o each scena io, he
olumes and con ac su aces we e ob ained and subs i u ed in o Equa ion (1). In Table 2,
he alues o he s udied cubic and py amidal cases a e summa ized. These alues we e
ob ained in a simila way o he p edic ion made o he sphe ical pa icles.
Table 2. Calcula ed olume and con ac su ace in he unc ion o he pa icle shape.
Pa icle Shape Volume Con ac Su ace
Cubic, wi h a nucleus on one ace V=π
3 3·(2+cos(θ))·(1−cos(θ))2S12 =2π 2·(1−cos(θ))
S23 =π 2·cos(θ)2
Cubic, wi h a nucleus on he e ex
θ≥45◦V=4π
3 3·h1(θ)S12 =4π 2h2(θ)
S23 =3 2
2h3(θ)
Py amidal V=4π
3 3−√2
12 3S12 =h5π−3·a cos1
3i· 2
S23 =3·√3
4· 2
He e, o a cubic pa icle wi h a nucleus on he e ex,
hn(θ)=an+bncosθ+cncos2
,
as ollows:
n a b c
1 0.8815967897 −0.654080314 −0.5033064535
2 0.8665609048 −0.276518036 −0.4422067834
3 0.3570758524 0.7369844695 1.179334371
The de eloped ma hema ical ela ionships o he c i ical Gibbs ene gy and o m ac o
in he unc ion o he pa icle shape a e summa ized in Table 3. These equa ions we e
ob ained om he geome ies shown in Figu es 1–5and ollow he me hodology p e iously
de ailed in he case o sphe ical pa icles.
Figu e 6shows he di e ence in he Gibbs ene gy cu e due o he in oduc ion o
TiC and Al
2
O
3
nanopa icles. In he case s udied, i was assumed ha he we ing angle
be ween hese pa icles and he AlSi10MnMg alloy was
θ
= 90
◦
. Rega ding pa icle shape,
no e ha : (1) TiC nanopa icles a e cubic, and (2) Al
2
O
3
nanopa icles a e sphe ical. In bo h
cases, he pa icles used ha e an app oxima e size o 80 nm (equi alen diame e ).
Me als 2022,12, 855 7 o 12
Table 3. Calcula ed c i ical Gibbs ene gy and o m ac o in he unc ion o he pa icle shape.
Pa icle Shape C i ical Gibbs Ene gy Fo m Fac o
Sphe ical ∆G∗
sph =16πγ3
12
3∆G2
·Fsph(m,x)
Fsph(m,x)=1
21−mx−1
g3
+x32−3x−m
g+x−m
g3
+3mx2x−m
g−1o
being g=√1+x2−2mx and x=R/ *; m=cosθ
Cubic, wi h a nucleus on one ace ∆G∗
c: ace =16πγ3
12
3∆G2
·Fc: ace(θ)Fc; ace(θ)= (θ)=1
42−3cosθ+cos3θ
Cubic, wi h a nucleus on he e ex
θ≥45◦∆G∗
c: e ex =16πγ3
12
3∆G2
·Fc: e ex(θ)Fc: e ex(θ)=3−1
12 cu: e ex(θ)·h1(θ)+h2(θ)
−3
8h3(θ)·cosθ· 2
c: e ex (θ)
Py amidal ∆G∗
py amidal =16πγ3
12
3∆G2
·Fpy amidal (θ)Fpy amidal(θ)=√2(8√2π−1)
16π· 3
py amidal(θ)
He e c: e ex(θ)=8·h2(θ)−3h3(θ)·cosθ
8·h1(θ)and py amidal (θ)=√2h12a an√2
2−π(cosθ+10)i
1−8√2π.
Me als 2022, 12, x FOR PEER REVIEW 7 o 12
Table 3. Calcula ed c i ical Gibbs ene gy and o m ac o in he unc ion o he pa icle shape.
Pa icle Shape C i ical Gibbs Ene gy Fo m Fac o
Sphe ical ∆G
∗ = 16
3∆
·
(,)
(,)= 1
2 1−−1
+2−3−
+−
+3−
−1
being =
√
1+−2 and x = R/ *; m = cosθ
Cubic, wi h a nucleus
on one ace ∆G:
∗= 16.
3∆
·
:() ;()=
()=1
4 (2−3+)
Cubic, wi h a nucleus
on he e ex
θ ≥ 45° ∆G:
∗= 16
3∆
·
:.() :()=3−1
12
:()
·
()+()
−3
8()
·
·
:
()
Py amidal ∆G
∗= 16
3∆
·
() ()=
√
2 (8
√
2−1)
16
·
()
He e
:
()=
·
()
()·
·
()
()=
√
√
()
√
.
Figu e 6 shows he di e ence in he Gibbs ene gy cu e due o he in oduc ion o
TiC and Al
2
O
3
nanopa icles. In he case s udied, i was assumed ha he we ing angle
be ween hese pa icles and he AlSi10MnMg alloy was θ = 90°. Rega ding pa icle shape,
no e ha : (1) TiC nanopa icles a e cubic, and (2) Al
2
O
3
nanopa icles a e sphe ical. In bo h
cases, he pa icles used ha e an app oxima e size o 80 nm (equi alen diame e ).
Figu e 6. In luence o he in oduc ion o TiC and Al
2
O
3
nanopa icles on he Gibbs ene gy.
Figu e 6. In luence o he in oduc ion o TiC and Al2O3nanopa icles on he Gibbs ene gy.
As can be seen om he p e ious Figu e 6, a a simila pa icle olume, cubic pa icles
a e mo e a o able o he e ogeneous solidi ica ion han sphe ical pa icles. Based on he
o m ac o s shown in Table 3, he same e ec was ob ained wi h a py amidal pa icle.
3.3. In luence o he Nanome ic Pa icles in he Solidi ica ion P ope ies o he Alloys
Figu e 7shows he non-equilib ium solidi ica ion cu e o he AlSi10MnMg (base
alloy). The calcula ion was pe o med using he The mo-Calc so wa e, using he elemen al
composi ion ob ained in Table 1. In he diag am, h ee di e en cu es a e plo ed, ob ained
wi h he classic Scheil model (Scheil calcula o 1), he Scheil model wi h back di usion in
he p ima y phase (Scheil Calcula o 2), and he equilib ium model (do ed line).
Me als 2022,12, 855 8 o 12
Me als 2022, 12, x FOR PEER REVIEW 8 o 12
As can be seen om he p e ious Figu e 6, a a simila pa icle olume, cubic pa icles
a e mo e a o able o he e ogeneous solidi ica ion han sphe ical pa icles. Based on he
o m ac o s shown in Table 3, he same e ec was ob ained wi h a py amidal pa icle.
3.3. In luence o he Nanome ic Pa icles in he Solidi ica ion P ope ies o he Alloys
Figu e 7 shows he non-equilib ium solidi ica ion cu e o he AlSi10MnMg (base
alloy). The calcula ion was pe o med using he The mo-Calc so wa e, using he
elemen al composi ion ob ained in Table 1. In he diag am, h ee di e en cu es a e
plo ed, ob ained wi h he classic Scheil model (Scheil calcula o 1), he Scheil model wi h
back di usion in he p ima y phase (Scheil Calcula o 2), and he equilib ium model
(do ed line).
Figu e 7. Non-equilib ium solidi ica ion cu es o AlSi10MnMg alloy, ob ained using di e en
solidi ica ion models.
The inpu da a o he Scheil model wi h back di usion had a cooling a e o 0.2 °C/s
and a seconda y dend i e a m-spacing o 20 µm. The da a we e expe imen ally ob ained
ia DTA and image analysis. As he nuclea ion o he nanome ic ce amic pa icles
occu ed du ing he i s s ages o solidi ica ion, an analysis o he cooling cu es in he
i s s ages o solidi ica ion was conduc ed. All he cu es exhibi ed he same solidi ica ion
pa e n when he empe a u e was o e 572 °C. The liquidus poin was a 598 °C in all he
cu es. A his empe a u e, he AlSi3Ti2 compound was p ecipi a ed om he liquid
phase. A 592 °C, he Al15Si2(Mn,Fe,C )4 phase was p ecipi a ed. Then, when he
empe a u e cooled down o 588 °C, he p ecipi a ion o he α-Al phase was ini ia ed. A
574 °C, a sha p down u n was obse ed in he cu es. This is co ela ed wi h he
p ecipi a ion o he eu ec ic Si pa icles. The equilib ium model (do ed line) di e ged
om he o he ones when he Si phase was s abilized; his di e gence was mo e isible a
Figu e 7.
Non-equilib ium solidi ica ion cu es o AlSi10MnMg alloy, ob ained using di e en
solidi ica ion models.
The inpu da a o he Scheil model wi h back di usion had a cooling a e o 0.2
◦
C/s
and a seconda y dend i e a m-spacing o 20
µ
m. The da a we e expe imen ally ob ained
ia DTA and image analysis. As he nuclea ion o he nanome ic ce amic pa icles occu ed
du ing he i s s ages o solidi ica ion, an analysis o he cooling cu es in he i s s ages o
solidi ica ion was conduc ed. All he cu es exhibi ed he same solidi ica ion pa e n when
he empe a u e was o e 572
◦
C. The liquidus poin was a 598
◦
C in all he cu es. A his
empe a u e, he AlSi
3
Ti
2
compound was p ecipi a ed om he liquid phase. A 592
◦
C,
he Al
15
Si
2
(Mn,Fe,C )
4
phase was p ecipi a ed. Then, when he empe a u e cooled down
o 588
◦
C, he p ecipi a ion o he
α
-Al phase was ini ia ed. A 574
◦
C, a sha p down u n
was obse ed in he cu es. This is co ela ed wi h he p ecipi a ion o he eu ec ic Si
pa icles. The equilib ium model (do ed line) di e ged om he o he ones when he Si
phase was s abilized; his di e gence was mo e isible a 573
◦
C. A e his empe a u e,
he solidi ica ion pa h o his model was comple ely di e en om he o he ones.
The e o e, he non-equilib ium solidi ica ion sequence o AlSi10MnMg a he begin-
ning o he solidi ica ion is as ollows:
•Liquid (L)→L + AlSi3Ti2a 598 ◦C
•L + AlSi3Ti2+ Al15Si(Mn,Fe)4a 591 ◦C
•L + AlSi3Ti2+ Al15Si(Mn,Fe)4+α-Al a 588 ◦C
•L + AlSi3Ti2+ Al15Si(Mn,Fe)4+α-Al + Si a 575 ◦C
As can be seen, in he base alloy, he e a e only wo compounds ha p omo e he
e inemen o he mic os uc u e, he AlSi
3
Ti
2
and Al
15
Si(Mn,Fe)
4
in e me allic phases.
These phases can be clea ly dis inguished in he non-equilib ium solidi ica ion cu es
(Figu e 7), as hey p esen an almos e ical line shape.
Me als 2022,12, 855 9 o 12
The expe imen al solidi ica ion p ope ies o he alloys s udied we e ob ained by
means o DTA o he solidi ica ion cu es, using he i s and second de i a i es o he
phase-change cu es. As an example, in Figu e 8, he cooling cu e and he i s de i a i e
o AlSi10MnMg (base alloy) a e ep esen ed. The pou ing empe a u e (T
p
), liquidus
empe a u e (T
L
), and he empe a u e p ecipi a ion o he Al-Si eu ec ic (T
Al-Si
) and solidus
empe a u e (TSol) we e also ma ked in he cu e.
Me als 2022, 12, x FOR PEER REVIEW 9 o 12
573 °C. A e his empe a u e, he solidi ica ion pa h o his model was comple ely
di e en om he o he ones.
The e o e, he non-equilib ium solidi ica ion sequence o AlSi10MnMg a he
beginning o he solidi ica ion is as ollows:
• Liquid (L)→ L + AlSi
3
Ti
2
a 598 °C
• L + AlSi
3
Ti
2
+ Al
15
Si(Mn,Fe)
4
a 591 °C
• L + AlSi
3
Ti
2
+ Al
15
Si(Mn,Fe)
4
+ α-Al a 588 °C
• L + AlSi
3
Ti
2
+ Al
15
Si(Mn,Fe)
4
+ α-Al + Si a 575 °C
As can be seen, in he base alloy, he e a e only wo compounds ha p omo e he
e inemen o he mic os uc u e, he AlSi
3
Ti
2
and Al
15
Si(Mn,Fe)
4
in e me allic phases.
These phases can be clea ly dis inguished in he non-equilib ium solidi ica ion cu es
(Figu e 7), as hey p esen an almos e ical line shape.
The expe imen al solidi ica ion p ope ies o he alloys s udied we e ob ained by
means o DTA o he solidi ica ion cu es, using he i s and second de i a i es o he
phase-change cu es. As an example, in Figu e 8, he cooling cu e and he i s de i a i e
o AlSi10MnMg (base alloy) a e ep esen ed. The pou ing empe a u e (T
p
), liquidus
empe a u e (T
L
), and he empe a u e p ecipi a ion o he Al-Si eu ec ic (T
Al-Si
) and solidus
empe a u e (T
Sol
) we e also ma ked in he cu e.
Figu e 8. The cooling cu e and he i s de i a i e dT/d o AlSi10MnMg alloy.
In Figu e 9, he sec ion o he solidi ica ion cu e ela ed o he unde cooling is
highligh ed. F om he p e ious igu e, i is possible o de e mine he e ec i eness o he
added pa icles in e ms o he e ogeneous nuclea ion and, he e o e, g ain e inemen .
This is pe o med by analyzing he cooling cu e in de ail (Figu e 9). The beha io o he
cooling cu e in he highligh ed sec ion gi es a good indica ion o he e ec o pa icle
in oduc ion on he solidi ica ion o he alloy [17]. I is well known ha he cu es exhibi
li le unde cooling when many nuclea ing pa icles a e p esen . I is also assumed ha i
he e a e ew nuclea ing pa icles, he e will be less unde cooling. The e o e, he
subcooling pa ame e (ΔT) can be calcula ed using he ollowing equa ion:
∆=− (7)
The ollowing pa ame e s ep esen a i e o unde cooling a e de ined, hus: (T
M
) is
he empe a u e a which newly nuclea ed c ys als ha e g own o such an ex en ha he
eleased la en hea goes ou o equilib ium; (T
G
) ep esen s he s eady-s a e g ow h
Figu e 8. The cooling cu e and he i s de i a i e dT/d o AlSi10MnMg alloy.
In Figu e 9, he sec ion o he solidi ica ion cu e ela ed o he unde cooling is
highligh ed. F om he p e ious igu e, i is possible o de e mine he e ec i eness o he
added pa icles in e ms o he e ogeneous nuclea ion and, he e o e, g ain e inemen .
This is pe o med by analyzing he cooling cu e in de ail (Figu e 9). The beha io o he
cooling cu e in he highligh ed sec ion gi es a good indica ion o he e ec o pa icle
in oduc ion on he solidi ica ion o he alloy [
17
]. I is well known ha he cu es exhibi
li le unde cooling when many nuclea ing pa icles a e p esen . I is also assumed ha i
he e a e ew nuclea ing pa icles, he e will be less unde cooling. The e o e, he subcooling
pa ame e (∆T) can be calcula ed using he ollowing equa ion:
∆T=TG−TM(7)
Me als 2022, 12, x FOR PEER REVIEW 10 o 12
empe a u e o he mol en me al; (T
N
) is he s a o he nuclea ion empe a u e, he
nuclea ion powe o he pa icles p esen in he liquid me al. These p e ious pa ame e s
a e mos easily ecognized by a sudden change in he i s de i a i e, as shown in Figu e
9a. The pe iod equi ed o his hea ing is called he ecalescence pe iod (
REC
). A e his
ime, he mol en me al hea s up o he s eady-s a e g ow h empe a u e.
Figu e 9. Cooling cu es in he ange o unde cooling o (a) base alloy and (b) base + TiC and base
+ Al
2
O
3
alloys.
The abo e-men ioned solidi ica ion p ope ies o he s udied alloys a e summa ized
in Table 4. In e ms o he e ec i eness o he pa icles, i is obse ed ha he in oduc ion
o he Al
2
O
3
and TiC nanopa icles p omo es a dec ease in he unde cooling pa ame e .
Fu he mo e, he TiC nanopa icles p o ed o be mo e e ec i e.
Figu e 9. Con .
