1
TITLE:
The Clus e Model: a Simula ion o he Ae ogel S uc u e as a Hie a chically
O de ed A angemen o Randomly Packed Sphe es.
V. Mo ales-Flo ez
*
, N. de la Rosa-Fox, M. Piñe o, L. Esqui ias
Depa amen o de Física de la Ma e ia Condensada, Facul ad de Ciencias,
Depa amen o de Física Aplicada, CASEM
Uni e sidad de Cádiz. 11510 Pue o Real, Spain.
ABSTRACT:
A new s uc u al model based on he p emises widely used o desc ibing he ae ogels
s uc u e has been in oduced. These s uc u es ha e been desc ibed as an assembly o
andom-packed sphe es in se e al hie a chically-o de ed le els. A new algo i hm o
building ou models by Compu e simula ion ha e been de eloped om hese p emises.
Subsequen ly, some cha ac e izing applica ions o ob aining he ex u al pa ame e s as
he speci ic su ace, speci ic po ous olume o he appa en densi y o he sys ems, based
on he Mon e Ca lo echnique and in geome ical conside a ions ha e been simula ed o
es ing he abili y o he models in explaining he s uc u e o some eal TMOS and TEOS
ae ogels. As a i s app oach o he s udy o he mechanical p ope ies o he ae ogels
hese models ha e been applied as well. Resul s suppo he idea ha hese models a e a
good way o explaining he s uc u e o he ae ogels.
I. INTRODUCTION
Silica ae ogels a e chemically ine , highly po ous, nanos uc u ed ma e ials, syn hesized
by he well-known sol-gel me hod [
1
], and d ied by he supe c i ical d ying p ocess
*
To whom co espondence should be add essed: ic o .mo ale[email p o ec ed]
2
concei ed by S. Kis le [
2
] o a oiding c acking. This way we ob ain he silica ae ogels,
mo e po ous ma e ials han he con en ionally-d ied gels, also known as xe ogels. I s
pa icula s uc u e is esponsible o he mos in e es ing p ope ies o he ae ogels, such
as low he mal conduc i i y o e y high speci ic su ace a ea wha can each alues like
1000 m2/g o mo e. By he way, nowadays an ae ogel is he solid wi h he lowes densi y
e e syn hesized [
3
], wi h a alue o 1.9 mg/cm3. Sonogels a e ob ained exposing a
mix u e o alkoxide and wa e o in ense ul asound [
4
,
5
,
6
]. This me hod does no equi e
adding a common sol en (gene ally, me hyl o e hyl alcohol) o mix homogeneously he
sys em alkoxide-wa e . These gels a e dense and hei s uc u e is ine and homogeneous,
because o he absence o sol en o he sol ob aining and, mainly, by he ini ial c oss-
linked s a e o e icula ion induced by ul asound. Gela ion occu s in en hs o seconds.
O he special cha ac e is ic o hese gels a e d ying is ha sonogels esul in a pa icula e
s uc u e, con a y o gels ob ained by hyd olysis o me allo ganic compounds unde acid
ca alys wi hou applying ul asound. Sonogels ha e a e y na ow po e size dis ibu ion,
e y high bulk densi y and su ace/ olume a io, wo o h ee imes highe han gels
p epa ed in alcohol solu ions. These gels, do no ul ill he au osimila i y condi ion along
one o de o magni ude 
7
.
The s uc u e o he ae ogels has been desc ibed as an assembly o andom-packed
sphe ical pa icles in se e al hie a chically-o de ed le els [
8
,
9
]. The knowledge abou he
ae ogel s uc u e has been app oached using compu e simula ion echniques ha ake
inpu s om se e al opics, like he unde s anding o he sol-gel p ocess, he s uc u e
o ma ion p ocess o he ela ionship be ween he s uc u e and he mechanical
p ope ies. The s uc u e o ma ion p ocess has been s udied by Molecula Dynamics
Technique [
10
] since Ga o alini i s applied i o he sol-gel p ocess in 1994 [
11
] using he
Feus on-Ga o alini po en ial [
12
], concluding ha he s uc u e o ma ion s a s wi h a
3
slow g owing p ocess o he clus e s, ollowed by he as e g owing o he s uc u e due
o he clus e -clus e agg ega ion. A. Hasmy has gone deepe in his aspec s udying he
beha iou o he cha ac e is ic clus e size and he in luence o he simula ion box size
[
13
]. O he au ho s as Gelb and Gubbins mainly ha e add essed hei wo k o de elop
cha ac e iza ion applica ions based on he Mon e Ca lo echnique o he po ous s uc u es
gene a ed by simula ion [
14
]. They ha e wo ked wi h he Lenna d-Jones po en ial o
each elemen , and he Lo enz-Be helo ules o mixing he in e -elemen po en ial.
O he opic o in e es consis s o ep oducing he o ma ion and g owing p ocesses o he
ae ogels by compu e , using he eac ion o di usion limi ed clus e agg ega ion (RLCA o
DLCA) algo i hms, o some modi ica ion o hem [
15
], o he ballis ic clus e -clus e
agg ega ion [
16
]. E en simula ion echniques ha e been used o es he alidi y o he
BET [
17
] o he BJH [7] me hods o analysing he adso p ion/deso p ion iso he ms.
Wo king in he s uc u e-mechanical p ope ies ela ionship, Sche e [
18
] ha e used
s uc u es gene a ed wi h DLCA-modi ied algo i hms cha ac e izing hem by hei ac al
dimension, o achie e he powe law exponen and hey ha e p esen ed some models o
explain he s uc u e-p ope ies ela ionship [
19
,
20
]. As o Woignie e col., hey ha e
wo ked wi h DLCA-gene a ed s uc u es [
21
], in oducing a new echnique o
cha ac e izing his po ous sys ems [
22
]. They conclude ha he po e size dis ibu ion and
he hyd oxyl con en a e ele an o desc ibing and unde s anding he mechanical
p ope ies o hese ma e ials [
23
]. In a p e ious wo k, Woignie and Phallipou p oposed
one app oach s a ing om a cubic s uc u al model [
24
] and using he Rump exp ession
o he ensile s eng h o a igid assembly o cohesi e sphe es [
25
]. Emme lig and F icke
s udied his p oblem, exac ly elas ici y and conduc i i y, h ough he scaling p ope ies
ob ained by hei simula ed ae ogel s uc u es [8].
4
In his we a e p oposing a new algo i hm based on he p emise o andom-packed sphe es
in se e al hie a chically-o de ed le els o building he Clus e Models, oge he wi h an
app oach o he mechanical p ope ies o hese ma e ials based on hese models. The aim
o his echnique is o build s uc u al models o he eal sys ems. I s bes pe o mance is
i s e sa ili y: uning he geome ic pa ame e s o he model we can ob ain e y di e en
assemblies o andom-packed sphe es o ep esen ing e y di e en sys ems. The main
s uc u al pa ame e s in his model a e he elemen a y pa icle adius, he numbe o
hie a chical le els and he con ac dis ance and shells o each le el. The densi y is jus a
e e ence o es ima e he numbe o hie a chic le els since he densi y is s ongly
dependen o his pa ame e . Howe e , sys ems made by his p ocedu e a e no supposed
o desc ibe he g owing p ocess o eal sys ems, bu hey belong o wha has been called
s a ic models [
26
] in he sense ha hese models desc ibe he inal s a e o he eal
sys ems, p o iding a new ool o he s uc u al s udies.
II. CLUSTER STRUCTURAL MODEL
F om he p emise ha he ae ogel s uc u e can be desc ibed by an assembly o andom-
packed sphe es in se e al hie a chically-o de ed le els, we de eloped an algo i hm o
building s uc u al models. We ha e made use o an AMD A hlon 1700 (1.46GHz)
p ocesso ha spen ew seconds in building hose sys ems. Along his wo k he pa icle
diame e has been used as a educed uni o desc ibe he models.
We disca ded using cubic simula ion boxes o building he models in spi e o being he
mos ecommended echnique, bu we buil a sphe ical sys em. This is because he
algo i hm p emise o sel -simila i y in se e al hie a chically-o de ed le els is easy o
implemen wi hin a sphe ical symme y simply subs i u ing each sphe e o he sys em o
a sphe ical assembly o sphe es. Cubic simula ion boxes do ha e been used o hose
5
cha ac e izing applica ions ha a e bounda y-dependen and ini e size-dependen o
pe mi pe iodic bounda y condi ions be applied. To ob ain a cubic box o cha ac e izing
he sys em, we jus c opped he bigges cubic box inside ou sphe ical sys em.
1. Algo i hm
The Clus e model algo i hm wo ks his way: i s we place one
elemen a y
sphe e o
diame e 1 in he cen e o ou sys em. Then we place andomly o he elemen a y sphe es
coa ing he i s one’s su ace so he i s andom shell is buil . Any sphe e has o sa is y
one condi ion o be placed: i has o be in con ac a leas wi h ano he one. The c i e ion
o be in con ac is unde s ood as o be a a dis ance be ween he minimum and maximum
con ac dis ances p e iously de ined, hus a oiding he exis ence o ee sphe es. Wi h his
pu pose, he dis ance wi hin he con ac ange is chosen andomly. We le i g ow as
many shells o andom placed sphe es as we conside necessa y o building ou wished
model. Once inished his p ocess, his agg ega e is aken as he
basic agg ega e
. I s size
is measu ed and ano he agg ega e is buil wi h
seconda y
sphe es o diame e equal o
he diame e o he basic agg ega e. A e building his new agg ega e, each seconda y
sphe e is eplaced by one basic agg ega e ob aining a wo-le el hie a chically-o de ed
assembly o andom-packed sphe es. Then, he sys em size is measu ed again and i s size
is aken as he diame e o one
e ia y
sphe e. An agg ega e o
e ia y
sphe es is buil
hen and, inally, each
e ia y
sphe e is eplaced by one wo-le el sys em, ob aining his
way a h ee-le el hie a chically-o de ed sys em (Figu e 1). This p ocess can be epea ed
as many imes as necessa y. Typical alues o ou models a e 60.000 pa icles o ganised
in 2 shells o andom-packed sphe es and h ee hie a chical le els; hei con ac dis ances,
d
, a e ound in he in e al (0.9D <
d
< 1.0D), D being he pa icle diame e (Figu e 2).

6
Al hough au osimila i y is po en ially p esen in he Clus e Models as a consequence o i s
gene a ion algo i hm, in he p esen case we ha e no gone in a ac al desc ip ion
because he s uc u e o sonogels is no au osimila along on o de o magni ude. In he
u u e we will emula e ac al s uc u e o hose ae ogels ha does p esen a ac al
dimension well de ined.
2. Cha ac e iza ion echniques
Some applica ions o cha ac e izing he models ha e been de eloped o calcula e ex u al
pa ame e s o he simula ed s uc u es. The compa ison o he calcula ed alues wi h hei
ac ual coun e pa s checks he alidi y o he models. Along his wo k we y o build
Clus e models wi h he same s uc u al pa ame e s han he eal ae ogels. We ake a eal
sys em as a a ge and we wo k uning he geome ic pa ame e s in he building algo i hm
in o de o ob ain i s co esponding model, ha is, he model wi h he same ex u e han
he eal sys em.
In his wo k we p esen esul s om his s a egy applied o eal sys ems om p e ious
wo ks.
The pa ame e s ha we ied o ep oduce a e:
Densi y: we conside ou sys em o med by an assembly o pu e silica sphe es o densi y
2.2 g/cm3, so once known he numbe o sphe es, i is known he sys em speci ic mass. In
some iden i ied cases, when we a e ying o emula e a sys em whose elemen al pa icles
a e desc ibed o ha e a de e mined densi y [20,
27
] we conside he mass o ou
elemen al sphe e wi h his pa icula alue (2.09 g/cm3, 1.85 g/cm3) ins ead o he
egis e ed densi y o he bulk silica. This di e ence may be caused by longe Si-O bond
dis ances [
28
] o no ha ing de ec ed some kind o mic opo osi y by he cha ac e iza ion
me hod used. On he o he hand, we conside he olume o e lapped be ween sphe es as
7
coun ed wice in he mass calcula ion ( olume sha ed by h ee sphe es is negligible).
Consequen ly we sub ac once he o e lapped mass.
Speci ic su ace: he heo y desc ibes he eal physiso p ion expe imen s a ing wi h
he o ma ion o a ni ogen monolaye on he su ace o he sys em o cha ac e ize. This
monolaye does no co e he whole ex e nal su ace o he ma e ial, bu only he
accessible su ace o he ni ogen. Taking his in o accoun , among he di e en
de ini ions o su ace a ea [17], he one calcula ed in his wo k is called he accessible
su ace a ea. We conside ed a sphe ical model o he ni ogen molecule o 16.2 Å2 o
c oss sec ion wha gi es a adius o 0.227 nm, and we de ined he educed adius o he
elemen al silica sphe e in e e ence o his. Then, we ob ained by Mon e Ca lo me hod he
ex e nal accessible su ace o he ni ogen molecule in ou sys em. This is a widely used
me hod [14,17,
29
,
30
] o cha ac e izing s uc u al models o po ous ma e ials.
Speci ic po ous olume and po osi y: we calcula ed by Mon e Ca lo he olume
accessible o a ni ogen sphe e inside ou sys em. In his poin we had o conside he
ini e olume co ec ion p esen ed by Sand a Ga alda [
31
]: he olume ob ained by his
echnique is lowe han he expec ed accessible olume due o he omission o he olume
be ween he cen e o he ni ogen sphe es and he su ace o ou sys em. Fo ixing his,
Sand a Ga alda p oposed adding he olume calcula ed con en ionally by Mon e Ca lo o
he esul ing olume om mul iplying he speci ic su ace by he ni ogen sphe e adius.
Po osi y is ob ained au oma ically nex o his pa ame e , educing he alues and
exp essing hem in he pe cen age no occupied by he sys em.
Appa en densi y: Since ou sys em is de ined in se e al hie a chical le els, we know
he numbe o sphe es in ol ed in building any o he le els and he olume occupied by
hose sphe es ha a e o ming i . Consequen ly, we ob ain he densi y a he di e en
8
le els, om he lowes – he elemen a y pa icle – o he highes , also called appa en
densi y.
III. RESULTS AND DISCUSSION
We applied his simula ion echnique o build se e al sys ems o explaining he s uc u e
o some eal sys ems. As a i s applica ion, we ook om a p e ious wo k [27] he ex u e
pa ame e s o wo ae ogels and we buil hei co esponding hie a chical models. As a
second applica ion, we ace he p oblem be ween he s uc u e and mechanical p ope ies,
simila o wha has been done by Woignie [20].
1. Simula ion o s uc u es
In [24] he s udied i ems we e wo ae ogels p epa ed om TEOS. Di e en clus e models
o desc ibing hose ae ogels’ s uc u es we e gene a ed. Bo h se s o da a a e shown in
Table 1. The models co esponding o he i s ae ogel was a h ee hie a chical le el
a angemen o packing sphe es. The elemen a y pa icles o his sys em we e desc ibed
in he o iginal wo k as sphe es o adius o 1.1 nm wi h a densi y o 2.09 g/cm3. We
conside ed hese alues o de ining ou sys em, so he esul ing models we e based on
he eal da a.
The goal o his pa o he wo k was o build success ully he co esponding models o he
eal sys em, s a ing om he expe imen al s uc u al pa ame e s. The p esen ed models
ep oduce he ex u al alues o he eal sys ems, as i was expec ed. We can see how
models buil as an assembly o andom packed sphe es o hie a chically a anged can
ep oduce qui e well he ex u e o he eal ae ogels. Pa ame e s o he esul ing models
a e also shown in Table 1.
9
2. Mechanical p ope ies
In [24], a simple s uc u al model was applied o explain he mechanical p ope ies o
hese ma e ials. A s udy abou he ela ionship be ween he no malized s eng h and he
po osi y was p esen ed.
The no malized s eng h o ae ogels om TMOS as silica p ecu so was ob ained by h ee-
poin lexu al es s and diame al comp ession es s (also known as “B azilian es ”, ASTM
#D3967 [
32
]). Fo explaining he beha iou o his pa ame e and i s dependence wi h he
po osi y, hey used a s uc u al model o cubic cells in which he edges a e o med by
sphe ical silica beads. The cohesion o he sys ems is explained as a unc ion o he
o e lapping olume be ween neighbou sphe es, aking in o accoun he Rump ’s
exp ession o he ensile s eng h o a igid assembly o cohesi e sphe es o adius
R
[25]:
2
R32
KF9




Equa ion 1
whe e

is he olume ac ion o solid, ela ed o he po osi y
P
as
(1-P)
, and
K
is he
mean coo dina ion numbe . The ac o
F
, gi en by Equa ion 2, is he bonding o ce
be ween wo o e lapped sphe es o dense silica wi h an o e lapping neck adius
a
:
2
0aF


Equa ion 2
whe e
σ0
is he mechanical s eng h o dense silica glass. Thus, he ensile s eng h is
no malized as ollows:
16

17
Figu e 3
: Compa a i e esul s o he no malized s eng h om expe imen al es s, Woignie ’s heo e ical
model, Clus e model wi h he o iginal Rump ’s exp ession and he modi ied exp ession. Values and hei
e o ba s in Clus e model da a a e he esul o he a e age o a leas 5 epea s o he same sys em.
80 84 88 92 97
0
2
4
6
8
/0 *102
Expe imen al
Woignie 's model
Clus e models
Clus e models-modi ied exp ession
Po osi y (%)
18
Figu e 4
: Two-dimensional diag am o he sphe ical cap, wi h heigh
h
, and he base adius o o e lapping
neck adius
a
, and sphe e adius
R
.
19
TABLES
Table 1. S uc u al pa ame e s o he eal ae ogels and o i s co esponding clus e models.
Table 2. S uc u al pa ame e s o he Woignie ’s ae ogels (le ) and o hei co esponding clus e models.
E o s in models’ esul s conce n o s anda d e o om a leas 10 i e a ions.
EXPERIMENTAL
MODELS
Po osi y (%)
Speci ic
su ace
(m2/g)
Densi y
(g/cm3)
Po osi y (%)
Speci ic
su ace
(m2/g)
Densi y
(g/cm3)
78
450
0.41
77±1
459±5
0.41±0.02
80
400
.36
81±3
404±2
0.36±0.02
82
250
0.33
83±2
253±9
0.34±0.02
88
350
0.23
88±2
340±3
0.20±0.01
90
300
0.19
90±5
307±4
0.19±0.02
REAL SYSTEM
Appa en densi y: 0,83 g/cm3
Speci ic su ace: 387-407 m2/g
Speci ic po ous olume: 0,73-0,74 cm3/g
MODELS
Appa en
densi y
(g/cm3)
Speci ic
su ace
(m2/g)
Po ous
olume
(cm3/g)
0,80
384
0,72
0,81
376
0,88
REAL SYSTEM
Elemen al sphe e adius: 1,2 nm
Fi s agg ega e adius: 4,5 nm
Speci ic su ace: 640 m2/g
MODELS
Agg ega e
adius (nm)
Speci ic su ace
(m2/g)
4,5
612
4,4
669
20
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