The cluster model: A simulation of the aerogel structure as a hierarchically-ordered arrangement of randomly packed spheres

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 REFERENCES 1 J. B inke and G. Sche e , “Sol–Gel Science: The Physics and Chemis y o sol–gel P ocessing.” Academic P ess, San Diego, CA, 1990. 2 S. Kis le . J.Phys.Chem. 36(1), (1932) 52 3 Law ence Li e mo e Na ional Labo a o y: h p://www-cms.llnl.go /s- /ae ogels_guinness.h ml 4 J. Za zycki, He e ogeneous Chemis y Re iews , 1, (1994) 243. 5 E. Blanco, L. Esqui ias, R. Li án, M. Piñe o, M. Ramí ez-del-Sola and N. de la Rosa-Fox Appl. O ganome al. Chem. , 13 (1999) 399 6 N. de la Rosa-Fox, M. Piñe o, M.J. Mosque a and L. Esqui ias ‘O ganic-Ino ganic Hyb id Ma e ials om Sonogels’ in “ Encyclopaedia o Nanoscience and Nano echnology ” Ed. S.H. Nalwa. Ame ican Scien i ic Publishe s, Ca., 2003, p.241. 7 J. Za zycki, J. Non-C ys . Solids, 121 (1992) 8 J. Za zycki. J. Non-C ys . Solids 147&148 (1992) 176-182 9 L. Esqui ias, J. Rod iguez-O ega, C. Ba e a-Solano, N. de la Rosa-Fox. J. Non-C ys . Solids. 225 (1998) 239-243 10 K. Yamaha a, K. Okazaki. Fluid Phase Equilib ia 144 (1998) 449-459. 11 S. H. Ga o alini, G .E. Ma in. J. Phys. Chem. 98 (1994) 1311-1316 12 B. P. Feus on, S. H. Ga o alini. J. Phys. Chem. 94 (1990) 5351-5356 13 A. Hasmy, R. Jullien. J. Non-C ys . Solids 186 (1995) 342-348 14 L. D. Gelb, K.E. Gubbins. Langmui 15, (1999) 305-308 15 A. Emme ling, J. F icke. J. Sol-Gel Sci. and Tech. 8 (1997) 781-788 16 M. G zego czyk, M. Rybaczuk, K. Ma uszewski. Chaos, Soli ons & F ac als 19 (2004) 1003-1011 17 L. D. Gelb, K. E. Gubbins. Langmui 14 (1998) 2097-2111 18 H. Ma, J.H. P e os , G.W. Sche e . In e na ional Jou nal o Solids and S uc u es 39 (2002) 4605-4614 19 H. Ma, A.P. Robe s, J.H. P e os , R. Jullien, G.W. Sche e . J. Non-C ys . Solids 277 (2000) 127-141 20 H. Ma, J.H. P e os , R. Jullien, G.W. Sche e . J. Non-C ys . Solids 285 (2001) 216-221 21 T. Woignie , J. Reynes, A.H. Alaoui, I. Beu oies, J. Phallipou. J. Non-C ys . Solids 241 (1998) 45-52 22 J. P ime a, A. Hasmy, T. Woignie . J. Sol-Gel Sci. and Tech . 26 (2003) 671-675 23 T. Woignie , F. Despe is, A. Alaoui, P. E ienne, J. Phalippou. J.Sol-Gel Sci. and Tech. 19 (2000) 163-169 24 T. Woignie , J. Phalippou. J. Non-C ys . Solids 100 (1988) 404-408 25 A. Rump . Chem. Ing. Tech. 30 (1958) 144 26 L.T. To, Z.H. S achu ski. J. Non-C ys . Solids 333 (2004) 161-171 27 M.C. Ba e a-Solano, N. de la Rosa-Fox, L. Esqui ias. J. Non-C ys . Solids 147&148 (1992) 194-200 28 L. Esqui ias, C. Ba e a-Solano, N. de la Rosa-Fox, F.L. Cumb e a, J. Za zycki, in “ Ul as uc u e P ocessing o Ad anced Ma e ials ” edi ed by D.R. Uhlmann, D.R. Ul ich. John Wiley & Sons, Inc. (1992) 315-325 29 K.T.Thomson, K.E. Gubbins. Langmui 16 (2000) 5761-5773 30 S. Ga alda, K.E. Gubbins, Y. Yanzawa, K. Kaneko, K.T. Thomson. Langmui 18 (2002) 2141-2151 31 S.Ga alda, K. Kaneko, K.T. Thomson, K.E. Gubbins. Colloids and Su aces A 187-188 (2001) 531-538 32 Ame ican Socie y o Tes ing and Ma e ials. h p://www.as m.o g