Radial Distribution Functions of the Structures Built through Fractional Deposition of Hard Spherical Particles

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1 Raal Dstbuton Functons of the Stuctues Bult though Factonal Deposton of Ha Sphecal Patcles Panu Danwanchakul Abstact Deposton of lage patcles such as colloal o bo-patcles on a sol suface s usually moele by the anom sequental asopton (RSA. The moel was pevously escbe by the ntegal-equaton theoy whose valty was pove by Monte Calo smulaton. Ths eseach genealze the moel to nclue the concentaton effect of ae patcles on the suface. The facton of patcles nsete was vae by the numbe ensty of 0.05, 0.1, an 0.. It was foun that the mofe ntegal-equaton theoy yele the esults n goo accoance wth the smulaton. When the facton of patcles ae was ncease, the aal stbuton functon has hghe peak, ue to the coopeatve an entopc effects. Ths wok coul bge the gap between equlbum asopton, whee all patcles may be consee movng an RSA, whee thee s no movng patcles. Inex Tems Collo, Deposton, Integal Equaton Theoy, Monolaye Flms. I. INTRODUCTION To unestan the onset of evesblty, t s vey useful to genealze the asopton moel by conseng the tme scales that chaacteze ts vaous steps. Thus, the genealze asopton pocess wll epen on at least thee stnct chaactestc tmes, assumng that the actvaton eneges fo esopton ae much hghe than fo the othe pocesses. Those tmes ae a suface elaxaton o ffuson tme ( τ measung tme fo movng one step, a eacton tme ( τ measung the tme fo evesble attachment to the suface, an an asopton tme ( τ a measung the sepaaton between two consecutve asopton events. The elatve magntue of the thee tmes efne above lea to vey ffeent physcal stuatons. In the equlbum case thee s no eacton wth the suface an the asobe molecules equlbate aply between successve atons such that τ a. In anothe lmtng case, Ranom Sequental Asopton (RSA, the asobe molecules eact wth the suface nstantaneously but not ffuse, so that τ a. In the th lmt, Manuscpt eceve Decembe 8, 007. Ths wok was fnancally suppote by the offce of Natonal Reseach Councl of Thalan n the fscal yea 008. Panu Danwanchakul s wth the Depatment of Chemcal Engneeng, Faculty of Engneeng, Thammasat Unvesty, 99 Moo 18 Paholyothn R., Khlong Nung, Khlong Luang, Pathumthan, 110 Thalan (e-mal: panu@eng.tu.ac.th. Sequental Quenchng, the sepaaton n the chaactestc tmes such that τa. A smla scusson about elatve magntue of the chaactestc tmes was also gven by Schaaf an Talbot [1]. Usually, the asopton of lage patcles such as colloal patcles an potens on a suface s evesble. Fee an Gaeve [] epote that the monolaye stuctues of asobe fetn on cabon suface coul be explane by the moel of anom sequental asopton (RSA. Onoa an Lnge [3] also foun that the confguatons of polystyene sphees on a glass suface fnally eache the jammng lmt of RSA. Howeve, thee have been many nect an ect evences showng that among the mmoble patcles thee exsts moble facton of patcles. Repots of suface coveages sgnfcantly geate than the RSA jammng lmt pove nect evence fo lateal ffuson [4]. An ealy stuy of suface ffuson of asobe potens was cae out by Mchael et al. n 1980 [5]. In the expement the stbuton of bovne seum albumn (BSA asobe on glass was mage by autoaogaphy. They saw no evence of esopton an notce that a poten font mgate ove stances popotonal to the squae oot of tme, as expecte fo ffuson. Late, Bughat an Axelo [6] publshe a quanttatve stuy of BSA suface ffuson on quatz. They use an ellptcal spot fluoescence ecovey afte photobleachng (FRAP n a total ntenal eflecton confguaton to measue smultaneous suface ffuson an exchange between asobe an ssolve potens n soluton. They foun that asobe BSA exsts n thee stnct states: evesble, slowly evesble an aply evesble. Tlton et al. [7], [8] use FRAPP base on ntefeence of two coheent beams n total ntenal eflecton at a sol-lqu nteface. In the stues, BSA was asobe on polyme sufaces: spn-cast polymethyl-methacylate (PMMA flms an coss-lnke spn-cast flms of polymethylsloxane (PDMS. They foun coexstence of a moble an an mmoble populaton of BSA. Tlton et al. [8] also foun that the moble facton oes not epen on suface concentaton. The coexstence of moble an appaently mmoble potens appeas to esult not fom aggegaton of asobe BSA but fom a change n confomaton o oentaton of the asobe poten. Othe expements on poten asopton, such as that of DNA olgonucleotes on APTES glass [9] an of BSA on slca-ttana sufaces [10], have also eveale suface ffuson n the asobe state. If the poten-poten nteactons ae favoable, suface ffuson wll lea to clusteng [11], [1]. (Avance onlne publcaton: 0 Novembe 008

2 The moel of sequental quenchng (SQ has been stue to escbe the effect of suface ffuson n evesble asopton on monolaye stuctues [13], [14]. Nonetheless, only one patcles movng at one tme s an ealstc stuaton. In ths wok, the effect of moble facton of ha patcles movng among the mmoble ones on the suface was nvestgate n the pocess calle factonal eposton. We vae the numbe of ae patcles at each tme an they ae allowe to ffuse une the effect of othe pevously asobe patcles. In Secton II, the etal of smulaton metho s pove an the smulaton esults togethe wth the scusson ae gven n Secton III. The atcle s then conclue n Secton IV. II. AN INTEGRAL-EQUATION THEORY FOR FRACTIONAL DEPOSITION coesponng total coelaton functons h( whch ae fee fom noal ponts. These cluste ntegals ae best epesente n gaphcal fom; a evew of gaphcal notaton can be foun n classcal efeences [17]. The sub-nces c an b enote the so-calle connecte an blockng pats of the 1-1 coelatons, stngushng whethe all paths between the oots n the gaphcal epesentaton pass though a matx patcle o not. B. Bnay-Mxtue Appoxmaton An evolvng sequentally o ffeentally quenche system can be vewe as a bnay mxtue of the pevously quenche patcles (enote by the nex 0 an the (nfntely lute newly ae patcles (enote by 1, of numbe enstes ρ an ρ, espectvely as shown n Fg 1. The set of equatons n (1 becomes A. Asopton wthn Dsoee Matces A flu wthn n a soee poous matx can be vewe as a bnay mxtue of quenche patcles (component 0 an equlbate o anneale patcles (component 1, as shown n Fgue 1. h ( = c ( + ρc ( h ( h ( = c ( + ρc ( h ( + ρc ( h ( h ( = c ( + ρc ( h ( + ρc ( h ( h ( = c ( + ρc ( h ( + ρc ( h ( ρc ( h ( b11 h ( = c ( + ρ c 11( hc 11( 1 c ( These must be supplemente wth an appoxmate closue such as the Pecus-Yevck (PY elatons Fg 1: Flu patcles (1 ae asobe wthn a quenche matx (0 [ + 01 ] = 01 [ + ] [ + 11 ] = 11 [ + ] [ + ] = [ + ] f ( 1 h ( c ( 1 f ( f ( 1 h ( c ( 1 f ( f ( 1 h ( c ( 1 f ( (3 ( Let ρ00 ( enote pa ensty functon of the quenche ( stuctue, let g ( = ρ ( / ρ be ts pa coelaton functon an let h00 ( = g00 ( 1 be the esual o total coelaton functon between two patcles sepaate by a fxe stance. Smla coelatons may be efne fo 0-1 an 1-1 pas. The Replca Onsten-Zenke equatons accong to Maen an Glant [15] an late mofe by Gven an Stell [16] fo such a system ae o any altenatves, whee f( s the Maye functon whch s exp( u( 1 / kt -1, u( s the pa nteacton whch n ths wok s ha sphee potental, k s the Boltzmann constant, an T s the tempeatue. In factonal eposton, when the total ensty of the system nceases fom ρ to ρ + ρ, a balance of patcle pas yels the change n the pa ensty functon h ( = c ( + ρ c ( h ( h ( = c ( + ρ c ( h ( + ρ c ( h ( h ( = c ( + ρ c ( h ( + ρ c ( h ( h ( = c ( + ρ c ( h ( + ρ c ( h ( ρ c ( h ( 1 b11 ( = ( + ρ1 h c c ( h ( whee the symbol enotes a convoluton ntegal. The c( n the above equatons ae the ect coelaton functons,.e. the sums of all coeffcents ( agams o cluste ntegals n the ensty expansons of the (1 ( ρ + ρ g ( ; ρ + ρ = ρ g ( ; ρ ρ ρ g ( ; ρ + ( ρ g ( ; ρ The fst tem on the ght-han se epesents the pe-exstng pas whle the secon coespons to the atonal 0-0 pas ceate upon quenchng of equlbum patcles an the th tem s fom the contbuton of the nteacton among ae moble patcles. Equaton (4 togethe wth ts ntal conton escbes the evoluton of a stuctue bult though factonal eposton. As pevously stue, fo RSA the th tem was neglgble snce ρ s compaatvely small. The ntal conton s (4 (Avance onlne publcaton: 0 Novembe 008

3 g ( ; ρ = 0 = 1 h ( ; ρ = 0 = exp( βu( ( Equaton (5 an ts PY closue, must be solve fo g ( an g ( at each ensty n oe to compute g ( 00 though ntegaton of the ffeental equaton. pas N collecte fom n confguatons, at the sepaaton angng fom an + fom the aveage patcle to the coesponng numbe n a system of anomly place patcles. C. Numecal Integaton The stuctue evolves as the ensty of the system nceases, whch we enote as ρ. On takng the Foue tansfom on both ses of ( an usng the fact that total coelaton functon, h(, s compose of ect pat, c(, an nect pat, b(,.e. h(= c(+ b(, we obtan algebac elatons, h% ( k; ρ ρ ( k; ρ = 1 + ρhˆ ( k; ρ ρh% ( k; ρ h% 11( k; ρ + ρ ( k; ρ h% ( k; h11 ( k; ρ % ρ 1 + ρ hˆ ( k; ρ (6 We use the ntal conton to obtan the fst h% (. Ths 00 an h % c 11 (, togethe wth appopate PY appoxmaton (3 yele the value of h% (. Subsequently, h% ( an 01 h % ( 01 c, 11 togethe wth PY appoxmaton yele the value of h% (. 11 In each calculaton, Pca teaton algothm was apple to convegence. The nvese Foue tansfoms wee then apple to obtan h ( an h (, fom whch the pa coelatons g ( an g ( wee calculate. Each set of g ( an g ( was then use to compute the next g ( 00 fom (4, wtten n scetze fom as (7. N g( + = π ρ N n whee, N s the numbe of patcles n the ncement +, N s the total numbe of patcle n the system, an ρ s the numbe ensty of the system. To make t mensonless, the * euce numbe ensty s efne as ρ = ρ. The etals of the smulatons ae as the followng. Patcles ρ ( h% 00 ( k; ρ 00 ( k; ρ h% 01( k; ρ ae ae one by one onto a 0 x 0 suface, whee s the 01( k; ρ = ha-coe amete. The numbe of patcles ae s 0, 40, 1 + ρhˆ ( k; ρ o 80 coesponng to ρ * =0.05, 0.10, o 0.0, espectvely. ρh% ( k; ρ h% 01( k; ρ ρhˆ ( k; ρ h % 01( k; h% ( k; h % 01( k; 01( k; ρ ρ ρ ρ ρ ρ ρ 11( k; ρ = ρ hˆ ( k; ρ If the ae patcles ovelap wth pevously quenche patcles, they ae emove an new atons ae attempte untl the nseton s successful. Ths step s followe by classcal Metopols splacement steps, allowng the patcles to each equlbum une the effect of all othe patcles. The Makov geneates a anom walk such that the pobablty of vstng a patcula pont s popotonal to the patcle s Boltzmann facto at that locaton, exp( βu(. Thus, the move fom to was accepte wth pobablty ' { β } acc( o n = mn(1,exp U ( U( (9 Each patcle pefoms a mnmum of 1000 accepte moves wth a maxmum splacement of 0.5 befoe t s quenche n place. The peoc bounay conton s also apple. Ou fnal esults, aveage ove 100 ealzatons, ae pesente n the followng secton. IV. RESULTS AND DISCUSSIONS (8 g ( ρ g ( ; ρ + ρ ρ g ( ; ρ + ρ g ( ; ρ ( ; ρ+ 1 = ρ+ 1 The computaton s epeate at successvely lage values of the ensty, wth ρ =0.05, 0.10, o 0.0,.e. ρ+ 1 = ρ + ρ, untl the eque ensty s eache. III. MONTE CARLO SIMULATION We pefome Monte Calo smulatons to vefy the accuacy of the numecal esults fom the Onsten-Zenke equatons. The calculaton of the aal stbuton functon g( + / s, as customay, base on a hstogam fo small ncements of wth. The aveage stbuton functon s then tvally obtane as the ato between numbe of (7 The esults fom the ntegal-equaton theoy ae compae wth the ones fom smulatons. The monolaye was fome by factonal eposton of patcles on a homogeneous suface. Each ae facton s equvalent to the numbe ensty of 0.1. The examples ae shown fo the euce * system ensty, ρ, of 0., 0.3, an 0.4 n Fgues, 3, an 4, espectvely. As can be seen n Fg., at a low system ensty of 0., the smulaton gves esults n goo acco wth the theoetcal ones. (Avance onlne publcaton: 0 Novembe 008

4 Fg : The aal stbuton functon fo the system euce ensty of 0. geneate by epostng a facton of patcles whose ae ensty s 0.1. The theoetcal esults ae compae wth the smulaton ones. Fg 3: Smla to Fg., but the compason of the esults at the euce numbe ensty of 0.3 s mae. Fg 4: Smla to Fg., but the compason of the esults at the euce numbe ensty of 0.4 s mae. Fg 5: The aal stbuton functons fom the ntegal-equaton theoy, showng the ncease of shot-ange oe wth nceasng ρ *. Moe evatons ae obseve when the system ensty nceases. Ths s attbutable to the PY appoxmatons, n whch some agams of hghe oe of ensty ae neglecte. Howeve, the consstency between those s acceptable. Ths poves the valty of the ntegal-equaton theoy n explanng the evoluton of stuctues bult though factonal eposton. Snce the nteacton between a pa of patcles s fom a smple ha sphee moel, the stuctue of the aal stbuton functon shows only the shot-ange nteacton, epesente by only one pomnent peak. Ths peak s clealy seen hghe when the system ensty nceases, esultng fom the effect of patcle packng on the suface,.e. n a late aton, the patcles wll be eposte close to one anothe. Nomally, the eposton flux epens on the patcle concentaton n suspenson an the avalable space on the suface. Theefoe, the flux ung the monolaye gowth shoul not be assume constant as has been one n ths wok. Howeve, Tlton et al. [8] foun that the self-ffusvty of BSA asobe on PMMA was stongly epenent on suface concentaton an that between 15 mn. an 7 h. t was nepenent of asopton tme. Convesely, the moble facton oes not epen on suface concentaton but oes epen on asopton tme. The coexstence of moble an appaently mmoble potens appeas to esult not fom aggegaton of asobe BSA but fom a change n confomaton o oentaton of the asobe poten. The mpotant fnng n ths stuy s shown n Fg. 5, whee the effect of numbe of ae patcles s compae. The system ensty s kept constant at 0.4, wheeas the facton of patcles ae s 0.05, 0.10, an 0.0. It s shown that when the facton of ae patcles s ncease, even though thee s no enthalpc effect at all n the system, the entopc effect togethe wth the coopeatve effect of all movng patcles leas to hghe peak of the aal stbuton functon. Thus, the system of lage ρ * s a lttle moe compact an oee, as was pevously foun n [18]. The oee stuctue contans geate fee aea fo eposton than soee stuctues, theeby possessng hghe aea entopy whle losng confguatonal entopy. (Avance onlne publcaton: 0 Novembe 008

5 If the eposton contnues, the evesble natue of asobe patcles woul lea to the jammng, whee thee s no avalable space on the suface to accommoate a patcle. The jammng coveage fo the asopton fo ffeent ae moble facton wll be ffeent an t coul be etemne fom Monte Calo smulatons. The values shoul be geate than jammng coveage fo RSA an less than the satuaton coveage of equlbum asopton. Theeby, the factonal eposton coul be a smple moel fo bgng the gap between totally evesble as n RSA an completely evesble as n equlbum. V. CONCLUSION Ths wok popose the ntegal-equaton theoy base on the Onsten-Zenke equatons to escbe the gowth of monolaye stuctues by factonal eposton. The theoetcal esults ae n goo ageement wth the smulaton esults. Moeove, the theoy s supeo to Monte Calo smulaton snce the calculaton s faste. Howeve, to etemne the jammng coveage, the ntegal-equaton theoy fal to pect such hgh ensty as was scusse befoe. The Monte Calo smulaton woul then be the helpful tool. The ntegal-equaton theoy s amenable to nclue the vaaton of ρ * along the gowng pocess nstea of fxng the same numbe of ae patcle at one tme as was one n ths stuy. It s also flexble to use wth any types of ntemolecula potentals, ethe attactve o epulsve ones. [1] J. J. Ramsen, G. I. Bachmanova, an A. I. Achakov, Knetc evence fo poten clusteng at a suface, Phys. Rev. E. vol. 50, 1994, [13] P. Danwanchakul an E. D. Glant, Sub-Monolaye Gowth by Sequental Deposton of Patcles, J. Collo Inteface Sc. vol. 94, 006, [14] P. Danwanchakul an E. D. Glant, Contnuty Between Oe an Dsoe n the Sequental Deposton of Patcles, Chemcal Engneeng Communcatons. vol. 19, 005, 1-19 [15] W. G. Maen an E. D. Glant, Dstbuton functons fo flus n anom mea, J. Stat. Phys., vol. 51, 1988, [16] J. A. Gven an G. Stell, Comment on: Flu stbutons n two-phase anom mea: Abtay matces, J. Chem. Phys., vol. 97, Sep. 199, [17] J. P. Hansen an I. R. McDonal, Theoy of Smple Lqus (Acaemc, New Yok, 1986 [18] P. Danwanchakul, Revesblty n Patcle Deposton: The Effect of Moble Facton of Patcles on Monolaye Stuctues, J. Collo Inteface Sc., vol. 318, 008, REFERENCES [1] P. Schaaf an J. Talbot, Suface Excluson Effects n Asopton Pocesses, J. Chem. Phys., vol. 91, 1989, pp [] J. Fee an I. Gaeve, Asopton of fetn, J. Collo Inteface Sc., vol. 78, Nov. 1980, pp [3] G. Y. Onoa an E. G. Lnge, Expemental etemnaton of the anom-pakng lmt n two mensons, Phys. Rev. A, vol. 33, 1986, [4] W. Noe an J. Lyklema, The asopton of human plasma albumn an bovne panceas bonuclease at negatvely chage polystyene sufaces: I. Asopton sothems. Effects of chage, onc stength, an tempeatue, J. Collo Inteface Sc., vol. 66, Sep. 1978, [5] I. Mchael, D. R. Absolom, an C. J. van Oss, Dffuson of asobe poten wthn the plane of asopton, J. Collo Inteface Sc., vol. 77, Oct. 1980, [6] T. P. Bughat an D. Axelo, Total ntenal eflecton/fluoescence photobleachng ecovey stuy of seum albumn asopton ynamcs, Bophys. J., vol. 33, Ma. 1981, [7] R. D. Tlton, C. R. Robetson, an A. P. Gast, Lateal ffuson of bovne seum albumn asobe at the sol-lqu nteface, J. Collo Inteface Sc., vol. 137, Jun. 1990, [8] R. D. Tlton, A. C. Gast, an C. R. Robetson, Suface Dffuson of Inteactng Potens: Effect of Concentaton on the Lateal Moblty of Asobe Bovne Seum Albumn, Bophys. J., vol. 58, 1990, [9] V. Chan, D. J. Gave, P. Fotna, an S. E. Mckenze, Asopton an suface ffuson of DNA olgonucleotes at lqu/sol ntefaces, Langmu, vol. 13, 1997, [10] R. Kuat, J. E. Penosl an J. J. Ramsen, Knetcs of Human an Bovne Seum Albumn Asopton at Slca Ttana Sufaces, J. Collo Inteface Sc. vol. 185, 1997, 1-8 [11] C. T. Shbata an A. M. Lenholf, TIRF of salt an suface effects on poten asopton : I. Equlbum, J. Collo Inteface Sc. vol.148, 199, (Avance onlne publcaton: 0 Novembe 008