ISSN 06-933X Colloid oural 0 Vol 73 No 3 pp 406 47 Pleiades Publishig Ltd 0 Origial Russia Text AK Shhei S Kshevetsiy OS Pelevia 0 published i Kolloidyi Zhural 0 Vol 73 No 3 pp 404 46 iellizatio Kietis with Allowae for Fussio ad Fissio of Spherial ad Cylidrial ielles: Set of Noliear Equatios Desribig Slow Relaxatio A K Shhei S Kshevetsiy ad O S Pelevia Faulty of Physis St Petersburg State Uiversity ul Ul yaovsaya Petrodvorets St Petersburg 98504 Russia Reeived uly 00 Abstrat Based o the geeral ieti equatio that desribes the aggregatio ad fragmetatio of surfatat moleular aggregates a losed set of oliear equatios is derived for the slow relaxatio of surfatat moomer oetratio ad the total oetratios of oexistig spherial ad ylidrial mielles to the equilibrium state of a miellar solutio Both the trasitios aompaied by the emissio ad apture of surfatat moomers by mielles ad the trasitios resultig from the fussio ad fissio of mielles are tae ito aout The derived set of equatios desribes all stages of the slow relaxatio from the iitial perturbae to the fial equilibrium state of a miellar solutio DOI: 034/S06933X0304 INTRODUCTION The ability of amphiphili surfatat moleules to aggregate with the formatio stable aggregates (mielles) i aqueous solutios attrats great iterest from the viewpoit of both diverse physiohemial ad tehologial appliatios ad the peuliarities of miellizatio mehaism itself This iterest is to a high extet due to the polymorphism of mielles ie the possible formatio of aggregates with differet shapes eg spherial ylidrial or threadlie At a fixed temperature of a miellar solutio trasitios from oe shape of mielle to aother are stimulated by a irease i the overall surfatat oetratio i the solutio For example at a overall oetratio above the first ritial mielle oetratio (CC ) spherial mielles are formed i a solutio while above the seod ritial mielle oetratio (CC ) exteded ylidrial mielles are formed with otieably larger aggregatio umbers tha those of the spherial mielles At preset there is various experimetal ad theoretial evidee that whe the polymorphism of mielles taes plae the wor of moleular aggregate formatio (aggregatio wor) i surfatat solutios with oetratios above CC demostrates a maximum ad a miimum i the aggregatio umber axis whih are related to ritial ulei of mielles ad stable spherial mielles respetively ad at higher aggregatio umbers it exhibits a seod maximum ad a wide flat regio of a liear rise i whih ylidrial mielles are aumulated [ 3] The latter irumstae auses the appearae of substatial peuliarities i the miellizatio ietis as ompared with the situatio below CC i whih oly a maximum ad miimum exist for the wor of spherial aggregate formatio Experimetal studies of relaxatio pheomea i miellar solutios were always importat soures of iformatio o the struture ad properties of mielles Relaxatio proesses relevat to the trasitios from premiellar moleular aggregates to mielles ad ba or trasitios betwee mielles of differet polymorphous modifiatios are assiged to slow proesses aompayig miellizatio The term fast relaxatio implies a loal rearragemet i the distributio of mielles over aggregatio umbers toward the most probable (quasi-equilibrium) distributio that proeeds withi relatively short time periods without variatios i the total umbers of mielles at urret values of system parameters Alog with overomig the formatio wor barrier betwee spherial ad ylidrial mielles by meas of solitary (step-by-step) variatios i the aggregatio umber of aggregates upo the apture ad emissio of surfatat moleules as the oetratio of mielles ireases the forward ad ba umps over the barrier due to the fussio of spherial mielles ad the fissio of ylidrial oes beome possible [4 5] All of these fats must be tae ito aout whe formulatig the ietis of miellizatio ad the theory of relaxatio i miellar solutios Variatios i the distributio of mielles via the apture ad emissio of moomers by surfatat moleular aggregates will be regarded as a moleular mehaism of substae exhage betwee aggregates ad solutios The mehaism of variatios i aggregate distributio through the fussio ad fissio of mielles themselves will be referred to as the fussio fissio mehaism At small deviatios from quasi- 406
ICELLIZATION KINETICS WITH ALLOWANCE FOR FUSSION AND FISSION 407 equilibrium or true equilibrium the moleular mehaism of the fast ad slow relaxatio of a miellar solutio (liear relaxatio) was osidered i [6 0] while i [ ] it was ivestigated at arbitrary iitial deviatios (oliear relaxatio) The approah to the developmet of the liear relaxatio theory for the fussio fissio mehaism was proposed by Wato [5] ad Kahlweit [4] The problem oerig the simultaeous realizatio of the moleular ad fussio fissio mehaisms has ot bee systematially disussed i terms of the miellar relaxatio theory I this wor the ieti desriptio of the miellar relaxatio proeedig uder the oditios of oexistee spherial ad ylidrial mielles will be developed based o the geeral ieti equatio for the aggregatio ad fragmetatio of surfatat moleular aggregates This equatio whih geeralizes the Beer Dörig ad Smoluhowsi ieti equatios taes ito aout the realizatio of both the moleular ad fussio fissio mehaisms It is the moleular mehaism that govers the establishmet of idividual quasi-equilibrium distributios of spherial ad ylidrial mielles whih represet the startig distributios for the slow relaxatio i miellar solutios I this paper a losed set of oliear equatios will be obtaied to desribe the slow relaxatio proess for surfatat moomer oetratio ad the total oetratios of oexistig spherial ad ylidrial mielles from a iitial arbitrary perturbae to the fial equilibrium state of a miellar solutio with simultaeous allowae for the moleular ad mielle fussio fissio mehaisms GENERAL INFORATION Assume that we have a solutio of a oioi surfatat i a polar solvet The solutio is supposed to be ideal with respet to the surfatat moomers ad surfatat moleular aggregates beig formed The umber of surfatat moleules i a moleular aggregate (aggregatio umber) i will be applied as the harateristi of its iteral state Let us write a li of aggregate trasitios alog the aggregatio umber axis that result from the apture ad emissio of a surfatat moomer (ie at the moleular mehaism of aggregatio) as follows: a i {} i + {} { i + } i 3 bi + Here { i} refers to aggregates otaiig i moleules a i is the umber of moomers aptured by aggregate { i} from a solutio with moomer oetratio for uit time ad b i+ is the umber of moomers emitted by aggregate { i+} ito the solutio for uit time Deotig the oetratio of aggregates (the umber of aggregates i uit volume of the solutio) with aggregatio umber i as i the developmet of the aggregatio with time through the moleular mehaism is desribed by the followig ieti equatio: i ( i i ) t where t is the time ad i is the flux of aggregates alog the aggregatio umber axis Flux i is here speified with regard to the meaig of above-itrodued rates ai ad b i+ by the followig relatio: i a i ( t) i( t) bi + i + ( t) (3) Expressio alog with the defiitio of the flux (3) is referred to as the Beer Dörig ieti equatio I additio to the moleular mehaism the umber of aggregates i a miellar solutio a vary due to the fussio ad fissio of moleular aggregates themselves (ie through the fussio fissio mehaism) The li of the forward ad ba trasitios that are realized i this ase may be represeted as follows: a {} i i + { } { i + } b i (4) Here as before { i } { } ad { i + } refer to aggregates omposed of i ad i+ moleules respetively; ai is the umber of the evets of additio of aggregates omprisig moleules to aggregates omposed of i moleules for uit time; ad b i is the umber of fissio evets of aggregates otaiig i+ moleules ito aggregates osistig of i ad moleules for uit time It is obvious that the aggregatio through the fussio fissio mehaism omprises the moleular mehaism as a partiular ase i whih or i Developmet of the aggregatio through the fussio fissio mehaism a be desribed by the followig geeralized ieti equatio (see [3] Eq (44) i [4]): i i t t i i (5) The values i ai ( t) i( t) bi i + ( t) (6) represet the fluxes of aggregates alog the aggregatio umber axis upo the fussio of aggregates with aggregatio umbers i ad ad the fissio of aggregates with aggregatio umber i+ ; ai ( t) i( t) ad bi i + ( t) are the umbers of fussio evets of aggregates { i } { } fissio evets of aggregates { i + } for uit time i uit volume of the system I the literature Eq (5) is also referred to as the oagulatio fragmetatio equatio or the geeralized Smoluhowsi equatio I the theory of miellizatio it is referred COLLOID OURNAL Vol 73 No 3 0
408 SHCHEKIN et al W W W W 0 W s W s to as the mielle fussio fissio equatio; here ad below we shall use this ame Beause the proess of the fussio ad fissio of surfatat aggregates i a solutio is idepedet of the fat whether a ith aggregate is added to a th aggregate or vie versa the th aggregate is added to a ith aggregate ad aalogously the fissio of the ( i + ) th aggregate ito ith ad th aggregates is equivalet to the fissio ito th ad ith aggregates flux i must be symmetri with respet to the permutatio of the subsripts ie s i i s 0 Fig Typial form of aggregatio wor W as a futio of aggregatio umber for surfatat moomers at surfatat solutio oetratios above CC Supersripts ad deote orrespodig poits of maxima ad miima i wor W Subsripts s ad refer to stable ad ritial aggregates respetively Idex 0 idiates oset of liear W depedee (7) The symmetry of the fussio a i ad fissio b i oeffiiets also follows from this irumstae: ai a i bi b i (8) I the fial state of the aggregatio equilibrium of a miellar solutio all fluxes i must be redued to zero I this situatio the followig detail balae relatios tae plae: ai i bi i + (9) Here ad below the tilde refers to the values relevat to the fial state of the omplete equilibrium I additio to the form (5) of the fussio fissio equatio whih is disrete with respet to aggregatio umbers a otiual form of this equatio is also ow The properties of the otiual fussio fissio equatio were desribed i the literature (see eg [5]) I this ommuiatio we ofie ourselves to the disrete form (5) of the fussio fissio equatio 3 QUASI-EQUILIBRIU CONCENTRATIONS OF ONOERS PREICELLAR AGGREGATES SPHERICAL ICELLES AND CYLINDRICAL ICELLES Note that all surfatat moleular aggregates preset i a solutio may be divided ito the groups omprisig premiellar aggregates spherial mielles ad ylidrial mielles We assume that i eah group of aggregates the fast relaxatio proeeds via the exhage of moomers whih is eded by the establishmet of the aggregatio quasi-equilibrium withi a give group at a urret moomer oetratio Hee the fussio ad fissio of mielles are relatively ifrequet evets; therefore durig the slow establishmet of the omplete aggregatio equilibrium the detail balae relatios will be realized i eah group via the moleular mehaism I this ase whe desribig the slow relaxatio i eah group of aggregates (whe i ad i+ ) i Eq (6) belog to the same group) at by aalogy with (9) we obtai the followig relatio from Eq (6): ai ( t) i( t) bii + ( t) (3) At > ad arbitrary i values equality (9) is oly fulfilled i the fial equilibrium state Figure qualitatively demostrates aggregatio wor W as a futio of aggregatio umber for the ase i whih the overall surfatat oetratio i a solutio is higher tha CC ad spherial ad ylidrial mielles oexist Let us itrodue W W s W values Aggregatio wor W at the poit of its first maximum i the aggregatio umber axis determies the height of the ativatio barrier for the formatio of spherial mielles aggregatio wor W s at the poit s of its first miimum haraterizes the depth of the potetial well i whih spherial mielles are aumulated; the differee betwee aggregatio wor W at the poit of its seod maximum ad W determies the height of the ativatio barrier for the formatio of ylidrial mielles Aggregatio wor W0 is determied at the poit 0 that orrespods to the left-had boudary of the aggregatio umber rage for ylidrial mielles i whih the depedee of W o is already liear [8] Aordig to the aforesaid i Fig the rage of the aggregatio umbers is divided ito three rages: < < ad The first rage orrespods to premiellar aggregates (we shall deote the orrespodig group of aggregates as group 0 ) the seod rage omprises ylidrial ad traylidrial mielles (group ) ad the third rage o- tais ylidrial mielles (group ) Taig ito aout Eq (3) the distributios of premiellar aggregates spherial mielles ad yli- COLLOID OURNAL Vol 73 No 3 0
ICELLIZATION KINETICS WITH ALLOWANCE FOR FUSSION AND FISSION 409 drial mielles represet the followig quasi-equilibrium Boltzma distributios [8]: (3) where At ad Bt are ormalizatio oeffiiets whih are idepedet of aggregatio umber The arrows i relatios (3) deote the limits to whih the distributios ted i the fial equilibrium sate Stritly speaig at the stage of the slow relaxatio relatios (3) are ivalid i the viiities of poits ad orrespodig to the aggregatio wor maxima I these viiities at the stage of the slow relaxatio the quasi-statioary distributios of aggregates develop whih are osidered i greater detail i Appedix However the amout of aggregates with aggregatios umbers appearig i the viiities of poits ad orrespodig to the aggregatio wor maxima is small to ompare with the total amouts of premiellar aggregates ad spherial ad ylidrial mielles With regard to this fat with orrespodig reservatios we will igore the iauray of formulas (3) i the viiities of poits ad wheever possible Expressio (3) may be rewritte i the followig form: where W t W t t e e 0 W t W t A t e e W t W t B t e e t X t e (33) ( t) 0 X t A t (34) B t oreover we tae ito aout the fat that aggregatio wor W ( t) i a ideal solutio of surfatat moomers a be preseted as follows [7 8]: (35) where W is the part of aggregatio wor that depeds oly o aggregatio umber I this ase the expressio for aggregate oetratio (33) a be writte as follows: X W e (36) ad aordigly for fial equilibrium oetratio we obtai the equatio W( t ) W t W ( l ) ( t) ( ) e W (37) Substitutig distributio (36) ito relatio (6) for flux i ad usig detail balae relatio (9) i ombiatio with distributio (37) we arrive at X i + i ai i XX i (38) Note that similar to distributio (3) expressio (38) for flux i is oly valid i the rages where the quasi-equilibrium of aggregates taes plae ad this expressio is violated whe aggregatio umbers i or i+ fall ito the viiities of poits ad orrespodig to the maxima Relatios desribig fluxes i i the viiities of ad are disussed i Appedix I the state of the omplete aggregatio equilibrium the paretheti expressio i the right-had side of Eq (38) is redued to zero This expressio depeds o the groups to whih aggregates {i} {} ad {i + } belog rather tha o speifi values of i ad As follows from Eq (38) there are three ases i whih flux i is idetially redued to zero (uder the oditio that the values of i ad i + do ot lie i the viiities of poits ad ) The followig evets of aggregate fussio fissio orrespod to these ases: premiellar premiellar + aggregate aggregate premiellar aggregate premiellar spherial + aggregate aggregate spherial aggregate premiellar ylidrial 3 + aggregate aggregate ylidrial aggregate Let us pass from oetratios to physially measurable parameters The total aggregate oetratio the average aggregatio umber ad the dispersio of aggregatio umbers are ommoly applied as these parameters For eah group of aggregates i ( i 0 ) total oetratio i i average size COLLOID OURNAL Vol 73 No 3 0
40 i ad dispersio Δ at arbitrary time momet t are determied as follows: ( 0) ( 0) t 0 t 0 0 ( 0) ( 0) ( Δ ) ( 0) ( ) t 0 t t (39) ( Δ ) t t t ( Δ ) t Igorig the otributios from the viiities of poits ad orrespodig to the maxima to the values that are determied by relatios (39) ad taig ito aout expressios (36) ad (34) we fid that is a futio of A ad ; is a futio of B ad ; ad other values are futios of aloe Differetiatio of oetratio i ( i 0 ) with respet to time t usig expressios (34) ad (36) ad defiitios (39) we derive the followig relatios: ( 0) dl ( 0) dl (30) d l l d l d A + ( ) (3) d l l d l d B + ( ) (3) The first of these equatios idiates that time variatios i are exlusively determied by the behavior (0) of moomer oetratio ( t) Aordigly Eq (30) is ot of self-depedet iterest for us (0) beause we a always restore from a ow ( t) futio The right-had sides of the other equatios omprise uow values A B ad their futios Now we should ote that the left-had sides of Eqs (3) ad (3) a be foud usig ieti fussio fissio equatio (5) whih will be show i subsequet setios I additio to relatios (3) ad (3) we should also osider the law of oservatio of surfatat amout i a solutio For defiiteess we will osider a isolated system i whih the amout of a surfatat i uit volume of a solutio is ostat ad equal to overall oetratio I this ase we have SHCHEKIN et al t + t + t 0 or with regard to defiitios (39) COLLOID OURNAL Vol 73 No 3 0 (33) 0 0 + + (34) Differetiatig the left- ad right-had sides of equality (33) with respet to time t usig expressios (34) ad (36) ad defiitios (39) ad exludig da ad db derivatives usig relatios (3) ad (3) we derive a equatio that expresses the derivative of moomer oetratio ( t) via derivatives d ad d i the followig form: dl d d (35) + ( 0) ( 0) 0 + Δ + Δ + Δ ( ) This equatio speifies the relatio betwee A ad B whih additioally determies the set of Eqs (3) ad (3) The etire system will be losed provided that the d ad d derivatives are determied as futios of A ad B 4 EQUATION FOR TOTAL CONCENTRATION OF SPHERICAL ICELLES I order to desribe variatios i the total umber of spherial mielles for uit time i uit volume of a solutio let us sum the left- ad right-had sides of Eq (5) over the rage of spherial mielles (ie over all i values from to ) Thus we obtai i i t t i i i i i (4) With allowae for from Eq (39) this relatio may be rewritte as follows: d i i i i i (4) I the subsequet aalysis of this equatio we shall distiguish betwee the two followig ases: ad > ( ) Let us first osider the ase i whih ( ) It is oveiet to graphially aalyze the otributios to the right-had side of Eq (4) For this ase the positios of the summed terms i the first ad seod sums of Eq (4) i the ( i ) ad ( i ) plaes is illustrated i Fig The otributios from the regios similarly deoted i Figs a ad b
ICELLIZATION KINETICS WITH ALLOWANCE FOR FUSSION AND FISSION 4 (а) (b) D 0 D D D0 D D C 0 C0 C D C D B 0 B 00 A 000 B B 0 B 0 C 0 D 0 B 00 A 000 i B B 0 C0 D0 i Fig Pael (a): positios of fluxes i i the i plae uder the oditios i < ad < i ad pael (b): positios of fluxes i i the (i) plae uder the oditios i ad The shaded regios refer to the positios of the summed terms i the right-had side of Eq (4) at ( ) oiide with oe aother Here A lm Blm Clm ad D lm refer to the sum of all i fluxes over a orrespodig regio i Fig while the subsripts deote the ollisios of aggregates of the th type with aggregates of the l th type that yield aggregates of the mth type (0 refers to premiellar aggregates deotes spherial ad traspherial aggregates ad idiates ylidrial mielles) Let us rewrite Eq (4) for the ase ( ) usig the deotatios of the regios preseted i Fig ad taig ito aout the orrespodig oeffiiets of Eq (4) As a result Eq (4) taes the followig form: d ( B00 + B + B0 + B0) ( B + B + C + C + D ) 0 0 (43) Reduig the right-had side of Eq (43) we arrive at d ( B00 + B0 B0 B) ( C + C + D ) 0 (44) Taig ito aout the symmetry of Eq (7) it is easy to show that B B C C D D lm lm lm lm lm lm Therefore Eq (44) a be rewritte as follows: (45) d ( B00 B) ( C0 + C + D) ( ) ( ) (46) Note that the regios of B0 ad B0 fell out of osideratio Hee the fussio ad fissio of aggregates ourrig i these regios have o effet o variatios i the oetratio of spherial mielles Taig ito aout the meaig of deotatios B00 B C0 C D ad returig to the sums of the fluxes Eq (46) a be writte i the followig expliit form: d i i i i i i i i i (47) Now we shall osider the ase i whih > ( ) For this ase the positios of the summed terms i the first ad seod sums of Eq (4) i the ( i ) ad ( i ) plaes are preseted i Fig 3 The otributios from the regios similarly deoted i Figs 3a ad 3b oiide with eah other Here A lm Blm Clm ad Dlm refer to the sum of all i fluxes over the orrespodig regio i Fig 3; moreover the subsripts deote the ollisios of aggregates of the COLLOID OURNAL Vol 73 No 3 0
4 SHCHEKIN et al (а) (b) D 0 D D D 0 D D B 0 C 0 C D C 0 C D B 0 C 00 C 00 B 00 C 0 D 0 B 00 C 0 D 0 A 000 A 000 B 0 B 0 i i Fig 3 Same as i Fig but shaded regios refer to positios of summed terms i the right-had side of Eq (4) for the ase > ( ) th type with aggregates of the l th type that yield aggregates of the mth type (0 refers to premiellar aggregates deotes spherial ad traspherial aggregates ad idiates ylidrial mielles) Note that although the boudaries of some regios similarly deoted i Figs ad 3 are differet the sums orrespodig to these regios desribe the fussio ad fissio of aggregates of the same type As will be show i Appedix these sums a be writte uiformly For the osidered ase > ( ) Eq (4) may be rewritte by applyig the deotatios of the regios i Fig 3 ad taig ito aout orrespodig oeffiiets of Eq (4) As a result relatio (4) taes the followig form: d ( B00 + B0 + B0) (48) ( B0 + C0 + C + D ) Trasformig the right-had side of Eq (48) we arrive at d ( B00 + B0 B0) ( C0 + C + D) (49) With allowae for the symmetry of Eq (45) formula (49) may be rewritte as follows: d B00 ( C 0 +C +D ) > ( ) (40) Note that as before the regios of B0 ad B0 fell out of osideratio With regard to the meaig of symbols B00 C0 C ad D ad returig to the sums of the fluxes Eq (40) a be writte i the followig expliit form: d i i i i + i i (4) The right-had sides of Eqs (47) ad (4) deped oly o A ad B values This olusio follows from expressio (38) expressio (A8) ad ommets preseted i Appedix Thus these equatios supplemet the set of equatios (3) (3) ad (35) for the two opposite ases uder osideratio To lose the system it remais oly to derive a expressio for d This will be doe i the ext setio 5 EQUATION FOR TOTAL CONCENTRATION OF CYLINDRICAL ICELLES I order to desribe variatios i the total umber of ylidrial mielles for uit time i uit volume of a solutio let us sum the left- ad right-had sides of Eq (5) over the regio of ylidrial mielles (ie over all values i ) Thus we obtai the followig: COLLOID OURNAL Vol 73 No 3 0
ICELLIZATION KINETICS WITH ALLOWANCE FOR FUSSION AND FISSION 43 (а) (b) D 0 +D D D 0 +D D C D 0 +D D 0 +D C i i Fig 4 Pael (a) positios of fluxes i i the i plae uder the oditios i ad i ad pael (b): positios of fluxes i i the ( i ) plae uder oditios i ad Shaded regios refer to positios of summed terms i right-had side of Eq (5) i (t) i i i t i i i (5) With allowae for the determiatio of from (39) this relatio may be rewritte i the form of the followig equatio: d i i i i i (5) The positios of the summed terms i the first ad seod sums of Eq (5) i the ( i ) ad ( i ) plaes is illustrated i Fig 4 The otributios from the regios similarly deoted i Figs 4a ad 4b oiide with oe aother Here C ad Dlm refer to the sum of all i fluxes over orrespodig regio i Fig 4 while the subsripts deote the ollisios of aggregates of the th type with aggregates of the l th type that yield aggregates of the mth type (0 refers to premiellar aggregates deotes spherial ad traspherial aggregates ad idiates ylidrial mielles) Note that regio C represets the sum of all regios of the C lm type that were previously represeted i Fig ( ) or i Fig 3 ( > ) Let us rewrite Eq (5) usig the deotatios of the regios i Fig 4 ad taig ito aout the orrespodig oeffiiets of formula (5) Thus Eq (5) taes the followig form: d ( C + D0 + D + D0 + D + D) (53) ( D + D + D ) 0 Reduig the right-had side of Eq (53) with regard to the symmetry of relatio (45) we obtai d ( C D) (54) Note that the regios of D0 D0 D ad D fell out of the osideratio This fat idiates that the fussio ad fissio of aggregates relevat to these regios do ot affet variatios i the oetratio of ylidrial mielles For the ase (see Fig ) Eq (54) is trasformed ito the followig relatio: d C 0 + ( C D) (55) Aordigly for the ase > (see Fig 3) Eq (54) yields the followig: d C0 + ( C00 + C D ) > ( ) (56) COLLOID OURNAL Vol 73 No 3 0
44 SHCHEKIN et al Now returig to Eq (54) taig ito aout the meaig of symbols C ad D ad omig ba to the sums of the fluxes we arrive at d i i i i i (57) The right-had side of Eq (57) depeds oly o A ad B This fat follows from expressios (38) ad (A9) ad ommets give i Appedix Thus this equatio supplemets the set of Eqs (3) (3) (35) (47) ad (4) ad loses it 6 CONCLUSIONS Let us tae all equatios for uow ( t ) A( t) ad B( t) values together ad trasform them ito a losed set of oliear equatios that desribe the slow relaxatio The expressios give below follow from Eqs (39) (3) (3) ad (35): W e dla dl l d W e W e l dl d B dl W e (6) (6) dl W l dl d A e + B e P W (63) where W P ( ) e + 0 W e W + A e + W ( ) e (64) W e W +B e W ( ) e I view of relatios (46) (40) (55) ad (56) we have ( B00 B) ( C 0 +C +D ) d ( ) (65a) B00 ( C 0 +C +D ) ( > ) C 0 + ( C D ) ( ) d C 0 + ( C 00 +C D ) (65b) ( ) ( > ) As will be show i Appedix the right-had sides of these equatios represet ow futios of A B ad Thus the oliear system of equatios that COLLOID OURNAL Vol 73 No 3 0
ICELLIZATION KINETICS WITH ALLOWANCE FOR FUSSION AND FISSION 45 desribes the slow relaxatio is etirely determied with respet to the A B ad values I the right-had side of Eqs (65) let us isolate the otributio of the moleular mehaism whih is realized via the apture ad emissio of sigle surfatat moleules The results preseted i Appedix suggest that this otributio is oly due to oeffiiets B00 ad C0 Taig ito aout oly this otributio Eqs (65) (A) ad (A4) yield the followig relatios: d (66a) d (66b) As will be show i Appedix ad where ad are the quasi-statioary fluxes through the first ad seod maxima of the aggregatio wor for the moleular mehaism of miellizatio With regard to expressios (A8) ad (A9) Eqs (66) may be writte as d A A B ' i ' i W a ( ) e i a e i d A B i ' a ( ) e i (67a) (67b) Equatios of this type were earlier aalyzed [7 8] usig the otiual form of the Beer Dörig equatio APPENDIX QUASI-STATIONARY DISTRIBUTIONS OF AGGREGATES AND FLUXES i IN THE VICINITIES OF THE AXIA OF AGGREGATION WORK Let us osider the flux i of aggregates betwee sizes {} i ad { i + } Aordig to Eq (6) ad with regard to detail balae relatio (9) we have a (A) I the viiities of the maxima of aggregatio wor flux i ad oetratio i must be quasi-statioary beause they relate the quasi-equilibrium distributios of aggregates o the left ad right of the maxima W i i + i i i i i + W Solvig Eq (A) relative to i+ we obtai i i i+ i+ (A) i a i i The from formula (A) we may obtai the followig expressio for arbitrary oetratio ( > i) by reursio: i i i a (A3) i ( > i) Relatio (A3) maes it possible to fid both the quasi-statioary oetratios of aggregates i i the viiities of the maxima of aggregatio wor W i ad quasi-statioary flows i through the orrespodig maxima By virtue of the quasi-statioarity of oetratios i i the viiities of the maxima of aggregatio wor W i igorig fluxes i (whih a be ustified whe the oetratio of moomers substatially exeeds the oetratio of ay other aggregates) from Eq (5) we obtai the followig: (A4) i i + i + i + where is the quasi-statioary flux of aggregates through a orrespodig maximum of the aggregatio wor With allowae for this relatio distributio (A3) taes the followig form: i i i a (A5) i ( > i) I order to determie the value of quasi-statioary flux let us perform the mathig of quasi-statioary distributio (A5) with quasi-equilibrium distributios (3) o the left ad the right of the maximum poit Poits loated at the boudary of the appliability domai of formulas (3) at whih the oditios of the quasi-equilibrium are still fulfilled are tae as the poits of mathig Let the left poit of mathig oiides with poit i while the right poit of mathig is deoted as i ' The substitutig quasi-equilibrium relatios (36) for i ad i ' ito Eq (A5) taig ito aout equality (37) ad determiig flux from the resultig expressio we arrive at i ' i X i X a ( ) e (A6) Let us determie mathig poits i i ' ad i i ' for the first ad the seod maxima respetively i the i' W COLLOID OURNAL Vol 73 No 3 0
46 urve desribig the aggregatio wor (Fig ) by the followig oditios: i W W W W i SHCHEKIN et al (A7) ie by the oditios that at the mathig poits o the left ad the right of the maxima the wor dereases by uity Aordig to relatios (35) ad (A7) the mathig poits deped oly o surfatat moomer oetratio ad parameters of the aggregatio wor As a result for quasi-statioary flux of aggregates through the fist maximum of the aggregatio wor with regard to relatios (34) ad (A6) we have Aordigly for the seod maximum we fid (A8) (A9) Quasi-statioary oetratios i the viiities of the poits for the maxima of the wor are foud by substitutig relatios (A8) or (A9) ito Eq (A5) at ii or aordigly ii I tur the relatio thus obtaied maes it possible to determie quasi-statioary fluxes i usig Eqs (6) ad (36) at aggregatio umbers i or i + lyig i the viiities of the maxima It is obvious that these fluxes will deped o time via A ad B i' W W W W A i i i i ' ' i W a ( ) e i i' i A B i i i ' ' i W a ( ) e i i ( i ) B i i i + ai i i i i + B C 00 i ( i ) C00 i + i i ai i 0 0 i ( i ) C0 i+ i i + C C i ai i i max ( + ) i C i ( i ) C i max ( i) i ai i (A) (A3) (A4) (A5) (A6) (A7) i i+ ai i i i i + With regard to relatios (36) (A5) (A8) ad (A9) ad ommets preseted i Appedix the right-had sides of relatios (A) (A7) deped o time oly via A ad B values i+ i i + i+ i i + D D i ( i ) D i i+ ai i i i i + D i ( i ) D APPENDIX EXPRESSIONS FOR COEFFICIENTS B C D lm Let us derive expressios for the sums of fluxes that orrespod to the regios of Blm Clm ad Dlm i Figs ad 3 We shall oly osider the regios that eter ito the right-had sides of formulas (46) (40) (55) ad (56) With allowae for relatios (6) ad (9) ad the symmetry of oeffiiets i Eq (45) we have the followig: B 00 i ( i ) B00 mi i i i lm ( ) i i ai i lm + i i + (A) ACKNOWLEDGENTS This wor was supported by the Program o 6 for Basi Researh Divisio of Chemistry ad aterials Siee Russia Aademy of Siees Chemistry ad Physiohemistry of Supramoleular Systems ad Atomi Clusters ad the departmetal aalytial goal-orieted program Developmet of the Sietifi Potetial of the Higher Shool (009 00) proet o /4430 REFERENCES Porte G Poggi Y Appell ad aret G Phys Chem 984 vol 88 p 573 ay S ad Be-Shaul A Phys Chem B 00 vol 05 p 630 COLLOID OURNAL Vol 73 No 3 0
ICELLIZATION KINETICS WITH ALLOWANCE FOR FUSSION AND FISSION 47 3 Kshevetsiy S ad Shhei AK Kolloid Zh 005 vol 67 p 363 4 Kahlweit Pure Appl Chem 98 vol 53 p 069 5 Wato G Phys Chem B 997 vol 0 p 977 6 Aiasso EAG ad Wall SN Phys Chem 974 vol 78 p 04 7 Kui F Rusaov AI Shhei AK ad Grii AP Zh Fiz Khim 005 vol 79 p 967 8 Kui F Shhei AK Rusaov AI ad Grii AP Lagmuir 006 vol p 534 9 Kui F Shhei AK Grii AP ad Rusaov AI Kolloid Zh 005 vol 67 p 497 0 Kshevetsiy S Shhei AK ad Kui F Kolloid Zh 008 vol 70 p 49 Shhei AK Kui F ad Shahov KS Kolloid Zh 008 vol 70 p 70 Kshevetsiy S ad Shhei AK Chem Phys 009 vol 3 p 03093 3 Ball G ad Carr G G Stat Phys 990 vol 6 p 03 4 Wattis AD Phys D (Amsterdam) 006 vol p 5 Dubovsii PB ad Stewart IW i athematial ethods i the Applied Siees 996 vol 9 p 57 6 Nagaraa R Surfatat Si Ser 003 vol p COLLOID OURNAL Vol 73 No 3 0