An Algorithm for Emulsion Stability Simulations: Account of Flocculation, Coalescence, Surfactant Adsorption and the Process of Ostwald Ripening

Transcription

1 Int. J. Mo. Sc. 9,, 76-84; do:.339/ms376 evew OPEN ACCESS Internatona Journa of Moecuar Scences ISSN An Agortm for Emuson Stabty Smuatons: Account of Foccuaton, Coaescence, Surfactant Adsorpton and te Process of Ostwad penng German Urbna-Vaba Insttuto Venezoano de Investgacones Centífcas (IVIC), Centro de Físca, Lab. Fscoquímca de Coodes, Aptdo. 63, Caracas, Venezuea; E-Ma: Te ; Fax: eceved: 8 November 8; n revsed form: January 9 / Accepted: 3 January 9 / Pubsed: 6 February 9 Abstract: Te frst agortm for Emuson Stabty Smuatons (ESS) was presented at te V Conferenca Iberoamercana sobre Equbro de Fases y Dseño de Procesos [Lus, J.; García-Sucre, M.; Urbna-Vaba, G. Brownan Dynamcs Smuaton of Emuson Stabty In: Equfase 99. Lbro de Actas, st Ed., Too J., Arce, A., Eds.; Soucon s: Vgo, Span, 999; Voume, pp ]. Te former verson of te program conssted on a mnor modfcaton of te Brownan Dynamcs agortm to account for te coaescence of drops. Te present verson of te program contans eaborate routnes for tme-dependent surfactant adsorpton, average dffuson constants, and Ostwad rpenng. Keywords: Foccuaton, Coaescence, Adsorpton, Surfactant, Drops, Ostwad, Smuatons, Emuson, Deformaton.. Introducton Emusons are dspersons of two mmscbe quds, knetcay stabzed by te acton of a surface-actve substance known as a surfactant. Te roe of te surfactant n emuson stabty s decsve. Even te externa pase of te emuson resutng from te mxng of o (O) and water (W) depends on te surfactant soubty [-5]. In te absence of surfactants, drops aggregate rapdy as a consequence of te van der Waas force. Surfactants adsorb to te nterface of te drops creatng a repusve barrer tat deceerate aggregaton and Ostwad rpenng. Tey aso favor te occurrence of mmobe nterfaces, deayng te dranage of te ntervenng fm between foccuated drops [6].

2 Int. J. Mo. Sc. 9, 76 Conversey, surfactants ower te nterfaca tenson of tese O/W/O fms, favorng te appearance of surface oscatons and oes. Dependng on ter nterfaca propertes, tese fms can dran and rupture or reman stabe for ong perods of tme [7-]. In order to understand te roe of te surfactant n te stabty of emusons, smuatons can be a vauabe too. However, tere are severe computatona restrctons for te smuaton of emusons apart from te arge number of processes nvoved. Frst, te number of surfactant moecues n a typca system s very g even at dute concentratons. Hence, t s not possbe to smuate te movement of surfactant moecues expcty aong wt te movement of te drops. Second, te tme step of te smuaton as to be very sma n order to sampe appropratey te potenta of nteracton between te drops. Trd, drops of sma sze exbt Brownan moton due to ter terma nteracton wt te sovent []. Ts as to be taken nto account nto ter equaton of moton. Fourt, t s not possbe to smuate te beavor of drops usng a constant potenta as t s done n moecuar dynamcs. Te potenta of nteracton canges wt tme as a consequence of surfactant adsorpton and te progressve decrease of nterfaca area of te emuson due to te coaescence of drops. Ts demands te frequent cacuaton of te nterfaca propertes tat determne te forces between te partces. In ts revew we concentrate on te evouton of o-n-water (O/W) emusons composed of nondeformabe drops. Te next secton ntroduces te tecnque of Brownan Dynamcs smuatons and te most reevant aspects of te cassca teory of Deragun-Landau-Verwey-Overbeek (DLVO). Secton 3 descrbes te agortm of Emuson Stabty Smuatons (ESS) n deta. In secton 4 some ustratve resuts of ESS are dscussed. Foowng, te modfcatons of te former agortm requred for te smuaton of deformabe drops are outned. Te paper fnses wt a bref concuson and te bbograpy.. Brownan Dynamcs (BD) Smuatons Te movement of one sma partce ( nm μm) dffusng n a quescent meda s te resut of externa and random forces. andom forces represent te effect of mons of cosons occurrng between te sovent moecues and te partce surface. In te absence of externa forces te energy for te dspacement of a sperca partce s provded by te sovent, wc at te same tme takes away some energy from te partce n te form of frcton. Te woe process s a manfestaton of te Fuctuaton-Dsspaton teorem tat s expressed concsey n te form of te Stokes-Ensten equaton [-]: ~ D f = k T () In ts equaton, D ~ and f stand for te dffuson tensor and te resstance tensor of te partce (or a set of partces), and k B s te Botzmann constant. In te case of a sperca partce te dffuson tensor s dagona and ts tree components are equa (D). Accordng to Equaton (), te absoute temperature of te reservor (T ) s kept constant we te movement of te partce occurs, atoug tere s a contnuous excange of energy between te partce and te sovent. Ts stocastc process can be descrbed anaytcay n terms of te Langevn equaton [3]. Te souton of Langevn s equaton for one sperca partce provdes expct expressons for ts mean square dspacement and B

3 Int. J. Mo. Sc. 9, 763 ts dffuson constant (D). Te dffuson constant turns out to be proportona to te tempora correaton of te random fuctuatons of te partce movement. Te overa effect of te nteracton between a suspended partce and te surroundng moecues s a random drft n te traectory of te partce, wose dspacement sows a Gaussan dstrbuton wt zero mean and a standard devaton equa to D Δt aong eac co-ordnate axs. Here Δ t s te apse of tme consdered and D s te dffuson constant of a sotary partce [3]. ~ Te dffuson tensor can be evauated from te resstance tensor: D = f k B T for a partce movng troug te qud as a consequence of externa, ydrodynamc or nterpartce forces. In turn, te resstance tensor can be obtaned from te drag force experenced by te partce wen t moves troug te fud at a constant veocty v [4]: F = f v () Te specfc form of te dffuson constant of a spere depends on te veocty of te qud at te partce surface. If te surface s rgd and smoot, te veocty of te fud becomes zero at te surface. In ts case te dffuson constant comes out to be D = D = k B T 6π η, were: η and are te sear vscosty of te sovent and te radus of te partce, respectvey. In te case of qud drops, porous speres, and btumen drops, te dffuson constant can be expressed as: () D = D fcorr (3) () were f corr s a correcton functon dependng on te caracterstcs of te O/W nterface [4]. Te addton of more partces to te system ncreases te compexty of te probem consderaby. Frst, te random movement of te partces must be connected n suc a way tat tey fuf te Stokes-Ensten reaton. Second, te movement of eac partce generates fuxes (dsturbances) n te sovent wc affect te movement of te surroundng partces and ts own. Tus, t s necessary to account for ydrodynamc nteractons between te partces. Trd, te partces nteract wt determnstc forces oter tan externa forces. Tus, ter movement s a combnaton of determnstc ydrodynamc and random forces. One of te most wdey used agortms for Brownan Dynamcs smuatons s te one of Ermak t Δt s equa to: and McCammon [5]. In ts formasm te poston of a partce at tme t Δ t, ( ) r ( t Δt) = r ( t) D r [ D F k B T ] Δt ( Δt) were te superscrpt ndcates tat te varabe s evauated at te begnnng of te tme step. Subscrpts and run over te partce coordnates (, 3 N ). F s te sum of nterpartce and externa forces actng on drecton. D are te components of te dffuson tensor. Te gradent of te dffuson tensor (second term on te rgt and sde) can usuay made equa to zero seectng a tensor tat depends on te dstances between te partces and not on te partces coordnates. Te trd term on te rgt and sde stand for determnstc contrbutons and te fourt term correspond to te random dspacements. A random devate as te form: r ~ (4)

4 Int. J. Mo. Sc. 9, 764 ~ ( Δt) = σ X (5) X Here X stands for a random varabe samped from a Gaussan devate generator: X =, X = δ Δt, were δ s te Kronecker deta. Te wegtng factors are gven by: / = σ D σ k (6) k = σ / = D σ k σ k σ k = (7) Accordng to te Ermak and McCammon equaton (4) s compatbe wt a Fokker Panck descrpton of te probem n te pase space. In te past, our group carred out te mpementaton of te above-mentoned agortm wt dfferent tensors ncudng Oseen, otne-prager, and Batceor s [6] fndng severa mtatons of ts tecnque for emuson stabty cacuatons. Equatons (5)-(7) ~ descrbe a partcuar metod for takng te square root of te dffuson tensor ( D Δt ). However, te metodoogy suggested (Coesky decomposton [7]) ony works wt partces of equa rad, and dute systems. Oter metods found n te bbograpy ke te Q decomposton aso fa n smuatons of poydspersed concentrated systems. Ts faure s caused by te assumpton of par wse addtve ydrodynamc nteractons. Tese scemes overestmate te ydrodynamc fuxes generated between te one centra partce and ts negbours. Tey do not take nto consderaton tat te fux comng from a partce ocated n te second coordnaton ayer s partay screened by te nner negbours. Dcknson suggested tat te assumpton of par wse ydrodynamc nteractons mgt even ead to negatve dffuson constants [8]. Wen an average dffuson constant s used nstead of a tensor, ts vaue s de-couped from te random devates. Usng ts approxmaton and D = D, we quantfed te knetc energy of mcronsze partces produced by te random fuctuatons of te Box-Muer agortm [7]. Te knetc energy was computed from te tme step of te smuaton and te dspacement of eac partce n te absence of determnstc and externa forces. It was observed tat fuctuatons as g as, k B T must be aowed n order to reproduce a mean square dspacement equa to: r = 6D Δt. Cut off tresods n te vaue of te devates correspondng to knetc energes of k B T and k B T fa to reproduce te correct vaue of r. It was evdent tat vaues of te dspacement correspondng to te outskrts of te Gaussan dstrbuton are necessary n order to smuate te Brownan movement of te partces correcty [9]. Te above consderatons are profoundy reated to te outcome of ESS cacuatons and ts dscusson s not a mere tecncaty. Te most famous teory of cooda stabty, Deragun-Landau- Verwey-Overbeek s DLVO teory [] s based on te probem of dffusve passage over a potenta barrer. Te presence of a strong repusve force between te partces generates a potenta barrer and two mnma at eac sde of te barrer (Fgure ).

5 Int. J. Mo. Sc. 9, 765 Fgure. Cassca DLVO potenta. Te curve corresponds to te nteracton potenta between two 3.9-μm drops of dodecane (A H = 5.3 x - J) suspended n water. Te drops are partay covered wt sodum dodecy sufate (C s = -4 M, z s = -.57) usng a omogeneous surfactant dstrbuton (Secton 3). 4 3 epusve Barrer V/k B T Prmary Mnmum Secondary Mnmum r (nm) On te one and, prmary mnmum foccuaton occurs at very sma dstances of separaton and s assumed to be rreversbe. Irreversbe meanng tat te aggregates formed do not separate f one owers te onc strengt of te souton, ncreasng te repusve force. Ts beavour s due to te strong van der Waas force experenced by te partces at sort dstances of separaton. On te oter and, secondary-mnmum foccuaton s usuay reversbe for sma partces and coud be rreversbe for mcron sze drops []. Irreversbe meanng n ts case tat te mnmum s deep enoug to prevent te partces from movng n and out of te potenta we. Te occurrence of prmary mnmum foccuaton depends on te dffusve passage of te partces over te potenta barrer. Accordng to Candrasekar [] and Kramer [3] te probabty of umpng over te barrer decreases exponentay wt te sze of te barrer. In order to account for te effect of te barrer on prmary mnmum foccuaton, Fucs [4] defned a Stabty rato W. Te compete formua of W was deduced ater by McGown and Parftt [5]: W = k fast f k sow f = [ f () r r ] ( V k T ) dr f ( r) T B [ r ] exp( VA kb T ) exp dr (8) Here k f stands for te foccuaton rate. It can be eter fast ( k ) or sow ( k fast f sow f ) f te potenta of nteracton between te partces s ony due to attractve forces (V A ) or caused by a combnaton of attractve and repusve (V ) contrbutons (V T = V A V ). In Equaton (8) te dependence of te frcton on te dstance between te partces (r) as been remarked. Numerca evauatons of Equaton (8) ead Preve and uckensten [6] to a very usefu reaton between te egt of te repusve barrer (ΔV) and te stabty rato: ( ) =.4 ( ΔV k T ) og W B (9)

6 Int. J. Mo. Sc. 9, 766 Fucs demonstrated tat te teory of Smoucowsk [7] for fast foccuaton coud be convenenty modfed n order to ncorporate te average effect of a repusve barrer troug te stabty rato. Tus, te cange n te tota number of aggregates of a dsperson per unt voume (n), can be wrtten as: n fast () t n ( k n t W ) = f () were n = n(t = ). Accordng to Equatons (8)-() a repusve barrer produces arge vaues of W wc retard te attanment of prmary mnmum foccuaton. A measure of cood stabty towards foccuaton s gven by te af fetme of te dsperson t /. Ts s te tme requred for a decrease of n / n te nta number of aggregates: t/ n fast sow = W k f n = k f () In te case of non-deformabe drops, prmary mnmum foccuaton mpes coaescence. Drops tat ump over te repusve barrer necessary coaesce snce tere s no oter repusve force to prevent t. Due to te rreversbe nature of te coaescence process te correct smuaton of te dffuson tensor and te random devates s crtca. 3. Emuson Stabty Smuatons (ESS) 3.. Equaton of Moton Emuson Stabty Smuatons start from a cubc box tat contans N drops randomy dstrbuted. Te partces move wt an equaton of moton smar to te one of Brownan Dynamc Smuatons [8-3]: r r r r r ( t Δt) = ( t) F F ext, Deff, ( d, φ ) k B T Δt Deff, ( d, φ ) Δt [ Gauss] () In Equaton () subscrpts and refer to partces and. Te dspacement of partce durng v r tme Δt: r ( t Δt) ( t), s te resut of two contrbutons. Te second term on te rgt and sde of Equaton () represents te effect of determnstc forces actng on partce. It s composed of nter partce F r r and externa forces F ext,. Tese forces move wt a constant veocty r r F Fext, Deff, ( d, φ ) k B T durng tme Δt, were D eff, ( d,φ) s an effectve dffuson constant. In te case of non-deformabe partces D eff, ( d,φ ) s cacuated foowng te metodoogy descrbed n ef. [3]. At every step durng te smuaton te space around partce s dvded n tree sperca regons. If at east one negbour partce reaces te nterna regon: r < d = rnt v r (were: r = r () t ( t) ), te formua of Hong et a. [3] s used to compute te dffuson constant of. Oterwse te voume fracton of partces around ( r nt < r < rex t ) s used to evauate an emprca expresson of D eff, ( d,φ ) [3]. Partces ocated at r > rext do not contrbute to te () ydrodynamc nteractons of. For sperca partces, D eff, ( d,φ ) s equa to D ( d φ ) = D f, eff,, corr

7 Int. J. Mo. Sc. 9, 767 () () were D = D fcorr (Equaton (3)). Te frst correcton term ( f corr ) takes nto account tose factors tat cange te expresson of te dffuson constant of a partce at nfnte duton (D) [4]. Te second correcton term f () corr takes nto account ydrodynamc nteractons caused by te movement of te surroundng qud as te partces move [3]. () () Te program ncudes severa forms for f corr and f corr. However, n amost a smuatons of nondeformabe dropets prevousy reported, we assumed dea sperca partces wt zero tangenta () veocty at ter surfaces, for wc f =. Tus, we concentrated on te effect of te ydrodynamc () nteractons generated by te negbour partces of, D ( d φ ) = D f corr..734φ.9φ eff,, corr () f corr = for r < r < rext, wt: nt, (3) And: ( 6 u 4u) ( 6 u 3. ) () f corr = u for rnt r (4) In Equaton (3) φ stands for te oca voume fracton of o around a centra partce, and ~ ( r ) u = (5) ~ were s te radus of partce, and a radus of a reference. Te trd term on te rgt and sde of Equaton () gves te random devates of te movng partces. Te stocastc vector G r auss stands for a set of random numbers generated by te Box-Muer agortm. Te caracterstc mean square dspacement of te Brownan movement s obtaned mutpyng eac random devate D, d,φ Δ. by ( ) t eff In ESS, non-deformabe drops coaesce f te dstance between ter centres of mass gets smaer tan te sum of ter rad: r < (6) Wen ts occurs a new drop s created at te centre of mass of te coaescng drops. Te radus of te new drop resuts from te conservaton of voume: = (7) new 3.. Interacton Forces and te surface excess In ESS te attractve force between te drops s determned by te effectve Hamaker constant of two o drops separated by water [3-33]. Ts constant (A H ) s often of te order of - J for ydrocarbons and attces. In te case of Btumen some od expermenta evdence ndcated a vaue of A H ~ -9 J, but recent evauatons suggest a muc ower vaue (A H ~ - J). Te van der Waas potenta for two sperca drops of dfferent radus (, ) s equa to [3]: ( A )[ y ( x xy x) y ( x xy x y) n( x xy x) ( x xy x y )] V A = VvdW = H (8) ere: x =, y =, and = r. Tere s aways an attractve force between two drops of smar composton suspended n aqueous meda. However, te repusve force depends on te caracterstcs of te O/W nterface and te type

8 Int. J. Mo. Sc. 9, 768 of surfactant adsorbed. O drops often exbt an eectrostatc surface potenta between - mv (octacosane [34]) and -35 mv (benzene [35]) wen suspended n water. Ts carge orgnates from te preferenta adsorpton of OH - ons to te o/water nterface. If ts s te case, an nta vaue of te surface carge per unt area (σ ) can be ntroduced as nput n order to reproduce te tota vaue of te surface carge σ T n te absence of surfactants. Wen onc surfactants are present, tey add ter carge to te nta surface carge of te drop (σ ): ( 4 π ) = σ A z e Γ ( π ) = σ z Γ σ T = σ zs e N s 4 s e (9) Here N, A, z s, e stand for te number of surfactant moecues attaced to drop, te area of te drop: A = 4 π, te effectve vaence of a surfactant, and te unt of eectrostatc carge (.6 x -9 Cou.). Γ stands for number of surfactant moecues per unt area at te o/water nterface. Te nterfaca area of one surfactant moecue at maxmum packng (A s = Γ max ) s typcay of te order of 5 Å [36]. It depends on te adsorpton tme and on te way te surfactant partton between te mmscbe pases and te nterface. Te effectve carge of an onc surfactant moecue (q = z s e) can be cacuated from te zeta potenta (ζ) of a drop saturated wt surfactant moecues. In te most usua stuaton z s s vared unt te expermenta vaue of ζ s reproduced. Ts cacuaton s done assumng tat σ =. In ts case, te effectve carge of te surfactant contans te contrbuton comng from te ydraton ayer. It s aso possbe to use a fnte vaue of σ to reproduce te surface potenta of an o drop n water, and ten vary z s to ft te vaue of ζ. Te vaue of z s resuts from apportonng te effectve carge of a drop to a dscrete number of surfactant moecues (.7 z s.3 [37]). Te estmaton of q requres knowedge of te maxmum number of surfactant moecues n eac drop. Ts number s equa to: N max s = As (), 4π Te current verson of te program contans four anaytca expressons for te cacuaton of te eectrostatc force [38-4]. Te formasm of Sader et a. [39-4] was used n prevous smuatons of non-deformabe drops. Accordng to tese autors, te eectrostatc potenta between two sperca drops s gven by Equaton (): were: V E k B B T [ C B( ) B( ) r ] expκ = () C B ε = 4π k T ε e () () = ( Φ 4 γ Ωκ ) ( Ωκ ) P γ = tan ( Φ 4) 3 ( Φ 4γ ) ( γ ) Ω = P (5) In Equatons ()-(5), ε s te permttvty of vacumm, ε te deectrc constant, κ - te Debye engt, and Φ P = Ψ e kb T te reduced eectrostatc potenta of te partce at ts surface. Te reaton between te surface carge and te surface potenta s gven by: P (3) (4)

9 Int. J. Mo. Sc. 9, 769 ( sn ( Φ ) Φ ) Q σ e κ ε k T = Φ Φ κ κ (6) T B P P P P were: Q ( Φ 4) Φ [ sn( Φ ) Φ ] 4tan P P κ P P (7) = 4 Notce tat knowedge of σ T and te onc strengt of te souton aow te numerca evauaton of te surface potenta from Equaton (6), as we as te rest of te varabes (Equatons ()-(5)) of te potenta (Equaton ()). Snce te onc strengt s set by te expermenta condtons, and σ T depends on te surface excess, ony Г s requred n order to cacuate te eectrostatc potenta between suspended drops. Surfactants can be onc, non-onc, zwtteronc, and ave a compex moecuar structure. Hence, te nteracton potenta between te drops can vary ampy. Te program of ESS ncudes expressons for van der Waas, eectrostatc, oscatory, depeton, ydraton, and sterc forces. In te case of non-onc surfactants te stabzaton force between emuson drops s assumed to be sterc [43-44]. Sterc forces are composed of an osmotc (mxng) contrbuton and an eastc one: ( ) V ( ) Vsterc = Vosm east (8) were stands for te mnmum dstance between te surfaces of te drops. Te mxng contrbuton s generated by te cross nkng of te ydropc cans of te surfactants of eac drop. Te eastc force comes from te drastc modfcaton of te surfactant conformaton at cose dstances of separaton between te partces. We deveoped expressons for bot types of contrbutons modfyng te equatons reported by Vncent [45] and Bagc [46]. On te one and, Vncent used smpe anaytca formuas to descrbe te dstrbuton of poymer monomers perpendcuar to a panar nterface. Foowng e used te Deragun approxmaton [47] to evauate te expresson of te free energy of mxng deduced by Fory and Krgbaum [48], for two speres stabzed wt poymer moecues. On te oter and, Bagc cacuated te exact voume of overap between te poymer ayers of two codng drops, but used te Fory-Huggns [49] expresson for te cacuaton of te free energy. Lozsán et a. [43] sowed tat te metodoogy of Vncent wt te exact cacuaton of te voume of overap s equvaent to te metodoogy of Bagc f te free energy s evauated wt te expresson of Fory-Krgbaum. Te ESS code as severa expressons for te cacuaton of te sterc nteractons [43] ncudng te expresson of De Gennes [5]. An exampe of tese expressons s gven by Equaton (9) for dstances between δ < < δ [43-44]: V osm ( ) = ( 4 k T 3V )[ φ φ ]( / χ )( δ ) B ( 3( ) δ 3( ) ( ) w In Equaton (9) V w stands for te moar voume of te contnuous pase, χ s te Fory-Huggns sovency parameter, and φ stands for te voume fracton of surfactant moecues n te nterfaca ayer of drop. In order to estmate φ t s usuay assumed tat te ydropobc can of te surfactant s dssoved n te o pase, and ony ts ydropc can es n te outer regon of te drop. If te densty of ydropc cans n te nterfaca ayer s assumed to be constant, φ can be expressed n terms of te surface excess [44]: x (9)

10 Int. J. Mo. Sc. 9, ( ) ) φ = δ (3) 3 Γ ρ were: ρ stand for te densty of te ydropc can. As n te case of te eectrostatc potenta, te sterc potenta depends on some parameters and s a functon of te surface excess Surfactant Dstrbuton (Evauaton of te surface excess) Te vaue of te surface excess of a surfactant at te nterface of emuson drops cannot be measured drecty. It s extrapoated from te varaton of te nterfaca tenson n systems wt a macroscopc O/W nterface. Unfortunatey, emusons are constanty evovng, contnuousy cangng ter drop sze dstrbuton and tota nterfaca area, A T : A = T A As a resut, te surface excess s not constant and te nteracton potenta between te drops canges as a functon of tme. Ionc surfactants are not soube n te o pase. Tey can mgrate to te o pase n te form of nverse mcees f te sat concentraton of te system s unusuay g. In te typca case, onc surfactants adsorb to te O/W nterface before tey form mcees. Te amount of surfactant requred for te compete coverage of te nterface can be arger or equa to te crtca mcee concentraton (CMC). It vares wt te voume fracton of o (φ) and A T. Te arger te vaues of φ and A T, te ger te surfactant concentraton requred for te compete coverage of te drops. In te case of non-onc surfactants te stuaton s more unpredctabe. Te partton of non-onc moecues depends on te affnty of ts popc and ydropc moetes for te o and water pases. For exampe, te partton coeffcent of akypeno ogomers between water and n-akanes s equa to [5]: og K m = m.45 SACN. 3 ACN (3) were m s te number of etyene oxde unts n te surfactant moecue, SACN s te number of carbon atoms n te aky can of te surfactant, and ACN s te number of carbon atoms n te n- akane moecue. Te adsorpton of surfactant moecues depends on tme and on severa formuaton varabes ke te number of metyene groups of te o moecue (Equaton (3)), te sat concentraton, te presence of acoo moecues n te system, etc. Furtermore, adsorpton can be reversbe or rreversbe [5-53]. Te routnes of surfactant dstrbuton attempt to recreate te most common expermenta stuatons. Te strategy of ESS s to apporton te surfactants to te nterfaces of te drops n suc a way tat t reproduces te varaton of te surface excess n te expermenta system. Consequenty, ony te movement of te drops s consdered expcty n Equaton (). Some of te routnes for surfactant dstrbuton ave a forma teoretca background [54-55]. Some oters are ony practca approxmatons to te very compex probem of surfactant dffuson and adsorpton [56-57]. Tese routnes resembe te cases of Homogeneous and Non-omogeneous surfactant dstrbutons. Te effect of non-omogeneous dstrbutons resuts from an ncompete (3)

11 Int. J. Mo. Sc. 9, 77 mxng of te emuson components. Most recent studes concern te cases n wc te surfactant s eveny dstrbuted among te surfaces of te drops. Tree exampes of tese strateges are gven beow Homogeneous Surfactant Dstrbuton wt fast and rreversbe surfactant adsorpton Ts s te smpest metodoogy. It conssts n ascrbng to eac drop a number of surfactant moecues ( N, ) proportona to ts nterfaca area. Here N s, T s max = A AT ( N s, N s, ) (33) N s, N s, T A A = N s, T stands for te tota number of surfactant moecues n te system. Equaton (33) can aways be apped regardess of te tota surfactant concentraton avaabe. It resembes te cases n wc: (a) te surfactant adsorpton s very fast n comparson to te coson of drops; (b) te mxng condtons are omogeneous; (c) te adsorpton s rreversbe. If te number of drops decreases as a consequence of coaescence, Equaton (33) can be used to recacuate te number of surfactant moecues adsorbed to te remanng drops. Te number of surfactant moecues of eac drop ncreases as te cacuaton progresses because te tota nterfaca area of te emuson decreases as te drops coaesce. However, te nterfaca area of one surfactant moecue cannot be ower tan ts mnmum cross-sectona area at te O/W nterface ( A s ). Hence, max N, cannot exceed N s, = A As. If te number of surfactants n te system surpass te amount s requred for te compete coverage of a te drops, te vaue of N, s set equa to N. s max s, Homogeneous Surfactant Dstrbuton wt tme-dependent surfactant adsorpton In te case tat te adsorpton s bascay controed by te dffuson of surfactant moecues from te buk to te subsurface [58], te surface excess s a functon of te tota surfactant concentraton C T, te dffuson constant of te surfactant D s, and te tme: () t = C ( D t π ) / Γ (34) T Te mecansm of surfactant adsorpton can be very nvoved, ncudng barrers of adsorpton, reorentaton at te surface, etc. However, Lgger et a. [59] demonstrated tat most process of mxed adsorpton knetcs can be reformuated n terms of a dffuson controed mecansm (Equaton (34)) f D s s substtuted by an apparent dffuson constant (D app ). Furtermore, ts equaton s aso compatbe wt te fndngs of Hua and osen [6-6] for a arge number of surfactants wt dfferent moecuar structures. Accordng to tese autors, te surface tenson of most surfactants sows four caracterstc regons of cange known as: te nducton perod, te fast-fa regon, te mesoequbrum regon and te equbrum regon. Te tme requred for te surface pressure to drop to af ts vaue at mesoequbrum, s found to foow Equaton (35). Ts equaton aows estmatng te vaue of D app requred for te evauaton of Equaton (34). og( t) s ( Γ( t) C ) og ( 4 ) / T π D = og app (35)

12 Int. J. Mo. Sc. 9, 77 Te routne of tme dependent surfactant adsorpton uses D app, and C T as nput, estmatng te vaue Γ t = C D t π /. Te number of surfactant moecues of drop s of te surface excess from () ( ) cacuated accordng to Equaton (36) [55]: Here () t T app ( t) = A Γ( t) = Γ( t) N s, = A As 4π (36) A s stands for an effectve area per surfactant moecue at te o/water nterface. As Γ ( t) ncreases, A s () t decreases, ncreasng te number of surfactant moecues adsorbed to eac drop Equbrum Surfactant Dstrbuton (Gbbs) Equbrum soterms are ony attaned after ong perods of tme. In ts routne we assume tat: a) te adsorpton of surfactant s very fast, and b) equbrum adsorpton s obtaned nstantaneousy. Gbbs soterm ony appes to tat range of surfactant concentraton between te begnnng of te decrease of te nterfaca tenson C T = C, and te Crtca Mcee Concentraton (CMC). Wen C T C cmc te maxmum number of surfactants per drop s aready adsorbed: N, = N (Equaton ()). Wen C T C te cange n te nterfaca tenson often sows a sma maxmum. We dsregard ts feature of te expermenta curve. Instead we approxmate te ow concentraton mt eter by a) usng an addtona Gbbs soterm between C T = and C T = C ; or b) cacuatng an average surface excess ncudng te pont C T =, γ = γ. In terms of fnte dfferentas te equaton of Gbbs s equa to: ~ γ = γ c k B T Γ og( C C ) (37) were c = or dependng on te type of surfactant and te onc strengt of te souton. Assumng proportonaty between te suggested vaue of te tenson and te number of surfactant moecues adsorbed: γ max ( γ γ )[ N s N ] = γ ( γ < γ ) (38), s, Equaton (38) s very convenent because t expresses te tenson n terms of te surfactant popuaton of eac drop. Accordng to Equatons (37)-(38): ~ max N s, = ( c k B T Γ N s, [ γ cmc γ ] ) og( CT C ) γ > γ cmc (39) Equaton (39) gves te number of surfactants attaced to drop as a functon of te tota surfactant concentraton n te system. Here, γ stands for te vaue of te nterfaca tenson at C. Te vaue of Г s obtaned from Equaton (39) dvdng N, by A. Notce tat tere mgt be cases n wc te s number of surfactant moecues n te system mgt not be enoug to cover te drops accordng to Equaton (39). In tose cases a omogeneous surfactant dstrbuton s apped (Equaton (33)) despte te nta seecton of te Gbbs routne Ostwad penng Atoug a arge number of numerca tecnques are avaabe for te smuaton of Ostwad rpenng [6-66] tey are based on popuaton baance equatons and cannot be ncorporated nto te agortm of ESS. In order to smuate te process of Ostwad rpenng we ncuded te agortm of s max s,

13 Int. J. Mo. Sc. 9, 773 De Smet et a. [67-69] n our ESS code. Te fundamenta equaton of ts metod s derved from te Fck s aw and te Kevn s equaton assumng α << : ~ dno, () t dt = 4π Do C( ) α ( ( t) c ( t) ) (4) Here, n, stands for te number of moecues of o n partce. o D ~ o refers to te dffuson constant of an o moecue. C ( ) stands for te aqueous soubty of te o n te presence of a panar O/W nterface. c s te crtca radus of te emuson equa to [7]: () t = c N Here N s te tota number of drops. Parameter α s te so caed capary engt, defned as: α = γ T ˆ (4) In Equaton (4) V m s te moar voume of te o and ˆ te gas constant. Defnng: V m () t = ( t) ( t) (4) P c (43) and: C( ) t M = 4 π Dm α Δ, (44) A smpe equaton for te excange of o moecues s obtaned: no, ( t Δt) = no, ( t) M ( t) P ( t) (45) At any step of te smuaton tere exsts a crtca radus c of te emuson. Partces wt < c, dssove we partces wt > c grow. Partces wt te same radus as te crtca radus = c preserve ter sze. Te number of moecues excanged by partce s equa to te product M ( t) P ( t). In ESS te vaue of M(t) s set once te tme step of te smuaton s cosen, M () t = M. Wen te smaest partce = sma contans fewer moecues tan te number t soud ose accordng to Equaton (45), M(t) s substtuted by: M ( t) n M = o, sma (46) ecenty we mpemented a new procedure to avod a substanta decrease n te number of partces [7]. Te smuaton starts from a gven Drop Sze Dstrbuton (DSD) of N drops, and evoves unt t reaces N(t = t ) = N =. At ts pont, a new Drop Sze Dstrbuton (DSD) wt N = N drops s but. Ts can be aceved approxmatng te probabty dstrbuton of partce szes by te reatve number of drops of eac sze exstng at tme t = t : ~ P( ) = N(, t' ) N (47) were (,t') sze n te new dstrbuton, '(, t' ) N s te number of drops wt radus exstng at tme t. Tus, te number of drops of N' N, s equa to: (, t' ) N N(, t' ) N N(, t ) ( N N ) = = ' (48) Use of Equatons (47)-(48) produces a new DSD wt N drops tat exacty matces te od one. Te auxary code aso cacuates te new sze of te smuaton box, wc s requred n order to

14 Int. J. Mo. Sc. 9, 774 preserve te nta voume fracton of o wt te new set of partces. Once te nput fe s modfed and te new dstrbuton read from an externa fe, te program generates a new set of co-ordnates for te new partces. As n te begnnng of te smuaton te partces are dstrbuted at random avodng overap. At ts pont te code resumes te cacuaton of te man cyce (see beow) Te Agortm of ESS Te agortm for ESS s sown n Fgure n te form of a fowcart. Fgure. Fowcart of Emuson Stabty Smuatons. EAD O GENEATE COODINATES AND PATICLE SIZE DISTIBUTIONS BUILDS A SIMILA DOP SIZE DISTIBUTION WITH N PATICLES DISTIBUTE SUFACTANT MOLECULES AMONG THE DOPS NO YES CALCULATE INTEFACIAL PAAMETES AND DIFFUSION CONSTANTS YES YES IS THE NUMBE OF PATICLES EQUAL TO N? CALCULATE COECTIONS TO THE DIFFUSION CONSTANTS NO TIME-DEPENDENT SUFACTANT ADSOPTION? NO COALESCENCE? COMPUTE EXTENAL AND INTEPATICLE FOCES MOVE THE PATICLES EXCHAGE OIL MOLECULES At te begnnng of te smuaton te code reads or generates a drop sze dstrbuton and a set of co-ordnates for eac partce. Addtonay, te tme step(s) of te smuaton (snge or doube) must be specfed. A combnaton of a sma tme step and a arge one s used for te cacuaton of dute dspersons [7]. For ts purpose a dstance of cosest approac (d mn ) must aso be seected. Te maxmum range of te nteracton potenta usuay approxmates ts dstance (5 nm d mn nm). A doube tme-step cacuaton mpes te use of a onger tme step (Δt L ) wen te partces are

15 Int. J. Mo. Sc. 9, 775 far apart from eac oter, r > d mn. If te dstance between two partces becomes equa or ower tan d mn, a partces are returned to ter prevous postons, and a sma tme step (Δt S ) s used n Equaton () unt Δt S = ΔtL. Ts tecnque s effcent n tose cases were te repusve potenta does not prevent coaescence. Oterwse te drops aggregate makng r aways ower tan d mn. Ts causes te contnuous use of Δt S after te frst foc s formed. After te creaton of te DSD, te program dstrbutes te surfactant moecues among te drops. Tere are routnes to cover te most common expermenta stuatons. At ts pont te nterfaca tenson of eac drop can be determned. Te cacuaton of te dffuson constants s aso executed ere, because some of ts expressons depend on nterfaca parameters. Frst, te program assgns te dffuson constant of Stokes to a partces accordng to ter radus. Second t makes te correctons necessary to account for te effect of te nterfaca propertes and ydrodynamc nteractons on te dffuson constant ( f and f ). Foowng, te forces are cacuated and te drops are moved accordng to Equaton (). If te Ostwad rpenng mode s seected, te o moecues are excanged at ts pont, and te program foows te pat ndcated by te tn sod arrows n Fgure. Oterwse te program foows te pat ndcated by te tn broken-ne arrows. In te atter case, te program cecks for coaescence after te partces are moved. In te former case te program cecks for coaescence after te excange of o moecues takes pace. If coaescence occurs, te tota nterfaca area of te emuson canges aong wt te voume of one (or severa) drop(s). Terefore te surfactant popuaton ad to be redstrbuted among te drops. Te same occurs f Ostwad rpenng appens. However, n te atter case a mnmum number of drops must be mantaned. Terefore, te program cecks f te remanng number of drops s ger tan a fxed vaue (N(t = t ) > N ). If ts s te case, te program proceeds wt te redstrbuton of surfactant moecues. Oterwse, te program stops. An auxary program s used to bud a new drop sze dstrbuton wt N = N = N (t = ) drops. Ts program aso cacuates te new sze of te box necessary to keep te voume fracton of o constant wt te new set of partces. Te new dstrbuton s read from a fe by te program of ESS. Te co-ordnates of te new partces are ten generated and te man cyce of ESS contnues. If te Ostwad rpenng process s not seected, te program neter excange o moecues nor does t re-buds te drop sze dstrbuton wen N(t = t ) N. In ts case te code foows dfferent routes dependng on te outcome of te coaescence ceck. If coaescence occurs t s necessary to reassgn te surfactant popuaton. If t does not occur tere are two possbtes. If one of te routnes of tme-dependent adsorpton s used, t must recacuate te surfactant popuaton anyway. If ts s not te case, te program proceeds wt te evauaton of te ydrodynamc nteractons and a new cyce begns. It s mportant to remark at ts pont tat te cange of te number of partces durng te smuatons s ony equa to te varaton of te number of aggregates n te absence of a repusve force. Ts was demonstrated quanttatvey n ef. [73]. Wenever a repusve barrer s present, te cange n te number of aggregates as a functon of tme as to be cacuated at te end of te smuaton usng a dfferent program [, 73-74]. Ts addtona code uses as nput te postons of te partces produced durng te smuaton and a fxed () corr () corr

16 Int. J. Mo. Sc. 9, 776 foccuaton dstance. Te foccuaton dstance s usuay approxmated by te poston of te secondary mnmum of te nteracton potenta. Wen tese computatons are fnsed, statstca data regardng aggregates, focs, radus of gyraton, etc, s obtaned. 4. esuts Te genera resuts of te smuatons can be cassfed nto tree categores dependng on te tota surfactant concentraton of te system [74]. 4.. Low surfactant concentraton If te surfactant concentraton s not enoug to stabze te nta drop sze dstrbuton, drops coaesce as soon as tey code wt eac oter. Hence, te cange n te number of partces s equa to te cange n te number of aggregates [73]. In ts case, te systems foow te dynamcs of Smoucowsk, n te sense tat te number of aggregates canges as predcted by Equaton () wt W = [7-73]: n fast () t n ( k n t ) = f (49) fast Te vaue of k f depends on te mean free pat between te drops and ter attractve force. For voume fractons between -5 φ.3 and a Hamaker constant A H ~ - J [7]: In te case of A H ~ -9 J: k fast 8 ( 6.44 ) exp( 8.4 φ) f = x m 3 /s (5) k fast 8 ( 9.55 ) exp( 5.4φ) f = x m 3 /s (5) fast Notce tat te vaue of k f dffers from te teoretca estmaton of Smoucowsk ( ks = 4kB T 3η ~ 6. x -8 m 3 /s) based on te Brownan moton of te partces ony. Te attractve force between te drops ncreases te vaue of k fast f we te ydrodynamc forces decrease t. For A H ~ - J te effect of te attractve potenta s sma and te ydrodynamc nteractons domnate (see Fgure 8 n ef. [9]). Tese are te cases of ydrocarbon-n-water emusons and atex dspersons. For A H ~ -9 J (poar os, meta sats) te attractve force domnates. In eter case, te effect of te voume fracton remans. As φ ncreases te mean free pat between te drops decreases. Ts dmnses te tme of dffuson between te cosons of te partces, ncreasng te vaue of k. fast f Notce tat Equaton (49) was ntended to expan te penomenon of rreversbe foccuaton of sod partces. Danov et a. [75] demonstrated tat Equaton (49) aso appes to emusons subect to te smutaneous processes of foccuaton and coaescence. For ts demonstraton no dstncton was made between aggregates formed by te coson of smaer focs, and tose of te same sze resutng from te parta coaescence of foccuated drops. Smary, no dstncton was made between snge partces formed troug coaescence, and tose ntay present at t =. Tese are te same assumptons we use for determnng te varaton of te number of te aggregates as a functon of

17 Int. J. Mo. Sc. 9, 777 tme. Danov et a. aso supposed a constant coson kerne, but dd not make any specfc ypotess regardng te coaescence rates. Te fact tat Equaton (49) ods for coaescng emusons s reated to te suppostons foowed by Smoucowsk for determnng te foccuaton rate. He pctured te case n wc one partce was fxed n space we te oters coded wt t as a consequence of a gradent of concentraton. Ts gradent was estabsed as soon as t > between te coson radus of te fxed partce and te buk of te qud. However, te densty of partces at te coson radus was assumed to be equa to zero durng te woe aggregaton process. Hence, te fxed partce acted as a perfect snk. Every partce tat coded wt te fxed partce was assumed to dsappear at te moment of te coson. Moreover, te coson effcency of a custer composed of snge partces was estmated usng te same procedure, but empoyng te average ydrodynamc radus of te aggregate to approxmate ts coson radus. As a resut of tese assumptons, te process of foccuaton descrbed by te teory of Smoucowsk ncudes te possbe occurrence of nstantaneous coaescence. Instantaneous meanng tat te tme requred for coaescence s neggbe n comparson to te tme requred for foccuaton. Inomogeneous surfactant dstrbutons as we as reversbe adsorpton, aso ead to fast aggregaton and to te coaescence of drops, favourng te vadty of Equaton (49) [54]. Te same occurs f te surfactant does not adsorb rapdy to te O/W nterface (D app < - m /s) [55]. Fgure 3. Smoucowskan decrease of te number of aggregates as a functon of tme. Subscrpts a, n agg and agg stand for: te number of aggregates pus snge partces N, te tota number of partces n aggregates ( g)an, ( ) a respectvey (A H =.4 x -9 J). nn, and te number of focs ( ) agg 5 N a N 5 N n agg N agg 5 5 TIME (s) Te data of Fgure 3 corresponds to system # n efs. [9, 73]. Te surfactant concentraton n te system (C s = -5 M) s not enoug to prevent te coaescence of drops durng te course of te smuaton. Hence, te number of snge partces s equa to te tota number of aggregates ( N a ). Te number of aggregates consttuted by two or more partces ( N agg ) s zero, as we as te number of partces n aggregates ( N ng)a.

18 Int. J. Mo. Sc. 9, 778 In genera, te nta nterfaca area of an emuson s ger tan te one tat can be stabsed wt te surfactant concentraton avaabe. In ts case a Smoucowskan drop n te number of partces occurs at sort tmes. In te preparaton of emusons, cemca metods usuay empoy a arger surfactant concentraton tan mecanca ones. Ts favours te formaton of a repusve barrer as soon as te drops are formed. In ts case, te dynamc of aggregaton s not expected to foow Equaton (49). 4.. Intermedate surfactant concentraton. Insuffcent surfactant moecues to prevent te nta coaescence of drops Accordng to ESS of non-deformabe dropets, te varaton of te number of aggregates n te presence of an apprecabe repusve barrer conforms to a remarkaby smpe expresson [73]: [ A ( k n t) B ( k n t) ] n a = n exp (5) were A, B, k and k are constants and B = - A. Te second term n Equaton (5) resuts from te consderaton of te coaescence rate as a frst order process, dependng on te number of foccuated doubets. We used a Smoucowskan term to represent foccuaton and an exponenta term to represent coaescence. Hence, k and k were formery ascrbed to te foccuaton rate ( k sow f ) and te coaescence rate (k c ), respectvey [73]. Coeffcents A and B measure te extent of tese two processes durng te smuaton. Te curve descrbed by Equaton (5) generay sows a pronounced nta decrease foowed by a muc sower cange n te number of aggregates per unt voume ( na = N a V ). See Fgure 4. Te nta decrease corresponds to te combned processes of foccuaton and coaescence (FC perod). Durng te FC nterva, te tota nterfaca area of te emuson decreases. Ts causes a substanta redstrbuton of surfactant moecues amongst te remanng drops f a omogeneous surfactant dstrbuton s assumed. As a resut tere s a progressve ncrease n te repusve potenta between te drops, wc sows down bot destabsaton processes. Equaton (5) was abe to reproduce te beavour of 34 smuatons, ncudng dfferent DSDs, surfactant concentratons, number of partces, voume fractons (. φ.3), and spata dstrbutons. Ts s remarkabe snce te structure of te aggregates and ter spata dstrbutons canged consderaby from one system to anoter. We ave aso compared te predcton of Equaton (5) wt some unpubsed expermenta data on dodecane-n-water nano-emusons (φ =., T = 5 C) obtanng good resuts. For ts purpose te average radus of te emuson was measured as a functon of tme. Te number of partces was estmated dvdng te tota voume of o by te average voume of a drop at eac tme. Based on te resuts of te prevous secton (4.), we approxmated te number of aggregates by te number of partces durng te FC perod, and used Equaton (5) to ft te resuts. Te fttng s sown n Fgure 5 aong wt te expermenta data (green crces). Notce tat te expermenta curve sows a sarp decrease n te FC rates after a pronounced nta drop n te number of partces. Equaton (5) aso appears to ft te data beyond te FC nterva.

19 Int. J. Mo. Sc. 9, 779 Fgure 4. Beavor of system #3 from efs. [9, 73-74]. Te subscrpts are te same of Fgure 3. (φ =., C s = 8.65 x -5 M, A H =.4 x -9 J). 8 N a N 6 4 N n agg N agg 5 TIME (s) Fgure 5. Varaton of te number of partces as a functon of tme for a dodecane n water nano-emuson stabzed wt Br 3 (φ =.). 8 n x -5 m ,,4,8,,6,,4,8 t x -5 s Seres expansons of te exponenta term and te foccuaton functon of Equaton (5) around t =, sowed tat tese two functons are very smar. In fact, t s possbe to excange te par of coeffcents correspondng to eac process (A, k ) (B, k ) and obtan a fttng of smar quaty. Ts demonstrated tat te processes of foccuaton and coaescence are mxed n te knetc rates of Equaton (5). Moreover, t was aso observed tat Equaton (5) was abe to ft a termna aggregaton stage n tose systems n wc te number of partces stabses after a certan perod of tme. It was aso found tat te nterfaca area of te emuson after ~ seconds was proportona to te number of surfactant moecues. Te sope of ts curve provdes a measurement of te area per surfactant moecue tat s requred (~ 36 Å ) n order to avod an nta pronounced drop n te number of partces. Usng te effectve carge of a surfactant moecue (.e), te correspondng

20 Int. J. Mo. Sc. 9, 78 vaue of te surface carge can be obtaned (σ = 3.95 mcou/m ). Te eectrostatc surface potenta can aso be cacuated (Ψ ~44 mv) usng te reaton between σ and Ψ (Equaton (6)). Unexpectedy, tese vaues do not correspond to a postve magntude of te potenta energy for te referred systems. Instead, tey correspond to te appearance of a souder at negatve vaues of te potenta energy (see Fgure 6-7). Ts souder orgnates te secondary mnmum (for wc F = V r = ), aong wt a sma potenta barrer. Yet, ts barrer occurs at negatve vaues of te nteracton potenta were an attractve force between te partces exsts. Fgure 6. DLVO potenta (red ne) between two sperca drops of btumen ( =.39 μm) wt a surface carge of σ = 3.95 mcou/m (Ψ ~44 mv, A H =.4 x -9 J). Te onc strengt of te souton s equa to.4 x - M. Te van der Waas potenta s sown n bue. - - V/k B T r (nm) Fgure 7. Potenta energy between two sperca drops of btumen ( = 3.9 μm) wt a surface carge of σ = 3.95 mcou/m (Ψ ~44 mv, A H =.4 x -9 J). Te onc strengt of te souton s equa to.4 x - M V/k B T r (nm) From te varaton of n a vs. t (or N a vs. t) n te systems studed (see Fgure 4) t s cear tat a consderabe stabsaton s reaced sorty after te FC nterva. At ts pont, severa factors may favour te decrease of te foccuaton rate. Te number of remanng drops after te FC perod s smaer and te mean free pat between te drops s arger tan t was at te begnnng of te

21 Int. J. Mo. Sc. 9, 78 smuaton. Moreover, te dffuson constant of te remanng drops s smaer snce t s nversey proportona to te radus of te drops. However, tese factors are key to produce a progressve decrease of te FC rate as s observed n Fgure 8. Instead a sarp cange n te beavour of n a was observed n most systems studed n ef. [73]. We beeve tat te drastc cange n te sope of n a vs. t s bascay caused by te ncrease of te repusve force between te remanng partces of te emuson. In ts regard t soud be notced tat wen te partces execute Brownan moton, ter knetc energy s ost n a sort perod of tme. As a resut, te tota energy of te drops soud approac cosey te potenta energy curve. Hence, te absence of a drvng force at te secondary mnmum ( F = V r = ) soud sow down consderaby te movement of sma drops. In ts stuaton a sma repusve force coud be enoug to prevent prmary mnmum foccuaton (coaescence). Fgure 8. Beavor of system # from efs. [9,73-74]. Te subscrpts are te same of Fgure 3. ( φ =.6, C s = 4. x -5 M,, A H =.4 x -9 J). 7 6 N N n agg N a N agg TIME (s) A more conventona effect of te repusve barrer was observed n te smuaton of exadecanen-water nanopartces ( = nm) stabzed wt nony peno etoxyated (NPE) surfactants. In tese cacuatons te repusve potenta was assumed to be sterc, and te nterfaca area of te surfactant moecues was kept constant durng te woe smuaton. Tese cacuatons sowed tat tere s an anaytca reatonsp between te egt of te repusve barrer of te nteracton potenta and te stabty rato: ( ) =.493( ΔV k T ) og W B (53) Ts equaton s very smar to Equaton (9) except for te metod of evauaton of ΔV and W. Equaton (9) was deduced from te systematc evauaton of Equaton (8) for te case of sod partces suspended n water. Instead Equaton (53) was deduced from emuson stabty smuatons of qud drops exposed to foccuaton and coaescence. In te present case Δ V was measured from te base of te secondary mnmum up to te top of te potenta barrer. In te former case te egt of te barrer was measured from V =. Moreover, Equaton (8) was deduced assumng te vadty of te formasm of Smoucowsk (Equaton ()). However, te pot n a vs. t does not foow te Equaton

22 Int. J. Mo. Sc. 9, 78 () n te presence of a szeabe repusve barrer. Instead, we redefned W n terms of te affetme of te dsperson: fast W t sow / t/ = (54) Equaton (54) resuts from te nverse proportonaty between te af-fetme and te foccuaton rate suggested by Equaton (). However, t does not depend on te partcuar form of te curve of n a vs. t Hg surfactant concentratons Equaton (8) does not mpose any restrcton to te vaue of W: te ger te potenta barrer te ger te stabty rato. However, accordng to ESS of non-deformabe drops [44], te number of partces of te emuson s preserved wen te repusve barrer between te drops s ger tan.7 k B T. In ts case te dynamcs of te system depend on te dept of te secondary mnmum of te nteracton potenta []. Accordng to Verwey and Overbeek [] a 5-k B T barrer s necessary for a 7-day stabzaton of a concentrated emuson (n ~ 4 m -3 ). Ts corresponds to W = 9. Muc ger vaues of te stabty rato ( 7 ) are found n te terature pubsed between 97 and 94 (see efs. [, 76] and ctatons teren). Tese g stabty ratos mosty correspond to sos of very sma partces (between nm and nm). Muc ower vaues of W ( W ) are found for atex suspensons (see Tabe n ef. []). Te wde range of stabty ratos reported durng te past century s partay caused by te nadequate defnton of W. Te frst expresson of W deduced by Fucs [4] compared te actua vaue of te foccuaton rate wt te teoretca vaue predcted by Smoucowsk. Tese stabty ratos tend to be g snce Smoucowsk dd not ncude te effect of te attractve force n te cacuaton of k fast f ( ks = 4kB T 3η ). However, even wen te correct expresson of W s used (Equaton (8)), ts vaue rarey surpasses for partces between.5 μm and a few mcrons. Tese ow vaues of W do not agree wt te fact tat most of tese partces exbt a g repusve barrer aganst prmary mnmum foccuaton (Equatons (9)). We studed ts probem n ef. [] usng severa partce szes and nteracton potentas. Varyng te surface carge of te partces and te onc strengt of te souton, t was evdent tat partces between nm and nm usuay sow very saow secondary mnma or no secondary mnma at a. Mcron sze partces nstead sow deep secondary mnma. Te smuatons ndcated tat te systems wt barrer egts ger tan k B T preserve te number of partces (coaescence does not occur durng te extent of te smuaton). In tese cases te dept of te secondary mnmum determnes te evouton of te system. On te one and, saow mnma do not ead to aggregaton [,74]. Hence, te tota number of aggregates (n a ) oscates frequenty around te number of partces ndcatng te sporadc formaton of doubets and ter quck dssouton. On te oter and, deep secondary mnmum promotes fast foccuaton. In ts case rreversbe secondary-mnmum foccuaton occurs. Te random fuctuaton of te number of aggregates prevousy observed wt saow mnma does not occur. Te partces do not go n and out te secondary mnmum as t appened n te former case. Te curves of n a vs. t sow a