CHAPTER 1 PLANAR FLUID INTERFACES

Transcription

1 Planar Flud Intrfacs Chaptr n th book: P.A. Kralchvsky and K. Nagayama, Partcls at Flud Intrfacs and Mmbrans (Attachmnt of Collod Partcls and Protns to Intrfacs and Formaton of Two-Dmnsonal Arrays) Elsvr, Amstrdam, 00; pp CHAPTER PLANAR FLUID INTERFACES An ntrfac or mmbran s on of th man actors n th procss of partcl-ntrfac and partcl-partcl ntracton at a flud phas boundary. Th lattr procss s nfluncd by mchancal proprts, such as th ntrfacal (mmbran) tnson and th surfac (Gbbs) lastcty. For ntrfacs and mmbrans of low tnson and hgh curvatur th ntrfacal bndng momnt and th curvatur lastc modul can also bcom mportant. As a rul, thr ar surfactant adsorpton layrs at flud ntrfacs and vry frquntly th ntrfacs bar som lctrc charg. For ths rasons n th prsnt chaptr w pay a spcal attnton to surfactant adsorpton and to lctrcally chargd ntrfacs. Our purpos s to ntroduc th basc quantts and rlatonshps n mchancs, thrmodynamcs and kntcs of flud ntrfacs and surfactant adsorpton, whch wll b furthr currntly usd throughout th book. Dfntons of surfac tnson, ntrfacal bndng momnt, adsorptons of th spcs, surfac of tnson and qumolcular dvdng surfac, surfac lastcty and adsorpton rlaxaton tm ar gvn. Th most mportant quatons rlatng ths quantts ar drvd, thr physcal manng s ntrprtd, and approprat rfrncs ar provdd. In addton to known facts and concpts, th chaptr prsnts also som rcnt rsults on thrmodynamcs and kntcs of adsorpton of onc surfactants. Four tabls summarz thortcal xprssons, whch ar rlatd to varous adsorpton sothrms and typs of lctrolyt n th soluton. W hop ths ntroductory chaptr wll b usful for both rsarchrs and studnts, who approach for a frst tm th fld of ntrfacal scnc, as wll as for xprts and lcturrs who could fnd hr a somwhat dffrnt vwpont and nw nformaton about th factors and procsss n ths fld and thr ntrconncton.

2 Chaptr.. MECHANICAL PROPERTIES OF PLANAR FLUID INTERFACES... THE BAKKER EQUATION FOR SURFACE TENSION Th balanc of th lnar momntum n flud dynamcs rlats th local acclraton n th flud to th dvrgnc of th prssur tnsor, P, s.g. Rf. []: d v P dt (.) Hr s th mass dnsty of th flud, v s vlocty and t s tm; n fact th prssur tnsor P quals th strss tnsor T wth th oppost sgn: P = T. In a flud at rst v 0 and Eq. (.) rducs to P 0 (.) whch xprsss a ncssary condton for hydrostatc qulbrum. In th bulk of a lqud th prssur tnsor s sotropc, P PB U (.3) as statd by th known Pascal law (U s th spatal unt tnsor; P B s a scalar prssur). Indd, all drctons n th bulk of a unform lqud phas ar quvalnt. Th lattr s not vald n a vcnty of th surfac of th flud phas, whr th normal to th ntrfac dtrmns a spcal drcton. In othr words, n a vcnty of th ntrfac th forc actng across unt ara s not th sam n all drctons. Corrspondngly, n ths rgon th prssur tnsor can b xprssd n th form [,3]: P P ( ) P (.4) T x x y y N z z Hr x, y and z ar th unt vctors along th Cartsan coordnat axs, wth z bng orntd normally to th ntrfac; P N and P T ar, rspctvly, th normal and th tangntal componnts of th prssur tnsor. Du to th symmtry of th systm P N and P T can dpnd on z, but thy should b ndpndnt of x and y. Thus a substtuton of Eq. (.4) nto Eq. (.3) ylds on non-trval quaton: P N z 0 (.5)

3 Planar Flud Intrfacs 3 In othr words, th condton for hydrostatc qulbrum, Eq. (.3), mpls that P N must b constant along th normal to th ntrfac; thrfor, P N s to b qual to th bulk sotropc prssur, P N = P B = const. Lt us tak a vrtcal strp of unt wdth, whch s orntd normally to th ntrfac, s Fg... Th nds of th strp, at z = a and z = b, ar supposd to b locatd n th bulk of phass and, rspctvly. Th ral forc xrtd to th strp s b (ral) F T PT ( z) dz a (.6) On th othr hand, followng Gbbs [4] on can construct an dalzd systm consstng of two unform phass, whch prsrv thr bulk proprts up to a mathmatcal dvdng surfac modlng th transton zon btwn th two phass (Fg..). Th prssur vrywhr n th dalzd systm s qual to th bulk sotropc prssur, P B =P N. In addton, a surfac tnson Fg... Sktch of a vrtcal strp, whch s normal to th boundary btwn phass and.

4 4 Chaptr s ascrbd to th dvdng surfac n th dalzd systm. Thus th forc xrtd to th strp n th dalzd systm (Fg..) s F (dalzd) T b a P N dz (.7) Th rol of s to mak up for th dffrncs btwn th ral and th dalzd systm. Sttng F T (dalzd) F T (ral) surfac tnson: from Eqs. (.6) and (.7) on obtans th Bakkr [5] quaton for th P P N T dz (.8) Snc th boundars of ntgraton z = a and z = b ar locatd n th bulk of phass and, whr th prssur s sotropc (P T = P N ), w hav st th boundars n Eq. (.8) qual to. Equaton (.8) mans that th ral systm wth a planar ntrfac can b consdrd as f t wr composd of two homognous phass sparatd by a planar mmbran of zro thcknss wth Lqud, phas I Gas, phas II z-z v, Angstroms Fg... Ansotropy of th prssur tnsor, P, plottd vs. th dstanc to th qumolcular dvdng surfac, zz v, for ntrfac lqud argon-gas at 84.3 K; Curvs and ar calculatd by th thors n Rfs. [8] and [0].

5 Planar Flud Intrfacs 5 tnson gvn by Eq. (.8). Th lattr quaton gvs a hydrostatc dfnton of surfac tnson. Not that ths dfnton dos not dpnd on th xact locaton of th dvdng surfac. Th quantty P P N P T (.9) xprsss th ansotropy of th prssur tnsor. Th functon P(z) can b obtand thortcally by mans of th mthods of th statstcal mchancs [6-9]. As an llustraton n Fgur. w prsnt data for P vs. zz v for th ntrfac lqud argongas at tmpratur T = 84.3 K; z v s th poston of th so calld qumolcular dvdng surfac (s Scton.. blow for dfnton). Th mpty and full ponts n Fg.. ar calculatd by mans of th thors from Rfs. [8] and [0], rspctvly. As sn n Fg.., th wdth of th transton (P N - P T ) x0-7, dyn/cm () (3) () z-z v, Angstroms () () (3) Fg..3. Ansotropy of th prssur tnsor, P, plottd vs. th dstanc to th qumolcular dvdng surfac, zz v, calculatd by th thory n Rf. [0] for th phas boundars n-dcangas (curv ), gaswatr (curv ) and n-dcanwatr (curv 3).

6 6 Chaptr zon btwn th lqud and gas phass (n whch P 0) s of th ordr of 0 Å. On th othr hand, th maxmum valu of th ansotropy P(z) s about 0 8 dyn/cm,.. about 00 atmosphrs, whch s an mprssv valu. Th ara blow th full ln n Fg.. gvs th surfac tnson of argon at that tmpratur, = 3.45 mn/m, n accordanc wth Eq. (.8). Curvs, and 3 n Fg..3 prsnt P(z) calculatd n Rf. [0] for th ntrfacs n- dcan/gas, gas/watr and n-dcan/watr, rspctvly. On s that P(z) typcally xhbts a sngl maxmum for a lqud-gas ntrfac, whras P(z) xhbts a loop (maxmum and mnmum) for a lqud-lqud ntrfac. For all curvs n Fg..3 th wdth of th ntrfacal transton zon s of th ordr of 0 Å... INTERFACIAL BENDING MOMENT AND SURFACE OF TENSION To mak th dalzd systm n Fg.. hydrostatcally quvalnt to th ral systm w hav to mpos also a rqurmnt for quvalnc wth rspct to th actng forc momnts (n addton to th analogous rqurmnt for th actng forcs, s abov). Th momnt xrtd on th strp n th ral systm (Fg..) s b (ral) M PT ( z) a zdz (.0) Lkws, th momnt xrtd on th strp n th dalzd systm s []: M (dalzd) b P z dz z B (.) a N 0 0 Hr z = z 0 s th poston of th dvdng surfac and B 0 s an ntrfacal bndng momnt (coupl of forcs), whch s to b attrbutd to th dvdng surfac n ordr to mak th dalzd systm quvalnt to th ral on wth rspct to th forc momnts. Sttng (dalzd) (ral) M M from Eqs. (.8), (.0) and (.) on obtans an xprsson for th ntrfacal bndng momnt:

7 Planar Flud Intrfacs 7 PN PT ( z0 z) B0 dz (.) As n Eq. (.8) w hav xtndd th boundars of ntgraton to. From th vwpont of mchancs postv B 0 rprsnts a forc momnt (a coupl of forcs), whch tnds to bnd th dvdng surfac around th phas, for whch z s an outr normal (n Fg.. ths s phas ). Th comparson of Eqs. (.8) and (.) shows that unlk, th ntrfacal bndng momnt B 0 dpnds on th choc of poston of th dvdng surfac z 0. Th lattr can b dfnd by mposng som addtonal physcal condton; n such a way th qumolcular dvdng surfac s dfnd (s Scton.. blow). If onc th poston of th dvdng surfac s dtrmnd, thn th ntrfacal bndng momnt B 0 bcoms a physcally wll dfnd quantty. For xampl, th valus of th bndng momnt, corrspondng to th qumolcular dvdng surfac, for curvs No., and 3 n Fg..3 ar, rspctvly [0]: B 0 =.,.3 and 5. 0 N. On possbl way to dfn th poston, z 0, of th dvdng surfac s to st th bndng momnt to b dntcally zro: B 0 z 0 z s 0 (.3) Combnng Eqs (.8), (.) and (.3) on obtans [] z s P P N T zdz (.4) Equaton (.4) dfns th so calld surfac of tnson. It has bn frst ntroducd by Gbbs [4], and t s currntly usd n th convntonal thory of capllarty (s Chaptr blow). At th surfac of tnson th ntrfac s charactrzd by a sngl dynamc paramtr, th ntrfacal tnson ; ths consdrably smplfs th mathmatcal tratmnt of capllary problms. Howvr, th physcal stuaton bcoms mor complcatd whn th ntrfacal tnson s low; such s th cas of som mulson and mcromulson systms, lpd blayrs and bommbrans. In th lattr cas, th surfac of tnson can b locatd far from th actual transton rgon btwn th two phass and ts usag bcoms physcally mannglss.

8 8 Chaptr Indd, for 0 Eq. (.4) ylds z s. Thrfor, a mchancal dscrpton of an ntrfac of low surfac tnson nds th usag of (at last) two dynamc quantts: ntrfacal (surfac) tnson and bndng momnt. In fact, B 0 s rlatd to th so calld spontanous curvatur of th ntrfac. In Chaptr 3 w wll com to ths pont agan...3. ELECTRICALLY CHARGED INTERFACES As a rul, th boundars btwn two phass (and th bommbrans, as wll) bar som lctrc charg. Oftn t s du to th dssocaton of surfac onzabl groups or to adsorpton of chargd amphphlc molculs (surfactants). It should b notd that vn th boundars watr-ar and watr-ol (ol hr mans any lqud hydrocarbon mmscbl wth watr) ar lctrcally chargd n th absnc of any surfactant, s.g. rfs. [] and [3]. If th surfac of an aquous phas s chargd, t rpls th coons,.. th ons of th sam charg, but t attracts th countrons, whch ar th ons of th oppost charg, s Fg..4. Thus a nonunform dstrbuton of th onc spcs n th vcnty of th chargd ntrfac appars, whch s known as lctrc doubl layr (EDL), s.g. Rf. [4]. Th convntonal modl of th EDL stms from th works of Gouy [5], Chapman [6] and Strn [7]. Th EDL s consdrd to consst of two parts: (I) ntrfacal (adsorpton) layr and (II) dffus layr. Th ntrfacal (adsorpton) layr ncluds chargs, whch ar mmoblzd (adsorbd) at th phas boundary; ths ncluds also adsorbd (bound) countrons, whch form th so calld Strn layr, s Fg..4. Th dffus layr conssts of fr ons n th aquous phas, whch ar nvolvd n Brownan moton n th lctrcal fld cratd by th chargd ntrfac. Th boundary, whch sparats th adsorpton from th dffus layr, s usually calld th Gouy plan. Th convntonal thory of th lctrc doubl layr s brfly prsntd n Scton..4 blow. For our purposs hr t s suffcnt to tak nto account that th lctrc potntal vars across th EDL: = (z). Th thcknss of th dffus EDL could b of th ordr of hundrd (and vn thousand) nm,.. t s much gratr than th thcknss of th ntrfacal transton zon (cf. Fgs.. and.3). Ths fact rqurs a spcal approach to th thortcal dscrpton of th

9 Planar Flud Intrfacs 9 chargd ntrfacs, whch can b basd on th xprsson for th Maxwll lctrc strss tnsor [8]: c z Fg..4. Sktch of th lctrc doubl layr n a vcnty of an adsorpton monolayr of onc surfactant. (a) Th dffus layr contans fr ons nvolvd n Brownan moton, whl th Strn layr conssts of adsorbd (bound) countrons. (b) Nar th chargd surfac thr s an accumulaton of countrons and a dplton of coons, whos bulk concntratons ar both qual to c. P k ( P o E 8 ) k 4 E E k (, k,,3) (.5) Hr k s th Kronckr symbol (th unt matrx), s th dlctrc prmttvty of th mdum (usually watr), E s th -th componnt of th lctrc fld,

10 0 Chaptr E, x E 3 E, (.6) x = x, y = x and z = x 3 ar Cartsan coordnats, and P o s an sotropc prssur, whch can vary across th EDL du to th osmotc ffct of th dssolvd onc spcs. As alrady mntond, n th cas of plan ntrfac w hav = (z), and thn Eq. (.5) rducs to th followng two xprssons: P N P zz P o 8 d dz (.7) P T P xx P yy P o 8 d dz (.8) Eqs. (.7) and (.8) can b appld to dscrb th prssur tnsor wthn th dffus part of th lctrc doubl layr. Now, lt us locat th plan z = 0 n th Gouy plan sparatng th dffus (at z > 0) from th adsorpton layr. Thn by mans of th Bakkr quaton (.8) on can rprsnt th surfac tnson as a sum of contrbutons from th adsorpton and dffus layrs: a d (.9) whr 0 a ( P P ) dz, ( P P ) dz N T d 0 N T (.0) Substtutng Eqs. (.7) and (.8) nto th abov quaton for d, on obtans a gnral xprsson for th contrbuton of th dffus layr to th ntrfacal tnson [9,0]: d 4 0 d dz dz (.) Equaton (.) shows that th contrbuton of th dffus lctrc doubl layr to th ntrfacal tnson, d, s always ngatv,.. th ntractons n th dffus layr tnd to dcras th total

11 Planar Flud Intrfacs ntrfacal tnson. Explct xprssons for d, obtand by mans of th doubl layr thory for varous typs of lctrolyts, can b found n Tabl.3 blow...4. WORK OF INTERFACIAL DILATATION Lt us consdr an magnary rctangular box contanng portons of phass and, and of th ntrfac btwn thm. As bfor, w wll assum that th ntrfac s paralll to th coordnat plan xy, and th sds of th rctangular box ar also paralll to th rspctv coordnat plans. Movng th sds of th box on can crat a small chang of th volum of th box, V, wth a corrspondng small chang of th ntrfacal ara, A. Th work W carrd out by th xtrnal forcs to crat ths dformaton can b calculatd by mans of a known quaton of flud mchancs []: W ( P : D)dV V (.) Hr D s th stran tnsor (tnsor of dformaton) and : dnots doubl scalar product of two tnsors (dyadcs): ( AB) : ( CD) ( A D)( B C) (.3) Snc w consdr dsplacmnts of th sds of our rctangular box along th normals to th rspctv sds, th stran tnsor has dagonal form n th Cartsan bass [,]: D x x ( dx) ( dy) ( dz) y y z z dx dy dz (.4) Hr (dx) dnots th xtnson of a lnar lmnt dx of th contnuous mdum n th cours of dformaton. Equaton (.4) shows that th gnvalus of th stran tnsor ar th rlatv xtnsons of lnar lmnts along th thr axs of th Cartsan coordnat systm. Substtutng Eqs. (.4) and (.4) nto Eq. (.) on can drv []: ( dx) ( dy) ( dz) ( dx) ( dy) W PN dxdydz PN PT dxdydz (.5) V dx dy dz V dx dy Th ncrmnts of th lmntary volum and ara n th procss of dformaton ar

12 Chaptr ( dv ) dydz ( dx) dxdz ( dy) dxdy ( dz), ( da) dy ( dx) dx ( dy) (.6) Combnng Eqs. (.8), (.5) and (.6) on fnally obtans W PN V A (.7) Hr P N V xprsss th work of changng th volum and A s th work of ntrfacal dlataton. Equaton (.7) gvs a conncton btwn th mchancs and thrmodynamcs of th flud ntrfacs... THERMODYNAMICAL PROPERTIES OF PLANAR FLUID INTERFACES... THE GIBBS ADSORPTION EQUATION Lt us consdr th sam systm as n scton..4 abov. Th Gbbs fundamntal quaton, combnng th frst and th scond law of thrmodynamcs, s [,4] du TdS P dv N da dn, (.8) whr T s th tmpratur; U and S ar th ntrnal nrgy and ntropy of th systm, rspctvly; and N ar th chmcal potntal and th numbr of molculs of th -th componnt (spcs); th summaton n Eq. (.8) s carrd out ovr all componnts n th systm. Equaton (.8) stats that th ntrnal nrgy of th systm can vary bcaus of th transfr of hat (TdS) and/or mattr ( dn ), and/or du to th mchancal work, W, carrd out by xtrnal forcs, s Eq. (.7). Followng Gbbs [4], w construct an dalzd systm consstng of two bulk phass, whch ar unform up to a mathmatcal dvdng surfac modlng th boundary btwn th two phass. Snc th dvdng surfac has a zro thcknss, th volums of th two phass n th dalzd systm ar addtv: V V () V () (.9)

13 Planar Flud Intrfacs 3 W assum that th bulk dnsts of ntropy, s (k), ntrnal nrgy, u (k), and numbr of molculs, n (k), ar known for th two nghborng phass (k =,). Thn th ntropy, ntrnal nrgy and numbr of molculs for phas k of th dalzd systm ar: S ( k ) s ( k ) V ( k ) ; U ( k ) u ( k ) V ( k ) ; N ( k ) n ( k ) V ( k ) ( k,) (.30) Each of th two unform bulk phass has ts own fundamntal quaton [,4]: du du () () TdS TdS () () P dv B P dv B () () dn dn () () (.3) It s prsumd that w dal wth a stat of thrmodynamc qulbrum, and hnc th tmpratur T and th chmcal potntals ar unform throughout th systm [3]; n addton, P N = P B = const., s Eq. (.5) abov. Nxt, w sum up th two quatons (.3) and subtract th rsult from Eq. (.8); thus w obtan: du ( s) T ds ( s) ( s) da dn, (.3) whr U ( s) U U () U S S S S N N () ( s) () () ( s) () (),, N N (.33) ar, rspctvly, surfac xcsss of ntrnal nrgy, ntropy and numbr of molculs of th -th spcs; ths xcsss ar consdrd as bng attrbutd to th dvdng surfac. Equaton (.3) can b ntrprtd as th fundamntal quaton of th ntrfac [4, 4]. Snc th ntrfac s unform, thn du (s), ds (s) (s and dn ) can b consdrd as amounts of th rspctv xtnsv thrmodynamc paramtrs corrspondng to a small porton, da, of th ntrfac; thn Eq. (.3) can b ntgratd to yld [,4]: U ( s) TS ( s) ( s) A N, (.34) Fnally, w dffrntat Eq. (.34) and compar th rsult wth Eq. (.3); thus w arrv at th Gbbs [4] adsorpton quaton:

14 4 Chaptr d S A ( s) dt d (.35) whr N ( s) A z0 () () n ( z) n dz n ( z) n z 0 dz (.36) s th adsorpton of th -th spcs at th ntrfac; n (z) s th actual concntraton of componnt as a functon of th dstanc to th ntrfac, z, cf. Eq. (.33); z 0 dnots th poston of th dvdng surfac. Fgur.5a shows qualtatvly th dpndnc n (z) for a nonamphphlc componnt,.. a componnt, whch dos not xhbt a tndncy to accumulat at th ntrfac; f phas s an aquous soluton, thn th watr can srv as an xampl for a non-amphphlc componnt. On th othr hand, Fgur.5b shows qualtatvly th dpndnc n (z) for an amphphlc componnt (surfactant), whch accumulats (adsorbs) at th ntrfac, s th maxmum of n (z) n Fg..5b.... EQUIMOLECULAR DIVIDING SURFACE As dscussd n scton.. abov, th dfnton of th dvdng surfac s a mattr of choc. In othr words, on has th frdom to mpos on physcal condton n ordr to dtrmn th poston of th dvdng surfac. Ths can b th condton th adsorpton of th -th componnt to b qual to zro [4]: z z 0 (qumolcular dvdng surfac) (.37) 0 v Th surfac thus dfnd s calld qumolcular dvdng surfac wth rspct to componnt. In ordr to hav 0 th sum of th ntgrals n Eq. (.36) must b qual to zro. Ths mans that th postv and ngatv aras, whch ar comprsd btwn th contnuous and dashd lns n Fg..5a,b and dnotd by (+) and (), must b qual.

15 Planar Flud Intrfacs 5 Fg..5. Illustratv dpndnc of th dnsty n of th -th componnt on th dstanc z to th ntrfac for (a) non-amphphlc componnt and (b) amphphlc componnt; z v dnots th poston of th qumolcular dvdng surfac; n () and n () ar th valus of n n th bulk of phass and. As sn n Fg..5a, f componnt s non-amphphlc (say th watr as a solvnt n an aquous soluton), th qumolcular dvdng surfac, z = z v, s rally stuatd n th transton zon btwn th two phass. In contrast, f componnt s an amphphlc on, thn th qumolcular dvdng surfac, z = z v, s locatd far from th actual ntrfacal transton zon (Fg..5b). Thrfor, to achv a physcally adquat dscrpton of th systm, th qumolcular dvdng surfac s usually ntroducd wth rspct to th solvnt; t should nvr b ntroducd wth rspct to an amphphlc componnt (surfactant)...3. THERMODYNAMICS OF ADSORPTION OF NONIONIC SURFACTANTS A molcul of a nononc surfactant (lk all amphphlc molculs) conssts of a hydrophlc and a hydrophobc moty. Th hydrophlc moty (th hadgroup ) can b a watr solubl polymr, lk polyoxthyln, or som polysacchard [5]; t can b also a dpolar hadgroup, lk thos of many phospholpds. Th hydrophobc moty (th tal ) usually conssts of on or two hydrocarbon chan(s). Th adsorpton of such a molcul at a flud ntrfac s accompand wth a gan of fr nrgy, bcaus th hydrophlc part of an adsorbd molcul s xposd to th aquous phas, whras ts hydrophobc part contacts wth th non-aquous (hydrophobc) phas. Lt us consdr th boundary btwn an aquous soluton of a nononc surfactant and a hydrophobc phas, ar or ol. W choos th dvdng surfac to b th qumolcular dvdng

16 6 Chaptr surfac wth rspct to watr, that s w = 0. Thn th Gbbs adsorpton quaton (.35) rducs to d d (T = const.) (.38) whr th subscrpt dnots th nononc surfactant. Snc th bulk surfactant concntraton s usually rlatvly low, on can us th xprsson for th chmcal potntal of a solut n an dal soluton [3]: (0) kt ln c (.39) whr c s th concntraton of th nononc surfactant and (0) s a standard chmcal potntal, whch s ndpndnt of c, and k s th Boltzmann constant. Combnng Eqs. (.38) and (.39) on obtans d kt d ln c (.40) Th surfactant adsorpton sothrms, xprssng th conncton btwn and c ar usually obtand by mans of som molcular modl of th adsorpton. Th most popular s th Langmur [6] adsorpton sothrm, Kc Kc (.4) whch stms from a lattc modl of localzd adsorpton of non-ntractng molculs [7]. In Eq. (.4) s th maxmum possbl valu of th adsorpton ( for c ). On th othr hand, for c 0 on has Kc ; th adsorpton paramtr K charactrzs th surfac actvty of th surfactant: th gratr K th hghr th surfac actvty. Tabl. contans th 6 most popular surfactant adsorpton sothrms, thos of Hnry, Frundlch, Langmur, Volmr [8], Frumkn [9], and van dr Waals [7]. For c 0 all othr sothrms (xcpt that of Frundlch) rduc to th Hnry sothrm. Th physcal dffrnc btwn th Langmur and Volmr sothrms s that th formr corrsponds to a physcal modl of localzd adsorpton, whras th lattr to non-localzd adsorpton. Th Frumkn and van dr Walls sothrms gnralz, rspctvly, th Langmur and Volmr sothrms for th cas, whn thr s ntracton btwn th adsorbd molculs; s th paramtr,

17 Planar Flud Intrfacs 7 Tabl.. Th most popular surfactant adsorpton sothrms and th rspctv surfac tnson sothrms. Hnry Frundlch Langmur Volmr Frumkn van dr Waals Surfactant adsorpton sothrms (for nononc surfactants:a s c ) Ka Ka Ka s s F s / m Ka s xp xp kt Ka s xp kt Ka s Hnry Surfac tnson sothrm 0 ktj d (for nononc surfactants: d 0) J Frundlch Langmur Volmr J J m ln J Frumkn van dr Waals J ln J kt kt

18 8 Chaptr whch accounts for th ntracton. In th cas of van dr Waals ntracton can b xprssd n th form [30,3]: u( r) kt xp kt r0 r0 u( r) rdr whr u(r) s th ntracton nrgy btwn two adsorbd molculs and r 0 s th dstanc btwn th cntrs of th molculs at clos contact. Th comparson btwn thory and xprmnt shows that th ntracton paramtr s mportant for ar-watr ntrfacs, whras for ol-watr ntrfacs on can st = 0 [3,33]. Th lattr fact, and th fndng that > 0 for ar-watr ntrfacs, lads to th concluson that taks nto account th van dr Waals attracton btwn th hydrocarbon tals of th adsorbd surfactant molculs across ar (such attracton s mssng whn th hydrophobc phas s ol). What concrns th paramtr K n Tabl., t s rlatd to th standard fr nrgy of adsorpton, f (0) (0) s, whch s th nrgy gan for brngng a molcul from th bulk of th watr phas to a dlutd adsorpton layr [34,35]: K xp (0) kt (0) s (.4) Hr s a paramtr, charactrzng th thcknss of th adsorpton layr, whch can b st (approxmatly) qual to th lngth of th amphphlc molcul. Lt us consdr th ntgral J c 0 dc c 0 d ln c d d (.43) Th drvatv d ln c / d can b calculatd for ach adsorpton sothrm n Tabl., and thn th ntgraton n Eq. (.43) can b carrd out analytcally. Th xprssons for J, obtand n ths way, ar also lstd n Tabl.. Th ntgraton of th Gbbs adsorpton sothrm, Eq. (.40), along wth Eq. (.43) ylds 0 ktj, (.44) whch n vw of th xprssons for J n Tabl. prsnts th surfactant adsorpton sothrm, or th two-dmnsonal (surfac) quaton of stat.

19 Planar Flud Intrfacs 9 Tabl.. Exprssons for th Gbbs lastcty of adsorpton monolayrs (vald for both nononc and onc surfactants), whch corrspond to th varous typs of sothrms n Tabl.. Typ of surfac tnson sothrm Gbbs lastcty E G Hnry E G kt Frundlch kt m E G Langmur E G kt Volmr E G kt Frumkn van dr Waals kt kt E G kt E G kt An mportant thrmodynamc paramtr of a surfactant adsorpton monolayr s ts Gbbs (surfac) lastcty: EG T (.45) Exprssons for E G, corrspondng to varous adsorpton sothrms, ar shown n Tabl.. As an xampl, lt us consdr th xprsson for E G, corrspondng to th Langmur sothrm; combnng rsults from Tabls. and. on obtans E G ktkc (for Langmur sothrm) (.45a) On ss that for Langmuran adsorpton th Gbbs lastcty grows lnarly wth th surfactant concntraton c. Snc th concntraton of th monomrc surfactant cannot xcd th crtcal mcllzaton concntraton, c c CMC, thn from Eq. (.45a) on obtans

20 0 Chaptr E G EG kt KcCMC (for Langmur sothrm) (.45b) max Hnc on could xpct hghr lastcty E G for surfactants wth hghr c CMC ; ths concluson s consonant wth th xprmntal rsults [36]. Th Gbbs lastcty charactrzs th latral fludty of th surfactant adsorpton monolayr. For hgh valus of th Gbbs lastcty th adsorpton monolayr at a flud ntrfac bhavs as tangntally mmobl. Thn, f a partcl approachs such an ntrfac, th hydrodynamc flow pattrn, and th hydrodynamc ntracton as wll, s approxmatly th sam as f th partcl wr approachng a sold surfac. For lowr valus of th Gbbs lastcty th so calld Marangon ffct appars, whch can consdrably affct th approach of a partcl to a flud ntrfac. Ths aspcts of th hydrodynamc ntractons btwn partcls and ntrfacs ar consdrd n Chaptr 6 blow. Th thrmodynamcs of adsorpton of onc surfactants (s Scton..5 blow) s mor complcatd bcaus of th prsnc of long-rang lctrostatc ntractons n th systm. As an ntroducton, n th nxt scton w brfly prsnt th thory of th lctrc doubl layr...4. THEORY OF THE ELECTRIC DOUBLE LAYER. Boltzmann quaton and actvty coffcnts. Whn ons ar prsnt n th soluton, th (lctro)chmcal potntal of th onc spcs can b xprssd n th form [3] (0) kt ln a Z (.46) whch s mor gnral than Eq. (.39) abov; hr s th lmntary lctrc charg, s th lctrc potntal, Z s th valncy of th onc componnt, and a s ts actvty. Whn an lctrc doubl layr s formd n a vcnty a chargd ntrfac, s Fg..4, th lctrc potntal and th actvts of th onc spcs bcom dpndnt on th dstanc z from th ntrfac: = (z), a = a (z). On th othr hand, at qulbrum th lctrochmcal potntal,, s unform throughout th whol soluton, ncludng th lctrc doubl layr (othrws dffuson fluxs would appar) [3]. In th bulk of soluton (z) th lctrc potntal tnds to a constant valu, whch s usually st qual to zro; thn on can wrt

21 Planar Flud Intrfacs lm ( z) 0 z d lm 0 z dz (.47) (.48) Sttng qual th xprsson for at z and that for at som fnt z, and usng Eqs. (.46) and (.47), on obtans [3]: Z ( z) a ( z) a xp kt (.49) whr a dnots th valu of th actvty of on n th bulk of soluton. Equaton (.49) shows that th actvty obys a Boltzmann typ dstrbuton across th lctrc doubl layr (EDL). If th actvty n th bulk, a, s known, thn Eq. (.49) dtrmns th actvty a (z) n ach pont of th EDL. Th studs on adsorpton of onc surfactants [3,33,0] show that a good agrmnt btwn thory and xprmnt can b achvd usng th followng xprsson for a : a c (.50) whr c s th bulk concntraton of th rspctv on, and th actvty coffcnt s to b calculatd from th known smmprcal formula [37] log A Z Z B d I I bi (.5) whch orgnats from th Dby-Hückl thory; I dnots th onc strngth of th soluton: I Z c (.5) whr th summaton s carrd out ovr all onc spcs n th soluton. Whn th soluton contans a mxtur of svral lctrolyts, thn Eq. (.5) dfns for ach sparat lctrolyt, wth Z + and Z bng th valncs of th catons and anons for ths lctrolyt, but wth I bng th total onc strngth of th soluton, accountng for all dssolvd onc spcs [37]. Th log n Eq. (.5) s dcmal, d s th damtr of th on, A, B, and b ar paramtrs,

22 Chaptr whos valus can b found n th book by Robnson and Stoks [37]. For xampl, f th onc strngth I s gvn n mols pr ltr (M), thn for solutons of NaCl at 5C th paramtrs valus ar A = 0.55 M /, Bd =.36 M / and b = M. Intgraton of Posson-Boltzmann quaton. Th Posson quaton rlatng th dstrbuton of th lctrc potntal (z) and lctrc charg dnsty, (z), across th dffus doubl layr can b prsntd n th form [4] d 4, (.53) d z Lt us choos componnt to b a coon, that s an on havng lctrc charg of th sam sgn as th ntrfac. It s convnnt to ntroduc th varabls ( z) Z ( z) ~,, kt Z z k Z Z k (k =,,...N) (.54) For symmtrc lctrolyts and ~ thus dfnd ar always postv rrspctv of whthr th ntrfac s postvly or ngatvly chargd. Combnng Eqs. (.49), (.53) and (.54) on obtans d d z c ~ N c z a xp( z ) (.55) whr Z (.56) kt 8 c As usual, th z-axs s drctd along th normal to th ntrfac, th lattr corrspondng to z = 0. To obtan Eq. (.55) w hav xprssd th bulk charg dnsty n trms of ffctv concntratons,.. actvts, Z a (z), rathr than n trms of th nt concntratons, (z) (z) Z c (z). For not-too-hgh onc strngths thr s no sgnfcant quanttatv dffrnc btwn ths two xprssons for (z), but th formr on consdrably smplfs th mathmatcal drvatons; morovr, th formr xprsson has bn combnd wth

23 Planar Flud Intrfacs 3 Eq. (.49), whch s rgorous n trms of actvts (rathr than n trms of concntratons). Intgratng Eq. (.55) on can drv d dz N c a xp( z ) (.57) whr th boundary condtons 0 z and / d z z 0 d hav bn usd, cf. Eqs. (.47), (.48) and (.54). Not that Eq. (.57) s a nonlnar ordnary dffrntal quaton of th frst ordr, whch dtrmns th varaton of th lctrc potntal (z) across th EDL. In gnral, Eq. (.57) has no analytcal soluton, but t can b solvd rlatvly asly by numrcal ntgraton. Analytcal soluton can b obtand n th cas of symmtrc lctrolyt, s Eq. (.65) blow. Furthr, lt s b th surfac lctrc charg dnsty,.. th lctrc charg pr unt ara of th ntrfac. Snc th soluton, as a whol, s lctronutral, th followng rlatonshp holds [4]: 0 s ( z) dz (.58) Substtutng (z) from Eq. (.55) nto Eq. (.58) and ntgratng th scond drvatv, d / dz, on drvs d ~ s c s, ~ s d z (.59) Z z0 Th combnaton of Eqs. (.57) and (.59) ylds a conncton btwn th surfac charg dnsty, s, and th surfac potntal, s (z=0), whch s known as th Gouy quaton [5,38]: / ~ N s a xp( z s ), c Z s s (.60) kt Not that bcaus of th choc componnt to b a coon, th sgn of s and ~ s s always postv and that s th rason why n Eq. (.60) w hav takn sgn + bfor th squar root.

24 4 Chaptr To obtan an xprsson for calculatng th dffus layr contrbuton to th surfac tnson, d, w frst combn Eqs. (.) and (.54): d kt c 0 d dz dz kt c s 0 d d dz (.6) A substtuton of Eq. (.57) nto Eq. (.6) ylds d kt s N a c 0 / xp( z ) d (.6) Exprssons for d, obtand by mans of Eq. (.6) for solutons of surfactant and varous lctrolyts, can b found n Tabl.3 blow, as wll as n Rf. [0]. Analytcal xprssons for Z :Z lctrolyt. Analytcal xprsson for (z) can b obtand n th smplr cas, whn th soluton contans only symmtrc, Z :Z lctrolyt, that s Z = Z (Z = 0 for > ). In ths cas Eq. (.57) can b rprsntd n th form d dz snh (Z :Z lctrolyt) (.63) whr N c z a (.64) s known as th Dby scrnng paramtr. Th ntgraton of Eq. (.63) ylds an analytcal xprsson for th varaton of th lctrc potntal (z) across th EDL [4]: s ( z ) 4arctanhtanh xp(-z) (Z :Z lctrolyt) (.65) 4 Equaton (.65) shows that th lctrc potntal, cratd by th chargd ntrfac, dcays xponntally n th dpth of soluton, that s (z) xp(z) for z. Th nvrs Dby paramtr,, rprsnts a dcay lngth, whch charactrzs th thcknss of th EDL. Th Gouy quaton (.60), gvng th conncton btwn surfac charg and surfac potntal, also smplfs for Z :Z lctrolyt:

25 Planar Flud Intrfacs 5 ~ 4 s s a snh (Z :Z lctrolyt) (.66) c whr and ar th adsorptons of th onc componnts and, rspctvly. For th sam cas th ntgraton n Eq. (.6) can b carrd out analytcally and th followng smplr xprsson for th dffus layr contrbuton to th surfac tnson can b drvd [9,38,39]: 8kT s d a cosh (Z :Z lctrolyt) (.67) c Th abov quatons srv as a bass of th thrmodynamcs of adsorpton of onc surfactants...5. THERMODYNAMICS OF ADSORPTION OF IONIC SURFACTANTS Basc quatons. Combnng Eqs. (.46), (.47) and (.49) on obtans a known xprsson for th chmcal potntal: (0) kt ln a. Th substtuton of th lattr xprsson nto th Gbbs adsorpton quaton (.35) ylds [9,33,40,4]: N ~ d kt d ln (T = const) (.68) a Hr wth ~ w dnot th adsorpton of th -th componnt; ~ rprsnts a surfac xcss of componnt wth rspct to th unform bulk soluton. For an onc spcs ths mans that ~ s a total adsorpton, whch nclud contrbutons from both th adsorpton layr (surfactant adsorpton layr + adsorbd countrons n th Strn layr, s Fg..4) and th dffus layr. Lt us dfn th quantts ~ [ a ( z) a ] dz, 0 (.69) and can b ntrprtd as contrbutons of th dffuson and adsorpton layrs, rspctvly, nto th total adsorpton ~. Usng th thory of th lctrc doubl layr and th dfntons (.69) on can prov (s Appndx A) that th Gbbs adsorpton quaton (.68) can b prsntd nto th followng quvalnt form [0]

26 6 Chaptr N d kt d ln a (T = const) (.70) a s whr a = d s th contrbuton of th adsorpton layr nto th surfac tnson, d s th contrbuton of th dffus layr, dfnd by Eq. (.), and a s a xp( z ), s Z z, (.7) Z s th subsurfac actvty of th -th onc spcs. Th comparson btwn Eqs. (.68) and (.70) shows that th Gbbs adsorpton quaton can b xprssd thr n trms of, ~ and a, or n trms of a, and a s. In Appndx A t s provn that ths two forms ar quvalnt. To drv xplct adsorpton and surfac tnson sothrms, blow w spcfy th typ of onc surfactant and non-amphphlc salt n th soluton. Surfactant and salt ar : lctrolyts. W consdr a soluton of an onc surfactant, whch s a symmtrc : lctrolyt, n th prsnc of addtonal common symmtrc : lctrolyt (salt). Hr w assum that th countrons du to th surfactant and salt ar dntcal. For xampl, ths can b a soluton of sodum dodcyl sulfat (SDS) n th prsnc of NaCl. W dnot by c, c and c 3 th bulk concntratons of th surfac actv ons, countrons, and coons, rspctvly. For th spcal systm of SDS wth NaCl c, c and c 3 ar th bulk concntraton of th DS, Na + and Cl ons, rspctvly. Th rqurmnt for th bulk soluton to b lctronutral mpls c = c + c 3. Th multplcaton of th last quaton by, whch accordng to Eq. (.5) s th sam for all monovalnt ons, ylds a = a + a 3 (.7) Th adsorpton of th coons of th non-amphphlc salt s xpctd to b qual to zro, 3 = 0, bcaus thy ar rplld by th smlarly chargd ntrfac (howvr, 3 0: th ntgral n ~ Eq. (.69) gvs a ngatv 3, s Fg..4; hnc 0 ). Thn th Gbbs adsorpton quaton (.70) can b prsntd n th form 3 3 d a kt ( d ln as d ln a s ) (.73)

27 Planar Flud Intrfacs 7 Th dffrntals n th rght-hand sd of Eq. (.73) ar ndpndnt (on can vary ndpndntly th concntratons of surfactant and salt), and morovr, d a s an xact (total) dffrntal. Thn accordng to th Eulr condton [3] th cross drvatvs must b qual, vz. as as ln ln (.74) A surfactant adsorpton sothrm, (a s, a s ), and a countron adsorpton sothrm, (a s,a s ), ar thrmodynamcally compatbl f thy satsfy Eq. (.74). Intgratng Eq. (.74) on obtans ln J a s (.75) whr w hav ntroducd th notaton J a s ( aˆ 0 s, a s daˆ ) aˆ s s (.76) To dtrmn th ntgraton constant n Eq. (.75) w hav usd th condton that for a s = 0 (no surfactant n th soluton) w hav = 0 (no surfactant adsorpton) and = 0 (no bndng of countrons at th hadgroups of adsorbd surfactant). Th ntgral J n Eq. (.76) can b takn analytcally for all popular surfac tnson sothrms, s Tabl.. Dffrntatng Eq. (.76) on obtans J / ln as. Th substtuton of th lattr quaton, togthr wth Eq. (.75) nto Eq. (.73), aftr ntgraton ylds a 0 ktj, (.77) whr 0 s th valu of for pur watr. Combnng Eqs. (.9) and (.77) on obtans th surfac tnson sothrm of th onc surfactant: 0 ktj d, (.78) whr d s gvn by Eq. (.67) and xprssons for J, corrspondng to varous adsorpton sothrms, ar avalabl n Tabl.. Not that for ach of th sothrms n Tabl. dpnds on th product Ka, that s = ( Ka ). Thn Eq. (.76) can b transformd to rad s s

28 8 Chaptr Kas dx J ( X ) X 0 (.79) Dffrntatng Eq. (.79) on can brng Eq. (.75) nto th form [0] ln K ln a s (.80) whch holds for ach of th surfactant adsorpton sothrms n Tabl.. Not that Eq. (.80) s vald for a gnral form of th dpndnc K = K( a s ), whch xprsss th dpndnc of th qulbrum constant of surfactant adsorpton on th concntraton of th salt n soluton. Lt us consdr a lnar dpndnc K = K( a s ), that s K K Kas (.8) whr K and K ar constants. Th physcal manng of th lnar dpndnc of K on a s n Eq. (.8) s dscussd blow, s Eqs. (.8)(.8) and th rlatd txt. Th substtuton of Eq. (.8) nto Eq. (.80) ylds [0] K a K K s a s (.8) Equaton (.8) s n fact a form of th Strn sothrm [7,38]. On can vrfy that th Eulr condton (.74) s dntcally satsfd f s substtutd from Eq. (.8) and s xprssd by thr of th adsorpton sothrms n Tabl.. In fact, Eq. (.8) s th ncssary and suffcnt condton for thrmodynamc compatblty of th Strn sothrm of countron adsorpton, Eq. (.8), wth thr of th surfactant adsorpton sothrms n Tabl.. In othr words, a gvn sothrm from Tabl., say th Langmur sothrm, s thrmodynamcally compatbl wth th Strn sothrm, f only th adsorpton paramtrs K, K and K n ths sothrms ar rlatd by mans of Eq. (.8). Th constants K and K hav a straghtforward physcal manng. In vw of Eqs. (.4) and (.8) K xp kt (0) (.83)

29 Planar Flud Intrfacs 9 ( 0) whr has th manng of standard fr nrgy of adsorpton of surfactant from dal dlut soluton to dal adsorpton monolayr n th absnc of dssolvd non-amphphlc salt; th thcknss of th adsorpton layr s about nm for SDS. Not that th Langmur and Strn sothrms, Eqs. (.4) and (.8), hav a smlar form, whch corrsponds to a statstcal modl consdrng th ntrfac as a lattc of quvalnt, dstngushabl, and ndpndnt adsorpton sts, wthout ntractons btwn bound molculs [7]. Consquntly, an xprsson, whch s analogous to Eq. (.83), holds for th rato K /K [th lattr s a countrpart of K n Eq. (.4)]: K K xp kt (0) (.84) whr s th thcknss of th Strn layr (c.a. th damtr of a hydratd countron) and (0) has th manng of standard fr nrgy of adsorpton (bndng) of a countron from an dal dlut soluton nto an dal Strn layr. In summary, th paramtrs K and K ar rlatd to th standard fr nrgs of surfactant and countron adsorpton. Th abov quatons form a full st for calculatng th surfac tnson as a functon of th bulk surfactant and salt concntratons (or actvts), = ( a, a ). Thr ar 6 unknown varabls:, s, a s,, a s and. Ths varabls ar to b dtrmnd from a st of 6 quatons as follows. Equaton (.49) for =, provds quatons. Th rmanng 4 quatons ar: Eqs. (.66), (.78), (.8) and on surfactant adsorpton sothrm from Tabl., say th Langmur sothrm. Comparson of thory and xprmnt. As llustraton w consdr an ntrprtaton of xprmntal data by Tajma t al. [4,43] for th surfac tnson vs. surfactant concntratons at two concntratons of NaCl: c 3 = 0 and c 3 = 0.5 M, s Fg..6. Th onc surfactant usd n ths xprmnts s trtatd sodum dodcyl sulfat (TSDS), whch s : lctrolyt (th radoactvty of th trtum nucl hav bn masurd by Tajma t al. to dtrmn drctly th surfactant adsorpton). Procssng th st of data for th ntrfacal tnson c, c ) as a functon of th bulk concntratons of surfactant ons, c, and ( countrons, c, on can dtrmn th surfactant adsorpton, ( c, c ), th countron

30 30 Chaptr Fg..6. Surfac prssur at ar-watr ntrfac, 0, vs. th surfactant (TSDS) concntraton, c, for two fxd NaCl concntratons: 0 and 0.5 M; th symbols ar xprmntal data from Rfs. [4] and [43]; th contnuous lns rprsnt th bst ft by mans of th thory from Rf. [0]. adsorpton, ( c, c ), and th surfac potntal, s ( c, c ). To ft th data n Fg..6 th Frumkn sothrm s usd (s Tabl.). Th thortcal modl contans four paramtrs,,, K and K, whos valus ar to b obtand from th bst ft of th xprmntal data. Th paramtrs valus can b rlably dtrmnd f only th st of data for ( c, c) contans xprmntal ponts for both hgh and low surfactant concntratons, and for both hgh and low salt concntratons; th data by Tajma t al. [4,43] satsfy th lattr rqurmnt. (If ths rqurmnt s not satsfd, th mrt functon xhbts a flat and shallow mnmum, and thrfor t s practcally mpossbl to dtrmn th bst ft [0]). Th valu of, obtand n Rf. [0] from th bst ft of th data n Fg..6, corrsponds to / = 37.6 Å. Th rspctv valu of K s 56 m 3 /mol, whch n vw of Eq. (.83) gvs a ( 0 standard fr nrgy of surfactant adsorpton ) =.8 kt pr TDS on, that s 3.3 kj/mol. Th dtrmnd valu of K /K s 8. m 3 /mol, whch aftr substtuton n Eq. (.84) ( 0 ylds a standard fr nrgy of countron bndng ) =.64 kt pr Na + on, that s 4.04 kj/mol.

31 Planar Flud Intrfacs 3 Fg..7. Plots of th calculatd adsorptons of surfactant / (th full lns), and countrons / (th dottd lns), vs. th surfactant (TSDS) concntraton, c. Th lns corrspond to th bst ft of th data n Fg..6 obtand n Rf. [0]. Th valu of th paramtr s postv ( /kt = +0.8), whch ndcats attracton btwn th hydrocarbon tals of th adsorbd surfactant molculs. Fgur.7 shows calculatd curvs for th adsorptons of surfactant, (th full lns), and countrons, (th dottd lns), vs. th TSDS concntraton, c. Ths lns rprsnt th varaton of and along th two xprmntal curvs n Fgur.6. On ss that both and ar markdly gratr whn NaCl s prsnt n th soluton. Th hghst valus of for th curvs n Fg..7 ar 4.30 mol/m and 4.0 mol/m for th solutons wth and wthout NaCl, rspctvly. Th lattr two valus compar wll wth th saturaton adsorptons masurd by Tajma [4,43] for th sam systm by mans of th radotracr mthod, vz. = 4.33 mol/m and 3.9 mol/m for th solutons wth and wthout NaCl. In Fg..8 th occupancy of th Strn layr, = /, s plottd vs. th surfactant concntraton for th curvs n Fg..7. For th soluton wthout NaCl / rss from 0.5

32 3 Chaptr Fg..8. Calculatd occupancy of th Strn layr by adsorbd countrons, /, vs. th surfactant (TSDS) concntraton, c, for two fxd NaCl concntratons: 0 and 0.5 M. Th lns corrspond to th bst ft obtand n Rf. [0] for th data n Fg..6. up to 0.74 and thn xhbts a tndncy to lvl off. As t could b xpctd, th occupancy / s hghr for th soluton wth NaCl; vn at TSDS concntraton 0 M th occupancy s about 0.40; for th hghr surfactant concntratons lvls off at / = 0.74 (Fg..8). Th lattr valu s consonant wth data of othr authors [4447], who hav obtand valus of / up to for varous onc surfactants; pronouncd vdncs for countron bndng hav bn obtand also n xprmnts wth solutons contanng surfactant mclls [4853]. Ths rsults mply that th countron adsorpton (bndng) should b always takn nto account. Th ft of th data n Fg..6 gvs also th valus of th surfac lctrc potntal, s. For th solutons wth salt th modl prdcts surfac potntals varyng n th rang s = mv wthn th xprmntal ntrval of surfactant concntratons, whras for th soluton wthout salt th calculatd surfac potntal s hghr: s = mv (not that for TSDS s has a ngatv sgn). Thus t turns out that masurmnts of surfac tnson, ntrprtd by mans

33 Planar Flud Intrfacs 33 of an approprat thortcal modl, provd a mthod for dtrmnng th surfac potntal s n a broad rang of surfactant and salt concntratons. Th rsults of ths mthod could b compard wth othr, mor drct, mthods for surfac potntal masurmnt, such as th lctrophortc -potntal masurmnts [,3,54,55], or Volta (V) potntal masurmnts, s.g. Rf. [56]. Surfactant s : lctrolyt, salt s Z 3 :Z 4 lctrolyt. In ths cas w wll numbr th onc componnts as follows: ndx surfactant on, ndx countron du to th surfactant, ndx 3 coon du to th salt, and ndx 4 countron du to th salt. As bfor, w assum that th coons du to th salt do not adsorb at th ntrfac: 3 = 0. Th countrons du to th surfactant and salt ar consdrd as sparat componnts, whch can xhbt a compttv adsorpton n th Strn layr (s Fg..4). Th analogs of Eqs. (.8) and (.8) for th cas undr consdraton ar [0]: K K Kas K4a4s K a s K Kas K 4 a 4s (.85) ( =,4) (.86) whr K, K and K 4 ar constants. All xprssons for surfactant adsorpton sothrms and surfac tnson sothrms gvn n Tabl. ar vald also n th prsnt cas. Dffrnt ar th forms of th Gouy quaton and of th xprsson for d, whch dpnd on z 3 and z 4 n accordanc wth Eqs. (.60) and (.6). In partcular, th ntgraton n Eq. (.6) can b carrd out analytcally for som typs of lctrolyt. Tabl.3 summarzs th xprssons for th Gouy quaton and d, whch hav bn drvd n Rf. [0] for th cass, whn th salt s :, :, : and : lctrolyt. (Hr : lctrolyt mans a salt of bvalnt countron and monovalnt coon.) On may chck that n th absnc of salt (a 4 = 0) all xprssons n Tabl.3 rduc thr to Eq. (.66) or to Eq. (.67). Mor dtals can b found n Rf. [0]. Gbbs lastcty for onc surfactants. Th dfnton of Gbbs (surfac) lastcty s not wll lucdatd n th ltratur for th cas of onc surfactant adsorpton monolayrs. That s th rason why hr w dvot a spcal dscusson to ths ssu.

34 Chaptr 34 Tabl.3. Spcal forms of th Gouy quaton (.60) and of th xprsson for d, Eq. (.6), for solutons of surfactant whch s : lctrolyt, and salt whch s :, :, : and : lctrolyt. Typ of salt Exprssons obtand from Eqs. (.60) and (.6) snh s c a a : cosh 8 4 s c d a a kt 4 ) ( g y y I c ; xp ; ; s y I a a a I : ln 3 3 g y g y y I kt c d ; g y / 4 ) ( g u u I c ; xp ; ; s u I a a a I : ln 3 3 g u g u u I kt c d ; g u / 4 snh 4 q a s c ; cosh s q : ln 4 q q q q a kt c d ; 4 4 a a

35 Planar Flud Intrfacs 35 Th physcal concpt of surfac lastcty s th most transparnt for monolayrs of nsolubl surfactants. Th changs of and n th xprsson E G = (/ ) corrspond to varatons n surfac tnson and adsorpton durng a ral procss of ntrfacal dlataton. In th cas of a solubl nononc surfactant th dtctd ncras of n a ral procss of ntrfacal dlataton can b a pur manfstaton of surfac lastcty only f th prod of dlataton, t, s much shortr than th charactrstc rlaxaton tm of surfac tnson, t <<. Othrws th adsorpton and th surfac tnson would b affctd by th dffuson supply of surfactant molculs from th bulk of soluton toward th xpandng ntrfac. Th dffuson transport tnds to rduc th ncras of surfac tnson upon dlataton, thus apparntly rndrng th ntrfac lss lastc and mor flud. To dscrb th varaton of th surfac tnson aftr an ntal dlataton on s to solv th dffuson quaton usng an approprat ntal condton (s Scton.3. for dtals). In such a cas th Gbbs lastcty, E G, ntrs th thortcal xprssons through ths ntal condton, whch corrsponds to an nstantanous dlataton of th ntrfac (that s t << ), s.g. Rf. [57]. Ths nstantanous dlataton dcrass th adsorptons and th subsurfac concntratons c s of th spcs (th subsurfac s prsumd to b always n qulbrum wth th surfac), but th bulk concntratons c rman unaffctd [58,59]. Ths ntally cratd dffrnc btwn c s and c furthr trggrs th dffuson procss. Now, lt us try to xtnd ths approach to th cas of onc surfactants. In th cas of soluton of an onc surfactant, a non-unform dffus lctrc doubl layr (EDL) s formd n a vcnty of th ntrfac; ths s th major dffrnc wth th cas of nononc surfactant. Th man quston s whthr or not th lctrc fld n th EDL should b affctd by th ntal nstantanous dlataton of th ntrfac. Ths problm has bn xamnd n Rf. [60] and t has bn stablshd that a varaton of th lctrc fld durng th ntal dlataton lads to thortcal rsults dvod of sns. Ths s du to th followng two facts: () Th spd of propagaton of th lctrc sgnals s much gratr than th charactrstc rat of dffuson. () Evn a small ntal varaton n th surfac charg dnsty s mmdatly gvs rs to an lctrc potntal, whch s lnarly ncrasng wth th dstanc from th ntrfac (potntal of a planar wall). Thus a small ntal prturbaton of th ntrfac would

36 36 Chaptr mmdatly affct th ons n th whol soluton; of cours, such an ntal condton s physcally unaccptabl. In ralty, a lnarly growng lctrc fld could not appar n th onc soluton, bcaus a varaton of th surfac charg dnsty would b mmdatly supprssd by xchang of countrons, whch ar abundant n th subsurfac layr of th soluton (s Fg..4). Th thortcal quatons suggst th sam: to hav a mathmatcally manngful ntal condton for th dffuson problm, th ntal dlataton must b carrd out at constant surfac charg dnsty s ( s = const. mans also s = const., s Eq..66). Thus w can conclud that th ntal nstantanous ntrfacal dlataton, whch s rlatd to th dfnton of Gbbs lastcty of a solubl onc surfactant, must b carrd out at s = const. From Eq. (.9) on obtans d s d a s d d s (.87) W rcall that a and d ar, rspctvly, th contrbutons of th adsorpton and dffuson layrs to th total ntrfacal tnson,. An ntrfacal dlataton at constant s and s dos not altr th dffus part of th EDL, and consquntly, 0 d d s. Snc, a 0 ktj, th xprssons for J n Tabl. show that a dpnds only on at constant tmpratur. Thn th dfnton of Gbbs lastcty of nononc adsorpton layrs, Eq. (.45), can b xtndd to onc adsorpton layrs n th followng way: E G T, s a T (.88) Th dpndnc of on for nononc surfactants s th sam as th dpndnc of a on for onc surfactants, s th surfac tnson sothrms n Tabl.. Thn Eqs. (.45) and (.88) show that th xprssons for E G n Tabl. ar vald for both nononc and onc surfactants. Th ffct of th surfac lctrc potntal on th Gbbs lastcty E G of an onc adsorpton monolayr s mplct, through th qulbrum surfactant adsorpton, whch dpnds on th lctrc proprts of th ntrfac. To llustrat ths lt us consdr th cas of Langmur sothrm; combnng xprssons from Tabls. and. w obtan E G ktka s. Furthr, usng Eqs. (.49) and (.8) w drv s kta K K a EG (for Langmur sothrm) (.89)

37 Planar Flud Intrfacs 37 Equaton (.89) vsualzs th ffct of salt on E G : whn th salt concntraton ncrass, a also ncrass, but th (dmnsonlss) surfac potntal s dcrass; thn Eq. (.89) prdcts an ncras of E G wth th salt concntraton. Not also that th valus of E G, calculatd from th fts, lk that n Fg..6, dpnd on th typ of th usd adsorpton sothrm; for xampl, th Frumkn sothrm gvs valus of E G, whch ar systmatcally largr than thos gvn by th van dr Waals sothrm. Th lattr s prfrabl for flud ntrfacs nsofar as t corrsponds to th modl of non-localzd adsorpton. Th dfnton of Gbbs lastcty gvn by Eq. (.88) corrsponds to an nstantanous (t << ) dlataton of th adsorpton layr (that contrbuts to a ) wthout affctng th dffus layr and d. Ths wll caus an ntal chang n th subsurfac concntratons c s of th spcs, whch wll furthr trggr a dffuson transport of componnts across a changng lctrc doubl layr. Thus w rach agan th subjct of th adsorpton kntcs, whch s consdrd n th nxt scton..3. KINETICS OF SURFACTANT ADSORPTION Whn a collodal partcl approachs an ntrfac from th bulk of soluton, or whn an attachd partcl s movng throughout th ntrfac, th surfactant adsorpton layr s locally dsturbd (xpandd, comprssd, shard). Th surfactant soluton has th proprty to damp th dsturbancs by dffuson of surfactant molculs from th bulk to th ntrfac (or n th oppost drcton). If th partcl moton s slow nough (compard wth th rlaxaton tm of surfac tnson ) th ntrfac wll bhav as a two-dmnsonal flud and surfac lastc ffcts wll not ars. On th contrary, f th charactrstc tm of th procss of partcl moton s comparabl wth or smallr than, th moton of th partcl wll b accompand by surfac lastc ffcts and adsorpton dynamcs. Th crtron, showng whn th lattr ffcts would appar, s rlatd to th rlaxaton tm of th surfac tnson. Our attnton n th prsnt scton wll b focusd on th thortcal rsults about obtand for varous typs of surfactant adsorpton, as follows: () adsorpton undr dffuson control, () adsorpton undr lctro-dffuson control, () adsorpton undr barrr (kntc) control,

38 38 Chaptr (v) adsorpton from mcllar solutons, (v) adsorpton from protn solutons. Our purpos s to gv a brf rvw and rlatd rfrncs n th contxt of th subjct of ths book; dtald nformaton about th varty of xprmntal mthods and thortcal approachs can b found lswhr [58-66]..3.. ADSORPTION UNDER DIFFUSION CONTROL Insofar as w ar ntrstd manly n th rlaxaton tm, w wll rstrct our consdratons to a physcal stuaton, n whch th ntrfac s nstantanously xpandd at th ntal momnt t = 0 and thn (for t > 0) th dffuson transport of surfactant tnds to saturat th adsorpton layr, and vntually to rstor th qulbrum n th systm. In othr words, th ntrfacal xpanson happns only at th ntal momnt, and aftr that th ntrfac s quscnt and th dynamcs n th systm s du only to th dffuson of surfactant. Th adsorpton procss s a consqunc of two stags: th frst on s th dffuson of surfactant from th bulk to th subsurfac and th scond stag s th transfr of surfactant molculs from th subsurfac to th surfac. Whn th frst stag (th surfactant dffuson) s much slowr than th scond stag, and consquntly dtrmns th rat of adsorpton, th procss s trmd adsorpton undr dffuson control; t s consdrd n th prsnt scton. Th oppost cas, whn th scond stag s slowr than th frst on, s calld adsorpton undr barrr (or kntc) control and t s prsntd n Scton.3.3. If an lctrc doubl layr s prsnt, th lctrc fld to som xtnt plays th rol of a slant barrr; ths ntrmdat cas of adsorpton undr lctro-dffuson control, s prsntd n Scton.3.. Hr w consdr a soluton of a nononc surfactant, whos concntraton, c c (z,t), dpnds on th poston and tm bcaus of th dffuson procss. As bfor, z dnots th dstanc to th ntrfac, whch s stuatd n th plan z = 0. Th surfactant adsorpton and th surfac tnson vary wth tm: (t), (t). Th surfactant concntraton obys th quaton of dffuson: c c D (z > 0, t > 0) (.90) t z