Langmuir 1994,lO)

Transcription

1 Langur 994,lO) lctrcal Doubl Layr ntracton btwn Dsslar Sphrcal ollodal Partcls and btwn a Sphr and a Plat: Th Lnarzd PossonBoltzann Thory Stvn L. arn,* Drk Y.. han, and Jas S. Gunnng Dpartnt of Mathatcs, Unvrsty of Mlbourn, Parkvll, Vctora 35, Australa Rcvd March 4, 994@ Th lnarzd PossonBoltzann thory s usd to calculat th lctrcal doubl layr ntracton fr nrgy, as wll as th doubl layr forc, btwn unqual sphrcal collodal partcls. Rsults ar gvn for ntracton undr condtons of constant surfac potntal, for constant surfac charg, and for th cas n whch charg rgulaton du to th dssocaton of surfac groups ay b odld by a lnar rlatonshp btwn th surfac charg and th surfac potntal. Th ntracton nrgy and forc btwn a sphrcal collodal partcl and a plat, whch has partcular rlvanc to forc asurnts wth th atoc forc croscop and partcl dposton studs, ar also calculatd. Th rsults ar usd to tst th accuracy of th lnar Dryagun approxaton whch fors th bass of th HoggHalyFurstnau forula. Th valdty of th lnarzd PossonBoltzann thory s tstd aganst a nurcal soluton of th nonlnar PossonBoltzann quaton n th spcal cas of sphrcal partcls of th sa sz but oppost surfac potntals.. ntroducton Although th fundantal thory for th lctrcal doubl layr ntracton btwn collodal partcls has bn stablshd for or than half a cntury,lr analytc xprssons for th forcs and fr nrgs of ntracton, vn for sphrcal partcls, ar only avalabl as approxat xprssons such as th lnar suprposton approxaton3 (whch s vald whn th partcls ar far apart and th doubl layr ovrlap s rlatvly wak). Altrnatvly, on can us th Dryagun constructon (vald for thn doubl layrs rlatv to th partcl sz) to drv th ntracton fr nrgy btwn sphrs fro that btwn paralll plat^.^ Th advnt of th atoc forc croscop (AFM) has gvn collod scntsts th ablty to asur drctly th forc btwn a collodal partcl and a planar ~urfac.~ n ordr to ntrprt such asurnts, t would b usful to b abl to coput th xpctd forc curv, gvn paratrs dscrbng th collodal partcl and th surfac. n ths papr, w prsnt rsults for th lctrcal doubllayr forc and ntracton fr nrgy, both for unqual sphrcal partcls and for th sphr/plat gotry, basd on th lnarzd PossonBoltzann quaton, but wthout any furthr rstrctons rgardng partcl sz, sparaton, and rlatv agntuds of th surfac potntals. Part of th dffculty of calculatng th forc and ntracton fr nrgy s du to th nonlnar natur of th PossonBoltzann thory whch rndrs th probl analytcally ntractabl xcpt for th splst gotrs. Howvr, n any practcal probls, th surfac potntal of th partcls ar coparabl to or only slghtly largr than th thral potntal (ktl). onvntonally, th lnarzd PossonBoltzann thory s thought to b rstrctd to systs wth lctrc potntals lss than th thral potntal (KT), corrspondng to about 5 Abstract publshd n Advanc AS Abstracts, July 5, 994. () Vnvy,. J. W.; Ovrbk, J. Th. G. Thory of th Stablty of Lyophobc ollods; lsvr: Nw York, 948. () Dwan, B. V.; Landau, L. D. Acta Phys.h. USSR 94, 4, 633. (3) Bll, G. M.; Lvn, S.; Mcartny, L. N. J. ollod ntrfac Sc. 97, 33, 335. (4) Hogg, R.; Haly, T. W.; Furstanau, D. W. Trans. Faraday SO. 986, 6, 638. (5) Duckr, W. A.; Sndn, T. J.; Pashly, R. M. Natur 99,353, /94/4993$4.5/ V at roo tpratur. Rcnt rsults for th forc btwn two dntcal sphrcal collodal partcls usng th nonlnar PossonBoltzann thory6 suggst that, n so crcustancs, th lnarzd PossonBoltzann quaton gvs farly accurat rsults for potntals up to about 4 V, that s, th rgon of applcablty of th lnarzd PossonBoltzann quaton s largr than s usually supposd. As a consqunc, t s dsrabl to hav an accurat and convnnt way to calculat th lctrcal doubl layr ntracton basd on th lnarzd PossonBoltzann thory. Th thortcal frawork for th analytc soluton of th lnarzd PossonBoltzann quaton has bn avalabl for so t.7 n ths papr w provd an ffcnt thod for calculatng th doubl layr ntracton fr nrgy and forc btwn two dsslar sphrcal collodal partcls by solvng th lnarzd Posson Boltzann quaton usng a two cntr xpanson thod. Hr w consdr (a) partcls whch antan a unfor fxd surfac potntal durng ntracton, th constant potntal odl, (b) partcls that antan a unfor furd surfac charg dnsty durng ntracton, th constant charg odl, and (c) partcls that, as a rsult of th chcal onzaton of surfac groups, antan a known rlaton btwn th surfac potntal and surfac charg, th rgulaton odl. For ths last odl, th surfac chargpotntal rlatonshp s n gnral nonlnar for aphotrc or zwttronc surfacs. Ths nonlnarty s nconsstnt wth th lnarzd PossonBoltzann thory. Howvr, w hav shown lswhr8 that undr approprat condtons, t s possbl to rplac ths chargpotntal rlaton by a lnar quaton btwn th surfac charg and th surfac potntal so that th thod of soluton for th cass of constant potntal and constant charg odls can b radly xtndd to th lnarzd rgulaton odl to calculat th ntracton fr nrgy and forc. Prvous work for dntcal sphrcal partcls has usd th two cntr xpanson thod to calculat th forc (6) arn, S. L.; han, D. Y..; Stankovch, J. J. ollod ntrfac Sc. 994, 65, 6. (7) Glndnnng, A. B.; Russl, W. B. J. ollod ntrfac Sc. 983, 93, 95. (8) arn, S. L.; han, D. Y.. J. ollod ntrfac Sc. 993,6, Arcan hcal Socty

2 994 Langur, Vol., No. 9, 994 btwn th partcls for th constant potntal and constant charg odls' and th lnarzd rgulaton odlg and th ntracton fr nrgy for all th abov surfac odls.loj Th only slar work w ar awar of for unqual sphrs obtand th forc for sphrs at constant potntal and thn ntgratd nurcally to obtan th ntracton fr nrgy.lz Ohsha has rcntly obtand an xplct (albt nvolvd) analytc srs soluton for th ntracton fr nrgy for unqual sphrs at constant p~tntal.'~ W ar unawar of any prvous work n th sphr/plat gotry. n our prvous work on dntcal sphrs, w allowd for th possblty of th partcls havng arbtrary dlctrc constant p whch s rlvant to cass b and c dscrbd abov. For th coon cas cp < 5 n a hghly polar solvnt, th nrgy and forc ar wll approxatd by thos for p = (for an llustraton of ths, s rf 6, and so w follow ths splfcaton hr. Th or gnral cas can b dalt wth usng th thods n rf, f dsrd. Our thod of solvng th lnrazd PossonBoltzann quaton for th lctrostatc potntal nar two ntractng sphrs s basd on a two cntr xpanson n trs of sphrcal haroncs. Th coffcnts n ths xpanson ar found by solvng a syst of lnar quatons that ars fro atchng boundary condtons at th partcl surfacs. n scton, w drv ths syst of lnar quatons for varous cobnatons of boundary condtons on ach ntractng sphrs. n sctons and V, w gv th forulas for th forc and ntracton fr nrgy n trs of th xpanson coffcnts. For th ntracton btwn a sphr and a plat, a slghtly dffrnt xpanson for th lctrostatc potntal s rqurd. Ths s bcaus th srs xpanson of th potntal around two sphrs cass to convrg as th radus of on sphr gos to nfnty to for a plat. n scton V, w furnsh ths nw xpanson for th lctrostatc potntal for th ntracton btwn a sphr and a plat and drv th syst of quatons that has to b solvd to fnd th coffcnts of ths xpanson. xprssons for th forc and ntracton fr nrgy ar gvn n sctons V and V for th sphr/plat ntracton. Nurcal rsults ar prsntd n scton V and conclusons n scton.. Th Potntal around ntractng Dsslar Sphrs W now outln th thod of dtrnng th lctrostatc potntal nar th two ntractng sphrs. n th lnarzd PossonBoltzann odl th lctrostatc potntal ly satsfs th quaton v 7) = K 7) (outsd th sphrs) () n th lctrolyt charactrzd by K, th nvrs Dby lngth. Wth th assupton of zro dlctrc constant for th partcls, cp =, t s only ncssary to dtrn th potntal n th rgon xtror to th sphrs. W st up a coordnat syst wth th orgn locatd dway btwn th cntrs of th sphrs of radus a and az, as shown n Fgur. (9) Krozl, J. W.; Savll, D. A. J. ollod ntrfac Sc. 99, 5, 365. () arn, S. L.; han, D. Y.. J. ollod ntrfac Sc. 993,55, 97. () Rnr,. S.; Radk,. J. Adv. ollod ntrfac Sc. 993,47, 59. () K, S.; Zukowsk,. F. J. ollod ntrfac Sc. 99,39,98. (3) Ohsha, H. J. ollod ntrfac Sc. 994, 6, 487. arn t al. Fgur. oordnat syst for two dsslar sphrs. n th lctrolyt (q > a and r > ad, th soluton of () can b wrttn as a two cntr xpanson usng sphrcal haroncs7 whr ', r( (3) (n+v)!(n+v+ n +nv+)x 3 (n v)! ( v)! Y! (4) Th dstanc btwn th cntrs of th two sphrs s R, T(z) s th Gaa functon, P,(x) s a Lgndr polynoal of ordr n, and,(x) and k,(x) ar, rspctvly, odfd sphrcal Bssl functons of th frst and thrd knd4 and ar rlatd to odfd Bssl functon of half ntgr ordr n+/(~) and K,+/Z(X) by,(x) = ( d~)~/~,+/~(x) and k&) = (~/X)'K,+/d~). Th unknown coffcnts {a,} and {b,} ar found by applyng th approprat boundary condtons on th surfacs of th sphrs. W now gv n dtal th quatons that nd to b solvd to fnd th coffcnts. t wll bco clar that th constant charg odl s actually a spcal cas of th lnarzd rgulaton odl. Snc th two sphrs ar not forcd to hav th sa surfac odl, ths ans w hav thr dstnct cassach sphr can hav thr constant potntal or lnarzd rgulaton boundary condtons: (a) constant surfac potntal odl on both sphrs; (b) constant potntal on on sphr and lnarzd rgulaton or constant charg on th othr; (c) both sphrs wth lnarzd rgulaton boundary condtons. Th constant charg boundary condton s a spcal lt of th lnarzd rgulaton boundary condton. W now gv th quatons that ar ndd to solv for th unknown coffcnts {a,} and {b,} that appar n th sphrcal haronc xpanson for th lctrostatc potntal n q for th thr cass lstd. (4) Abraowtz, M.; Stgun,. A. Handbook of Mathatcal Functons; Dovr: Nw York, 965.

3 lctrcal Doubl Layr ntracton Fr nrgy (a) onstant Surfac Potntal Modl on Both Sphrs. Whn th potntal on th surfac of both sphrs s unfor and rans fxd at q and q, rspctvly, th coffcnts a, and b, can b found by applyng th condton q = q at rl = a and q = q at r = a. Usng qs 4 ths gvs a syst oflnar quatons a + Lb = v Ma + b = v (5a) (5b) whr th coponnts of th vctor of coffcnts ar gvn by aj = ajkj(ka,) (64 bj = bjkj(ka) th atrx lnts of L and M ar and Ljn = (j + )B,jZj(KZl)/k,(KU) Mj, = (j + )Bnjj(KU)/k,(KU,), j = O.= {, j > O (6b) (7a) (7b) Thr should b no confuson btwn th coffcnts {a,} and th rad a and a. Th coffcnts always hav gnrc subscrpts. t can asly b sn that, for th cas of sphrs of qual sz and surfac potntal, w rcovr th prvous rsults. A syst of quatons for th offcnts {a,} and {b,} can also b obtand by collocatng th frst for of q at a st of ponts on th surfac of ach sphr, as don n rf. W hav trd both thodsthy gv dntcal rsults and rqur about th sa sz atrcs so thr ss to b no prssng rason to choos on thod ovr th othr. (b) onstant Potntal on On Sphr and Lnarzd Rgulaton or onstant harg on th Othr. W can, wthout loss of gnralty, choos sphr to hav constant surfac potntal and sphr to hav th lnarzd rgulatng boundary condton. Th ost gnral for of a lnar rlaton connctng th surfac charg dnsty, u, and th surfac potntal, v, that odls surfac onzaton s gvn by Langur, Vol., No. 9, whr = s th product of th prttvty of fr spac, O, and th rlatv prttvty of th solvnt, *, and n s th outward surfac noral. Applyng th boundary condtons (9) togthr wth () at 7 = a and sttng q = ~ at r = al, w obtan th syst of lnar quatons for th coffcnts {a,} and (bn} a + Lb = +, (lla) whr aj = ajkj(ka,) (4 bj = [(a&dc)kj(ka) ~U,k)(~a,)]b~ (b) Lj, = (j + l)b~~(ka,)/[(a&d )k,(kuz) Ka~k ~(Ka) (c) Mj, = (j + )B~[(a&d)j(Ka) KUZ)(K~)]/k,(KUl) (4 n th scond qualty of q llb, w hav lnatd th quantty S n favor of th surfac potntal of sphr whn t s n solaton, q ~, and th rgulaton capacty of sphr, K. Th cas n whch sphr has a constant charg boundary condton can b obtand by takng th lt Kz. (c) Both Sphrs wth Lnarzd Rgulaton Boundary ondtons. Applyng th boundary condtons 9 and on th surfacs of both sphrs, w obtan th syst of lnar quatons for th coffcnts {a,} and {bn} whr a + Lb = w,( + KZ~ + ak ) (3a) whr th sgn of th constant S s th sa as th sgn of th surfac charg whn th partcl s n solaton. Th constant K s always postv: ts agntud rflcts th ablty of th surfac onzaton ractons to antan a constant surfac charg. Th lt K = corrsponds to a constant charg surfac. For latr rfrnc, w call th constant K, th rgulaton capacty snc t has th dnson of lctrostatc capactanc pr unt ara (F/ ). Prvously,lo w usd th sybols K and Kz for S and K, rspctvly; hr, for obvous rasons, w rsrv th subscrpts and to rfr to th sphrs. For th cas c,, =, th surfac charg satsfs th quaton u = cvpn at r = a, or r = a () and and ~ ar th surfac potntals of th sphrs whn thy ar nfnty far apart. For ach of th cass whr th surfac s odld wth lnarzd rgulaton quatons, th spcal cas of a constant charg surfac can b obtand by sply sttng th approprat valu of K to zro. Provdd R > (a + a), th coffcnts of th xtror soluton {a,} and {b,} ar thn found by truncatng th atrx quatons (5,, or 3) at an approprat uppr lt and solvng th syst of quatons by drct nurcal thods. W dfr dscusson of th choc of th cutoff sz of th syst of lnar quatons untl latr n ths papr. W now procd to drv xprssons for

4 996 Langur, Vol., No. 9, 994 arn t al. th forc and ntracton fr nrgy n trs of th coffcnts (a,> and (b,} whch appar n th sphrcal haronc xpansons of th potntal outsd th sphrs as gvn by q. thn th forc n q bcos. Th Forc btwn Dsslar Sphrs W can calculat th forc actng on th partcls by ntgratng th strss tnsor T = (n + 5c) (6) ovr a sutabl surfac. Hr = Vly s th lctrc fld and n s th dffrnc n th local osotc prssur fro that n th bulk lctrolyt soluton. n th lnarzd PossonBoltzann thory ths dffrnc n osotc prssur s rlatd to th lctrostatc potntal by f w want to valuat th forc on sphr (du to th prsnc ofsphr, w can choos any convnnt surfac S that ncloss sphr and prfor th followng surfac ntgral to gt th vctor forc on sphr f = LTa ds (8) whr th unt noral n at th surfac S s drctd toward sphr. Hr w calculat th forc on sphr by choosng S to b th surfac of sphr (rl = al) so that n wll b th radus vctor drctd away fro th cntr of sphr th outward surfac noral. By sttng th orgn of a artsan axs syst at th cntr of sphr and th z axs to l along th ln jonng th cntr of th sphrs, th agntud of th forc on sphr n th z drcton can b xprssd as ntgrals ovr functons of th r and 8 coponnts of th lctrc fld: n g f l n what follows w shall prsnt rsults for a nondnsonal forc f (5).. w scal by th thral potntal rathr than any of th partcl surfac potntals as was don prvou~ly~,~j~ bcaus w want to hav th frdo to lt thr partcl hav zro surfac potntal. Ths ans that n=o whr th potntals apparng n th govrnng quatons (5 or ) ar to b ntrprtd as (VkT), that s, th potntal scald by th thral potntal. (b) Sphr a Lnarzd Rgulatng Surfac. For th othr two cass, q 9 for th rpulsv forc btwn th partcls can b rwrttn, usng, = ol = (S KV), as p=o whr th ntgrand s valuatd on th outsd surfac of sphr (rl = al+) and th ntgraton varabl p s p = cos 8. Postv valus offply rpulson btwn th sphrs. Th valuaton of q 9 procds dffrntly for (a) a constant potntal sphr as copard to (b) a sphr wth lnarzd rgulaton boundary condtons. W shall consdr ths two cass sparatly. (a) Sphr at onstant Surfac Potntal. For thr th constant potntal odl or th cas whr sphr s at constant surfac potntal and sphr s a lnarzd rgulatng surfac, q 9 for th rpulsv forc btwn th partcls splfs to xplctly, th nondnsonal forc s f = ~Z? f',;p dp () f w wrt th potntal fro q n th for whr n=o dn(krl) = unkn(krl) + (n + l)zn(~rl) bb, () =O whr, ( + ) ( + 3)( + )' n=+l ( + ) (9) n= ( + )( )' n t f l

5 lctrcal Doubl Layr Zntracton Fr nrgy Langur, Vol., No. 9, ( + l)( ) n=+l ( + 3)( + ) ' ( l)( + ) n= ( + )( ) ' (3), n+fl Agan, th quantty 5' can b lnatd n favor of and K, s q llb. Thus havng solvd th st of lnar quatons gvn n scton for th coffcnts {a,} and (b,}, th rsults can b substtutd nto th abov quatons to calculat th forc. W. Th ntracton Fr nrgy btwn Dsslar Sphrs Whl th xprssons for th forc btwn th partcls ar sowhat cubrso, th xprssons for th ntracton fr nrgy tak on vry spl fors. W calculat th ntracton fr nrgy for all cass by a throdynac ntgraton othrws known as a couplng constant ntgraton or chargng procdur. Ths sch has a vry spl graphcal rprsntaton n trs of th surfac chargsurfac potntal rlat~nshps.~~*j~ For th lnarzd PossonBoltzann odl, th throdynac ntgratons ar trval. Furthror, consdrabl splfcaton s affordd by judcous us of th j = quaton n th lnar systs for th unknown coffcnt a, and b, gvn n qs 5,, or 3 to lnat trs lk &,B,o. Th ntracton fr nrgy btwn th two sphrs wth a dstanc of closst approach h = R a a can b wrttn as a su of surfac ntgrals ovr ach sphr: whn th sphrs ar at a sparaton h. Whl (3a) s gnral for constant surfac potntal ntractons, (3b) s only vald for th lnarzd PossonBoltzann odl. Usng () and (), w gt whr th scond tr cos fro th contrbuton at h =. Usng thj = quaton n thr (5a) or (lla), ths splfs to whr U s th nondnsonal nrgy (35) whch agan s dffrnt to our prvous scalng. As for th forc, th potntal ly n (34) for th nondnsonal ntracton fr nrgy should b ntrprtd as potntals scald by th thral potntal. f sphr has a constant surfac potntal, w sply chang th subscrpts n (34) fro to and us bo n plac of ao. (b) Lnarzd Rgulaton Sphr. Th ntracton fr nrgy pr unt ara for a sphr wth lnarzd rgulaton condton as a functon of surfac poston p = cos 8 s gvn by8 Th quantty uk(h,p) (wth k =,) s th ntracton fr nrgy pr unt ara at poston p = cos 8 on th surfac of th sphr k whn th sphrs ar at a dstanc h apart. Th surfac odl for ach sphr dtrns th approprat xprsson for ul(h) and uz(h). W prsnt dtals of th drvaton of ul(h), th corrspondng xprsson for uz(h) can b found by ntrchangng th ndcs and xchangng for th approprat coffcnt as dtald blow. Thr ar two dffrnt fors for th ntgrands ul(h,pl) and uz(h,pz) dpndng on whthr th constant potntal or lnarzd rgulaton boundary condton appls at th sphr. Th constant charg boundary condton s a spcal cas of th lnarzd rgulaton boundary condton. W now drv xprssons for ul(h,pl) for ths two dffrnt boundary condtons. (a) onstant Surfac Potntal Sphr. Th ntracton fr nrgy pr unt ara whn th sphr s hld at constant surfac potntal ~ s gvn by whr q(h,p) s th surfac potntal at poston p = cos 8 on th surfac of th sphr whn th sphrs ar at a dstanc h apart. Usng quaton (, w gt u,(h) = JUlSl x whr agan th scond tr cos fro th contrbuton at h = 3. Usng th frst quaton (j = ) n (3a) for th coffcnt ao, th nondnsonal ntracton fr nrgy splfs to whr a(h,p) s th surfac charg at p = cos 8 on sphr (5) han, D. Y.. n Gochcal Procsss at Mnral Surfacs; Davs, J. A., Hays, K. F., ds.; AS Syposu Srs No. 33; Arcan hcal Socty: Washngton, D, 986; p 99.

6 998 Langur, Vol., No. 9, 994 arn t al. 'ag' +/ ntud n th potntal but oppost n sgn wll produc zro potntal on th plat and th choc yjo = qp wll satsfy th boundary condton on th plat. So for th cas of a constant potnta plat w hav R l Fgur. oordnat syst for a sphr and a plat. whr w hav lnatd 5' n trs of V, th potntal of th sphr at nfnt sparaton. Th potntal n q 38 should b ntrprtd as th nondnsonal potntal (V/kZ'). Agan, f sphr has a lnarzd rgulatng surfac, U can b found fro (38) by changng subscrpts fro to and usng bo n plac of ao, Ths coplts th spcfcaton of th forc and ntracton fr nrgy for unqual sphrs. V. Th Potntal n a SphrPlat Gotry W now consdr th soluton of th lctrostatc potntal for th ntracton btwn a sphr and a flat plat. Th addton thor for Bssl functons that ls at th hart of th potntal rprsntaton n q bcos nvald n th cas whr on of th partcl rad bcos nfnt, that s, th sphr/plat gotry. Ths ans w ust look for an altrnatv rprsntaton for th potntal. n ordr to otvat our choc, frst consdr th cas of a plat at constant zro surfac charg. Nar th sphr th potntal s rprsntd by th sa xpanson n K,(Krl)P,(cos 6) as bfor. Th lctrc fld noral to th surfac ust b zro at th plat du to th zro surfac charg boundary condton posd. Ths condton can b antand by placng an ag sphr of th sa proprts at a poston rflctd n th plat, th cntr of th ag sphr at a dstanc R bhnd th plat. Thn th potntal has th for w whr th coordnats (r,ol),($), and (r,bz), as wll as th dstanc R ar dfnd n Fgur. Ths rprsntaton for th potntal satsfs th fld quaton outsd th sphr and plat and satsfs th boundary condton at th plat. Th coffcnts {a,) ar found by applyng th approprat boundary condton on th sphr, as don n scton. W now consdr rprsntatons for th potntal for th cass n whch th plat obys constant charg, constant potntal, or lnarzd rgulaton boundary condtons. f th surfac charg on th plat u f, w can add th tr QO " to th potntal. Ths addtonal tr stll satsfs th fld quaton, q, and f w choos th constant yjo = (u//k), th boundary condton on th plat, q, wll also b t. So for th cas of a constant charg plat w hav ou q = ~a,(kn(krl)p,(cos,) + K,(K~,)P,(cos,)} + n=o (u/ K)KZ (39) f th plat s at constant potntal VP nstad of at constant charg, an ag sphr wth th sa ag R w q = a,{k,(krl)pn(cos,) ~,(K~,)P,(cos,)} + n=o VJp= (4) n th cas whr th plat obys th lnarzd rgulaton boundary condton, w can us th followng rprsntaton for th potntal w VJ = x{a,k,(kr,)p,(cos n=o 6,) + b,k,(~r~)p,(cos 6,)) f q" (4) whr th constant,/jo s yt to b dtrnd. n th spcal cass of constant charg plat and constant potntal plat, w know b, = a, (constant charg plat) b, = an (constant potntal plat) (4) as wll as th valu of th constant Z~O. W us th spcal fors n qs 39 and 4 whn applcabl snc thy halv th nubr of unknowns to solv for; n th ost gnral cas, w ust us th for (4). n all cass w ust satsfy th approprat boundary condton on th sphr. Ths s ad possbl by usng th xpanson4,kz = ~R K(nR) = ~R xp(kr, OS,) to xprss = n trs of th coordnats cntrd about th sphr. W now gv th quatons that dtrn th coffcnts {a,} and {b,} that ars fro atchng boundary condtons at th sphr. f th plat s hld at constant charg or constant potntal, w only hav on st of unknown coffcnts {a,} to dtrn, but f th plat obys th lnarzd rgulaton boundary condton, w nd to dtrn two sts of coffcnts: {a,) and {b,} plus th constant VO. W shall consdr ths two cass sparatly. (a) onstant Potntal or onstant harg Plat. For ths cas, w only hav on st of coffcnts {a,} to dtrn. Th syst to b solvd dpnds on th boundary condton on th sphr. As n th prcdng scton, th constant surfac charg condton on th sphr s tratd as a spcal cas of lnarzd rgulaton by sttng K,, th rgulaton capactanc of th sphr, to. So w hav two cass to consdr: constant potntal sphr and lnar rgulatng sphr. onstant Surfac Potntal Sphr. Applyng th condton?/j = wb, th constant potntal on th sphr, at rl = a and usng q 39 for a constant charg plat or 4 for a constant charg plat togthr wth q 43, w obtan th lnar quatons for {a,} ( f L) a = Vs qokrc (44) wth th approprat choc ofth constant QO as dscussd abov. Th uppr sgn appls to a constant charg plat

7 lctrcal Doubl Layr ntracton Fr nrgy Langur, Vol., No. 9, and th lowr sgn to a constant potntal plat. Th xplct fors of th vctor and atrx lnts ar Lnarzd Rgulatng Sphr. Applyng th condtons n qs 9 and at rl = a and usng q 39 (or 4) wth 43, w gt whr and agan th uppr sgn appls to a constant charg plat and th lowr sgn to a constant potntal plat togthr wth th approprat choc for VO dpndng whthr th plat s at constant charg or constant potntal. (b) Lnarzd Rgulatng Plat. n ths cas, w ust not only choos th coffcnts to satsfy th boundary condton on th sphr but also (9) and () on th plat. W frst consdr th splr cas S, = for whch VO = n th gnral soluton gvn by q 4. Th or gnral cas of S, * s covrd by rtanng th tr (VO "*) wth vo = (S~K). Fro q 4, wth ly =, w s that th potntal on th plat, z =, s and th surfac charg s gvn by whr A, (K, K)/(K, + K). quaton 5 ust hold for all valus of@. Th constant potntal lt corrsponds to A, = (K, = a) and th constant charg cas corrsponds to A, = (K, = ). Unfortunatly, rcurrnc rlatons drvd fro q 5 connctng th coffcnts (a,} and (b,} provd not to b nurcally hlpful. nstad w us collocaton at ponts on th plat gvn by qually spacd valus of th angl 8 dfnd by cos 6 = R/(R Ths wll gv on st of quatons that rlats th coffcnts {an} and Th scond st of auatons for (a,} and (b,} cos fro applyng th aproprat boundary condtons on th sphr. Th rsultng quatons dffr dpndng whthr th sphr s at constant potntal or has lnarzd rgulaton boundary condton. onstant Surfac Potntal Sphr. For a constant potntal sphr (surfac potntal V,), applcaton of th boundary condton on th sphr gvs whr a, L and c ar gvn by q 45 and bj = bjkj(~a) (5) Lnarzd Rgulatng Sphr. For a lnar rgulatng sphr (a = S, K,V), applcaton of th boundary condton on th sphr gvs a + Lb = as, yo KR c = as, K whr a, L, and c ar gvn by q 47 and ~ sp KR c (53) bj = [(UKJ )kj(ku) KU k>(ku)]bj (54) Ths coplts th spcfcaton of th lnar quatons that ndd to b solvd to obtan th coffcnts n th xpanson of th lctrostatc potntal n th lctrolyt. V. Th Forc btwn a Sphr and a Plat As n th unqual sphr cas, w valuat th forc by ntgratng th total strss ovr th surfac of th sphr. Th forc s thn agan gvn by q 6 or 8, dpndng on th boundary condton on th sphr, but wth KU rplacng Kal and q rplacd by So, substtutng ths rsults nto q 9, wth S, =, w obtan th followng st of quatons for {a,} and {b,} wth th approprat choc for Po dpndng on boundary

8 3 Langur, Vol., No. 9, 994 arn t al. condtons of th plat. Of cours, q 4 should b usd to splfy q 55 for constant potntal or constant charg condtons on th plat. V. Th ntracton Fr nrgy btwn a Sphr and a Plat As n th cas of ntractng sphrs, th ntracton fr nrgy conssts of two parts: on fro a surfac ntgraton ovr th sphr and th othr fro an ntgraton ovr th surfac of th plat (cf. qs 3 and 3 or 36). Th contrbuton fro th ntgraton ovr th sphr has th sa for as n th unqual sphr cas dscussd arlr. Th contrbuton to th nondnsonal ntracton fr nrgy fro th sphr, Vu, s gvn by (34) or (38)) dpndng on th boundary condton on th sphr, but wth all subscrpts changd fro to s (to dnot th sphr). All that rans s to spcfy th contrbuton fro ntgratng ovr th plat. Th tr proportonal n q 39, 4, or 4 dos not contrbut to th ntracton fr nrgy snc ts contrbuton s ndpndnt of sparaton, s qs 3 and 3 or 36. (a) onstant Surfac Potntal Plat. n ths cas, th ntracton fr nrgy pr unt ara s gvn by (3b) so, usng (4) and (49, w hav for th contrbuton to nondnsonal ntracton fr nrgy fro th ntgraton ovr th plat, n n th absnc of analytc xprssons for ths ntgrals, w us GaussLagurr quadratur to valuat th ntgrals n (46) snc th ntgrand dcays as "Q for larg K. Th potntal qp should b ntrprtd as th potntal on th plat scald by (kt/). (b) Lnarzd Rgulatng Plat (ncludng onstant Surfac harg) Plat. n ths cas, th contrbuton to th nondnsonal ntracton fr nrgy fro th ntgraton ovr th plat s gvn by q 36 and ths togthr wth th xprsson for th potntal gvn by q 48 gvs whr Ap (Kp K)/(K~ + K) and Kp s th rgulaton capacty of th plat. Th constant qo should b ntrprtd as th potntal scald by (kt/). Th cas of constant surfac charg corrsponds to th lt Ap =, Kp =. And usng (4) to lnat {bj, w fnd th xplct xprsson for th contrbuton to dnsonlss ntracton fr nrgy fro a constant charg plat Both ntgrals n qs 57 and 58 ar valuatd nurcally by GaussLagurr quadratur. V. Rsults For ntractng dsslar sphrs or ntractng sphr/ plat, thr ar any cobnatons of partcl szs and boundary condtons on th two surfacs to consdr. Suppos w charactrz th charg stat of ach surfac by th surfac potntal of ach surfac whn thy ar n solatonat a larg dstanc (h = apartw call ths potntals q$o and ~'". W can now splfy th stuaton sowhat by obsrvng that both th forc and th ntracton fr nrgy n th lnar PossonBoltzann thory hav th blnar for whr th functonsfl(h),fz(h), andfdh) ar ndpndnt of vlso and lyzs. Thrfor t would b possbl to consdr just thr canoncal cass corrspondng to th potntal ratos ( q~~~~/q~~~~) =,, and. All othr cobnatons and 7 / ~ can ~ ~ thn ~ b constructd fro th rsults of vls of ths thr cass. W usd ths thr cass n ordr to dtrn th rqurd szs of th atrcs n qs 5,,3,44,46,5, and 53. For dsslar sphrs, th coffcnts {a,} and {bn} ar truncatd at ndcs ZV and N, rspctvly (th atrcs ar of sz Nl + by Nz + ). As n arlr work,l prcally t was found that fourfgur accuracy for all boundary condtons was obtand for 5 fhla > N = { [7(a/h)/] f hla < (59) for =, and whr [X dnots th ntgr part of th ral nubr x. For th sphr/plan gotry, th coffcnts {a,} and {bn}(whr ncssary) ar both truncatd at ndx N whr, for fourfgur accuracy 5 fhla > f hla (6) N = { [6(~lh)'/~] s suffcnt. n what follows, howvr, w wll prsnt rsults for four or physcally usful cass havng potntal ratos ( q ~ ~ ~ qual / ~ to ~ ~ 3, ~, ), and 3. Snc th cas of dntcal doubl layrsa potntal rato of wth qual szshas bn wll studd?" w wll pay or attnton to th othr cass. Snc w ar usng th lnar Posson Boltzann quaton, th cass wth potntal rato 3 and 3 should b thought of as rprsntng sphrs wth potntals of, say, 3 V and f V. n prvous coparsons wr ad wth th lnar suprposton approxaton (LSA) and th Dryagun approxaton. Although th LSA has bn workd out for dsslar sphr^,^ ts applcablty s ltd to larg sparatons and s not consdrd hr. Th Dryagun approxaton for dsslar sphrs at constant (sall) potntal was frst drvd by Hogg, Haly, and Furstnau3 and w us DA to dnot th Dryagun approxaton for any boundary condton. Th approprat xprsson for lnarzd rgulatng surfacs has bn gvn only rcntly.sj

9 lctrcal Doubl Layr ntracton Fr nrgy Langur, Vol., No. 9, h P c W. V. z ntracton Fr nrgy. *,,. 3: ;;:, ntracton Fr nrgy 4:, d. w,= w,= 3 53 O. O. Ka = KO = j h% uv kh a? U. V. U z kh Fgur 3. (a, top) Nondnsonal ntracton fr nrgy, q 35, of sphrs wth qual solatd potntals, = q~~~~ =, and unqual rad, Kul=, K U = ~, for nn cobnatons of boundary condtons. Th boundary condton on sphr s dnotd by th ln styl: constant charg, ; lnarzd rgulaton, ; constant potntal, a. Th boundary condton on sphr s dnotd by th sybol: constant potntal, U, lnarzd rgulaton, ; constant charg,. (b, botto) Th nondnsonal forc, q 5, for th sa paratrs as (a). Dryagun Approxaton (DA). Forc (nwtons): nc~a,a, fda = (a, + a,) ~v~v,so ntracton fr nrgy (jouls): UDA(h) = (a, + a,) [A(~P) + A(qls) (6) AlA, whr A and A ar dfnd by A = (K K)/(K~ + K), =,. Th approprat xprssons for th sphrplat gotry ar obtand sply fro qs 6 and 6 by takng on of th rad to nfnty n th prfactors. n Fgurs 3 and 4 w show so rsults for th full rang of boundary condtons covrd n th prvous X Fgur 4. (a, top) Nondnsonal ntracton fr nrgy of sphrs wth unqual solatd potntals, vll8o = l,/)so = 3, and unqual rad, Kul=, KU~ =, for nn cobnatons of boundary condtons. Sybols and styls ar as n Fgur 3 so that th top thr curvs hav sphr wth constant charg, th ddl thr curvs hav sphr wth a rgulatng surfac, and th botto thr curvs hav sphr wth constant potntal. (b, botto) Th nondnsonal forc for th sa paratrs as (a). sctons. To lt th nubr of cass, for lnarzd rgulatng surfacs th quantty A or A or Ap s st to zroths s th canoncal ntrdat cas btwn constant charg and constant potntal.8 vn so, for sphrs of unqual sz, ncludng a sphr and a plat, ths lavs nn possbl cobnatons of boundary condtonsach surfac can b thr constant potntal, ntrdat)), or constant charg. For sphrs of qual sz, dgnracs rduc ths to sx cass. Fgur 3 shows rsults prtanng to sphrs of unqual sz: Kal =, Ka =, but wth qual solatd potntalsth ntracton fr nrgy n Fgur 3a, th forc n Fgur 3b, scald accordng to qs 35 and 5, rspctvly. For th nrgy, th nn cobnatons of boundary condtons covr th whol rang of bhavor fro constant charg to constant potntal. That s, whn both sphrs ar at constant charg, th rpulsv ntracton nrgy s th largst, whl f both sphrs ar at constant potntal, th rpulsv nrgy s th sallst for all sparatons down to ~h =.. Slarly whn th sphrs hav qual but oppost solatd potntals (not shown), th cas of both sphrs at constant charg gvs th last attractv ntracton nrgy whl that wth both sphrs at constant potntal gvs th ost attractv ntracton. Agan ths obsrvaton s vald for sparatons down to =.. W hav not xplord th bhavor for sallr sparatons as such rgons ar physcally lss ntrstng. Howvr, graphcal argunts suggst that

10 3 Langur, Vol., No. 9, 994 arn t al. 8 " kl P w. V. z al LL. V. z 6' Forc ly =3 ly=l KO = KO = Fgur 5. (a, top) Nondnsonal ntracton fr nrgy of sphrs wth qlso =, qzso = 3 and unqual rad Kal=, Ka = for nn cobnatons of boundary condtons. Th top thr curvs hav sphr wth constant charg, th ddl thr curvs hav sphr wth a rgulatng surfac, and th botto thr curvs hav sphr wth constant potntal. (b, botto) Th nondnsonal forc for th sa paratrs as (a). ths rarks ar vald for all sparatons.lsj6 Th sa s not tru for th forc curvs (Fgur 3b) although t s volatd only at sparatons sallr than %.. Th sa faturs ar sn n rcnt work on ntractng plats usng th nonlnar PossonBoltzann quaton' Nvrthlss, th gnralzaton that doubl layr ntractons ar boundd by th constant potntal and constant charg cass s a usful approxaton n ost cass. Fgur 4 shows nrgy and forc curvs agan for a sz rato of but for a potntal rato of 3. Both sts ofnn curvs splt nto thr groups of thr curvs xcpt for th forc at sall sparaton. Th splttng s dtrnd by th boundary condton on th sphr wth th potntal of sallr agntud (sphr ). Th botto thr curvs all hav constant potntal boundary condtons on sphr, th ddl thr hav rgulaton condtons on sphr, and th top thr constant charg condtons on sphr. Th boundary condtons on sphr, whch has a hghr valu of ~ ~ hav ~ ~ a lssr, ffct and dtrn th splttng wthn ach group of curvs. Fgur 5 shows th sa splttng and wth th sa ordrng for a potntal rato of 3 and Fgur 6 a splttng wth dffrnt ordrng for a potntal rato of 3. n Fgur 6 th largr sphr (sphr ) has th sallr valu of qlsol and so th potntal on sphr dtrns th ajor splttng, wth th boundary condton on sphr (6) han, D. Y.. J. ollod ntrfac Sc. 983,95, 93. (7) Mcorack, D.; arn, S. L.; han, D. Y.. J. ollodlntrfac Sc., n prss., * 6 Fgur 6. (a, top) Nondnsonal ntracton fr nrgy of sphrs wth ql = 3, yjzm' = and unqual rad Kal =, Ka = for nn cobnatons ofboundary condtons. Th top thr curvs hav sphr wth constant charg, th ddl thr curvs hav sphr wth a rgulatng surfac, and th botto thr curvs hav sphr wth constant potntal. (b, botto) Th nondnsonal forc for th sa paratrs as (a). controllng th splttng wthn ach group. Agan, slar ffcts ar sn for pats.l' Th gnral concluson s that th sphr or plat wth th sallr valu of vsol prturbd proportonally or du to th prsnc of th othr surfac wth a largr valu of llysol and so t s th boundary condton on th sphr wth th sallr valu of vl that prarly dtrns th for and agntud of th nrgy or forc curv. Notc that th curvs bco ndpndnt of boundary condtons for > so that ffctv surfac potntals could b xtractd fro that part of th forc curv. Th ffcts of dffrnt boundary condtons ar vdnt at sallr sparaton, <. Th splttng bhavor shown n Fgurs 36 for sphrs of sz rato s also sn for othr sz ratos, ncludng a sz rato of, and for th sphrlplat gotry. n what follows w shall gnrally b contnt to xan only th constant charg and constant potntal cass wth th knowldg that othr cass wll xhbt bhavor ntrdat btwn ths two cass. Wthn th contxt of DLVO thory, th only approxaton w ar akng s that of lnarzng th PossonBoltzann quaton. t s prtnnt to ask thrfor, how rlabl s th lnar thory for gvn valus ofparatrs such as Ka,, and v? n prvous work,6 w hav found that for dntcal doubl layrs, th lnarzd thory gvs forc curvs wthn % of th nonlnar rsult for surfac potntals up to 354 V undr constant potntal condtons but s only qualta

11 lctrcal Doubl Layr ntracton Fr nrgy Langur, Vol., No. 9, rror n lnar thory for constant potntal forc y,=l y = qual sz 5 t 3 4.~ / 4 rror n lnar thory! qual sz 5' kh 5 % rror n lnar thorv for constant ~otntal foc qual sz ZXA 3 l.lo 5 U tvly corrct undr constant charg condtons. Howvr, for sall partcls (Ka, th lnar thory s rlabl for both constant potntal and constant charg boundary condtons. For th spcal cas of sphrs of qual sz and oppost solatd potntals, th forc curv for th nonlnar thory can b calculatd usng th thods of rf 6 wth two nor changs. Th frst s that, at th dplan, th potntal vanshs rathr than th radal drvatv. Th scond dffrnc occurs n th xprsson for th strss at th dplan whr th osotc prssur and transvrs lctrc flds vansh, lavng contrbutons only fro th radal lctrc flds. For dntcal surfacs th prforanc of th lnar thory can b xpland by obsrvng that n th constant potntal cas th potntal s fxd at th boundars and so thr s rlatvly lttl frdo for th potntal profl to chang wth surfac sparaton. Howvr, n th constant charg cas th surfac potntals rs as th sphrs approach so that th lnarzaton bcos unjustfabl. For surfacs wth potntals qual n agntud but oppost n sgn, howvr, slar argunts would suggst that thr s or roo for opts. For surfacs havng qual but oppost surfac potntals ntractng undr constant potntal, th potntal s agan furd at th surfacs and as th sphrs approach th potntal profl vars alost lnarly btwn th surfacs. Ths suggsts that, provdd transvrs gradnts ar not portant, th lnar and nonlnar rsults should b clos spcally at sall sparatons. For th constant charg cas, th surfac potntals actually fall as th sphrs approach whch suggsts that th lnar. 5 ~ qual sz LU Kk Fgur 7. Prcntag rror n th lnar thory for th forc Fgur 8. Prcntag rror n th lnar thory for th forc undr constant potntal.,fro top to botto, Ka =, 5, 3, undr constant charg., Fro top to botto, Ka =,3,,5, and and. (a, top) qlso = t)zls =. (b, botto) qlao = q = ~ ~. ~ (a, ~ top) qlls = qzuo =. (b, botto) qlls = qzls =. 3. thory should gt bttr at sall sparatons, vn for farly larg solatd potntals. Nurcal rsults for th nonlnar cas show that ths xpctatons ar at last partally fulflld. Th rato of lnar to nonlnar rsults for sphrs of qual sz hld at constant potntal s shown n Fgur 7 for varous valus of Ka and zyso. oparson wth th corrspondng rsults n rf 6 shows that th lnar thory s n fact or accurat for sphrs of qual sz but oppost potntal than for dntcal sphrs. Th rror s lss than % vn for Ka = at = 5 V for sparaton and vn for 75 v for sall sparaton ( ). n ths cas, our opts has bn justfd. As bfor, th lnar thory provs for sallr partcls. Rsults for th constant charg cas for qual szd sphrs ar shown n Fgur 8. n ths cas, our hops ar only partally fulflld, n that th lnar thory s vry good for qs = 5 V (rror wthn 5% for Ka = ) but starts to dtrorat for lyao = 5 V (rror s alost %). Nvrthlss, t s a sgnfcant provnt ovr th corrspondng prforanc for dntcal sphrs whr th rror vars fro 5% to 5% for slar paratr valus. Havng obtand xact rsults for th cas of qualszd sphrs wth oppost potntals, w can also tst how th nonlnar Dryagun approxaton prfors n ths cas. Rsults for th constant potntal cas ar shown n Fgur 9 and for th constant charg cas n Fgur. Th nonlnar Dryagun rsults ar obtand by applyng th Dryagun constructon to solutons of th nonlnar PossonBoltzann quaton for paralll plats. Th rsults ar slar to th dntcal sphr cas6 n that n th rang th rror s wthn % provdd Ka

12 34 Langur, Vol., No. 9, 994 arn t al. ' Dryagun approxaton for ra = SO! constant potntal forc ly = ly = qual sz 4: '.O/./ c/o rror n nonlnar Dryagun approxaton for constant charg forc 4 / nlt t g! 6 % rror n nonlnar. Dryagun approxaton for SO; constant potntal forc,! + = 4 ly, = 4 4 ol 3 t qual sz Fgur 9. Prcntag rror n th nonlnar Dryagun approxaton for th forc undr constant potntal. Fro top to botto,, =, 3, 5, and. (a, top) qls = =. (b, botto) qlls = *Z'SO = 4. 4[ / : % rror n nonlnar 3[ Dryagun approxaron for constant charg forc rdt Fgur. Prcntag rror n th nonlnar Dryagun approxaton for th forc undr constant charg. Fro top to botto at ~h = ~,,Ku =,3,5, and. (a, top) lyllso = lyzso =. (b, botto) qllso = ppo = 4. forc croscop. For ths rason w lt ourslvs to 5 and provs as th solatd potntal s rasd. Th forc curvs n ths scton. n ths gotry thr s only rror s slar for both constant potntal and constant on valu of Ka, and by varyng ths paratr, w study charg boundary condtons. t s notcabl fro Fgur th ffct of th curvatur of th sphrcal partcl. n la that th rror n th Dryagun rsult appars to scal Fgur w show rsults for Ka = whr a pror w as (Ka)l. Ths suggsts that a corrcton to th Dryagun would xpct th Dryagun approxaton to prfor xprsson to th nxt ordr n a (Kal xpanson should wll. For th coparson n Fgur, th Dryagun b worthwhl. Fgur llb shows th rsult of addng rsult s gvn by q 6, whch s obtand by applyng th a corrcton of ths ordr to th lnar Dryagun Dryagun constructon to th lnarzd PossonBoltztr8v7although a worthwhl provnt t s clar ann rsult for paralll plats8w call ths th lnar that th corrcton s only vald for th rg ~h << Ka. Dryagun approxaton. For th cass whr both n any cas, a hrustc odfcaton of th Dryagun surfacs hav constant potntal or ar rgulatng, th xprsson usng Lvn's surfac dpol thod for th accuracy of th lnar Dryagun approxaton s conconstant potntal cas gvs good accuracy for a wd frd by th data. Howvr th constant charg cas s rang of condtons.lg nvr accurat for ~h < wth th xcpton of th cas Ths rsults, togthr wth thos of rf 6, suggst that wth potntal rato qual to. Ths spcal cas s th th lnar thory has a surprsngly wd rgon of only cas whr th lnar Dryagun approxaton gvs applcablty, at last for th forc curvs. t ust b sad, a forc that s always attractvn vry othr cas, th howvr, that th only vdnc cos fro th two rathr constant charg curv bcos rpulsv at suffcntly spcal cass of qualszd sphrs wth dntcal (qual sall sparaton. Slarly, th lnar Dryagun apbut oppost) potntals. n partcular, th forc s always proxaton for th constant potntal cas s always anfstly rpulsv (attractv) for all boundary condrpulsv only for dntcal surfacst bcos attractv tons so that non of th rch bhavor sn for doubl for suffcntly sall sparaton n all othr cass. layrs wth potntals of dffrnt agntud (s Fgurs Howvr, th lnar Dryagun approxaton s good for 46) has bn drctly tstd. Th thods of rf 6 can constant potntal surfacs vn n ths attractv b xtndd farly radly to thos cass, whch w hop cass. to pursu n th futur. Th ffct of th partcl curvatur can b sn n Fgur Th nxt rsults w prsnt ar for th sphrdplat 3 whr, n on cas, th sphr has th potntal of gotry slar to forc asurnts on th atoc lowr agntud (Fgur 3a) and, n th othr, t s th plat (Fgur 3b). Snc t s th surfac of lowr (n (8)Ohsha,H.;han,D.Y..;Haly,T.W.;Wht,L.R.J.ollod ntrfac Sc. 983, agntud) potntal that controls th ntracton, as n 9, 3. (9) Sadr, J.; han, D. Y..; arn, S. L. To b subttd for th splttng sn n Fgurs 35, th lnar Dryagun publcaton. approxaton s notcably wors n dscrbng th frst

13 lctrcal Doubl Layr ntracton Fr nrgy Langur, Vol., No. 9, Rlav rror (scald by Ka) n nonlnar Dyagun approxaton 5 5 ooot sool 6r Forc n sphrplat gotq y,=l ly = KO = 9c rror n lnar Dryagun \ approxaton + corrcon \ for constant pontal forc u/ = ly = qual sz v Fgur. (a, top) rror n th forc fro th nonlnar Dryagun approxaton undr constant potntal ultpld by KU, for KL =, 3, 5, and vllso = qlso =. (b, botto) Th prcntag rror n th corrctd forc (th lnar Dryagun forc plus th corrcton fro rf 8) copard to th lnar thory undr constant potntal for vls = vs =. Fro top to botto, KU =, 5, 3, and. cas, whr th portant surfac has K =, than th scond cas. Nvrthlss, th prforanc s good xcpt for constant charg surfacs. Fgur 4 shows slar rsults for a sphr wth KU =. As xpctd, th lnar Dryagun approxaton prfors wors for such a sall partcl but thr ar svral notabl faturs of th curvs. Frstly, th constant potntal and rgulatng cass ar stll dscrbd rarkably wll. t s as f havng on surfac as a plat wth nfnt radus of curvatur has xtndd th applcablty of th lnar Dryagun approxaton to low valus of ~a. A hurstc xplanaton of ths obsrvaton s gvn n rf 9. Scondly, th constant charg rsults ar agan good only for a potntal rato qual to ; all othr cass ar poorly handld. n partcular, th bhavor for a potntal rato of 3 s not vn qualtatvly corrct at KU = wth th Dryagun approxaton gvng qut strong rpulson copard to a ld attracton n th xact calculaton. n th cas of rgulatng surfacs wth As = Ap =, th lnar Dryagun approxaton rducs to th lnar suprposton approxaton8ths ay xplan th rlatvly good prforanc for ths cas for a wd varty of condtons. n Fgurs 5 and 6 w show rsults for xd boundary condtonson surfac has constant charg and th othr constant potntal. Agan, th prforanc s rarkably good at KU = and dtrorats sowhat at KU =. Onc agan th surfac wth th lowr (n agntud) potntal controls th ntracton so, f t has 8. s. n ly.zoo\ 4 6.lOOO Forc n sphrpla goy u/ = ly = Ka = O.5 s Fgur. Forc curvs n th sphr/plat gotry for KU = and thr sts ofboundary condtons: fro top to botto, both surfacs constant charg, both surfacs rgulatng, both surfacs constant potntal. Rsults of th lnar thory ar shown as sall sold sybols, corrspondng rsults for th lnar Dryagun approxaton ar shown as larg opn sybols. (a, top) y:so =, ljlplso = 3. (b, ddl) =, vpsa =. (c, botto) v,so =, *ps =. constant potntal, th curv s constantpotntallk.g. th lowr par of curvs n Fgur 5, corrspondng to th sphr n parts a and b of Fgur 5 and th plat n Fgur 5c. Agan, by far th worst prforanc s for a constantcharglk curv wth potntal rato qual to 3 at low KU (Fgur 6b). To suarz, th sphr/plat gotry has th happy rsult of xtndng th applcablty of th lnar Dryagun approxaton to valus of KU as low as prhaps but crtanly 3 for any boundary condtons xcpt constant charg. n that cas, t consstntly ovrstats th forc xcpt for a potntal rato qual to. For a potntal rato of 3 at low KU, th lnar Dryagun

14 36 Langur, Vol., No. 9, P._ P Forc n sphrlplat gotry yl= yl =3 KO =.4 5, a Forc n sphrlplat gotry ys=3 yl = KU = Fgur 3. As for Fgur but yth (a, top) ) 3 and (b, botto) v,sso = 3, l/lp =. 4 Forc n sphrplat gotry yl,= yl =3 Q 3 ra=l L A..5,5!? =, vpso = arn t al. approxaton s qualtatvly wrong. Our confdnc n ths statnt s bolstrd by th good prforanc of th lnar thory at low Ka copard to th nonlnar thory as w hav sn hr and n rf 6. Havng dalt so xhaustvly wth th sphr/plat gotry t s only ncssary to nton th nw faturs xhbtd by dsslar sphrs. Bcaus htrocoagulaton studs (for xapl, rf ) tnd to focus on th ntracton nrgy, w wll prsnt th ntracton fr nrgy curvs n what follows. Wth both surfacs now bng of collodal dnson, w would xpct th Dryagun approxaton to b lss accurat than n th abov dscusson. n Fgurs 7 and 8 w show rsults for sphrs of qual sz. Th nrgy curvs at KU = show all th charactrstcs of th forc curvs abov: good prforanc for constant potntal but not for constant charg, xcpt at a potntal rato of. At KU = th agrnt s uch wors n trs of th rlatv rror spcally at larg a fatur that dos not show up n th forc curvs abov. A ky ngrdnt n provng th prforanc ofth Dryagun approxaton for th nrgy s to ncorporat th corrct asyptotc bhavor at larg,h.9 By coparng Fgur 8 aganst Fgurs 9 and, w s th ffct of ncrasng th sz rato of th partcls fro to 5 to nfnty wth on partcl hld at Ka =. Th prforanc of th Dryagun approxaton n th rang ~h 5 provs n all cass. Th constant charg cas s nvr vry accurat and th constant potntal curv at a potntal rato of 3 provs draatcally. Th prforanc of th constant potntal curv at a potntal rato of 3 s or problatct appars that th agrnt at Kal = ~ a = ay b fortutous. 'r Forc n sphrlplat gotry yl,=l = Ka = Q 8 d, a 4._ 6. g 8. Forc n sphrplat gotry yl,=l y= KU = p 3 b Forc n sphrlplat gotry y=l w=3 Ul=l! Fgur 4.,As q,,,:o =, l/rplso =, and (d, botto rght) qso =, qplso = 3. for Fgur but wth KU =, and (a, top lr) vysso =, lypso = 3, (b, top rght) v,so =, v,pso =, (c, botto lft)

15 lctrcal Doubl Layr ntracton Fr nrgy _"., Forc n sphrplat gotry 4 P for xd boundary condtons? ly = ly =3 r F 3r F KO = P P 5 4 " Langur, Vol., No. 9, Forc n sphrlplat gotry for xd boundary condtons ly,= ly =3. = KO =. o. j.5.5 j 5 5 h%.lo,..' o c s o... Y.. c....4 for xd boundary condtons y,=l w=3 KO = t. Forc n sphrplat gotry for xd boundary condtons 5! 3 ' Forc n sphrplat gotry for xd boundary condtons rca =.5.5 Fgur 5. Forc curvs n th sphr/plat gotry for KU = and two sts of"xd" boundary condtons: crcls dnot th cas of a constant charg plat and a constant potntal sphr, squars dnot th cas of a constant potntal plat and a constant charg sphr. As bfor, rsults of th lnar thory ar shown as sall sold sybols and corrspondng rsults for th lnar Dryagun approxaton ar shown as larg opn sybols. (a, top) q:so =, qplso,= 3. (b, ddl) lyss =, qplso = 3. (c, botto) qss = 3, qplso =. X. onclusons W hav shown how to obtan solutons for th doubl layr forc and ntracton fr nrgy for dsslar collodal sphrs as wll as btwn a sphr and a plat, accordng to th lnarzd PossonBoltzann quaton. Nurcal calculatons hr and n rf 6 suggst that such rsults ar rasonabl, spcally for constant potntal surfacs, for surfac potntals approachng 4 V. For xprntalsts usng th atoc forc croscop, th ost sgnfcant fndng s that, xcpt f th surfacs h% L 5' 5 5 Fgur 6. As for Fgur 5 but wth K a =. (a, top), *'pso = 3. (b, botto) qss =, qps = 3. undr study ar known to b constant charg surfacs, th Dryagun approxaton gvs an accurat pctur of th doubl layr forc btwn a sphr and a plat for partcl szs currntly usd n th AFM. Snc rsults fro th nonlnar PossonBoltzann quaton6 show that th Dryagun approxaton provs at hghr potntals, ths strongly suggsts that th Dryagun approxaton should b adquat for th analyss of sphr/plat forc curvs n ost crcustancs. vn n th constant charg cas, th Dryagun approxaton consstntly provds an uppr bound on th tru forc. For htrocoagulaton studs, th pctur s slar xcpt that th nrgy curvs start to dtrorat onc both partcl szs fall blow KU = 5. For vry sall prcursor partcls (Ka < ) th lnarzd Posson Boltzann quaton s known to b good6 and so th prsnt thod should b rlabl. Ths thod s not nurcally dauntngach forc or nrgy curv prsntd hr only taks at ost a fw nuts to calculat on a Macntosh ntrs 65. Ndlss to say, th coparatvly good prforanc of th Dryagun approxaton copard to an xact soluton of th lnarzd PossonBoltzann quaton found hr for partcls of qut dffrnt sz and solatd surfac potntals s n arkd contrast to th clad poor prforanc by Barouch t al Th thortcal flaws n that work hav bn loquntly statd by Ovrbkt suffcs to say hr that our nurcal calculatons support th argunts of Ovrbk n vry rspct. () Barouch,.; Matjvc,.; Rng, T. A.; Fnlan, J. M. J. ollod ntrfac Sc. 978, 67,, and latr paprs ctd n rf. () Ovrbk, J. Th. J. h. SO., Faraday Trans. 988, 84, 379. =

16 ~ Ka 38 Langur, Vol., No. 9, 994 j 5, % F 4? t 3; A * &._ U p ntracton fr nrgy for qual szd sphrs ly = ly =3 Ka = F p., ntracton fr nrgy for qual szd sphrs ly,=l y=3 = arn t al., d d ntracton fr nrgy for qual szd sphrs. yf=l ly=l Ka = lo.l 5 ' 3 ttracron fr nrgy for qual szd sphrs yf = ly = =l 5 5 x P )._ v)._ U z" zoo/ 4. d ntracton fr nrgy for qual szd sphrs ly,=l ly=3 = v 3 5 4O * o ntracton fr nrgy for qual szd sphrs ly, = ly, = 3 KO = 5~.6;.5 "" Fgur 7. ntracton fr nrgy curvs for sphrs of qual sz K U = ~ KUZ = and thr sts of boundary condtons, fro top to botto, both surfacs constant charg, both surfacs rgulatng, both surfacs constant potntal. Rsults of th lnar thory ar shown as sall sold sybols; corrspondng rsults for th lnar Dryagun approxaton ar shown as larg opn sybols. (a, top) vlw =, vzso,= 3. (b, ddl) vls, &fso =. (c, botto) vlso 3. =, 7)p = Th on ranng doubt n ths work s how rlabl ar ths conclusons for partcls wth larg surfac potntals. n two spcal cass, qual sz sphrs wth qual or oppost surfac potntals, w hav answrd thos doubts. "hos rsults strongly suggst that th Dryagun approxaton provs at larg surfac potntals and so th gnral conclusons rgardng th ffcts of sz rato and curvatur ar corrct. Dfntv vdnc howvr awats nurcal rsults of th nonlnar PossonBoltzann quaton for dsslar sphrs and th sphrlplat gotry, a goal w now blv s n sght. Fgur 8. As for Fgur 7 but wth KU = KUZ = and two sts of boundary condtons, fro top to botto, both surfacs constant charg, both surfacs constant potntal. (a, top) ljtlso, vz'? = 3. (b, ddl) vllso =, v~~~~ =. (c, botto) vlso =, qp = 3. S, K f F h U U Glossary sphr rad coffcnts n th two cntr xpanson of th potntal paratrs for th lnarzd rgulaton odl forc n nwtons =flh)/(~(kt/)) nondnsonal forc shortst dstanc btwn th sphrs or btwn sphr and plat =u(h)/(&t/)k) nondnsonal ntracton fr nrgy ntracton fr nrgy n jouls