Poisson-Boltzmann Theory with Non-Linear Ion Correlations

Transcription

1 Posson-oltzmann Thory wth Non-Lnar Ion Corrlatons Mao Su ( 苏茂 ) 1,,3, Zhj Xu ( 徐志杰 ) 3 *, Yantng Wang ( 王延颋 ) 1, * 1 CAS Ky Laboratory of Thortcal Physcs, Insttut of Thortcal Physcs, Chns Acadmy of Scncs, 55 East Zhongguancun Road, P. O. ox 735, jng , Chna School of Physcal Scncs, Unvrsty of Chns Acadmy of Scncs, 19A Yuquan Road, jng , Chna 3 Computatonal Mathmatcs Group, Pacfc Northwst Natonal Laboratory, Rchl, Washngton 9935, USA ASTRACT: Th Posson-oltzmann (P) thory s wdly usd to dpct onc systms n a man-fld mannr wthout consdrng th corrlatons btwn ons. A modfd thory ncludng on corrlaton ffct whl rtanng th smplcty of th orgnal P thory s dsrd. Although rcnt thortcal fforts succssfully ncorporat on corrlatons nto th P quaton as th slf-nrgy, th rsults dvat sgnfcantly from our molcular dynamcs (MD) smulaton rsults du to th lnar form of th slf-nrgy. W hrn rntrprt th slf-nrgy trm from a physcs pont of vw, on th bass of whch w prsnt a nw way for ncorporatng on corrlatons nto th orgnal P quaton by utlzng th Grn s functon wth a non-lnar form of th slf-nrgy. Justfd by our MD smulaton rsults, our modfd P thory wth non-lnar on corrlatons surpasss lnarly formd modfd P thors. I. INTRODUCTION Rlabl thortcal modls for onc systms ar of grat fundamntal mportanc for basc rsarch, spcally for th study of soft-mattr [1, ]. As a man-fld thory, th Posson-oltzmann (P) modl grasps th man faturs of onc systms n th absnc of on corrlatons [3]. y assumng that th on dnsty satsfs th oltzmann dstrbuton, along wth th lctrostatc potntal satsfs th Posson s quaton, th orgnal P quaton s whr z z s xp -z kt, (1) s th dlctrc constant, s th unt charg, ar th valanc bulk dnsty of th th on s spcs, rspctvly, k s th oltzmann constant, T s th tmpratur. Th P thory has many applcatons n varous rsarch flds, rangng from nvstgatng th lctrostatc proprts of chargd molculs onc solutons [4] to studyng th structurs flxblts of mmbrans [5, 6], as wll as calculatng charg dstrbutons on th surfac of bomolculs such as DNA, RNA protns [7, 8]. In ts applcaton, t s wll known that th P thory wll qualtatvly fal whn on corrlatons bcom sgnfcant [9-1], so a varty of thortcal modls, whthr basd on P thory or not, hav bn dvlopd to ncorporat th on corrlaton ffct. Th ntgral quaton mthod wth th Hyprnttd Chan (HNC) approxmaton s known to b vry accurat for onc systms [13, 14], but 1 thortcally th physcal pctur bhnd ths approxmaton s vagu, practcally th HNC quatons can b hard or vn mpossbl to solv n crtan cass [1, 15]. Othr mthods lk th dnsty functonal thory [16, 17] nvolv laborat mathmatcal xprssons that ar too complx for practcal applcatons. Thrfor, som xact man-fld thors ar dvlopd to yld accurat rsults whl kpng thr mathmatcal xprssons as smpl as th Dby-Hückl (DH) quaton, whch s a lnarzd nstanc of th P quaton, wrttn as z s kt. () For nstanc, by mplmntng ffctv chargs nstad of ral chargs, th drssd-on thory [18] ylds a Yukawa potntal as th soluton to th abov DH quaton, th molcular Dby-Hückl thory [19, 0] utlzs a lnar combnaton of Yukawa potntals. Howvr, mthods of ths typ hav th drawback that th dlctrc functon of th systm must b known n advanc to dtrmn ky paramtrs. Rcntly, th fld-thortc () approach provds an lgant way of solvng th gr partton functon, ladng to a corrcton trm namd th slf-nrgy [1]. low w confn ourslvs to th cas of havng only two on spcs whos charg valancs ar z z, thn th slf-nrgy n th approach at poston r can b dfnd as: 1 l u z lm G( r, r '), r' r r r' (3)

2 whr G( r, r ') s a Grn s functon to b dtrmnd l 4 kt s th jrrum lngth. Intrstngly, svral dffrnt mthods, such as th varatonal mthod [1-5] th loop-ws xpanson [6], nd up wth th sam xprsson for G( r, r ') : G( r, r ') 4 l z z G( r, r ') 4 l ( r r '), corrspondngly th P quaton s modfd as (4) z u z u 4 l z z + x, (5) whr th dmnsonlss potntal smplcty, a fxd on dstrbuton gnralty. Th on dnsty s s usd for kt s ncludd for x z u, whr v s th fugacty of ons wth bng th chmcal potntal v bng a volum scal n th partton functon [1]. Not that f th on dnsty s a constant, thn Eq. (4) rducs to th orgnal DH quaton. W thrfor call t a DH-lk quaton for th Grn s functon. Although th DH-lk quaton Eq. (4) looks lnar, t s actually non-lnar bcaus t dpnds mplctly on th slf-nrgy. Howvr, bttr accuracy can b xpctd by rplacng th DH-lk quaton by a crtan P-lk quaton for th Grn s functon, snc th orgnal DH quaton s just a lnar approxmaton of th orgnal P quaton [7-9]. On th othr h, numrcal solutons to Eqs. (3)-(5) ar ntnsvly studd [30-3], but no molcular smulatons hav bn prformd to drctly justfy th accuracy of thos mthods. As w wll show blow by our molcular dynamcs (MD) smulaton, ths DH-lk quaton svrly lmts ts accuracy at fnt concntratons. To ncorporat th on corrlaton ffct mor accuratly nto th orgnal P thory, w hrn prsnt a modfd P thory wth non-lnar on corrlatons, whos slf-nrgy s rntrprtd wth a P-lk quaton from a physcs pont of vw. As justfd by our MD smulaton rsults, our nw mthod has a bttr accuracy than th xstng lnarly formd mthods wthout notcabl xtra computatonal cost. II. METHODS In ths work, w start from th orgnal two-spcs P quaton wth th dmnsonlss potntal z P z P P 4 l z s zs, (6) x whr th subscrpt P sts for th soluton to th orgnal P quaton, s s ar avrag dnsts of th two on spcs. To add n on corrlatons, w now nsrt a tst on at poston r, whos bar potntal G0 r, r ' at poston r s gvn by th Grn s functon as, ' 4 ' G0 r r lz r r. (7) Whn th systm s fully rlaxd to a nw qulbrum stat aftr th nsrton of th tst on, th avrag potntal s prturbd to b G ( r, r ') P, whr G ( r, r ') s th ncrmntal potntal du to th tst on. Th corrspondng quaton for such a systm s zp zg ( r, r ') G ( r, r ') 4 l [ z P s ( ')]. zp zg ( r, r ') zs x z r r In contrast to th orgnal P quaton Eq. (6), w fnd th followng quaton for G ( r, r ') by a smpl subtracton: zp s zgr, r ' z r ' G r r l z, ' 4 [ 1 1 ' ]. zp zs z r r Eq. (9) ndcats that two factors contrbut to G ( r, r ') : th prsnc of th tst on th chang of surroundng ons du to th prsnc of th tst on. Th potntal nducd by th chang of surroundng ons rsults from on corrlatons: 0 rr ' rr ' (8) (9) u ( r ') lm u ( r, r ') lm G ( r, r ') G ( r, r '), (10) whch s xactly th slf-nrgy by dfnton. Wth th corrcton of th slf-nrgy, th on dnsts bcom ( ) z r z r u r ( ) ( ) 0, (11) whr s th nw lctrc potntal to b solvd, 0 zu s wth u bng th slf-nrgy at nfnty to nsur that () r approachs s whn r gos to nfnty. Th P quaton s thn modfd to b z z u z z u 4 l z 0 z 0 x, (1) th corrspondng Grn s functon s dtrmnd by zgr, r ' z r ' G r r l z r, ' 4 [ ( ) 1 z 1 z r r ' ]. (13) Not that now Eq. (13) s dffrnt from Eq. (9) by th oltzmann factor zp z zu. Ths s th P-lk quaton for Grn s functon. As xpctd, our thory rducs to th rsults of th approach f on appls a lnar approxmaton, whos drvatons ar shown n th Appndx. Howvr, such a lnarzaton s n fact r dvrgs as mathmatcally problmatc snc r gos to r, whl th lnarzaton s accptabl only r s clos to zro. As a rsult, our thory whn

3 th approach should lad to qualtatvly dffrnt rsults. III. A. Modl RESULTS AND DISCUSSIONS In ths scton w justfy th accuracs of both our thory th approach by MD smulaton. To avod th dvrgnc problm du to th sngularty n lctrostatc ntractons, w trat ach on as a soft ball nstad of a pont charg by ntroducng btwn ons th van dr Waals (VDW) ntracton as vr () 1 6 4, (14) kt r r whr vr () s th Lnnard-Jons (LJ) potntal rscald by a factor of 1 kt, ε s th dpth of th potntal wll σ s th dstanc at whch th LJ potntal s zro. For th sak of smplcty, w assum that both on spcs hav th sam VDW paramtrs, th VDW ntractons ar also tratd n a man-fld way. Consquntly, by subtractng Eq. (7) from Eq. (13) ncludng th VDW ntractons, th slf-nrgy quaton bcoms z u r, r ' G0 r, r ' v r r ' u r, r ' 4 l [ z 1 z u r, r ' G0 r, r ' v r r ' z 1 ]. (15) In gnral, Eq. (15) can only b solvd numrcally. Eq. (7) s also solvd numrcally by rplacng th Drac dlta functon by th Kronckr dlta functon to avod dvrgnc. Th boundary condton for Eq. (15) s l s r r' lm u( r, r ') 1 snc Eq. (13) should r r r' rduc to th orgnal DH quaton at larg r. Wth th hlp of Eq. (7), Eq. (15) can b tratvly solvd to dtrmn u r, r ' by mployng th Fnt-Dffrnt mthod, thn lmt r r'. r u can b obtand by takng th Th structur of th onc systms s oftn dscrbd by th radal dstrbuton functon (RDF), whch s th dnsty dstrbuton wth rspct to a rfrnc partcl can b drctly calculatd n MD smulatons. In ordr to calculat th RDF by th modfd P thors, w add a fxd on as th rfrnc partcl, whch s qual to st x () r n Eq. (5) Eq. (1). Thn th RDF s calculatd as z r z u r vr. g r (16) caus ths systm s sotropc wth rspct to r 0, th locaton of th rfrnc on, th potntal th slf-nrgy ur ( ) bcom on-dmnsonal n th sphrcal coordnats. Th slf-nrgy functon u( r, r ') s symmtrcal wth rspct to th ln connctng r 0 r r'. Thrfor, Eq. (15) s rducd to two-dmnsonal by transfrrng nto th cylndrcal coordnats. Th Succssv Ovr-Rlaxaton (SOR) mthod s usd to solv Eq. (15).. Smulatons To compar wth th modfd P thors whch trat th solvnts mplctly, an mplct-solvnt aquous onc soluton s modlld n th smulatons. A cubc box wth a sd lngth of 0 nm s usd n th smulatons, 190 sngl-chargd on pars ar put n th box, rsultng n a dnsty of 0.4 M (smulatons wth othr concntratons rangng from 0.1 mol/l to.0 mol/l asymmtrc salts yld smlar rsults). Th prodc boundary condtons ar appld n all thr dmnsons. Watr molculs ar not xplctly prsntd n th smulaton stup, nstad th ffct of th aquous solvnt s mplctly modlld by rscalng th on chargs from ±1 to ±0.11. Th Partcl-Msh Ewald mthod s usd to dal wth lctrostatc ntractons. Th VDW ntractons btwn ons hav th sam forc-fld paramtrs ε = kj/mol σ = nm. Th smulaton runs for 10 ns at T = 330 K wth a tm stp of fs a samplng ntrval of 0. ps. Th smulaton s prformd wth th Gromacs MD smulaton packag [33]. Th RDFs obtand from th MD smulaton ar compard wth thos calculatd accordng to Eq. (15) for th orgnal P thory, th fld-thortc approach, our thory, rspctvly. Th countr-on co-on RDFs ar dnotd as g () r g () r, rspctvly. caus th postv ngatv chargs ar symmtrc, g () r s -- always th sam as g () r. Th dffrncs of th RDFs btwn th thors th MD smulaton g g gmd, ar shown n FIGs. 1. For th countr-on RDFs, both modfd thors mak clar corrctons to th P thory, whl our thory obvously ylds bttr rsults as dmonstratd by th fgurs. To fnd th solutons, on can start wth ur ( ) 0, thn solv Eqs. (1) (15) tratvly untl th solutons convrg. 3

4 FIG. 1. Dffrncs of countr-on RDFs btwn th thors th MD smulaton. Th orgnal countr-on RDFs ar shown n th nst. Although our thory prdcts mor accurat RDFs than th fld-thortc approach, thr ar stll small dvatons from th MD smulatons, whch may b attrbutd to thr factors. Th frst on s th man-fld approxmaton for th VDW ntractons. To tst f th non-lctrostatc ntracton plays an mportant rol, a bnchmark run s prformd by turnng off th on chargs comparng ths smpl approach wth th MD smulatons. Th rsults gvn n th FIG. 3 show that th dffrnc of RDFs btwn th thory th MD smulaton n th charg-fr cas s nglgbl compard wth th onc cas, so th non-lctrostatc ntracton s not a major sourc of rror. Th scond on s that th hghr ordr lctrostatc corrlaton ffcts ar not consdrd n our thory. Fnally, dffrnt approachs for calculatng th RDFs ar dffrnt: n th thortcal approach on on s fxd to captur th on dstrbutons, whras n th MD smulatons no ons ar fxd, so fluctuatons may attrbut to th dffrnc n RDF rsults. Snc t s not th goal of ths work to calculat RDFs vry accuratly, ths approxmatons ar accptabl should not nflunc our qualtatv conclusons. FIG.. Dffrncs of co-on RDFs btwn th thors th MD smulaton. Th orgnal co-on RDFs ar shown n th nst. It s ntrstng to not that th fld-thortc approach wth Eq. (4) lads to an vn wors co-on RDF than th orgnal P thory although th dvatons ar small. Ths mans that th DH-lk P-lk quatons for th Grn s functon lad to qualtatvly dffrnt rsults. In FIGs. 1, t s clar that th orgnal P thory undrstmats th countr-on RDF pak hght ovrstmats th co-on RDF pak hght. Howvr, for th symmtrc cas,.., z z, th DH-lk quaton of th approach has th slf-nrgs for postv ngatv ons dntcal accordng to Eqs. (3) (4), whch rsults n vn wors co-on RDF paks than th orgnal P thory. Not that th approach s basd on a varatonal mthod, a rfrnc acton whch s hard to dtrmn s rqurd to prform th drvatons. Th most commonly usd rfrnc acton has th Gaussan form, howvr th valdty of ths Gaussan assumpton s not asy to quantfy [34]. Thrfor, th problm of th DH-lk quaton may b attrbutd to th qustonabl assumpton of th Gaussan rfrnc acton. 4 FIG. 3. Th RDFs of a smpl VDW systm obtand by th thortcal approach th MD smulatons. In th cas of calculatng th RDF of a homognous systm, our modfd P thory only provds quanttatv corrctons to th orgnal P thory. Howvr, th concpt of slf-nrgy s mportant n many applcatons. For xampl, n th lctrc doubl layr problm, th mag charg du to dlctrc dscontnuty can b mathmatcally dscrbd by th slf-nrgy. Th mag charg ntracton can b domnant at th boundary of dlctrc dscontnuty, ladng to qualtatvly dffrnt rsults from th orgnal P quaton vn n th wak-couplng lmt [30, 3, 35]. IV. CONCLUSIONS In ths work, w ntroduc a slf-nrgy functon from a physcs pont of vw to obtan a novl modfd P thory by utlzng th Grn s functon. Th on corrlaton ffct s succssfully dscrbd by th slf-nrgy. Our modfd P thory s found to rduc to th rsults of th approach by applyng a lnar approxmaton. MD smulatons ar prformd to tst th valdty of th thors by comparng th RDFs of an mplct-solvnt onc systm.

5 oth thors mak clar corrctons to th orgnal P thory for th countr-on RDFs, our thory provds bttr rsults as xpctd. Not that th approach lads to an vn wors co-on RDF than th orgnal P thory, whch may b attrbutd to th ad hoc Gaussan assumpton for th rfrnc acton usd n th drvaton. Although w hav only tstd th valdty of our modfd P thory wth a two-spcs onc soluton, t s apparntly applcabl to onc systms wth multpl on spcs. Our thory s spcally usful for studyng systms wth fnt on dnsts, such as mult-valnc onc solutons, chargd collods on-dstrbutd bomolcular surfacs. ACKNOWLEDGMENTS Ths work was fundd by th Natonal Natural Scnc Foundaton of Chna (Nos ). Y. W. also thanks th fnancal support through th CAS ophyscs Intrdscplnary Innovaton Tam Projct (No ). Allocatons of computr tm from th Constanc Clustr of PNNL th HPC clustr of ITP-CAS ar gratfully acknowldgd. APPENDIX: DEVIATIONS FROM OUR THEORY TO THE APPROAH Y TAKING A LINEAR APPROXIMATION Startng from th P-lk quaton for th Grn s functon (Eq. (13) n th man txt): zgr, r ' z r ' G r r l z r, ' 4 [ ( ) 1 z 1 z r r ' ]. Th lnar approxmaton lads to: z r ' (A1) 1 z r ', ' +4 ( ) ( ), ' G r r l z r z r G r r + =4 l z r r '. Th fld-thortc () Grn s functon s G( r, r ') 4 l z z G( r, r ') 4 l ( r r '), (A) (A3) Not that th trms on th rght h sd of th abov two quatons, G ( r, r ') (n th P-lk quaton), should corrspond to z G( r, r ') (n th DH-lk quaton,.., th approach). Dfn th slf-nrgy: u ( r ') lm G ( r, r ') G ( r, r '), (A4) 0 rr' whr G ( r, r ') 0 s dfnd n Eq. (7) n th man txt, 5 1 l u z lm G( r, r '), r' r r r' (A5) Frst not that w prsnt all th physcal quantts as th functons of r r wth th lattr bng a constant rprsntng th locaton of th tst on. Snc G ( r, r ') G ( r, r ') 0 ar th potntals at r du to th on at r, th slf-nrgy at r can b obtand by takng th lmt r r'. Onc w fnsh calculatng th slf-nrgy for ach r, w obtan th slf-nrgy functon ur ( '), thn ur ( ') s rplacd by ur ( ) whn solvng th modfd P quaton. Th coffcnts n th dfntons of slf-nrgy ar dffrnt, th coffcnts bfor th slf-nrgy ar also dffrnt n th modfd P quatons. Th modfd P quatons n our thory n th thory ar z z u z z u 4 l z 0 z 0, (A6) z u z u 4 l z z, (A7) rspctvly, whr 0 ar th sam as dscrbd n th man txt rfrncs thrn. Now t s asy to prov th quaton st (A)(A4)(A6) of our nonlnar P thory s qual to th quaton st (A3) (A5)(A7) of th approach, xcpt for a factor of 1/ n th dfntons of th slf-nrgy. Th major dffrnc btwn our quatons th quatons s th valancs z, whch always appar as z z, for a symmtrc lctrolyt,.., z z, th slf-nrgs u u ar th sam. Howvr, ths rsult s nconsstnt wth th MD smulaton as dclard n th man txt. Morovr, such a lnarzaton s n fact r dvrgs as r mathmatcally problmatc snc gos to r, th lnarzaton s accptabl only whn r s clos to zro. *Corrspondng authors: Zhj.Xu@pnnl.gov, wangyt@tp.ac.cn. [1] J. N. Isralachvl, Intrmolcular Surfac Forcs (Acadmc Prss, London, 011), 3rd dn. [] R. Mssna, J. Phys.: Condns. Mattr 1, 18, (009). [3] P. Dby E. Huckl, Physk. Z. 4, 185 (193). [4] P. Grochowsk J. Trylska, opolymrs 89, 93 (008).

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