Strong coupling electrostatics for randomly charged surfaces

19 Pages·2015·1.1 MB·English

by  Malihe Ghodrat

Soft Matter
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Li
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ort Strong coupling electrostatics for randomly
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0 charged surfaces: antifragility and effective
AM. cial 3. Citethis:SoftMatter,2015,11,3441 interactions
9 er
1m
1:m
2:1Co Malihe Ghodrat,a Ali Naji,*a Haniyeh Komaie-Moghaddama and Rudolf Podgornikbc
1n
23 No
20n- We study the effective interaction mediated by strongly coupled Coulomb fluids between dielectric
3/3/utio surfacescarryingquenched,randommonopolarchargeswithequalmeanandvariance,bothwhenthe
d on Attrib Coulomb fluid consists only of mobile multivalent counterions and when it consists of an asymmetric
des ionic mixture containing multivalent and monovalent (salt) ions in equilibrium with an aqueous bulk
an
nlomo reservoir.Weanalyzetheconsequencesthatfollowfromtheinterplaybetweensurfacechargedisorder,
owom dielectric and salt image effects, and the strong electrostatic coupling that results from multivalent
DC
uary 2015. a Creative cbaroeetuwneateettrnrioatcnhtseedsounrtfoathwceeasr.ddIsinstartibhdueietiloescnutrrfoiacfcaeltlyshehsowemithioogneasneasoninudgsutshlyaesrt,eemffde,iswcotreidvesehr-oiniwntdetruhacacettdiothnepomprtueelnstsitvuiaarlleentththacetoyudnmitveeerdrioiganetess
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Article. PublisheThis article is lic psroaurnnerfdstaheocnemecssleyuirsocfahfrceaaermngsoedidvniestspedulrafrbayfaucsctiaienlmsdguaieseltnlievceacorlatmerlincpatahdrceiiogsdcuhonlwyntiettnirhnoioununi-tnsymi,foaotrrhnemeolstyotsiinlcnlihgcaaucrblcgaeuerhmdabvueoihlonaaretvesaid.osTramhoefufucinnchcotetmuironaoncterteoiroifonstnthrpoedrneeignsnsltyseuitrrcy-elsouaastrceftatinhctogee
s separationwitha prominentregime of attractionat small to intermediate separations. This attractionis
s
e
cc caused directly by the combined effects from charge disorder and strong coupling electrostatics of
A
n multivalentcounterions,whichdominatethesurface–surfacerepulsionduetothe(equal)meancharges
e
Op onthetwosurfacesandtheosmoticpressureofmonovalentionsresidingbetweenthem.Theseeffects
Received21stDecember2014 canbequitesignificantevenwithasmalldegreeofsurfacechargedisorderrelativetothemeansurface
Accepted23rdFebruary2015
charge. The strong coupling, disorder-induced attraction is typically much stronger than the van der
DOI:10.1039/c4sm02846e
Waals interaction between the surfaces, especially within a range of several nanometers for the inter-
www.rsc.org/softmatter surfaceseparation,wheresucheffectsarepredictedtobemostpronounced.
I. Introduction charge attractioninduced bymultivalent counterions between
charged surfaces of the same sign. The like-charge attraction
The counter-intuitive electrostatic effects produced by mobile phenomena stand at odds with the weak coupling (WC) para-
multivalent counterions in the vicinity of charged macromo- digmbasedtraditionallyonthemean-eldPoisson–Boltzmann
lecular surfaces (such as biopolymers like DNA, membranes, (PB) theory.9–13 They require the framework of the strong
colloids,nano-particlesandvirus-likenano-capsids)havebeen coupling(SC)theories1–7,14–19thatincorporatestrongion–surface
studied extensively within the counterion-only models and in correlations,totheleadingorder,andion–ioncorrelations, to
the case of non-disordered, homogeneous surface charge subleading orders, ubiquitous in the situation where electro-
distributions(see,e.g.,recentreviewsinref.1–8andreferences static interactions are strong; this is realized at low tempera-
therein).Theseeffectsinclude,mostnotably,theso-calledlike- tures,solventswithlowdielectricconstant,and/orwithhighly
chargedmacromolecularsurfacesbut,mostprominently,with
multivalent counterions. Usually, however, charged bio-so
aSchool of Physics, Institute for Research in Fundamental Sciences (IPM), Tehran systems with multivalent counterions include also a nite
19395-5531,Iran.E-mail:[email protected]
amountofmonovalentsaltionsthatcoupleweaklytocharged
bDepartmentofTheoreticalPhysics,J.StefanInstitute,SI-1000Ljubljana,Slovenia
surfaces. A few experimental examples of such systems are
cDepartmentofPhysics,FacultyofMathematicsandPhysics,UniversityofLjubljana,
furnishedbythecondensationofDNAbymultivalentcationsin
SI-1000Ljubljana,Slovenia
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the bulk20–26 or in viruses and virus-like nano-capsids,27–29 and In the most realistic model of the interaction between
formation of large aggregates (or bundles) of highly charged chargedmacromolecules,onewouldthusneedtoconsidernot
biopolymers such as DNA,30,31 F-actin32 and microtubules.33 In only an asymmetric conned Coulomb uid between macro-
these situations, one deals with a difficult problem in which molecular surfaces, with a mixture of mono- and multivalent
e.
c neithertheWCnortheSClimitinglawscanbeappliedwithout salt ions, but also surface charges that exhibit a disordered
n
Lice reservations. For such asymmetric Coulomb uids, a general- component.Whileoffhandthisseemstobelikeaformidable
d izeddressedmultivalent-ionapproach,thatbridgesthetwolimits task,detailedcalculationshaveshownthatthenalresultsare
e
ort in one single theoretical framework, has been introduced7,34 rather intuitive and revealing.56–64 First of all, it became clear
p
n
U and tested against extensive explicit- and implicit-ion that, for quenched charge disorder, which is of interest in this
AM. cial 3.0 simIunlactoiounnst.e7r,3i4o–3n8-only systems and in the case of uniformly psoalpuetrio,nansdubinphthaseeaabnsdentcheeomfadcireolemctorlieccauslaymrsmureftarcyeb,ethtweereenistnhoe
12:11:19 nCommer occohfuatphrgleienedglespcuatrrrfaoamscteeastteiwrciXitnh¼tesurqar2cf‘atic/omenc¼shi2asrpgdqee3s‘dcer2nisbs.e1i4tdyItb(cid:1)iyssteth0h,eetehrlaeetcsitotrrooesnftagthttihec SdCinsdolsrimdaeilrto-niangnd-druacinengdetdheff,edesicsatomirndeetgrh-eeinnWedruaCclerrdeeggaiitmmtreae.c5t6oi,5fo8npH,aworawhmeicvehetrem,rsiun,sottnhbeee
2023 n-No rescaled Bjerrum length,Bq2lB with ‘BB ¼ e02/(4p330kBT), that addedtotherepulsiveforcebetweenequallychargedsurfaces.56
3/3/utio measures the strength of electrostatic interactions between It has also been shown that in the linear WC regime with
ded on s Attrib lceonugntthe,rimon¼s1o/(f2pchlBaqrsg)e, tvhaaltenmceyasqu,reasndthethsetreGnoguthy–oCfhealpecmtraon- Gquaeunsscihaenddeilsdo-rduecrtuwaotiuolndslaeraodutnodrtehpeumlsievaeno-realdttrsaocltuitvieonin,ttehre-
oaon staticinteractionbetweenacounterionandthesurfacecharge actions provided that the system has an inhomogeneous
nlm
wm inunitsofthermalenergyk T(conventionally,wetakes>0and dielectricconstantormonovalentsaltdistribution(inthiscase,
oo B
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Article. PublisheThis article is lic cqaau2naDSsbuice-btrswgyeseoqtn-audelei.rm1na–l7tei,1zsn4e–ts1udi6od,1t9nioeasml,aoslrstreoocnroegmvleypacllioecdrartetehldaattaednthdleiqrWeuaiCldi–sltSaiCcyespri,atuoraardteiiovgenmns ssenuttoaerttrfiisacs,cetieaXcbacllayhndaldierssgtscrei,bdvuaraartreisiotainvnce.c5rm6ey–5a8wsniOimntbheivrilianooruntbshluteyht,etGthhvaeaeultselwsnaiotcatynecroouAfdpnetlshpianeetngzcdopfsuoanrriantmeitras--
s usually encountered in an experimental context, with an ions.Justastheoverallfeaturesofthebehaviorofasystemwith
ces asymmetricCoulombuidcomprisingamixtureofmono-and homogeneous charge distributions depend on the value of X,
c
n A multivalent ions in the vicinity of charged macromolecular the behavior of a disordered system depends in a very funda-
pe interfaces. Neither the WC nor the SC paradigm can be mentalmannernotonlyontheexactvalueofc,butalsoonthe
O
consistentlyappliedtothis situationwherethesystemiselec- presence of salt ions and dielectric inhomogeneities (“image
trostaticallypartstronglyandpartweaklycoupled.Inthiscase charges”)inthesystem.59–64
the dressed multivalent-ion theory, based on piecewise applica- Here,westudytheeffectiveinteractionmediatedbystrongly
tionoftheWCandSCformalism,hasbeenshowntopossessa coupled Coulomb uids between two plane-parallel dielectric
robust regime of validity.7,34–38 Other cases demanding an surfaces carrying quenched, random monopolar charges with
extension and/or modication of the WC–SC framework have equal mean and variance, both when the Coulomb uid
beenrecentlyreviewedinref.7. consists only of mobile multivalent counterions and when it
Aseparateissue,ofthenatureofthechargedistributionon consists of an asymmetric ionic mixture. We analyze the
macromolecularsurfaces,hasalsoreceivedprogressivelymore consequences that follow from the interplay between surface
focused attention. In many cases, these surfaces are not only chargedisorder,dielectricandsaltimageeffects,andthestrong
inhomogeneous in terms of their charge conguration, but electrostaticcouplingthatresultfrommultivalentcounterions
exhibitafundamentallydisordereddistributionofcharges2,39–72 on the distribution of these ions and the effective interaction
dictatedeitherbythemethodofsamplepreparationand/orby pressurethey mediatebetweenthesurfaces. Inadielectrically
the inherent structural properties of the surface materials.39–48 homogeneoussystem,themultivalentcounterionsarefoundto
These disordered systems, with either thermalized, annealed be attracted towards the surfaces with a singular, disorder-
surfacecharges50,51,59,60,66–71orstructurallydisordered,quenched inducedpotentialthatdivergeslogarithmicallyonapproachto
surface charges2,47–67 (or even partially annealed or partially the surfaces, creating a singular but integrable counterion
quenchedones58,72)havereceivedamorerigorousattention,as density prole that exhibits an algebraic divergence at the
it is becoming clear that the nature of charge disorder can surfaces with an exponent given by the disorder coupling
interfere fundamentally with the behavior of the system parameter.Thiseffectdrivesthesystemtowardsastateoflower
approximatedwithxed,homogeneoussurfacecharges. thermal‘disorder’,onethatcanbedescribedbyarenormalized
effective temperature, exhibiting thus a remarkable
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antifragility73 reported in a previous work by the present chargedistributionr(r),characterizedbyaGaussianprobability
authors.64Wethusndthat,inthisgeneralcontextofstrongly weight
ð
coupled,disorderedsystems,anti-fragilebehaviorisubiquitous ​ (cid:2) (cid:3)
e. and stems from the interplay of structural and thermal P½r(cid:4):¼Ce(cid:1)12 drg(cid:1)1ðrÞ rðrÞ(cid:1)r0ðrÞ 2: (1)
c disorder,withtheformerengenderingadecreaseinthetrans-
n
e
Lic lationalentropyofmultivalentcounterionsasaconsequenceof
d the fact that the disordered charge distribution generates a Here, C is a normalization factor and r (r) and g(r) are the
e 0
ort nite degree of (non-thermal) congurational entropy. In the mean and the variance of the disordered charge distribution,
p
Un presenceofaninterfacialdielectricdiscontinuity,thesingular respectively.Forthespecicmodelconsideredhere,weassume
0
AM. cial 3. bbeuhtamviuolrtiovafltehnetccoouunntteerriioonnsdaernesisttyilaltacthcuemsuurlfaatceedsmisurcehmmovoerde tahreatstthateiscthicaarlglyeiddiesntrtiibcualtiaonndsognivethnebiynnersurfacesoftheslabs
19 mer stronglyclosetorandomlychargedsurfacesascomparedwith
2:11:Com uniformlychargedones.Theinteractionpressurebetweenthe r0(r)¼(cid:1)se0[d(z+d/2)+d(z(cid:1)d/2)], (2)
1n surfacesdisplaysingeneralahighlynon-monotonicbehavioras
2023 n-No a function of the inter-surface separation with a prominent g(r)¼ge02[d(z+d/2)+d(z(cid:1)d/2)], (3)
oaded on 3/3/ons Attributio ciraoehttngarisarm,gcweteiohdonificshaoitsrtddroaecmcartuiiaosnnenaddtaeStdtChsirmeeecalsetulllcyrttfroaobcsiyent–attstehiurcemrsfaecocodfemiamrtbeeupinlsuteeilvdspaiaolreenaffntdetiocucnteossut.ofnTrtothehmires- w(innhiegtreeend,eiwsraoitlrh,dtehnreovsaulrorisafasnccoeesf,mwgehaniyceahralswloiteyh,sahwvaeellntcoaoknneestigdceh$rarin0gedabneutdatiolsne>llsye0a-
nlm where74).Wedevelopthemainformalismforanarbitraryshape
wm (equal) mean charges on the two surfaces and the osmotic
Article. Published on 23 February 2015. DoThis article is licensed under a Creative Co stpecmSdtchhuhofferisoeeecnraeoTfsdtrsccairshgeeitoocdudsefflenenereaeidoercrnpIcreleriIeoutrsdgIdsSeof.saqsecirminTuosocndoihuintfoeuztiSerareonneasetno(tdsanicveiovilitdsagrItsianIiorntlosdirenioaaitnioinubfnnsn-ItcdcutocdVshausineteniaesB)plodalyst.rprtnnieeeocaTiidslnvoaopnaheensfetnenSediisrnmvetibiweirecssoeeuStlitehiuseatlsoothicctsaridnuttvtvrfaiithsdaiooinhocleislnIgeeerlmVonmodibVownrtaCefe.eihlsnctat–ltw:ooihnIdcwSVmueeaeeseenlgocuEentirtgfffre,nieotfeorefteharinorncmoceooetlenmlidIifaovtVsculeisw.iehucisTAesinaemrnhdfraotateganehbucsirerndeeeeyr.- actsmoedsrchellaoxfiaimpeueasmttbTlermlheuts(cid:1)opchi(eclpdhvpoettdaaicrluars/veirlha2ercsceaegcdaelonlatce#erfitoabotgnchtsfvncseathszaaeodlsuqellautmye#msabr:,tnntenwoo1matectdeunnodyasru/3noda2sipmloovdaslt.tltaonifuraacvTdlomsrnoeamhibfnlnaeaceeieburstrtsodna,eul1usttiipwllinmnat:kcavaohsitoa1brnpescilutsusrcoeegbinicahnrnmdeouamlttcetnewnleeisaore-cdmenociraafoorntbsbttmunircohylbluanoysemeeetlbntirkcndreomesstaercoentanasicreodoupbaotcnbmnlrpio0ntcniieslvscnowey.t5epnbn6Ci.tTretaiteoeoahlrnhdevasutaseeqthlowbianiooeo>ldlimtuentnleis0sihtedbsmpinlio.beune0fincecrstaTtuotiiinrnnhhmiotiddoggeeccf
ss entering the outer regions, |z| > d/2 + b, by enclosing these
e
Acc II. The model regions in semi-permeable membranes). This assumption has
en noimpactonourresultsinalargepartoftheparameterspace,
Op We consider two plane-parallel dielectric slabs of innite e.g.,forslabthicknesseslargerthantheDebyescreeninglength,
surface area S, nite thickness b and dielectric constant 3p, andespeciallyforsemi-inniteslabsthatwillbeofmaininterest
placed perpendicular to the z axis with the inner bounding
inthiswork.
surfaces separated by a distance d (see Fig. 1). The outer
boundingsurfacesoftheslabsareassumedtobeneutral,while
III. The formalism
theinneronescarryaquenched,spatiallyuncorrelated,random
A. Dressedmultivalent-iontheory
The problem of an asymmetric Coulomb uid containing
monovalent and multivalent ions next to charged macromo-
lecularsurfacesisacomplicatedone,mainlybecausedifferent
components of the Coulomb uid couple differently to the
surface charges: multivalent counterions tend to couple
strongly, and therefore generate non-mean-eld effects,
whereasmonovalentanionsandcationscoupleweakly,andare
thus expected to follow the standard mean-eld paradigms as
they would in the absence of multivalent ions.9–13 Strong
Fig.1 Schematicviewoftwoinfinite,plane-paralleldielectricslabsof coupling effects due to multivalent counterions within the
thickness b and dielectric constant 3p in a bathing ionic solution of counterion-onlymodelscanbedescribedwellusingtherecent
dielectricconstant 3 , containinga monovalent and multivalent salt
m SC theories that have been studied thoroughly over the last
mixture(withmultivalentcounterionsshownbylargebluespheresand
several years (see, e.g., ref. 1–8 and references therein). In the
monovalentsaltanionsandcationsshownbysmallorangeandblue
spheres).Themultivalentcounterionsareconfinedintheslitbetween caseofanasymmetricCoulombuid,althoughitisnotapriori
therandomlychargedinnersurfacesoftheslabs. clear if a single approximation can be introduced in order to
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describethehybridnatureoftheelectrostaticcouplingsinthe function can be virial-expanded in terms of the counterion
presence of charged boundaries, it turns out that a simple fugacity,l ,as
c
generalizationofthecounterion-onlySCtheory,5–7,14–18dubbed
dressedmultivalent-iontheory,providesaverygoodapproxima- Z[r]¼Z0[r]+lcZ1[r]+O(lc2), (7)
e.
c tioninalargepartoftheparameterspaceasdiscussedindetail
d Licen tinherepfa.r7titainodn3fu4n–3c8ti.oTnhoisvearptphreoadcehgrieseosbotafifnreededboymrasstsotrcaiacitnedg wleintht-itohnethresotrtywoonterthmeslceoardreinsgpoonrddienrg. tTohtehezedrroetshs-eodrdmerultteirvma-
orte withmonovalentionswithinalinearizationapproximationand corresponds to the electrostatic interaction of xed charged
p
n objectsintheabsenceofmultivalentcounterions,
U then virial expanding it with respect to the fugacity of multi-
19 AM. mercial 3.0 viscuaalffiellnyc3ti4ecnootnulylnytleianrrigotehnesc.ocBuaonstetherotifohnhesigevhasllteyenpacssyycmqanmanbedterijpcursmotviiixdeteuddrseysthstwaetimthtahtae- Z0(cid:2)r(cid:3)¼e(cid:1)12lndetG(cid:1) b2 ðdrdr0rðrÞGðr;r0Þrðr0Þ; (8)
12:11:nCom bsmulakllc(aosncisenotraetniotnheocfasmeuinltievxapleenritmceonutanltesryisotnems siscornetlaaitnivienlgy adnudettohemurltsitv-oalrednetrctoeurmntegriivoenssthe single-particle contribution
3/3/2023 ution-No iaosnymicmspeetcriicesioinnitchmesixotluurteiosn20–c3a3)n,nboectabuestereoattheedrwonisesuthcheddiiffffeerreenntt Z1[r]¼Z0[r]ÐdrU(r)e(cid:1)bu(r;[r]).
ded on s Attrib lreevveielsw.edThreecednrtelys7sebdridmgeuslttihvaelegnapt-ibonetwteheenortyhedweesacrkibceoduplainndg Here,u(r;[r])isthesingle-particleinteractionenergythathas
oaon DH theory and the counterion-only SC theory and reproduces
nlm theform
uary 2015. Dowa Creative Com tphaermamaestetwrso,rsepsepceicatlilviemlyi.tingcasesatlargeandsmallscreening where Gium(cid:4)(rr;,½rr(cid:4))(cid:5)¼isqteh0eðgder0nGeðrra;lriz0Þerdðr0BÞoþrnq22een02eGrgimyðrc;ornÞ;tributio(n9),
Article. Published on 23 FebrThis article is licensed under atBIprinesa.oarnflttoiaiGhztllalie-eooitnnwinodetsnefrr:uaeg36lsnroasffcelotdeifrooxmZntrmem(cid:2)ororunff(cid:3)(las¼tatlheinveeeca,a(cid:1)hliesena12.yngrltmneg.t-rerimdaeosecfnet.trtGi7(rorti5nð)ch–eDcC7foar9ofernuyaee,(cid:1)lnboebdenStmh½ferwbe;errg(cid:4)fr:eiygtrurteeaindnnc,dwe-isicntathhnaaeorfneuixincn(e4acd))l-, wziGibdnnaee0hdt(eiiIricoqcnn,vahnnierltd)dch(,eiuu7seffwta)lhaegcliwtrtcaecheoetsnosdrueueGeg(rono0fsharf(thrtroece(cid:1),otrmodhwri3uo)e0npnn3rt“hemutstiepoerm(irVerngielaso2eysacgnne(cid:1)rebh-eneoyorcetknaihnm2tlnt)hyaeeGgordesagg0ty(eedhsfrnsrti,eneee”erim)elfo0t,eo)-euscisr¼.pctm,seroat.ib,dhancca(tGeetricaroitikmn(cid:1)Gbwngd(urorrr(e/to,s0oeil)ueroe.rn)lannf's¼ds-da)tieo;lnfGtntugth(ehpnrirte,ocsegtlrrfycai)rmoareo(cid:1)nniesf-
ss energy,constitutingtheSCtheory.5–7,14–18Thehigher-orderterms
e
cc contain subleading contributions from multi-particle interac-
A
pen Here,b¼1/(kBT),f(r)istheuctuatingelectrostaticpoten- tions that become important only in the crossover regime
O tialandS[f,r]istheeffective“eld-action”,i.e., betweentheWCandSCregimes.Inthemoregeneralcontextof
(cid:2) (cid:3) 1 ð ð thedressedmultivalent-iontheory,theleading-ordervirialterms
S f;r ¼ drdr0fðrÞG(cid:1)1ðr;r0Þfðr0Þþi drrðrÞfðrÞ cangenerateboththeSCandWCbehaviorsofthesysteminthe
2
ð limits of small and large screening parameters, respectively.34
(cid:1)kBTlc drUcðrÞe(cid:1)ibqe0f; (5) The predictions of this theory were analyzed for uniformly
chargedsurfaces(andalsoforstrictlyneutralsurfaces)andwere
wherel ¼c isthefugacity(orbulkconcentration)ofmultivalent comparedwithextensivenumericalsimulationselsewhere.7,34–38
c 0
counterions, U(r) is an indicator function that determines the Inthepresentstudy,thechargedistributionofthedielectric
c
regionofspacethatisaccessibletomultivalentcounterions(e.g., surfaces,r(r),hasaquenched,Gaussiandisorderedcomponent
inthepresentmodel,wehaveU(r)¼(q(z+d/2)(cid:1)q(z(cid:1)d/2)),and withanitevariancearoundthemeanchargedensityr (r).One
c 0
G(cid:1)1(r,r0)istheoperatorinverseofGreen'sfunctionG(r,r0)that,in thereforeneedstoaveragethethermodynamicquantities over
thedressedmultivalent-iontheory,satisestheDHequation differentrealizationsofthechargedisorderaswell.Hence,for
instance,thefreeenergyfollowsfrom
(cid:1)30V$3(r)VG(r,r0)+303(r)k2(r)G(r,r0)¼d(r(cid:1)r0). (6)
bF ¼(cid:1)hhlnZ[r]ii. (10)
Here,k(r)istheDebye(orsalt)screeningparameterwhichis Ð
non-zero only outside the region occupied by the dielectric Thedisorderaverageisgivenbyhh.ii¼ Dr(.)P[r]andthe
slabs, i.e., k2 ¼ 4pl n , where n ¼ 2n + qc is the bulk Gaussianweightbyeqn(1).TheaveragedquantityhhlnZ[r]iican
B b b 0 0
concentrationduetoallmonovalentions.Therefore,theroleof becalculated in general by employing the Edwards–Anderson's
monovalent ions is incorporated into the Green's function on replicaAnsatz.56Sinceweareinterestedonlyintheleadingvirial
theDHlevel. terms,wecanalsodirectlyaveragetheleading-orderfreeenergy
Asnotedbefore,inthepresenceofmultivalentcounterions termsoverthequenchedchargedisorderweight,yielding
with sufficiently small bulk concentration, the partition
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(cid:6)(cid:6) (cid:7)(cid:7)
Z (cid:4) (cid:5) slabs (Section2),inwhichcasethe Green'sfunctionG(r, r0)¼
bF ¼(cid:1)hhlnZ0ii(cid:1)lc Z1 þO lc2 : (11) G(r,r0;z,z0),withr¼(x,y)andr0¼(x0,y0)beingthetransverse
0
(in-plane) coordinates, is onlya functionof |r (cid:1) r0|. Thus,its
Fourier-BesseltransformGˆ(Q;z,z0)canbedenedby
e. The averages in the above expression can be carried out ð
d Licenc smtrualitgivhatlfeonrwt-iaorndlfyreaenednewregyonfdthtehseys(tgermanads-canonical) dressed Gðr;r0Þ¼ 0NQ2dpQG^ðQ;z;z0ÞJ0ðQ|r(cid:1)r0|Þ: (15)
e ð
Unport bF ¼b2 drdr0 r0ðrÞGðr;r0Þr0ðr0Þþb2Tr½gðrÞGðr;r0Þ(cid:4) Usingstandardmethods,wend
0 ð (cid:8)
9 AM. ercial 3. (cid:1)lc drUðrÞe(cid:1)buðrÞ; (12) G^ðQ;z;z0Þ¼23013mg e(cid:1)g|z(cid:1)z0|þ1(cid:1)2ee(cid:1)(cid:1)2g2dgdYYðQ2ðbQÞbÞ (cid:9)
2:11:1Comm wthheeraebtsheencerstoafndmuthlteivsaelceonntdctoeurmntseraiorenst,herecpornetsreibnutitniognsthine (cid:5)(cid:4)egd coshgðzþz0ÞþYðQbÞcoshgðz(cid:1)z0Þ(cid:5) ; (16)
1n
23 No interaction free energies due to the mean surface charge
ded on 3/3/20s Attribution- dsiwneeedcnaousknciltydeyd,ctrofeo0ur(rmprc)le,esdaanrbiCdestoewtsuheleoeamnldsiobdsoisirnodureidtdrhe;v5re7ea,5dra9i–na6s3naulcriytefsa,igcsgiev(sreo)sf,inratehvsnaepcoeuncu-tuvimvcaetnuloiyasr.thiiToninhngae- whereg2¼k2+Q2,YðQbÞ¼D1s(cid:1)(cid:4)1D(cid:1)s2ee(cid:1)(cid:1)22QQbb(cid:5); (17)
an
nlomo contribution only in inhomogeneous systems with a nite and
wm
Article. Published on 23 February 2015. DoThis article is licensed under a Creative Co sdstoeipninreamWglttehilceaeert-leprllnypeiacoarrietdtneusdiicheðnitlrsnoehgÞcmtao(si¼sntnotittnhtgqeh(cid:1)ieegerne0nalucebðceo-itoptqdrinyou2asrt2et0nrsrG0atit2ibtedðecðrunrli;detsmtehr)tirr0or0eÞlggiienrbynðv06burfeð40reortlÞoi,0q(cid:2)uÞomiGnnnnþðd(wmr1oqi;nh32fru2eg0)iÞl0cst(cid:3)o2hai2svrGl:utaiugilrm(ieirfonða)nrncait;setsrces.ÞtoshTueafhnrneoetffdem/ertocih(ortt1iihnrv3daees) kG^nðoQFwo;nrz;seze0xÞmp¼rie-is2þns3Di0o13nsnmciotges(cid:8)hsel(cid:1)gagDðb|zzss(cid:1)(cid:1)z¼(0|bzþ330/Þmm(cid:5)1gg(cid:9)(cid:1)2:(cid:1)þNDDs33)e,spp2(cid:1)QQoe2(cid:1)gn:d2egdc(cid:5)an(cid:4)ergedcocoveshr tghðezþw((11ezl890lÞ))-
ss interaction of multivalent counterions with the mean surface Note that the information about the image charge effects,
e
c
Ac charge density, the second term from the self-interactions of which result from the inhomogeneous distribution of the
en individualcounterions(withtheirownimagecharges)andthe dielectric constant or the bathing salt solution in the system,
p
O third term is due to the presence of surface charge disorder. enters here through the parameter D. In the absence of salt
s
Thistermisproportionaltothedisordervarianceandshowsan ions (k ¼ 0), D reduces to the bare dielectric discontinuity
s
explicit temperature dependence and a quadratic dependence parameter
oq;nitthaeriGserseefrno'smfuthnectsioamnapnled-ttoh-seammupllteivaulecntutaiotinonchsa(orgrevavarilaennccey), D¼33mm(cid:1)þ33pp: (20)
ofthesingle-particleinteractionenergy(9).64
In the SC limit or within the multivalent dressed-ion
theory,14,16,34thedensityproleofmultivalentcounterionscan Inthecasestreatedbelow,relevantforaqueoussolventsin
be derived solely in terms of the effective single-particle inter- the presence of bounding slabs of low dielectric constant, we
delimit ourselves to D $ 0, which gives repulsive image
actionenergyas
interactions.
c(r)¼lcU(r)e(cid:1)bu(r). (14) The effective single-particle interaction energy in the two-
slabsystemcanbewrittenusingeqn(13)as
Thisconcludestherecapitulationofthemultivalentdressed- uðzÞ¼(cid:1)qse02(cid:2)G^ð0;z;(cid:1)d=2ÞþG^ð0;z;d=2Þ(cid:3)
ion theory that forms the basic framework of our analysis of ð
electrostatic coupling and quenched charge disorder in what þq2e02 NQdQG^ðQ;z;zÞ
4p
follows. 0 ð
(cid:1) bq2ge04 NQdQ(cid:2)G^2ðQ;z;(cid:1)d=2ÞþG^2ðQ;z;d=2Þ(cid:3):
4p
C. Speciccaseofplanardielectricslabs 0
(21)
Thefreeenergyexpression(12)isvalidregardlessoftheshape
of the boundaries. In the rest of this paper, however, we shall
delimitourselvestothespecicexampleoftwoplanardielectric
Thisjournalis©TheRoyalSocietyofChemistry2015 SoftMatter,2015,11,3441–3459 | 3445
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Thedensityproleofcounterionsintheslitregion,(cid:1)d/2#z N
#d/2,followsas(seeeqn(14)) lc ¼ ðd=2 : (26)
S dze(cid:1)buðzÞ
c(z)¼lce(cid:1)bu(z), (22) (cid:1)d=2
e.
c
n
ce andthegrand-canonicalfreeenergy(perkBTandperunitarea,S) TheGreen'sfunctionofthecounterion-onlymodelreduces
d Li canbewrittenas tothe bare Coulombinteraction, G0(r, r0)¼ 1/(4p303m|r (cid:1) r0|),
Unporte bSF ¼4p‘kBs2cothðkd=2Þþg‘Bfðk;d;DÞ(cid:1)lcðd=2 dze(cid:1)buðzÞ: iarnredletvhaenteffaedcdtiivtievesincoglnes-ptaanrtticalendintinerarcetsiocanleedneurgnyits()upfotlolowans
0 (cid:1)d=2 straightforwardlyfromeqn(13)or(21)as
9 AM. ercial 3. (23) bu(cid:4)z~(cid:5)¼d~þcln(cid:10)d~2(cid:1)z~2(cid:11): (27)
1m 2 4
2:11:Com In the above expression, the different contributing terms
1n appearinthesameorderasineqn(12)andwehave
ded on 3/3/2023 s Attribution-No wherewehavefoðmk;idt;teDdÞhirrðe0NlevQadnQtagdD(cid:4)desð2i1dtgivþ(cid:1)eDtDesÞrs22m(cid:5);sthatarein(2d4e)- gisnuedorNfimavoicedteteuryatclhhoaafctrogtuwhenesh,tiemlerti,hoodebneeslcc,aofturuhosnemetreeoriftiohsntehnseo(oespnxyppemtoemrsaiiettetentrlrcayiecctsipoiaglnnanneaaedc-t)tptiranamrgcaetlioalvennel
oaon pendentofthesurfaceseparation,d.Theinteractionfreeenergy potentialduetothequenchedsurfacechargedisorder,whichis
nlm
wm followsfromeqn(23)bysubtractingallsuchtermsor,inother givenbythesecondtermineqn(27).
oo
3 February 2015. Dunder a Creative C DIbwtyo.isrrdecDsso,icnmatvhleeiennngrsieeionftehntrelteeonssmpcsearatefkirpaeerleeucsseoeenoneortrdfagatiyniodaonitfmetsehnwesisitoyhsnttleehmsesfGsoeortudoyf–/qCuhaNanp.tmitiaens obtTahineedrebcs~y(cid:4)cza~u(cid:5)leshdin2gdpceeð‘zn~qBÞssni2t(y¼22p(cid:10))raod2~s(cid:11)leC0o(cid:1)f1ðccoÞu(cid:10)n14t(cid:1)eridoz~~2n2(cid:11)s(cid:1)icn=2;the sli(t28is)
ss Article. Published on 2 This article is licensed lap8oecpfancrmqgoa2trmu‘hdBleitcmnit0ve,g¼arwllyeh1,an/ien(ct.2dghcp.o,iaqdstu~hnlrnB¼eestfaei)ednrnra/itmraoseel,nr~ord~k-sgs¼t,uo¼oiur.raef/ksa.smlm,cy.t,ehOc~dsetceehrp¼eeansrrccepaacdtalmiero.dlanembn,utgehlttkheercssosaacnltarcelseecrn,retesrcecacnat2(iil2one5¼ndg) wherewehavedeC0nðceÞd¼2(cid:1)1þcpffipffiffiffi GG(cid:10)(cid:10)132(cid:1)(cid:1)c2c2(cid:11)(cid:11): (29)
ce In the no-disordered case (c ¼ 0), the above expression
n Ac The rescaling of the Bjerrum length leads to the dimen- reducestothestandardSCdensityprole~c(~z)¼2/d~.3,5–7,14–16As
pe sionless electrostatic coupling parameter X ¼ q2‘ /m ¼ showninFig.2,thedensityproleofmultivalentcounterionsis
O B
2pq3‘ 2s, associated with the mean surface charge density,14 strongly modied by the surface charge disorder and exhibits
B
while the surface charge variance appears in the dimen- algebraic singularities close to the randomly charged bound-
sionless disorder coupling (or disorder strength) parameter, aries at z ¼ (cid:6)d/2 for any nite value of the disorder coupling
c ¼ 2pq2‘ 2g.56 The rescaling of the density prole and the parameter, in clear violation of the contact-value theorem
B
interactionpressurewillbediscussedlater. established for uniformly charged surfaces.80–83 This kind of
behaviorwasdiscussedinourrecentworkonthedistributionof
IV. Results multivalent counterions next to a single, randomly charged
interface64 and we shall not delve further into the details,
A. Distributionofcounterions:thecounterion-onlymodel
makingonlyafewremarksinwhatfollows.
Letusrstconsiderthedistributionofmultivalentcounterions Thesingularbehaviorofthecounteriondensityprolecomes
inthespecialcaseofthecounterion-onlymodel,wherethesalt directly from the logarithmic disorder term in the single-particle
screeningandimagechargeeffectsaresettozerobyassuminga energy in eqn (27), which drives the counterions towards the
homogeneoussystemwithk¼0andD¼0,orequivalently,3 ¼ surfaces. The density of counterions away from the surfaces
p
3 (this case was considered in a previous work,56 which decreasesasthedisordercouplingparameterincreases(Fig.2)and,
m
however focused only on the effective interaction between the eventually,ittendstozero~c(~z)/0forany|~z|<d~/2whenc/2(cid:1)
surfaces and did not investigate the distribution of counter- sinceC (c)/N.Counterionsarethusdenselyaccumulatedinthe
0
ions). The two impermeable, randomly charged surfaces are immediatevicinityofthe surfaces.Thisbehavior must be distin-
placedataseparationdistance,d,conningintheslitbetween guishedfromthesurfaceadsorptionorcounterioncondensation
them a xed number of multivalent counterions, N, which is phenomena7as,inthepresentcontext,onecanshowsystematically
givenbythemeanchargeonthetwosurfacesthroughtheglobal that the mean surface charge is not renormalized by the surface
electroneutrality condition, Nq ¼ 2Ss. Hence, the fugacity of accumulationofcounterions(seeAppendixCinref.58).
counterionsisgivenby14,56
3446 | SoftMatter,2015,11,3441–3459 Thisjournalis©TheRoyalSocietyofChemistry2015
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Another notable point is that the renormalization of the
repulsiveentropictermbyafactor1(cid:1)ccanbeinterpretedalso
asarenormalizationoftheeffectivetemperatureofthesystem
to a lower value. However, this interpretation should be
ce. considered with caution because rst, this particular form of
n
e
Lic the renormalization of the repulsive pressure is found only in
d the SC limit of the two-surface counterion-only model and,
e
ort secondly, one can show, by inspecting the thermodynamic
Unp quantitiesofthesystem,thattherenormalizingterm,(cid:1)2c/d~,in
0
9 AM. ercial 3. Fciogu.n2terRioensc-aolneldydmeondseitlyapsraoffiulensctoiofnmouflttihvealreenstccaoleudnnteorrimonaslpwoitshitiinonth,e~z eeqnntro(p3i0c)coinndtreiebdutcioonmse.s from both electrostatic energy and
1:1mm ¼ z/m, between two randomly charged surfaces located at inter-
2:1Co surface separation d~ ¼ d/m ¼ 4. Different curves correspond to B. Distributionofcounterions:saltscreeningandimage
23 1Non differentvaluesofthedisordercouplingparameterasshownonthe chargeeffects
20n- graph.
3/3/utio Inthenextstep,weassumethat,inadditiontothemultivalent
ded on s Attrib The accumulation of counterions in the vicinity of the csaolutnotferbiuolnksctohnacteanrteraitniotnrocd0u,ctehdeitohnroicugmhixatunreasaylmsomceotnritcaiqns:1a
ownloaommon csthaatregeodf lboowuenrdtahreiremsafulr‘tdhiseormrdoerre’,dsirnivcees, tahseosnyestceamn tsohwoawr,dtshae mDeobnyoevsaclreenetnsianlgtpoafrbaumlketceorniscekn¼tr(a4tpiolnBnnb0).1/A2swnitohtendb¼be2fno0re+,qthc0e.
Article. Published on 23 February 2015. DThis article is licensed under a Creative C let‘o(tcgqtoasrhhoxenuaetrteennteedneereesnrsemraearnlrcraasqtneeahayttfutdllciepe.ioe’o-sde(fnnd5msqrodaat6aciuasflpigeesoteiafeenolronnri,ondrcterttydfwhierrdt7eoortehe3htrppwdih.eycyoa()iTenhtnfsohdhcotdfiadeiintsstsnhmmih-osatdboeerohsulrnedsreilfhcsoidetvrrcoyiaeromevaasrvsmraltrsieateleoohedlfmsttr)urnhhepewr,rteecocecmoahncedinsoneasdoidffyunradisggnisneewytmuesgrrntehrogeemiaanncryindmctiiotiiioahoessnrinttthxnceeo-rspadaaiediwnlbrlslidlieasituyznerosertatochdmsnrirtrdiodetsieoeroonuefaonrni.rc,aspnee6iseyntdo4wssor;mfgtihfwcanatattoihbsisahhctsrliihnhaeeeeecs Ftio((owstn21rhuffoii26etcrrmmhI))fan–nadtalui(huaclnet1tyleeelhnt9dptsihi(es)vetvbb.(oiiaasaatmym¼lykelbeemeesnopanNettgtarbthnoek)caeeccnoiaienosnuebnlmeugoognldsan,ufuatcietssawlmskelru,reeriedipsroio.fooieeanarfsln.rlesc,titsmdsaseho3ccnpeeaccsacaauhnh¼alesberaposbdvlfr3pwuereegmromswteiocmonaiapdeatnalhrFicetlsdiniluhaaogtoitlcatrh.oaehnhdt3enta-eeeatmsotxrhbd;.f,ptueonThftrtnlrhohhesekootseeyeemsstsvddotlitaaaoweeenellbnmnnsuoqisocssesnbiiafitttbocrsh(yryo02euea)dpp1hmn.tsrri)adeehTooavelmiehienqnorllcinidgeeesr--
ccess disoTrhdeerreedsccaoluednteinritoenra-ocntiloynmpordeesls)u.re acting on each of the bneahmaevliyo,rthisesdauletstcoretehneiningteeffrpelcatyanbdettwheee“nsatlwtoimdaigstei”necfftefcatc.tTorhse,
n A surfaces,P~ ¼bP/(2p‘ s2),canbeobtainedas56 former dominates at intermediate to large distances from the
e B
p surfaces that are comparable to or larger than the Debye
O P~¼(cid:1)1þ2ð1(cid:1)~ cÞ: (30) screening length ~k(cid:1)1, while the latter dominates at small to
d intermediate distances from the surfaces, causing a partial
which, by setting c ¼ 0, reduces to the standard SC pressure depletionofmultivalentcounterionsfromtheproximityofthe
betweentwonon-disorderedlike-chargedsurfaces14 surfaceboundaries.Thesaltimageeffectisgeneratedbecause
of the inhomogeneous distribution of salt ions that are not
2
P~0¼(cid:1)1þ ~: (31) allowedtopermeateintothewallregions(i.e.,|~z|>d~/2),leading
d
in turn to a discontinuous change in the polarization of the
medium at the interfacial boundaries, with the slit region
The attractive (electrostatic) and repulsive (entropic) terms having a larger polarizability response than the slab region.
inthepressuregiverisetoanequilibriumbound-statesepara- Thisgivesrisetorepulsive“saltimage”effectsthatshowsome
tion of d~ ¼ 2 between two identical, uniformly charged qualitativesimilaritiestothe“dielectricimage”effects.
*
surfaces.5,6,14–16 As is clear from eqn (30), the surface charge When the bounding surfaces carry a nite degree of
disorder gives an additive attractive contribution to the total quenched charge randomness (c > 0), the multivalent coun-
interactionpressure,whichrenormalizestheentropictermand terions exhibit a strong attraction towards them and, again,
leadstoamorecloselypackedboundstatewiththeequilibrium generate a singular density prole with a diverging contact
inter-surfaceseparationd~ ¼2(1(cid:1)c).Thisseparationtendsto value(dashedcurvesinFig.3a).Infact,thecounteriondensity
*
zeroforc/1(cid:1),beyondwhichthetwosurfacescollapseintoa prole exhibits the same algebraic singularity on approach to
primary minimum. This is because the entropic contribution thesurfacesasinthecounterion-onlycase,eqn(28).64Thus,at
(secondtermineqn(30))changessignatc¼1and,hence,the smalldistancesfromthesurfaces,thedisorder-inducedeffects
pressurebecomesattractiveatallseparations(anddivergesfor overcomebothsaltimageandsaltscreeningeffects.
d~/0)intheregimec>1. Thesefeatureschangequalitativelywhenthesystemisdie-
lectrically inhomogeneous, exhibiting a nite dielectric
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SoftMatter Paper
e.
c
n
e
c
Li
d
e
ort
p
n
U
0
M. al 3.
Aci
19 mer Fig.3 Rescaleddensityprofilesofdressedmultivalentcounterionsasafunctionoftherescalednormalposition,~z,betweentherandomly
23 12:11:NonCom locehncattrrhgiceeadgllyrianipnnhhesor.ms(uco)rfgiasecnteheseoousfasmtswyeostaessme(mbs)i(-Dbinu¼fitnw0it.ie9th5s)filaaxbtesfidxpcelad¼cpe2adraaanmtdreevtseacrrayvlinealdguesslseap~kba¼trha0tici.o3knnaedn~sds¼Xb~1¼¼05i5n0,1(aa0n),dd1i5vealaernicodturiNscad.lilsyohrdoemrocgoeunpelionugsp(aDra¼me0t)earsndas(bsh)odwien-
20n-
3/3/utio
ded on s Attrib discontinuity at the interfacial boundaries at z ¼ (cid:6)d/2. In populatedregionsintheslit.(Suchbimodalproleshavealso
oaon Fig. 3b, we show the counterion density proles for D ¼ 0.95 beenfoundforthecounteriondensitybetweenheterogeneous
nlm
wm (corresponding to the dielectric discontinuity at the water/ butregularlypatterned,planarchargedsurfaces84andalsofor
oo
d on 23 February 2015. Densed under a Creative C hddnbsouryiiyeeffrdaalfeerreaborncctescthheenra.aeintrcTnbcsihvceoumaeenroldsafufeagbecpesfoseeeuursaaerontfkpwfuadsuircattelehhrsasyietcovahwnaradaensinitrsiqhlgsoeiueharn3adidtmdnleeiitgtrrsoa¼moctcarioeovd8zdueu0eoinlprany,talteeisenanriolmdigtofdhnici3plosopataudmrar¼geniaphnnmcls2eettie)hthtetyfeeaer,dnropfd.emdorpielSsllsaotoeerertwoanfdieoneccewnhdger tutllmdaeheninrdsuegitglftaetotoinhmrviamcn.aoepTltelenphynbroei-ctesasmhturkwcearaiorfenrtaugedscennudedotffieeftsnphrmecislpeoiaioeatnnprnyrspaaetrlhtayoickosoafluslnano,rrsgfpgdaiabeoczcselpadey,lcsieqsoc.la8oehum5scr)aadtpnaTnreanoghtrrileeeyvasitdaenneedrwaithalhicobnutehhgycmdaetoipisdhnsfseotesrrtaishnbrrcbeeeuirnleetae‘tigeotwxonirpnvediteenehoeclngeyrf-
s Article. Publishe This article is lic gabgimteeutnnlrakaeegcrreraateittneeesffegsdreavlbcaotelrsiacgrraecguiraensnrtdeoneb(unutehmsniedltbiiekeesseslrci,ttor.hiefbseTmephsdeueaclliittinainvilmltatyeelrearafngmatetcsticeahoffoleufedncppetteesopa)riliknetothtni-reolesinkgdfeiriozeoonlimemnsc,eattrhbgiieecys cp(ssfhioaonnorrgawtl~kiment-¼uehhtouee0umr.r’3sep,iscnud~hplratt¼hosnego8lebepathantaotaidnsaaeeDddttho¼rfrauoen0brss.lh9teihot5-ihel)od,unttmhvhcaripoelsusnoqhetnueooexal.fdtn.IctnvAi*atsFylxuissgeeh.e0co4n.w*b38saian5snaaFfdsrihofgcuma,.nrw4pcaae-,
ces charges (especially for large D x 1);36 these images repel the tionofthedielectricdiscontinuityparameter,D(atxedd~¼5),
n Ac counterions with a singular image potential (second term in and as a function of the rescaled inter-surface distance, d~ (at
pe eqn.(13)and(21))thatbehavesapproximatelyastherst-image xed D ¼ 0.95), respectively. The region below (above) the
O interaction potential bu x XD/4(d~/2 (cid:6)~z) at small distances curvesinthesegurescorrespondstotheparametervaluesfor
im
from the boundaries, thus overcoming the logarithmic, whichthedensityproleisuni-(bi-)modal.AsseeninFig.4b,at
disorder-inducedattractionexperiencedbythecounterions. larger values of the dielectric discontinuity or at larger salt
Weshouldalsonotethatthenitenessoftheslabthickness screening parameters, a larger disorder coupling parameter
hastypicallyonlyasmalleffectonthecounteriondistribution, (disorder variance) is required in order to counteract the
especiallywhenitiscomparabletoorlargerthanthescreening counterion-imagerepulsionsfromtheboundariesandcreatea
length, kb T 1, which is in fact oen the case in realistic bimodalstructure.Thesameistrueforasystemwithasmaller
systems. For instance, in Fig. 3c, we show the results for the inter-surfaceseparation,seeFig.4c.
sameparametersasinFig.3bbutwithc¼2anddifferentslab Thesaltscreeninghasareverseeffectatsmallorlargeinter-
thicknessesintherangeb~$5(whichcoverstherangeb>1nm surface separations: whereas at small separations it increases
inactualunits,seetheDiscussion).Thedensityofcounterions thevaluesofc ,atlargeseparationsitdecreasesthem.Another
*
in the slit is slightly increased but saturates quickly when the pointtobenotedhereisthat,fortheparametervaluesusedin
slab thickness is increased to innity. The smaller counterion Fig.4bandc,thereisaplateau-likeregionwithc ¼0,where
*
densityfoundinthecaseofthinnerslabsisduetothefactthat thedensityproleisbimodalforanyvalueofc.
the overall surface attraction experienced by counterions
becomessmallerforsmallerslabthicknessesand,atthesame
C. Interactionpressure
time, the salt ions in the outer region behind the slabs (see
Fig.1)alsocontributemorestronglytothescreeningeffects. In the most general case, where the system is immersed in an
An interesting effect seen in the above results is that the asymmetricelectrolytebath,theinteractionpressure(equivalent
competition between disorder-induced attraction and image totheosmoticordisjoiningpressure)actingontheslabsfollows
repulsion leads to a highly pronounced bimodal prole with fromthedifferenceintheslitpressureandthebulk(electrolyte)
two humps that correspond to two distinct counterion- pressure, i.e., P ¼ P (cid:1) P . The slit pressure is obtained by
b
S
3448 | SoftMatter,2015,11,3441–3459 Thisjournalis©TheRoyalSocietyofChemistry2015
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Paper SoftMatter
e.
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e
c
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M. al 3.
Aci
19 mer Fig.4 (a)Rescaleddistancebetweenthepeaks(humps)inthedensityprofileofmultivalentcounterionsasafunctionofthedisordercoupling
12:11:nCom bpiamraomdeatlesrh,acp,ef,oars~ka¼fu0n.c3t,iod~n¼o8ftahneddDiel¼ec0tr.i9c5d.i(sbc)oTnhtienuthitryepshaoraldmveatelure,Dc,*a,twfihxeerded~th¼e5coaunndtedriffioenrednetnvsaitlyuepsroofifl~keacshsahnogwesnforonmthaeugnraimpho.d(acl)tTohea
23 No thresholdvaluec asafunctionoftherescaledinter-surfacedistance,d~,atfixedD¼0.95anddifferentvaluesof~k.In(b)and(c),theregionbelow
20n- (above)thecurve*scorrespondtotheparametervaluesforwhichthedensityprofileisuni-(bi-)modal.Inallthesecases,theslabsaresemi-
3/3/utio infinite.
d on Attrib
des
an
oo
wnlmm differentiatingthefreeenergyexpression(23)withrespecttothe whereQ~¼Qmandg~¼gm.Thiscontributioncanbeattractiveor
DoCo inter-surface separation as P ¼ (cid:1)vF/(Svd), where all other repulsive and will be non-vanishing only in inhomogeneous
uary 2015. a Creative (tpnhabera+mmco0en)tkeoBrvTsaalwerinetthkienopbnts¼ax2sendd0,ea+nnqdecS0dthbebeebifnuoglrket.hpIertebssushulokruecldoisnbgceievnnetnortabetdyioPntbhoa¼tf tsmwysootgeecmnonseotwruiibstuhdtiiasotnrisbnuittoteiotdnhieeolfeincsttaerlirtcaicodtniisoscno(inp.etr.ie,nswushiuteryenawDnidsl/lsobre0a)np.rTeinhsheensoet-
d on 23 Febrensed under ctmhouenltstriliivbtaupletrineotsn-siuofnrreofmorebettaheineneeordgsmyfrwootimtichtprheresepdsseiffuctreetroeondftidmaotoieonsnonvooaftltcehonentdtiaoriennssstheinde raengaTalryhdzeelecdsosinntordifebtutahtiileofnomsrPs~ueclmtaivnia-dilneP~nmtnointce,oousnlnatbtehsreiionontrhse;ef.rt5hh7aeynandh,dar5ev9pe–re6bs3ee.ennt
cess Article. Publishe This article is lic td(2hepenW‘ssBiletisty2,d)oP,efmicnmoontmoo,nwpiotohsvPs~iacdelh¼iefftnhcPet~aersnieorn+bnetsesP~cc.c3dao6ailmselc+dpuolPi~annctete+endrtaPis~ncmattiosoenrn,mpsroefsstuhreem, P~id-¼pl(ab3nP2e)/ sfotioohtfrnlealFsotihoigwaneshnsamtdlfelfooyffrmrto,eiwmcoP~cantmpriovdotrevhnelayesilsssbeiutnnychrtgaerirllesecdascu-cloptlatmaaeilrotirpntemniodgcsnl,etfierhrnenoeismtnepsptqeaetfncrrhrtaaoiecm(vm2teci3elooyt)mnne.rwPt~usaecilctntiihtnisev-vargeu˜agliqel(vy~uznne,)ent(u˜2otic(h1nb~zo))etu.,taoenirrantemneemrdds-
c
A in terms of the total mid-plane density of monovalent ions,
pen where which can be estimated here through the relationship n(z) ¼
O (cid:10) (cid:11)
P~s¼csch2 ~k2d~ ; (33) iln+exrepf[.(cid:1)3b6u,(zw)]h|qe¼r1e+l+l(cid:1)¼expn[0(cid:1)abnud(z)l]|(cid:1)q¼¼(cid:1)1na0s+disqccu0sasreedtihnedbetualikl
concentrations of monovalent cations and anions, respec-
(cid:4) (cid:5)
P~dis ¼(cid:1)cvf~~k;d~~;D ; (34) tdievelnye;dthaesn˜r(e0s)c¼alend(~zm¼id0)-p/nla.ne density in eqn (36) is then
vd b
" #
ð
P~c ¼c~4c2 vvd~ (cid:1)d~=d~=22dz~e(cid:1)u~ðz~Þ(cid:1)1 ; (35) DW.e Inrstetrcaoctnisoindeorftnhoen-idnitseorradcetiroendspurerfsascuerse between two non-
P~mon ¼q22~k2(cid:4)n~ð0Þ(cid:1)1(cid:5): (36) dmisuoltridvearleendt-i(ounnitfohremorlyy. Tchhaerginedte)rascutirofancepsrewssiuthreinactthinegdornessthede
slabsinthiscasefollowsfromeqn(32)–(36)bynotingthatinall
theexpressionsinvolvedweneedtosetc¼0;hence,inparticular
Thersttermineqn(32),P~s,followsfromthersttermin wehaveP~dis¼0.Weshallprimarilyfocusonthecaseofsemi-
eqn(23)andrepresentstherepulsivepressureduetothemean inniteslabs(b¼N)andxtheelectrostaticcouplingparameter
charges on the inner surfaces of the slabs. The second term, at X ¼ 50, which can be achieved with tetravalent counterions
P~dis, gives the contribution from the surface charge disorder (q¼ 4)andthe meansurface chargedensitys¼ 0.24nm(cid:1)2in
(second term in eqn (23)) with ~f(~k, d~, D) being dened using water(3 ¼80)andatroomtemperatureT¼293K(seeTable1).
m
eqn(24)as Unless otherwise specied, we take D ¼ 0.95, which is appro-
ð
f~(cid:4)~k;d~;D(cid:5)h 0NQ~dQ~ g~D(cid:4)esð21d~g~þ(cid:1)DDsÞs22(cid:5); (37) priaTtheeforerswualttserf/ohrytdhreorceasrcbaolnedinintetrefraaccetsio.npressureareshownin
Fig.5aasafunctionoftherescaledinter-surfaceseparationfor
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afewdifferentvaluesoftherescaledscreeningparameter,i.e.,~k because of the contributions P~ and P~ as discussed above
s mon
¼ 0.25, 0.27, 0.28, 0.29, 0.30, and c~ ¼ 0.15 (in actual units, (see Fig. 6b and also Fig. 5b and c). However, the multivalent
c
~
theseparametervaluescanbeobtained,forinstance,bytaking counterions pressure component, P , again shows a non-
c
saltbulkconcentrationsn x105,123,133,143,153mMandc monotonicbehaviorwithbothattractiveandrepulsiveregions
ce. x 2.5 mM). At sufficientl0y large ~k (e.g., high monovalent sal0t (Fig.6b).
n
e ~
Lic concentration), the pressure is repulsive and decays mono- The non-monotonic behavior of Pc can be understood by
d tonically with the distance. In this regime, the salt screening inspecting the average number of multivalent counterions in
nporte effectsaredominantandtheSCeffectsduetothemultivalent theslitbetweentheslabs,N(cid:1) ¼Sðd=2 dzcðzÞ,which,inrescaled
U counterionsarestronglysuppressed;therepulsivepressureisin
12:11:19 AM. nCommercial 3.0 aflfaarnotcdtmtAesrdthtec~khetoeeinorsptmsrrdmieibensocusetrtudieicroaemnspecrabdeoies,mnsitlnuphygroeebntypheorneftehtsmlseasruionsgnrueeoFrrvfidoagaec.flvee5ten–hblstoe.upirotswfnaaosc,en(aP~orsmenpco-namu)nlsowibnoieotnhtso(etnP~heisnec) unitsandperunitarNS(cid:1)~ea¼,is8c~pgcXi2veðn(cid:1)d~=d~b=2(cid:1)y2d=d2z~c~(cid:4)z~(cid:5): (38)
oaded on 3/3/2023 ons Attribution-No fbiDanuleeltrhybetharyt-veeusiruros,nrrcifsnrawedcieineintchtaiosnteaiganpgaplnerraneaogtgsniattoorhtnouivns(neFgciweggadil.totth5rblaaioan)ccl.taiTmalvhemirinpsaiirnnmelogismucesamuulcrmeomamisnianpd~ktiaumiricnsauetbmdedlreebmceertvetoeewdaniestatheeutndee- cadoliorrenTrcehgtsilwpysoirqtnehudlaaisttnsettdocitoyttrohrieesvsNs(cid:2)rpeh/odvond~wddanainssihgonenpdFreiecgcsu.asrun7vresfeoeicreno~kbmFy¼ipgco.o0n5m.2ea7pn(atnarP~onicntdetghc~teahcqta¼nitnP~(0f2c.a21ci)5st,
wnlmm the slabs. For ~k ¼ 0.27 (red dashed curve), the minimum (35) and (38)). At large separations and upon decreasing the
3 February 2015. Dounder a Creative Co aratPhettxttperrlaaoc(cid:1)ccthttt2iieovv8dieecbepaaprsroer)fse.tsshFtushoeurererre~kieas¼dciP~tsu0sxa.o2all5ip(cid:1)pdp(a7brr5colaaux(mcorikmvree,tsaPeotirelnxildvyaF(cid:1)cliuuP~g7re.8vs0xe5mcbin)a,e(cid:1)rFn)t.i2thgiN.eo.7on5tmae(eodwitrnhh,aiamibctwohutivhtmeihes, iitsasnnhuetttpertrefahabrarce-cuasteuslitkvilroiecftsnao(oincsrlereiuisegntdliaacoditstrnieietvoc,aaenrnws)eecihande(sia,tcedtehtdtruhr-iaesesfcuutntcorriafuotaauhnmcsel)eaebr,drpetghrrtbeeahroystesf,uaupmnnirnteuiaunmcklttcoeuibrmveroeanafprls,etoenhonedtefnesccnhmeootauiuuP~onnnlcnct.ttieseevAsrrafsiirloooettnhhnnm–eest
d on 2ensed nlikeetactthraarcgteiodn. hTehriesaipspebaercsaduessepiotef tthheefaScCttheafftetchtse,suwrhfaicchesaarree counterions in the slit reaches a maximum value. The multi-
Article. PublisheThis article is lic ptFainirogodn.d5eubncce)etcedaornmbtdhyersetohxquheugilbhietiaettdshliaeanrgpgnersoeunsars-tfmuasrcomeen–acocoltolomusnnpcirtcoeenrebieneonhinntagcPv~oicop.rrraTerthlahaimtasitoectnwoesner5st–sr7,i(h1bs4ae–u1le6l- vsttchhtaoreelmoennsspllgtoiittlnycwdeorenhuectepnrndeetelualtrseehiedoetson;bssmutyhruatfehaltytceiaevisrasrmeliceamoenlvlmateegcrneeotuciucnlnahottlesaeleryrr-gistcoeuoosnrmcsfaaotpncnhldeetuatstecshcltey.ehpiTaearhnjrenaegctuetipmeosrdnsebissgfesrrnuoaimrraneet
s consider later in more detail. The non-monotonic behavior of (cid:2)
pen Acces tsnhteeenmtnss,edwtiirptehrcetP~slysubfrreeoimnwgittthhheetimhnetoesritpnletaesyrs-ebsnuettriwfaaleceoennseie.tspadriaffteiorennt(Fciogm. p5oa)- cttho0keBthlToe)cwabthuioelknnwpd~rh/eesrseu0Nr(etrheP~iasccl/himesi(cid:1)tainc~mgc2av/x4aimlu(ouerm,isinannoadtc,dteuivsaeclneurtnuniaibtllslye,,aPtetcnt/hdes
O c rangeofscalesshowninFig.7).
SimilarbehaviortothoseshowninFig.5acanbeseenupon
We should also note that the intermediate attractive-pres-
increasing the rescaled bulk concentration of multivalent
counterions, c~ (Fig. 6), and upon decreasing the dielectric sure regime seen in Fig. 5a and 6a is followed by a weakly
c
repulsiveregime(hump)atlargerseparationsbutthereisavery
discontinuity parameter, D (not shown) with the minimum
shallow local minimum at large separations that gives the
attractivepressureturningouttobequitesensitivetotheexact
values of these parameters. Fig. 6a shows the results for c~ ¼ pressurecurvesaweaklyattractivelongtail.Thisisshownina
c
0.09,0.11,0.13,0.15and~k¼0.27(corresponding,forinstance, close-upviewintheinsetofFig.6aforthecurvesthatappearin
to choosing the bulk concentrations as c x 0.9, 1.3, 1.9, 2.5 the main set. Comparing the different pressure components
0
mMwithn x127,126,125,123mM,respectively).Thepres- aroundthislarge-separationminimum(Fig.6c)showsthatthe
0 contribution from the surface–surface repulsion (P~ ) nearly
sure still remains strongly repulsive at very small separations s
Table1 Afewtypicalexamplesfortheactualvaluesofthebulkconcentrationsn andc ,whichcancorrespondtothetypicalvalueswehavechosen
0 0
fortherescaledparameters~kandc~ .Here,wehavefixedtheotherparametervaluesasq¼4(tetravalentcounterions),l ¼0.71nm(correspondingto
c B
waterwith3 ¼80atroomtemperature,T¼293K)andsurfacechargedensitys¼0.24nm(cid:1)2,whichgivem¼0.23nmandX¼50.Wealsoshowthe
m
actualvaluesofthedisordervarianceg,whichcancorrespondtoafewtypicalvaluesofthedisordercouplingparameterc(seethetextfordefinitions)
~k¼0.2 0.25 0.3 0.4 c g(nm(cid:1)2)
n ¼68mM 108mM 156mM 279mM c ¼1.1mM c~ ¼0.1 0.5 0.01
0 0 c
65mM 105mM 153mM 277mM 2.5mM 0.15 1.0 0.02
62mM 101mM 150mM 273mM 4.4mM 0.20 2.0 0.04
57mM 96mM 145mM 268mM 6.9mM 0.25 4.0 0.08
3450 | SoftMatter,2015,11,3441–3459 Thisjournalis©TheRoyalSocietyofChemistry2015