Partially Annealed Disorder and Collapse of Like-Charged Macroions

23 Pages·2008·0.57 MB·English

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JStatPhys
DOI10.1007/s10955-008-9635-7
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4 Partially Annealed Disorder and Collapse
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of Like-Charged Macroions
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YevgeniS.Mamasakhlisov·AliNaji·RudolfPodgornik
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Received:21March2008/Accepted:10October2008
15 ©SpringerScience+BusinessMedia,LLC2008
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18 Abstract Charged systems with partially annealed charge disorder are investigated using
19 field-theoreticandreplicamethods.Chargedisorderisassumedtobeconfinedtomacroion
20 surfaces surrounded by a cloud of mobile neutralizing counterions in an aqueous solvent.
21 Ageneralformalismisdevelopedbyassumingthatthedisorderispartiallyannealed(with
22 purely annealed and purely quenched disorder included as special cases), i.e., we assume
23 ingeneralthatthedisorderundergoesaslowdynamicsrelativetofast-relaxingcounterions
24 makingitpossiblethustostudythestationary-statepropertiesofthesystemusingmethods
25 similartothoseavailableinequilibriumstatisticalmechanics.Byfocusingonthespecific
26 caseoftwoplanarsurfacesofequalmeansurfacechargeanddisordervariance,itisshown
27 thatpartialannealingofthequencheddisorderleadstorenormalizationofthemeansurface
28 chargedensityandthusareductionoftheinter-platerepulsiononthemean-fieldorweak-
29 couplinglevel.Inthestrong-couplinglimit,chargedisorderinducesalong-rangeattraction
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32 Y.S.Mamasakhlisov
Dept.ofMolecularPhysics,YerevanStateUniversity,1Al.Manougianstr.,Yerevan375025,Armenia
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34
A.Naji
35 MaterialsResearchLaboratory,UniversityofCalifornia,SantaBarbara,CA93106,USA
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(cid:2)
37 A.Naji( )
Dept.ofChemistryandBiochemistry,UniversityofCalifornia,SantaBarbara,CA93106,USA
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e-mail:[email protected]
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40 A.Naji·R.Podgornik
41 KavliInstituteforTheoreticalPhysics,UniversityofCalifornia,SantaBarbara,CA93106,USA
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R.Podgornik
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Dept.ofPhysics,FacultyofMathematicsandPhysics,UniversityofLjubljana,1000Ljubljana,
44 Slovenia
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46 R.Podgornik
Dept.ofTheoreticalPhysics,J.StefanInstitute,1000Ljubljana,Slovenia
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48
R.Podgornik
49 Lab.ofPhysicalandStructuralBiology,NationalInstitutesofHealth,Bethesda,MD20892,USA
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Y.S.Mamasakhlisovetal.
51 resultinginacontinuousdisorder-drivencollapsetransitionforthetwosurfacesasthedis-
52 ordervarianceexceedsathresholdvalue.Disorderannealingfurtherenhancestheattraction
53 and,inthelimitoflowscreening,leadstoaglobalattractiveinstabilityinthesystem.
54
55 Keywords Classicalchargedsystems·Like-chargeattraction·Chargedisorder·Partial
56 annealing
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59 1 Introduction
60
61 Interactionofchargedmacromolecules(macroions)isessentialforsoftandbiologicalma-
62 terialsinordertomaintaintheircomplexstructureanddistinctfunctioning.Inmanycases,
63 chargepatternsalongmacromolecularsurfacesareinhomogeneousandexhibitahighlydis-
64 orderedspatialdistribution.DNAmicroarrays[1,2],surfactant-coatedsurfaces[3–6],ran-
65 dompolyelectrolytesandpolyampholytes[7,8]presentexamplesofsuchdisorderedcharge
66 distributions.Thechargepatterncanbeeithersetandquenchedintheprocessofassembly
67 of these surfaces, or can exhibit various degrees of annealing when interacting with other
68 macromolecules in aqueous solutions. Disorder annealing in charged systems may result
69 fromdifferentsources;e.g.,finitemobilityandmixingofchargedunits(lipidsandproteins)
70 inlipidmembranes[9],conformationalrearrangementofDNAchainsinDNAmicroarrays
71 [1,2]andchargeregulationofcontactsurfacesbearingweakacidicgroupsinaqueousso-
72 lutions[10–12],tonameafew,allleadtoannealingeffects.Inreality,onemaydealwitha
73 morecomplexsituationwherethesurfacechargepatterndisplaysanintermediatecharacter
74 [3–6]andthusmayneitherbeconsideredaspurelyquenched(i.e.,withfixedrandomspatial
75 distribution)noraspurelyannealed(i.e.,thermallyequilibratedwiththebulksolution).
76 Charge disorder appears to produce electrostatic features that are remarkably differ-
77 ent from those found in non-disordered systems. Mounting experimental evidence shows
78 thatlike-chargedphospholipidmembranesandfluidvesicles,whichprimarilycontainmo-
79 bile surface charges, may undergo aggregation and fusion in the presence of multivalent
80 cations[9].Asimilarbehaviorisobservedwithnegativelychargedmicasurfacesexposed
81 toasolutionofpositivelychargedsurfactants[3–6];hereformationofarandommosaicof
82 surfactant patches on apposing surfaces (after the surfactant is adsorbed from the bathing
83 solutionontothesurfaces)leadstoalong-rangeattractionandthusaspontaneousjumpto
84 acollapsedstate.Althoughthisbehaviorisakintothetransitiontotheprimaryminimum
85 withinthestandardDLVOtheoryofweaklychargedsystems[13,14],theattractiveforcesat
86 workhereexceedtheuniversalvan-der-Waalsforces[15]incorporatedintheDLVOtheory
87 byafewordersofmagnitude[3–6].
88 Infact,theemergenceofaninstabilityisnotcapturedbythestandardtheoriesofcharged
89 systemsthatincorporatestatic,non-disorderedchargedistributionsformacromolecularsur-
90 faces.Thesetheoriescoverbothmean-field[13,14,16]orweak-couplinglimit(including
91 theGaussian-fluctuationscorrectionaroundthemean-fieldsolution[17–19])aswellasthe
92 strong-coupling(SC)limit[20–24],wherethecentralthemeistheabsenceoremergenceof
93 electrostaticcorrelationsinducedbyneutralizingcounterionsinthesystemthatgiveriseto
94 attractiveinteractionsbetweenlike-chargedobjects.Bothuniform[13,14,16,22]aswellas
95 modulated[25,26]chargedistributionshavebeenconsideredinthiscontext.Inthemean-
96 fieldregime,like-chargedobjectsalwaysrepel.Whiletheoppositelimitofstrongcoupling
97 (realized, e.g., with high valency counterions, highly charged macroions, low medium di-
98 electricconstantorlowtemperature[20–22])isdominatedbycorrelation-inducedattractive
99 forcesthatcanbringtwoapposinglike-chargedsurfacestoverysmallseparationdistances.
100
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PartiallyAnnealedDisorderandCollapseofLike-ChargedMacroions
101 Atsmallseparations,however,auniversalrepulsionduetotheconfinemententropyofin-
102 terveningcounterionssetsinandstabilizesthesysteminacloselypackedboundstatewith
103 a finite surface-surface separation [20–22]. This is true even for surfaces of opposite (un-
104 equal) uniform charge distribution [27]. Therefore, other mechanisms have to be at work
105 thatwouldleadtoattractionsstrongenoughtocounteractsuchrepulsiveforcesandleadto
106 collapseorinstabilityinasystemofchargedmacroions.Onesuchmechanismweproposeis
107 thedisorderofthechargedistributionalongmacromolecularsurfacesthatturnsouttobeas
108 significantasthecounterioniccorrelationsandcouldprovideanewparadigminthetheory
109 ofchargedsoftmatter.
110 Previousstudiesofchargedisorderonmacromolecularsurfaceshaveinvestigatedboth
111 typesofquenched[7,8,28–31]andannealed[7,8,30–36]disorder(including,specifically,
112 theclassicalworkoncharge-regulatingsurfaces[10–12]).Theymainlydealwithsituations
113 where the system is in equilibrium and, on the question of electrostatic interaction [34–
114 36],focusprimarilyontheweak-couplingregime,wheredisorderedsurfacesofequalmean
115 chargealwaysrepelandnocollapseorinstabilityarises.1Asystematicanalysisofquenched
116 disorder effects is presented in the previous works of two of the present authors [28, 29],
117 whereitwasshownthatnotonlycanelectrostaticinteractionsbetweenlike-chargedobjects
118 turnfromrepulsivetoattractiveduetocounterioniccorrelations,butalsothedisorderofthe
119 surfacechargeitselfcangiverisetoanadditivelong-rangeattraction.Thisismostclearly
120 demonstrated by attraction induced between disordered surfaces of zero mean charge but
121 withafinitevarianceofthedisorderedchargedistribution[29].IntheSClimit,thequenched
122 disorder-inducedattractionmaybesostrongthatitcandominatetheentropicrepulsionat
123 smallseparationsandcontinuouslyshrinktheSCsurface-surfaceboundstate[20–22]upon
124 increasingthequencheddisordervariance,predictingthusacontinuouscollapsetransition
125 betweenastableandacollapsedphasebeyondathresholddisordervariance[28].Sincethe
126 experimentalsituationmaybemorecomplex[3–6],anditmightnotallowforstraightfor-
127 warddifferentiationofthechargepatternintopurelyquenchedandpurelyannealedcases,
128 wenextsetourselvestoexplorepossibleeffectsfrompartialannealingofthesurfacecharge.
129 Ifthereisafingerprintofthepartiallyannealedsurfacechargedisorderonthenatureand
130 magnitudeofsurfaceinteractions,thiswouldhelpinassessingwhethertheexperimentally
131 observed interactions can be interpreted in terms of disorder-induced interactions or not.
132 Thisisthemotivationwithwhichweventureonthisexploration.
133 In this paper, we present a general formalism for charged systems with partially an-
134 nealeddisorderbyinvokingfield-theoreticandreplicamethods.Wethenfocusonthecase
135 of two interacting planar charged surfaces as a model system and examine explicitly the
136 effectsofdisorderontheinter-surfaceinteractioninthissystem.Partiallyannealeddisorder
137 ingeneralariseswhenacoupledmotionofslowandfastvariables(correspondinghereto
138 surfacechargesandcounterions,respectively)ispresentinthesystem.Itrepresentsanon-
139 equilibriumsituation,whoseinvestigationrequiressuitablemethods.Thepreviouslystud-
140 iedcasesofstatic,non-disorderedsurfacechargedistribution[20–22]aswellasquenched
141 [28] and annealed surface chargedistribution followas special cases from ourformalism.
142 In the SC limit, we find that the system of two like-charged planar surfaces with neutral-
143 izing counterions becomes globally unstable upon annealing the quenched surface charge
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146 1Anotableexceptioninthecourseofprevious studiesofquencheddisorder, whichhoweverwillnotbe
consideredhere,isthecasewheredielectricdiscontinuities atbounding surfaces(orthepresenceofsalt
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inbetweentheboundingsurfaces)aretakenintoaccount[29].Inthiscaseattractionemergeseveninthe
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weak-couplingregime.Anotherandsimilarsituationinwhichweak-couplinginstabilitiesmayariseiswhen
149 membraneundulationsareallowed;see,e.g.,[16,37–39].
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Y.S.Mamasakhlisovetal.
151 and collapses into contact regardless of other system parameters due to strong attractive
152 forcesfromtheannealingeffects.Hence,thequenchedphasebehaviorisnotstableagainst
153 small annealing perturbations and is dramatically changed. However, stability may be re-
154 storedinthissystembyaddingafiniteamountofaddedsaltthatscreensoutthelong-range
155 Coulomb interactions. In this case, we recover the continuous collapse transition between
156 astabilizedcloselypackedboundstateofthetwosurfacesandacollapsedstatewherethe
157 surfacesareincontact.Thisisqualitativelysimilartothepurelyquenchedcase[28].How-
158 ever,thepartiallyannealedboundstateshowsasignificantlylargerattractionandasmaller
159 optimalsurfaceseparationascomparedtothequenchedcase.Inotherwords,allowingfor
160 rearrangementsofthemacroionchargesleadstoconfigurationsoflowerfreeenergy.Since
161 thepresentformalism isquitegeneral, weshallalsostudythemean-field limit,where(in
162 contrasttothequenchedcasewherenodisordereffectsarefound[28,30,31])thedisorder
163 annealing appears to suppress the mean-field repulsion significantly by renormalizing the
164 surfacechargetosmallervalues.Hence,besidesthepreviouslyestablishedmechanismsof
counterion-induced [22–24] and quenched disorder-induced [28, 29] correlations, we find
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that the annealing of macroion charges provides another mechanism enhancing the like-
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chargeattraction.
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The organization of the paper is as follows: We start with the general formalism that
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allowsustodefineandtodealwiththepartiallyannealeddisorderintermsofan“effective
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partitionfunction”thatisobtainedintheformofafunctionalintegraloverafluctuatinglocal
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electric potential field. The structure of this field theory is too complicated to allow for a
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generalsolution.Wethusderiveasymptoticsolutionsinthemean-fieldlimit(corresponding
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tothePoisson-BoltzmanntheoryoftheclassicalDLVOframework)aswellasthestrong-
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couplinglimitviaanapplicationofthereplicaformalism.Wefinallyevaluateandanalyze
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theinter-surfaceinteractionsforthespecificcaseofplanarchargedsurfacesinbothlimits
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andcomparethem.Weconcludebypositioningourresultsinthegrowingframeworkofthe
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weak–strongcouplingformalismforchargedmacromolecularinteractions.
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2 GeneralFormalism
180
181 Letusconsiderasystemoffixed macroionswithdisorderedchargedistribution,ρ(r),im-
182 mersed in an aqueous medium of dielectric constant, ε, along with their point-like neu-
183 tralizingcounterionsofvalencyq.Inwhatfollows,weshalldevelopourformalismforan
184 arbitrary ensemble of fixed macroions but for explicit calculations, we shall delimit our-
185 selvestoamodelsystemoftwoapposedchargedplanarsurfaceswithρ(r)representingthe
186 chargedistributionalongbothplanarsurfaces(seeFig.1).Inthequenchedlimit,ρ(r)isas-
187 sumedtobestaticandonlycounterionsaresubjecttothermalfluctuations.Intheannealed
188 limit,bothcounterionsandmacroionchargesaresubjecttofluctuationsofcomparabletime
189 scales(i.e.,τ ∼τ respectively)andthusmutuallyequilibrate.Theintermediatesituation
ci s
190 ofpartiallyannealeddisorderbydefinitionoccurswhenthereisamacroscopicseparation
191 oftimescalesbetweentheso-calledfastandslowvariablesasfrequentlyobservedinglassy
192 systems[43,44].
193 Inthepresentcontext,counterionscomprisethefastvariablesastheyaredispersedand
194 freely fluctuate in the bulk. Macroion surface charges are assumed to constitute the slow
195 variableswiththetimescaleτ (cid:3)τ duetotheirdisorderednature(as,e.g.,theyarecon-
s ci
196 finedtypicallywithincloselypackedorquasi-two-dimensionaldisorderedregionssuchas
197 inlipidbilayers[9],surfactant-coatedsurfacesorsurfacehemimicelles[3–6,45–47]).
198 Undertheseconditions,themutualequilibrationoffastandslowvariablesishindered.
199 CounterionsrapidlyattaintheirequilibriumatbulktemperatureT andthus,becauseofthe
200
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PartiallyAnnealedDisorderandCollapseofLike-ChargedMacroions
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Fig.1 Schematicviewofasystemofmacroionswithpartiallyannealeddisorderedchargedistributionρ(r)
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andq-valencycounterionsatbulktemperatureT.Surfacechargesmayexhibitadifferenteffectivetempera-
210 tureT(cid:4)duetotheirdisorderednatureandslowdynamicsrelativetothefast-relaxingcounterions.Asamodel
211 system,westudytwoapposedplanarmacroionsurfaceswithdisorderedsurfacechargedistributions(speci-
212 fiedinthetext)locatedatz=±aattheseparationdistanced=2a.Weneglectthedielectricdiscontinuity
attheboundariesorthepresenceofaddedsaltinthesystem[29](seealso[40–42])
213
214
215 widetime-scalegap,theirequilibriumfreeenergyactsasadrivingforcepushingtheslow
216 dynamics of the surface charges to reach a non-equilibrium stationary state at long times.
217 Thisscheme,knowngenerallyastheadiabaticeliminationoffastvariables[48–50],isin-
218 vestigatedinagrowingnumberofworks,forinstance,inthecontextoffar-from-equilibrium
219 stationarystatesandthermodynamicsoftwo-temperaturesystems[51–54].Ithasbeenap-
220 plied in particular to study spin glasses with partially annealed disorder of the spin-spin
221 couplingstrength[43,44,55–64].Ithasbeenshowningeneralthatthestationarystateof
222 suchsystemsmaybedescribedbyaBoltzmann-typeprobabilitydistributionfeaturingthe
223 temperatureoffastdegreesoffreedomT aswellasaneffectivetemperatureT(cid:4) associated
224 withthedisorder.Thispeculiartwo-temperaturerepresentationclearlyreflectstheintrinsi-
225 callynon-equilibriumnatureofpartialannealing.Obviously,theequilibriumfreeenergyof
226 fast variables (i.e., counterions in our case) will show up explicitly in the aforementioned
227 probabilitydistribution(hereweshallnotconsidertherelaxationaldynamicsofthesystem
228 andfocusonlyonstationary-stateproperties).
229 Ingeneral,onemaythusdefineaneffective“partitionfunction”,Z,inanalogywiththe
230 equilibriumpartitionfunctionthatgreatlyfacilitatestheanalysisofthesystemfarfromequi-
231 librium[43,44,52–64].ThisprocedureisdiscussedinAppendixAbyadoptingasimple
232 dynamicalmodelforachargedsystemwithsurfacechargedisorderandbyidentifyingits
233 stationary-stateprobabilitydistribution.Wethusfind
234 (cid:2)
(cid:3) (cid:4)
235 Z= Dρexp −β(cid:4)W[ρ] , (1)
236
237
238 whereβ(cid:4)= kB1T(cid:4) andthedensityfunctionalW[ρ]canbecastintotheform
239 (cid:2)
1
240 β(cid:4)W[ρ]= drg−1(r)[ρ(r)−ρ (r)]2−nlnZ [ρ], (2)
241 2 0 ci
242
where
243
T
244 n= . (3)
245 T(cid:4)
246 Equation(2)includesstatisticsofbothcounterionsandthedisorderedchargesonmacroion
247 surfaces.Thefirsttermisthecontributionofthedisorder.Itcanbeinterpretedasageneral
248 effectivedisorderpotentialexpandedtothesecondorderaroundatypicalvalueρ (Appen-
0
249 dix A), which is always possible if one interprets g(r) as playing the role of an effective
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Y.S.Mamasakhlisovetal.
251 disorder“compressibility”.ThisexpansionleadstothestandardGaussiandisorderweight
252 with the mean value ρ0(r) and variance g(r) that can be handled most conveniently by
253 replicatechniques[43,44].Namely,
254 (cid:2)
(cid:3) (cid:4) (cid:5)(cid:5)(cid:3) (cid:4) (cid:6)(cid:6)
255 Z= DρP[ρ] Z [ρ] n= Z [ρ] n , (4)
ci ci
256
(cid:7)
257 where double-brackets denote the average (cid:5)(cid:5)···(cid:6)(cid:6) = DρP[ρ](···) with respect to the
258
Gaussianprobabilitydistribution
259
(cid:8) (cid:2) (cid:9)
260 1
P[ρ]=Cexp − drg−1(r)[ρ(r)−ρ (r)]2 (5)
261 2 0
262
263 withC beinganormalizationfactor.
264 The second term in (2) is the equilibrium free energy of a system of counterions at a
265 fixedrealizationofdisorderedmacroioncharge,ρ=ρ(r).Itfollowsbyintegratingoverthe
266 counterionicdegreesoffreedomequilibratedattemperatureT.Ingrand-canonicalensem-
267 ble,thefixed-ρ partitionfunction,Zci[ρ],canbecastintoaformofafunctionalintegralas
268 [17–19,22]
(cid:2)
269 Dφ
270 Zci[ρ]= Z e−βH[φ,ρ], (6)
v
271
272 whereφ(r)isthefluctuatingelectrostaticpotentialfield,β= 1 and
kBT
273 (cid:2) (cid:10) (cid:11)
εε (cid:3) (cid:4)
274 H= dr 0 ∇φ 2+iρφ−λk T(cid:8)(r)e−iβqe0φ (7)
275 2 B
276
is the effective Hamiltonian of the system comprising Coulomb interaction v(x) =
277
278 (4πεε0|x|)−1 betweenallchargedunits(thefirsttwo√terms)aswellastheentropyofcoun-
terions(thelastterm).Hereλisthefugacity,Z = detβv(r,r(cid:4)),and(cid:8)(r)isageometry
279 v
functionthatspecifiesthefreevolumeavailabletocounterions,i.e.,thespacebetweenthe
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twoapposedplanarsurfacesinthemodelsysteminFig.1.
281
The partition function (1) can be evaluated by using the replica trick [43, 44], i.e., by
282
taking n an integer number and then standardly extending the results to real axis (for any
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real value of n=β(cid:4)/β) by analytical continuation. Thus by using (6) and averaging over
284
ρ(r),wearriveatthedisorder-averagedexpression
285
(cid:2) (cid:8) (cid:9)
286 (cid:12)n Dφ
287 Z= a e−S[{φa}], (8)
Z
288 a=1 v
289
wherethen-replicaeffectiveHamiltonianreads
290
(cid:2)
291 1(cid:13)
292 S[{φa}]= drdr(cid:4)φa(r)Dab(r,r(cid:4))φb(r(cid:4))
2
293 a,b
(cid:2)
294 (cid:13) (cid:14) (cid:15)
+ dr iβρ (r)φ (r)−λ(cid:8)(r)e−iβqe0φa(r) . (9)
295 0 a
296 a
297 ThekernelD (r,r(cid:4))introducedaboveisdefinedas
ab
298
299 D (r,r(cid:4))=βv−1(r,r(cid:4))δ +β2g(r)δ(r−r(cid:4)), (10)
ab ab
300
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PartiallyAnnealedDisorderandCollapseofLike-ChargedMacroions
301 wherev−1(r,r(cid:4))=−εε0∇2δ(r−r(cid:4)).
302 Equation (8) carries complete information about the mutual coupling between counte-
303 rions and the surface charge disorder. The grand-canonical “free energy” of the partially
304 annealedsystemisthenobtainedas
305
306 F=−kBT(cid:4)lnZ. (11)
307
308 The special cases of purely quenched and purely annealed disorder follow from (11) for
309 n→0andn=1,respectively(seeAppendixB).
310 Notethatherethenumberofreplicas,n=T/T(cid:4),hasadirectphysicalmeaningoftemper-
311 atureratio[43,44,52–64].Acloseexaminationof(9)indicatesthatthepartiallyannealed
312 disordergivesrisetoquadraticsurfacetermsoftheformg(r)φa(r)φb(r).Itmaythuslead
313 torenormalizationofthemeansurfacecharge(AppendixC)ascanbeseenmostclearlyby
314 lookingatthemean-fieldequationswhichweshallderivenext.
315
316
317 3 Mean-FieldLimit
318
319 The mean-field or Poisson-Boltzmann (PB) equation [13, 14, 16] (which becomes exact
320 in the limit of small coupling parameters corresponding, for instance, to low counterion
321 valency or low surface charge density [20–22]) follows from the saddle-point equation of
322 thefunctionalintegral(8)as
323 (cid:13)
324 εε ∇2φ¯ =iλqe (cid:8)(r)e−iβqe0φ¯a(r)+iρ (r)+βg(r) φ¯ (r). (12)
0 a 0 0 b
325 b
326
We shall assume no preferences among different replicas on the saddle-point level, thus
327 φ¯ (r)=φ¯(r) for a=1,...,n (replica symmetry ansatz). In this way we arrive at the PB
328 a
equationforthereal-valuedmean-fieldpotentialϕ (r)=iφ¯(r)as
329 PB
330 εε ∇2ϕ (r)=−λqe (cid:8)(r)e−βqe0ϕPB(r)−ρ (r), (13)
331 0 PB 0 eff
332
where
333
334 ρ (r)≡ρ (r)−β(cid:4)g(r)ϕ (r) (14)
eff 0 PB
335
is the effective (renormalized) macroion charge distribution (Appendix C). It is therefore
336
seenthatinthequenchedlimit(norβ(cid:4)→0),thedisordereffectscompletelyvanishonthe
337
mean-fieldlevelandthePBtheorycoincidesexactlywiththatofanon-disorderedsystem
338
of bare charge distribution ρ (r) [28]. This is however not true for the partially annealed
339 0
disorder(n>0).
340
ToproceedwiththePBtheory,weshallconsiderthespecificcaseoftwoparallelcharged
341
plateslocated(normalto z-axis)at z=−a and z=+a attheseparationdistance d =2a
342
(Fig.1).Wetakethemeanchargedistributionanditsvarianceas
343
(cid:14) (cid:15)
344
ρ (r)=−σe δ(z+a)+δ(z−a) (15)
345 0 (cid:14)0 (cid:15)
346 g(r)=ge2 δ(z+a)+δ(z−a) , (16)
0
347
348 whereg≥0andwithoutlossofgeneralityweassumethatσ ≥0(andthusq≥0).Counte-
349 rionsareassumedtobeconfinedinbetweentheplates(i.e.,(cid:8)(r)=1for|z|<a andzero
350
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Y.S.Mamasakhlisovetal.
351 elsewhere),where(13)admitsthewell-knownsolution[13,14,16]
352
1
353 ϕ (z)= lncos2(Kz) (17)
354 PB βqe0
355
356 with K2=2π(cid:13)Bq2λ tobedeterminedfromtheelectroneutralityconditionstipulatingthat
357 thetotalchargeonthetwosurfacesshouldbeequaltothetotalchargeofthecounterions.
358 ThisleadstotheequationforK oftheform
359
σ
360 Kμtan(Ka)=1+γlncos2(Ka)≡ eff. (18)
σ
361
362
Here μ=1/(2πq(cid:13) σ) is the Gouy-Chapman length, (cid:13) =e2/(4πεε k T) the Bjerrum
363 B B 0 0 B
length,and
364
365 σ =σ +β(cid:4)ge ϕ (a) (19)
eff 0 PB
366
367 therenormalizedsurfacechargedensity(14).Thelatterexpressionclearlyreflectsthemixed
368 boundary conditions encountered here, resembling the situation in the classical charge-
369 regulationproblems[10–12].Thedimensionlessparameter
370
ng
371 γ = (20)
qσ
372
373
gives a measure of the disorder annealing and is obviously proportional to the ratio n=
374
T/T(cid:4)ofthecounterionsandsurfacedisordertemperatures.
375
ThePBpressure,P ,actingbetweentheplatesisobtainedfromthestandarddefinition
376 PB
βP =n (z )− 1βεε (dϕ /dz)2| [13,14,16]foranarbitrary|z |<aas
377 PB PB 0 2 0 PB z0 0
378
βP
379 PB =(Kμ)2. (21)
2π(cid:13) σ2
380 B
381
382 The counterion number density profile between the plates, nPB(z) = λe−βqe0ϕPB(z)
[13,14,16],isobtainedas
383
(cid:8) (cid:9)
384 n (z) Kμ 2
385 PB = . (22)
2π(cid:13) σ2 cosKz
386 B
387 It follows from (18) that the mean-field renormalized surface charge density is always
388 smaller than the bare value and tends to zero but never changes sign as γ increase (0≤
389 σ ≤σ).Therefore,thesurfacesareeffectivelyneutralizedandthepressureaswellasthe
eff
390 counterionnumberdensityprofiletendtozeroasγ increasestoinfinity(seeFig.2).This
391 picturereliesontheassumptionthatthenumberofsurfacechargedunitsisnotfixedandcan
392 respondtochangesofthesurfacepotential.Imposingtheconstraintthatfixesthisnumber
393 obviously rules out surface charge renormalization and one observes no effects from the
394 disorderannealinginagreementwith[30,31].
395 In the limit γ →0, we recover the non-disordered [22] or quenched [28] mean-field
396 resultswiththefollowingasymptoticbehaviorforthepressure,
397
(cid:16)
398 βPPB (cid:13) 2μ/d d/μ(cid:14)1, (23)
399 2π(cid:13) σ2 π2μ2/d2 d/μ(cid:3)1.
B
400
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PartiallyAnnealedDisorderandCollapseofLike-ChargedMacroions
Fig.2 RescaledPBpressure
401
betweentwochargedplatesasa
402 functionoftheirseparationdfor
403 γ=0,1,10and102.Insetshows
404 thePBcounteriondensityprofile
405 ford/μ=4.Forγ=0,we
recoverthenon-disordered
406
resultswiththepressuredecaying
407 as∼1/d2[22].Forγ(cid:3)1,the
408 pressuredecaysas∼1/(γd2)
409
410
411
412
413
414
415
416
Forlargeγ (cid:3)1,wefindthat(Ka)2(cid:13)(μ/a+γ)−1andthus
417
(cid:16)
441189 2πβP(cid:13)PσB2 (cid:13) 24μμ/2/d(γd2) γγdd//μμ(cid:14)(cid:3)11,. (24)
420 B
421 Thesmallseparationexpressionaboveisnothingbuttheideal-gasosmoticpressureofcoun-
422 terionsthatdominatesovertheenergeticcontributions.Atlargeseparationsthepressureis
423 foundtodecayasymptoticallyas∼1/(γd2).Thepressureremainsalwaysnon-negativeand
424 thesurface-surfaceinteractionisthusalwaysrepulsiveinthemean-fieldlimit.Choosingthe
425 non-disordered system as the reference, however, the decrease in the interaction pressure
426
uponincreaseofthesurfacedisorderannealingcanbeinterpretedasbeingduetoaneffec-
427
tivedisorder-inducedattractionwhoseasymptoticformcouldbedescribedby
428
429 1−γ 1
(cid:15)P ∼ (25)
430 PB γ d2
431
432 forlargeγ.Thisasymptoticformagainatteststothefactthatthewaythedisorderactson
433 themean-fieldinteractionbetweenthetwoapposedsurfacesisviaarenormalizationofthe
434 surfacechargedensity.
435
436
437 4 Strong-Coupling(SC)Limit
438
439 Nextweshallinvestigatetheasymptoticstrong-couplinglimitwhichiscomplementaryto
440 themean-fieldlimitandwherecounterion-inducedcorrelationsbecomedominant.Weem-
441 ploythestandardstrong-couplingschemereviewedextensivelyin[20–22]inordertostudy
442 thepartialannealingeffectsintheSClimit.Theso-calledasymptoticSCtheoryisobtained
443 fromtheleadingordertermsofanon-trivialvirialexpansion(inpowersofthefugacity)of
444 thepartitionfunction(8),i.e.
445
446 Z=Z +λZ +O(λ2). (26)
0 1
447
448 It becomes exact in the limit of large coupling parameters corresponding, for instance, to
449 highcounterionvalencyorhighsurfacechargedensity[20–22].
450
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Y.S.Mamasakhlisovetal.
451 Thezeroth-order(nocounterion)term,Z0,andthefirst-order(singlecounterion)term,
452 Z1,followfrom(8)as
453 (cid:2) (cid:8) (cid:9)
454 Z = (cid:12)n Dφa e−S0, (27)
455 0 Z
a=1 v
456 (cid:2) (cid:2) (cid:8) (cid:9)
457 Z =(cid:13)n dR(cid:8)(R) (cid:12)n Dφa e−S0−iβqe0φb(R), (28)
458 1 Z
b=1 a=1 v
459
460 where
(cid:2) (cid:2)
461 (cid:13) (cid:13)
1
462 S0= 2 drdr(cid:4)φa(r)Dab(r,r(cid:4))φb(r(cid:4))+iβ drρ0(r)φa(r). (29)
463 a,b a
464 Wethusneedtocalculateboththesetermsforanarbitrarynumberofreplicas,n.Indoing
465 so,weshallmakeuseofsomemathematicalrelationsthatwebrieflydiscussbelow.
466
467 4.1 MathematicalPreliminaries
468
469 First, it turns out that the most convenient way to carry out the calculations is to replace
470 the long-range Coulomb interaction v(x)=1/(4πεε |x|) with the exponentially screened
0
471 Yukawainteraction
472 e−κ|x|
473 v (x)= (30)
s 4πεε |x|
474 0
475 byintroducingafinitescreeninglengthκ−1.Intheend,weshalltakethelimitκ→0.Note
476 thatnotonlyisthisprocedureoftechnicalconveniencebutitisalsoofphysicalrelevance
477 forthepresentproblem.Itcorrespondstoaddingabackgroundsalttothesystem,leading
478 tothescreenedCoulombinteractionbetweenchargedunits.Itistobenotedhoweverthat
479 the salt effects are taken into account in this way only on the linear Debye-Hückel level.
480 (Further studyoftheroleofaddedsaltintheSClimitispresentedelsewhere[42].)Gen-
481 eralization of (8)–(10) in the presence of Yukawainteraction is immediate as v−1(r,r(cid:4)) is
482 simplyreplacedby
483
v−1(r,r(cid:4))=εε (−∇2+κ2)δ(r−r(cid:4)). (31)
484 s 0
485 Second,incalculatingZ andZ oneneedstoevaluatethedeterminantandtheinverse
0 1
486 of the block-matrix D (r,r(cid:4)). These calculations are straightforward and may be carried
ab
487 outmosteasilybyemployingpropertiesofblock-matricesandtheoperatoralgebradefined
488 over the Hilbert space {|r(cid:6)}. We shall use the compact notation by defining the operators
489 (cid:5)r|vˆ |r(cid:4)(cid:6)=v (r,r(cid:4)),(cid:5)r|gˆ|r(cid:4)(cid:6)=g(r)δ(r,r(cid:4))(forthescreenedCoulombinteractionanddisor-
s s
490 dervariance),and(cid:5)r|Dˆ |r(cid:4)(cid:6)=D (r,r(cid:4))via(10),wherethelatterisdefinedasanelement
ab ab
491 ofthen×noperatormatrix
492
493 Dˆ =βe⊗vˆ−1+β2u⊗gˆ (32)
s
494
495 (cid:7)witheab=δab anduab=1.Also,weshalluse(cid:5)r|ρ0(cid:6)=ρ0(r)andthewell-knownnotation
496 r,r(cid:4)ρ0(r)vs(r,r(cid:4))ρ0(r(cid:4))=(cid:5)ρ0|vˆs|ρ0(cid:6),etc.
497 OnecanprovethefollowingidentitiesforDˆ
498 (cid:3) (cid:4) (cid:3) (cid:4)
499 detDˆ = detβvˆ−1 ndet 1ˆ+nβgˆvˆ , (33)
s s
500