Table Of ContentSpringerBriefs in Physics
Ralf Blossey
The Poisson-Boltzmann
Equation
An Introduction
SpringerBriefs in Physics
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Ralf Blossey
The Poisson-Boltzmann
Equation
An Introduction
RalfBlossey
CNRS
UniversityofLille
Villeneuved’AscqCedex,France
ISSN 2191-5423 ISSN 2191-5431 (electronic)
SpringerBriefsinPhysics
ISBN 978-3-031-24781-1 ISBN 978-3-031-24782-8 (eBook)
https://doi.org/10.1007/978-3-031-24782-8
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Preface
ThisshortbookisintendedtoprovideaconciseintroductionintoPoisson-Boltzmann
theory(oftenshort:PBtheory).Theoryseemsabigwordhere,andthefirstquestion
asked may be whether Poisson-Boltzmann theory actually is a theory, or rather a
modeling approach. The readers will find my answer upon reading and working
throughthebook.
WhatisPoisson-Boltzmanntheoryabout?Thereaderswillseethatwhatthisbook
contains(tolowestorder)isessentiallyasetoftoolsthatallowtosolvetheMaxwell
equation
∇·D=(cid:2)
wherebyDisthedielectricdisplacementfield,and(cid:2)thedensityofcharges.Wewill
mostly consider simple geometries, typically of planar type, like a single wall or
a slitor channel containing electric charges with density (cid:2) that will be both fixed,
typically on the system boundaries, and ‘mobile’—dissolved charges in the bulk
liquid—aprototypicalelectrolyte.Inthelattercase,thechargedensity(cid:2) generally
turnsintoanonlinearfunctionoftheelectrostaticpotential,andtheensuingequation
becomesaninvolveddifferentialequationoftheelectrostaticpotential.
Thisverybasicsystemsetupis,atthesametime,ofanincrediblegeneralityin
softmatter,physico-chemicalandbiophysicalsystems.Electrolytesystemsbounded
by charged surfaces are in a sense ‘elementary building blocks’ in these contexts.
Ontheotherhand,despitethisgenerality,theyarealsoofanenormousrichnessand
diversity,necessitatingextensionsofthebasicPoisson-Boltzmanntheorywhichhas
notseenanend,andmostlikelywon’t.
The material in this book is arranged in three chapters. Chapter 1 introduces
thePoisson-Boltzmannequationstraightawayforthecaseof(1:1)salt,i.e.,anelec-
trolytecontainingdissolvedmonovalentions,e.g.,thecaseofsodiumchloride,NaCl,
dissolvinginNa+ andCl–.Herewediscusstherelevantphysicallengthsscalesand
thebehaviorofthesolutionstothePBequationinplanar,cylindricalandspherical
geometries. We also show how Poisson-Boltzmann equations can be obtained for
vii
viii Preface
morecomplexfluidswithinvolvedequationsofstate,andwewillshowthatspatial
densityvariations,astheyoccur,e.g.,inionicliquids,giverisetoPoisson-Boltzmann
equationsofhigherorder.
Chapter 2 goes one step further. Here, we now derive the Poisson-Boltzmann
equation from a systematic statistical physics approach which allows us to show
thatitisthesaddlepointofthefield-theoreticpartitionfunction.Thisapproachwill
enableustogobeyondmeanfieldandconsidertheone-loopcorrectiontoitssolution.
Asanapplication,wecomputethesurfacetensionofanair/electrolyteinterface,asa
classicprobleminthefieldofsoftmatterelectrostaticswhichwasfirstdiscussedin
the1930s.Wethenintroduceavariationalmethodthatallowstoderiveafluctuation-
correctedPoisson-Boltzmannequationandprovideafirstillustrativeapplicationof
themethodfortheinteractionofachargedpolymerwithalike-chargedmembrane.
Chapter 3 addresses the problem of going beyond the structureless solvent
described solely by its macroscopic dielectric constant in PB theory. To make
the solvent properties ‘explicit’, we first introduce a phenomenological approach
by introducing a wave vector-dependent dielectric function or permittivity. Subse-
quently, we discuss a microscopic model of a Poisson-Boltzmann equation with
explicitsolventmodeledbypointdipoles,beforeturningthepointdipolesintofinite-
sizemolecules.Thisapproachallowsustodetermine,e.g.,amodeleffectivedielectric
function,firstinamean-fieldapproach.Wethenapplythevariationalapproachfrom
Chap. 2 to a Hamiltonian adapted to the slit geometry already discussed in stan-
dardPoisson-BoltzmanntheoryinChap.1.Ultimately,weprovidesolutionstothe
variationalequationsforthisHamiltonianinspecialcasesanddiscusstheresulting
physics.And,finally,weconcludeonthecontentsofthisbook.
Theauthorofthisbookofcoursehopesthatitwillbeusefultoitsreaderswho
wish to start with Poisson-Boltzmann theory but also like to go beyond the most
basiclevelsalready.Anunderstandingofthermodynamicsandstatisticalphysicsat
anadvancedlevelisrequired.Workingthroughitshouldbeseenasamountaineering
effort:thechallengegrowswitheachchapter,sometimessomemoreeasygoingpaths
emerge,butthenitmightbecomesteepagain.Reachingthetopisarewardwellworth
takingthepath.Forthosewhoarewillingtoevenlookonforthenextchallenge,I
haveprovidedseveralsuggestionsforfurtherreadingattheendofeachchapter.
Lille,France RalfBlossey
November2022
Acknowledgements
Itisapleasuretothankmycollaboratorsandcolleaguesforshapingmyviewofthe
subject.IprofitedfrommanydiscussionsovertheyearswithHélèneBerthoumieux,
Markus Bier, Sahin Buyukdagli, Fabrizio Cleri, Guillaume Copie, Marc Delerue,
RalfEverarers,AndreasHildebrandt,MarcLensink,AnthonyMaggs,ArghyaMajee,
RolandNetz,HenriOrland,FabienPaillusson,RudiPodgornik,Emmanuel Trizac
andmanyothers.AndreasHildebrandtistobeparticularlycreditedwiththework
onphenomenologicalnonlocalelectrostaticsperformedduringhisPh.D.atSaarland
University,asisSahinBuyukdagliforresultsonthesolvent-explicitdipolarnonlocal
electrostaticsdiscussedinthisbook,obtainedduringanextraordinarilyproductive
postdoctoralstayinLille.
ix
Contents
1 ThePoisson-BoltzmannEquation ................................. 1
1.1 LengthScales .............................................. 1
1.2 ThePoisson-BoltzmannEquationfor(1:1)Salt ................. 3
1.3 Solution of the Poisson-Boltzmann Equation
fortheSingle-PlateGeometry ................................ 4
1.4 SlitGeometry .............................................. 7
1.5 ThePoisson-BoltzmannEquationinCylindricalandSpherical
Geometry ................................................. 12
1.6 GeneralizedPoisson-BoltzmannEquations ..................... 15
1.7 AHigher-OrderPoisson-BoltzmannEquation .................. 17
1.8 Summary ................................................. 23
1.9 FurtherReading ............................................ 24
References ...................................................... 24
2 Poisson-BoltzmannTheoryandStatisticalPhysics .................. 27
2.1 ThePartitionFunctionofaCoulombGasofInteractingIons ..... 27
2.2 Mean-FieldTheory ......................................... 31
2.3 One-LoopCorrection ....................................... 33
2.4 TheFreeEnergyandtheSurfaceTension ...................... 38
2.5 TheVariationalMethod ..................................... 42
2.6 A Charged Polymer Interacting with a Like-Charged
Membrane ................................................. 45
2.7 Summary ................................................. 50
2.8 FurtherReading ............................................ 51
References ...................................................... 52
3 Poisson-BoltzmannTheorywithSolventStructure ................. 53
3.1 NonlocalElectrostatics ...................................... 53
3.2 TheDipolarPoisson-BoltzmannEquation:APoint-Dipole
Theory .................................................... 57
3.3 Finite-SizeDipoles ......................................... 59
xi
