Beyond Poisson-Boltzmann: fluctuations and fluid structure in a self-consistent theory

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by  Sahin Buyukdagli

Beyond Poisson-Boltzmann: fluctuations and fluid structure in a self-consistent theory
S. Buyukdagli1 and R. Blossey2
1 Department of Physics, Bilkent University, Ankara 06800, Turkey
2University of Lille 1, CNRS, UMR 8576 UGSF - Unit´e de Glycobiologie Structurale et Fonctionnelle, 59000 Lille, France
Poisson-Boltzmann (PB) theory is the classic approach to soft matter electrostatics which has
been applied to numerous problems of physical chemistry and biophysics. Its essential limitations
are the neglect of correlation effects and of fluid structure. Recently, several theoretical insights
have allowed the formulation of approaches that go beyond PB theory in a systematic way. In this
topicalreviewweprovideanupdateonthedevelopmentsachievedinself-consistentformulationsof
correlation-correctedPoisson-Boltzmanntheory. Weintroducethecorrespondingsystemofcoupled
nonlinear equations for both continuum electrostatics with a uniform dielectric constant and a
structured solvent, a dipolar Coulomb fluid, including nonlocal effects. While the approach is only
6
approximate and also limited to corrections in the so-called weak fluctuation regime, it allows to
1
0 include physically relevant effects, as we show for a range of applications of these equations.
2
PACSnumbers:
l
u
J
I. INTRODUCTION Therefore, in order to remedy this deficit, methods to
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include fluctuation effects have been devised. In this
2
Poisson-Boltzmann (PB) theory is the cornerstone of Topical review, we deal exclusively with one such ap-
] soft matter electrostatics, but in recent years several proach, which relies on a variational formalism, leading
ft shortcomings of this theory have also been clearly re- to self-consistently coupled equations of the electrostatic
o vealed. PB theory is a mean-field theory, hence it ne- potential and its correlation function. The formulation
.s glects all fluctuation or correlation effects, and as a sim- that we base ourselves on was originally introduced by
at ple continuum theory it also ignores the structure of sol- NetzandOrland[3], followingearlierworkbyAvdeevet
m vent and ions. In the presence of ions of high valency, al.[4]. More precisely, Netz and Orland used the varia-
prominent in particular in biological systems, the the- tional approach in order to calculate the mean-field level
-
d ory fails even qualitatively. A systematic field-theoretic charge renormalisation associated with the non-linearity
n approach to soft matter electrostatics developed on the of the Poisson-Boltzmann approach, without consider-
o counter-ion case allowed the identification of a coupling ingthecorrelationeffectsembodiedintheself-consistent
c
[ parameter [1], equations.
In recent years, the variational approach has how-
2 Ξ≡ q3|σ|e4β2 =q2 (cid:96)B (1) ever seen a number of physically relevant applications,
v 8πε2 (cid:96)CG covering different charge geometries, and even dynami-
3
2 where q is the valency of the counter-ions, σe is the sur- cal situations such as flow-related effects in nanopores.
5 face charge density with electronic charge e, ε is the di- Hatlo et al. considered the variational formulation of
0 electric constant, and β = 1/k T. Ξ thus is essentially inhomogeneous electrolytes by introducing a restricted
B
0 the ratio of the Bjerrum length (cid:96) = e2/(4πε ε k T) self-consistent scheme [5]. In Refs. [6, 7], one of us
1. and the Gouy-Chapman length (cid:96)B = 1/(2π0(cid:96) wq|σB|). (SB) introduced a numerical scheme for the exact so-
GC B s
0 Poisson-Boltzmann theory is the weak coupling limit lution of the variational equations in slit and cylindri-
6 Ξ→0 of the more general theory, while for Ξ→∞, the cal nanopores. At this point, one should also mention
1 strongcouplingcase,asingle-particlepictureemerges[2]. theone-looptreatmentofchargecorrelationsthatallows
v: Even within the weak coupling limit, or for intermedi- an analytical treatment of inhomogeneous electrolytes.
i atevaluesofthecouplingparameter,Poisson-Boltzmann Netz introduced the one-loop calculation of ion partition
X
theory does not fully describe electrostatic phenomena atmembranesurfacesincounterion-onlyliquids[8]. This
ar in soft matter systems. Being a mean-field theory, it en- wassubsequentlyextendedbyLau[9]toelectrolytessym-
tirelylackscorrelationeffectsbetweenthecharges. These metrically partitioned around a thin charged plane. In
are, however, crucial in many physical settings. For the Ref.[6],weintegratedtheone-loopequationsofacharge
case of electrostatics near macromolecular surfaces or liquid in contact with a thick dielectric membrane. Fi-
membrane interfaces - one of the most basic situations nally, in Ref. [10, 11], we considered the role played by
encountered in soft matter, this omission does not allow charge correlations on the electrophoretic and pressure-
to treat image charge effects of solvated ions. Another driven DNA translocation through nanopores.
crucialeffectisthechargereversalofmacromoleculesin- In addition, while the original self-consistent equa-
duced by the overscreening of their bare charge by mul- tions have only covered the case of systems that can
tivalent counterions. This effect can indeed modify the be described by macroscopic dielectric constants, mod-
interactionsbetweenchargedobjectseveninaqualitative ified equations have been derived that can also include
way. effects from fluid structure. The first dipolar Poisson-
2
Boltzmann theory including solvent molecules as point- larityoftheresultingimage-chargepotentialdoesnotal-
dipoles was introduced in Ref. [12]. Abrashkin et al. in- low the one-loop expansion of the grand potential. As in
corporated into this model excluded volume effects [13]. many other branches of physics, variational approaches
One of us (SB) derived a Poisson-Boltzmann equation thus come as an often fruitful alternative. In this vein,
thatrelaxesthepoint-dipoleapproximationandincludes thestartingpointistheGibbsvariationalprocedurethat
solvent molecules as finite size dipoles [14]. This model consists in minimizing the variational grand potential in
that also accounts for the ionic polarizability was shown the form [3] Ω = Ω +(cid:104)H −H (cid:105) /Ξ, where H [φ] is a
v 0 0 0 0
to contain the non-locality of electrostatic interactions trialHamiltonianfunctionalandthebracket(cid:104)·(cid:105) denotes
0
observed in Molecular Dynamics simulations. Finally, the field-theoretic average with respect to this Hamilto-
we derived the dipolar self-consistent equations of this nian. For this Hamiltonian, the most general functional
model and generalized in this way Netz-Orland’s varia- ansatz is a gaussian one including the mean electrostatic
tional equations to explicit solvent liquids [15]. potential Φ and the covariance of the field expressed via
Our ambition in this topical review is to provide a its Green’s function G as variational parameters
quick technical introduction to the method and the re-
sults that have been achieved with this approach. We
1(cid:90) (cid:90)
have attempted to make it accessible to a newcomer to H [φ]= [φ(r)+iΦ(r)](ΞG(r,r(cid:48)))−1[φ(r(cid:48))+iΦ(r(cid:48))] .
0 2
theapproachbygivingsufficientamountoftechnicalde- r r(cid:48)
(2)
tail for the simpler cases. This level of detail then, by
For definiteness, we now consider the case of monovalent
necessity, diminishes for the more complex ones that fol-
ions with charges ±q confined to a region Ω in presence
low,butwehopethatbythattimeareaderwillingtogo
of a fixed charge density (cid:37) . The Hamiltonian is then,
through about the first third of the equations in a step- f
following [16]
wisemannerwillhavenodifficultyinfollowingtherestof
the paper. For the latter part, as in all reviews, we refer
our readers to the original articles for further details. 1 (cid:90) (cid:20)(∇φ)2 Λ (cid:21)
The material presented in the review is organized as H[φ]= 2π 2 +i(cid:37)fφ− 2eΞG0(r,r)/2cosφ ,
follows. In Section 2 we derive the model equations r
(3)
which have been called either variational PB equations,
where Λ is the fugacity of the ions, and G (r,r(cid:48)) =
self-consistent field equations, or fluctuation-enhanced 0
1/|r−r(cid:48)| is the bare Coulomb potential. The introduc-
Poisson-Boltzmann equations (FE-PB), for the case of a
tion of G at this level takes care of the regularization of
systemwith1-1salt. Wealsodiscussthelimitsofvalidity 0
theGreen’sfunctionG(r,r(cid:48))inthefinalequationsasthe
thatcanbeexpectedfromtheequations. InSection3we
ionic self-energy corresponding to the equal-point corre-
reviewresultsforsystemswhosedielectricpropertiescan
lation function G(r,r) diverges in the present dielectric
beproperlycoveredbydielectricconstants. Inparticular,
continuum formalism. In eq.(3), this factor shifts the
we discuss situations of high current interest, the appli-
chemical potential of the ions. With this ansatz, we can
cation of the approach to nanopore geometries. Section
compute the grand potential Ω from which the sought
4 contains very recent extensions of the approach, the
equations follows after extremization with respect to the
fluctuation-enhanced Poisson-Boltzmann equations for a
functions Φ and G. These self-consistent equations read
dipolarsolvent,theDPBL-equation,aswellasanonlocal
as
version of the latter. Section 5 presents a brief summary
and outlook. We finally note that we explain, wherever
possible at present, the theoretical results closely in re- ∇2Φ(r)−Λe−Ξc(r)/2sinhΦ(r)=−2(cid:37) (r), (4)
f
lation to experimental findings. This is obviously the (cid:104) (cid:105)
ultimate way to validate a theory, and the reader is in- ∇2−Λe−Ξc(r)/2coshΦ(r) G(r,r(cid:48))=−4πδ(r−r(cid:48)),(5)
vited to see how far the self-consistent approach to soft c(r)= lim [G(r,r(cid:48))−G (r,r(cid:48))] (6)
0
matter electrostatics is carrying so far. r→r(cid:48)
Equation (4) is a modified Poisson-Boltzmann equa-
II. THE FLUCTUATION-ENHANCED
tion, augmented by the correlation function (6) in the
POISSON-BOLTZMANN EQUATIONS
exponential. The correlation function fulfills a modi-
fied Debye-Hu¨ckel (DH) equation, eq.(5), in which the
A. Derivation. usual inverse Debye-length κ2 is replaced by a nonlin-
ear function of both c(r) and φ(r). These equations
The fluctuation-enhanced Poisson-Boltzmann equa- are one realization (for the case of 1-1 salt, and with-
tions result from the observation that a simple pertur- out further specification of the fixed charge geometries)
bative treatment of the non-linear Poisson-Boltzmann of the self-consistent or fluctuation-enhanced Poisson-
equationhasonlypoorconvergenceproperties. Thisisin Boltzmannequations. OneshouldalsonotethatEqs.(4)-
particularthecaseofelectrolytesincontactwithlowper- (5) can be easily generalized to an asymmetrical elec-
mittivitymacromoleculesormembraneswherethesingu- trolyte (see e.g. Ref. [7]).
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B. Validity. ofhighervalency,wherePB-theoryisknowntofail. Fur-
ther applications of the equations in the case of a dielec-
tric continuum concerned the effect of image charges on
The self-consistent equations (4) - (6) are, by their
macro-ions[19]andontheelectricaldoublelayer[20,21].
very construction, only approximate. Their validity ulti-
Ionsizeeffectsuponionicexclusionfromdielectricinter-
mately rests on the validity of the Gaussian assumption
faces and slit nanopores were treated in [22–24]. The
to begin with, and this is, as usual in variational ap-
modification of ion polarizabilities from the gas phase to
proaches,notalwayseasytoquantify. Onecan,however,
solvation in polar liquids were discussed in [25].
identify qualitatively the validity regime by considering
chargecorrelationsinabulkelectrolyte. Inthiscase,the
electrostatic potential φ vanishes and we are left with a
Debye-Hu¨ckeltypeequationfortheGreen’sfunction[17] A. One-loop expansion of SC equations and charge
reversal
−∇2G(r,r(cid:48))+Λe−Ξc(r)/2G(r,r(cid:48))=4πδ(r−r(cid:48)). (7)
Webeginthediscussionwiththeone-loop(1-(cid:96))expan-
If we define the screening parameter as κ2 ≡Λe−Ξc(r)/2, sionoftheelectrostaticSCequationsvalidexclusivelyfor
the Green’s function becomes dielectricallycontinuoussystemsε(r)=εw. Weconsider
a symmetric electrolyte composed of two ionic species
e−κ|r−r(cid:48)| with valencies ±q and bulk density ρ . In order to facil-
G(r,r(cid:48))= . (8) b
|r−r(cid:48)| itate the link to the research literature, in passing from
Eq. (4) to the subsequent Eq. (10) we will introduce the
Inserting Eq. (8) into Eq.(6), one finds c = −κ. We are definition of the new average potential ψ(r) = −qΦ(r)
thus led to a first self-consistency condition given by and the Green’s function v(r,r(cid:48)) = ΞG(r,r(cid:48)). For this
case, the SC equations read as
κ2 =ΛeΞκ/2. (9)
∇2ψ(r)−κ2be−Vw(r)−q22δv(r,r)sinh[ψ(r)]= −4π(cid:96)Bqσ(r)
This equation ceases to have a solution for large values (10)
of the coupling parameter Ξ, clearly indicating that the
stoelfm-coodnesrisatteentcheaqrugaetcioonrrsel(a5t)i-o(n6s).aTrehiosnclyonvdailtiidonfocranwebaek, (cid:26)∇2−κ2be−Vw(r)−q22δv(r,r)cosh[ψ(r)](cid:27)v(r,r(cid:48))=
inturn,quantifiedthroughEq.(1)intermsofthemodel
−4π(cid:96) δ(r−r(cid:48)), (11)
parameters. B
Ininhomogeneousliquids,thevaliditylimitoftheself-
where the ionic self-energy (or renormalized equal-point
consistent equations (5)-(6) is not so easy to assess, as
correlation function) is defined by
they are, due to their highly non-linear character, not
amenable to analytic solutions, even for simple geome- δv(r)≡δv(r,r)=(cid:96) κ +v(r,r)−vb(0), (12)
B b c
tries. Numerical methods have been recently developed
to solve them [6, 7, 16, 18]. In particular, by comparison with the DH screening parameter κ2 = 8π(cid:96) q2, and the
b B
with MC simulations, Refs. [6] and [7] identified the va- functionV (r)istheionicstericpotentialaccountingfor
w
lidity regime of the equations for liquids confined to slit the rigid boundaries in the system.
and cylindrical nanopores, respectively. In the follow- The1-(cid:96)expansionoftheseequationsconsistsinTaylor-
ing we will discuss approximate solutions of Eqs. (4)-(6) expanding Eqs. (10)-(11) in terms of the electrostatic
and their modifications to physically relevant situations Green’s function v(r,r(cid:48)). Splitting the average potential
whichtosomeextentalsopermitanalyticalcalculations. intoitsMFand1-(cid:96)componentsasψ(r)=ψ (r)+ψ (r),
0 1
In particular, we confront these solutions to data from oneobtainsfortheMFpotentialandthe1-(cid:96)electrostatic
experiment and simulations. Green’s function the equations
∇2ψ (r)−κ2e−Vw(r)sinh[ψ (r)]=−4π(cid:96) σ(r), (13)
0 b 0 B
III. THE VARIATIONAL EQUATIONS FOR A
DIELECTRIC CONTINUUM
(cid:110) (cid:111)
∇2−κ2e−Vw(r)cosh[ψ (r)] v(r,r(cid:48))=
b 0
In this section we turn to the application of the SC
−4π(cid:96) δ(r−r(cid:48)), (14)
equations to some specific physical situations. First, we B
discuss ion correlations and charge reversal at a planar
and for the 1-(cid:96) correction to the average potential
interface,andthenmoveontodynamiceffectsassociated
withDNAtranslocationthroughmembranepores. These (cid:110) (cid:111)
∇2−κ2e−Vw(r)cosh[ψ (r)] ψ (r)=
examples were selected as they allow to convey the main b 0 1
insights on correlation effectsthat can be gained from q2
the variational approach, and in conjunction with ions − 2 κ2be−Vw(r)δv(r)sinh[ψ0(r)]. (15)
4
Inordertosolvethesystemofdifferentialequations(13)- whereweintroducedtheparameters=κ µandtheaux-
b √
(15), we first invert Eq. (14) and recast it in a more iliaryfunctionsz =−ln[γ(s)]/κ andγ(s)= s2+1−
0 b
practical form for analytical evaluation. By using the 1. In the DH-limit of weak surface charge or strong salt,
definition of the Green’s function the potential (21) becomes ψ0(z)(cid:39)(2/s)e−κbz. The pa-
rameter s thus corresponds to the inverse magnitude of
(cid:90)
dr v−1(r,r )v(r ,r(cid:48))=δ(r−r(cid:48)), (16) the surface potential.
1 1 1
Takingadvantageoftheplanarsymmetryinthe(x,y)-
planeonecanexpandtheGreen’sfunctionintheFourier
onecaninvertEq.(14)andexpresstheelectrostaticker-
basis as
nel as
(cid:90) ∞ dkk (cid:16) (cid:17)
v−1(r,r(cid:48))=−4π1(cid:96) (cid:110)∇2r−κ2be−Vw(r)cosh[ψ0(r)](cid:111)δ(r−r(cid:48)), v(r,r(cid:48))= 0 2π J0 k|r(cid:107)−r(cid:48)(cid:107)| v˜0(z,z(cid:48)), (22)
B
(17)
where r is the position vector in the (x,y)-plane and
in terms of which Eq. (15) can be written as (cid:107)
J the Bessel function of order zero. Substituting the
0
(cid:90) expansion (22) together with the MF-potential (21) into
dr1v−1(r2,r1)ψ1(r1)=q4ρbe−Vw(r2)δv(r2)sinh[ψ0(r2)]. the kernel equation (11), the latter takes the form
(18)
∂2(cid:2)1−θ(z)(cid:8)p2+2κ2csch2[κ (z+z )](cid:9)(cid:3)v˜(z,z(cid:48),k)=
MultiplyingEq.(18)withthepotentialv(r,r )andinte- z b b b 0
2
grating over the variable r , one finally gets the integral
2
relation for the 1-(cid:96) potential correction completing the
−4π(cid:96) δ(z−z(cid:48)) (23)
equations (13)-(14) B
(cid:112)
(cid:90) where we defined the function p = k2+κ2. The solu-
ψ1(r)=ρbq4 dr1e−Vw(r1)v(r,r1)δv(r1)sinh[ψ0(r1)]. tion of Eq. (23) satisfying the cobntinuity of tbhe potential
v˜(z,z(cid:48))andthedisplacementfield∂ v˜(z,z(cid:48))atz =0and
(19) z
z =z(cid:48) reads
We also note that within 1-(cid:96)-theory the ion number den-
sities are given by
2π(cid:96) p
v˜ (z,z(cid:48))= B b [h (z)h (z(cid:48))+∆h (z)h (z(cid:48))] (24)
(cid:26) q2 (cid:27) 0 k2 + − − −
ρ (r)=ρ e−Vw(r)∓ψ0(r) 1− δv(r)∓ψ (r) . (20)
± b 2 1 for 0 ≤ z ≤ z(cid:48), where the homogeneous solutions of
Eq. (23) are given by
Wenowhavethemainsetofequationsreadyandturnto
theanalyticalsolutionsofEqs.(13)-(15)forasimplepla- (cid:26) κ (cid:27)
nargeometryinordertoinvestigatethechargeinversion h±(z)=e±pbz 1∓ pb coth[κb(z+z0)] , (25)
b
phenomenon [7]. Then we will couple these equations
with the Stokes equation and show that charge inversion and the ∆-function reads
gives rise to the reversal of the electrophoretic DNA mo-
bility[10]andthehydrodynamicallyinducedioncurrents ∆= κ2bcsch2(κbz0)+(pb−k)[pb−κbcoth(κbz0)].
through cylindrical pores [11]. κ2csch2(κ z )+(p +k)[p +κ coth(κ z )]
b b 0 b b b b 0
(26)
For z ≥ z(cid:48), the solution of Eq. (23) can be obtained by
1. Ionic correlations and charge reversal at planar interchanging the variables z and z(cid:48) in Eq. (24). From
interfaces.
now on we switch to the non-dimensionalized coordinate
z¯=κ z. Rescalingaswellthewave-vectoroftheFourier
b
In this part we consider the charge reversal effect in expansion (22) as k →u=p /κ and inserting the func-
b b
the case of a negatively charged planar interface located tion (24) into Eq. (12), the ionic self-energy accounting
at z = 0. The electrolyte occupies the half-space z > 0 for charge correlations takes the form
while the left half-space at z < 0 is ion-free. In the SC-
equations this corresponds to a steric potential Vw(r) = δv (z¯) = Γ(cid:90) ∞ du (cid:8)−csch2[z¯−lnγ (s)] (27)
∞ if z < 0 and Vw(r) = 0 for z > 0. The wall charge 0 1 u2−1 c
distribution function is σ(r) = −σ δ(z). By introduc- (cid:111)
s +∆¯ (u+coth[z¯−lnγ (s)])2e−2z¯u ,
ing the Gouy-Chapman length (cid:96) ≡ µ = 1/(2πq(cid:96) σ ) c
GC B s
that corresponds to the thickness of the interfacial coun-
terionlayer,thesolutionofEq.(13)satisfyingGauss’law where we defined the electrostatic coupling parameter
ψ(cid:48)(z =0)=2/µ reads as Γ=(cid:96)Bκb and
0
√ √
(cid:0) (cid:1)(cid:0) (cid:1)
ψ0(z)=−2ln(cid:20)11+−ee−−κκbb((zz++zz00))(cid:21), (21) ∆¯ = 11++ss(cid:0)ssuu+−√ss22++11(cid:1)(cid:0)uu−+√uu22−−11(cid:1). (28)
5
(a)
Figure 2: (Color online) Charge renormalization factor η(s),
see Eq. (33) against s−1, s = κ µ, for several values of the
b
coupling parameter Γ=(cid:96) κ .
B b
where the MF-ion densities are given by ρMF(z¯) =
±
ρbe∓ψ0(z¯). The top plot (a) illustrates the density cor-
rection at a weakly charged membrane (s = 1000). In
(b) thiscase,theabsenceofionsinthelefthalf-spaceresults
in the ionic screening deficiency close to the membrane
Figure1: (Coloronline)One-loopcorrectionstothecounter- surface. Asaresult,ionsprefertomoveawayfromthein-
ion (solid lines) and co-ion densities (dashed lines) from terface towards the bulk where they are more efficiently
Eq.(31),fortwodifferentvaluesoftheparameters=κbµ,the screened and possess a lower free energy. This trans-
ratio of the Gouy-Chapman and the screening length. Case lates in turn into a positive ionic self-energy δv (z¯) > 0
0
(a): s = 1000 (a weakly charged membrane) and case (b):
(see the inset) and a decrease of the MF-level co-ion and
s = 0.75 (a strongly charged membrane). The inset shows
counter-ion densities (main plot) by charge correlations.
theionicself-energyandtheone-loopcorrectiontotheexter-
nal potential for these parameters. In the plot of Figure 1 (b), we consider a strongly
chargedmembrane(s=0.75). Inthisparameterregime,
the strong counterion attraction results in an interfacial
chargeexcess. Ionsbeingmoreefficientlyscreenedinthe
InsertingintoEq.(19)theMF-potential(21),theGreen’s vicinity of the membrane surface, they tend to approach
function (24), the self-energy (27), and carrying out the the interface. This effect is reflected in the attractive
spatial integral, the 1-(cid:96)-correction to the average poten- ionic self-energy δv (z¯) < 0 (inset) and the amplifica-
0
tial takes the form tion of the MF-density of both coions and counterions
(main plot). In Ref. [6], it was shown that the transition
q2 (cid:90) ∞ du
ψ (z¯)= Γcsch[z¯−lnγ (s)] F(z¯,u), (29) between these two regimes with the surface self-energy
1(cid:96) 4 c u2−1
1 δv0(0) switching from positive to negative occurs when
the size of the interfacial counterion layer becomes com-
where we introduced the auxiliary function
parable to the ionic screening radius in the bulk region,
F(z¯,u) = √2+s2 −∆¯ (cid:18)1 +2u+ 2√+3s2 (cid:19) it.iec.maetmµbra(cid:39)neκc−bh1a.rgTehσi∗s =eq(cid:112)ua2liρty/(yπie(cid:96)ld)satbhoevcehwarhaicchtertihse-
s 1+s2 u s 1+s2 s b B
interfacial screening dominates the bulk screening.
+∆¯e−2uz¯+(cid:0)∆¯e−2uz¯−1(cid:1)coth[z¯−lnγ ((s3)0]). For a negatively charged membrane, the MF-potential
u c
is negative (see Eq. (18)). Furthermore, in the inset of
Figure 1(b) where we plotted the 1-(cid:96)-correction Eq. (29)
We note that the correlation-corrected ion densities (20) to the average potential, one sees that the former is pos-
are fully characterized by the potentials (21), (27), itive. This stems from the interfacial screening excess.
and(29). Onealsoseesthatthesefunctionsdependsolely Thus, at strongly charged membranes, correlations at-
on the parameters s and Γ. tenuate the magnitude of the negative MF-potential.
InFigure1, weshowthe1-(cid:96)-correctiontotheionden- Wewillnowshowthatthispeculiarityistheprecursor
sities of the charge reversal effect. In order to illustrate this
point, we note that at large distances from the charged
∆ρ (z¯)≡ ρ±(z¯)−ρM±F(z¯) =−q2δv (z¯)∓ψ (z¯), (31) plane z¯(cid:29) 1, the MF-potential (21) behaves as ψ0(z¯) (cid:39)
± ρMF(z¯) 2 0 1(cid:96) −4γ(s)e−z¯. Expanding the 1-(cid:96)-correction to the average
±
6
electrostatic potential (30) in the same limit, one finds as a charged cylinder with radius a = 1 nm, translo-
thatthecorrelation-correctedaveragepotentialψ (z¯)= cating through a nanopore of cylindrical geometry with
1(cid:96)
ψ (z¯)+ψ (z¯) takes the form radiusd=3nm. TheconfigurationisdepictedinFigure
0 1
3. The solution of the 1-(cid:96) Eqs. (13)-(15) in a cylindrical
2
ψ (z¯)(cid:39)− η(s)e−z¯, (32) geometry is similar to their solution in planar geometry.
1l s Thus, we will skip the technical details here and refer to
Refs. [7] and [10] for details. The DNA molecule has a
where we introduced the charge renormalization factor
negative surface charge distribution σ(r)=−σ δ(r−a),
accounting for electrostatic many-body effects p
with σ = 0.4 e/nm2 and r the radial distance of ions
p
(cid:20) q2Γ (cid:21) from the symmetry axis of the cylindrical polymer. In
η(s)=2sγ(s) 1− I(s) , (33)
8 thegeneralcase,thepermittivityofthenanoporeεm and
DNA may differ from the water permittivity ε . How-
w
with the auxiliary function ever, in order to simplify the technical task, we assume
thatthereisnodielectricdiscontinuityinthesystemand
I(s)= (34)
set ε =ε =ε =80.
m p w
As the electrophoretic translocation of DNA under an
(cid:90) ∞ du (cid:26) 2+s2 (cid:18)1 2+3s2 (cid:19)(cid:27) external potential gradient ∆V corresponds to the col-
√ −1−∆¯ +2u+ √ . lective motion of the electrolyte and the DNA molecule,
u2−1 s 1+s2 u s 1+s2
1 we need to derive first the convective fluid velocity u (r)
c
In Figure 2, we plot Eq. (33) versus s−1 for various given by the Stokes equation
coupling parameters Γ. In the MF-limit Γ = 0, and
∆V
the charge renormalization factor η(s) accounts for non- η∇2u (r)+eρ (r) =0, (37)
c c L
linearities neglected by the linearized PB-theory. As
the latter overestimates the actual value of the electro- withtheviscositycoefficientofwaterη =8.91×10−4 Pa·
static potential at strong charges, with increasing sur-
s, the nanopore length L, and the liquid charge density
face charge from left to right, the correction factor η(s)
drops from unity to zero. Moreover, one sees that at the (cid:88)
ρ (r)= q ρ (r), (38)
c i i
coupling parameter Γ = 2, the charge renormalisation
i
factor passes from positive to negative. In other words,
at large distances from the interface, the total average where ρ (r) is the number density of the ionic species i
i
potential (32) switches from negative to positive. This with valency q ,
i
is the signature of the charge inversion effect. As the
adsorbed counterions overcompensate the negative fixed ρ (r)=ρ e−qiψ0(r)(cid:20)1−q ψ (r)− qi2δv(r)(cid:21). (39)
surface charge, the interface acquires an effective posi- i ib i 1 2
tivecharge. Forsmalls(orlargesurfacechargedensity),
Eq. (33) takes the asymptotic form By making use of the Poisson equation ∇2ψ1(cid:96)(r) +
4π(cid:96) ρ (r) = 0 in Eq. (37), the latter can be written
B c
(cid:20) (cid:21)
η(s)=2s 1−Γπ−4ln(2s) +O(cid:0)s2(cid:1). (35) explicitly in the cylindrical geometry as
16
η ∂ ∂ k T ∆V ∂ ∂
r u (r)− B rε(r) ψ (r)=0. (40)
Settingtheequality(35)tozero,onefindsthatthecharge r∂r ∂r c er L ∂r ∂r 1(cid:96)
reversal takes place at the particular value
In order to solve Eq. (40), we first note that the 1-(cid:96)-
1 (cid:18)π 4(cid:19) potential ψ (r) satisfies Gauss’ law φ(cid:48)(a) = 4π(cid:96) σ at
1(cid:96) B p
s (Γ)= exp − . (36)
c 2 4 Γ theDNAsurface. Inthesteady-stateregimewhereDNA
translocatesatconstantvelocityv, thelongitudinalelec-
Havingestablishedthechargereversaleffectintheplanar tric force per polymer length F = −2πaeσ ∆V/L will
e p
geometry,wewillnowturntoitsinfluenceonioncurrents compensate the viscous friction force F = 2πaηu(cid:48)(a),
v c
and polymer mobilities through cylindrical nanopores, that is F +F = 0. Finally, we impose the boundary
e v
which is a problem of current interest in the field of soft conditionsu (d)=0(no-slip)andu (a)=v. Integrating
c c
matter physics. Eq. (40) and imposing the above-mentioned conditions,
we get the convective flow velocity
2. Electrophoretic DNA mobility reversal by multivalent ∆V
u (r)=−µ [φ(d)−φ(r)], (41)
counterions. c e L
and the DNA translocation velocity
In this part we discuss the effect of charge inversion
induced by multivalent ions on the electrophoretic mo- ∆V
bility of a DNA molecule [10]. The molecule is modeled v =−µe L [φ(d)−φ(a)], (42)
7
Figure 3: (Color online) Nanopore geometry: A cylindrical
polyelectrolyte of radius a = 1 nm, surface charge −σ , and
p
dielectric permittivity ε is confined to a cylindrical pore of
p
radius d=3 nm, with wall charge −σ , and membrane per-
m
mittivity εm. The permittivity of the electrolyte is εw =80. Figure5: (Coloronline)Cumulativechargedensitiesrescaled
with the bare DNA charge (top plots) and solvent velocities
(bottomplots)atfixedK+ andvaryingSpd3+ concentration
in(a)and(b),andfixedSpd3+andvaryingK+concentration
in (c) and (d). In each column, the same colour in the top
andbottomplotscorrespondstoagivenbulkcounterioncon-
centrationdisplayedinthelegend. Theremainingparameters
are the same as in Figure 4.
and quadrivalent spermine ions, and with the increase of
the multivalent ion density, the DNA velocity switches
fromnegativetopositive. Inotherwords,atlargemulti-
valent counter-ion concentrations, the molecule translo-
cates parallel with the field. Finally, one notes that be-
yond the mobility reversal density, the positive DNA ve-
locity in spermidine and spermine liquids reaches a peak
and drops beyond this point.
In order to explain the reversal of the DNA mobility
by multivalent counterions, we plot in Figure 5 (a) the
Figure 4: (Color online) Polymer translocation velocity
cumulative charge density of the KCl+Spm3+Cl fluid
againstthedensityofthemultivalentcounterionspeciesIm+ 3
(see the legend) in the electrolyte mixture KCl+ICl . K+ including the polyelectrolyte charge
m
densityisfixedatρ+b =0.1M.Theconfineddouble-stranded (cid:90) r
DNA has surface charge σp =0.4 e/nm2. The potential gra- ρtot(r)=2π dr(cid:48)r(cid:48)[ρc(r(cid:48))+σs(r(cid:48))] (44)
dientis∆V =120mV.Circlesmarkthechargeinversion(CI) a
point of the DNA molecule.
forvariousbulkSpd3+ concentrations. Wealsoillustrate
in Figure 5 (b) the cumulative liquid velocity (42) at the
samedensities. AtthelowSpd3+densityρ =2×10−3
with the reduced electrophoretic mobility 3+b
M, moving from the DNA surface towards the pore
k Tε wall, the total charge density rises from the net DNA
µe = Beη w. (43) charge towards zero. This corresponds qualitatively to
the MF-picture where counterions gradually screen the
We now consider an electrolyte mixture of composi- DNA charge. Hence, the negatively charged fluid and
tionKCl+ICl includinganarbitrarytypeofmultivalent DNA move oppositely to the field, that is u (r) < 0
m c
counterions Im+. The electrophoretic DNA velocity (42) and v = u (a) < 0. Increasing now the Spd3+ density
c
against the multivalent ion density ρ is displayed in to ρ = 7 × 10−3 M, the cumulative charge density
mb 3+b
Figure 4 for various multivalent ion types. With diva- switches from negative to positive at r (cid:38) 1.3 nm. This
lent Mg2+ ions, the DNA translocation velocity is nega- is the signature of charge reversal where due to correla-
tive. This corresponds qualitatively to the classical MF- tion effects induced by Spd3+ ions, counterions locally
transport where the negatively charged DNA molecule overcompensate the DNA charge. Consequently, the liq-
moves oppositely to the applied electric field. However, uid flows parallel with the field (u (r)>0) in the region
c
in the mixed electrolytes containing trivalent spermidine r > 1.3 nm. However, because the hydrodynamic drag
8
force is not sufficiently strong to dominate the electro- in hydrodynamically-induced transport experiments, the
staticcouplingbetweentheDNAmoleculeandtheexter- externally applied voltage difference in Figure 3 is re-
nalelectricfield,themoleculeandtheliquidaroundcon- placed with a pressure gradient ∆P > 0 at the pore
z
tinue to move in the direction of the field. At the larger edges. Theresultingioniccurrentthoroughthenanopore
Spd3+ densityρ =1.2×10−2 Mwherechargereversal of length L = 340 nm and radius d = 3 nm is given by
3+b
becomes more pronounced, the drag force compensates the number of charges flowing per unit time through the
exactly the electric force on the DNA molecule. As a re- cross section of the channel,
sult, DNA stops its translocation, i.e. v =u (a)=0. At
larger Spd3+ concentrations, the DNA moleccule and the (cid:90) d∗
I =2πe drrρ (r)u (r). (45)
electrolyte move parallel with the field. We emphasize str c s
a∗
that an important challenge in DNA translocation con-
sistsintheminimizationoftheDNAtranslationvelocity InEq.(45),weintroducedtheeffectiveporeandpolymer
for an accurate sequencing of its genetic code [26]. The radiid∗ =d−ast anda∗ =a−ast whereast =2˚Astands
present result suggests that this can be achieved by tun- for the Stern layer accounting for the stagnant ion layer
ing trivalent or quadrivalent ion densities in the liquid. in the vicinity of the charged pore and DNA surfaces.
The charge density function ρ (r) is defined by Eq. (38).
To summarize, it is found that the inversion of the c
The streaming current velocity is given in turn by the
DNA mobility is induced by charge reversal. However,
solution of the Stokes equation with an applied pressure
charge inversion should also be strong enough for mo-
gradient,
bility reversal to occur. This can also be seen in Fig-
ure 4 where we display the charge inversion points by
∆P
open circles. One notes that charge inversion precedes η∆us(r)+ Lz =0. (46)
the mobility reversal that occurs only with trivalent and
quadrivalent ions. We now focus on the peak of the ve- Solving Eq. (46), we impose the boundary conditions
locity curves in this figure. One notes that at the largest u (d∗)=0(no-slip)andu (a∗)=v. Wealsoaccountfor
s s
Spd3+ concentration in Figure 5 (a), the first positive the fact that the viscous friction force F =2πa∗ηu(cid:48)(a∗)
v s
peak of the cumulant charge density is followed by a vanishes in the stationary state of the flow. Integrat-
well. The corresponding local drop of the cumulative ing Eq. (46) under these conditions, the streaming flow
charge density stems from the attraction of Cl− ions to- velocity follows in the form of a Poisseuille profile,
wardsthechargeinvertedDNA.AthigherSpd3+concen-
∆P (cid:104) (cid:16) r (cid:17)(cid:105)
trations, the stronger chloride attraction attenuates the u (r)= z d∗2−r2+2a∗2ln . (47)
charge reversal effect responsible for the mobility inver- s 4ηL d∗
sion. This explains the reduction of the inverted charge
The knowledge of the charge density (38) and liquid ve-
mobility at large multivalent densities in Figure 4. Fi-
locity (47) completes the calculation of the ionic current
nally, we consider the effect of potassium concentration
of Eq. (45).
that can be easily tuned in translocation experiments.
In Figure 6 (a), we report streaming currents of the
In Figures 5 (c) and (d), we show the charge density
electrolytemixtureKCl+ICl againstthereservoirden-
andvelocityatfixedSpd3+ concentrationforvariousK+ m
sity of the multivalent cation species Im+ in a neutral
densities. Starting at the charge inverted density values
pore. At weak multivalent ion densities the current is
ρ =1.2×10−2 M and ρ =10−2 M, and raising the
3+b +b positive. This corresponds to the MF-regime where the
bulk potassium concentration from top to bottom, the
negatively charged translocating DNA attracts cations
cumulativechargedensityisseentoswitchfrompositive
into the pore. Increasing the bulk magnesium concen-
to negative. This drives the DNA and electrolyte veloci-
tration in the KCl + MgCl liquid, in agreement with
ties from positive to negative. Thus, K+ ions cancel the 2
MF ion transport picture, the streaming current drops
DNA mobility inversion by weaking the charge reversal
slightlybutremainspositive. However,intheliquidscon-
effect induced by ion correlations.
tainingspermidineandspermineions,atacharacteristic
In the next paragraph, we will discuss the effect of
multivalent ion density, the current turns from positive
charge reversal on ion currents induced by a pressure
to negative. It is noteworthy that this streaming cur-
gradient.
rent reversal has been previously observed in nanofluidic
transportexperimentsthroughchargednanoslitswithout
DNA [27].
3. Inversion of hydrodynamically induced ion currents The positive ion currents of Figure 6 (a) indicating
through nanopores. a net negative charge flow through the pore cannot
be explained by MF-transport theory. In order to ex-
We now investigate the effect of charge correlations plain the underlying mechanism behind the current re-
on streaming currents during hydrodynamically-induced versal we plot in Figure 6 (b) the electrostatic cumu-
polymer translocation events [11]. The DNA-membrane lative charge density ρ (r) = 2π(cid:82)rdr(cid:48)r(cid:48)ρ (r(cid:48)) and
cum a c
geometry including the electrolyte mixture is the same the hydrodynamic cumulative charge density ρ∗ (r) =
as in the previous section. The only difference is that 2π(cid:82)r dr(cid:48)r(cid:48)ρ (r(cid:48)) of the KCl+SpdCl liquid nocrummalized
a∗ c 3
9
pore induced in turn by the DNA charge reversal. It
is important to note that, similar to the electrophoretic
DNA transport of the previous section, the observation
of streaming current reversal necessitates a strong DNA
charge inversion for the adsorbed anions to bring the
dominantcontributiontothehydrodynamicflow. Thisis
againillustratedinFigure6(a)wherethechargereversal
densities (open circles) are seen to be lower than current
inversion densities by several factors.
B. Solving SC equations in dielectrically
inhomogeneous systems
In the previous paragraphs, we have considered the
consequencesofthechargereversaleffectindifferentset-
tings. In the next section, we will focus on the image-
charge effects in dielectrically inhomogeneous systems.
1. Inversion scheme for the solution of the SC equations.
Figure 6: (Color online) (a) Streaming current curves at a In this subsection we present the solution of the SC
pressure gradient ∆Pz = 1 bar against the reservoir den- Eqs. (10)-(11) in dielectrically inhomogeneous systems.
sity of the multivalent counter-ion species listed in the leg- The technical details of the solution scheme that can
end. Open circles mark the DNA charge reversal (CR)
be found in Refs. [6, 7] will be briefly explained for the
points. (b) Electrostatic (black curves) and hydrodynamic
case of neutral interfaces. In this case where the av-
(red curves) cumulative charge densities, and (c) Cl− densi-
erage potential ψ(r) is zero, the fluctuation-enhanced
tiesintheKCl+SpdCl liquidatthereservoirconcentrations
3 Poisson-Boltzmann Eq. (10) vanishes. This leaves us
ρ =0.01M(dashedcurves)and0.1M(solidcurves). The
3+b
neutral nanopore (σ =0) contains a double-stranded DNA with Eq. (11) to be solved in order to evaluate the ion
m
molecule of charge density σ = 0.4 e/nm2, with a bulk K+ density
p
density ρ =0.1 M in all figures.
+b
ρi(r)=ρbe−q22δv(r), (48)
with the ionic self-energy δv(r) defined by Eq. (12). The
by the DNA charge. The hydrodynamic charge density iterativesolutionstrategyconsistsinrecastingthediffer-
accounts only for the mobile charges contributing to the ential equation (11) in the form of an integral equation.
streaming flow. Figure 6 (c) displays the chloride den- To this aim, we reexpress Eq. (14) as
sities between the DNA and pore surfaces. At the bulk
(cid:104) (cid:105)
spermidine concentration ρ3+b = 0.01 M, Figure 6 (b) ∇2−κ2be−Vw(r) v(r,r(cid:48)) = −4π(cid:96)Bδ(r−r(cid:48))=
shows that the electrostatic cumulative charge density
exceeds slightly the DNA charge. This is the sign of a
DNA charge reversal effect. This in turn leads to a weak −4π(cid:96) δn(r)v(r,r(cid:48)) (49)
Cl− excess ρ (r) > ρ between the pore and the DNA B
− −b
(see Figure 6 (c)). However, because the charge reversal wherewedefinedthenumber-densitycorrectionfunction
and the resulting chloride attraction is weak, the hydro-
(cid:20) (cid:21)
dtiyvne,ami.eic. flρ∗ow c>ha0rgfeordoam∗i<narte<d bdy∗.coAunntienrciroenasseisopfotshie- δn(r)=2q2ρbe−Vw(r) 1−e−q22δv(r) . (50)
cum
spermidine density to ρ = 0.1 M where one arrives
3+b
at the inverted current regime in Figure 6 (a), the in- Introducing now the DH-kernel
tensified DNA charge reversal results in a much stronger
Cl− attraction into the pore (see Figures 6 (b) and (c)). v−1(r,r(cid:48))=− 1 (cid:110)∇2−κ2e−Vw(r)(cid:111)δ(r−r(cid:48)) (51)
This strong anion excess leads in turn to a negative hy- 0 4π(cid:96)B b
drodynamic charge density ρ (d) < 0 and a negative
cum andusingthedefinitionoftheGreen’sfunction(16),one
streaming current through the pore.
can invert Eq. (49) and finally obtain
To conclude, these calculations show that ionic cur-
rent inversion during hydrodynamically induced DNA (cid:90)
v(r,r(cid:48))=v (r,r(cid:48))+ dr(cid:48)(cid:48)v (r,r(cid:48)(cid:48))δn(r(cid:48)(cid:48))v(r(cid:48)(cid:48),r(cid:48)). (52)
translocation events results from the anion excess in the 0 0
10
permittivity is ε = 1. We emphasize that this config-
m
uration is also relevant to the water-air interface whose
surfacetensionwasfirstcalculatedbyOnsagerandSama-
ras [28]. Because their calculation was based on the uni-
form screening approximation corresponding in our case
to the DH approach, the latter is called as well the On-
sagerapproximation. Theseparationdistanceisrescaled
by the ion size a , introduced in order to stabilize the
i
MC simulations. At the salt density ρ = 0.01 M (top
b
plot), the SC result exhibits a very good agreement with
MC simulations while the DH-theory slightly deviates
from the MC data, although the error is minor. At this
bulk salt concentration where the electrostatic coupling
parameter Γ = κ (cid:96) ≈ 0.2 corresponds to the weak-
b B
coupling regime, the accuracy of the DH-theory is ex-
pected. At the much higher salt density ρ = 0.2 M
b
(bottom plot), the SC-theory exhibits a good agreement
with MC data but the DH-result overestimates the ion
density over the whole interfacial regime. The weak de-
viation of the SC theory from the MC data is likely to
result from excluded volume effects related to ion size
but absent in the SC theory. Although the inclusion of
ionsizeisstillanopenissue,theexcludedvolumeeffects
can be included into the present SC theory via repulsive
Yukawa interactions (see e.g. Ref. [24]).
TheinaccuracyoftheDH-resultisduetothefactthat
theiondensityρ =0.2Mcorrespondstotheintermedi-
b
atecouplingregimeΓ≈1. Theoverestimationoftheion
Figure 7: (Color online) Ion density profiles at the dielectric
density stems from the non-uniform salt screening of the
interface against the distance from the surface with ε = 1,
m image-charge potential at the interface. The mobile ions
ε = 80, and ion diameter a = 4.25 ˚A at the bulk ion con-
w i
that screen this potential being also subject to image-
centration(a)ρ =0.01Mand(b)ρ =0.2M.Theredlines
b b
are MC simulation data, the blue lines the DH theory, and charge forces, the interfacial charge screening is lower
the black lines are from the SC scheme (52). The theoretical than the bulk screening. As the DH-theory assumes a
curves and MC data are from Ref. [6]. constant screening parameter κ , it cannot account for
b
the reduced screening at the interface. In the SC-theory,
thiseffectistakenintoaccountbythesecondtermonthe
rhs ofEq. (52), correctingthe uniform screeningapprox-
Eq. (52) expresses the solution of the SC-kernel Eq. (11) imation of the DH-theory. We also note that in the close
as the sum of the Debye-Hu¨ckel potential v0(r,r(cid:48)) solu- vicinity of the interface, the SC-theory slightly deviates
tiontoEq.(49)andacorrectiontermassociatedwiththe from the MC result. This may be due to ionic hard-core
non-uniform charge screening induced by image-charge effects neglected so far in the SC-formalism.
forces. The iterative solution scheme of Eq. (52) con-
sists in replacing at the first iterative step the potential
v(r,r(cid:48))ontherhsbytheDHpotentialv (r,r(cid:48)),inserting
0 3. Image-charge effects on ion transport through
the output potential v(r,r(cid:48)) into the rhs of the equation
α-Hemolysin pores.
atthenextiterativelevel,andcontinuingthiscycleuntil
numerical convergence is achieved. The solution scheme
The most significant implication of image-charge cor-
for charged interfaces/nanopores is based on the same
relations are found in electrophoretic charge transport
inversion idea but technically more involved. The more
through strongly confined α-hemolysin pores. The par-
general scheme can be found in Refs. [6, 7].
ticularly low conductivity of these pores cannot be ex-
plained by the MF electrophoresis. As in the previous
section,wewillmodeltheporeasaneutralcylinderwith
2. Image-charge induced correlations at planar interfaces. radius d = 8.5 ˚A and length L = 10 nm [26] (see Figure
3). In the most general case, the pore may be blocked
In Figure 7, we display the monovalent ion density by a single-stranded DNA molecule with radius a = 5.0
profiles obtained from the DH and the SC theories that ˚A[26]. Theporealsocontainsthemonovalentelectrolyte
we compare with MC simulations [6]. The dielectric in- solution KCl. Under an external potential gradient ∆V,
terface located at z = 0 is neutral and the membrane thetotalvelocityofthepositiveornegativeionicspecies