Dipolar depletion effect on the differential capacitance of carbon-based materials

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Dipolar depletion effect on the differential capacitance of carbon-based materials
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  Dipolar depletion effect on the differential capacitance of carbon based materials Sahin Buyukdagli 1 ∗ and T. Ala-Nissila 1 , 2 † 1 Department of Applied Physics and COMP center of Excellence,Aalto University School of Science, P.O. Box 11000, FI-00076 Aalto, Espoo, Finland  2 Department of Physics, Brown University, Providence, Box 1843, RI 02912-1843, U.S.A. (Dated: March 13, 2012)The remarkably low experimental values of the capacitance data of carbon based materials incontact with water solvent needs to be explained from a microscopic theory in order to optimizethe efficiency of these materials. We show that this experimental result can be explained by thedielectric screening deficiency of the electrostatic potential, which in turn results from the interfacialsolvent depletion effect driven by image dipole interactions. We show this by deriving from themicroscopic system Hamiltonian a non-mean-field dipolar Poisson-Boltzmann equation. This canaccount for the interaction of solvent molecules with their electrostatic image resulting from thedielectric discontinuity between the solvent medium and the substrate. The predictions of theextended dipolar Poisson-Boltzmann equation for the differential capacitance are compared withexperimental data and good agreement is found without any fitting parameters. PACS numbers: 03.50.De,05.70.Np,87.16.D- I. INTRODUCTION New generation supercapacitors are used for a broadrange of applications in nanoscopic scale technologies. Inwater purification technology, capacitive desalination isan efficient candidate that might replace the current lead-ing technics such as reverse osmosis, a membrane basedpurification method known to suffer from the membranefouling phenomenon [1]. Supercapacitors are also usedas low cost and long life energy storage devices with con-siderably higher energy densities than conventional elec-trolytic capacitors [2]. A through understanding of thedouble layer structure of these devices is thus necessaryto optimize their efficiency.The understanding of the double layer structure waslimited for several decades to the Gouy-Chapman-Sternmodel [3]. This model was later completed by consider-ing additional effects specific to electrolyte solutions, toname but a few, the steric layer associated with the size of solvent molecules as well as the dipolar alignment close tothe interface [4], non-local effects in electrolytes at metal-lic interfaces [5], ionic crowding [6] and overscreening [7]. Supercapacitors are commonly fabricated from carbonbased materials with a dielectric permittivity  ε m  ≈ 2 − 5much lower than the permittivity of the water solvent ε w  = 78. The polarization of the interface resulting fromthis large dielectric discontinuity can drastically changethe physics of the double layer. Image dipole interac-tions were considered in Ref. [8] for a metallic interface.However, the work accounted exclusively for the effectof image interactions on the dipolar orientation withoutconsidering their role on the interfacial dipole density.Furthermore, it was recently shown in Ref. [9] that the ∗ email:  sahin buyukdagli@yahoo.fr † email:  Tapio.Ala-Nissila@aalto.fi Gouy-Chapman (GC) capacitance largely overestimatesthe experimental data obtained for carbon based mate-rials. The failure of the GC capacitance was explainedby the unability of the Poisson Boltzmann formalism toaccount for non-local dielectric effects.In order to gain insight about the contribution of sur-face polarization effects on the capacitance of low dielec-tric substrates, we introduce in this work a first micro-scopic modeling of solvent molecules beyond the MF levelapproximation. Namely, we derive an extended dipolarPB (EDPB) equation that can self-consistently take intoaccount the interfacial solvent depletion. This depletionresults from the interaction of solvent molecules (modeledas dipoles) with their electrostatic image, an effect absentin the mean-field level DPB equation [11, 12]. The predic-tion of the EDPB equation is shown to agree well with ex-perimental data for the differential capacitance of carbonbased materials. However, it is also shown that the DPBformalism yields the same result as the PB equation, thatis, it overestimates the experimental data by one orderof magnitude. These observations strongly suggests thatthe dielectric discontinuity between the substrate and thesolvent can solely explain the observed low values of thedifferential capacitance of carbon based materials, unlikethe conclusion of Ref. [9] where it was argued that thesurface polarity does not play a major role in the differen-tial capacitance. Our results are also in agreement withthe experimental work in Ref. [10], where the surface hy-drophobicity was actually shown to strongly reduce thecapacitance of carbone nanotubes. II. EXTENDED DIPOLAR POISSONBOLTZMANN (EDPB) EQUATION We will present in this part the derivation of an ex-tended dipolar Poisson-Boltzmann formalism. The field-theoretic partition function of ions immersed in a dipo-   a  r   X   i  v  :   1   2   0   3 .   2   2   8   5  v   1   [  c  o  n   d  -  m  a   t .  s  o   f   t   ]   1   0   M  a  r   2   0   1   2  2lar liquid was derived in Ref. [11] as a functional integralover a fluctuating electrostatic potential  φ ( r ) in the form Z   = ￿   D φ e − H  [ φ ] , where the Hamiltonian functional isgiven by H  [ φ ] = 󿿿   d r 􏿿 [ ∇ φ ( r )] 2 8 π￿ B ( r )  − iσ ( r ) φ ( r ) 󟿿  (1) − 󿿿   d r d Ω 4 π λ d e E  d − V  w ( r )+ i ( p ·∇ φ ) −  i λ i 󿿿   d r e E  i − V  w ( r )+ i [ q i φ ( r )] . The first integral term of the Hamiltonian (1) is com-posed of the Maxwell tensor associated with a freely prop-agating electric field  ∇ φ ( r ) in the air, and a second partthat couples the corresponding potential  φ ( r ) to a fixedsurface charge distribution  σ ( r ). The second and thirdintegrals respectively account for the presence of solventmolecules (point dipoles) and ions of different species de-noted by the index  i . Moreover,  r  = ( x,y,z ) is the config-urational space and  Ω  = ( θ,ϕ ) stands for the solid anglecharacterizing the orientation of solvent molecules, with θ  the angle between the dipole and the  z  axis. We notethat the external wall potential  V  w ( r ) in Eq. (1) restrictsthe space accessible to the particles, and in the case of thesingle dielectric interface located at  z  = 0, it is of the form V  w ( z <  0) = ∞ and  V  w ( z >  0) = 0. Furthermore,  λ d  and λ i  are respectively the fugacity of dipoles and ions,  p  thedipole moment vector, and  q  i  stands for the valency of ions for the species  i . The heterogeneous Bjerrum lengthis defined as  ￿ B ( r ) =  e 2 / [4 πε ( r ) k B T  ], where  e  is the ele-mentary charge,  T   = 300 K is the ambient temperature,and  ε ( r ) =  ε 0 θ ( z )+ ε m θ ( − z ) is the dielectric permittivityof the medium in the absence of solvent molecules for thesame single planar interface geometry. More precisely,  ε 0 and  ε m  denote respectively the dielectric permittivity of the air (the subspace in  z >  0) and the low dielectricsubbstrate located at  z <  0. From now on, the dielec-tric permittivities will be expressed in units of   ε 0 . Wealso note that the Bjerrum length in the air is  ￿ B  ≈  55nm. Finally, the self energy of ions and polar moleculesthat are substracted from the potential and electrostaticfield respectively read  E  i  =  q 2 i 2  v bc ( r − r ￿ ) | r = r ￿  and  E  d  = 12 ( p ·∇ r )( p ·∇ r ￿ ) v bc ( r − r ￿ ), where the Coulomb operatorin the air is defined as  v bc − 1 ( r , r ￿ ) = − k B Tε 0 e 2  ∆ δ  ( r − r ￿ ).In this work, we aim at investigating the model of Eq. (1) beyond the MF approximation where surface po-larization effects are absent [11, 12]. One way to progress consists in opting for a variational minimization proce-dure that aims at finding the upper boundary for thedimensionless Grand potential of the system Ω = − ln Z  by minimizing the variational Grand potential defined asΩ v  = Ω 0  + ￿ H  − H  0 󿿿 0 , where the reference Hamiltonianis a Gaussian functional of the form H  0  = 12 󿿿  r , r ￿ [ φ ( r ) − iφ 0 ( r )] v − 10  ( r , r ￿ )[ φ ( r ￿ ) − iφ 0 ( r ￿ )] . (2)Furthermore,  φ 0 ( z ) is a variational external potential andthe electrostatic trial kernel is chosen in the same formas in Refs. [13–15], v − 10  ( r , r ￿ ) =  k B T e 2  −∇ ( ε v ( r ) ∇ ) +  ε v ( r ) κ 2 c ( r )  δ  ( r − r ￿ ) , (3)where the piecewise variational dielectric permittivity isdefined as  ε v ( r ) =  ε w θ ( z )+ ε m θ ( − z ) and the trial screen-ing length is given by  κ c ( r ) =  κ c θ ( z ). After performingthe functional integrals over  φ ( r ), one getsΩ v  = Ω 0  +  k B T  2 e 2 󿿿   d r d r ￿ δ  ( r − r ￿ ) ×  [ ε ( r ) − ε v ( r )] ∇ r ·∇ r ￿  − ε v ( r ) κ 2 c ( r )  v 0 ( r , r ￿ )+ 󿿿   d r  σ ( r ) φ 0 ( r ) −  k B T  2 e 2  ε ( r )[ ∇ φ 0 ( r )] 2   (4) −  i 󿿿   d r ρ i ( r ) − 󿿿   d r d Ω 4 π  ¯ ρ d ( r , Ω ) , where the gaussian contribution reads Ω 0  = − ln ￿   D φ e − H  0 [ φ ] . We also defined above the localion density ρ i ( r ) =  λ i e E  i − V  w ( r ) e − q i φ 0 ( r ) − q 2 i 2  v 0 ( r , r ) (5)and the local density of dipoles with orientation  Ω ¯ ρ d ( r , Ω ) =  λ d e E  d − V  w ( r ) (6) × e − p ·∇ φ 0 ( r ) − 12 ( p ·∇ r )( p ·∇ r ￿ ) v 0 ( r , r ￿ ) | r ￿ = r . By taking the derivative of the variational Grand po-tential Eq. (4) with respect to  κ c  and  ε v , one gets κ 2 c  = 4 π￿ w  i ρ b,i q  2 i  and  ε v  =  ￿ B /￿ w  =  ε w  = 1 + 4 π 3  ￿ B  p 20 ρ bd . These two relations respectively introducethe Debye-Huckel screening parameter and the Debye-Langevin form for the bulk dielectric permittivity of thewater medium  ε w . The additional variational equationfor  φ 0 ( z ), i.e.  δ  Ω v /δφ 0 ( r ) = 0 yields ∂ ∂  z  ˜ ε ( z ) ∂φ 0 ( z ) ∂  z  + 4 π￿ B σ ( z ) + 4 π￿ B  ρ i ( z ) q  i  = 0 ,  (7)where we took into account the translational symmetryof the electrostatic potential within the ( x,y ) plane. Wenote that the variational minimization left us in Eq. (7)with a spatially varying dielectric permittivity of theform˜ ε ( z ) = 1 −  4 π￿ B φ ￿ 0 ( z ) 󿿿   d Ω 4 π  ¯ ρ d ( z, Ω )  p z ,  (8)where  p z  =  p 0  cos θ  stands for the component of thedipolar moment vector  p  in the  z  direction. We willcall Eq. (7) the Extended Dipolar Poisson Boltzmann(EDPB) equation.The fugacity of dipoles and ions can be related to theirbulk density in the limit  z  → ∞  of the equations (5)  3and (6). By injecting the obtained relations for the fu-gacities with the inverse of the kernel Eq. (3) [13] into Eqs. (5) and (6), the local density functions take the form ρ i ( z ) =  ρ b,i e − V  w ( z ) e − q i φ 0 ( z ) − V  c ( z ) (9)¯ ρ d ( z, Ω ) =  ρ bd e − V  w ( z ) e − p ·∇ φ 0 ( z ) − V  d ( z, Ω ) ,  (10)where we defined the following ionic and dipolar poten-tials, V  c ( z ) =  q  2 ￿ w 2 󿿿   ∞ 0 d kk ρ c ∆ e − 2 ρ c z (11) V  d ( z, Ω ) =  U  d ( z ) +  T  d ( z )cos 2 θ ,  (12)with the functions U  d ( z ) =  ￿ w  p 20 4 󿿿   d kk 3 ρ c ∆ e − 2 ρ c z (13) T  d ( z ) =  ￿ w  p 20 4 󿿿   d kk ρ c (2 ρ 2 c  − k 2 )∆ e − 2 ρ c z .  (14)and ∆ = ( ρ c  −  η k ) / ( ρ c  +  η k ),  η  =  ε m / ε w , and ρ c  =   κ 2 c  +  k 2 . Carrying out the integral over  θ  inEq. (8) with the dipole density Eq. (10) and the potential Eq. (12), the local dielectric permittivity takes the form˜ ε ( z ) = 1 + 4 π 3  ￿ B  p 20 ρ db e − V  w ( z ) e − U  d ( z ) J  ( z ) ,  (15)where we defined the function J  ( z ) = 3 √  π 8 T  3 / 2 d  ( z ) e p 20 φ ￿ 20 ( z )4 T d ( z ) { Erf [Ψ + ( z )] + Erf [Ψ − ( z )] } . − 3 e − T  d ( z ) 2 T  d ( z )sinh[  p 0 φ ￿ 0 ( z )]  p 0 φ ￿ 0 ( z )  ,  (16)and the potentialsΨ ± ( z ) = 2 T  d ( z ) ±  p 0 φ ￿ 0 ( z )2   T  d ( z ) .  (17)The EDPB Eq. (7) has to be solved numerically withthe ionic density profiles of Eq. (9) and the dielectricpermittivity profile of Eq. (15).The second order differential equation (7) should besolved with the boundary conditions  φ 0 ( z  → ∞ ) = 0and  φ ￿ 0 ( z  →  0 + ) = 2 ε w /µ , where the second boundarycondition valid over the parameter domain 0 ≤ ε m  ≤ ε w follows by integrating Eq. (7) in the close neighborhoodof the interface, and noting that according to the dipo-lar potentials of Eqs. (13) and (14), one has  ρ d (0) = 0and ˜ 󿿿 (0) = 1 on the boundary. We also note that in thelimit where the potentials  V  c ( z ),  U  d ( z ), and  T  d ( z ) van-ish, EDPB equation. (7) reduces to the mean field DPBequation of Refs. [11, 12].We finally note that the orientation averaged densityof solvent molecules is obtained according to  ρ d ( r ) = ￿   d Ω 4 π  ¯ ρ d ( r , Ω ). Evaluating the integral over  θ , the solventdensity takes the form ρ d ( z ) =  ρ bd √  π 4 √  T  d ( z ) e − V  w ( z ) e − U  d ( z ) e p 20 φ ￿ 20 ( z )4 T d ( z ) (18) ×{ Erf [Ψ + ( z )] + Erf [Ψ − ( z )] } . III. NUMERICAL RESULTS We will investigate in this part the EDPB Eq. (7) fora symmetric electrolyte composed of two ion species of bulk densities  ρ b,i  =  ρ bi  and valency  q  i  =  q  . All numer-ical results will be derived for monovalent ions ( q   = 1)in contact with a negatively charged planar surface, i.e. σ ( z ) =  − σ s δ  ( z ) with  σ s  >  0. We also note that withinthe convention adopted in this article, the surface charge σ s  is expressed in units of the elementary charge  e . More-over, the model parameters  ρ db  and  ε w  are taken thesame as in Ref. [9]. Namely, the bulk density of sol-vent molecules is  ρ db  = 50 . 8 M, which yields with thedipole moment  p 0  = 1 ˚A the bulk dielectric permittivity ε w  = 71.The potential profile obtained from the numerical solu-tion of the EDPB Eq. (7) for the parameters  ρ bi  = 0 . 1 M, ε m  = 1 and  σ s  = 0 . 01 nm − 2 is reported in Fig. 1.a. Onenotices that the potential profile is composed of three re-gions, namely two successive layers close to the interfacewhere  φ 0 ( z ) behaves as a linear function of   z , and a thirdlayer over which  φ 0 ( z ) exponentially decays.In order to understand the form of the potential profile,we illustrate in Fig. 1 the form of the dielectric permittiv-ity ˜ 󿿿 ( z ) and the screening parameter  κ 2 ( z ) =  κ 2 c e − V  c ( z ) in the vanishing surface charge limit of Eq. (7). It isseen that with increasing distance from the surface, thedielectric permittivity increases from the air permittiv-ity ˜ 󿿿 ( z ) = 1 to the bulk permittivity ˜ 󿿿 ( z ) =  ε w  over adistance  h ≈ 2 ˚A. We note that this dipolar exclusion ef-fect is mainly due to the interaction of solvent moleculeswith their electrostatic images. Then, one sees that thissolvent depletion regime is followed by an ionic depletionregime of thickness  d ≈ 6 ˚A, an effect known to srcinatefrom image charge interactions [13].Inspired by the behaviour of ˜ 󿿿 ( z ) and  κ ( z ) that re-sults from the interfacial depletion of solvent moleculesand ions, we will introduce a restricted variational ansatzbased on a piecewise trial solution for the electrostaticpotential. We assume that  φ 0 ( z ) is the solution of Eq. (7) in the linear limit of weak surface charge, with˜ 󿿿 ( z ) =  θ ( h − z )+ ε w θ ( z − h ) and  κ ( z ) =  κ c θ ( z − d ), wherethe dipolar and ionic depletion lengths  h  and  d  are trialparameters that will be obtained from a numerical op-timization procedure of the Grand potential of Eq. (4).The solution of Eq. (7) with the above piecewise dielec-tric permittivity and ion density profiles, and satisfyingthe continuity of the potential  φ 0 ( z ) and the displace-ment field  D ( z ) = ˜ 󿿿 ( z ) φ ￿ 0 ( z ) at  z  = 0,  z  =  h , and  z  =  d ,  4 (a) 0 5 10 15 20 25 30-0.20-0.15-0.10-0.050.00                 0        (        k        B        T        ) z(A) 0 1 2 3-0.5-0.4-0.3-0.2-0.10.0                 0        (        k        B        T        ) z(A) (b) 02460.00.20.40.60.81.0 z(A)  2 (z)  c2  (z)   w  eff  (z)   w FIG. 1: (Color online) (a) Electrostatic potential profile( σ s  = 0 . 01 nm − 2 ) and (b) renormalized density and dielectricpermittivity profiles for  ε w  = 71 and  ρ bi  = 0 . 1 M. The redline in (a) is from the restricted variational ansatz Eq. (19)and the dashed black line corresponds to the solution of theEDPB equation. reads φ 0 ( z ) =  −  2 µκ c [1 +  κ c ( d − h )] + 2 ε w µ  ( z − h ) ,  0  < z  ≤ h φ 0 ( z ) =  −  2 µ κ c + 2 µ ( z − d ) , h ≤ z  ≤ d  (19) φ 0 ( z ) =  −  2 µ κ c e − κ c ( z − d ) , z  ≥ d. Numerical optimization yields  h  = 0 . 6 ˚A and  d  = 2 . 3˚A. We note that due to the piecewise nature of the trialpotential in Eqs. (19), these values correspond approxi-mately to half saturation densities. Figure 1 shows thatthe potential profile obtained from the numerical opti-mization agrees very well with the general form obtainedfrom the numerical solution of the EDPB equation. InEqs. (19), the first linear regime at 0  < z  ≤ h  correspondsto the solvent depletion layer resulting from image dipoleinteractions. This layer associated with dielectric screen-ing deficiency is responsible for an amplification of thePB prediction of the surface potential by a factor of five.The second and third intervals correspond respectivelyto the usual ionic depletion and diffuse layers [13]. The (a) 1E-3 0.01 0.1 1110100    C    (          F   /  c  m    2    )  bi (M) PBMPBMDPBEq. 21DPB (b) 1E-3 0.01 0.1 110100  m =1  bi (M)    C    (          F   /  c  m    2    ) PB  m =60  m =70  m =  w 20 40 600.00.20.40.60.8    h   (   A   )  m FIG. 2: (Color online) (a) Differential capacitance against thebulk ion concentration for  σ s  = 0,  ε m  = 1, and  ε w  = 71.The black circles are the experimental data, the red solidcurve is the result of the EDPB equation, the red squaresare from Eq. (21), the dashed blue line is the MPB result,the dotted black line is the GC capacitance, and the blacksquares correspond to the prediction of the DPB equation.(b) The same plot as in (a) for various  ε m . The inset displaysthe evolution of the dipolar depletion length  h  (solid curve)and  l d  (dashed curve) as a function of   ε m  for  ρ bi  = 0 . 1 M. contribution of these layers to the differential capacitancewill be investigated below.The differential capacitance of the double layer is de-fined as C  d  =  qe 2 k B T   ∂σ s ∂φ s  ,  (20)where  φ s  =  φ 0 ( z  = 0) is the surface potential. Fig. 2.acompares the differential capacitance computed withEq. (7) in the vanishing surface charge limit with ex-perimental data obtained for several types of monova-lent electrolytes at various concentrations (for details seeRef. [9] where the data were taken from). We also re-port in this figure the prediction of various formalisms.As stressed in Ref. [9], the PB result largely overesti-mates the experimental data. Furthermore, the result of the modified PB (MPB) equation (see Ref. [13]) that canexclusively take into account the ionic depletion effect  5brings a very small correction to the PB result. How-ever, the EDPB result that additionally contains the sur-face depletion effect of solvent molecules exhibits a goodagreement with the experimental data. We finally notethat in the vanishing surface charge limit considered inthis part, the DPB equation yields the same result as thePB one (see black squares in Fig. 2.a).The physics of the EDPB prediction for the capac-itance can be understood within the restricted self-consistent scheme of Eq. (19), where the differential ca-pacitance in Eq. (20) takes in the limit  σ s  → 0 the simpleform C  d  =  ε w κ c 1 +  κ c ( d − h ) +  ε w κ c h .  (21)We note that the prediction of this equation reportedin Fig. 2 (red squares) fits very well the numerical re-sult of the EDPB equation. The inverse capacitance of Eq. (21) is composed of three parts. The first contri-bution from the diffuse layer is the inverse GC capaci-tance  C  − 1 d 1  = ( ε w κ c ) − 1 corresponding to the PB resultin Fig. 2.a. The second part  C  − 1 d 2  = ( d − h ) / ε w  asso-ciated with the ionic depletion layer is shown to dropthe differential capacitance to the MPB curve. Finally,the third contribution from the solvent depletion layer C  − 1 d 3  =  h  characterized by the dielectric screening defi-ciency brings the most important correction to the totalcapacitance by dropping the latter to the correct orderof magnitude.In order to estimate the order of magnitude of the dipo-lar depletion length  h , we will compute the asymptoticlimit of ˜ 󿿿 ( z ) far from the interface, where the dipolar po-tentials in Eqs. (13) and (14) become weak, by expanding Eq. (15) in  U  d ( z ),  T  d ( z ), and  p 0 φ ￿ 0 ( z ). Furthermore, wenote that in the limit  ε m  = 0, the dipolar potentials aregiven by the closed form expressions U  d ( z ) =  ￿ w  p 20 1 + 2 κ c z 16 z 3  e − 2 κ c z (22) T  d ( z ) =  ￿ w  p 20 1 + 2 κ c z (1 + 2 κ c z )16 z 3  e − 2 κ c z .  (23)Renormalizing all lengths by the length scale  l d  =( ￿ w  p 20 / 10) 1 / 3 according to ¯ κ c  =  κ c l d  and ¯ z  =  z/l d , andtaking into account that  ε w  􏿿 1, the asymptotic form of Eq. (15) far from the dielectric interface reads˜ 󿿿 ( z ) ε w ≈ 1 −  1 + 2¯ κ c ¯ z  + 3¯ κ 2 c ¯ z 2 / 2¯ z 3  e − 2¯ κ c ¯ z .  (24)We now note that for a bulk permittivity  ε w  = 71 andionic concentration  ρ bi  = 10 − 1 M, one has  κ c l d  = 8 . 10 − 2 .Hence, in the regime  z  ∼  l d  = 0 . 9 ˚A, the terms inEq. (24) that depend on the screening length becomenegligible. This simple calculation fixes  l d  as the charac-teristic length over which the local permittivity tends toits bulk value according to an inverse cubic power law,i.e. ˜ 󿿿 ( z ) / ε w  ≈  1 − l 3 d /z 3 . We note that an inverse cubiclaw for the dielectric permittivity profile was derived inRef. [16] in the strict limit of a single dipole (i.e.  ε w  = 1)and without salt. In the dilute salt limit  κ c  →  0, onecan actually extend the estimation of   h  to finite values of  ε m  by noting that the dipolar potentials possed the closeform expression  U  d ( z ) =  T  d ( z ) =  ￿ w  p 20 ∆ 0 / (16 z 3 ), where∆ 0  = ( ε w − ε m ) / ( ε w  +  ε m ). Following the same steps asabove, one obtains for the characteristic dipolar depletionlength the more general expression  l d  = (∆ 0 ￿ w  p 20 / 10) 1 / 3 .Hence, for biological solvent concentrations and diluteelectrolytes with bulk density  ρ bi  ￿ 0 . 1 M, the dielectricscreening is mainly responsible for the decay of the imagedipole interactions and solely determines the region overwhich a reduced dielectric permittivity is observed. Forlarger ion concentrations, Eq. (24) shows that the screen-ing of image dipole interactions by surrounding ions pos-itively adds to the dielectric screening of these forces.We display in the inset of Fig. 2.b the evolution of   h as a function of   ε m  together with  l d , while the main plotshows  C  d  for various values of the membrane permittiv-ity from  ε m  = 1 to  ε m  =  ε w . One notices that within therange 1 ≤ ε m  ≤ 60,  h  exhibits a slow linear decrease withincreasing  ε m  while  C  d  remains within the same order of magnitude as the experimental capacitance data. How-ever, with an increase of   ε m  from 60 to  ε w , the dipolardepletion length quickly drops to zero and consequently, C  d  approaches the PB result. One also sees in the insetthat although  l d  is slightly higher than the dipolar deple-tion length  h , it can reproduce the correct trend of thelatter as a function of   ε m . These observations suggestthat image dipole interactions are mainly responsible forthe low values of the experimental capacitance data inFig. 2.a. We emphasize that this result is in agreementwith the experimental observation of a strong reductionof the double layer capacitance with increasing surfacehydrophobicity [10].In addition to the dipolar depletion effect, the orienta-tion is also expected to play some role in the form of theinterfacial dielectric permittivity profile. The measureof the dipolar orientation is defined in the literature as µ m ( z ) =   p 2 z  / [  p 20 ρ d ( z )], where   p 2 z   = ￿   d Ω 4 π  ¯ ρ d ( z, Ω )  p 2 z .The function  µ m  was studied in Ref. [17] for multipolarions and it was found invariably below the free dipolevalue 1/3 in the SC limit (i.e. dipolar alignment parallelto the wall) and above this value in the WC limit (align-ment along the electrostatic field). We show in Fig. 3that for a neutral interface, one has  µ m ( z )  <  1 / 3 as inthe SC limit, i.e. the solvent molecules exhibit a ten-dency to align parallel to the wall over a distance ≈ 2 ˚A,that is, until image dipole forces vanish. We now definean effective dielectric permittivity function of the form˜ ε eff  ( z ) = 1+4 π￿ B  p 20 ρ d ( z ) / 3 that solely accounts for thedipolar depletion. The comparison of ˜ ε eff  ( z ) in Fig. 1.awith ˜ ε ( z ) shows that the main effect of the dipolar align-ment close to the interface is a slight reduction of thelocal dielectric permittivity. However, it is seen that thiseffect is largely dominated by the solvent depletion.In the presence of a finite surface charge, Fig. (3) showsthat interestingly, the function  µ m ( z ) exhibits a non-
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