Electrolyte solutions at curved electrodes. I. Mesoscopic approach
aa r X i v : . [ c ond - m a t . s o f t ] M a r Electrolyte solutions at curved electrodes. I. Mesoscopic approach
Andreas Reindl, ∗ Markus Bier, † and S. Dietrich Max-Planck-Institut f¨ur Intelligente Systeme, Heisenbergstr. 3, 70569 Stuttgart, Germany andIV. Institut f¨ur Theoretische Physik, Universit¨at Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany (Dated: 28 March, 2017)Within the Poisson-Boltzmann (PB) approach electrolytes in contact with planar, spherical, andcylindrical electrodes are analyzed systematically. The dependences of their capacitance C on thesurface charge density σ and the ionic strength I are examined as function of the wall curvature.The surface charge density has a strong effect on the capacitance for small curvatures whereas forlarge curvatures the behavior becomes independent of σ . An expansion for small curvatures givesrise to capacitance coefficients which depend only on a single parameter, allowing for a convenientanalysis. The universal behavior at large curvatures can be captured by an analytic expression. I. INTRODUCTION
An electrical double layer capacitor or supercapaci-tor basically consists of electrodes which are insulatedby a separator and which are in contact with an elec-trolyte. Supercapacitors are used as alternative electri-cal energy storage devices and combine the properties ofconventional batteries, with high energy but low powerdensities, and conventional capacitors with the oppositecharacteristics [1]. They are used in electric vehicles andmobile phone equipments. Moreover, in search of sus-tainable energy systems there is still growing interest indouble layer capacitors. The capacitive behavior is de-termined by the nature of the electrode material, e.g., itsporosity and accessible surface area. Often carbon is theelectrode material of choice and especially ordered carbonallotropes have received much attention because theirmicro-texture influences the electronic properties. Differ-ent kinds of carbon nanostructured materials, includingcarbon nanotubes, carbon nanorods, spherical fullerenes,and carbon nano-onions, have been used as electrodes[1, 2]. Fiber-shaped supercapacitors exhibit low weightand high flexibility and thus are promising candidatesfor power sources in wearable electronics [3]. In contrastto conventional capacitors with smooth electrode mor-phologies, supercapacitors exhibit highly curved surfacesin order to obtain large specific areas, i.e., high porosity.This poses the problem of understanding the propertiesof electric double layers at curved geometries. A suit-able method to model an electric double layer is given bythe Poisson-Boltzmann (PB) theory. Within this meso-scopic approach the focus is on length scales larger thanthe ions or solvent molecules because electrolyte solu-tions are taken to consist of pointlike ions dissolved ina homogeneous solvent which is described by its electricpermittivity only. The PB theory has been pioneered byGouy [4] and Chapman [5] in the 1910s and sometimesit is referred to as the Gouy-Chapman theory. Althoughthe model is simple, reliable predictions can be expected ∗ Electronic address: [email protected] † Electronic address: [email protected] to hold for low ionic strengths (below 0 . . /ℓ )and low electrode potentials (below 80 mV), in the caseof aqueous solutions and monovalent salts [6]. For thatreason and due to its simplicity the PB theory is usedfrequently. Under certain circumstances it even allowsfor exact solutions, e.g., for electrolyte solutions at pla-nar electrodes [6, 7]. Recently, exact results have beenpresented for an electrolyte bounded by parallel plates orinside a cylindrical charged wall if only counterions areconsidered [8]. This setup might be used as a descrip-tion for ions confined in a charged nanotube or pore.However, for the corresponding spherical system, so fara solution in closed form has not been found. In Ref. [9]the same authors presented an expansion for the solutionof the PB equation in spherical and cylindrical geome-tries with large radii of curvature, which might resemblecharged macromolecules surrounded by an electrolyte so-lution and which comes closest to an analytic solution ofthe full PB equation for these geometries. Within theframework of the linearized Gouy-Chapman-Stern the-ory in Ref. [10] a model for an arbitrary surface mor-phology was developed. This facilitates, for example, thecalculation of capacitances of nanostructured electrodes,the study of which might contribute to the developmentof efficient energy generating and storage devices. Butalso for numerical studies the PB equation often is themodel of choice because its simplicity allows for fast cal-culations: In order to understand the properties of thediffuse double layer at charged nanoelectrodes or carbonnanotubes the PB equation was solved in Refs. [11, 12]for spherical and cylindrical electrodes. The potentialand capacitance were analyzed for various values of theelectrode radius. The evolution of capacitance modelsfor supercapacitors gave rise to the study in Ref. [13] inwhich cylindrical and slit pores were considered withinthe Gouy-Chapman-Stern model to address, inter alia,the issue of how the pore shape affects the capacitance.However, to our knowledge, so far the dependence ofthe capacitance on the geometry has only been addressedon a sample basis, i.e., for particular choices of systemparameters. The intention of the present work is to studythe curvature dependence of the capacitance systemati-cally within the entire, relevant parameter space.In the present study the PB equation is solved for elec-trolytes surrounding spherical and cylindrical electrodes(see Sec. II). In addition to presenting results for a varietyof parameter choices, a thorough overview of the spec-trum of solutions is given. To that end the dependenceof the differential capacitance on the various parametersis analyzed in detail within the PB theory for these ge-ometries. We are able to discuss the limiting behaviorsystematically, i.e., the dependence on only one parame-ter or analytically. This facilitates the understanding ofthe essential behavior of the data of interest which in thepresent case is the differential capacitance as a functionof the wall curvature. In Sec. III A a short overview ofthe exact results within the linearized theory is given be-fore in Sec. III B the full PB equation for various choicesof the parameters is solved. In the subsequent Secs. III Cand III D general trends for large and small radii of theelectrodes are worked out. Corresponding technical de-tails are discussed in Appendices A and B. Summary andoutlook are given in Sec. IV. II. MODEL
Consider an electrolyte composed of pointlike, monova-lent ions, i.e., particles without volume carrying positiveor negative elementary charge ± e . Due to local chargeneutrality the number densities of both ion species inthe bulk are equal to the ionic strength I . The solventis regarded as a dielectric continuum with homogeneousrelative permittivity ǫ . The electrostatic potential Φ inthis system obeys the Poisson-Boltzmann (PB) equation[6, 7] ∆Φ( r ) = 2 eIǫ ǫ sinh[ βe Φ( r )] , (1)where ∆ is the Laplace operator, r ∈ R denotes a po-sition in three-dimensional space, ǫ is the vacuum per-mittivity, β = ( k B T ) − with the Boltzmann constant k B and the absolute temperature T . The electrolyteis assumed to be in contact with a convex electrode ofplanar, spherical, or cylindrical shape. The electrode isdescribed as a homogeneously charged hard wall withsurface charge density σ . Under these assumptions thepotential Φ in Eq. (1) depends on a single spatial vari-able Φ( r ) := Φ( r ) = Φ( x, y, z ) where the meaning of r depends on the geometry: • A planar wall occupies the half space z < r := z , • a spherical wall x + y + z < R of radius R gives rise to a dependence of the potential on r := p x + y + z , and for a • cylindrical wall x + y < R of radius R the po-tential depends on r := p x + y .By introducing the parameter d in order to distinguishthe three geometries the PB equation (1) may be formu- lated in a one-dimensional fashion:1 r d ∂∂r (cid:20) r d ∂ Φ( r ) ∂r (cid:21) = 2 eIǫ ǫ sinh[ βe Φ( r )] ,d = , planar wall , , cylindrical wall , , spherical wall . (2)Solutions of Eq. (2) are subject to boundary conditionsat the wall surface r w and in the bulk r → ∞ :Φ ′ ( r ) (cid:12)(cid:12)(cid:12) r = r w = − σǫ ǫ , r w = ( , d = 0 ,R, d ∈ { , } , Φ ′ ( r ) (cid:12)(cid:12)(cid:12) r = ∞ = 0 . (3)For the considered geometries the electric field exhibitsonly a component E ( r ) = − Φ ′ ( r ) in direction normal tothe wall surface. The value of the component at the sur-face r w is linked to the surface charge density σ by thefirst boundary condition. The second boundary conditionensures global charge neutrality. In the following we ad-ditionally demand Φ( ∞ ) = 0 so that the lower boundarycondition is fulfilled and the arbitrary integration con-stant of Φ is set.For small values of the dimensionless potential βe Φ( r ) → r d ∂∂r (cid:20) r d ∂ Φ( r ) ∂r (cid:21) = κ Φ( r ) , κ := s e Iβǫ ǫ , (4)with the inverse Debye length κ . III. DISCUSSION
The differential capacitance is defined by [7] C := ∂σ∂ Φ( r w ) (5)as the change of the surface charge density σ upon chang-ing the potential at the wall Φ( r w ). Here the theoreticalresults are presented in terms of this measurable quantityin order to facilitate comparison with experiments. Ourexaminations focus on the dependence of the capacitance C on the curvature 1 /R of a spherical and a cylindricalwall. A. Linearized PB equation
The linearized PB equation (4) can be solved analyti-cally for the geometries under consideration: • At a planar wall the potential at the electrode isΦ(0) = σǫ ǫκ , (6)and the capacitance is given by the double-layercapacitance ǫ ǫκ [7] Cǫ ǫκ = 1 . (7)This quantity will be used as a reference in orderto define dimensionless capacitances. • For a spherical wall one hasΦ( R ) = σǫ ǫκ κRκR + 1 , (8)and Cǫ ǫκ = 1 + 1 κR , (9)a polynomial of linear order in the dimensionlesscurvature ( κR ) − . • In case of a cylindrical wall the electrode potentialΦ( R ) = σǫ ǫκ K ( κR )K ( κR ) , (10)and the capacitance Cǫ ǫκ = K ( κR )K ( κR )= 1 + 12 1 κR −
18 1( κR ) + O (cid:20) κR ) (cid:21) (11)are given by a ratio of modified Bessel functions[14]. For large radii κR ≫ → σ . Strictly speaking thefull PB equation has to be considered as soon as systemswith non-vanishing Φ or σ are of interest. This will bethe focus in the following sections. B. Full non-linear PB equation
The solution of the PB equation (2) at the planar wallis available in closed form and the capacitance is givenby [7] Cǫ ǫκ = cosh (cid:20) βe Φ(0) (cid:21) , (12)where the potential at the wall Φ(0) depends on the sur-face charge density σ asΦ(0) = 2 βe arsinh (cid:18) βeσ ǫ ǫκ (cid:19) . (13)For spherical and cylindrical geometries the PB equa-tion (2) is solved numerically. If lengths, charges, and en-ergies are measured in units of the Debye length 1 /κ , theelementary charge e , and the thermal energy 1 /β = k B T ,respectively, the present model of a monovalent salt solu-tion is specified by the following three dimensionless, in-dependent parameters: I/κ , κR , and σ/ ( eκ ). In Figs. 1and 2 results for the reduced capacitance are shown fortwo cases A and B corresponding to the choices I/κ ≈ . I/κ ≈ . T = 300 K with relativepermittivity ǫ = 77 . /κ ≈ .
600 ˚A[30 .
36 ˚A], and ionic strength I = 0 . .
01 M] in case A[B]. In Figs. 1 and 2 the reduced capacitance C/ ( ǫ ǫκ )is plotted as function of the dimensionless wall curvature1 / ( κR ) for various values of the reduced surface chargedensity σ/ ( eκ ). The value 1 / ( κR ) = 0 corresponds tothe planar wall result in Eq. (12). Larger values on thehorizontal axes are equivalent to larger curvatures andhence to smaller radii of the wall. Since the PB equationoriginates from a classic theory the results for large cur-vatures should be treated with caution. In case A, forexample, the Debye length is about 1 /κ ≈
10 ˚A. Thismeans that for 1 / ( κR ) >
10 the wall radius is smallerthan the atomic length scale of 1 ˚A. Within this rangethe particle size, which is not captured by the PB the-ory, should play a role. In the case of the spherical wall(Fig. 1) the curvature dependence of the capacitance forthe smallest chosen value of σ almost coincides with astraight line. This is in accordance with the analyticresult in Eq. (9) which renders a linear polynomial in1 / ( κR ). For increasing σ the capacitance in the planarlimit 1 / ( κR ) = 0 increases, whereas for large curvaturesthe curves seem to converge from above to the graph for σ →
0. This indicates that the linear theory becomes themore valid the larger the curvature is chosen. In betweenthe limits of high and low curvatures the capacitance ex-hibits a minimum the position of which shifts with σ .Also the slope of the graph for small curvatures dependson σ . For σ → C / ( ǫ ǫ κ ) / ( κR ) (a) spherical wall, case A σ C / ( ǫ ǫ κ ) / ( κR ) (b) spherical wall, case B σ σ/ ( eκ ) ≈ . σ/ ( eκ ) ≈ . σ/ ( eκ ) ≈ . σ/ ( eκ ) ≈ . σ/ ( eκ ) ≈ . σ/ ( eκ ) ≈ . σ/ ( eκ ) ≈ . σ/ ( eκ ) ≈ . σ/ ( eκ ) ≈ . σ/ ( eκ ) ≈ . FIG. 1: Reduced differential capacitance C/ ( ǫ ǫκ ) as a func-tion of the dimensionless curvature 1 / ( κR ) of spherical elec-trodes. The data are obtained by solving the PB equation (2)for two cases (A and B) of the bulk parameter choices (seethe main text). Each curve corresponds to a constant valueof the reduced surface charge density σ/ ( eκ ). The verticalarrow points in the direction of increasing σ . resemble the ones at spherical walls in Fig. 1. The resultsfor cylinders look like the results for spheres stretched inhorizontal direction. However, for σ → R ) at the cylindricalelectrode agrees well with the expression for the surfacepotential in Ref. [15] [Eqs. (3) and (4) therein] within thespecified range of validity, i.e., for not too small curva-tures and line charge densities.The linearized PB equation (4) corresponds to the low-est curves in Figs. 1 and 2. Thus important features, par-ticularly in the range of small curvatures, are neglected,whereas for large curvatures a description based on thelinear theory appears to be sufficient. In the solution ofthe full equation (2) the surface charge density affects thecapacitance for small curvatures to a large extent whereasfor large curvatures the behavior becomes more and moregeneral and independent of σ . This phenomenon will be C / ( ǫ ǫ κ ) / ( κR ) (a) cylindrical wall, case A σ C / ( ǫ ǫ κ ) / ( κR ) (b) cylindricalwall, case B σσ/ ( eκ ) ≈ . σ/ ( eκ ) ≈ . σ/ ( eκ ) ≈ . σ/ ( eκ ) ≈ . σ/ ( eκ ) ≈ . σ/ ( eκ ) ≈ . σ/ ( eκ ) ≈ . σ/ ( eκ ) ≈ . σ/ ( eκ ) ≈ . σ/ ( eκ ) ≈ . FIG. 2: Same as Fig. 1 for cylindrical electrodes. addressed in the following sections.
C. Limit of large wall radii
Within this subsection we focus on walls with largeradii κR ≫ / ( κR ) ≪
1. It hasbeen shown before that in this limit the capacitance asfunction of the curvature varies strongly with the surfacecharge density σ (see Figs. 1 and 2 and the discussion inthe previous Sec. III B). In order to examine this obser-vation in more detail, the capacitance is taken as a powerseries in terms of small curvatures ( κR ) − ≪ C = ǫ ǫκ ∞ X n =0 C n ( κR ) n , (14)where ǫ ǫκC is the capacitance of a planar wall[Eq. (12)]. In Appendix A the calculation of the dimen-sionless coefficients C n is explained in detail. In Fig. 3 thelowest order coefficients C , , of the curvature expansionin Eq. (14) are plotted as function of the dimensionless t (c) cylindrical wall C (1 − | t | ) − C C C C FIG. 3: Lowest order coefficients C , , of the curvature ex-pansion [Eq. (14)] as function of the parameter t defined inEqs. (15) and (A6). The entire information about the in-fluences of T, ǫ, I , and σ is contained in the dependence on t ∈ [ − ,
1] (see the discussion in Appendix A). For clarity thecoefficient C , describing the contribution of the planar wallto C [panel (a)], is displayed separately from the coefficients C and C for spherical (b) and cylindrical (c) walls. parameter t := tanh (cid:20)
12 arsinh (cid:18) βeσ ǫ ǫκ (cid:19)(cid:21) ∈ [ − , , (15)which is a combination of T, ǫ, I , and σ such that the signof t agrees with the sign of σ [see also Eq. (A6)]. Apart from the geometry captured by d , the coefficients C n de-pend only on t . Thus within PB theory every parameterchoice can be assigned to Fig. 3. Since the solutions Φ( r )of Eqs. (2) and (3) are odd functions of σ , the capaci-tance in Eq. (5) and hence the coefficients C n are evenfunctions of t , i.e., C n ( t ) = C n ( − t ); it is therefore suffi-cient to only discuss the range t ≥
0. For the planar wall[Fig. 3(a)] the formulation in terms of scaled variables is C = 1 + t − t (16)which diverges for σ → ±∞ ⇔ t → ± − | t | ) − .The coefficient C [Figs. 3(b) and (c)] exhibits the samequalitative behavior for both curved walls: its value inthe spherical case is twice of that in the cylindrical case.At t = 0, C attains a positive maximum; for t > t -axis at t = 0 . C corresponds to theslope for small curvatures. Indeed the slope changes frompositive to negative with increasing σ , i.e., increasing t .The roots of C in Figs. 3(b) and (c) correspond to a spe-cial combination of parameters for which the initial slopein Figs. 1 and 2 would be exactly zero. For C qualitativedifferences between the curved wall shapes occur. In caseof the spherical wall [Fig. 3(b)] C is zero at t = 0 whichis consistent with Fig. 1 showing a straight line for σ → t leads to a somewhat oscillatory behavior of C . Positive values of C correspond to a convex function(from below) in Fig. 1 for small curvatures and interme-diate values of σ whereas negative values of C for t → C is relatively small so that the concave be-havior is less pronounced. However, the latter is visiblein Fig. 1(b) for large σ . For the crossover value for t be-tween convex and concave we obtain t = 0 . C is negative at t = 0 and consequently in Fig. 2 the concave behavior forsmall curvatures and small σ is visible. Upon increasing t the coefficient C changes sign from negative to positiveand remains positive for values of t larger than the rootat t = 0 . σ . This analysis shows that even forvery large σ no concave behavior can be expected as inthe case of spherical walls. The coefficients [see Eqs. (9)and (11)] obtained within the linearized theory [Eq. (4)]are covered by the present analysis and correspond to thevalues at t = 0 ⇔ σ = 0.At this stage an excursion to morphometric thermo-dynamics (MT) is appropriate. Within that approachthe interfacial tension γ takes a very simple form withrespect to the dependence on the geometry of the sur-rounding walls. For the geometries of the current studythe dependences on the radius R can be formulated as γ = γ + γ s R + γ s R , spherical wall ,γ + γ c R , cylindrical wall . (17)Within MT there are no higher order terms and the coef-ficients γ c,sn are independent of the radius R . (For furtherdetails see Ref. [16].) The connection with the presentstudy is given by the Lippmann equation [7] σ = − ∂γ∂ Φ( r w ) (18)and hence C = − ∂ γ∂ Φ( r w ) . (19)Since differentiation of Eq. (17) with respect to the elec-trode potential Φ( r w ) does not change the form of theequations, MT predicts the same truncated curvature de-pendence for the capacitance C . However, examinationsin terms of the capacitance have the advantage that thisquantity is uniquely defined contrary to the interfacialtension. In a previous study [17] we examined the im-plications of various interface conventions concerning theaccuracy of MT in terms of the interfacial tension. Wefound that the quality of the approach as an approxima-tion depends to a large extent on the interfacial positionwhich in principle may be chosen arbitrarily. Follow-ing the prediction of MT in the case of cylindrical wallsthe coefficient C should be zero. However, in agree-ment with earlier work on curved interfaces using gradi-ent expansion approaches [18, 19], already linear theory[Eq. (11)] exhibits a nonzero coefficient and the full solu-tion Fig. 3(c) reveals that C is nonzero for most choicesof the parameters. Therefore MT is not an exact ap-proach, which is not surprising because even for simplefluids its precision has been doubted recently (see, e.g.,Refs. [17, 20–22]). Therefore, MT has the status of anapproximation. For example, when discussing cylindricalwalls the necessary restriction | C | ≪ | C | might be ade-quate to truncate the curvature expansion in accordancewith MT which is the case for values of t far away fromthe root of C . It is remarkable that this is the case for | t | → ⇔ | σ | → ∞ , i.e., for highly charged electrodes. Inany case this quality criterion depends on t and thereforeon the surface charge density. In general the curvaturecoefficients are properties of the fluid and the wall-fluidinteraction [16]. As a consequence, for simple fluids, thecoefficients are fixed once a certain wall-fluid system hasbeen chosen. However, in the case of electrode-electrolytesystems, the wall-fluid interaction is typically not fixedbut can be adjusted via the surface charge density. Forsuch cases the dependence of the coefficients C n on σ hasto be known. This further complicates and reduces theapplicability of MT. D. Limit of small wall radii
For spherical walls and large curvatures a somewhatgeneral behavior is observed (Fig. 1): all curves shownapproach the straight line which corresponds to the re-sults for small σ and which is in accordance with the result of the linearized theory [Eq. (9)]. In the case ofcylindrical walls (Fig. 2) a similar behavior is visible;however, the degree of convergence towards the curvecorresponding to small σ is inferior to that for sphericalwalls, at least within the shown curvature interval.Indeed, it is possible to show analytically that non-linear contributions to the solution of the PB equation[Eq. (2)] are negligible for sufficiently large curvature,e.g., if 1 κR ≫ s √ βe | σ | ǫ ǫκ (20)for a spherical wall (see Appendix B for details). Thisfinding explains the general behavior encountered inFig. 1 because for any finite σ there is a range of (large)curvatures for which the inequality in Eq. (20) holds.Cylindrical electrodes (Fig. 2) exhibit curvature depen-dent capacitances which resemble the spherical results(Fig. 1), stretched in horizontal direction. This finding issupported by the linearized theory. In the limit of smallradii κR → | Φ( R ) | of the electrode po-tentials [Eqs. (8) and (10)] are monotonically increasingfunctions of R and the one at the cylindrical electrodewith the same R is larger than the corresponding one atthe spherical wall. On one hand this means that in thecase of cylindrical walls smaller radii or larger curvaturesare necessary in order to get the same value of Φ( R ) asin the spherical case; this also holds for the capacitance.On the other hand the linearized theory is based on smallvalues of the dimensionless potential βe | Φ | ≪
1. Thus,in the case of cylindrical walls the linearized theory turnsinto a reliable description at smaller radii or larger curva-tures as compared to the spherical wall. From the com-parison of Figs. 1 and 2 it follows that a correspondingestimate like the one in Eq. (20) would lead to wall radiibelow molecular sizes (see the discussion in Sec. III B)and which therefore would be of no practical use.
IV. SUMMARY AND OUTLOOK
In terms of the Poisson-Boltzmann (PB) equation[Eq. (1)] we have analyzed electrolytes in contact withelectrodes of planar ( d = 0), cylindrical ( d = 1), orspherical ( d = 2) shape. The differential capacitance C [Eq. (5)] was calculated for various ionic strengths I ,surface charge densities σ , and electrode radii R . Thefocus was on examining the dependence of the capaci-tance on the curvature 1 /R of the electrode as displayedin Figs. 1 and 2. In all cases the surface charge densityhas a strong effect on the capacitance for small curva-tures whereas for large curvatures the behavior becomesindependent of σ . These limits have been analyzed in de-tail. For small curvatures (see Sec. III C) we found that acurvature expansion of the capacitance [Eq. (14)] revealsthe behavior in a very convenient way because the corre-sponding expansion coefficients C n depend on the singleparameter t ∈ [ − ,
1] [Eqs. (15) and (A6)] and on the ge-ometry d ∈ { , , } only. Therefore, within PB theory,the influence of any conceivable combination of systemparameters on the lowest order coefficients C n can be in-ferred from Fig. 3. For large curvatures (see Sec. III D)an analytic discussion provides the insight that the lin-earized PB theory becomes reliable, if the radius of thespherical wall is chosen to be small enough; this explainsthe general behavior visible in Fig. 1.In the present study the mesoscopic structure of elec-trolyte solutions at curved electrodes has been discussedsystematically in terms of the capacitance within PB the-ory (i) because this approach is widely used in variousresearch fields, and (ii) because it offers to judge less in-tegral, microscopic approaches such as the one presentedin part II of this study [23]. Appendix A: Limit of large wall radii
We assume that for large radii R → ∞ the solution ofthe PB equation (2) can be expanded in terms of powersof the curvature such that the dimensionless potential y := βe Φ takes the form y ( r = R + z ) = ∞ X n =0 y n ( z )( κR ) n , (A1)where z ∈ [0 , ∞ ) measures the distance from the wall.For all radii the boundary conditions in Eq. (3) translateinto an inhomogeneous condition at the wall z = 0, y ′ ( z ) (cid:12)(cid:12)(cid:12) z =0 = − βeσǫ ǫ , y ′ n> ( z ) (cid:12)(cid:12)(cid:12) z =0 = 0 , (A2)and a homogeneous one at z = ∞ : y n ≥ ( ∞ ) = 0 . (A3)The ansatz in Eq. (A1) gives rise to a curvature expansionof the PB equation (2). In the following, we take intoaccount orders up to and including 1 / ( κR ) . The lowestorder leads to y ′′ ( z ) = κ sinh[ y ( z )] , (A4)which is the PB equation (2) with d = 0 for the dimen-sionless potential y ( z ) at the planar wall. The orderswhich are linear and quadratic in the curvature 1 / ( κR )correspond to differential equations for the spatially vary-ing expansion coefficients y , ( z ): y ′′ n ( z ) κ − cosh[ y ( z )] y n ( z ) == n = 1 : − d y ′ ( z ) κ ,n = 2 : − d y ′ ( z ) κ + zdy ′ ( z )+ 12 sinh[ y ( z )] y ( z ) ,d ∈ { , } , n ∈ { , } . (A5) Within the curvature expansion given by Eq. (A1) thecontribution of the planar wall is entirely captured bythe coefficient y . Higher order coefficients y n> occursolely at curved walls ( d = 0). The solution of Eq. (A4)is given by (see Ref. [7]) y ( z ) = 4 artanh[ t exp( − κz )] ,t := tanh (cid:20)
12 arsinh (cid:18) βeσ ǫ ǫκ (cid:19)(cid:21) = 2 ǫ ǫκβeσ s (cid:18) βeσ ǫ ǫκ (cid:19) − ∈ [ − , . (A6)In Eq. (A6) the dependence on z maps onto the scaledspatial variable x := t exp( − κz ) , | x | ∈ [0 , | t | ] , with f n ( x ) := y n ( z ( x )) , (A7)such that, e.g., the planar wall result takes the simpleform f ( x ) = 4 artanh( x ) . (A8)The differential equations for f , ( x ) are given by x f ′′ n ( x ) + xf ′ n ( x ) − cosh[ f ( x )] f n ( x ) == n = 1 : xdf ′ ( x ) ,n = 2 : xdf ′ ( x ) + d ln (cid:16) xt (cid:17) xf ′ ( x )+ 12 sinh[ f ( x )] f ( x ) ,d ∈ { , } , n ∈ { , } , (A9)subject to the boundary conditions f , ( x = 0) = 0 ,f ′ , ( x ) (cid:12)(cid:12)(cid:12) x = t = 0 . (A10)From Eqs. (A9) and (A10) it follows that the scaled po-tentials f , depend parametrically on d ∈ { , } and t .Moreover, the differential equations (A9) are to be solvedwithin a finite domain of values of x [Eq. (A7)] so that thewhole parameter space can be scanned rapidly. Finally,the capacitance follows as C = (cid:20) ∂ Φ( r w ) ∂σ (cid:21) − = " βe ∂∂σ X n y n ( z = 0)( κR ) n − = " βe ∂∂σ X n f n ( x = t )( κR ) n − = ǫ ǫκ "X n
14 (1 − t ) t ∂f n ( x = t ) ∂t κR ) n − =: ǫ ǫκ X n C n ( κR ) n , (A11)which defines the dimensionless expansion coefficients C n of the differential capacitance C . Alternatively, the coef-ficients C , can be determined from the expressions forthe surface potential of spherical and cylindrical surfaceswhich are given in Ref. [24]. In Eq. (A11) the expression ∂f n ( x = t ) /∂t refers to the derivative of f n with respectto t after evaluation at x = t ; the dependence on d and t is transferred to the coefficients C n . Apart from the in-fluence of the geometry via d , the whole parameter spacegiven by T, ǫ, I , and σ is contained in t ∈ [ − , Appendix B: Limit of small wall radii
Here we investigate under which conditions the non-linearities of the full PB equation (2) may be neglected.To this end, with the dimensionless potential y := βe Φwe consider the equation1 r ∂ ∂r [ rλy ( r )] = κ sinh[ λy ( r )]= κ (cid:20) λy ( r ) + 16 λ y ( r ) + O (cid:0) λ (cid:1)(cid:21) (B1)in spherical geometry, where λ ∈ C is an arbitrary com-plex parameter. For λ = 1 the PB equation (2) is re-covered, whereas for | λ | ≪ y ( r ) = y ( r ) + λ y ( r ) + O (cid:0) λ (cid:1) . (B2)In lowest order O (cid:0) λ (cid:1) the linearized PB equation (4)with d = 2 is recovered for which the spatially vary-ing potential y ( r ) and the electrode potential y ( R ) aregiven by y ( r ) = A exp( − κr ) r ,A := sR κR exp( κR ) , s := βeσǫ ǫ , and y ( R ) = sR κR , (B3) respectively. The next higher order O (cid:0) λ (cid:1) leads to adifferential equation for the dominant non-linear contri-bution y ( r )1 r ∂ ∂r [ ry ( r )] = κ y ( r ) + κ y ( r ) , (B4)where the inhomogeneity is given by the solution ofthe linearized PB equation [Eq. (B3)]. The solution ofEq. (B4) takes the form y ( r ) = B exp( − κr ) r + f ( r ) r , where f ( r ) := − κA ∞ Z R d r ′ exp( − κ | r − r ′ | ) exp( − κr ′ ) r ′ ,B := f ( R ) κR − κR + 1 exp( κR ) . (B5)The electrode potential y ( R ) is given by y ( R ) = − y ( R ) 16 (cid:16) sκ (cid:17) h (4 κR ) κR (cid:0) κR (cid:1) , with h ( z ) := z exp( z ) ∞ Z z d x exp( − x ) x . (B6)From these results one infers that the contribution fromthe linearized PB equation is the dominant one if theleading non-linear term y ( R ) is much smaller than thelinear one y ( R ). Since h ( z > ≤
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