Electrochimica Acta 204 (2016) Contents lists available at ScienceDirect. Electrochimica Acta

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1 Electrochimica Acta Contents lists available at ScienceDirect Electrochimica Acta journal homepage: Diffuse electric ouble layer in planar nanostructures ue to Fermi-Dirac statistics Mitja Drab, Veronika Kralj-Iglič Laboratory of Clinical Biophysics, Faculty of Health Sciences, University of Ljubljana, Zravstvena pot 5, SI- Ljubljana, Slovenia article info abstract Article history: Receive 4 November 5 Receive in revise form 5 February 6 Accepte 9 April 6 Available online April 6 A ouble nanocapacitor moelle by two equally charge planar surfaces that confine oppositely charge quanta subjecte to Fermi-Dirac statistics is consiere theoretically. A global thermoynamic equilibrium was foun by minimization of the Helmholtz free energy satisfying constraints that require electroneutrality an fixe total number of confine quanta. The solution obtaine by using the Euler Lagrange metho yiels self consistent quantities: istribution of quanta within the pore, electric potential, equilibrium free energy an ifferential capacitance. Within real values, a rigorous numerical solution an an approximate analytical solution for electrons in the low temperature limit was foun. The Fermi Dirac constraints on the wave functions in the nanopore inuce an effect of a iffuse electrical ouble layer near both charge surfaces. This effect is comparable to the corresponing effect of entropy at finite temperatures an for classical particles, as escribe by the acknowlege Poisson Boltzmann theory. At small istances an small surface charges, the electrons are almost evenly istribute within the pore, while at larger istances they conense to the charge surfaces, shieling the electric fiel. The force between the charge surfaces is repulsive an monotonously ecreases with increasing istance between surfaces. The energies store in the nanocapacitor are up to 5 ev/nm. 6 Elsevier Lt. All rights reserve.. Introuction The electric ouble layer EDL is a crucial phenomenon in many technological an biological applications. Primarily of interest in electrochemistry, the EDL has foun use in unerstaning the interactions of biological membranes [,], while in recent years EDL also plays a pivotal role in energy storage technologies. Electrochemical nanocapacitors manufacture with nanoporous materials an a variety of electrolytes are yieling increasing energy an power ensities, briging the gap between batteries an classic capacitors [3 5]. Electric ouble layer nanocapacitors were compose from multiple layere separate by a nanometer istance carbon nanotubes in contact with electrolyte [6]. It was inicate that electrochemical ouble layer capacitors can store large amounts of energy an eliver high peak power [6]. It was foun alreay a century ago that the capacitance of a system forme by the charge surface an an oppositely charge iffuse layer varies with electrolyte properties, its concentration, applie potential an charge on the surfaces [7,8]. Since then, the Corresponing author. aresses: mitja.rab@zf.uni-lj.si M. Drab, veronika.kralj-iglic@zf.uni-lj.si V. Kralj-Iglič. EDL theory was constantly avancing, incorporating ever novel ways of moelling [9]. One of the first things taken into account were steric effects inuce by the finite sizes of ions [ ], while later moels inclue the ipole nature of water in the electrolyte solution leaing to a coorinate-epenent ielectric permittivity [3,4]. Some moels investigate a shift in the capacitance curve ue to the asymmetric sizes of ions [5] an others consiere the electron wave functions in the electroe interface [6]. Interest for nanostructure materials [7 ] requires that theoretical moels of EDL are revisite with possible quantum effects taken into account. We attempt this in the present paper, where the effects of quantum statistics are stuie through a simple moel of charge quanta confine between two planar charge surfaces. For this purpose we generalize a previously evelope ensity functional metho for escription of the electrical ouble layer with the local Helmholtz free energy of the system, an its minimization by solving the corresponing Euler Lagrange variational problem [].. Theory The moel system consists of a single type of negatively charge quanta electrons confine between two large planar yz surfaces at x = an at x = Fig.. Each surface carries a uniformly / 6 Elsevier Lt. All rights reserve.

2 M. Drab, V. Kralj-Iglič / Electrochimica Acta entropy contribution an we calculate the Helmholtz free energy of a slice ue to Fermi-Dirac wave function symmetry constraints to electrons ıf sym as h ıf sym = ıe = 3 5 8m 3 /3 /3 N N. 5 V Summing all of the slices together we obtain F sym = 3 h 3 /3 A n 5/3 xx m Here, we introuce a volume ensity of quanta, n = N Aıx. 7 Charge particles an surfaces create an electric fiel that also contributes to the free energy of the system, namely the electrostatic energy Fig.. A schematic of the moel. A Probability ensities of charge fermions trappe in a nanopore oppositely charge at its surfaces. B Two symmetrically positione capacitors are forme, each consisting of a charge surface at x = an x =, respectively, an a iffuse layer create by fermions of the opposite charge represente by the hue. The light line represents the electric fiel profile, an parameter.95 a boun which encloses 95% of all the quanta of the respective half of the system represents the effective thickness of the iffuse layer. istribute positive charge with surface charge ensity. Generalization to positively charge quanta an negatively charge surfaces is straightforwar. The entire system is electrically neutral. The quanta are subjecte to the constraint on available eigenenergy states implie in the Fermi-Dirac statistics while the effect of the electric fiel on the wavefunctions an on the energy states is not consiere. These states can be either occupie or unoccupie, with the average number of quanta in the n-th energy state s n = + expε n /kt, where is the Lagrange coefficient for the constraint requiring fixe total number of quanta in the ensemble, k is the Boltzmann constant an T is absolute temperature []. It is taken that each quantum is confine in an infinite three imensional square potential well so that its energy is ε n = n h, n =,, 3,... 8ma Here, h is the Planck constant, m is the quantum mass an a is the extension of the potential well. As there are many quanta in the system, the energies are assume to lie close together an the summation in the statistical averages can be replace by integrals. The average number of quanta is N = /8 an the average energy is E = /8 4n + expε /kt 4εn + expε /kt n, 3 n. 4 The factor comes from specifying quanta as electrons an consiering their state egeneration. The system is ivie into thin slices with the area A an with ıx, parallel to the yz plane an all fiels are consiere constant within the slice []. In the low temperature limit T, only the lowest energies up to the Fermi energy are occupie. There is no F el = ε A E xx, 8 where Ex is the magnitue of the electric fiel strength an ε is the permittivity of vacuum. For simplicity we assume that the electric fiel oes not explicitly influence the solutions of the Schröinger equation, therefore the total free energy is simply the sum F = F sym + F el. 9 To obtain the global thermoynamic equilibrium, the latter function is minimize with respect to the unknown functions nx an Ex. The constraints of the system are the valiity of Gauss law at each x, E ε x = e nx, an electroneutrality e nxx =, where e is the elementary charge. For convenience, imensionless quantities marke by tile are use: x = x/, ñ = n/n, Ẽ = E/E, where n =, e = ε an E = ε. 3 With this substitution in place, we rop the tile an subsequently consier all expressions evoi of imension. The free energy ensity values are normalize with the area energy ensity of a parallel plate capacitor in a vacuum, which reas f =. 4 ε The imensionless Euler-Lagrange function of the system reas L = n 5 Ex 3 x + E x x + nx + nx, 5 x where x an are the local an global Lagrangian multipliers, respectively. The imensionless constant is equal to = /3 h ε me 5/

3 56 M. Drab, V. Kralj-Iglič / Electrochimica Acta The variational problem ı Ln, E, E x,xx = 7 is expresse by the system of Euler Lagrange equations L =, 8 n L E L =, 9 x E x an the above constraints. Consiering E = /x reveals that the local Lagrange coefficient is the electric potential while the Gauss law yiels the ifferential equation for the potential /3 ˇ 5 x = 4 4. Here, ˇ = /4. 5 Electric fiel at the miline vanishes ue to the symmetry of the system x x=/ =, with electric potential there being constant. We set the constant value to zero at the miline x = / =. 3 The other bounary conition follows from electroneutrality of the system x x= =. 4 We consier only the real branch of Eq., 3 ˇ 5 x = 4 4 3/. 5 We make use of the ientity x x =. 6 x x Introucing a new variable u = 4, 7 multiplying both sies of this equation by /x, an integrating yiels u ˇu /, u x / 8 where we approximate the power obtaine from integration 5/ by by assuming that the above expression, which is analytically integrable, iffers from the original function by a negligible amount. Here, u / is the reuce potential at x = /. Consiering that u 5/ / u 5/, the simplifie integral Eq. 8 can be solve analytically. The constant u / = 9 ˇ sinh ˇ is etermine by consiering the reference value of the potential. The final result for potential in the nanopore yiels x = ˇ sinh ˇ coshˇx. 3 The variable parameters in ˇ are the with of the nanopore, the surface charge of the surfaces an the properties of the quanta, which we take to be electron mass an charge. Density of particles is erive from the Euler Lagrange equations an is in irect relation to the potential, nx = x 3. Results an iscussion = ˇ coshˇx. 3 sinh ˇ Eq. 5 was also solve rigorously by the shooting numerical metho with Mathematica software Wolfram Research, Inc., Mathematica, Version 9., Champaign, IL. Fig. shows that potential monotonously ecreases from the charge surface towars the miline x = / panel A an that ue to electrostatic attraction electrons accumulate near each of the charge surfaces panel B. As they cannot all attain the same energy state near the surface, they are force to states of higher energies or to locations further away from the charge surfaces. Holing the istance between the charge surfaces at a constant value an changing the surface charge, we fin that at larger istances an high surface charge ensities the potential oes not change much, while the particle ensities are significant only near the surfaces. This particle accumulation accounts for effective screening of the electric fiel. At lower values of the electron ensity is almost constant throughout the pore Fig. A, but variation of potential is larger ue to the bulk charge accumulate between the surfaces Fig. B. Differences between the numerical an analytical results are negligible Fig. A, inset. Matching is best at x = /. Helmholtz free energy of the EDL is of consierable importance in colloi an surface sciences as it is use to escribe phenomena such as the spreaing pressure of charge monolayers an forces acting between the charge surfaces []. In our moel system, the equilibrium free energy is the integral sum of the contribution ue to symmetry restrictions upon the wave functions of the electrons an the contribution of the electrostatic fiel, f f = n 5/3 x + E x x n x + x x 3 The power approximation 5/3 is use for purposes of convenient analytic treatment while exact rigorous integration was also performe numerically. Inserting Eq. 3 an Eq. 3 into Eq. 3 an integrating yiels f 3 /3 ˇ + sinh ˇ = + f 5 ˇ sinh ˇ ˇ. 33 5/3 sinh ˇ }{{ ˇsinh ˇ }}{{} sym. el. For small ˇ, we use series expansion to obtain f/f =. Fig. C shows that the electrostatic energy increases as the surfaces are brought apart, an limits to zero at small istances. The energy ue to symmetry constraints iverges towars positive infinity at small istances. The analytical results coincie well with their numerical counterparts. As mentione in the introuction, a pivotal role concerning energy storage of EDL capacitors is resume by the ifferential capacitance, resulting in many technological efforts which are irecte towars maximizing its value [3,4]. In a classical approach, the ifferential capacitance of the EDL is efine as [5,6] C = x =. 34

4 M. Drab, V. Kralj-Iglič / Electrochimica Acta Fig.. A Normalize electric potential / in epenence on x for pore size = nm an ifferent values of surface charge ensity. Full line:.3 C/m, ashe line:.3 C/m, otte line:.3 C/m. Inset A Absolute ifferences of analytical an numerical solutions. B Normalize electron number ensity n/n in epenence on x corresponing to the parameter values in panel A. C Free energy of the system f Eq. 33, full line an its contributions: energy ue to antisymmetric wavefunctions of electrons full gray line an electrostatic energy ashe gray line. Also shown is free energy calculate using the Poisson-Boltzmann theory for T = 3 K Eq. 43, ashe line. In both cases surface charge ensity is =.3 C/m. Inset C A etail of the full black an ashe curves in a log-log scale fitte with the function ax b with specifie power as inicate in the inset. D Analytic ifferential capacitance /x = Eq. 38, full line an the corresponing ifferential capacitance calculate by the Poisson - Boltzmann theory Eq. 44, ashe line at pore size =nmant = 3 K. Electric potential an electron number ensity are shown for one half of the system, the epenencies are symmetric with respect to the miline. Here x = is the potential at x = in unit V. By inserting x = into Eq. 3, we obtain the epenence of x = on ˇ. Since we want to ifferentiate by, we write ˇ = à 5/4 /4, 35 where à is efine by universal constants, 3/4 me 5/3 à = 4 5 h. 36 ε The potential at x = in unit V is K 3/4 x = = cosh K /4. 37 K sinh K /4 Here, K = /ε is in unit V, while K = à 5/4. We make use of implicit ifferentiation by /x =, x = = 6K cosh K /4 4 /4 6K sinh K /4 + K K /4. 38 Numerically, we transpose the epenency x = an ifferentiate. We compare the results with the results of the acknowlege Poisson-Boltzmann theory where at finite temperature the quanta are istribute over the energies accoring to the Boltzmann istribution [] s n = exp ε n /kt an the entropic term is consiere in the free energy. This is reflecte in the partition functions an, accoringly, the Lagrange function of the system. Using an analogous approach an the same constraints upon the system as above we erive the ifferential equation for the electric potential x B exp 4 =, 39 where = 4kTε e, 4 an B is a constant. We o not present the proceure to solve Eq. 39 in etail here as a very similar formalism has previously been consiere in an electrolyte [7]. The imensionless electric potential in this case is x = 4 ln + tan x, 4 where it was chosen that x = / =. The constant was etermine from the bounary conition at the charge surface, which yiels the equation tan =, 4 which was solve numerically for. The equilibrium free energy is f K = ln f e + 43 sin where K = h /mkt 3/ /ee. The ifferential capacitance is x = = e tan kt. 44 In nee of a measure of particle ensity near the charge surface, we efine an effective thickness of the electric ouble layer at ω within which there is ω% of all electrons. Since the number ensity function nx is symmetrical to the miline x = /, the effective thickness is calculate by fining the upper boun of the integral ω. We chose ω =.95 in line with the efinition of electronic orbitals, ω nxx = ω. 45

5 58 M. Drab, V. Kralj-Iglič / Electrochimica Acta Fig. 3. A Electric potential at the charge surface x = in units V, B number ensity of electrons at the charge surface nx =, C free energy of the electric ouble layer f an D ifferential capacitance of the electric ouble layer /x =, in epenence on the area ensity of charge. All quantities are calculate for two pore sizes = nm - full lines an =. nm - otte lines. Black lines pertain to the moel taking into account the quantum statistics an gray lines pertain to the Poisson - Boltzmann theory at T = 3 K. It can be seen in Fig. C that the values of free energy of the system subjecte to quantum statistics in the low temperature limit are of the same orer of magnitue as the corresponing values obtaine by the Poisson-Boltzmann theory at room temperature. Moreover, the quantum statistics effects inicate higher energy store in the capacitor. Due to quantum constraints, the electrons are less effective in shieling the electric fiel so that the effective thickness of quantum EDL is larger an the electric fiel in the capacitor is on average larger. The ifferential capacitance of the two moels shows a monotonous increase with the potential at the charge surface Fig. D an values within the same orer of magnitue in both moels, however, the curves have ifferent shapes ue to ifferent epenencies of the electric fiel on the surface charge ensity an the istance between the surfaces. Fig. C shows that the electrostatic energy increases as the surfaces are brought apart, an approaches zero at zero istance. The energy ue to symmetry constraints behaves ifferently: at =it iverges towars positive infinity. The analytical results coincie well with their numerical counterparts Fig.. Both analytical an numerical functions are symmetric aroun the miplane x = /. For small ˇ, we use series expansion of the hyperbolic sine to obtain f/f =. For large ˇ, the expression limits to zero. However, at chosen, the slope of the f curve is ifferent in both moels inset of Fig. C. Fig. 3C shows epenencies of the electric potential at the charge surface x = in units V Panel A, number ensity of electrons at the charge surface nx = Panel B, free energy of the electric ouble layer f Panel C an ifferential capacitance of the electric ouble layer /x = Panel D, in epenence on the surface charge ensity. All these quantities monotonously increase with increasing surface charge ensity. The antisymmetry of the wavefunctions imposes stronger constraints upon the accumulation of the quanta near the charge surface an consequently prevents them to shiel the electric fiel. Therefore, the electric fiel protrues further in the irection from the charge surface Fig. 4. As larger surface charge means larger number of electrons in the system to satisfy electroneutrality, also the energy of the system increases with increasing. It can be seen in Figs. 3B, C an D that the curves pertaining to the Poisson-Boltzmann theory are almost ientical, owing to the conensation of quanta to the surfaces. As almost all quanta are in the near vicinity of the charge surface, shieling of the charge surface is very effective an the contribution of the space aroun the miplane if enlarge beyon certain istance to the relevant quantities is negligible. Unlike in the ieal gas moel, where the particles are explicitly consiere inepenent, the particles in a system obeying quantum statistics are necessarily interacting: fermions are subjecte to the Pauli exclusion principle ue to the asymmetry of the wavefunctions, which has been taken into account in the moel by the formulation of the free energy. In the moel, the electrostatic interaction is consiere within the mean fiel approximation. This means that the charge surfaces as well as the spatially istribute charges have effects through the solution of the variational problem incluing the Gauss law an the bounary conitions. The effect of the electric fiel on the wavefunctions is not taken into account. As we wante to focus on the effect of quantum statistics, we consiere the electric interaction by simplest possible means. This enable almost analytical an therefore transparent solution of the problem as regars the epenencies of the relevant quantities on the surface charge, istance between the charge surfaces an quanta mass. The next step woul be consieration of the effect of the electric fiel on the solution of the Schröinger equation, other possible confinement potentials an finite temperatures. By using a square well moel we have shown that nano - size iffuse electric ouble layer which stores energy can be forme ue to constraints on the eigenfunctions of fermions electrons. Consieration of antisymmetry of wave functions expans the electric ouble layer, causes the electric fiel to protrue further from the charge surface, evens the istribution of quanta an increases the effective thickness of the electric ouble layer. The effect on the ifferential capacitance is however more complex an epens also on other moel parameters. Incluing anisotropy of the potential well, effects of the electric fiel on the solution of the Schröinger equation an finite temperatures woul improve the escription but woul not change this major qualitative result.

6 M. Drab, V. Kralj-Iglič / Electrochimica Acta Fig. 4. Electric fiel in the irection perpenicular to the charge surface E in epenence on x for A =.3 C/m an B =.3 C/m, where in both cases, =nm. Effective thickness of the electric ouble layer.95 in epenence on C surface charge ensity for two pore sizes = nm - full lines an =. nm - otte lines, an on the istance between charge surfaces D for two surface charge ensities =.3 C/m - full lines an =.3 C/m - broken lines. Here, = nm. Black lines pertain to the moel taking into account the quantum statistics an gray lines pertain to the Poisson - Boltzmann theory at T = 3 K. Electric fiel is shown for one half of the system, the epenencies are antisymmetric with respect to the miline see Fig.. Acknowlegements Authors are inebte for ARRS grants Nos. P3-388, J-678 an J References [] J.A. Manzanares, S. Mafe, J. Bisquert, Electric ouble layer at the membrane/solution interface: istribution of electric potential an estimation of the charge store, Ber. Bunsenges. Phys. Chem [] I. Bivas, Electrostatic an mechanical properties of a flat lipi bilayer containing ionic lipis, Collois Surf. Physicochem. Eng. Asp [3] S.L. Canelaria, Y. Shao, W. Zhou, X. Li, J. Xiao, J.-G. Zhang, et al., Nanostructure carbon for energy storage an conversion, Nano Energy 95. [4] Y. Chen, F. Trier, T. Kasama, D.V. Christensen, N. Bovet, Z.I. Balogh, et al., Creation of high mobility two-imensional electron gases via strain inuce polarization at an otherwise nonpolar complex oxie interface, Nano Lett [5] R. Costa, C.M. Pereira, A.F. Silva, Insight on the effect of surface moification by carbon materials on the ionic liqui electric ouble layer charge storage properties, Electrochimica Acta [6] R. Signorelli, D.C. Ku, J.G. Kassakian, J.E. Schinall, Electrochemical ouble-layer capacitors using carbon nanotube electroe structures, Proc. IEEE [7] M.G. Gouy, Sur la constitution e la charge electrique à la surface un electrolyte, Journal e Physique et Le Raium [8] D.L. Chapman, LI. A contribution to the theory of electrocapillarity, Philos. Mag. Ser [9] R. Parsons, The electrical ouble layer: recent experimental an theoretical evelopments, Chem. Rev [] O. Stern, Zur Theorie er elektrolytischen Doppeltschicht, Z.Elektrochem [] V. Kralj-Iglič, A. Iglič, A simple statistical mechanical approach to the free energy of the electric ouble layer incluing the exclue volume effect, J. Phys. II [] I. Borukhov, D. Anelman, H. Orlan, Steric effects in electrolytes: A moifie Poisson-Boltzmann equation, Phys. Rev. Lett [3] E. Gongaze, U. van Rienen, V. Kralj-Iglič, A. Iglič, Langevin Poisson-Boltzmann equation: point-like ions an water ipoles near a charge surface, Gen. Physiol. Biophys [4] E. Gongaze, A. Iglič, Asymmetric size of ions an orientational orering of water ipoles in electric ouble layer moel - an analytical mean-fiel approach, Electrochimica Acta [5] A. Velikonja, V. Kralj-Iglič, A. Iglič, On asymmetric shape of electric ouble layer capacitance curve, Int J Electrochem Sci [6] N.D. Lang, W. Kohn, Theory of metal surfaces: charge ensity an surface energy, Phys. Rev. B [7] A. Mioek, G. Castillo, T. Hianik, H. Korri-Youssoufi, Electrochemical aptasensor of human cellular prion base on multiwalle carbon nanotubes moifie with enrimers: a platform for connecting reox markers an aptamers, Anal. Chem [8] P. Kowalczyk, A. Ciach, A.P. Terzyk, P.A. Gauen, S. Furmaniak, Effects of critical fluctuations on asorption-inuce eformation of microporous carbons, J. Phys. Chem. C [9] R. Imani, M. Pazoki, A. Tiwari, G. Boschloo, A.P.F. Turner, V. Kralj-Iglič, et al., Ban ege engineering of TiO@DNA nanohybris an implications for capacitive energy storage evices, Nanoscale [] M. Kulkarni, Y. Patil-Sen, I. Junkar, C.V. Kulkarni, M. Lorenzetti, A. Iglič, Wettability stuies of topologically istinct titanium surfaces, Coll. Surf. B: Biointerfaces [] T.L. Hill, An Introuction to Statistical Thermoynamics, Dover Publications Inc., New York, 986, pp [] D.Y. Chan, D.J. Mitchell, The free energy of an electrical ouble layer, J.Colloi Interface Sci [3] A.A. Moya, The ifferential capacitance of the electric ouble layer in the iffusion bounary layer of ion-exchange membrane systems, Electrochimica Acta [4] M. Pazoki, A. Hagfelt, G. Boschloo, Stark effects in D35-sensitize mesoporous TiO: influence of ye coverage an electrolyte composition, Electrochimica Acta [5] A.A. Kornyshev, Double-layer in ionic liquis: paraigm change? J. Phys. Chem. B [6] V. Lockett, M. Horne, R. Seev, T. Roopoulos, J. Ralston, Differential capacitance of the ouble layer at the electroe/ionic liquis interface, Phys. Chem. Chem. Phys [7] S. Engström, H. Wennerström, Ion conensation on planar surfaces. A solution of the Poisson-Boltzmann equation for two parallel charge plates, J. Phys. Chem