Keywords: capacitance, electric double layer, Stern layer, relative permittivity, water ordering

Transcription

1 Int. J. Electochem. ci., 9 (14) Intenational Jounal of ELECTROCHEMICL CIENCE hot communication Chage Dependent Capacitance of ten Laye and Capacitance of Electode/Electolyte Inteface ljaž Velikonja 1, Ekateina Gongadze 1, Veonika Kalj-Iglič and leš Iglič 1,* 1 Faculty of Electical Engineeing, Univesity of Ljubljana, Tžaška 5, I-1 Ljubljana, lovenia Faculty of Health tudies, Univesity of Ljubljana, Zdavstvena 5, I-1 Ljubljana, lovenia * ales.iglic@fe.uni-lj.si Received: July 14 / ccepted: 8 ugust 14 / Published: 5 ugust 14 The influence of potential dependent elative pemittivity of ten laye on the capacitance of electode-electolyte inteface is studied theoetically. decease of capacitance at lage magnitudes of electode suface potential is pedicted. t lage electode suface potentials the capacitance of metal-electolyte inteface is detemined solely by the capacitance of ten laye, wheeas the contibution of the capacitance of use laye is negligible. Keywods: capacitance, electic double laye, ten laye, elative pemittivity, wate odeing 1. INTRODUCTION When metal electode comes into contact with electolyte solution an electic double laye (E), composed of chaged suface of electode and counte-ions is ceated (Figue 1). Counte-ions, i.e. ions with the sign of chage opposite to that of electode suface, ae accumulated at electode suface, while co-ions, having the same sign of the chage as the electode metal suface, ae depleted fom the metal-electolyte inteface. s a consequence the potential eence ove the metal-electolyte inteface is geneated [1,]. The dop of electic potential at metal-electolyte inteface occus mostly in electolyte [].

2 Int. J. Electochem. ci., Vol. 9, Figue 1. chematic figue of the ten laye ( x b) and use electic double laye (b x ). Oute Helmholtz plane (OHP) is located at the distance of closest appoach (b) which is appoximately equal to the hydated adius of the counte-ions. The wate dipoles ae oiented within ten laye as well as aound counte-ions and co-ions in the bulk solution. The symbol denotes the suface chage density of the metal suface. Most of the theoetical models of electolyte solution in contact with chaged suface assume that the elative (dielectic) pemittivity is constant eveywhee in the electolyte solution [-7] and do not conside the space dependence of elative pemittivity of electolyte solution nea the chaged suface [1]. Theefoe the classical Gouy-Chapman theoy of electic double laye [8,9] has been genealized by taking into account the wate polaization in electolyte solution nea the chaged suface esulting in spatial decay of elative pemittivity in close vicinity of the chaged suface [1-16]. In this wok we adopted ten model [17] as a combination of Gouy-Chapman and Hemholtz models [18] whee the oute Helmholtz plane (OHP) define a bode between ten laye and use laye (Figue 1). The elative pemittivity of ten laye is calculated fo eent values of suface chage density of the metal suface within a simple theoetical model of oientational odeing of wate. The total (eential) capacitance of the metal-electolyte inteface ( C ) is calculated using the fomula [19]: 1 1 1, (1.1) C C C

3 Int. J. Electochem. ci., Vol. 9, assuming that C is the capacitance of two capacitos in seies, i.e. ten laye capacito ( C ) and use laye capacito ( C ).. THEORETICL MODEL.1. Capacitance of metal-electolyte inteface In the above Equation 1.1 the influence of the final size of molecules is taken into account only by the distance of closest appoach ( b ) (Figue 1) in the fist tem fo the capacitance of ten laye C / b. Hee is the pemittivity of the fee space and the electic field dependent elative pemittivity of the Helmholtz/ten laye. The capacitance of the use laye ( x b) in the second tem ( C ) also depends on the size of molecules, which can be descibed within eent theoetical appoaches [-1,1,11,5]. Howeve, in this wok we would like to keep all fomulas analytical and simple. To this end C is calculated within Gouy-Chapman model fo point-like ions fom Gahame equation [1,16,19]: d 1 C e n cosh( e ( x b) / ). (.1) d( x b) Hence 1 b 1 C e n cosh( e ( x b) / ) 1. (.) Hee n is the bulk numbe density of salt ions, e is the unit chage, pemittivity of use electic double laye in the egion b x and 1/kT is the constant, whee kt is the themal enegy. Close to the chaged metallic suface, i.e. fo small values of x, the elative pemittivity stongly depends on the distance fom the chaged metallic suface [1,11,13-16]. Fo simplicity easons in this wok the pemittivity of use laye (fo x b) in Equation.1 ) is consideed independent on the electic field stength [1,19], while due to stong ( oientation of wate molecules in ten laye the dependence of elative pemittivity of ten laye ) on the electic field stength [15] is taken into account. This means that also the ( capacitance of ten laye (1 st electic field stength. tem on the ight hand side of the Equation.) depends on the.. Capacitance of Helmholtz/ten laye In this subsection we shall fist deive the electic field dependence of elative pemittivity in ten laye ( ). In the second step the dependence of capacitance of ten laye C b on the electic field stength will be detemined. /

4 Int. J. Electochem. ci., Vol. 9, In the model a single wate molecule is consideed as the sphee with the point-like igid (pemanent) wate dipole located at its cente [13,16]. The pemittivity of the sphee is n, whee n 1.33 is the optical efactive index of wate [13]. The polaization of wate molecules P in ten laye ( x b) can be expessed as [13,16] : n Pnw p L ( pe ), (.3) 3 whee n is the constant (bulk) numbe density of wate molecules, E is the magnitude of the w electic field stength, extenal dipole moment, n is the optical efactive index of wate, p is the magnitude of the wate L u coth u 1/ u is the Langevin function and n /. In ten laye ( x b) the magnitude of electic field stength E is constant fo given, theefoe also elative pemittivity ( ) is constant fo given. The elative pemittivity in ten laye can then be witten as (see also [13]) : P n w p n L( pe ) n n E 3 E, (.4) and finally the capacitance of ten laye in the egion x b: 1 b. (.5) C n L( pe ) n nw p 3 E Fo x b we assume constant elative pemittivity 78.5 at oom tempeatue..3. Bounday conditions The bounday conditions at x and x b ae : d dx d d,,, (.6) xb xb dx dx x xb whee is the suface chage density of the metallic suface at x. Due to constant electic d d field stength in ten laye it follows fom Equations.6 : dx dx d dx xb x xb xb. (.7) The bounday condition.7 can be used in the egion b x to solve Gouy-Chapman equation and calculate the value of electic potential ( x b). 3. REULT ND DICUION Bounday condition at x (see Equation.6) and Equation.4 togethe yields the nonlinea equation fo the magnitude of electic field ( E ) in ten laye :

5 Int. J. Electochem. ci., Vol. 9, E n n p w n 3 ( p E) L. (3.1) E Inseting the calculated value of E in Equation.4 yields the value of elative pemittivity in ten laye ( ) fo given suface chage density. Figue shows the elative pemittivity as a function of suface chage density. It can be seen that stongly deceases with inceasing magnitude of, which can be explained by satuation of oientational odeing of wate dipoles in stong electic field at lage values of [16,4,5]. Fo illustation Figue 3 shows the space dependence of elative pemittivity fo thee eent values of. Figue. The elative pemittivity in ten laye x b as a function of the magnitude of the suface chage density of electode suface ( ). The values of model paametes ae : distance of closest appoach b.4 nm, p 3.1 D and concentation of wate n w / N = 55 mol/l. Figue 3. The space dependence of elative pemittivity in ten ( x b) and use laye (b x ) calculated fo thee values of electode suface chage density ( ). The values of othe paametes ae the same as in Figue.

6 Int. J. Electochem. ci., Vol. 9, The electic potential dependence in the use egion b x is calculated fom Gouy- Chapman equation [1,3,4]: d en sinh( e ( x)) dx, (3.) whee the volume chage density of electolyte solution en sinh( e ( x)) is taken into account. To calculate fom Equation 3. the dependence ( x) fo x b, the bounday condition defined by Equation.7 should also be consideed. Figue 4. The calculated electic potential ( x) as a function of the distance fom the chaged metallic suface. The model paametes ae: suface chage density.1 s / m, bulk concentation of ions n / N =.1 mol/l, distance of closest appoach b.4 nm, p 3.1 D and concentation of wate n w / N = 55 mol/l. The inset shows the intedependence between the potentials ( x b) and ( x ) calculated fo eent. Figue 4 shows the space dependence of the electic potential ( x). The linea space dependence of electic potential in ten laye x b was detemined by integation fom the bounday conditions x (fist Equation.6) and not fom Eqution 3.. The inset in Figue 4 shows the intedependence between electic potentials at x and x b. Note that in Equation.1 fo capacitance of use laye C appeas the potential ( x b) [19] and not ( x ). Fo bette undestanding Figue 5 shows also the dependences of the suface potentials ( x ) and ( x b) on suface chage density.

7 Int. J. Electochem. ci., Vol. 9, Figue 5. The calculated electic potential ( x ) as a function of the electode suface chage density. Inset shows the dependence of ( x b) on. The values of model paametes n / N, b, p and n w / N ae the same as in Figue 4. Figue 6 shows the capacitances of ten ( C ) and use laye ( C ) and the total capacitance ( C ) as functions of the suface potential ( x ). It can be seen in Figue 6 that use laye capacitance C inceases, while ten laye capacitance C deceases with inceasing magnitude of ( x ). s the capacitance C is calculated by Equation 1.1, this explains why at lage magnitudes of suface potential ( x ) the contibution of C to C is negligible and C C as pesented in Figue 6. Figue 6. Diffeential capacitance of ten ( C ) and use laye ( C ( ) and the total capacitance C ) as a function of the suface potential ( x ). The values of model paametes n / N, b, p and n w / N ae the same as in Figue 4.

8 Int. J. Electochem. ci., Vol. 9, Figue 7 shows the calculated capacitance of electode-electolyte inteface ( C ) fo elative pemittivity in ten laye which depends on suface chage density of electode ( ( ) ), detemined as descibed above, and fo the case of constant value of elative pemittivity in ten laye ( 3 ). It can be seen that in the fist case of ( ) the capacitance C exhibit a maximum and subsequent monotonous deceasing of C with inceasing magnitude of ( x ). On the contay, fo constant 3 thee is no maximum in C and its decease at lage magnitudes of ( x ). The maximum in dependence of C on suface potential ( x ) and decease of C at lage magnitudes of ( x ) can be pedicted also at constant elative pemittivity by taking into account the finite size of ions (volume excluded effect) within Bikeman model [6,16,3]. In accodance, it was shown ecently [16], that combination of volume excluded effect [] and space dependence of elative pemittivity in electolyte solution [13] pedict stonge decease of C at lage magnitudes of ( x ) as pedicted within pue Bikeman model. Figue 7. Diffeential capacitance of metal-electolyte inteface ( C ) as a function of the suface potential ( x ) fo the case voltage/suface chage dependent elative pemittivity of ten laye (Figues and 3) and constant pemittivity in ten laye. The values of the model paametes ae the same as in Figue CONCLUION Wate is one of the most impotant molecules in biological systems [6]. In this wok we attempt to uncove the elation between the electic potential dependent oientational odeing of wate molecules in ten laye and the capacitance of metal electode-electolyte inteface. In ode to keep ou theoy analytical and appopiate fo analysis of expeimental esults as much as possible the finite volume of molecules was taken into account by the distance of closest

9 Int. J. Electochem. ci., Vol. 9, appoach only. In addition, also the tanspot phenomena in ten and use layes [7] ae totally neglected. To this end the influence of conductivity of use laye on the capacitance of ten laye can not be descibed within the pesented model. Obtaining the chage/potential dependence of elative pemittivity in ten laye, the capacitance of electode-electolyte inteface as a function of electode chage/potential is calculated. decease of capacitance at lage magnitudes of electode suface potential is pedicted. It is also shown that at lage electode suface potentials the capacitance of metalelectolyte inteface is detemined solely by the capacitance of ten laye, wheeas the contibution of the capacitance of use laye becomes negligible. CKNOWLEDGEMENT This wok was in pat suppoted by the lovenian Reseach gency (RR). Refeences 1. H. Butt, K. Gaf and M. Kappl, Physics and Chemisty of Intefaces, Wiley-VCH, Weinheim (3).. W. chmickle and E. antos, Intefacial Electochemisty, pinge, Heidelbeg (1). 3.. McLaughlin, nnual Review of Biophysics and Biophysical Chemisty, 18 (1989) G. Cevc, Biochimica et Biophysica cta, 131 (199) V. Kalj-Iglič and. Iglič, Jounal of Physics II, 6 (1996) M.Z. Bazant, M.. Kilic, B. toey and. jdai, dvances in Colloid and Inteface cience, 15 (9) I. Bivas and Y.. Emakov, dvances in Plana Lipid Bilayes and Liposomes 5 (7) M.G. Gouy, Jounal de Physique et Le Radium (Fance), 9 (191) D.L. Chapman, Philosophical Magazine, 5 (1913) C.W. Outhwaite, Molecula Physics, 31 (1976) C.W. Outhwaite, Molecula Physics, 48 (1983) T. Nagy, D. Hendeson and D. Boda, Jounal of Physical Chemisty B, 115 (11) E. Gongadze and. Iglič, Bioelectochemisty, 87 (1) R. Misa,. Das and. Mita, Jounal of Chemical Physics, 138 (13) E. Gongadze and. Iglič, cta Chimica lovenica, 61 (14) E. Gongadze,. Velikonja, Š. Peutkova, P. Kama,. Maček-Leba, V. Kalj-Iglič and. Iglič, Electochimica cta, 16 (14) O. ten, Zeitschift fü Elektochemie, 3 (194) H. Helmholtz, nnals of Physics, 7 (1879) V. Lockett, M. Hone, R. edev, T. Rodopoulos and J. Ralston, Physical Chemisty Chemical Physics, 1 (1) J. Bikeman, Philosophical Magazine, 33 (194) T. Gimley and N. Mott, Discussions of the Faaday ociety, 1 (1947) M. Eigen and E. Wicke, Jounal of Physical Chemisty, 58 (1954) Konyshev, Jounal of Physical Chemisty B, 111 (7), F. Booth, Jounal of Chemical Physics, 19 (1951) H. Föhlich, Theoy of Dielectics, Claendon Pess, Oxfod (1964).

10 Int. J. Electochem. ci., Vol. 9, Z. sov, M. Rappolt and J. Gdodolnik, ChemPhysChem, 1 (9) H. Wang and L. Pilon, Electochimica cta, 64 (1) The uthos. Published by EG ( This aticle is an open access aticle distibuted unde the tems and conditions of the Ceative Commons ttibution license (