Electrical double layer: revisit based on boundary conditions

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1 Electrcal double layer: revst based on boundary condtons Jong U. Km Department of Electrcal and Computer Engneerng, Texas A&M Unversty College Staton, TX , USA Abstract The electrcal double layer at nfnte flat sold surface was dscussed wth respect to the trple layer model. Charge densty at the outer Helmholtz plane s assumed to be zero usually. However, the assumpton causes the poston of the outer Helmholtz plane to be obscure. The charge densty was calculated and the condton that the charge densty s neglgble was dscussed. Keywords: Electrcal double layer; Stern layer; outer Helmholtz plane; trple layer model Correspondng author. E-mal address: jongkm@ee.tamu.edu 1

2 1. Introducton Understandng phenomena near the nterface of sold and an aqueous soluton s of mportance n electroknetcs, mcrofludcs, collodal dsperson, and electrochemstry. When sold surface contacts an aqueous soluton ncludng electrolyte, the sold surface becomes charged due to the dfference of electron (or on) affntes between the sold surface and the soluton or the onzaton of surface groups. In addton, the surface charges cause a specal structure at the nterface, so called the electrcal double layer (EDL) [1,]. Usually Gouy-Chapman-Stern model (GCS) s wdely used to descrbe the EDL. The GCS model conssts of two layers; Stern layer (SL) and dffuse layer (DL). The SL s the regon next to the surface, and ons n the SL are bound near the surface due to specally-adsorbng and Coulomb nteractons. The DL s the regon next to the SL and ons n the DL can move freely n any drecton. The SL has two planes; the nner Helmholtz plane (IHP) and the outer Helmholtz plane (OHP) as shown n Fgure 1. In general, the surface charge densty and the charge densty at the IHP are predcted n thermodynamc approach or ste-bndng models [1-4]. However, the charge densty at the OHP s assumed to be zero wthout any consderaton. In ths letter, we wll evaluate the charge densty at the OHP as a functon of the sum of the surface charge densty and the charge densty at the IHP and dscuss the condton that the charge densty at the OHP s neglgble.. Electrcal double layer Let s consder nfnte flat sold surface n contact wth an aqueous soluton ncludng electrolyte. The surface s unform. Fgure 1 shows a schematc dagram of the EDL. To easly understand the structure of the EDL, we ntroduce three types of ons n the soluton; potentaldetermnng, specfcally-adsorbed and ndfferent ons [5]. Potental-determnng ons are adsorbed at the surface drectly. Ther equlbrum dstrbuton between the surface and the

3 soluton determnes the surface potental relatve to potental n bulk soluton. The adsorbed potental-determnng ons form the surface charge densty. Indfferent ons are affected by Coulomb force of the surface charge. Thus, they are repelled by the same sgn surface charges whle they are attracted by the opposte sgn. Specfcally-adsorbed ons are strongly nteracted wth the surface through all nteractons other than purely Coulomb force. In the trple layer model, the IHP s located at the center of specfcally-adsorbed ons and the OHP s located at the center of ndfferent ons [1]. Here we consder only trple layer model whch has the specfc adsorbed ons. However, our concluson may be applcable to other EDL models. In the trple layer model, potentals n the SL consstng of the regon 1 and the regon shown n Fg. 1 satsfy ψ =. (1) If we apply the potental-related boundary condtons at the surface, the IHP and the OHP, the potental profles n the SL are ψ ( ) ψ + ( ψ ψ ) x β = at x β (regon 1) () 1 x and x β ψ ( x) = ψ + ( ζ ψ ) at β x δ (regon ), (3) δ β where ψ and ψ are, respectvely, the surface potental and the potental at the IHP, β and δ are, respectvely, the postons of the IHP and of the OHP from the surface, and ζ s the potental at the OHP. The above potental profles n the SL are obtaned through boundary condton based on potental but they also satsfy boundary condtons based on electrc dsplacement. Usually, t s assumed n the trple layer model that the charge densty at the OHP s zero. However, the poston of the OHP s obscure f there s no charge at the OHP. Therefore, the boundary 3

4 condton at the OHP need to nclude the charge densty at the OHP. The boundary condtons based on electrc dsplacement at the surface, the IHP and the OHP are, respectvely, gven by 1 = 1 at = x= x= β x (the surface), (4) = at x = β (the IHP), (5) x x= β and + = at x = δ (the OHP), (6) where, and dffuse OHP b x= δ x= δ OHP are the charge denstes at the surface, the IHP and the OHP, respectvely, and 1 and are the delectrc constants n the regon 1 and the regon, respectvely. These delectrc constants are generally dfferent. The delectrc constant n the regon 1 s dscussed by Sverjensky [4]. The delectrc constant n the DL s thought of as the delectrc constant of bulk water (77.78 at 3K). Summng Eqs. (4) through (6) gves dffuse + + OHP = b. (7) Snce the vrtual charge densty n the DL s defned as x= δ d dffuse = b, (8) x= δ Eq. (7) s rewrtten as + + OHP + d =. (9) Equaton (9) s electro-neutralty condton and s the same as that n Ref. [1] except ncludng OHP. For smplcty of notaton, let s ntroduce effectve surface charge densty (ESCD), sayng = +, and charge densty rato defned by OHP = γ s where < γ 1 [6]. Snce the s 4

5 OHP s not movable n the normal drecton of the surface, the sum of Coulomb force per area actng on the charges at the OHP due to the effectve surface charge, electrostrcton pressure of the water n the regon, and electrostrcton pressure of the soluton n the DL has to be zero. The Coulomb force actng on the charges at the OHP s F C ( y y ) j + ( z z ) ( y y ) + ( z ) γ s δ + dy dz 4 dydz π z = 3 k [ δ + ], (1) where, j and k are the unt vectors n x, y and z drectons, respectvely. Integratng the rght hand sde of Eq. (1) yelds FC A γ s =. (11) Ths force acts on the charges at the OHP n the negatve x drecton. That s, the water n the regon s compressed by the charges at the OHP. There are two knds of electrostrcton pressure [7]. One s sutable for constant volume system and the other s sutable for constant chemcal potental system. If water molecules freely enter or leave the regon, the latter should be used; otherwse, the former should. Snce the poston of the OHP s fxed, the volume of the water n the regon s constant. Thus, electrostrcton pressure for constant volume system s approprate for the EDL. Electrostrcton pressures of flud at constant volume system, P, satsfes [7] dp ( ) E de E T = ρ 1, (1) ρ, where s the delectrc constant of flud, ρ s densty, and E s electrc feld. The delectrc constant of water as a functon of electrc feld strength E s gven by [8] ( n + ) 73 E ( n + ) 7 ρ µ µ = n + L, (13) 3 73 E 6k BT 5

6 where µ s electrc dpole of a sngle water molecule (. Debye unts), n s the optcal refractve ndex of water (1.33 at 3K), and L ( x) s the Langevn functon gve by ( x) coth( x) x L = 1. Combnng Eq. (13) wth Eq. (1) and ntegratng both sdes from zero electrc feld to electrc feld n the regon yeld ( n 1) s P = P + (14) where P s the electrc feld-free pressure. Here the electrc feld n the regon s obtaned as E = by use of Eqs. (4) and (5). s Snce there s no pressure-drven flow n the DL, the pressure n the DL s constant. In addton, t s feld-free pressure n bulk soluton. Therefore, the electrostrcton pressure n the DL s equal to the feld-free pressure P. In order that the OHP does not move n the normal drecton of the surface, the pressure of the water n regon s equal to the sum of the Coulomb force per area and the pressure of bulk soluton; P + ( n 1) s γ s = + P, (15) or OHP γ = + n 1 =. (16) Fgure shows the delectrc constant of the water n the regon as a functon of the ESCD. The delectrc constant s numercally calculated by usng Eq. (13) and E = s. It s shown n Fg. that the delectrc constant n the regon decreases wth ncreasng the ESCD and that t s approxmately the delectrc constant of bulk water at the ESCD of less than 1 µm/cm. Fgure 3 shows the dependence of the charge densty rato γ on the ESCD. It s shown n Fg. 3 that the charge densty rato s approxmately.1 at the ESCD of smaller than 1 µc/cm. However, t s also shown that the charge densty at the OHP cannot be gnored at the 6

7 ESCD of bgger than 3 µc/cm. Therefore, only when the ESCD s smaller than 1 µc/cm, the charge densty at the OHP s neglgble. That s, the exstent trple layer model s avalable when the ESCD s smaller than 1 µc/cm. 3. Summary In summary, the charge densty at the OHP was calculated. It was shown that the charge densty at the OHP s neglgble at the effectve surface charge densty of smaller than 1 µc/cm. Acknowledgement The author would lke to acknowledge the support of Ebensbeger/Fouraker Graduate Fellowshp. 7

8 Reference 1. J. Lyklema, Fundamentals of Interface and Collod Scence Vol. II: sold-lqud nterface, Academc press, London, R. J. Hunter, Foundatons of collod scence, Oxford Unversty, New York, D. E. Yates, S. Levne and T. W. Healy, J. Chem. Soc. Faraday I 7 (1974) D. A. Sverjensky, Geochm. Cosmochm. Acta 69 (5) J. Lyklema, Pure Appl. Chem. 63 (1991) Snce the charges at the OHP are nduced by the surface charge through Coulomb force, the absolute value of the charge densty at the OHP should be smaller than that at the surface. 7. H. S. Frank, J. Chem. Phys. 3 (1955) In-Chul Yeh and M. L. Berkowtz, J. Chem. Phys. 11 (1999)

9 Fgure Captons Fg. 1. Schematc dagrams of the electrcal double layers wth respect to the trple layer model. The numbers 1 and represent, respectvely, regon 1 and regon, and the long-dashed lne represents potental profle when the surface charge s negatve. Fg.. Delectrc constant of the water n the regon as a functon of the absolute value of the effectve surface charge densty. Fg. 3. Charge densty rato γ as a functon of the absolute value of the effectve surface charge densty. 9

10 Fg. 1 1

11 Fg. 11

12 Fg. 3 1

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