Electrical double layer: revisit based on boundary conditions

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1 Electrical oule layer: reviit ae on ounary conition Jong U. Kim Department of Electrical an Computer Engineering, Texa A&M Univerity College Station, TX , USA Atract The electrical oule layer at infinite flat oli urface ha een icue with repect to ounary conition ae on potential an electric iplacement, repectively. It wa hown that the electrokinetic charge enity i equal to the um of the urface charge enity an the charge enitie at the inner an the outer Helmholtz plane. The ratio of the charge enity at the outer Helmholtz plane to the urface charge enity wa evaluate, an a new relation etween the urface charge enity an the zeta potential wa erive. Keywor: Electrical oule layer; Surface charge; Zeta potential; Electrokinetic charge Correponing author. are: jongkim@ee.tamu.eu

2 . Introuction Unertaning phenomena near the interface of oli an an electrolyte olution i of importance in electrokinetic, microfluiic, colloial iperion, an electrochemitry. When oli urface contact an electrolyte olution, the oli urface ecome charge ue to the ifference of electron (or ion) affinitie etween the oli urface an the olution or the ionization of urface group. In aition, the urface charge caue a pecial tructure at the interface, o calle the electrical oule layer (EDL) [-5]. Uually Gouy-Chapman-Stern moel (GCS) i wiely ue to ecrie the EDL. The GCS moel conit of two layer; Stern layer (or Helmholtz region) an iffue layer. The Stern layer i the region next to the urface where ion in the olution cannot move in the longituinal irection of the urface ue to pecially-aoring an Coulom interaction [6]. The iffue layer i the region next to the Stern layer. In the iffue layer, ion in the olution can move freely in any irection. The Stern layer ha two plane; the inner Helmholtz plane (IHP) an the outer Helmholtz plane (OHP) a hown in Figure a. In general, the charge enitie an the potential at the urface an the IHP are evaluate through equilirium contant of urface reaction, an they affect the charge enity an the potential at the OHP. Since the equilirium contant involve ion concentration in the olution, the charge enitie an the potential at the urface an two plane epen on the ion concentration in the olution [7-]. From the electrokinetic point of view, potential relate to the interface i name zeta potential. The zeta potential i the potential at hear plane within which the motion of flui i tationary. However, the zeta potential i aume to e the potential at the OHP ince the hear plane lie very cloe to the OHP []. The zeta potential i meaure y electroomotic moility, treaming potential, an electrophorei [,3]. Thee meaurement are relate irectly to the iffue layer. The electrokinetic charge amount correpon to total charge amount in the iffue layer [], an it i expree with repect to the zeta potential.

3 In thi note, we will icu the EDL with repect to ounary conition an rive a new relation etween the urface charge enity an the zeta potential.. Electrical oule layer Let conier infinite flat oli urface in contact with an electrolyte olution. Thu, phyical quantitie uch a potential, ion itriution, an electric iplacement are epenent on only oneimenional coorinate. Figure a how a etaile chematic iagram of the EDL. A mentione efore, it conit of the Stern layer an the iffue layer. To eaily unertan the tructure of the EDL, we introuce three type of ion in the olution; potential-etermining, pecificallyaore an inifferent ion [6]. Potential-etermining ion are aore at the urface irectly. Their equilirium itriution etween the urface an the olution etermine the urface potential relative to potential in ulk olution. The aore potential-etermining ion form the urface charge enity σ. Inifferent ion are affecte y Coulom force of the urface charge. Thu, they are repelle y the ame ign urface charge while they are attracte y the oppoite ign. Specifically-aore ion are trongly interacte with the urface through all interaction other than purely Coulom force [6]. By the efinition of ion type, the IHP i locate at the center of pecifically-aore ion an the OHP i locate at the center of inifferent ion. A a matter of fact, an ion can e inifferent or pecifically-aore ince the efinition of the type oe not epen on what it i ut on where it i. It i commonly aume that there i no charge etween the urface an the IHP an etween the IHP an the OHP. Since the urface charge enity an the charge enity at the IHP are etermine y equilirium contant of chemical reaction [7-] an we are interete in the relation etween the charge enity at the OHP an the zeta potential, we merge the urface an the IHP into a urface []. Thu, implifie EDL conit of the urface, the OHP an the iffue layer a hown in Fig.. Uing Gau law give a moifie urface charge enity σ = σ +, () σ i 3

4 where σ i i the charge enity at the IHP, an a moifie urface potential i regare a ψ = ψ i. () where ψ i the potential at the IHP. It i note that thi implification oe not affect our reult. i Since there i no ion etween the urface an the OHP, potential in the Stern layer atifie ψ =. (3) If we ue the urface potential an the zeta potential a the potential at the ounarie of the Stern layer, the potential profile in the Stern layer i ( x) = ψ + ( ζ ψ ) x δ ψ Stern at x δ, (4) where δ i poition of the OHP an ζ i the zeta potential. Here, it i aume that the potential at the OHP i the zeta potential ζ a mentione efore. To otain potential profile in the iffue layer, we ue Poion-Boltzmann equation for a electrolyte []; ψ q = n qψ inh k BT, (5) where q i the elementary charge, n i the concentration of ulk electrolyte olution, k B i the Boltzmann contant, an T i the aolute temperature. Solving the Poion-Boltzmann equation with the zeta potential give the potential profile in the iffue layer [4] ( κ( x δ )) tanh( qζ 4k T ) ( ( )) ( ) B κ x δ tanh qζ 4k BT k BT + exp ψ iffue ( x) = ln, (6) q exp where κ = q n k B T, i.e., the invere Deye length. Here, potential in ulk region i zero. The potential profile in the Stern layer an the iffue layer are otaine through the ounary conition ae on potential. However, ounary conition ae on electric iplacement can e ue. The ounary conition of the electric iplacement at the urface an at the OHP (refer to Figure.) are given y 4

5 ψ Stern σ = at = x= x (the urface), (7) an ψ Stern ψ + σ = at x = δ (the OHP), (8) iffue x= δ x= δ where ψ Stern i the potential in the Stern layer, ψ iffue i the potential in the iffue layer, σ i the charge enity at the OHP, i the vacuum permittivity, an an are ielectric contant in the Stern layer an in the iffue layer, repectively. The ielectric contant in the Stern layer i ifferent from that in the iffue layer, an the ielectric contant in the iffue layer i thought of a the ielectric contant of ulk water (77.78 at 3K). Comining Eq. (4) with Eq. (7) an (8) give ψ iffue σ + σ =. (9) It i worthy noting that Eq. (9) i ifferent from relation in literature [-4, 6-]. The term on the left han ie of Eq. (9) in the literature i only the charge enity at the OHP or only the negative urface charge enity intea of the um of the urface charge enity an the charge enity at the OHP. In aition, Eq. (9) i not compatile with electroneutrality conition [6]: x= δ σ + σ = σ + σ i + σ =. () That i to ay, if the electroneutrality conition i right, the electric iplacement at the OHP i zero, which i not zero experimentally. Although the electroneutrality conition i commonly ue, it i not appropriate for the EDL ince the electric iplacement at the OHP in the iffue layer i not zero. To aure Eq. (9), we erive it again with Gau law; D x = ρ ( x), () 5

6 where D i the electric iplacement an ( x) x ρ i charge enity per unit volume. Integrating Eq. () from a point a to a point ( > a ) give If a = an = δ in Eq. (), then D ( ) D ( a) = ( x ) D x x ρ x. () x δ ( δ ) ρ( x ) x = σ + σ a =. (3) Eq. (3) i exactly the ame a Eq. (9). In oth of Eq. (9) an Eq. (3), we aume that electric iplacement inie oli i zero. A well, when a = an i infinite, i.e., in ulk region, Eq. () ecome σ + σ σ ek = or σ ek = σ + σ, (4) where σ i the electrokinetic charge enity, i.e., σ = ρ ( x ) ek ek x. Eq. (4) i gloal electrical neutrality conition. However, thi gloal electrical neutrality conition nee moifying when charge exit inie the oli. δ 3. Charge enity at the outer Helmhotz plane If the ign of the urface charge i the ame a that of the zeta potential, the charge amount at the OHP i equal to or le than the urface charge amount, i.e., σ σ []. Let ue a new notation, σ = γ σ where < γ. Uing the new notation, Eq. (9) i rewritten a ψ iffue σ =. (3) ( γ ) Since the OHP i not movale in the normal irection of the urface, the um of Coulom force per area etween the urface charge an the charge at the OHP, electrotriction preure of the flui in the Stern layer an preure of the flui in the iffue layer i zero. The Coulom force etween the urface charge an the charge at the OHP i x= δ 6

7 F C ( y y ) j + ( z z ) ( y y ) + ( z ) γσ δ i + y z 4 yz π z = 3 [ δ + ] k, (4) where i, j an k are the unit vector in x, y an z irection, repectively. Integrating the right han ie of Eq. (4) yiel FC A γσ =. (5) Thi force act on the charge at the OHP in the negative x irection. Electrotriction preure of the flui in the Stern layer i [3] P = P ρ T ( ) E ρ, (6) where P i fiel-free preure in the Stern layer, ρ i enity, an E i electric fiel in the Stern layer. From Eq. (7), E = σ. Uner trong electric fiel, the ielectric contant of water i a function of electric fiel trength E a [4] ( n + ) 73 E ( n + ) 7 ρ µ µ = n + L, (7) 3 73 E 6k BT where µ i electric ipole of a ingle water molecule (. Deye unit), n i the optical refractive inex of water (.33 at 3K), an L ( x) i the Langevin function give y ( x) coth( x) x L =. Differentiating Eq. (7) with repect to enity an then multiplying it with enity give n ρ =. (8) ρ T Thu, comining Eq. (6) an (8) with E =, the electrotriction preure of the flui in the Stern layer i rewritten y σ 7

8 ( n ) σ P = P + (9) Since there i no preure-riven flow in the iffuion layer, the preure in the iffuion layer i contant an it i fiel-free preure in ulk region. Thu we aume that the preure in the iffuion layer i equal to the fiel-free preure P. In orer that the OHP oe not move in the normal irection of the urface, the preure of the flui in the Stern layer, Eq. (), i equal to the um of the Coulom force per area, Eq. (5) an the fiel-free preure in the iffuion layer; P + ( n ) σ γ σ = + P, (a) or γ = n. () Figure how the ielectric contant in the Stern layer a a function of the urface charge enity. The ielectric contant of water i numerically calculate y uing Eq. (7) an E = σ. It i hown in Fig. that the ielectric contant in the Stern layer ecreae rapily in µm/cm urface charge enity range. Figure 3 how the epenence of the ratio γ of the charge amount at OHP to the urface charge amount on the urface charge enity. It how clearly that the ratio i le than unity an it increae with the urface charge enity. Finally, The ratio γ i alo experimentally etermine y the zeta potential an the urface charge meaurement [6,5]. Comining Eq. (6) an (3) yiel γ qζ (4) ( ) σ = k T n k T B inh B Equation (4) i a relation etween the urface charge enity an the zeta potential. A mentione efore, the term on the right han ie i the electrokinetic charge enity. The 8

9 ifference etween the urface charge enity an the electrokinetic charge enity get igger with increaing the ratio γ. 4. Summary The ounary conition ae on electric iplacement wa taken into conieration in the icuion of the electrical oule layer. It wa hown that the electrokinetic charge enity i the um of the urface charge enity an the charge enity at the outer Helmholtz plane in the implifie electrical oule layer. A new relation etween the urface charge enity an the zeta potential wa erive. Acknowlegement The author woul like to acknowlege the upport of Eeneger/Fouraker Grauate Fellowhip. 9

10 Reference. R. J. Hunter, Zeta potential in colloi cience, Acaemic, New York, 98.. D. Li, Electrokinetic in microfluiic, Elevier, Lonon, B. J. Kiry, E. F. Haelrink Jr, Electrophorei 5 (4) W. B. Ruel, D. A. Saville an W. R. Schowalter, Colloial iperion, Camrige Univerity, Camrige, S. R. Morrion, Electrochemitry at emiconuctor an oxiize metal electroe, Plenum, New York, J. Lyklema, Pure Appl. Chem. 63 (99) A. Revil, P. A. Pezar an P. W. J. Glover, J. Geophy. Re. 94 (999). 8. D. E. Yate, S. Levine an T. W. Healy, J. Chem. Soc. Faraay I 7 (974) D. A. Sverjenky, Geochim. Comochim. Acta 69 (5) 5.. I. Laron an P. Attar, J. Colloi Interface Sci. 7, () 5.. In principle, our approach i applicale to any type of electrolyte.. Meanwhile, ince the charge at the OHP are inuce y the urface charge through Coulom force, the aolute value of the charge enity at the OHP houl e maller than that at the urface. 3. H. S. Frank, J. Chem. Phy. 3 (955) In-Chul Yeh an M. L. Berkowitz, J. Chem. Phy. (999) A. Foiy an J. Perello, The urface propertie of Silica, E. A. P. Legran, John wiley an Son, New York, 998.

11 Figure Caption Fig.. Schematic iagram of (a) etaile an () implifie electrical oule layer. The urface an the inner Helmholtz plane in the etaile electrical oule layer are merge into the urface in the implifie electrical oule layer. Fig.. Dielectric contant a a function of the aolute value of the urface charge enity. Fig. 3. Ratio of the charge enity at the OHP to the urface charge a a function of the aolute value of the urface charge enity.

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