Available online at www.sciencedirect.com
Physica A 326 (2003) 129 – 140
www.elsevier.com/locate/physa
Polarization between concentric cylindrical electrodes C. Chassagnea , D. Bedeauxa;∗ , J.P.M v.d. Ploega , G.J.M. Koperb a Leiden
Institute of Chemistry, P.O. Box 9502, 2300 RA Leiden, The Netherlands of Physical Chemistry, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands
b Laboratory
Received 7 March 2003
Abstract We consider an asymmetric electrolyte between two cylindrical concentric electrodes that are uncharged in the absence of an applied voltage di1erence. We calculate the dielectric response of this capacitor to an alternating voltage di1erence. The problem is solved using both classical boundary conditions and the new boundary conditions using excess densities to describe the charge build-up near the condensator plates as given in a previous article (C. Chassagne, D. Bedeaux and G.J.M. Koper, Colloid Surf. A 210 (2002) 137.). We verify that both boundary conditions give the same results. The advantage of the new boundary conditions lies in the possibility to extend in the future the analysis to real electrodes including reactions and speci5c ion adsorption. A comparison of the model predictions, involving no adjustable parameters with experimental dielectric spectroscopy data, is performed and excellent agreement is found. c 2003 Elsevier B.V. All rights reserved. PACS: 66.10.−x; 05.70.Lu; 82.65.Dp Keywords: Electrode; Polarization; Ionic conduction
1. Introduction Charge build-up in the electrolyte close to the electrodes, a phenomenon commonly referred to as “electrode polarization” in colloid science has to be accounted for, when dielectric spectroscopy measurements are done. This is especially so in the low-frequency regime which, in general, provides most of the information concerning ∗
Corresponding author. E-mail address:
[email protected] (D. Bedeaux).
c 2003 Elsevier B.V. All rights reserved. 0378-4371/03/$ - see front matter doi:10.1016/S0378-4371(03)00286-3
130
C. Chassagne et al. / Physica A 326 (2003) 129 – 140
the system in terms of dielectric enhancement and frequency relaxation. In a previous study (see Ref. [1]), we presented an analytical expression for the dielectric permittivity in the case of planar electrodes. In this paper, we will extend this theory to a cylindrical cell geometry which is the one we deal with during our own experiments. As we did in the planar cell geometry case, we will assume that there is neither reaction nor ion transfer at the electrode surfaces. We will derive the equations using both classical boundary conditions and new boundary conditions. The advantage of the new boundary conditions lies in the possibility to extend to more complicated situations including, for instance, speci5c ion adsorption. These new boundary conditions are based on the concept of excess quantities introduced by Albano et al. [2,3] and were introduced to calculate the dielectric permittivity of spherical particles [4]. We prove that both boundary conditions give identical results. In the last section, we will compare the theory to experiments, and consider the inFuence of ion valencies, ionic strength and ionic di1usion coeGcient.
2. Theory For symmetry reasons, cylindrical coordinates (r; ; z) are chosen. The two cylindrical electrodes have the same central axis and the same height l, long enough to allow negligence of end e1ects. The inner cylinder has a radius R1 and the outer cylinder a radius R2 , where (R2 − R1 ) = d is the electrode spacing. The liquid in between consists of an electrolyte solution of known concentration. An alternating voltage di1erence is imposed between the electrodes. The Fux of ions between the electrodes can be described by the Nernst–Planck equation ze i Ji = −Di ni (1) ∇ + ∇ni : kT The ions are carried along under the inFuence of the electric and thermodynamic forces via the gradient terms. Here, ni is the number of ions of type i, and of valence zi , per unit of volume, i = +; −. is the electric potential. Furthermore, Di is the ionic di1usion coeGcient. We will not consider the inFuence of ions j on the Fux of ions i, which corresponds to approximating the conductivity by its value at in5nite dilution. In the following, we will only consider strong electrolytes and low salt concentrations, in which case this approximation is justi5ed. Conservation of ions gives 9ni + ∇ · Ji = 0 : 9t The charge density is related to via Poisson’s equation: −1 zi eni ; O ≡ ∇2 = j1 j0 i where j1 is the relative dielectric permittivity of the solvent.
(2)
(3)
C. Chassagne et al. / Physica A 326 (2003) 129 – 140
Because of the electro-neutrality of the salt we have i zi = 0
131
(4)
i
where the i ’s are the stoichiometric coeGcients of the constituting ions. An oscillating voltage di1erence V0 exp(i!t), where ! is the radial frequency (! = 2f if f is the applied frequency) is imposed along the r-axis. In view of the linearity, all variations of the various quantities around equilibrium are proportional to exp(i!t). This factor will, for ease of notation, further be suppressed. We write ≡ eq + ; ni ≡ ni; eq + ni ;
(5)
where the subscript “eq” stands for “equilibrium values in the absence of applied 5eld”. We therefore have ni; eq = i n∞ , eq being constant and taken to be zero. The Nernst–Planck equation becomes to linear order z e n i i ∞ Ji = −Di ∇ (6) + ni : kT Substitution in the conservation law equation (2) gives 9ni zi e ni : (7) = Di ∇ · i n ∞ ∇ + 9t kT i n∞ Poisson equation (3) becomes −1 O = zi eni j1 j0 i
(8)
and subsequent substitution in Eq. (7) results in i! i zi e2 n∞ Oni = ni + zj nj : Di 0 1 kT j
(9)
This equation can be written more conveniently in matrix form
−1 2 + 02 + z+ 02 + z+ z− i!D+ n+ n+ = (10) O −1 2 n− n− 02 − z− z+ i!D− + 02 − z− with 02 =e2 n∞ (0 1 kT )−1 and 2 =02 i zi2 , where −1 is the Debye screening length. By contraction of (n+ ; n− ) with the left eigenvectors of the matrix, one obtains two independent solutions, nn and nc , which satisfy Onn = n2 nn
and
Onc = c2 nc : 2
(11)
For frequencies such that !D± , to which most experiments are restricted, the eigenvectors reduce to 5rst order to i! −1 −1 nn = n+ + n− 1 + 2 (D+ − D− ) ; z− i! −1 −1 nc = n+ + n− 1 − 2 (D+ − D− ) (12) z+
132
C. Chassagne et al. / Physica A 326 (2003) 129 – 140
and the eigenvalues to i! z+ =D− − z− =D+ n2 = i! ≡ with Re( n ) ¿ 0 ; z+ − z − Dn i! z+ =D+ − z− =D− c2 = 2 + i! ≡ 2 + with Re( c ) ¿ 0 : z+ − z − Dc
(13)
The independent solutions nn and nc decay over characteristic lengths given by
Dn =! and −1 , respectively. nc therefore decays over a typical Debye length and nn over a typical di1usion length. For the frequencies
under considerations, the diffusion length is much larger than the Debye length, Dn =!−1 . The solution nc will therefore be in good approximation zero at distances of a few Debye lengths away from the capacitor plates. In view of this property we will refer to the layers in which nc is unequal to zero as the double layers. Eq. (11) in cylindrical coordinates becomes ( n r)2
92 nn 9nn − ( n r)2 nn = 0 ; + ( n r) 9( n r)2 9( n r)
( c r)2
92 nc 9nc − ( c r)2 nc = 0 : + ( c r) 2 9( c r) 9( c r)
(14)
The solutions of Eq. (14) are the Bessel functions of second kind denoted by K0 and I0 (Ref. [5]), such that nn = Cn1 K0 ( n r) + Cn2 I0 ( n r) ; nc = Cc1 K0 ( c r) + Cc2 I0 ( c r) ;
(15)
where Cn1; 2 and Cc1; 2 are constants to be determined. We note that |arg( n r)|==4 and |arg( c r)| 0 and refer to Ref. [5] for the relations concerning these two Bessel functions that will be needed in further derivations. Inverting the equations, Eqs. (12) gives z+ ni = N s nc + n+ i ; z+ − z − i n+ i =
i Ni nn + + −
(16)
with N− = 1 −
i! −1 −1 z+ + z− (D+ − D− ) ; 2 z+ − z −
N+ = 1 −
i! −1 2z+ −1 (D+ − D− ) ; 2 z+ − z −
N+s = 1 −
i! −1 2z− −1 (D − D− ) ; 2 + z+ − z −
s = −N− : N−
(17)
C. Chassagne et al. / Physica A 326 (2003) 129 – 140
133
The superscript + indicates the value of the variable beyond the double layers (see Ref. [4]). The solution of the Poisson equation has the form = Ac nc + + ; + = An nn + F ln(r) + G ;
(18)
where F and G are constants and where An = =
−1 ezi i Ni j1 j0 n2 + + − ez+ z− i! −1 −1 −1 (D − D− ); j1 j0 n2 z+ − z− 2 +
ez+ −1 zi Nis 2 j1 j0 c z+ − z− −1 ez+ z− i! −1 −1 ez = − (D − D ) : + − j1 j0 c2 z+ − z − 2 +
(19)
Ac =
(20)
The resulting charge density is & = e(z+ n+ + z− n− ) = −j1 j0 c2 Ac nc + &+ ; &+ = −j1 j0 n2 An nn :
(21)
3. Boundary conditions 3.1. The standard boundary conditions Neither charge transfer nor adsorption takes place at the electrodes. The 5rst boundary condition therefore reads: Ji = 0
in r = Ri
with i = 1; 2 :
The second one follows from Gauss’s law: 9 −Q = with i = 1; 2 ; 9r r=Ri 2j0 j1 lRi
(22)
(23)
where Q = |Q| is the absolute charge on one or the other electrode. For convenience, the 5rst boundary condition (Eq. (22)) is replaced by two equivalent ones. Eq. (22) is summed over i to get the 5rst relation. 9ni =0 : (24) 9r Ri
134
C. Chassagne et al. / Physica A 326 (2003) 129 – 140
In order to get the second relation, Eq. (22) is 5rst multiplied by ezi then summed again over i and Eq. (23) is used: 9& Q = 2 9r Ri 2lRi 9nc 2 : (25) = −j0 j1 Ac c 9r Ri We now substitute the expression for ni found in Eq. (16). To 5rst order in !, and using Eqs. (24) and (25), this leads to, in r = Ri , i = 1; 2 1 9nc 9nn −z+ i! 1 = − ; 9r Ri z+ − z− 2 D+ D− 9r Ri Ac
9nc 9r
Ri
=−
c
2
Q : 2j1 j0 lRi
(26)
Combining these two last equations, we obtain 9nn i! Q An ; = 9r Ri 2j1 j0 lRi 2 Dt
2
Ac
9nc 9r
Dt−1
(1=D+ − 1=D− )2 z+ z− : ≡ z+ − z− z+ =D− − z− =D+
where
Ri
=
c
−Q ; 2j1 j0 lRi
From the Gauss equation (23), 9nc 9nn F −Q Ac + An + = 9r Ri 9r Ri Ri 2j1 j0 lRi
(27)
(28)
(29)
from which we deduce
2 −Q i! 1− F= + 2 2j1 j0 l c Dt =
−Q i! ; 2j1 j0 l 2 D0
(30)
where D0−1 ≡ Dc−1 + Dt−1 = (z+ − z− )=(z+ D+ − z− D− ) : From Eq. (27) we also have the relations: 9nn 9nn R1 = R2 ; 9r R1 9r R2 9nc 9nc R1 = R2 : 9r R1 9r R2
(31)
(32)
C. Chassagne et al. / Physica A 326 (2003) 129 – 140
135
Under experimental conditions considered here, −1 is of the order of nanometers; therefore, the condition | c Ri |1 holds if Ri 10−9 m. Moreover, D± is of the order of 10−9 m2 =s and f ¿ 1000 Hz implying that | n Ri |1 if Ri 10−3 m. As Ri is of the order of millimeters both conditions are satis5ed. We therefore get from the two last equations R2 [Cn2 I0 ( n R2 ) − Cn1 K0 ( n R2 )] = R1 [Cn2 I0 ( n R1 ) − Cn1 K0 ( n R1 )] ; R2 [Cc2 I0 ( c R2 ) − Cc1 K0 ( c R2 )] = R1 [Cc2 I0 ( c R1 ) − Cc1 K0 ( c R1 )] :
(33)
Using the tabulated relations relating the Bessel functions in Ref. [5], Eq. (33) reduces to R2 Cn2 I0 ( n R2 ) = −R1 Cn1 K0 ( n R1 ) ; R2 Cc2 I0 ( c R2 ) = −R1 Cc1 K0 ( c R1 ) :
(34)
From Eq. (27), we get Q 1 −1 i! ; Cn1 = n An 2 Dt 2j1 j0 lR1 K0 ( n R1 ) 2 Q 1 1 Cc1 = ; c Ac c 2j1 j0 lR1 K0 ( c R1 ) we also have nn (R2 ) − nn (R1 ) =
(35)
−(R1 + R2 ) 1 Cn K0 ( n R1 ) ; R2
−(R1 + R2 ) 1 Cc K0 ( c R1 ) : R2 To 5rst order in ! the potential di1erence between the electrodes reduces to nc (R2 ) − nc (R1 ) =
U = (R2 ) − (R1 )
(36)
R2 = An [nn (R2 ) − nn (R1 )] + Ac [nc (R2 ) − nc (R1 )] + F ln R1
2 −Q i! 1 R2 1 i! −1 −1 = R1 + R2 + : ln − 2j1 j0 l 2 D0 R1 c c n 2 Dt
(37) Finally, for electrode spacings such that R2 −R1 =dR1 we can de5ne a surface charge density: +0 ≈ 2R1 lQ ≈ 2R2 lQ and the expression reduces to
2 i! 2 2 −+0 d + 2 (38) 1− 1− 1− U = j1 j0 c c d Dt n d which is the expression found for the case of planar electrodes. We de5ne a complex dielectric permittivity j˜ such that −Q U ≡ (39) ˜ 2j0 jl
136
C. Chassagne et al. / Physica A 326 (2003) 129 – 140
from which follows, in the general case
2 i! 1 R2 1 i! −1 −1 (R1 + R2 ) : + ln − j˜ = j1 2 D0 R1 c c n 2 Dt
(40)
One can similarly calculate the electric 5eld E = −(r)=r in the solution for any r. It is found that beyond the di1use layer, at positions r larger than | n−1 | from both electrodes: −F Q i! E(r) = : (41) r 2j1 j0 lr 2 D0 For spacings such that R2 − R1 = dR1 , one may take r R1 and de5ne a surface charge density +0 ≈ 2R1 lQ ≈ 2R2 lQ, we recover the expression found in the planar electrode case, where E(r) is independent of the position r: +0 i! E : (42) j1 j0 2 D 0 3.2. Boundary conditions with excess quantities In this section we will use the boundary conditions introduced by Albano et al. [2,3] and presented in a previous article [4]. It will be shown that this alternative method reproduces the same results found above. These conditions read for this case: i!nsi (R1 ) = −Ji+ (R1 ) ; i!nsi (R2 ) = +Ji+ (R2 ) ; 9+ Q − −j0 j1 = &s (R1 ) ; 9r r=R1 2R1 l Q 9+ = &s (R2 ) : + j 0 j1 2R2 l 9r r=R2
(43)
(44)
The notation nsi (R1 ) or nsi (R2 ) is adopted in order to indicate to which electrode the excess nsi calculated corresponds to. Conditions (43) express conservation of ions in the double layer and conditions (44) express the continuity of the electric displacement 5eld. The superscript “s” indicates the excess of the corresponding quantity in the double layer, following the notation adopted by Albano et al. [2,3]. The excess densities of the ions are found by integration through the double layer after subtracting the extrapolated value in the region beyond the double layer [2,3] 1 r2 nsi (Ri ) ≡ [ni (r) − n+ (45) i (r)]r dr ; Ri r 1 where [r1 ; r2 ] = [R1 ; R1 + d=2] or [R1 + d=2; R2 ] depending on the electrode considered. Using Eq. (16), we 5nd Nis r2 z+ s ni (Ri ) = nc r dr : (46) z+ − z − R i r 1
C. Chassagne et al. / Physica A 326 (2003) 129 – 140
The resulting excess of the charge density of the double layer is r2 Ac nc r dr : &s (Ri ) = ezi nsi = −j1 j0 c2 Ri r 1 i
137
(47)
Similarly to what has been done in the previous section, the 5rst boundary conditions (Eq. (43)) are for convenience replaced by two equivalent ones. Using the same procedure, Eq. (43) is summed over i to get the 5rst relation. In order to get the second relation, Eq. (43) is 5rst multiplied by ezi , then summed again over i and Eq. (44) is used. We get, to 5rst order in !: ns (R1 ) 9n+ i i = i! 9r D i R1 i i i!
ns (R2 ) i
i
Di
9n+ i =− 9r R2 i
(48)
i! (ezi )nsi (R1 ) Q =− − &s (R1 ) Di 2R1 l 2 i Q i! (ezi )nsi (R2 ) = − &s (R2 ) : Di 2R2 l 2 i We now substitute Eq. (46) and 5nd that 2 R1 +d=2 z+ 1 c i! 1 9nn − n r dr = ; c z+ − z − 2 D + D − R 1 R1 9r R1 −Q i! s = ezi ni (R1 ) 1 + 2 D 2R i 1l i R1 +d=2 i! 1 = ez+ 1 + 2 nc r dr D + R 1 R1
(49)
(50)
(51)
and similar relations for the other electrode. From Eqs. (51) and (49) and the analogous relations for R2 , we 5nd R1 +d=2 R2 nc r dr = − nc r dr ; (52) R1
R1
9nn 9r
R1
= R2
R1 +d=2
9nn 9r
R2
:
(53)
From the integration of (52), we recover Eqs. (32). Moreover, we 5nd that R1 nsi (R1 ) = −R2 nsi (R2 ) and we recover Eq. (30) for F and Eqs. (39) and (40).
(54)
138
C. Chassagne et al. / Physica A 326 (2003) 129 – 140
4. Particular case If we now consider the case for which 2 D± !2D± =(−1 d) which is the case encountered in practice, then Eq. (40) can be expanded and it is found that
2 ! R2 −1 −1 −1 ; j = j1 (R1 + R2 ) ln 2 D0 R1 K = j0 j1 2 D0
R2 ln : R1
(55)
This corresponds to the frequency range usually investigated in the experiments (typically 109 !102 ). At a given frequency, the permittivity j increases with increasing 3=2 ). The conductivity K does, in good approximation, not depend ionic strength (j ∼ n∞ on frequency in this frequency range and is equal to the standard relation usually given for the conductivity of the electrolyte solution, which reads K1 = e2 zi2 i n∞ Di =kT divided by a factor depending on the radii of the cylinders. 5. Experimental results The experimental setup we used is described elsewhere (Ref. [6]). The cell consisted of two cylindrical electrodes of radius R1 = 0:88 cm and R2 = 2:04 cm. Salt solutions (KCl, MgCl2 and NaCOOCH3 ) were made using tri-distilled water (Millipore) and pro analysi corresponding salt from Merck. Measurements were done at 25◦ C (Figs. 1 and 2). The impedance of the system can be calculated according to Z = U=I where I = −dQ=dt = −i!Q. From Eqs. (39), (40), we 5nd −U 1 1 1 Z≡ = = : (56) ˜ 0 l K + i!jj0 2l i!Q i!2jj Therefore j=
1 Im(1=Z) ; 2l !j0
1 Re(1=Z) : 2l The ratio 1=2l is called the cell constant. The conductivities of the solution could be calculated according to i zi C s / ∞ K1 = i ; K=
(57)
(58)
where Cs is the given neutral salt concentration in mM (1 mM = 10−3 mol=l). i |zi |Cs is called the equivalent concentration of the solution. The ionic di1usion coeGcients, 2 Di (m2 =s), are related to the equivalent limiting ionic conductivities /∞ i (S m =mol) by ∞ 2 /i = |zi |Di Na e =kT where Na is Avogadro’s number. We have n∞ (m−3 ) = Cs (mM)Na (mol−1 ). The limiting ionic conductivities /∞ are given in Table 1 (see also Fig. 2).
C. Chassagne et al. / Physica A 326 (2003) 129 – 140 80
139
250
(A) 60
1mM 2mM 3mM 4mM
(B) 200 150
0.5mM 1mM 2mM 3mM 5mM
40 100 20
50
0 4 10
10
5
150
0 4 10
10
5
300
(C)
0.5mM 1mM 2mM
(D) 250
100
0.25mM 0.5mM 1mM
200 150
50
100 50
0 4 10
10
5
0 4 10
10
5
Fig. 1. The dielectric permittivity increment Oj as a function of frequency !=2 (Hz) for di1erent salts and salt concentrations, (A): NaCOOCH3 , (B) KCl, (C) MgCl2 and (D) KFe(CN)6 .
150
300 MgCl2 KCl NaCOOCH3
250
100
200
KFe(CN)6 MgCl2 KCl NaCOOCH3
150 50
100 50
0 4 10
10
5
0 4 10
10
5
Fig. 2. Comparison of the dielectric permittivity increment Oj as a function of frequency !=2 (Hz) for various salts: (left) Cs = 2 mM and (right) Cs = 1 mM.
From the conductivity measurements, and relation equation (55) the exact salt concentrations could be established and these concentrations were used in the 5t of the permittivity increments. These were normalized by subtracting from the permittivity value the one found at the highest frequency measured. Moreover, the cell constant was
140
C. Chassagne et al. / Physica A 326 (2003) 129 – 140
Table 1 Limiting conductivities used in the 5gures
/∞ (×10−4 S m2 =mol) i
K+
Na+
Cl−
COOCH− 3
Fe(CN)6−
Mg2+
74.5
50.9
75.5
40.1
111
53.9
calculated for each electrolyte solution using six di1erent lengths l. In the experiments, this height was varied by removing electrolyte solution from the cell using a syringe. The theoretical predictions were plotted (solid lines in both 5gures), using the above cited (known) parameters, according to ˜ − Re(j˜∞ )] ; Oj = [Re(j)
(59)
where Re(j˜∞ ) corresponds to the value at the highest frequency measured. 6. Discussion and conclusions Given the facts that no parameter adjustment has been made and that all parameters were determined independently, the agreement between experiments and predictions must be considered as very good. The enormous increments measured at low frequencies do not allow for a proper estimate of the experimental error; our current estimate is of the order of 15%. The measurements were also proven to be very sensitive to the properties of the surface of the electrodes. For instance, it has been shown in our experiments that the polarization strongly depends on the way the electrodes were cleaned and that extreme care was taken in this procedure. In this article, we con5rm the results of the planar cell geometry [1], namely that both classical and new boundary conditions give the same results and that the theory predicts the e1ects of electrode polarization for all kinds of electrolytes very well. References [1] [2] [3] [4] [5] [6]
C. Chassagne, D. Bedeaux, G.J.M. Koper, Colloid Interfaces A 210 (2002) 137. A.M. Albano, D. Bedeaux, J. Vlieger, Physica A 99 (1979) 293. A.M. Albano, D. Bedeaux, J. Vlieger, Physica A 102 (1980) 105. C. Chassagne, D. Bedeaux, G.J.M. Koper, Physica A 317 (2003) 321. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover Publications, New York, 1972. J.P.M. Van der Ploeg, M. Mandel, Meas. Sci. Technol. 2 (4) (1991) 389.
