- Email: [email protected]
Theory of electrode polarization: application to parallel plate cell dielectric spectroscopy experiments
Colloids and Surfaces A: Physicochem. Eng. Aspects 210 (2002) 137 /145 www.elsevier.com/locate/colsurfa
Theory of electrode polarization: application to parallel plate cell dielectric spectroscopy experiments C. Chassagne a,*, D. Bedeaux a, J.P.M. van der Ploeg a, G.J.M. Koper b a
b
Leiden Institute of Chemistry, Leiden University, PO Box 9502, 2300 RA Leiden, The Netherlands Laboratory of Physical Chemistry, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands
Abstract An extension of the model for electrode polarization of Cirkel et al. [Physica A 235 (1997) 269] is given. The problem is solved using both classical boundary conditions and the new boundary conditions using excess densities presented in a previous paper [J. Phys. Chem. B 105 (2001) 11743]. In the present paper, the electrodes are supposed to be ideal, meaning that charge transfer or adsorption are not considered. The advantage of the new boundary conditions lies in the possibility to extend to more complicated situations including for instance specific ion adsorption. We prove that the new boundary conditions and classical ones give the same results. A comparison of the model predictions, involving no adjustable parameters, experimental dielectric spectroscopy data is performed and fairly good agreement is found. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Electrode; Polarization; Adsorption
1. Introduction Electrode polarization is a phenomenon that hampers dielectric spectroscopy measurements of many systems such as dispersions, microemulsions or polymer solutions at low frequencies due to the presence of electrolyte. As in most of these systems the relaxation frequencies and permittivity enhancements are located in these low frequency regimes, it is important to compensate for this phenomenon. Various authors proposed empirical
* Corresponding author. Tel.: /31-71-527-4560; fax: /3171-527-4239 E-mail address: [email protected] (C. Chassagne).
models involving equivalent circuits [1 /3] to account for electrode polarization. Considering the behavior of ions under the influence of an alternating electric field, we derive here an analytical expression for the complex permittivity due to electrode polarization. Such an analytical expression has already been presented for the case of a symmetric electrolyte and planar electrodes [4]. In the present paper, this theory will be extended to the case of asymmetric electrolytes. We will also derive the equations using the new boundary conditions that were introduced for the calculation of the dielectric permittivity of spherical particles [5]. These new boundary conditions are based on the concept of excess quantities in the sense of Gibbs introduced by Albano et al. [6,7]. The advantage of the new
0927-7757/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 0 2 ) 0 0 3 7 9 - 5
138
C. Chassagne et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 210 (2002) 137 /145
boundary conditions lies in the possibility to extend to more complicated situations including for instance specific ion adsorption. The new and the standard boundary conditions are found to give the same results as, of course, they should. In the last section we demonstrate that our model fairly accurately describes the experimental situation. It is important to stress here that these predictions were done without the introduction of new parameters.
2. Theory The planar electrodes are assumed to be perpendicular to the x-axis and located at x /0 and x /d , where d is the electrode spacing. The liquid in between consists of an electrolyte solution of known concentration. An alternating current (AC) is imposed between the electrodes. The flux of ions between the electrodes can be described by the Nernst/Planck equation: ze Ji Di ni i 9C9ni (1) kT The ions are carried along under the influence of the electric and thermodynamic forces via the gradient terms. Here ni is the number of ions of type i of valence zi per unit of volume and C the electric potential. Furthermore, Di is the ionic diffusion coefficient. We restrict ourselves here to binary electrolytes, i //, /, but the model can be extended to more complicated cases. Also, we will not consider the influence of ions j on the flux of ions i, which corresponds to approximating the conductivity by the conductivity at infinite dilution. In the following, we will only consider strong electrolytes and low salt concentrations, in which case this approximation is justified. Conservation of ions gives: @ni @t
9Ji 0
(2)
The charge density is related to the electric potential C via Poisson’s equation:
DC 92 C
1 X zi eni o1 o0
(3)
where o 1 is the relative dielectric permittivity of the solvent. Due to the electro-neutrality of the salt we have: X n i zi 0 (4) i
where the ni ’s are the stoichiometric coefficients of the resulting ions. An oscillating electric field E0 exp(ivt), where v is the radial frequency (v /2pf if f is the applied frequency) is imposed along the x-axis. In view of the linearity all variations of the various quantities around equilibrium are proportional to exp(ivt). This factor will, for ease of notation, be suppressed. We, therefore, write: CCeq dC ni ni;eq dni
(5)
where the subscript ‘eq’ stands for ‘equilibrium values in the absence of applied AC-field’. We have ni ,eq /ni n, Ceq, being constant. The Nernst /Planck equation becomes to linear order: z en n (6) Ji Di 9 i i dCdni kT Substitution in the conservation law Eq. (2) gives: @ni ze dn ivdni Di 9 ni n 9 i dC i (7) kT @t n i n The Poisson equation becomes: DdC
1 X zi edni o1o0 i
(8)
Using Poisson in Eq. (7) results in: Ddni
iv n z e2 n X dni i i zj dnj Di e0 e1 kT j
(9)
This equation can be written more conveniently in a matrix form:
C. Chassagne et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 210 (2002) 137 /145
dn ivD 1 k20 n z2 D 2 dn k0 n z z dn dn
k20 n z z 2 2 ivD 1 k0 v z
with k20 /e2n(/e0 e1/kT ) 1 and k2 /k20 a ni z2i . By contraction of (dn, dn ) with the left eigenvectors of the matrix one obtains two independent solutions, dnn and dnc, which satisfy:
l2n
and
Ddnc l2c dnc
where and are the eigenvalues of the matrix. For frequencies such that v /D 9k2, to which most experiments are restricted, the eigenvectors reduce to first order to: iv 1 dnn dn dn 1 (D 1 D ) k2
z z
1
iv k2
1 (D 1 D )
(12)
z =D z =D z z
N 1
iv 1 2z (D D 1 ) 2 k z z
s N 1
s N 1
iv k2
iv k2
1 (D 1 D )
1 (D 1 D )
z z z z 2z
z z
(16)
The superscript, / indicates the value of the variable beyond the double layers (see [5]). The solution of the Poisson equation has the form:
(17)
where 2
k
iv Dc
(13)
dnc Cc;0 exp( lc x)Cc;d exp(lc (x d))
o 1 o 0 l2n An
dnn Cn;0 exp( ln x)Cn;d exp(ln (x d))
X
ezi ni
i
n n
Ni
ez z iv 1 (D D 1 ) z z k 2
ez X zi Nis z z i z iv ez 1 z z k2 1 1 (D D )
(18)
o 1 o 0 l2c Ac (14)
where Cn,0, Cn,d , Cc,0 and Cc,d are constants to be determined. The first eigenvector is a diffusion mode and decays over a distance of the order pffiffiffiffiffiffiffiffiffiffiffiffiffi Dm =v: The second eigenvector dnc decays over a typical Debye length due to the contribution k2 to the eigenvalue l2c . Beyond the double layer, dnc does, therefore, not contribute to the solution. Inverting the Eq. (12) gives: z Nis dnc dn i z z
iv 1 z z (D D 1 ) 2 k z z
dC An dnn FxG
The solution of Eq. (11) then becomes:
dni
N 1
dC Ac dnc dC
and the eigenvalues to: z =D z =D iv 2 ln iv Dn z z l2c k2 iv
(15)
(11)
l2c
dnc dn dn
ni Ni dnn n n
with, (10)
Ddnn l2n dnn
dn i
139
(19)
and where, F and G are constants that are determined by the boundary conditions. The resulting charge density is: dr e(z dn z dn ) o 1 o 0 l2c Ac dnc dr dr o 1 o 0 l2n An dnn
(20)
140
C. Chassagne et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 210 (2002) 137 /145
(18) that:
3. Boundary conditions
Ac Cc
3.1. The standard boundary conditions The electrodes are supposed to be ideal meaning that neither charge transfer nor adsorption takes place. We, therefore, have: Ji (0) Ji (d) 0
Using the Nernst/Planck equation for the ionic fluxes one may replace the first two boundary conditions (Eq. (21)) by two equivalent ones. Summing [Eq. (21)]/Di over i one obtains: X@dni X@dni 0 (23) @x x 0 @x x d i i Multiplying Eq. (21) by ezi /Di and then summing over i, using Eq. (22), gives: @dr @dr k2 s0 (24) @x x 0 @x x d Substituting Eq. (15) for dni into Eqs. (23) and (20) for dr into Eq. (24), and using Eq. (14), one obtains to first order in v/k2D 9: ln Cn;0
ln Cn;d
z
iv
z z k2 z
iv
z z
k2
1 s0
and (27)
iv
ln o 1 o 0 k2 Dt
where D 1 t
z z
(1=D 1=D )2
z z z =D z =D
(28)
From Gauss, Eq. (22), and using Eq. (27) we find: 2 s0 k iv F 1 lc k2 Dt o1 o0
s0 iv 1 s0 iv (Dc D 1 t ) 2 o1 o0 k o 1 o 0 k2 D0
(29)
where D 1 0
(z z ) (z D z D )
(30)
Using that the potentials are continuous in x /0 and x /d , the applied potential difference between the electrodes can be written, to first order in v / k2D 9, as: dU dC(d) dC(0) 2Ac Cc 2An Cn Fd
1 (D 1 D )lc Cc;0
(31)
This results in: 1 (D 1 D )lc Cc;d
Ac lc Cc;0 Ac lc Cc;d
2 k s0 lc o 1 o 0
dU
(25)
where we used that jlcd j/jlnd j/1. Under the experimental conditions considered, k 1 is of the order of nanometers, d is of the order of millimeters, D9 is of the order of 10 9 m2 s 1 and v /10 000 Hz we have that jlcdj /104 and jlnd j/104. It follows that: Cn;0 Cn;d Cn
2 k
lc o 1 o 0 lc
An Cn
(21)
Furthermore, it follows from Gauss’s law that: @dC @dC s0 (22) @x x 0 @x x d o0 o1
1 s0
and
Cc;0 Cc;d Cc
(26)
It then follows from Eq. (25) together with Eq.
s0 d o1 o0 2 k 2 iv 1 1 lc lc d k2 Dt 2 1 ln d
(32)
Defining the complex dielectric permittivity o˜ by: dU
s0 d o 0 o˜
we find that:
(33)
C. Chassagne et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 210 (2002) 137 /145
o˜ o 1
=
1
2 k 2 iv 2 1 1 lc lc d k2 Dt ln d (34)
One can similarly calculate the electric field E / /@d C(x )@x in the solution for any x . It is found that: E(x)
express the conservation of ions in the double layer. The last two are the conditions for the electric displacement field. 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 [6,7]:
s0
x2
dnsi
o1 o0 2 iv k 2 k D0 lc [exp( lc x) exp(lc (x d))]
141
iv k2 Dt
[exp( ln x) exp(ln (x d))]
s0 iv o 1 o 0 k2 D0
i
i (x)]dx
(39)
x1
(35)
It can be verified that E (0) /E (d ) /s0/o 1o 0 as follows from Eq. (22). Furthermore, beyond the diffusion layer, i.e. at positions more than jln 1j away from both electrodes, this leads to: E(x)# F
g [dn (x) dn
(36)
where [x1, x2] /[0, d /2] or [d/2, d ] depending on the electrode considered. Using Eq. (15) together with Eq. (14), and the fact that jlcdj/1, we find that: dnsi (0)
z
Nis
z z l c
dnsi (d)
Cc;0
and (40)
z Nis Cc;d z z l c
The resulting excess of the charge density of the double layer is: X drs (0) ezi dnsi (0) o 1 o 0 lc Ac Cc;0 i
3.2. Boundary conditions with excess quantities
drs (d)
X
ezi dnsi (d) o 1 o 0 lc Ac Cc;d
(41)
i
In this section we will prove that the standard boundary conditions are equivalent to the boundary conditions introduced by Albano et al. [6,7] and presented in a previous article [5]. These conditions read: ivdnsi (0) Ji (0) and ivdnsi (d) Ji (d) @dC s0 drs (0) o 0 o 1 @x x 0 @dC s0 o 0 o 1 drs (d) @x x d
(37)
(38)
Following the notation of Albano et al. [6], the superscript ‘s’ indicates the excess of the corresponding quantity in the double layer. The notation dnsi (0) or dnsi (d) is adopted in order to indicate to which electrode the excess dnsi calculated corresponds to. The first two conditions Eq. (37)
Similarly to what has been done in the previous section, the first two boundary conditions (Eq. (37)) are replaced by two equivalent ones. Using the same procedure, [Eq. (37)]/Di is summed over i to get the first relation. In order to get a second relation, Eq. (37) is first multiplied by ezi /Di and then summed over i, using Eq. (38). We get, to first order in v /k2D9: X dns (0) X@dn i i iv Di @x x 0 i i iv
X dns (d) i i
Di
X@dn i i
@x
(42) x d
and iv X (ezi )dnsi (0) s0 drs (0) k2 i Di
142
C. Chassagne et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 210 (2002) 137 /145
iv X (ezi )dnsi (d) s0 drs (d) k2 i Di
(43)
Substituting the excess ionic and charge densities given in Eqs. (40) and (41) and using Eq. (14) we find to first order in v/k2D 9. ln Cn;0
z iv 1 (D D 1 )lc Cc;0 z z k2
z iv 1 (D D 1 )lc Cc;d z z k2 2 k s0 Ac lc Cc;0 Ac lc Cc;d lc o 1 o 0
ln Cn;d
(44) (45)
We hereby used the equalities: P z iv ezi Nis o 1 o 0 l2c Ac z z k 2 Di iv s0 iv 2 lc Ac 1 ez 1 k2 D k2 D ez 2 k s0 lc o 1 o 0
and
drs (0) drs (d)
(46)
We furthermore recover Eq. (29) for F , Eq. (33) for the potential difference dU between the electrodes and Eq. (34) for the complex dielectric permittivity o:˜/
4. Complex permittivity of the system 4.1. General case The complex permittivity of a system may be written as follows: o˜o
K ivo 0
=
(48)
For very low frequencies, v /2D 9k /d /vc, we get: 2 2 kd kd v and K o 0 o 1 o o1 (49) 2 2 k2 D0 In practice, the characteristic frequency vc corresponds to a frequency smaller than 100 Hz, for which usually no reliable measurements can be done. If we now consider the case for which vc / v /k2D 9 then: 2 kd v o o1 and K o 0 o 1 k2 D0 (50) 2 k2 D0
=
From Eqs. (44) and (45) we recover the expressions Eqs. (26) and (27) for Cn,0, Cn,d, Cc,0 and Cc,d found using the normal boundary conditions. As a corollary we find that: dnsi (0) dnsi (d)
k2D 9 Eq. (34) can be expanded as: kd iv kd 2 o˜ o 1 2 k2 D0 2 v 2 kd 2 1 k2 D0 2
(47)
where o is the dielectric permittivity of the system and K its conductivity. Using the fact that v /
This corresponds to the frequency range investigated in most dielectric spectroscopy experiments (typically 102 /v /109). At a given frequency, the permittivity o increases with increasing ionic strength (o /n3/2 ) and decreases with increasing electrode spacing (o /d 1). The conductivity K does in good approximation not depend 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 /ai e2z2i ni n Di /kT .
4.2. Special case In the case of an electrolyte, with equal diffusion coefficients (D /D /D ), Eq. (34) reduces to: 2 k 2 o˜ o 1 1 1 lc lc d iv 2 iv o1 1 k2 D lc d k2 D
= =
with k2 /e2n(o 0o 1kT ) 1 a ni z2i and l2c /k2/iv / D . This equation corresponds to Eq. (8) of the article of Cirkel et al. [4] if one takes n /n / z / /z /1. We also have:
C. Chassagne et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 210 (2002) 137 /145
o o 1
=
kd v 2 and 2 k2 D
K o 0 o 1 k2 D
(51)
for vc /2Dk /d /v /k2D .
ence of potential between two points xa and xb located in this region (xa /xb /d2) may be estimated as follows. We note that F /dI/(SK1): (55)
dU2 #(d2 =S)dI=K1
From Eq. (36), we can also estimate the measured potential between the electrodes:
5. Experimental results
dU #(d=S)dI=K1 #(d=d2 )dU2 5.1. Material and methods The experimental setup we used is described in detail in [8]. The cell consists of two parallel electrodes made of platinum, each with a surface area of S /2.29 cm2. Their separation could be varied from 1.50 to 4.50 mm with an uncertainty of 9/0.01 mm. KCl and NH4Cl solutions were prepared using tri-distilled water (Millipore) and pro analysis corresponding salt from Merck. Measurements were done at T /21 8C. The cell in which the experiments were performed is modeled as a parallel circuit of a frequencydependant ‘capacitance’, C , and a frequencydependent ‘resistance’, R . The equivalent impedance of the circuit is: Z
R
(52)
1 iRCv
The impedance of the system can be calculated according to Z /dU /dI where dI / /dQ /dt / / d(s0S )/dt / /ivs0S , and where S is the surface area of the electrodes. From Eqs. (33) and (34), we find: Z
dU d ivs0 S ivoo ˜ 0S
(53)
d 1 C S o0
and
K
(56)
Doing so, we neglect the capacitive contribution due to the polarization of the electrodes. This is justified by the fact that the ratio between the real and imaginary part of Z is given by vRC #/vo 0o 1/ K1 /10 7 /10 10 (taking electrode polarization into account). Beyond the diffusion layer, the electric field is in good approximation equal to the measured electric field on the electrodes. 5.2. Experimental The experimental conductivity curves were plotted according to Eq. (54). As K #/K1 in the frequency range of interest (see Eq. (50)), the conductivity should not depend on the electrode spacing. The exact spacings of the electrodes were, therefore, determined by fitting these spacings such that the experimental conductivity curves for all four spacings superimposed. Consequently, these spacings were used as known parameters for the experimental permittivity increments curves and in the calculation of the theoretical predictions. The predicted conductivities of the solution were plotted according to: X K vo 0 Im(o)# ˜ n i zi C s L (57) i i
Consequently, it can be shown that the relations between the measured (frequency-dependent) capacitance and resistance and the corresponding experimental permittivity and conductivity are: o
143
d 1 SR
(54)
The ratio d /S is called the cell constant. From Eq. (36), it is shown that the electric field does not depend on x in the bulk region situated beyond the diffusion layer, i.e. at position more than jln 1j away from both electrodes. The differ-
where Cs is the given neutral salt concentration in mM (1 mM /10 3 mol l 1 /1 mol m 3). The equivalent concentration of the solution is ni zi Cs. The ionic diffusion coefficients, Di (m2 s 1), are related to the equivalent limiting ionic conductiv2 2 ities L mol 1) by L i (S m i /zi Di Nae /kT , where Na is Avogadro’s number. We have n / CsNa. As the limiting ionic conductivities of K and NH4 are equal at 298 K and differ from the limiting ionic conductivity of Cl by only a few percents, we took for all three the same value, and used L /74 /10 4 S m2 mol 1. From the
144
C. Chassagne et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 210 (2002) 137 /145
conductivity measurements the salt concentrations Cs could be established and these were used in the calculation of the corresponding permittivity increment predictions. The experimental permittivity increments curves were plotted according to: d 1 Do (C C ) S o0
(58)
where C corresponds to the capacity at the highest frequency measured and the theoretical predictions were plotted, using the above cited (known) parameters, according to: ˜ o˜ )] Do pred [Re(o) Re(
(59)
˜ at where, Re(o˜ ) corresponds to the value of Re(o) the highest frequency measured.
Fig. 2. Dielectric increment as a function of frequency for a 8.6 mM KCl solution. Symbols represent the experimental data for various electrode spacings and lines are predictions according to Eq. (59).
6. Discussion In Figs. 1/4, experimental permittivity data for two kinds of binary electrolytes at various strength are plotted together with the model predictions. Given the facts that no parameter adjustment has been made and that all parameters were determined independently, the agreement must be considered as very good. The enormous incre-
Fig. 3. Dielectric increment as a function of frequency for a 1.1 mM NH4Cl solution. Symbols represent the experimental data for various electrode spacings and lines are predictions according to Eq. (59).
Fig. 1. Dielectric increment as a function of frequency for a 2.6 mM KCl solution. Symbols represent the experimental data for various electrode spacings and lines are predictions according to Eq. (59).
ments measured at low frequencies do not allow for a proper estimate of the experimental error; our current estimate is of the order of 15%. It should be emphasized that the experimental results are very sensitive to the properties of the surface of the electrodes. For instance, from our experiment we know that the polarization strongly depends on the way the electrodes were cleaned and that extreme care has to be taken in this procedure. All figures shown were for well cleaned electrodes.
C. Chassagne et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 210 (2002) 137 /145
145
these allow for the introduction of surface properties such as specific ion adsorption. Also, these boundary conditions still hold if the a weak DCpotential is applied on the electrodes. For a weak DC-potential there is no influence on the ACcomponent. However, if a strong DC-potential is applied in addition to the AC-potential on the electrodes it can not be assumed any longer that, the boundary conditions are modified which leads to a modification of the AC-field.
References Fig. 4. Dielectric increment as a function of frequency for a 2 mM NH4Cl solution. Symbols represent the experimental data for various electrode spacings and lines are predictions according to Eq. (59).
7. Conclusions In this article we extend the model of Cirkel et al. [4] for the dielectric permittivity increment due to electrode polarization to the more general case including all binary electrolytes. We have performed these calculations with standard boundary conditions and with the boundary conditions with excess quantities derived in a previous article. Both boundary conditions give the same results. The advantage of the new boundary conditions is that
[1] J.P.M. Van der Ploeg, M. Mandel, Meas. Sci. Technol. 2 (4) (1991) 389. [2] H.P. Schwann, in: W.L. Nastuk (Ed.), Physical Technics in Biological Research, Part B, vol. 6, Academic Press, New York, 1963, pp. 323 /407. [3] H.P. Maruska, J.G. Stevens, IEEE Trans. Electr. Instrum. 23 (1988) 197. [4] P.A. Cirkel, J.P.M. v.d. Ploeg, G.J.M. Koper, Phys. A 235 (1997) 269. [5] C. Chassagne, D. Bedeaux, G.J.M. Koper, J. Phys. Chem. B 105 (2001) 11743. [6] A.M. Albano, D. Bedeaux, J. Vlieger, Physica 99A (1979) 293. [7] A.M. Albano, D. Bedeaux, J. Vlieger, Physica 102A (1980) 105. [8] F. Van der Touw, Dielectric properties of aqueous polyelectrolyte solutions, Ph.D. thesis, Leiden University, 1975.
Theory of electrode polarization: application to parallel plate cell dielectric spectroscopy experiments C. Chassagne a,*, D. Bedeaux a, J.P.M. van der Ploeg a, G.J.M. Koper b a
b
Leiden Institute of Chemistry, Leiden University, PO Box 9502, 2300 RA Leiden, The Netherlands Laboratory of Physical Chemistry, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands
Abstract An extension of the model for electrode polarization of Cirkel et al. [Physica A 235 (1997) 269] is given. The problem is solved using both classical boundary conditions and the new boundary conditions using excess densities presented in a previous paper [J. Phys. Chem. B 105 (2001) 11743]. In the present paper, the electrodes are supposed to be ideal, meaning that charge transfer or adsorption are not considered. The advantage of the new boundary conditions lies in the possibility to extend to more complicated situations including for instance specific ion adsorption. We prove that the new boundary conditions and classical ones give the same results. A comparison of the model predictions, involving no adjustable parameters, experimental dielectric spectroscopy data is performed and fairly good agreement is found. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Electrode; Polarization; Adsorption
1. Introduction Electrode polarization is a phenomenon that hampers dielectric spectroscopy measurements of many systems such as dispersions, microemulsions or polymer solutions at low frequencies due to the presence of electrolyte. As in most of these systems the relaxation frequencies and permittivity enhancements are located in these low frequency regimes, it is important to compensate for this phenomenon. Various authors proposed empirical
* Corresponding author. Tel.: /31-71-527-4560; fax: /3171-527-4239 E-mail address: [email protected] (C. Chassagne).
models involving equivalent circuits [1 /3] to account for electrode polarization. Considering the behavior of ions under the influence of an alternating electric field, we derive here an analytical expression for the complex permittivity due to electrode polarization. Such an analytical expression has already been presented for the case of a symmetric electrolyte and planar electrodes [4]. In the present paper, this theory will be extended to the case of asymmetric electrolytes. We will also derive the equations using the new boundary conditions that were introduced for the calculation of the dielectric permittivity of spherical particles [5]. These new boundary conditions are based on the concept of excess quantities in the sense of Gibbs introduced by Albano et al. [6,7]. The advantage of the new
0927-7757/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 0 2 ) 0 0 3 7 9 - 5
138
C. Chassagne et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 210 (2002) 137 /145
boundary conditions lies in the possibility to extend to more complicated situations including for instance specific ion adsorption. The new and the standard boundary conditions are found to give the same results as, of course, they should. In the last section we demonstrate that our model fairly accurately describes the experimental situation. It is important to stress here that these predictions were done without the introduction of new parameters.
2. Theory The planar electrodes are assumed to be perpendicular to the x-axis and located at x /0 and x /d , where d is the electrode spacing. The liquid in between consists of an electrolyte solution of known concentration. An alternating current (AC) is imposed between the electrodes. The flux of ions between the electrodes can be described by the Nernst/Planck equation: ze Ji Di ni i 9C9ni (1) kT The ions are carried along under the influence of the electric and thermodynamic forces via the gradient terms. Here ni is the number of ions of type i of valence zi per unit of volume and C the electric potential. Furthermore, Di is the ionic diffusion coefficient. We restrict ourselves here to binary electrolytes, i //, /, but the model can be extended to more complicated cases. Also, we will not consider the influence of ions j on the flux of ions i, which corresponds to approximating the conductivity by the conductivity at infinite dilution. In the following, we will only consider strong electrolytes and low salt concentrations, in which case this approximation is justified. Conservation of ions gives: @ni @t
9Ji 0
(2)
The charge density is related to the electric potential C via Poisson’s equation:
DC 92 C
1 X zi eni o1 o0
(3)
where o 1 is the relative dielectric permittivity of the solvent. Due to the electro-neutrality of the salt we have: X n i zi 0 (4) i
where the ni ’s are the stoichiometric coefficients of the resulting ions. An oscillating electric field E0 exp(ivt), where v is the radial frequency (v /2pf if f is the applied frequency) is imposed along the x-axis. In view of the linearity all variations of the various quantities around equilibrium are proportional to exp(ivt). This factor will, for ease of notation, be suppressed. We, therefore, write: CCeq dC ni ni;eq dni
(5)
where the subscript ‘eq’ stands for ‘equilibrium values in the absence of applied AC-field’. We have ni ,eq /ni n, Ceq, being constant. The Nernst /Planck equation becomes to linear order: z en n (6) Ji Di 9 i i dCdni kT Substitution in the conservation law Eq. (2) gives: @ni ze dn ivdni Di 9 ni n 9 i dC i (7) kT @t n i n The Poisson equation becomes: DdC
1 X zi edni o1o0 i
(8)
Using Poisson in Eq. (7) results in: Ddni
iv n z e2 n X dni i i zj dnj Di e0 e1 kT j
(9)
This equation can be written more conveniently in a matrix form:
C. Chassagne et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 210 (2002) 137 /145
dn ivD 1 k20 n z2 D 2 dn k0 n z z dn dn
k20 n z z 2 2 ivD 1 k0 v z
with k20 /e2n(/e0 e1/kT ) 1 and k2 /k20 a ni z2i . By contraction of (dn, dn ) with the left eigenvectors of the matrix one obtains two independent solutions, dnn and dnc, which satisfy:
l2n
and
Ddnc l2c dnc
where and are the eigenvalues of the matrix. For frequencies such that v /D 9k2, to which most experiments are restricted, the eigenvectors reduce to first order to: iv 1 dnn dn dn 1 (D 1 D ) k2
z z
1
iv k2
1 (D 1 D )
(12)
z =D z =D z z
N 1
iv 1 2z (D D 1 ) 2 k z z
s N 1
s N 1
iv k2
iv k2
1 (D 1 D )
1 (D 1 D )
z z z z 2z
z z
(16)
The superscript, / indicates the value of the variable beyond the double layers (see [5]). The solution of the Poisson equation has the form:
(17)
where 2
k
iv Dc
(13)
dnc Cc;0 exp( lc x)Cc;d exp(lc (x d))
o 1 o 0 l2n An
dnn Cn;0 exp( ln x)Cn;d exp(ln (x d))
X
ezi ni
i
n n
Ni
ez z iv 1 (D D 1 ) z z k 2
ez X zi Nis z z i z iv ez 1 z z k2 1 1 (D D )
(18)
o 1 o 0 l2c Ac (14)
where Cn,0, Cn,d , Cc,0 and Cc,d are constants to be determined. The first eigenvector is a diffusion mode and decays over a distance of the order pffiffiffiffiffiffiffiffiffiffiffiffiffi Dm =v: The second eigenvector dnc decays over a typical Debye length due to the contribution k2 to the eigenvalue l2c . Beyond the double layer, dnc does, therefore, not contribute to the solution. Inverting the Eq. (12) gives: z Nis dnc dn i z z
iv 1 z z (D D 1 ) 2 k z z
dC An dnn FxG
The solution of Eq. (11) then becomes:
dni
N 1
dC Ac dnc dC
and the eigenvalues to: z =D z =D iv 2 ln iv Dn z z l2c k2 iv
(15)
(11)
l2c
dnc dn dn
ni Ni dnn n n
with, (10)
Ddnn l2n dnn
dn i
139
(19)
and where, F and G are constants that are determined by the boundary conditions. The resulting charge density is: dr e(z dn z dn ) o 1 o 0 l2c Ac dnc dr dr o 1 o 0 l2n An dnn
(20)
140
C. Chassagne et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 210 (2002) 137 /145
(18) that:
3. Boundary conditions
Ac Cc
3.1. The standard boundary conditions The electrodes are supposed to be ideal meaning that neither charge transfer nor adsorption takes place. We, therefore, have: Ji (0) Ji (d) 0
Using the Nernst/Planck equation for the ionic fluxes one may replace the first two boundary conditions (Eq. (21)) by two equivalent ones. Summing [Eq. (21)]/Di over i one obtains: X@dni X@dni 0 (23) @x x 0 @x x d i i Multiplying Eq. (21) by ezi /Di and then summing over i, using Eq. (22), gives: @dr @dr k2 s0 (24) @x x 0 @x x d Substituting Eq. (15) for dni into Eqs. (23) and (20) for dr into Eq. (24), and using Eq. (14), one obtains to first order in v/k2D 9: ln Cn;0
ln Cn;d
z
iv
z z k2 z
iv
z z
k2
1 s0
and (27)
iv
ln o 1 o 0 k2 Dt
where D 1 t
z z
(1=D 1=D )2
z z z =D z =D
(28)
From Gauss, Eq. (22), and using Eq. (27) we find: 2 s0 k iv F 1 lc k2 Dt o1 o0
s0 iv 1 s0 iv (Dc D 1 t ) 2 o1 o0 k o 1 o 0 k2 D0
(29)
where D 1 0
(z z ) (z D z D )
(30)
Using that the potentials are continuous in x /0 and x /d , the applied potential difference between the electrodes can be written, to first order in v / k2D 9, as: dU dC(d) dC(0) 2Ac Cc 2An Cn Fd
1 (D 1 D )lc Cc;0
(31)
This results in: 1 (D 1 D )lc Cc;d
Ac lc Cc;0 Ac lc Cc;d
2 k s0 lc o 1 o 0
dU
(25)
where we used that jlcd j/jlnd j/1. Under the experimental conditions considered, k 1 is of the order of nanometers, d is of the order of millimeters, D9 is of the order of 10 9 m2 s 1 and v /10 000 Hz we have that jlcdj /104 and jlnd j/104. It follows that: Cn;0 Cn;d Cn
2 k
lc o 1 o 0 lc
An Cn
(21)
Furthermore, it follows from Gauss’s law that: @dC @dC s0 (22) @x x 0 @x x d o0 o1
1 s0
and
Cc;0 Cc;d Cc
(26)
It then follows from Eq. (25) together with Eq.
s0 d o1 o0 2 k 2 iv 1 1 lc lc d k2 Dt 2 1 ln d
(32)
Defining the complex dielectric permittivity o˜ by: dU
s0 d o 0 o˜
we find that:
(33)
C. Chassagne et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 210 (2002) 137 /145
o˜ o 1
=
1
2 k 2 iv 2 1 1 lc lc d k2 Dt ln d (34)
One can similarly calculate the electric field E / /@d C(x )@x in the solution for any x . It is found that: E(x)
express the conservation of ions in the double layer. The last two are the conditions for the electric displacement field. 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 [6,7]:
s0
x2
dnsi
o1 o0 2 iv k 2 k D0 lc [exp( lc x) exp(lc (x d))]
141
iv k2 Dt
[exp( ln x) exp(ln (x d))]
s0 iv o 1 o 0 k2 D0
i
i (x)]dx
(39)
x1
(35)
It can be verified that E (0) /E (d ) /s0/o 1o 0 as follows from Eq. (22). Furthermore, beyond the diffusion layer, i.e. at positions more than jln 1j away from both electrodes, this leads to: E(x)# F
g [dn (x) dn
(36)
where [x1, x2] /[0, d /2] or [d/2, d ] depending on the electrode considered. Using Eq. (15) together with Eq. (14), and the fact that jlcdj/1, we find that: dnsi (0)
z
Nis
z z l c
dnsi (d)
Cc;0
and (40)
z Nis Cc;d z z l c
The resulting excess of the charge density of the double layer is: X drs (0) ezi dnsi (0) o 1 o 0 lc Ac Cc;0 i
3.2. Boundary conditions with excess quantities
drs (d)
X
ezi dnsi (d) o 1 o 0 lc Ac Cc;d
(41)
i
In this section we will prove that the standard boundary conditions are equivalent to the boundary conditions introduced by Albano et al. [6,7] and presented in a previous article [5]. These conditions read: ivdnsi (0) Ji (0) and ivdnsi (d) Ji (d) @dC s0 drs (0) o 0 o 1 @x x 0 @dC s0 o 0 o 1 drs (d) @x x d
(37)
(38)
Following the notation of Albano et al. [6], the superscript ‘s’ indicates the excess of the corresponding quantity in the double layer. The notation dnsi (0) or dnsi (d) is adopted in order to indicate to which electrode the excess dnsi calculated corresponds to. The first two conditions Eq. (37)
Similarly to what has been done in the previous section, the first two boundary conditions (Eq. (37)) are replaced by two equivalent ones. Using the same procedure, [Eq. (37)]/Di is summed over i to get the first relation. In order to get a second relation, Eq. (37) is first multiplied by ezi /Di and then summed over i, using Eq. (38). We get, to first order in v /k2D9: X dns (0) X@dn i i iv Di @x x 0 i i iv
X dns (d) i i
Di
X@dn i i
@x
(42) x d
and iv X (ezi )dnsi (0) s0 drs (0) k2 i Di
142
C. Chassagne et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 210 (2002) 137 /145
iv X (ezi )dnsi (d) s0 drs (d) k2 i Di
(43)
Substituting the excess ionic and charge densities given in Eqs. (40) and (41) and using Eq. (14) we find to first order in v/k2D 9. ln Cn;0
z iv 1 (D D 1 )lc Cc;0 z z k2
z iv 1 (D D 1 )lc Cc;d z z k2 2 k s0 Ac lc Cc;0 Ac lc Cc;d lc o 1 o 0
ln Cn;d
(44) (45)
We hereby used the equalities: P z iv ezi Nis o 1 o 0 l2c Ac z z k 2 Di iv s0 iv 2 lc Ac 1 ez 1 k2 D k2 D ez 2 k s0 lc o 1 o 0
and
drs (0) drs (d)
(46)
We furthermore recover Eq. (29) for F , Eq. (33) for the potential difference dU between the electrodes and Eq. (34) for the complex dielectric permittivity o:˜/
4. Complex permittivity of the system 4.1. General case The complex permittivity of a system may be written as follows: o˜o
K ivo 0
=
(48)
For very low frequencies, v /2D 9k /d /vc, we get: 2 2 kd kd v and K o 0 o 1 o o1 (49) 2 2 k2 D0 In practice, the characteristic frequency vc corresponds to a frequency smaller than 100 Hz, for which usually no reliable measurements can be done. If we now consider the case for which vc / v /k2D 9 then: 2 kd v o o1 and K o 0 o 1 k2 D0 (50) 2 k2 D0
=
From Eqs. (44) and (45) we recover the expressions Eqs. (26) and (27) for Cn,0, Cn,d, Cc,0 and Cc,d found using the normal boundary conditions. As a corollary we find that: dnsi (0) dnsi (d)
k2D 9 Eq. (34) can be expanded as: kd iv kd 2 o˜ o 1 2 k2 D0 2 v 2 kd 2 1 k2 D0 2
(47)
where o is the dielectric permittivity of the system and K its conductivity. Using the fact that v /
This corresponds to the frequency range investigated in most dielectric spectroscopy experiments (typically 102 /v /109). At a given frequency, the permittivity o increases with increasing ionic strength (o /n3/2 ) and decreases with increasing electrode spacing (o /d 1). The conductivity K does in good approximation not depend 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 /ai e2z2i ni n Di /kT .
4.2. Special case In the case of an electrolyte, with equal diffusion coefficients (D /D /D ), Eq. (34) reduces to: 2 k 2 o˜ o 1 1 1 lc lc d iv 2 iv o1 1 k2 D lc d k2 D
= =
with k2 /e2n(o 0o 1kT ) 1 a ni z2i and l2c /k2/iv / D . This equation corresponds to Eq. (8) of the article of Cirkel et al. [4] if one takes n /n / z / /z /1. We also have:
C. Chassagne et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 210 (2002) 137 /145
o o 1
=
kd v 2 and 2 k2 D
K o 0 o 1 k2 D
(51)
for vc /2Dk /d /v /k2D .
ence of potential between two points xa and xb located in this region (xa /xb /d2) may be estimated as follows. We note that F /dI/(SK1): (55)
dU2 #(d2 =S)dI=K1
From Eq. (36), we can also estimate the measured potential between the electrodes:
5. Experimental results
dU #(d=S)dI=K1 #(d=d2 )dU2 5.1. Material and methods The experimental setup we used is described in detail in [8]. The cell consists of two parallel electrodes made of platinum, each with a surface area of S /2.29 cm2. Their separation could be varied from 1.50 to 4.50 mm with an uncertainty of 9/0.01 mm. KCl and NH4Cl solutions were prepared using tri-distilled water (Millipore) and pro analysis corresponding salt from Merck. Measurements were done at T /21 8C. The cell in which the experiments were performed is modeled as a parallel circuit of a frequencydependant ‘capacitance’, C , and a frequencydependent ‘resistance’, R . The equivalent impedance of the circuit is: Z
R
(52)
1 iRCv
The impedance of the system can be calculated according to Z /dU /dI where dI / /dQ /dt / / d(s0S )/dt / /ivs0S , and where S is the surface area of the electrodes. From Eqs. (33) and (34), we find: Z
dU d ivs0 S ivoo ˜ 0S
(53)
d 1 C S o0
and
K
(56)
Doing so, we neglect the capacitive contribution due to the polarization of the electrodes. This is justified by the fact that the ratio between the real and imaginary part of Z is given by vRC #/vo 0o 1/ K1 /10 7 /10 10 (taking electrode polarization into account). Beyond the diffusion layer, the electric field is in good approximation equal to the measured electric field on the electrodes. 5.2. Experimental The experimental conductivity curves were plotted according to Eq. (54). As K #/K1 in the frequency range of interest (see Eq. (50)), the conductivity should not depend on the electrode spacing. The exact spacings of the electrodes were, therefore, determined by fitting these spacings such that the experimental conductivity curves for all four spacings superimposed. Consequently, these spacings were used as known parameters for the experimental permittivity increments curves and in the calculation of the theoretical predictions. The predicted conductivities of the solution were plotted according to: X K vo 0 Im(o)# ˜ n i zi C s L (57) i i
Consequently, it can be shown that the relations between the measured (frequency-dependent) capacitance and resistance and the corresponding experimental permittivity and conductivity are: o
143
d 1 SR
(54)
The ratio d /S is called the cell constant. From Eq. (36), it is shown that the electric field does not depend on x in the bulk region situated beyond the diffusion layer, i.e. at position more than jln 1j away from both electrodes. The differ-
where Cs is the given neutral salt concentration in mM (1 mM /10 3 mol l 1 /1 mol m 3). The equivalent concentration of the solution is ni zi Cs. The ionic diffusion coefficients, Di (m2 s 1), are related to the equivalent limiting ionic conductiv2 2 ities L mol 1) by L i (S m i /zi Di Nae /kT , where Na is Avogadro’s number. We have n / CsNa. As the limiting ionic conductivities of K and NH4 are equal at 298 K and differ from the limiting ionic conductivity of Cl by only a few percents, we took for all three the same value, and used L /74 /10 4 S m2 mol 1. From the
144
C. Chassagne et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 210 (2002) 137 /145
conductivity measurements the salt concentrations Cs could be established and these were used in the calculation of the corresponding permittivity increment predictions. The experimental permittivity increments curves were plotted according to: d 1 Do (C C ) S o0
(58)
where C corresponds to the capacity at the highest frequency measured and the theoretical predictions were plotted, using the above cited (known) parameters, according to: ˜ o˜ )] Do pred [Re(o) Re(
(59)
˜ at where, Re(o˜ ) corresponds to the value of Re(o) the highest frequency measured.
Fig. 2. Dielectric increment as a function of frequency for a 8.6 mM KCl solution. Symbols represent the experimental data for various electrode spacings and lines are predictions according to Eq. (59).
6. Discussion In Figs. 1/4, experimental permittivity data for two kinds of binary electrolytes at various strength are plotted together with the model predictions. Given the facts that no parameter adjustment has been made and that all parameters were determined independently, the agreement must be considered as very good. The enormous incre-
Fig. 3. Dielectric increment as a function of frequency for a 1.1 mM NH4Cl solution. Symbols represent the experimental data for various electrode spacings and lines are predictions according to Eq. (59).
Fig. 1. Dielectric increment as a function of frequency for a 2.6 mM KCl solution. Symbols represent the experimental data for various electrode spacings and lines are predictions according to Eq. (59).
ments measured at low frequencies do not allow for a proper estimate of the experimental error; our current estimate is of the order of 15%. It should be emphasized that the experimental results are very sensitive to the properties of the surface of the electrodes. For instance, from our experiment we know that the polarization strongly depends on the way the electrodes were cleaned and that extreme care has to be taken in this procedure. All figures shown were for well cleaned electrodes.
C. Chassagne et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 210 (2002) 137 /145
145
these allow for the introduction of surface properties such as specific ion adsorption. Also, these boundary conditions still hold if the a weak DCpotential is applied on the electrodes. For a weak DC-potential there is no influence on the ACcomponent. However, if a strong DC-potential is applied in addition to the AC-potential on the electrodes it can not be assumed any longer that, the boundary conditions are modified which leads to a modification of the AC-field.
References Fig. 4. Dielectric increment as a function of frequency for a 2 mM NH4Cl solution. Symbols represent the experimental data for various electrode spacings and lines are predictions according to Eq. (59).
7. Conclusions In this article we extend the model of Cirkel et al. [4] for the dielectric permittivity increment due to electrode polarization to the more general case including all binary electrolytes. We have performed these calculations with standard boundary conditions and with the boundary conditions with excess quantities derived in a previous article. Both boundary conditions give the same results. The advantage of the new boundary conditions is that
[1] J.P.M. Van der Ploeg, M. Mandel, Meas. Sci. Technol. 2 (4) (1991) 389. [2] H.P. Schwann, in: W.L. Nastuk (Ed.), Physical Technics in Biological Research, Part B, vol. 6, Academic Press, New York, 1963, pp. 323 /407. [3] H.P. Maruska, J.G. Stevens, IEEE Trans. Electr. Instrum. 23 (1988) 197. [4] P.A. Cirkel, J.P.M. v.d. Ploeg, G.J.M. Koper, Phys. A 235 (1997) 269. [5] C. Chassagne, D. Bedeaux, G.J.M. Koper, J. Phys. Chem. B 105 (2001) 11743. [6] A.M. Albano, D. Bedeaux, J. Vlieger, Physica 99A (1979) 293. [7] A.M. Albano, D. Bedeaux, J. Vlieger, Physica 102A (1980) 105. [8] F. Van der Touw, Dielectric properties of aqueous polyelectrolyte solutions, Ph.D. thesis, Leiden University, 1975.