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Chronopotentiometric Response of Electrically Heterogeneous Permselective Surface: 3D Modelling of Transition Time and Experiment Semyon Aleksandrovich Mareev, Vladlen Stanislavovich Nichka, Dmitrii Yur‘evich Butylskii, Magomet Khuseevich Urtenov, Natalia D. Pismenskaya, Pavel Yurievich Apel, and Victor V. Nikonenko J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b03629 • Publication Date (Web): 27 May 2016 Downloaded from http://pubs.acs.org on June 6, 2016
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Chronopotentiometric Response of Electrically Heterogeneous Permselective Surface: 3D Modelling of Transition Time and Experiment S.A. Mareev a*, V.S. Nichka a, D.Yu. Butylskii a, M.Kh. Urtenov b, N.D. Pismenskaya a, P.Yu. Apel c, V.V. Nikonenko a a Department of Physical Chemistry, Kuban State University, 149 Stavropolskaya St., 350040 Krasnodar, Russia. E-mail:
[email protected]; Tel. +7(861)219-95-73. b Department of Computer Technology and Applied Mathematics, Kuban State University, 149 Stavropolskaya St., 350040 Krasnodar, Russia c Joint Institute for Nuclear Research, Dubna, Joliot-Curie, 6 Dubna, 141980 Moscow region, Russia d Dubna State University, 19 Universitetskaya St., 141982 Dubna, Moscow region, Russia Abstract: Understanding of the current density distribution over an electrically heterogeneous surface and its effect on ion transport represent an important issue in electrochemistry, composite materials, geophysics and some other domains. We report an approach for three-dimensional (3D) modelling (with cylindrical symmetry) of transient ion transfer across a surface composed of conductive and nonconductive areas. In the model formulation and solution we use the electrical current stream function. It allows setting the integral boundary condition for electric current at heterogeneous surface without any restrictions on the local current density distribution. A very good agreement is found between the numerical solution and the experimental transition time determined from chronopotentiograms. The use of a specially designed membrane allows computation without fitted parameters. We show that the application of specific simplifications for the current density distribution over the surface (uniform distribution throughout all the surface or its conductive area, neglect of tangential current density) results in essential deviations from experimental transition time.
Introduction The most mathematical descriptions of ionic transport in electrochemical systems are limited to 1D geometry1. The passage to 2D and especially 3D geometry sharply increases mathematical difficulties. However, there are a large number of cases where multidimensional effects are important: mass transfer at partially blocked electrodes2, 3, microelectrodes4, 5, proton-exchange membrane fuel cells and batteries6; flow electrochemical devices, such as electrolysers and electrodialysers7, 8, electrochemically heterogeneous graphene9 or carbon nanofilaments10, heterogeneous ion-exchange membranes11, 12, 13, 14 etc. Development of approaches for handling multidimensional modeling of interface phenomena represents an important issue.
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Electrochemical responses of materials with heterogeneous surface essentially depend on the parameters of surface heterogeneity11, 12, 13: on the size and shape2, 9, as well as the surface fraction2, 3, 11, 15 of conductive and nonconductive areas. Chronopotentiometry (CP) is a powerful instrument for studying interfacial ion transport and chemical reactions13, 16, 17, 18. However, 2D mathematical description of the CP response in the case of heterogeneous surface is complicated, since an integral condition of current intensity is needed at the membrane or electrode surface. This constitutes a problem as the current density distribution over the conductive area is not known a priori. Rubinstein et al.11, Green et al.19, 20 used the condition of uniform current density distribution, however, this condition was not verified using another approach. In this paper we propose for the solution of this problem to apply the electric current stream function approach. Stream functions are widely used in fluid mechanics. They present a significant advantage in analytical analysis and numerical simulations of fluid flow21. The main feature of this approach is that it allows calculation of the fluid velocity distribution when the volumetric flow rate is known in heterogeneous systems22. Another advantage is that any solution obtained with the use of stream function satisfies automatically the law of mass conservation. There is a strong mathematical analogy between the flow of fluid and the flow of electric current. Hence, it is possible to introduce a stream function based on the current density by analogy with that based on the fluid velocity. Apparently, the first who has introduced the stream function for modeling electric current were G. Taylor et al.23. They used the mathematical analogy between the flow of fluid and that of electric current to solve problems of fluid mechanics. Later on, a rather large number of studies have applied stream functions for simulation of electric current density distribution in different media: in semiconductors24, in superconductors25 in thin circuits26, on the earth's surface27, in electromagnetic devices28 and other systems. In membrane electrochemistry this method was applied only by Pismensky et al.29. The aim of this paper is to show how the advantages of the stream function approach can be used in modeling transient ion transfer at an electrically heterogeneous surface. We describe CP of a heterogeneous ion-exchange membrane, find the transition time and compare the results with experiment and with other known theories, namely, that by Sand30, by Choi and Moon12 and by Rubinstein et al.11.
Experiment A specially designed membrane with tailored structure is prepared for CP study. The substrate is a polyethylene terephthalate track-etched membrane (Fig. 1) of thickness d1=10 µm, produced at the Joint Institute for Nuclear Research, Dubna, Russia. The membrane has cylindrical (from 25.42 to 26.29 microns in diameter) randomly arranged pores with the density 1.47 104 2 ACS Paragon Plus Environment
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pores/cm2. We coated the membrane on one side with a Nafion® perfluorinated resin solution of thickness d2 about 3 µm, the Nafion® solution filled also the pores. Hence, we obtained a membrane with a smooth homogeneous permselective surface on one side and a heterogeneous surface on the other. The most of the unmodified surface is nonconductive, the circular conductive areas filled with the Nafion® material occupy about ≈7.7% of the surface according to the membrane manufacturer. Chronopotentiograms were measured with a 0.02 M NaCl solution using the flow cell and the procedure described in 13.
Figure 1 - Optical micrograph of the ion-exchange membrane surface under study. The circles are conductive areas, which are holes in a track-etched membrane filled with Nafion® material.
Theory Geometry The transition from random to hexagonal distribution of conductive areas over the membrane surface is shown in Figure 2. The unit cell on the surface can be considered as consisted of a conductive disk of radius R1 within a nonconductive ring of external radius R2. The area of a circle with radius R2 should be equal to the area of one hexagon. Then R1 and R2 are linked with the fraction of conductive area on the surface ε: ε = (R1/R2)2. Though the membrane under study is characterized by a very uniform pore size, the micrographs show a slight dispersion, R1 varies from 12.71 to 13.15 µm. R2, equal to 46.53 µm, is found from the known pore density 1.47 104 pores/cm2. Let the diffusion boundary layer thickness (DBL) be δ, identical from both membrane sides. Then the 3D unit cell is modelled as a cylinder of radius R2, involving a membrane, a part of which is conductive and the other (shown as a darkened rectangle) nonconductive, and two solutions of thickness δ (Fig. 2c). The smooth layer of Nafion® of thickness d2 faced the enriched DBL. Using cylindrical symmetry, the system under study may be presented in two coordinates r and z. 3 ACS Paragon Plus Environment
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a)
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b)
c) z
z
δ
Bulk solution Depleted DBL
R2
0
-d1 -d1-d2
R1
δ r
R1
R2 r
Membrane
Enriched DBL
-d1-d2-δ Bulk solution
Figure 2 – Distribution of round conductive areas over the membrane surface in hexagonal geometry (a). The DBL of thickness δ is assumed independent of lateral position (b). Conversion of the 3D system (b) into a 2D simulation problem (c).
Problem formulation The following simplifying assumptions are made, according to earlier theoretical studies 31, 32: - Only the faradaic current is considered, the charging current is not taken into account; - The membrane is considered as a (non-ideal) permselective ohmic resistance; - The temperature, pressure, activity coefficient and density gradients, as well as solvent fluxes (current-induced convection) and water splitting are ignored. Non-stationary ion transport in the depleted DBL adjacent to the membrane is described by the Nernst-Planck equation (1) in the conventional form, the mass (2) and the charge (5) conservation laws, the equation (4) of current flow, under the local electroneutrality assumption (3):
r r F Jk = zk Dk ck E − Dk ∇ck RT
(1)
r ∂ck = −divJ k ∂t
(2)
z1c1 = −z2c2 = c
(3)
r r j = F ∑ zk Jk r divj = 0
(4) (5) 4 ACS Paragon Plus Environment
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r here ck, Dk , J k , and zk are the concentration, the diffusion coefficient, the flux density and the r charge, respectively, k=1 for the cation (counterion), 2 for the anion (co-ion); E = −∇ϕ is the
r
electric field; ϕ is the electric potential, T is the absolute temperature; j is the current density; t is the time; F is the Faraday constant, R is the gas constant. Combination of Eqs. (1) and (4) under the electroneutrality condition, Eq. (3), gives: r E=
RT 1 r ( D1 − D2 ) RT 1 j+ ∇c F ( z1 D1 − z2 D2 ) c F ( z1 D1 − z 2 D2 ) c
(6)
2
Substituting Eq. (6) into Eq. (1) yields Eq. (7) (presented in the form where the electric potential is excluded33):
r r tk j J k = − D∇ck + zk F where D =
(7)
zk Dk ( z1 − z2 ) D1 D2 is the electrolyte diffusion coefficient; tk = is the transport z1 D1 − z2 D2 z1 D1 − z2 D2
number of kth ion. After substituting Eq. (7) in Eq. (2) with taking into account Eq. (3), we obtain:
∂c = D∆c ∂t
(8)
The initial condition for the concentration:
c(r , z , 0) = c0
(9)
The flux continuity condition at the boundaries between the solution and the conductive region of membrane reads: mb
J1 z
sol z =0 + , r =0.. R1
jT = z 1 , J1 z z1F
sol z =− d1 − d 2 , r =0.. R2
jT = − z 1 z1F
mb
(10)
In the solution at the boundary, at z=0+, the contributions of diffusion and migration into the overall normal flux, J1z , are explicitly taken into account via Eq. (7). In the membrane phase, at z=0-, these contributions are taken into account implicitly through the effective transport number T1, which is the fraction of the current carried by ion 1 by migration (the dominating transport) and diffusion. The boundary flux condition at the nonconductive region reads:
J1 z
sol z = 0 + , r = R1 .. R2
=0
(11)
The boundary condition in the bulk solution sets a known constant concentration c0:
c(r , δ , t ) = c(r , −d1 − d 2 − δ , t ) = c0
(12) 5
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In galvanostatic regime, the average current density across the surface, jav , is a known parameter. It is linked with the unknown local current density as:
1 jav = R2
R2
∫ 0
1 jz ( r , δ , t ) dr = R2
R2
∫ j ( r , −d z
1
− d 2 − δ , t ) dr = const
(13)
0
The conventional approach to solving problems with condition (13) consists in applying an assumption on the form of local current density distribution. In Refs.11, 19, the uniform current density distribution over the conductive areas is applied: jz ( r , 0+, t ) r =0..R = jav / ε
(14)
1
Electric current stream function In this paper, we use the electric current stream function23, 29 η , which does not require any assumption for jz ( r , 0, t ) . This function is defined as follows:
1 ∂η 1 ∂η = − jr , = jz r ∂z r ∂r
(15)
This definition constrains the current density to satisfy the charge conservation (continuity) equation (5). Note that Eq. (5) holds, if the rate of the space charge density ( ρe ) variation with time is negligible, that is
r ∂ρ e ∂ 1 ∂ = − divj = − ( rj r ) + jz = 0 ∂t ∂z r ∂r
(16)
We can see that Eq. (16) is satisfied at any function η, if only the condition for equality of mixed derivatives of η is met. Below we present some transformations of Eq. (6) leading to the electric current stream function. The curl of the electric field: r rotE =
RT 1r ( D1 − D2 ) RT 1 rot ( j ) + rot ( ∇c) F ( z1 D1 − z2 D2 ) c F ( z1 D1 − z2 D2 ) c 2
r 1 r 1r 1 j ) = − 2 ( ∇c, j ) + rot( j ) c c c r Since rot(E ) = rot(-∇ ϕ ) = 0 and ∇c ×∇c = 0 , Eqs. (17) and (18) lead to rot(
−
r 1 r 1 ∇c, j ) + rot( j ) = 0 . 2 ( c c
(17)
(18)
(19)
The curl of current density, expressed using the definition of the electric current stream function, Eq. (15), is: r 1 ∂ 2η ∂ 1 ∂η 1 rot ( j ) = + ( ) = ∆η , 2 r ∂z ∂r r ∂r r
(20) 6
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Substitution of Eqs. (15) and (20) into Eq. (19) gives Eq. (21):
1 ∆η = (∇η , ∇c) . c
(21)
Note that the form of Eq. (21) is identical for the 2D and 3D cases. In galvanostatic mode under the electroneutrality condition, a constant current intensity or a constant average current density, jav , is set at the external boundaries. The value of jav does not depend on the distance z:
1 jav = π R22
R2
∫ 2π rj (r, z, t )dr = const ,
(22)
z
0
According to Eq. (15), Eq. (22) leads to: R
2 jav R22 = ∫ rjz (r , z , t )dr = 2 0
R2
∫ 0
∂η (r , z , t ) dr = η ( R2 , z , t ) − η (0, z, t ) ∂r
Thus, the general boundary conditions for Eq. (21) will be:
∂η ( r , z , t ) ∂z
=
∂η ( r , z , t ) ∂z
z =− d1 − d 2 −δ , z =δ r∈[ 0, R2 ]
∂η ( r , z , t ) ∂r
z∈[ − d1 , 0] r = 0, r = R1
=
∂η ( r , z , t ) ∂z
z∈[ − d1 , − d1 − d 2 −δ ]∪[0, δ ] r = 0, r = R2
=0,
= 0,
(23)
z =− d1 , z = 0 r∈[ R1 , R2 ]
1 2
η ( 0, δ , t ) = η ( 0, −d1 − d 2 − δ , t ) = 0 , η ( R2 , δ , t ) = η ( R2 , −d1 − d 2 − δ , t ) = jav ( R2 ) . (24) 2
In the membrane bulk, we neglect the diffusion and assume concentrations constant. This assumption is well verified in commercial membranes when contacting dilute solutions34 (such as 0.02 M NaCl in our experiment). Besides, during the short times used in the experiment, concentration changes are insignificant. Then ∇c =0 in the membrane, and Eq. (21) reduces to:
∆η = 0
(25)
Function η is quite similar to the stream function ψ in fluid mechanics. The latter is defined as
∂ψ ∂ψ = − rv r , = rv z ∂z ∂r
(26)
r here v is the fluid velocity. Any function ψ satisfying Eq. (26) satisfies also the mass
conservation law:
r div v = 0
(27)
The physical meanings of both functions, η and ψ , are similar.
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The lines of constant ψ/η are the streamlines of the fluid flow/electric current density; that is, they are everywhere parallel to the local velocity/current density vector. No flow can exist normal to a streamline; thus, each selected ψ/η line can be interpreted as a boundary not permeable to the flow/current density. Furthermore, in 2D geometry, for any two points in the flow/current density field, the difference in values of ψ/η function represents the volume flow/current intensity between the points. The latter follows from the integration of ψ/η function.
Limiting current density and diffusion layer thickness
To solve the problem above, the diffusion layer thickness must be known. This value may be calculated in the case of steady laminar flow of a solution between two parallel plane homogeneous permselective walls, when a limiting current density occurs. In this case, Eq. (8) may be solved with the boundary condition setting nearly zero electrolyte concentration at the walls7, 8. The approximate formulas for the average limiting current density, jav0 lim , and the effective thickness of diffusion layer, δ Lev , read: 0 av lim
j
1/3 h 2V0 FDc0 = 1.47 − 0.2 h(T1 − t1 ) LD
(28)
1/3
δ Lev
LD = 0.71 h 2 h V0
(29)
where V0 is the average fluid velocity, h is the distance between the walls, L is the channel length. Eqs. (28) and (29) hold for relatively “short” channels ( L ≤ 0.02h 2V0 / D ). They were deduced by Lévêque in 1926 for heat transfer and adapted by Newman7 for electrode and by Gnusin et al.35 for membrane systems. 2D numerical solution of the diffusion-convection problem is widely discussed in recent papers36, 37.
Results and discussions Concentration and current density distribution Eqs. (8) and (21) under the initial (9) and boundary (10)-(12), (23), (24) conditions are solved numerically using Comsol Multiphysics 4.4. η (r, z, t) and c(r, z, t) are the unknown functions. The results presented in Fig. 3 are obtained with the following parameters: R1 = 13.15 µm, R2 = 46.53 µm, T1=1 (the membrane is assumed ideally permselective that holds in dilute solutions38). The value of δ=260 µm, entering Eqs. (23), and the limiting current density
jav0 lim =19.74 A/m2 are found using the Lévêque approximate solution35 (28) for the 8 ACS Paragon Plus Environment
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hydrodynamic and geometric parameters of the cell used in the experiments. In the case of wellrefined mesh, the computation of one chronopotentiometric curve takes 10 minutes, on average. The calculated current lines (Fig. 3a) are distributed non-uniformly over the conductive membrane area. They condense at the boundary with a nonconductive area, and become sparser in the conductive area center. The corresponding concentration profile is shown by curve 1 in Fig. 3b. Сurve 2 shows the concentration profile calculated using condition (14).
a
Figure 3 – Current lines (a) and concentration distribution (b) near the membrane surface calculated using the 3D model with condition (13) (curve 1) and the condition (14) (curve 2), at jav=1.5 jav0 lim , t=1.74 s, R1=13.15 µm, the other parameters are given in the text
Chronopotentiograms and transition time In the case of homogeneous electrodes or membranes, the chronopotentiograms have three well distinguished regions13. The first one occurring at small times presents a slow growth of the potential difference (PD) with time. It is followed by the transition region characterized by a sharp increase of PD. The third region relates to establishing a steady state, where the PD attains a constant (or nearly constant) value. These three regions occur in the case of the membrane under study, however, they are diffuse and overlap each other because of the low fraction of conductive areas. Experimental and simulated chronopotentiograms are shown in Fig. 4a. Their important characteristic is the transition time, which corresponds to the transition from the state where the ion transport occurs only by electrodiffusion to the state where current-induced mechanisms, such as electroconvection39, 40 and water splitting41, 42, contribute also. These 9 ACS Paragon Plus Environment
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additional mechanisms result in reducing the rate of system resistance increase and allow attaining a steady state under a finite potential difference, ∆ϕ. The transition time relates to the inflection point of the experimental chronopotentiogam (Fig. 4b), where the rate of increase of ∆ϕ reaches its maximum and the beginning of the slowdown of the potential growth occurs.
Figure 4 Theoretical (solid) and experimental (dashed) chronopotentiometric curves (a). Estimation of the transition time by the inflection point of theoretical (solid) and experimental (dashed) chronopotentiogram in differential form (b) at jav = 1.375 jav0 lim . Parameters of the model: δ=260 µm, R1=12.71 µm, R2=46.53 µm.
Chronopotentiograms with inflection may be simulated theoretically, if the current-induced transport is taken into account43. We apply a model, which considers electrodiffusion only. This transport mode occurs in the initial part of chronopotentiograms. We do not take into account the change of the transport mechanism, but we can determine the moment of beginning of this change by a threshold value43 of interfacial salt concentration cs, which is evaluated as 2×10–6 M and is much lower than its bulk value30, 44, c0. A small value of cs is needed for the onset of electroconvection45, but also for starting water splitting. Under a value of cs close to 10–5 M, H+ 10 ACS Paragon Plus Environment
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and OH– ions always present in water (with concentration of the order of 10–7 M) can compete with the salt counterion in the charge transfer across the membrane. It is the necessary condition for water splitting31. Electroconvection can start even at lower values of cs, if the membrane surface is specially tailored40. However, the choice of the cs value in the range from 10–7 to 10–4 M does not affect noticeably the calculated transition time, as the time required for cs to pass from 10–4 to 10–7 M,
t ′ , is very small. For example, at j = 1.375 jav0 lim , t ′ = 0.0802 s ≈ 4.6 × 10 –3τ Sand . The calculated curves ∆ϕ − t and d ∆ϕ / dt − t go nearly vertically with time (Fig. 4a,b). The calculated time related to the moment, when cs attains its threshold value, corresponds well to the maximum of experimental curve d ∆ϕ / dt (Fig. 4b). For the first time, a theoretical expression for the transition time, τ, was deduced by Sand30 under the assumptions of homogeneous ideally permselective surface and an infinitely thick DBL. Lerche and Wolf44 and later Krol et al.18 adapted the Sand equation to membrane transport by introducing the transport number in the membrane: 2
τ Sand
π D ck 0 zk F 1 = 2 . 4 Tk − tk jav
(30)
where ck 0 is the bulk concentration of k-th ion. It is known, that the transition time for heterogeneous membranes is lower than that for homogeneous ones12, 13, 18. However, the only mathematical description of the transition time for an electrically non-homogeneous permselective interface was made by Choi and Moon12. They had modified the Sand equation, Eq. (30), replacing the average current density, jav, by the local current density, jlocal = jav / ε , across the conductive areas: 2
τ Choi − Moon
2 π D ck 0 zk F ε = 2 4 Tk − tk jav
(31)
The relationship between the transition times by Sand and by Choi and Moon is expressed as
τ Choi − Moon = τ Sand ε 2 . The transition time calculated using our 3D model is in a good agreement with the experimental data (Fig. 5a). We used no fitting parameters. The curve computed for the lower radius value, R1=12.71 µm, is below the experimental points; that for the higher radius, R1=13.15 µm, is above. The transition times calculated using the Sand equation, Eq. (30), are essentially higher than the experimental ones, and those calculated using the Choi-Moon equation are much lower (Fig. 5a). The values of τ calculated using our 3D model, but under the condition (14) are lower too. In fact, Eqs. (30) and (31) give two extreme asymptotes of our 11 ACS Paragon Plus Environment
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model. The first one is verified, when the surface is homogeneous, δ and j are sufficiently high (Fig. 5b,c); for a homogeneous surface the minimum value of j where the Sand equation is verified is evaluated43, 46 as (1.5 ÷ 2) jav0 lim . The reason is in the fact that the Sand theory assumes an infinitely large DBL. In reality, when the current density is low and the transition time is great, the current-induced concentration changes go beyond the DBL. The Sand equation is verified also when ε
