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discharge of an electrochemical supercapacitor electrode pore; non-uniqueness of mathematical models
Electrochemistry Communications 9 (2007) 211–215 www.elsevier.com/locate/elecom
Charge/discharge of an electrochemical supercapacitor electrode pore; non-uniqueness of mathematical models Pehr Bjo¨rnbom
*
Department of Chemical Engineering and Technology, KTH – Royal Institute of Technology, SE-10044 Stockholm, Sweden Received 17 August 2006; received in revised form 30 August 2006; accepted 8 September 2006 Available online 13 October 2006
Abstract A thermodynamic analysis has been done to enhance understanding of the relation between various mathematical models for electrochemical supercapacitor pores. For the same capacitive charge/discharge experiment a variety of one-dimensional mathematical model equations concerning the transport of ions and double layer charge/discharge along the pore are shown to be indistinguishable. Some of those indistinguishable equations could be interpreted as derived from diffusional mechanisms while others appear as derived from migrational mechanisms. Ohmic resistivities and diffusivities obtained in such case are not contradicting results but characterize identical physical processes. The results are valid as long as the assumptions of irreversible thermodynamics of local equilibrium along the pore and of linearization of the flux equations hold. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Single pore model; Mathematical model; Irreversible thermodynamics; Ionic diffusion; Ionic migration; Electrochemical supercapacitor
1. Introduction Electrodes for electrochemical double layer capacitors (EDLCs) or electrochemical supercapacitors typically consist of porous materials such as carbon [1]. The transport properties for ions in the pores are essential for the performance of supercapacitor electrodes [1–6]. Transport parameters may be obtained from dynamic voltammetry experiments, especially potential steps and impedance spectroscopy. Such experiments are commonly evaluated using mathematical models based on principles originating from de Levie’s seminal paper from 1963 [1,7] where the main transport resistance is assumed to be the ohmic resistance to ionic migration in the pore electrolyte solution. However, recent work has shown that potential step experiments also may be successfully evaluated using models based on Fickian diffusion of ions in the pores
*
Tel.: +46 8790 8255. E-mail address: [email protected].
1388-2481/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.elecom.2006.09.009
with effective diffusivities as the main transport parameters [5,8–11]. The purpose of this work is to show that there is no contradiction in this situation. Such voltammetry experiments under frequently used experimental conditions cannot give unique information on the transport mechanisms for the ions. The fundamental thermodynamic analysis in this paper shows that for the same experiment a variety of mathematical model equations are equally valid. Some of those equivalently valid equations could be interpreted as derived from diffusional mechanisms while others appear as derived from migrational mechanisms. 2. Model assumptions 2.1. The model pore at equilibrium We assume a straight cylindrical pore with walls of an electrically conducting material e.g. carbon (see Fig. 1). The pore is filled with a 1,1-electrolyte solution, typically KOH or KF in water. The diameter of the pore is assumed to be substantially greater than twice the Debye
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P. Bjo¨rnbom / Electrochemistry Communications 9 (2007) 211–215
At equilibrium the electrolyte concentration and Galvani potential have the same values in the core bulk as outside the pore Double layer
Core bulk solution
Differential cells
At non-equilibrium the electrochemical potential has the same value all over each cell due to the local equilibrium assumption but differ between cells due to the axial gradient (= the driving force) Fig. 1. A schematic picture of a model pore.
length of the electrolyte. As a consequence there is a central part of the pore solution with constant ionic concentrations at equilibrium, while close to the pore wall, within the diffuse part of the double layer, concentrations vary with the radial coordinate. At typical electrolyte concentrations used for electrochemical supercapacitors the diffuse layer is collapsed and only 0.5 nm in thickness [12]. At equilibrium the Galvani inner potential of the electrolyte is constant within the central core of the pore solution while it varies with the radial coordinate in the diffuse part of the double layer. On the other hand at equilibrium the electrochemical potentials of the ions have the same value everywhere in the system including the diffuse double layer and ions adsorbed on the surface. The values of the Galvani potential, the electrochemical potentials and the concentrations in the central core of the pore and the corresponding values in the external solution outside the pore mouth are equal. The equilibrium relation between Galvani potential and concentrations of ions in the pore is discussed below. 2.2. The model pore during charge/discharge At non-equilibrium conditions we assume that the driving forces for the flow of ions are the gradients in the electrochemical potentials and that linearized flux equations according to classical irreversible thermodynamics (CIT) are valid (see e.g. [13]), i.e. flux equations like Eq. (16) below (note that Onsager’s reciprocal relations are not needed in this work). We assume a one-dimensional case where the gradients in electrochemical potentials have axial directions. Furthermore we adopt the local equilibrium assumption of CIT. In Fig. 1 differential (infinitesimal) cells along the pore have been illustrated. Local equilibrium assumption means that there is an assumed equilibrium in each cell, however different in different cells, although the whole pore is not in equilibrium. This is a standard postulate in CIT. The ohmic resistivity of the solid material is assumed negligible.
3. Mathematical model equations 3.1. Transport in the core bulk solution The following Eq. (1) give the fluxes of the ions in the core bulk electrolyte space. In the first place we assume that only cations are desorbed/adsorbed and transported during charge/discharge i.e. we assume that the transport number for anions equals zero. In that way we obtain a simplest possible example useful for illustrating the principles of the theory. A more general treatise then follows. Since we assume that the concentration and inner potential are constant over the cross section in this core we obtain a one-dimensional equation. In the first place we assume for simplicity that concentrations are so small that the Nernst–Einstein equation is valid. oc1b o/ J 1b ¼ D1b c1b u1 ox ox D1 c1b RT oc1b o/ o~ l1 þF ð1Þ ¼ ¼ L1b c1b ox ox RT ox where index 1 refers to cation and index b to core bulk. Those equations relate the phenomenological coefficients (Lik) of classical irreversible thermodynamics to dilute theory transport equations for very dilute electrolyte solutions. The CIT driving force is the gradient of the electrochemical potential. For higher concentrations the same classical irreversible thermodynamics linearized formulation holds as long as the interaction between cations and anions may be neglected or, as we have assumed, only cations are transported. However, the relation between L1b and diffusivity seen in Eq. (1) is of course no longer valid. 3.2. Surface-related transport In the following equation the flux is calculated per the cross sectional pore space. This is the linearized irreversible thermodynamics transport equations for the cations adsorbed in the double layer (in the sense that they reside either adsorbed on the surface or in the diffuse layer) i.e. we assume that the electrochemical potential gradient is the driving force. Transport is taking place only in the axial direction of the pore since the electrochemical potential gradient in the radial direction is assumed to be negligible. o~ l1 J 1d ¼ L1d ð2Þ ox where index d refers to the double layer. 3.3. Resulting transport equation We now add the flow of cations in the bulk solution and the surface related flow both calculated per cross sectional area of the pore. Ab Ab o~ l1 o~ l1 J 1 ¼ J 1b ¼ L1 þ J 1d ¼ L1b þ L1d Apore Apore ox ox ð3Þ
P. Bjo¨rnbom / Electrochemistry Communications 9 (2007) 211–215
3.4. The equilibrium relation
J 1 ¼ L1
The assumed equilibrium relations between the Galvani potential in the external electrolyte and adsorbed amounts of cations and anions are of the same form as found by Kastening and Heins, 2005 [14], for 0.3 M KF (see Fig. 2). In the first place we also assume that our experiments occur in a potential interval on the negative side where only cations are transported. If the potential step used in our experiment is sufficiently small a linear relation between adsorbed amount of cations and Galvani potential of the external electrolyte may be assumed. The following equations are valid at equilibrium for a constant electrolyte concentration outside the pore, i.e. the amount of adsorbed cations is a unique linear function of the Galvani potential of the external electrolyte (like in Fig. 2). The amounts of cations are calculated as concentrations per volume of the pore, i.e. c1 ¼
213
o~ l1 L1 F oc1 o/ ¼ ¼ L1 F a1/ ox ox ox
ð5Þ
The current in the pore per cross sectional pore area is: j ¼ FJ 1 ¼ L1 F 2
o/ L1 F 2 oc1 ¼ ox a1/ ox
ð6Þ
Eqs. (5) and (6) tell us that the complex transport process of cations in the pore, after taking the local equilibrium assumption into consideration, appears both as a migration process with the Galvani potential gradient in the core bulk solution as the driving force and as a diffusion process with the gradient in the concentration of cations as the driving force. The apparent diffusivity according to Eq. (5) and the apparent conductivity according to Eq. (6) are: Dapp ¼
L1 F a1/
ð7Þ
japp ¼ L1 F 2
n1b þ n1d c1b V b þ n1d ¼ : V pore V pore
3.6. The combination of charge balance, mass balance, transport equations and local equilibrium relations
The following equations follow from the derivative of cation concentration in the pore with respect to potential, being constant. The derivative of electrochemical potential with Galvani potential is of course constant at a constant electrolyte concentration. Thus, for a given electrolyte concentration we have in equilibrium: dc1 dc1 d~ l1 1 dc1 a1/ ¼ ¼ ð4Þ ¼ d~ l1 d/ d/ F d/ F
The net charge in the pore calculated per volume of the pore is equal to the difference in the amounts per pore volume of cations and anions multiplied by Faraday’s constant: q ¼ ðc1 c2 ÞF ð8Þ Derivation of Eq. (8) with respect to time and combination with Eq. (4) gives (we assume that changes in the amount of anions may be neglected):
which also gives the definition of a1/.
oq oc1 o/ o/ C SPV oc1 ¼F ¼ C SPV ¼ ¼ a1/ F ot ot ot ot a1/ ot
3.5. The combination of transport equations with the local equilibrium assumption
ð9Þ
oq where the capacitance per volume of pore o/ ¼ C SPV ¼ a1/ F . The charge balance for the pore gives:
We now use the basic CIT assumption of local equilibrium. In the following equations we also assume that the relative change in electrolyte concentration in the core bulk solution of the pore may be neglected. This latter assumption will be further discussed below. Thus we combine Eqs. (3) and (4):
oq oj ¼ ot ox
ð10Þ
The mass balance for the cation gives: oc1 oJ 1 ¼ ot ox
ð11Þ
400 350
m/μmol g-1
300 250
K
+
F
-
200 150 100 50 0 -0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
E/V [vs she]
Fig. 2. Equilibrium contents of potassium and fluoride ions as a function of electrode potential in the micropores of activated carbon for an external aqueous electrolyte concentration of 0.3 M KF (adapted from Kastening and Heins, 2005 [14]).
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Combining Eqs. (10), (9) and (6) gives: 2
2
o/ L1 F o / ¼ ot C SPV ox2 oc1 L1 F 2 o2 c1 ¼ ot C SPV ox2
ð12Þ
Using Eq. (7) we may write: o/ japp o2 / ¼ ot C SPV ox2 ð13a; bÞ oc1 o2 c 1 ¼ Dapp 2 ot ox Eq. (13a) is an equation that would suggest that the transport of ions occur via migration while Eq. (13b) would suggest transport by diffusion. However, both equations were derived without even specifying to what extent ions are transported in one way or another. Thus we have a nonuniqueness of mathematical models for this physical situation. We may derive Eq. (13b), with the same expression for Dapp, by combining the mass balance (11) with the appropriate flux expression from Eq. (5). We also find that: japp Dapp ¼ ð14Þ C SPV Note that this equation is rather different from the relation between the diffusivity of an ion and the conductivity in a dilute electrolyte solution. This is so because Eq. (14) is the result of a complex process involving local phase equilibrium (i.e. Eq. (4)). 3.7. A more general case In a more general case both cations and anions would be adsorbed and desorbed during charge/discharge (cf. Fig. 2). Thus both ions would be transported. It appears that the corresponding derivations with the more general equations give basically similar results as before. Thus, for example, Eq. (2) has to be substituted with: o~ l1 o~ l2 J 1d ¼ L11d L12d ox ox ð15Þ o~ l2 o~ l1 L21d J 2d ¼ L22d ox ox Instead of Eq. (3) we obtain: Ab o~ l1 o~ l2 L12 þ J 1d ¼ L11 J 1 ¼ J 1b Apore ox ox Ab o~ l2 o~ l1 L21 þ J 21d ¼ L22 J 2 ¼ J 2b Apore ox ox
ð16Þ
The equilibrium relations now must include both ions: dc1 dc1 d~ l1 1 dc1 a1/ ¼ ¼ ¼ d~ l1 d/ d/ F d/ F ð17Þ dc2 dc2 d~ l2 1 dc2 a2/ ¼ ¼ ¼ F d/ d~ l2 d/ d/ F
The combined equation corresponding to Eq. (5) become: o~ l1 o~ l2 o/ L12 ¼ ðL11 L12 ÞF J 1 ¼ L11 ox ox ox ðL11 L12 ÞF oc1 ðL11 L12 ÞF oc2 ¼ ¼ a1u a2u ox ox ð18Þ o~ l2 o~ l1 o/ L21 ¼ ðL22 L21 ÞF J 2 ¼ L22 ox ox ox ðL22 L21 ÞF oc2 ðL22 L21 ÞF oc1 ¼ ¼ a2u a1u ox ox and j ¼ ðJ 1 J 2 ÞF ¼ ðL11 þ L22 L12 L21 ÞF 2 ¼ ¼
o/ ox
ðL11 þ L22 L12 L21 ÞF 2 oc1 a1/ ox
ðL11 þ L22 L12 L21 ÞF 2 oc2 a2/ ox
ð19Þ
The final equations, found by combing the charge balance with Eq. (19), take similar forms: o/ japp o2 / ¼ ot C SPV ox2 oc1 o2 c 1 ¼ Dapp 2 ot ox oc2 o2 c 2 ¼ Dapp 2 ot ox
ð20a; b; cÞ
where CSPV = (a1/ + a2/)F Note that the same apparent diffusivity is valid for both ions in using Eq. (20). Eq. (14) is still valid while the apparent diffusivity and the apparent conductivity to be used in Eq. (20) are: Dapp ¼
ðL11 þ L22 L12 L21 ÞF a1/ þ a2/
japp ¼ ðL11 þ L22 L12 L21 ÞF
ð21Þ 2
Since Eq. (20b) and (20c) may also be derived by combining the mass balances for the ions with the appropriate flux equations according to Eq. (18) we arrive at the following interesting relations: Dapp ¼ ¼
ðL11 þ L22 L12 L21 ÞF ðL11 L12 ÞF ¼ a1/ þ a2/ a1/ ðL22 L21 ÞF a2/
ð22Þ
4. Discussion 4.1. Local equilibrium assumption In our derivations the local equilibrium assumption enhanced with the assumption that the relative change in
P. Bjo¨rnbom / Electrochemistry Communications 9 (2007) 211–215
electrolyte concentration in the core bulk solution of the pore may be neglected. Without this assumption Eq. (4) would not be valid locally and we would not be able to obtain our result regarding the non-uniqueness of mathematical models. Consider the core bulk volume (excluding the diffuse double layer of the pore) of the local cells illustrated in Fig. 1. In each such volume we assume uniform conditions with respect to concentrations and Galvani potential. The transfer of cations into the double layer results in a change of net charge in the local cell that is related to the Galvani potential. The Galvani potential is very sensitive due to change in net charge. Thus even a small transfer of cations into the double layer will cause a large driving force for cations to enter the local cell from the neighbour cell and for anions to leave the cell. The net charge caused by the transfer of cations into the double layer will therefore quickly be compensated for except for a small part corresponding to a final change in Galvani potential. This is the potential change corresponding to the new equilibrium state due to the change of amount of cations in the double layer. In that way Galvani potential gradients are built up along the pore. This includes that the potential satisfies the equilibrium relation with the concentration of adsorbed cations. Depending on the transport numbers there will be some change of the electrolyte concentration in the observed cell. If all current would be transferred by cations no change would occur while the greatest change would occur if all current would be transferred by anions. However, if we consider only small potential changes (e.g. a small potential step) the change in electrolyte concentration would be negligible in most cases, especially if high external electrolyte concentrations are used (e.g. 6 M KOH as used in supercapacitor applications). This means that the local equilibrium relation should remain the same as for the whole pore for the given electrolyte concentration outside the pore mouth. 4.2. Small micropores So far we have discussed pores that are wide enough that the diffuse double layer at opposite parts of the pore wall far from overlap. In the case of small micropores there may be no core bulk solutions such that the electrolyte concentration of which would be approximately equal to the external electrolyte concentration as in the previous cases. However, if we gradually decrease the pore size the diffuse double layer will overlap gradually more and more (although, as noted above, this would rarely happen for such high electrolyte concentrations as usually used in electrochemical supercapacitors). At equilibrium the concentration of ions at the centre of the pore would be different from the concentrations in the external electrolyte
215
but there is no reason why an argument about negligible concentration changes could not still be applied as long as the pore volume may be treated as a continuum. 5. Conlusions On the basis of fundamental principles of irreversible thermodynamics several equivalent mathematical model equations could be derived from one and the same assumed physical situation in the pore. Some equations appear as if they were derived from a migrational mechanism with ohmic resistance in the pore electrolyte as the main transport parameter. Other equations appear as if they were derived from a diffusional mechanism with diffusivities as the main transport parameters. However, the apparent diffusivities and conductivities in those equations are not diffusivities or conductivities in the ordinary sense as for a common electrolyte solution. As a consequence transport parameters obtained as ohmic resistivities and as apparent diffusivities characterize identical physical processes and can be directly compared using Eq. (14). The results are valid as long as potential changes are small and slow enough that the local equilibrium assumption and the linearization of the flux equations of irreversible thermodynamics remain valid. References [1] B.E. Conway, Electrochemical Supercapacitors, Kluwer Academic, N.Y., 1999. [2] E. Frackowiak, F. Be´guin, Carbon 39 (2001) 937. [3] H. Shi, Electrochim. Acta 41 (1996) 1633. [4] D. Qu, H. Shi, J. Power Sources 74 (1998) 99. [5] M. Zuleta, Electrochemical and ion transport characterisation of a nanoporous carbon derived from SiC. Ph.D Dissertation, KTH, Stockholm 2005. [6] M. Eikerling, A.A. Kornyshev, E. Lust, J. Electrochem. Soc. 152 (2005) E24. [7] R. De Levie, Electrochim. Acta 8 (1963) 751. [8] M. Zuleta, M. Bursell, P. Bjo¨rnbom, A. Lundblad, J. Electroanal. Chem. 549 (2003) 101. [9] M. Zuleta, A. Lundblad, P. Bjo¨rnbom, G. Nurk, H. Kasuk, E. Lust, J. Electroanal. Chem. 586 (2006) 247–259. [10] M. Zuleta, P. Bjo¨rnbom, A. Lundblad, J. Electrochem. Soc. 153 (2006) A48. [11] H. Malmberg, M. Zuleta, A. Lundblad, P. Bjo¨rnbom, J. Electrochem. Soc. 153 (2006) A1914. [12] B.E. Conway, Electrochemical Supercapacitors, Kluwer Academic, N.Y, 1999, p. 163. [13] J. Koryta, J. Dvorak, Principles of Electrochemistry, John Wiley & Sons, Chichester, 1987, Chapter. 2.. [14] B. Kastening, M. Heins, Elctrochim. Acta 50 (2005) 2487–2498.
Charge/discharge of an electrochemical supercapacitor electrode pore; non-uniqueness of mathematical models Pehr Bjo¨rnbom
*
Department of Chemical Engineering and Technology, KTH – Royal Institute of Technology, SE-10044 Stockholm, Sweden Received 17 August 2006; received in revised form 30 August 2006; accepted 8 September 2006 Available online 13 October 2006
Abstract A thermodynamic analysis has been done to enhance understanding of the relation between various mathematical models for electrochemical supercapacitor pores. For the same capacitive charge/discharge experiment a variety of one-dimensional mathematical model equations concerning the transport of ions and double layer charge/discharge along the pore are shown to be indistinguishable. Some of those indistinguishable equations could be interpreted as derived from diffusional mechanisms while others appear as derived from migrational mechanisms. Ohmic resistivities and diffusivities obtained in such case are not contradicting results but characterize identical physical processes. The results are valid as long as the assumptions of irreversible thermodynamics of local equilibrium along the pore and of linearization of the flux equations hold. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Single pore model; Mathematical model; Irreversible thermodynamics; Ionic diffusion; Ionic migration; Electrochemical supercapacitor
1. Introduction Electrodes for electrochemical double layer capacitors (EDLCs) or electrochemical supercapacitors typically consist of porous materials such as carbon [1]. The transport properties for ions in the pores are essential for the performance of supercapacitor electrodes [1–6]. Transport parameters may be obtained from dynamic voltammetry experiments, especially potential steps and impedance spectroscopy. Such experiments are commonly evaluated using mathematical models based on principles originating from de Levie’s seminal paper from 1963 [1,7] where the main transport resistance is assumed to be the ohmic resistance to ionic migration in the pore electrolyte solution. However, recent work has shown that potential step experiments also may be successfully evaluated using models based on Fickian diffusion of ions in the pores
*
Tel.: +46 8790 8255. E-mail address: [email protected].
1388-2481/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.elecom.2006.09.009
with effective diffusivities as the main transport parameters [5,8–11]. The purpose of this work is to show that there is no contradiction in this situation. Such voltammetry experiments under frequently used experimental conditions cannot give unique information on the transport mechanisms for the ions. The fundamental thermodynamic analysis in this paper shows that for the same experiment a variety of mathematical model equations are equally valid. Some of those equivalently valid equations could be interpreted as derived from diffusional mechanisms while others appear as derived from migrational mechanisms. 2. Model assumptions 2.1. The model pore at equilibrium We assume a straight cylindrical pore with walls of an electrically conducting material e.g. carbon (see Fig. 1). The pore is filled with a 1,1-electrolyte solution, typically KOH or KF in water. The diameter of the pore is assumed to be substantially greater than twice the Debye
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P. Bjo¨rnbom / Electrochemistry Communications 9 (2007) 211–215
At equilibrium the electrolyte concentration and Galvani potential have the same values in the core bulk as outside the pore Double layer
Core bulk solution
Differential cells
At non-equilibrium the electrochemical potential has the same value all over each cell due to the local equilibrium assumption but differ between cells due to the axial gradient (= the driving force) Fig. 1. A schematic picture of a model pore.
length of the electrolyte. As a consequence there is a central part of the pore solution with constant ionic concentrations at equilibrium, while close to the pore wall, within the diffuse part of the double layer, concentrations vary with the radial coordinate. At typical electrolyte concentrations used for electrochemical supercapacitors the diffuse layer is collapsed and only 0.5 nm in thickness [12]. At equilibrium the Galvani inner potential of the electrolyte is constant within the central core of the pore solution while it varies with the radial coordinate in the diffuse part of the double layer. On the other hand at equilibrium the electrochemical potentials of the ions have the same value everywhere in the system including the diffuse double layer and ions adsorbed on the surface. The values of the Galvani potential, the electrochemical potentials and the concentrations in the central core of the pore and the corresponding values in the external solution outside the pore mouth are equal. The equilibrium relation between Galvani potential and concentrations of ions in the pore is discussed below. 2.2. The model pore during charge/discharge At non-equilibrium conditions we assume that the driving forces for the flow of ions are the gradients in the electrochemical potentials and that linearized flux equations according to classical irreversible thermodynamics (CIT) are valid (see e.g. [13]), i.e. flux equations like Eq. (16) below (note that Onsager’s reciprocal relations are not needed in this work). We assume a one-dimensional case where the gradients in electrochemical potentials have axial directions. Furthermore we adopt the local equilibrium assumption of CIT. In Fig. 1 differential (infinitesimal) cells along the pore have been illustrated. Local equilibrium assumption means that there is an assumed equilibrium in each cell, however different in different cells, although the whole pore is not in equilibrium. This is a standard postulate in CIT. The ohmic resistivity of the solid material is assumed negligible.
3. Mathematical model equations 3.1. Transport in the core bulk solution The following Eq. (1) give the fluxes of the ions in the core bulk electrolyte space. In the first place we assume that only cations are desorbed/adsorbed and transported during charge/discharge i.e. we assume that the transport number for anions equals zero. In that way we obtain a simplest possible example useful for illustrating the principles of the theory. A more general treatise then follows. Since we assume that the concentration and inner potential are constant over the cross section in this core we obtain a one-dimensional equation. In the first place we assume for simplicity that concentrations are so small that the Nernst–Einstein equation is valid. oc1b o/ J 1b ¼ D1b c1b u1 ox ox D1 c1b RT oc1b o/ o~ l1 þF ð1Þ ¼ ¼ L1b c1b ox ox RT ox where index 1 refers to cation and index b to core bulk. Those equations relate the phenomenological coefficients (Lik) of classical irreversible thermodynamics to dilute theory transport equations for very dilute electrolyte solutions. The CIT driving force is the gradient of the electrochemical potential. For higher concentrations the same classical irreversible thermodynamics linearized formulation holds as long as the interaction between cations and anions may be neglected or, as we have assumed, only cations are transported. However, the relation between L1b and diffusivity seen in Eq. (1) is of course no longer valid. 3.2. Surface-related transport In the following equation the flux is calculated per the cross sectional pore space. This is the linearized irreversible thermodynamics transport equations for the cations adsorbed in the double layer (in the sense that they reside either adsorbed on the surface or in the diffuse layer) i.e. we assume that the electrochemical potential gradient is the driving force. Transport is taking place only in the axial direction of the pore since the electrochemical potential gradient in the radial direction is assumed to be negligible. o~ l1 J 1d ¼ L1d ð2Þ ox where index d refers to the double layer. 3.3. Resulting transport equation We now add the flow of cations in the bulk solution and the surface related flow both calculated per cross sectional area of the pore. Ab Ab o~ l1 o~ l1 J 1 ¼ J 1b ¼ L1 þ J 1d ¼ L1b þ L1d Apore Apore ox ox ð3Þ
P. Bjo¨rnbom / Electrochemistry Communications 9 (2007) 211–215
3.4. The equilibrium relation
J 1 ¼ L1
The assumed equilibrium relations between the Galvani potential in the external electrolyte and adsorbed amounts of cations and anions are of the same form as found by Kastening and Heins, 2005 [14], for 0.3 M KF (see Fig. 2). In the first place we also assume that our experiments occur in a potential interval on the negative side where only cations are transported. If the potential step used in our experiment is sufficiently small a linear relation between adsorbed amount of cations and Galvani potential of the external electrolyte may be assumed. The following equations are valid at equilibrium for a constant electrolyte concentration outside the pore, i.e. the amount of adsorbed cations is a unique linear function of the Galvani potential of the external electrolyte (like in Fig. 2). The amounts of cations are calculated as concentrations per volume of the pore, i.e. c1 ¼
213
o~ l1 L1 F oc1 o/ ¼ ¼ L1 F a1/ ox ox ox
ð5Þ
The current in the pore per cross sectional pore area is: j ¼ FJ 1 ¼ L1 F 2
o/ L1 F 2 oc1 ¼ ox a1/ ox
ð6Þ
Eqs. (5) and (6) tell us that the complex transport process of cations in the pore, after taking the local equilibrium assumption into consideration, appears both as a migration process with the Galvani potential gradient in the core bulk solution as the driving force and as a diffusion process with the gradient in the concentration of cations as the driving force. The apparent diffusivity according to Eq. (5) and the apparent conductivity according to Eq. (6) are: Dapp ¼
L1 F a1/
ð7Þ
japp ¼ L1 F 2
n1b þ n1d c1b V b þ n1d ¼ : V pore V pore
3.6. The combination of charge balance, mass balance, transport equations and local equilibrium relations
The following equations follow from the derivative of cation concentration in the pore with respect to potential, being constant. The derivative of electrochemical potential with Galvani potential is of course constant at a constant electrolyte concentration. Thus, for a given electrolyte concentration we have in equilibrium: dc1 dc1 d~ l1 1 dc1 a1/ ¼ ¼ ð4Þ ¼ d~ l1 d/ d/ F d/ F
The net charge in the pore calculated per volume of the pore is equal to the difference in the amounts per pore volume of cations and anions multiplied by Faraday’s constant: q ¼ ðc1 c2 ÞF ð8Þ Derivation of Eq. (8) with respect to time and combination with Eq. (4) gives (we assume that changes in the amount of anions may be neglected):
which also gives the definition of a1/.
oq oc1 o/ o/ C SPV oc1 ¼F ¼ C SPV ¼ ¼ a1/ F ot ot ot ot a1/ ot
3.5. The combination of transport equations with the local equilibrium assumption
ð9Þ
oq where the capacitance per volume of pore o/ ¼ C SPV ¼ a1/ F . The charge balance for the pore gives:
We now use the basic CIT assumption of local equilibrium. In the following equations we also assume that the relative change in electrolyte concentration in the core bulk solution of the pore may be neglected. This latter assumption will be further discussed below. Thus we combine Eqs. (3) and (4):
oq oj ¼ ot ox
ð10Þ
The mass balance for the cation gives: oc1 oJ 1 ¼ ot ox
ð11Þ
400 350
m/μmol g-1
300 250
K
+
F
-
200 150 100 50 0 -0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
E/V [vs she]
Fig. 2. Equilibrium contents of potassium and fluoride ions as a function of electrode potential in the micropores of activated carbon for an external aqueous electrolyte concentration of 0.3 M KF (adapted from Kastening and Heins, 2005 [14]).
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P. Bjo¨rnbom / Electrochemistry Communications 9 (2007) 211–215
Combining Eqs. (10), (9) and (6) gives: 2
2
o/ L1 F o / ¼ ot C SPV ox2 oc1 L1 F 2 o2 c1 ¼ ot C SPV ox2
ð12Þ
Using Eq. (7) we may write: o/ japp o2 / ¼ ot C SPV ox2 ð13a; bÞ oc1 o2 c 1 ¼ Dapp 2 ot ox Eq. (13a) is an equation that would suggest that the transport of ions occur via migration while Eq. (13b) would suggest transport by diffusion. However, both equations were derived without even specifying to what extent ions are transported in one way or another. Thus we have a nonuniqueness of mathematical models for this physical situation. We may derive Eq. (13b), with the same expression for Dapp, by combining the mass balance (11) with the appropriate flux expression from Eq. (5). We also find that: japp Dapp ¼ ð14Þ C SPV Note that this equation is rather different from the relation between the diffusivity of an ion and the conductivity in a dilute electrolyte solution. This is so because Eq. (14) is the result of a complex process involving local phase equilibrium (i.e. Eq. (4)). 3.7. A more general case In a more general case both cations and anions would be adsorbed and desorbed during charge/discharge (cf. Fig. 2). Thus both ions would be transported. It appears that the corresponding derivations with the more general equations give basically similar results as before. Thus, for example, Eq. (2) has to be substituted with: o~ l1 o~ l2 J 1d ¼ L11d L12d ox ox ð15Þ o~ l2 o~ l1 L21d J 2d ¼ L22d ox ox Instead of Eq. (3) we obtain: Ab o~ l1 o~ l2 L12 þ J 1d ¼ L11 J 1 ¼ J 1b Apore ox ox Ab o~ l2 o~ l1 L21 þ J 21d ¼ L22 J 2 ¼ J 2b Apore ox ox
ð16Þ
The equilibrium relations now must include both ions: dc1 dc1 d~ l1 1 dc1 a1/ ¼ ¼ ¼ d~ l1 d/ d/ F d/ F ð17Þ dc2 dc2 d~ l2 1 dc2 a2/ ¼ ¼ ¼ F d/ d~ l2 d/ d/ F
The combined equation corresponding to Eq. (5) become: o~ l1 o~ l2 o/ L12 ¼ ðL11 L12 ÞF J 1 ¼ L11 ox ox ox ðL11 L12 ÞF oc1 ðL11 L12 ÞF oc2 ¼ ¼ a1u a2u ox ox ð18Þ o~ l2 o~ l1 o/ L21 ¼ ðL22 L21 ÞF J 2 ¼ L22 ox ox ox ðL22 L21 ÞF oc2 ðL22 L21 ÞF oc1 ¼ ¼ a2u a1u ox ox and j ¼ ðJ 1 J 2 ÞF ¼ ðL11 þ L22 L12 L21 ÞF 2 ¼ ¼
o/ ox
ðL11 þ L22 L12 L21 ÞF 2 oc1 a1/ ox
ðL11 þ L22 L12 L21 ÞF 2 oc2 a2/ ox
ð19Þ
The final equations, found by combing the charge balance with Eq. (19), take similar forms: o/ japp o2 / ¼ ot C SPV ox2 oc1 o2 c 1 ¼ Dapp 2 ot ox oc2 o2 c 2 ¼ Dapp 2 ot ox
ð20a; b; cÞ
where CSPV = (a1/ + a2/)F Note that the same apparent diffusivity is valid for both ions in using Eq. (20). Eq. (14) is still valid while the apparent diffusivity and the apparent conductivity to be used in Eq. (20) are: Dapp ¼
ðL11 þ L22 L12 L21 ÞF a1/ þ a2/
japp ¼ ðL11 þ L22 L12 L21 ÞF
ð21Þ 2
Since Eq. (20b) and (20c) may also be derived by combining the mass balances for the ions with the appropriate flux equations according to Eq. (18) we arrive at the following interesting relations: Dapp ¼ ¼
ðL11 þ L22 L12 L21 ÞF ðL11 L12 ÞF ¼ a1/ þ a2/ a1/ ðL22 L21 ÞF a2/
ð22Þ
4. Discussion 4.1. Local equilibrium assumption In our derivations the local equilibrium assumption enhanced with the assumption that the relative change in
P. Bjo¨rnbom / Electrochemistry Communications 9 (2007) 211–215
electrolyte concentration in the core bulk solution of the pore may be neglected. Without this assumption Eq. (4) would not be valid locally and we would not be able to obtain our result regarding the non-uniqueness of mathematical models. Consider the core bulk volume (excluding the diffuse double layer of the pore) of the local cells illustrated in Fig. 1. In each such volume we assume uniform conditions with respect to concentrations and Galvani potential. The transfer of cations into the double layer results in a change of net charge in the local cell that is related to the Galvani potential. The Galvani potential is very sensitive due to change in net charge. Thus even a small transfer of cations into the double layer will cause a large driving force for cations to enter the local cell from the neighbour cell and for anions to leave the cell. The net charge caused by the transfer of cations into the double layer will therefore quickly be compensated for except for a small part corresponding to a final change in Galvani potential. This is the potential change corresponding to the new equilibrium state due to the change of amount of cations in the double layer. In that way Galvani potential gradients are built up along the pore. This includes that the potential satisfies the equilibrium relation with the concentration of adsorbed cations. Depending on the transport numbers there will be some change of the electrolyte concentration in the observed cell. If all current would be transferred by cations no change would occur while the greatest change would occur if all current would be transferred by anions. However, if we consider only small potential changes (e.g. a small potential step) the change in electrolyte concentration would be negligible in most cases, especially if high external electrolyte concentrations are used (e.g. 6 M KOH as used in supercapacitor applications). This means that the local equilibrium relation should remain the same as for the whole pore for the given electrolyte concentration outside the pore mouth. 4.2. Small micropores So far we have discussed pores that are wide enough that the diffuse double layer at opposite parts of the pore wall far from overlap. In the case of small micropores there may be no core bulk solutions such that the electrolyte concentration of which would be approximately equal to the external electrolyte concentration as in the previous cases. However, if we gradually decrease the pore size the diffuse double layer will overlap gradually more and more (although, as noted above, this would rarely happen for such high electrolyte concentrations as usually used in electrochemical supercapacitors). At equilibrium the concentration of ions at the centre of the pore would be different from the concentrations in the external electrolyte
215
but there is no reason why an argument about negligible concentration changes could not still be applied as long as the pore volume may be treated as a continuum. 5. Conlusions On the basis of fundamental principles of irreversible thermodynamics several equivalent mathematical model equations could be derived from one and the same assumed physical situation in the pore. Some equations appear as if they were derived from a migrational mechanism with ohmic resistance in the pore electrolyte as the main transport parameter. Other equations appear as if they were derived from a diffusional mechanism with diffusivities as the main transport parameters. However, the apparent diffusivities and conductivities in those equations are not diffusivities or conductivities in the ordinary sense as for a common electrolyte solution. As a consequence transport parameters obtained as ohmic resistivities and as apparent diffusivities characterize identical physical processes and can be directly compared using Eq. (14). The results are valid as long as potential changes are small and slow enough that the local equilibrium assumption and the linearization of the flux equations of irreversible thermodynamics remain valid. References [1] B.E. Conway, Electrochemical Supercapacitors, Kluwer Academic, N.Y., 1999. [2] E. Frackowiak, F. Be´guin, Carbon 39 (2001) 937. [3] H. Shi, Electrochim. Acta 41 (1996) 1633. [4] D. Qu, H. Shi, J. Power Sources 74 (1998) 99. [5] M. Zuleta, Electrochemical and ion transport characterisation of a nanoporous carbon derived from SiC. Ph.D Dissertation, KTH, Stockholm 2005. [6] M. Eikerling, A.A. Kornyshev, E. Lust, J. Electrochem. Soc. 152 (2005) E24. [7] R. De Levie, Electrochim. Acta 8 (1963) 751. [8] M. Zuleta, M. Bursell, P. Bjo¨rnbom, A. Lundblad, J. Electroanal. Chem. 549 (2003) 101. [9] M. Zuleta, A. Lundblad, P. Bjo¨rnbom, G. Nurk, H. Kasuk, E. Lust, J. Electroanal. Chem. 586 (2006) 247–259. [10] M. Zuleta, P. Bjo¨rnbom, A. Lundblad, J. Electrochem. Soc. 153 (2006) A48. [11] H. Malmberg, M. Zuleta, A. Lundblad, P. Bjo¨rnbom, J. Electrochem. Soc. 153 (2006) A1914. [12] B.E. Conway, Electrochemical Supercapacitors, Kluwer Academic, N.Y, 1999, p. 163. [13] J. Koryta, J. Dvorak, Principles of Electrochemistry, John Wiley & Sons, Chichester, 1987, Chapter. 2.. [14] B. Kastening, M. Heins, Elctrochim. Acta 50 (2005) 2487–2498.