# Membrane Potential Generated by Ion Adsorption

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Section

#### 2.1. Potential across the Anion Exchange Membrane

_{L}while that of right solution is represented by C

_{R}.

**Figure 1.**Experimental setup for measuring the potential between two KCl aqueous solutions separated through the intermediary of (

**a**) a anion exchange membrane; (

**b**) an Ag wire; and (

**c**) a surface-modified glassy carbon plate KCl concentrations in the left and right compartments are represented by C

_{L}and C

_{R}, respectively, for all the setups.

#### 2.2. Potential across the Ag Wire Coated with AgCl

#### 2.3. Potential across the Glassy Carbon Plate Covered with –COOH Groups

_{3}and H

_{2}SO

_{4}weight ratio 1:3 [14,15]. This process creates –COOH groups on the surface of glassy carbon. Three types of surface-modified glassy carbon plates were prepared and they are designated as GC1, GC2 and GC3 (see below), respectively.

## 3. Results and Theoretical Analysis

#### 3.1. Potential Generation across the Anion Exchange Membrane

_{L}was increased from 10

^{−5}M to 3.4 M, while that of C

_{R}was maintained constant at 0.1 M. According to the current membrane theory, the membrane potential is given by Equation (1), which is based on the GHK Equation.

_{K}and P

_{Cl}represent the permeability of membrane to K

^{+}and Cl

^{−}, respectively; [K

^{+}]

_{L}and [K

^{+}]

_{R}represent the K

^{+}concentration in the left and right compartments, respectively; and [Cl

^{−}]

_{L}and [Cl

^{−}]

_{R}represent the Cl

^{−}concentration in the left and right compartments, respectively; R and F represent gas constant and Faraday constant, respectively.

^{+}, Equation (1) is reduced into Equation (2).

^{−}]

_{R}/[Cl

^{−}]

_{L}. Since [Cl

^{−}]

_{R}= C

_{R}and [Cl

^{−}]

_{L}= C

_{L}, [Cl

^{−}]

_{R}/[Cl

^{−}]

_{L}= C

_{R}/C

_{L}.

_{10}([Cl

^{−}]

_{R}/[Cl

^{−}]

_{L}). The computational potential data was obtained by considering ion activity of KCl [16], while the its horizontal axis quantities were all computed under the assumption that the activity coefficients were 1 so that all the potential data in Figure 3 can be directly compared one another.

**Figure 3.**Potential generated between two KCl solutions vs. −log([Cl

^{−}]

_{R}/[Cl

^{−}]

_{L}). ○: an anion exchange membrane separator; ×: Ag wire separator; •: potential computed by employing the Goldman-Hodgkin-Katz (GHK) equation.

_{L}. However, the deviation of each experimental potential data from the average potential in Figure 3 is not so large. Potential with the standard deviation was −141 ± 8 mV even when C

_{L}was as low as 10

^{−5}M. Another potential, when C

_{L}was 3.4 M, was 72 ± 1 mV.

_{10}([Cl

^{−}]

_{R}/[Cl

^{−}]

_{L}), they are still in good agreement with each other. As long as the concept of the GHK equation is valid, the discrepancy could be amended by considering the concentration of ions not taken into account for the potential computation of Figure 3. Namely, the KCl solution contains low level of protons and hydroxide. Those ions could affect on the potential generation, especially at the low level C

_{L}. Hereafter, the experimentally observed potential across the anion exchange membrane shown in Figure 3 is designated as V

_{aem}, and the computational data set shown in Figure 3 is designated as V

_{com}.

#### 3.2. Potential Generation across the Ag Wire

#### 3.2.1. Experimental

_{Ag}. V

_{Ag}behavior is quite similar to V

_{aem}and even in better agreement with V

_{com}. Similarly to the procedure of potential measurements across the anion exchange membrane described in the previous section, once the potential reached the stable potential, experimental potential data was obtained. Compared with the potential measurement described in the previous section, the potential was quite stable and such stable potential was achieved much shorter period after supplying two KCl solutions in the left and right compartments of setup illustrated in Figure 1b. Measurements were carried out a number of times in order to assure the data reproducibility of potential. Hence, V

_{Ag}in Figure 3 is average potential. There was a trend that the potential reproducibility deteriorated gradually with the decrease in C

_{L}, which is similar trend to the trend described in the previous section. However, the deviation of each experimental potential data from the average potential becomes less compared with those of the potential data V

_{aem}, and the deviation of most of the V

_{Ag}from the average potential shown in Figure 3 is within the range of 6 mV. Potential with the standard deviation was −200 ± 6 mV even when C

_{L}was as low as 10

^{−5}M. Another potential, when C

_{L}was 3.4 M was 91 ± 1 mV.

_{Ag}shown in Figure 3.

_{Ag}considering ion adsorption on the surface of AgCl coat on the wire.

#### 3.2.2. Theoretical Analysis

^{−}adsorbs onto AgCl. Hence, we assume that AgCl at the ends of Ag wire illustrated in Figure 1b selectively adsorbs Cl

^{−}. Under this assumption, we derived an Equation explaining the potential behavior of V

_{Ag}. Figure 4 is an illustration at the interface between AgCl coat at the Ag wire end and KCl solution with a V,x-coordinate system, where V and x represent the potential and the distance from the adsorption site of AgCl surface, respectively. Some chloride ions are in the state of adsorption on the AgCl surface. Adsorption site s (see Figure 4) associates with Cl

^{−}. Hence, Equations (3)–(5) are derived using the concept of Langmuir isotherm, where K is an association constant.

**Figure 4.**The interface between AgCl coat at the Ag wire end and KCl solution. s represents the adsorption site on the AgCl surface. Some chloride ions are in the adsorbed state on the AgCl surface. Coordinate system is set as illustrated, where x = 0 represents the interface plane between the AgCl surface and KCl solution phase and x = d represents the plane of ion diffuse phase closest to the adsorption site of AgCl. V represents the potential, where V = 0 at x = ∞. Dotted line represents the potential behavior expected.

_{T}is given by Equation (4).

_{T}= [s] + [sCl

^{−}]

^{−}is given by Equation (5) using Equations (3) and (4).

^{−}] =

_{o}: concentration of K

^{+}and Cl

^{−}at x = ∞, F: Faraday constant, V: potential, R: gas constant, T: absolute temperature of experimental environment.

^{−}] + = 0

_{o}: vacuum permittivity, ɛ: relative permittivity of water.

^{−}] one another. Hence, [Cl

^{−}] in Equation (11) can be given as C

_{o}. Therefore, Equation (11) is written as Equation (12).

_{sur}. Hence, Equation (13) establishes.

_{sur}

_{sur}is given by Equation (14) owing to the Gauss’s law, and ɛ

_{low}in the Equation (14) represents the permittivity of water within the range 0 < x < d. ɛ

_{low}is expected to be much lower than ɛ due to the dielectric saturation of water [1].

_{sur}=

_{L}(x = ∞) and ϕ

_{R}(x = ∞), respectively, and the potential difference ∆ϕ defined by ∆ϕ = ϕ

_{L}(x = ∞) − ϕ

_{R}(x = ∞) corresponds to the potential we experimentally measured (see Figure 5). ϕ

_{L}(x = ∞) and ϕ

_{R}(x = ∞) are given by Equations (16) and (17), respectively.

_{L}(x = ∞) − ϕ

_{R}(x = ∞) gives us the theoretically expected potential based on the adsorption theory that the membrane potential is generated by the ion adsorption on the membrane (in concrete terms Ag wire separator) surface. By the use of Langmuir isotherm and Poisson-Boltzmann equation, the membrane potential is analytically given by ∆ϕ as so far described. Except for a quite small number of researchers like Drs. Ling and Cheng, almost no researchers have proposed the evaluation of membrane potential by employing such theories—Langmuir isotherm and Poisson-Boltzmann equation—instead of using GHK equation [1,2,9,10,11,12,13]. However, the basically same idea—potential evaluation by employing Langmuir isotherm and Poisson-Boltzmann equation—has been often used in the research fields outside the membrane potential research field for quantitatively estimating the potential behavior in electrolytic solution around charged surface [17,18,19,20]. Moreover, the quantitative potential evaluation around the charged surface in aqueous solution primarily using the Poisson-Boltzmann equation has been very common research for more than several decades [17,21,22,23,24,25,26,27,28,29,30,31,32]. Hence, the foundation of the theory the authors propose here has been already widely accepted and well discussed outside the research field of membrane potential for decades. However, such a well-established concept has never replaced the GHK equation.

_{Ag}was measured by increasing C

_{L}from 10

^{−5}M to 3.4 M, while maintaining C

_{R}at 0.1 M. By the use of same condition, theoretically expected potential was obtained by computing ∆ϕ. For the computation, four factors should have been determined experimentally but were unable to be determined. They were [s]

_{T}, K, d and ɛ

_{low}. Hence, we assumed quite plausible quantities for them as follows: [s]

_{T}was assumed to be 1.56 × 10

^{18}m

^{−2}by considering molecular dimension. It was guessed that K is quite low, since the experimentally obtained potential did not saturate by the increase of C

_{L}even up to the highest concentration as seen in Figure 3. Thus, K was assumed to be 1 × 10

^{−25}m

^{−3}. d was assumed to be 5 Å by considering interfacial structure illustrated in Figure 4. We could not find any decisive literature for determine ɛ

_{low}. However, some literature shows that hydrated ions permittivity is only a bit lower than the relative permittivity of water [33,34,35]. Hence, we chose 60 as ɛ

_{low}. The results are shown in Figure 6. Potential represented by × in Figure 6 was obtained by replotting the potential in Figure 3 represented by the mark × as a function of −log

_{10}C

_{L}. Computational potential represented by □ in Figure 6 represents ∆ϕ vs. log

_{10}C

_{L}and the computational potential was obtained by considering ion activity of KCl [16], while the its horizontal axis quantities were all computed under the assumption that the activity coefficients were 1 just like Figure 3. Although the computational potential did not take into consideration the permeability of Ag wire to ions unlike the GHK equation, it well agrees with the experimental potential.

#### 3.3. Potential Generation across the Glassy Carbon Plate

#### 3.3.1. Experimental

_{10}[K

^{+}]

_{L}(=log

_{10}C

_{L}) as shown in Figure 7. Similarly to the procedure of potential measurements so far described, once the potential reached the stable potential, experimental potential data was obtained. Compared with the potential measurement described in the previous sections, it took a quite long time until the potential reached stable state. It took 4–6 h for achieving the stable potential after supplying two KCl solutions in the left and right compartments of setup illustrated in Figure 1c. The reason for such a slow realization of stable potential could be due to the rough surface of the surface-modified glassy carbon plate. Due to the rough surface of surface-modified glassy carbon plate, it took several hours until the ions migrated into the deep area of surface of the surface-modified glassy carbon plate. Because of such a slow process of stable potential generation, we were unable to obtain a large number of potential data which was statistically meaningful. Therefore, we paid careful attention to the experimental setup while carrying out the measurements. For example, we managed to prevent the solution evaporation from both left and right compartments of setup (Figure 1c), carefully handled the setup in order to prevent any mechanical impact on the experimental setup, monitored the environmental temperature carefully, and so on. Hereafter, the potential data set is designated as V

_{GC}. We take log

_{10}[K

^{+}]

_{L}as x-axis of diagram in Figure 7 for the theoretical analysis of potential behavior to be described.

**Figure 6.**×: potential generated between two KCl solutions using a separator of Ag wire vs. log

_{10}C

_{L}(=log

_{10}[Cl

^{−}]

_{L}); □: computational potential ∆ϕ based on the adsorption theory vs. log

_{10}C

_{L}(=log

_{10}[Cl

^{−}]

_{L}); Data represented by • in this diagram is created by rearranging the data in Figure 3 represented by the same mark •.

**Figure 7.**Potential generated between two KCl solutions using a separator of GC1 (GC2, GC3) vs. log

_{10}C

_{L}(=log

_{10}[K

^{+}]

_{L}), where offset potentials are added to some data for decongesting the diagram. Open and closed marks represent the experimental and computational potentials, respectively. Circle marks: in case a GC1 used Triangle marks: in case a GC2 separator used (+25 mV offset potential is added to the actual data) Square marks: in case a GC3 separator used (+50 mV offset potential is added to the actual data).

#### 3.3.2. Theoretical Analysis

^{+}and H

^{+}are in the adsorbed state to –COO

^{−}. Equations (18) and (19) are derived for the dissociation of –COOH and that of –COOK, where K

_{h}and K

_{k}are dissociation constants for them, respectively.

**Figure 8.**Interface between the surface-modified glassy carbon plate and KCl solution. Some potassium ions are in the state of adsorption to –COO

^{−}. Coordinate system is set in the same way as Figure 4.

_{h}=

_{k}=

_{T}represents the total concentration of –COO

^{−}, –COOK and –COOH, and it is given by Equation (20).

_{T}= [–COO

^{−}] + [–COOH] + [–COOK]

^{−}] =

^{−}] + = 0

_{L}(x = ∞) − ϕ

_{R}(x = ∞). Computational potentials are all given in Figure 7 along with the experimental data V

_{GC}. For the computation, [H

^{+}] was assumed to be 2.41 × 10

^{21}m

^{−3}(=4 × 10

^{−6}M), since this experiment was carries out in the atmosphere. K

_{h}was assumed to be 2.41 × 10

^{22}m

^{−3}(=4 × 10

^{−5}M) based on the work described in the reference [32]. K

_{k}, d, ɛ

_{low}and [–COOX]

_{T}should have been determined experimentally but we could not determine them. Hence, we assumed quite plausible quantities for the first three as follows: Since –COOK is a salt, it is expected the dissociation constant of K

_{k}is by far higher than K

_{h}. K

_{k}was assumed to be 10-fold of K

_{h}, K

_{k}= 2.41 × 10

^{22}m

^{−3}. d was assumed to be 5 Å by considering the interfacial structure illustrated in Figure 8. We could not find any decisive literature for determining ɛ

_{low}. However, the relative permittivity of water around proteins is often assumed to be quite low for computer chemistry field or so [36]. It is possible to state one strong reason for that assumption: It is widely believed that water molecules surrounding protein form an ice-like structure and relative permittivity of ice is around 5 [37,38]. The surface-modified glassy carbon plate bears a number of carboxylic groups like proteins have. Hence, we chose 10 as ɛ

_{low}. As to [–COOX]

_{T}, the following quantities were assumed: [–COOX]

_{T}= 1.38 × 10

^{16}(≡[–COOX]

_{1}for GC1), 4.00 × 10

^{16}(≡[–COOX]

_{2}for GC2), 25.0 × 10

^{16}(≡[–COOX]

_{3}for GC3) m

^{−2}. All the computationally obtained potentials relatively well reproduce the experimental results. According to the fabrication process of the surface-modified glassy carbon plates, it is expected that the density [–COOX] increases in the order of GC1–GC2–GC3. Indeed, we can see the relationship of [–COOX]

_{1}< [–COOX]

_{2}< [–COOX]

_{3}as expected. The computational potential in Figure 7 was obtained by considering ion activity of KCl [16], while its horizontal axis quantities were all computed under the assumption that the activity coefficients were 1 just like Figure 3.

_{GC}exhibit unexpected behavior in low K

^{+}concentration regime, that is, all the experimental diagrams have the highest peak of potential around at log

_{10}C

_{L}= −4 or the highest plateau at and below log

_{10}C

_{L}= −4. Even such unexpected potential behavior is reproducible theoretically. Hence, the adsorption theory as a generation mechanism for the membrane potential is strongly validated, although some physical quantities needed for the potential computation based on the adsorption theory were given by assumption. However, still one might emphasize that the potential across an impermeable membrane is generated by ion adsorption onto the membrane but the potential across a permeable membrane is generated by ion transport through the membrane. In order to defy this emphasis, we carried out another experiment shown in the next section.

#### 3.4. Potential Generation across the Impermeable Ion Exchange Membrane

**Figure 9.**Structure of SES membrane and the experimental setup for measuring the potential across the SES membrane.

**Figure 10.**Potential generated between two KCl solutions vs. −log([Cl

^{−}]

_{R}/[Cl

^{−}]

_{L}). ○: Potential across the permeable membrane of Selemion AMV (same data as that shown in Figure 3); ∆: Potential across the impermeable membrane of SES membrane.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Tamagawa, H.; Morita, S.
Membrane Potential Generated by Ion Adsorption. *Membranes* **2014**, *4*, 257-274.
https://doi.org/10.3390/membranes4020257

**AMA Style**

Tamagawa H, Morita S.
Membrane Potential Generated by Ion Adsorption. *Membranes*. 2014; 4(2):257-274.
https://doi.org/10.3390/membranes4020257

**Chicago/Turabian Style**

Tamagawa, Hirohisa, and Sachi Morita.
2014. "Membrane Potential Generated by Ion Adsorption" *Membranes* 4, no. 2: 257-274.
https://doi.org/10.3390/membranes4020257