## Abstract

The Gouy–Chapman surface potential is a key parameter for many interfacial phenomena in physical, chemical and biological systems. Existing theoretical approaches allow the determination of the surface potential at a solid–liquid interface only in single electrolyte solutions; however, mixed electrolytes are often encountered in practical applications. The development of a theoretical approach for the determination of the surface potential in mixed electrolyte solutions is therefore a desirable goal. In this study, this important issue was resolved for the first time. Based on the analytical solutions of the nonlinear Poisson–Boltzmann equation in different mixed electrolyte solutions, corresponding mathematical relationships were developed between the surface potential and the mean ionic concentration in the diffuse layer. As the mean ionic concentration in the diffuse layer can be easily determined, the surface potential could be calculated using the newly derived equations. The effects of electrolyte composition on the surface potential were theoretically quantified in the new equations, while only counterionic type was taken into account for mixed electrolyte solutions in the current studies.

## 1. Introduction

The electrochemical properties of a solid–liquid interface are key features for colloidal systems in aqueous media. The electrical potential at the solid–liquid interface is a fundamental parameter of electrochemical properties. The surface potential (potential at the original plane of the diffuse layer) of charged particles is a critical parameter for research into the transport [1] and electrical properties [2] of colloidal particles, the interactions between ions and surface [3] and interactions between particles [4,5].

The simple nonlinear Poisson–Boltzmann theory can be applied to determine potential and ion distributions around a charged plane, cylinder or sphere [6–12] and the adsorption of ions on charged particle surfaces [13,14]. The theory for determination of surface potential in a single electrolyte solution is based on the classical nonlinear Poisson–Boltzmann equation, and was derived by Li *et al.* [15]. However, mixed electrolyte solutions (rather than a single electrolyte solution) are often encountered in scientific research and chemical engineering. Zhao *et al.* [16] provided a concise description of the Gouy–Chapman theory for NaCl/CaCl_{2} mixtures that is one of electrolyte mixtures with monovalent and divalent ions (co-ions and counterions). Actually, the interfacial behaviours described by the Gouy–Chapman theory depend on the electrolyte composition with different ion species. Furthermore, the ionic dispersion effects become important to surface potential at high electrolyte concentrations (greater than 0.1 mol l^{−1}) [17,18]. Although ionic steric effects are important in the Stern layer [19], they are not significant in the diffuse layer for relatively small inorganic ions under the application conditions encountered in practical application [13,20–22]. Ions can be strongly polarized under a strong electric field [3,23], and such a strong electric field exists at the interface between nano-colloidal particles and water [23]; the polarization of ions should therefore be considered in the Poisson–Boltzmann equation [24]. It is difficult to evaluate the contribution of ionic polarization to the surface potential using the polarizable Poisson–Boltzmann equation for mixed electrolyte solutions. However, the polarizability effects can be evaluated using the approach used to determine the surface potential based on the classical nonlinear Poisson–Boltzmann equation. The principles for the determination of surface potential in mixed electrolyte solutions therefore remain an important issue at moderate electrolyte concentrations. On the basis of this work, the polarizability effects of ions on surface potential will be evaluated in subsequent studies.

In this study, the relationships between the surface potential and the mean concentration of counterions in the diffuse layer were established, based on analytical solutions of the classical nonlinear Poisson–Boltzmann equation for mixed electrolytes solutions [25]. Once the mean concentration was determined, the surface potential could be calculated theoretically. In other words, principles for the determination of the surface potential of charged particles were derived for mixed electrolyte solutions.

## 2. Establishment of the relationships between the surface potential and mean ionic concentration in the diffuse layer

The mean concentration of ions in the diffuse layer is defined as
*N*_{i} (mol g^{−1}) is the number of *i* ions adsorbed in the diffuse layer, *S* (dm^{2} g^{−1}) is the specific surface area of the particles, *κ* (dm^{−1}) is the Debye–Hückel parameter.

Based on the Boltzmann equation, the expression for the concentration distribution is
*f*_{i} (mol l^{−1}) refers to the concentration of *i* ions in the bulk solution, *Z*_{i} is the valence of the *i* ions, *F* (C mol^{−1}) is the Faraday constant, *φ*(*x*) (V) is the potential distribution in the diffuse layer, *R* (J mol^{−1} K^{−1}) is the gas constant and *T* (K) is the absolute temperature.

Introducing equation (2.2) into (2.1), we get
*φ* (*x*)) for a mixed solution with monovalent and bivalent ions [25], the relationships between the surface potential and the mean ionic concentration in the diffuse layer could be established.

### (a) Relationship between the surface potential and the mean ionic concentration in 1:1+2:1 mixed electrolyte solutions

Introducing the obtained expression for the potential distribution *φ* (*x*) in 1:1 (*AB*) and 2:1 (*CD*_{2}) mixed electrolyte solutions [25] into equation (2.3), the mean concentration of monovalent counterions in the diffuse layer (cation *A*) was expressed as
*f*_{A} and *f*_{C} (mol l^{−1}) refer to the concentration of the *A* and *C* ion species in bulk solution, *ε* is the dielectric constant in the medium and *φ*_{0} (V) is the surface potential or potential at the original plane of the diffuse layer.

The mean concentration of bivalent counterions in the diffuse layer (cation *C*) could be expressed as
_{1} value could be calculated. Once the λ_{1} value was obtained, the surface potential was calculated based on equation (2.6)
*A*, equation (2.5) is not a general solution and may be invalid when the concentration of cation *C* is zero. In actual applications, the surface potential is typically determined by measuring the adsorption of bivalent counterion.

From equations (2.10) and (2.11), we can see that the relationship between the mean concentration of counterions in the diffuse layer and the surface potential is represented by a very complex mathematical expression. However, under some conditions, the complex expression could be approximated by a simple form.

When the mean concentration of counterions in the diffuse layer was much larger than that in the bulk solution—i.e. *φ*_{0} was larger than −0.07681 V, for a negative value—*e*^{Fφ0/(RT)} was smaller than 0.05 (≈0), and thus λ_{1} was approximately equal to

The data in table 1 show that the approximate value of the surface potential agree with the precise value determined using high *f*_{A}/*f*_{C} values. With *f*_{A}/*f*_{C}=1, a relative error of less than 7% was achieved for the surface potential calculated from the approximate expression of equation (2.14), in comparison to the value calculated from the precise expression of equation (2.11) when *f*_{A}/*f*_{C}=30, a relative error of less than 12% was achieved for the surface potential calculated from equation (2.14), in comparison to the value calculated from equation (2.11) when *f*_{A}/*f*_{C} value, and it decreases with decreasing magnitude of *f*_{A}/*f*_{C} for a fixed

If only a bivalent counterion species is present, i.e. *f*_{A}=0, equation (2.14) naturally reduces to the expression for the surface potential in a single 2:1 electrolyte [15]

### (b) Relationship between the surface potential and the mean ionic concentration in 1:1+1:2 mixed electrolyte solutions

For 1:1 (*AB*) and 1:2 (*E*_{2}*F*) mixed electrolyte solutions, when the expression obtained for the potential distribution *φ*(*x*) [25] was introduced into equation (2.3), the mean concentration of counterions in the diffuse layer could be expressed as
*k* represents *A* or *E*.

The integration of the above equation gives
_{2} value could be calculated. Once the λ_{2} value was obtained, the surface potential could be calculated based on equation (2.18)

Under high-surface-potential conditions—i.e. when *φ*_{0} is larger than −0.07681 V, for a negative value—e^{Fφ0/(RT)} was smaller than 0.05 (≈0), thus λ_{2} was therefore approximately equal to *f*_{A}+*f*_{E}, based on equation (2.18). Therefore, combined equations (2.17) and (2.19) produced a simple expression for calculating the surface potential:
*f*_{E}=0, *k* represents *A*, the above equation naturally reduces to the expression for the surface potential in a single 1:1 electrolyte [15]:
*f*_{A}=0, namely *k* represents *E*, equation (2.20) naturally reduces to the expression for the surface potential in a single 1:2 electrolyte [15]:

### (c) Relationship between the surface potential and the mean ionic concentration in 1:1+2:2 mixed electrolyte solutions

For 1:1 (*AB*) and 2:2 (*GH*) mixed electrolyte solution, introducing the obtained expression of potential distribution *φ*(*x*) [25] into equation (2.3), the mean concentration of monovalent counterions (cation *A*) in the diffuse layer could be expressed as:
_{3} value could be calculated. Once the λ_{3} value was obtained, the surface potential could be calculated based on equation (2.25):
*A*, equation (2.24) is not a general solution, and may be invalid when the concentration of cation *G* equals zero.

From equations (2.27) and (2.28), we can see that the relationship between the mean concentration of counterions in the diffuse layer and the surface potential is represented by a very complex mathematical expression. However, under some conditions, the complex expression also could be approximated by a simpler form.

When the mean concentration of counterions in the diffuse layer was much larger than that in the bulk solution—i.e. when *φ*_{0} was larger than −0.07681 V, for a negative value—e^{Fφ0/(RT)} was smaller than 0.05 (≈0), and λ_{3} was approximately equal to *f*_{A}=0, the above equation naturally reduces to the expression for the surface potential in a single 2:2 electrolyte [15]
*f*_{A}/*f*_{G} values. With *f*_{A}/*f*_{G}=1, a relative error of less than 5% was achieved for the approximate value calculated from equation (2.30), relative to the precise value determined using equation (2.28) when *f*_{A}/*f*_{G}=30, a relative error of less than 14% was achieved for the approximate value calculated from equation (2.30), relative to the precise value using equation (2.28) when *f*_{A}/*f*_{G} value, and decreases with decreasing of *f*_{A}/*f*_{G} value for a fixed

### (d) Relationship between the surface potential and the mean ionic concentration in 1:2+2:1 mixed electrolyte solutions

For 1:2 (*E*_{2}*F*) and 2:1 (*CD*_{2}) mixed salt solutions, introducing the expression for the potential distribution *φ*(*x*) [25] into equation (2.3) yielded an expression for the mean concentration of monovalent counterions (cation *E*) in the diffuse layer, as follows:
*C* in diffuse layer could be expressed as
_{4} value could be calculated. Once the λ_{4} value was obtained, the surface potential could be calculated based on equation (2.34):

When the mean concentration of counterions in the diffuse layer was much larger than that in the bulk solution—i.e. when *φ*_{0} was larger than −0.07681 V, for a negative value—e^{Fφ0/(RT)} was smaller than 0.05 (≈0), and λ_{4} was approximately equal to *f*_{E}=0) the surface potential of mixed 1:2 and 2:1 electrolytes in equation (2.39) is reduced to the surface potential for a single 2:1 electrolyte (equation (2.15)) [15].

The data in table 3 show that the approximate value of the surface potential agree with the precise value for the high *f*_{E}/*f*_{C} value. With *f*_{E}/*f*_{C}=1, a relative error of less than 5% was achieved for the approximate value calculated from equation (2.39) and deviates from the precise value from equation (2.37) when *f*_{E}/*f*_{C}=30, a relative error of less than 6% was achieved for the approximate value calculated from equation (2.39) and deviates from the precise value from equation (2.37) when *f*_{E}/*f*_{C} value, and decreases with decreasing *f*_{E}/*f*_{C} value for a fixed

The potential distributions of cylindrical and spherical colloids in single 1:2 and 2:1 electrolytes were derived respectively by Téllez & Trizac [26]. In the case of large curvature radii, the nano-colloidal particles theory could be of use in the present calculation for mixed electrolyte solutions.

### (e) Relationship between the surface potential and the mean ionic concentration in 2:2+2:1 mixed electrolyte solutions

For a 2 : 2 (*GH*) and 2:1 (*CD*_{2}) mixed electrolyte solution, introducing the expression of potential distribution *φ*(*x*) [25] into equation (2.3) yielded an expression for the mean concentration of counterions in the diffuse layer, as follows:
*k* represents *G* or *C*.

The integration of equation (2.40) is
_{5} value could be calculated. Once the λ_{5} value was obtained, the surface potential could be calculated based on equation (2.42):

Under high-surface-potential conditions—i.e. when *φ*_{0} was larger than −0.07681 V, for a negative value—e^{Fφ0/(RT)} was smaller than 0.05 (≈0), and λ_{5} was approximately equal to *f*_{G}=0), *k* represents *C*, and equation (2.45) naturally reduces to the expression for the surface potential in a single 2:1 electrolyte. If *f*_{C}=0, *k* represents *G*, and the surface potential for mixed 2:2 and 2:1 electrolytes in equation (2.45) reduces to the surface potential for a single 2:2 electrolyte.

### (f) Relationship between the surface potential and the mean ionic concentration in 2:2+1:2 mixed electrolyte solutions

The potential distribution in a 2:2+1:2 mixed electrolyte solution (i.e. *GH*+*E*_{2}*F*) can be calculated by the nonlinear Poisson–Boltzmann equation
*E*) in the diffuse layer could be expressed as
*G*) in the diffuse layer could be expressed as
_{6} value could be calculated. Once the λ_{6} value was obtained, the surface potential could be calculated based on equation (2.47)

It is clear from equations (2.51) and (2.52) that the relationship between the mean concentration of counterions in the diffuse layer and the surface potential is represented by a very complex mathematical expression. However, under some conditions, this complex expression could be approximated by a simpler form.

When the mean concentration of counterions in the diffuse layer was much larger than that in the bulk solution—i.e. when *φ*_{0} was larger than −0.07681 V, for a negative value—e^{Fφ0/(RT)} was smaller than 0.05 (≈0), and λ_{5} was approximately equal to *f*_{E}/*f*_{G} value. With *f*_{E}/*f*_{G}=1, a relative error of less than 5% was achieved for the approximate value calculated from equation (2.54) and deviates from the precise value from equation (2.52) when *f*_{E}/*f*_{G}=30, a relative error of less than 6% was achieved for the approximate value calculated from equation (2.54) and deviates from the precise value from equation (2.52) when *f*_{E}/*f*_{G} value, and decreases with decreasing *f*_{E}/*f*_{G} value for a fixed

The effects of electrolyte composition on the surface potential were theoretically quantified in the new equations, while only the counterionic type was taken into account for mixed electrolyte solutions in the current studies [16].

## 3. A general expression for the surface potential in mixed electrolyte solutions

Agreement between the values obtained using the approximate and precise expressions was achieved when the ratio of the concentration of one counterion species in the diffuse layer and in the bulk solution (*f*_{i}/*f*_{j} value, the relative error of the approximate value for the surface potential deviates from the precise value and decreases with increasing *f*_{i}/*f*_{j} value was too large, it was difficult for the *j* ion species to enter the inner layer of the electric double layer, leading to a distribution of *j* ions in the diffuse layer that deviated from that predicted by the Boltzmann equation. The approximate expressions used above for the calculation of the surface potential were therefore valid for relatively small values of the ratio of the concentrations of the counterions in the bulk solution, and this parameter could be regulated easily in experiments.

According to the approximate expressions for the surface potential given above, a general expression for the surface potential in different single and mixed electrolytes can be obtained
*M* is a function of electrolyte type in ionic mixtures and *AB* electrolyte) and 2:1 (*CD*_{2} electrolyte) mixed solutions, *AB* electrolyte) and 1:2 (*E*_{2}*F* electrolyte) mixed solutions,
*AB* electrolyte) and 2:2 (*GH* electrolyte) mixed solutions,
*E*_{2}*F* electrolyte) and 2:1 (*CD*_{2} electrolyte) mixed solutions,
*GH* electrolyte) and 2:1 (*CD*_{2} electrolyte) mixed solutions,
*GH* electrolyte) and 1:2 (*E*_{2}*F* electrolyte) mixed solutions. The subscript capital letters represent the counterion species (cations for negative charged surfaces) in the mixed electrolyte solutions. In particular, for a single electrolyte solution, *M*=2 in 1:1 or 2:2 electrolyte, or their mixture;

In order to evaluate the robustness of the present analytical approach, it is necessary to compare the results predicted by the analytical approach with numerical results. We select the excellent work of Bolt for the adsorption equilibrium in NaCl and CaCl_{2} (1:1 and 2:1 mixed electrolyte) [13] as an example. Based on PB equation, we can obtain the precise and numerical surface potential. The analytical surface potential as a function of ( *f*^{1/2}_{Ca}×*S*)/*N*_{Ca} term using equation (2.14) or (3.1). Figure 1 shows that the analytical results agree with the precise numerical results, and the former may slightly smaller than the latter (negative values), but the relative error is lower than 10%.

## 4. Conclusion

The simple nonlinear Poisson–Boltzmann theory has a wide range of applications, and theory for determination of the surface potential can be established based on the analytical solutions of the Poisson–Boltzmann equation in mixed electrolyte solutions. However, it is difficult to obtain analytical solutions. Fortunately, this important issue was resolved in our previous study [25], which provided a potential approach for the determination of the surface potential in mixed electrolyte solutions. In this study, the theoretical expressions for the determination of the surface potential were derived for different mixed electrolyte solutions, based on the corresponding analytical solutions of the nonlinear Poisson–Boltzmann equation. The surface potential in mixed electrolyte solutions depends on the mean concentration of counterions in the diffuse layer (

## Data accessibility

All the data used throughout this article are available at a publication with doi:10.1097/00010694-195504000-00004.

## Authors' contributions

X.L. drafted the manuscript. X.L., H.L. and R.Y. performed the data analysis. All authors contributed to design of the study and gave final approval for publication.

## Competing interests

We declare we have no competing interests.

## Funding

This work was supported by the National Natural Science Foundation of China (41371249, 41201223 and 41101223), Fundamental Research Funds for the Central Universities (XDJK2015C059) and China Postdoctoral Science Foundation (2015M570762 and 2015M572430).

## Acknowledgements

We thank the referees for useful suggestions.

- Received January 30, 2015.
- Accepted June 23, 2015.

- © 2015 The Author(s)

Published by the Royal Society. All rights reserved.