Simple approximate analytic expression for the electrophoretic mobility of a spherical colloidal particle in a mixed solution of 1:1 and 2:1 electrolytes

Abstract

Simple approximate analytic expressions are derived for the electrophoretic mobility of a spherical colloidal particle of radius a and zeta potential ζ in a mixed solution of 1:1 and 2:1 electrolytes with common anions on the basis of the general mobility expression previously derived by Ohshima (Colloids Surf A Physicochem Eng Asp 267:50, 2005). The obtained expressions, which are applicable for spheres of any ζ and large radii such that κa ≥ ca. 30 (where κ is the Debye-Hückel parameter), consist of Smoluchowski’s equation and the correction term taking into account the relaxation effect.

Introduction

The zeta potential of a charged colloidal particle in an electrolyte solution, which is usually estimated from its electrophoretic mobility, plays an essential role in determining its electrokinetic behaviors. A number of theoretical studies have been made on the electrophoretic mobility of spherical particles [1–22]. In particular, approximate mobility expressions for charged spheres derived in Ref. [18] are applicable for all values of zeta potentials and large particle radii a such that κa ≥ ca. 30 (κ is the Debye-Hückel parameter) and have been applied to analyze the mobility of spherical particles with high zeta potentials [23–25]. The corresponding mobility expression for a cylindrical particle has also been derived [26]. However, simple mobility expressions for a sphere in a mixed electrolyte solution, so far, have not been available, since the general mobility expression given in Ref. [18] involves cumbersome numerical multiple integration, if applied for the case of mixed electrolyte solutions. The purpose of the present paper is to derive simple analytic mobility expressions of a charged sphere in a mixed solution of 1:1 and 2:1 electrolytes on the basis of the general mobility expression previously derived [18].

Theory

Consider a spherical particle of radius a and zeta potential ζ moving with a velocity U under an external electric field E in a liquid containing a general electrolyte composed of N ionic species with valence z _i and bulk concentration (number density) n ^∞ _i and drag coefficient λ _i (i = 1, 2, …, N). The electrophoretic mobility μ of the particle is defined by μ = U / E, where U = |U| and E = |E|. In a previous paper [18], we have derived the following general mobility expression, which is correct to the order of exp(ze∣ζ∣ / 2kT) / κa, where z is the valence of counterions, e is the elementary electric charge, k is Boltzmann’s constant, T is the absolute temperature, and κ is the Debye-Hückel parameter, so that it is applicable for all values of ζ at large κa (κa ≥ ca. 30):

$$ \mu =-\frac{\varepsilon_{\mathrm{r}}{\varepsilon}_{\mathrm{o}} kT}{3\eta ae}{\displaystyle \sum_{i=1}^N{z}_i{n}_i^{\infty }{\phi}_i(a)}{\int}_0^{\overline{\zeta}}\left\{{\int}_0^y\frac{e^{-{z}_i{y}^{\prime }}-1}{\sqrt{{\displaystyle \sum_{j=1}^N{n}_j^{\infty }}\left({e}^{-{z}_j{y}^{\prime }}-1\right)}} dy^{\prime}\right\}\frac{1}{\sqrt{{\displaystyle \sum_{j=1}^N{n}_j^{\infty }}\left({e}^{-{z}_jy}-1\right)}} dy $$

(1)

with

$$ \begin{array}{l}{\phi}_i(a)=\frac{3a}{2}+\operatorname{sgn}\left(\zeta \right)\frac{\phi_i(a)}{\kappa a}{\left(\frac{1}{2}{\displaystyle \sum_{j=1}^N{z}_j^2{n}_j^{\infty }}\right)}^{1/2}{\displaystyle {\int}_0^{\overline{\zeta}}\frac{1-{e}^{-{z}_iy}}{\sqrt{{\displaystyle \sum_{j=1}^N{n}_j^{\infty }}\left({e}^{-{z}_jy}-1\right)}} dy}\hfill \\ {}+\operatorname{sgn}\left(\zeta \right)\frac{3{z}_i{m}_i}{4\kappa a}{\left(\frac{1}{2}{\displaystyle \sum_{j=1}^N{z}_j^2{n}_j^{\infty }}\right)}^{1/2}{\displaystyle {\int}_0^{\overline{\zeta}}\left[\left({e}^{-{z}_iy}-1\right)\right.{\displaystyle \sum_{j=1}^N{z}_j}}{n}_j^{\infty }{\phi}_j(a)\hfill \\ {}\times {\displaystyle {\int}_{\overline{\zeta}}^y\left\{{\displaystyle {\int}_0^{y\prime}\frac{e^{-{z}_jy\prime \prime }-1}{\sqrt{{\displaystyle \sum_{j=1}^N{n}_j^{\infty }}\left({e}^{-{z}_jy\prime \prime }-1\right)}}d{y}^{{\prime\prime} }}\right\}}\left.\frac{ dy^{\prime }}{\sqrt{{\displaystyle \sum_{j=1}^N{n}_j^{\infty }}\left({e}^{-{z}_jy\prime }-1\right)}}\right]\frac{ dy}{\sqrt{{\displaystyle \sum_{j=1}^N{n}_j^{\infty }}\left(e{-}^{z_jy}-1\right)}}\hfill \end{array} $$

(2)

$$ {m}_i=\frac{2{\varepsilon}_{\mathrm{r}}{\varepsilon}_{\mathrm{o}} kT}{3\eta {z}_i^2{e}^2}{\lambda}_i=\frac{2{N}_{\mathrm{A}}{\varepsilon}_{\mathrm{r}}{\varepsilon}_{\mathrm{o}} kT}{3\eta \left|{z}_i\right|{\varLambda}_i^{\mathrm{o}}} $$

(3)

where sgn(ζ) = +1 if ζ > 0 and −1 if ζ < 0, ϕ _i is a function that is related to the deviation δμ _i of the electrochemical potential μ _i of the i-th ionic species from its equilibrium value due to the applied electric field E, ϕ _i (a) is its value at the sphere surface, y = eψ /kT is the scaled equilibrium electric potential (ϕ is the equilibrium electric potential), $\overline{\zeta}$ = eζ/kT is the scaled zeta potential, λ _i is the drag coefficient of the i-th ionic species, m _i is its scaled drag coefficient, which is further related to the limiting conductance Λ^o _i of that ionic species, N _A is Avogadro’s number, ε _r and η are, respectively, the relative permittivity and the viscosity of the electrolyte solution, and ε _o is the permittivity of a vacuum.

Now, as a good approximation, we assume that the deviation of the electrochemical potential of the i-th ionic species is not influenced by those of other ionic species j ( j ≠ i). Under this approximation, the terms involving ϕ _j (a) (j ≠ i) are neglected so that terms involving small quantities m _i × m _j may be dropped. This approximation holds good when m _i × m _j « 1. We give below explicit expressions for the electrophoretic mobility μ of a spherical particle of radius a and zeta potential ζ in a mixed solution of 1:1 electrolyte of bulk concentration n ₁ and 2:1 electrolyte of bulk concentration n ₂ with common anions (where the Debye-Hückel parameter κ of this solution is given by $\kappa =\sqrt{2\left({n}_1+3{n}_2\right){e}^2/{\varepsilon}_{\mathrm{r}}{\varepsilon}_{\mathrm{o}} kT}$):

$$ \mu =\frac{2{\varepsilon}_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}}{3\eta}\left[\frac{\phi_{+1}(a)}{a}\left\{\frac{\zeta }{2}-\left(\frac{ kT}{e}\right) \ln \left(\frac{1+\beta }{2}\right)+\left(\frac{ kT}{e}\right)H\left(\zeta, a\right)\right\}+\frac{\phi_{+2}(a)}{a}\left\{\frac{\zeta }{2}-\left(\frac{ kT}{e}\right) \ln \left(\frac{1+\beta }{2}\right)-\left(\frac{ kT}{e}\right)H\left(\zeta, a\right)\right\}+\frac{2{\phi}_{-1}(a)}{a}\left(\frac{ kT}{e}\right)1\mathrm{n}\left(\frac{1+\beta }{2}\right)\right] $$

(4)

with

$$ \begin{array}{l}H\left(\zeta, \alpha \right)=\frac{\left(1-\alpha \right)}{2}\sqrt{\frac{3}{\alpha }}\left\{\left(\frac{ e\zeta}{ kT}\right) \ln \left[\left(\frac{1+\sqrt{\alpha /3}}{1-\sqrt{\alpha /3}}\right)\left(\frac{1-\beta }{1+\beta}\right)\right]\right.-2 \ln \left(\frac{\beta +\sqrt{\alpha /3}}{1+\sqrt{\alpha /3}}\right)\cdot \ln \left(\frac{1-\beta }{1+\beta}\right)\hfill \\ {}-\mathrm{L}{\mathrm{i}}_2\left(\frac{\beta +\sqrt{\alpha /3}}{1+\sqrt{\alpha /3}}\right)+\mathrm{L}{\mathrm{i}}_2\left(\frac{\beta -\sqrt{\alpha /3}}{1-\sqrt{\alpha /3}}\right)-\mathrm{L}{\mathrm{i}}_2\left(-\frac{\beta -\sqrt{\alpha /3}}{1+\sqrt{\alpha /3}}\right)+\mathrm{L}{\mathrm{i}}_2\left(-\frac{\beta +\sqrt{\alpha /3}}{1-\sqrt{\alpha /3}}\right)\hfill \\ {}-\mathrm{L}{\mathrm{i}}_2\left(-\frac{1+\sqrt{\alpha /3}}{1-\sqrt{\alpha /3}}\right)+\left.\mathrm{L}{\mathrm{i}}_2\left(-\frac{1-\sqrt{\alpha /3}}{1+\sqrt{\alpha /3}}\right)\right\}-\frac{ e\zeta}{2 kT}+ \ln \left(\frac{1+\beta }{2}\right)\hfill \end{array} $$

(5)

$$ \alpha =\frac{3{n}_2}{n_1+3{n}_2},\kern0.5em \beta =\sqrt{\frac{\alpha }{3}+\left(1-\frac{\alpha }{3}\right) \exp \left(\frac{ e\zeta}{ kT}\right)} $$

(6)

where Li₂(z) is a dilogarithm function (defined by Li₂(z) = ∑  ^∞ _k = 1 (z ^k/k ²)), which can easily be evaluated via, e.g., Mathematica.

Consider first the case of ζ > 0. In this case, the counterions are anions of valence −1 and we have

$$ \frac{\phi_{-1}(a)}{a}=\frac{3}{2\left(1+{F}_{-1}\right)},\kern0.5em \frac{\phi_{+1}(a)}{a}=\frac{\phi_{+2}(a)}{a}=\frac{3}{2} $$

(7)

with

$$ {F}_{-1}=\frac{2\left(1+3{m}_{-1}\right)}{\kappa a\sqrt{1-\alpha /3}}\left\{ \exp \left(\frac{ e\zeta}{2 kT}\right)-1\right\},\kern0.5em {m}_{-1}=\frac{2{\varepsilon}_{\mathrm{r}}{\varepsilon}_{\mathrm{o}} kT}{3\eta {e}^2}{\lambda}_{-1}=\frac{2{N}_{\mathrm{A}}{\varepsilon}_{\mathrm{r}}{\varepsilon}_{\mathrm{o}} kT}{3\eta {\varLambda}_{-1}^{\mathrm{o}}}, $$

(8)

where λ _-1 and Λ ⁰_− 1 are, respectively, the drag coefficient and the limiting conductance of the counterions (anions) of valence −1 and m ₋₁ is the corresponding scaled drag coefficient. Equation (4) thus becomes

$$ \mu =\frac{\varepsilon_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}}{\eta}\left\{\zeta -\frac{2{F}_{-1}}{1+{F}_{-1}}\left(\frac{ kT}{e}\right) \ln \left(\frac{1+\beta }{2}\right)\right\} $$

(9)

In the limit of n ₂ → 0 (α → 0, pure 1:1 electrolyte), Eq. (9) tends to

$$ \mu =\frac{\varepsilon_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}}{\eta}\left\{\zeta -\frac{2{F}_{-1}}{1+{F}_{-1}}\left(\frac{ kT}{e}\right) \ln \left[\frac{ \exp \left( e\zeta /2 kT\right)+1}{2}\right]\right\} $$

(10)

with

$$ {F}_{-1}=\frac{2}{\kappa a}\left(1+3{m}_{-1}\right)\left\{ \exp \left(\frac{ e\zeta}{2 kT}\right)-1\right\} $$

(11)

In the limit of n ₁ → 0 (α → 1, pure 2:1 electrolyte), on the other hand, Eq. (9) tends to

$$ \mu =\frac{\varepsilon_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}}{\eta}\left\{\zeta -\frac{2{F}_{-1}}{1+{F}_{-1}}\left(\frac{ kT}{e}\right) \ln \left[\frac{1}{2}+\frac{1}{2}\sqrt{\frac{2}{3} \exp \left(\frac{ e\zeta}{ kT}\right)+\frac{1}{3}}\right]\right\} $$

(12)

with

$$ {F}_{-1}=\frac{\sqrt{6}}{\kappa a}\left(1+3{m}_{-1}\right)\left\{ \exp \left(\frac{ e\zeta}{2 kT}\right)-1\right\} $$

(13)

Equations (10) and (12), respectively, agree with the previous results derived for pure 1:1 and 2:1 electrolytes [18].

Consider next the case of ζ < 0. In this case, the counterions are cations of valence +2 and those of valence +1 and we have

$$ \frac{\phi_{+1}(a)}{a}=\frac{3}{2\left(1+{F}_{+1}\right)},\kern0.5em \frac{\phi_{+2}(a)}{a}=\frac{3}{2\left(1+{F}_{+2}\right)},\kern0.5em \frac{\phi_{-1}(a)}{a}=\frac{3}{2} $$

(14)

with

$$ {F}_{+1}=\frac{2}{\kappa a}\left(1+3{m}_{+1}\right)\left\{ \exp \left(\frac{e\left|\zeta \right|}{2 kT}\right)-1\right\},\kern0.5em {F}_{+2}=\frac{\sqrt{3}}{\kappa a}\left(1+3{m}_{+2}\right)\left\{ \exp \left(\frac{e\left|\zeta \right|}{ kT}\right)-1\right\} $$

(15)

$$ {m}_{+1}=\frac{2{\varepsilon}_{\mathrm{r}}{\varepsilon}_{\mathrm{o}} kT}{3\eta {e}^2}{\lambda}_{+1}=\frac{2{N}_{\mathrm{A}}{\varepsilon}_{\mathrm{r}}{\varepsilon}_{\mathrm{o}} kT}{3\eta {\varLambda}_{+1}^{\mathrm{o}}},\kern1em {m}_{+2}=\frac{\varepsilon_{\mathrm{r}}{\varepsilon}_{\mathrm{o}} kT}{6\eta {e}^2}{\lambda}_{+2}=\frac{N_{\mathrm{A}}{\varepsilon}_{\mathrm{r}}{\varepsilon}_{\mathrm{o}} kT}{3\eta {\varLambda}_{+2}^{\mathrm{o}}} $$

(16)

where λ ₊₁ and Λ ^o_+ 1 are, respectively, the drag coefficient and limiting conductance of counterions (cations) of valence +1, λ ₊₂ and Λ ^o_+ 2 are those of counterions (cations) of valence +2, and m ₊₁ and m ₊₂ are the corresponding scaled drag coefficients. The electrophoretic mobility μ is thus given by

$$ \mu =\frac{\varepsilon_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}}{\eta}\left[\zeta +\left(\frac{1}{1+{F}_{+1}}+\frac{1}{1+{F}_{+2}}-2\right)\left\{\frac{\zeta }{2}-\left(\frac{ kT}{e}\right) \ln \left(\frac{1+\beta }{2}\right)\right\}+\left(\frac{1}{1+{F}_{+1}}-\frac{1}{1+{F}_{+2}}\right)H\left(\zeta, \alpha \right)\right] $$

(17)

Note that

$$ H\left(\zeta, \alpha \right)\to \frac{ e\zeta}{2 kT}- \ln \left(\frac{1+\beta }{2}\right)\kern0.62em \mathrm{as}\ {n}_2\to 0\ \mathrm{and}\ H\left(\zeta, \alpha \right)\to -\frac{ e\zeta}{2 kT}+ \ln \left(\frac{1+\beta }{2}\right)\kern0.62em \mathrm{as}\ {n}_1\to 0 $$

(18)

In the limit of n ₂ → 0 (α → 0, pure 1:1 electrolyte), in which case, Eq. (17) tends to

$$ \mu =\frac{\varepsilon_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}}{\eta}\left\{\zeta +\frac{2{F}_{+1}}{1+{F}_{+1}}\left(\frac{ kT}{e}\right) \ln \left[\frac{ \exp \left(e\left|\zeta \right|/2 kT\right)+1}{2}\right]\right\} $$

(19)

while in the limit of n ₁ → 0 (α → 1, pure 2:1 electrolyte), Eq. (17) tends to

$$ \mu =\frac{\varepsilon_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}}{\eta}\left\{\zeta +\frac{2{F}_{+2}}{1+{F}_{+2}}\left(\frac{ kT}{e}\right) \ln \left[\frac{1}{2} \exp \left(\frac{e\left|\zeta \right|}{2 kT}\right)+\frac{1}{2}\sqrt{\frac{1}{3} \exp \left(\frac{e\left|\zeta \right|}{ kT}\right)+\frac{2}{3}}\right]\right\} $$

(20)

Equations (18) and (19), respectively, agree with the previous results derived for pure 1:1 and 2:1 electrolytes [18]. Note that F ₊₁, F ₊₂, and F ₋₁ in the above equations correspond to Dukhin’s number, expressing the relaxation effect.

Results and discussion

We have derived approximate analytic expressions Eqs. (4) and (17) for the electrophoretic mobility μ of a spherical particle of radius a and zeta potential ζ in a mixed solution of 1:1 and 2:1 electrolytes with common anions. The relaxation effect, which becomes appreciable for high zeta potentials, is taken into account. Equations (4) and (17) are approximate equations applicable for all values of ζ at large κa. Since these equations are derived by neglecting terms of the order of 1 / κa, they are expected to be applicable for κa ≥ ca. 30 with tolerable errors. Some examples of the calculation of the electrophoretic mobility μ for a positively (Fig. 1) or negatively (Fig. 2) charged spherical particle for κa = 50 in an aqueous mixed solution of KCl and MgCl₂ at 25 °C for several values of the concentration ratio [MgCl₂] / [KCl] are shown in Figs. 1 and 2 in comparison with the results for pure 1:1 or 2:1 electrolytes. These figures show the scaled electrophoretic mobility E _m  = (3ηe/2ε _r ε _o kT)μ plotted as functions of the scaled zeta potential eζ/kT. For ζ > 0 (Fig. 1), in which case the counterions are Cl⁻ (Λ ⁰_− 1  = 76.3 × 10⁻⁴ m² Ω⁻¹ mol⁻¹ and m ₋₁ = 0.169), the mobility μ for a mixed solution slightly differs from those for pure electrolytes. For ζ < 0 (Fig. 2), however, in which case the counterions are K⁺ ions (Λ ⁰_+ 1  = 73.5 × 10⁻⁴ m² Ω⁻¹ mol⁻¹ and m ₊₁ = 0.176) and Mg²⁺ ions (Λ ⁰_+ 2  = 106.1 × 10⁻⁴ m² Ω⁻¹ mol⁻¹ and m ₊₂ = 0.061), the mobility μ for a mixed solution considerably differs from those for pure electrolytes. This difference comes from the fact that in this example, there are two kinds of counterions with different valences for ζ < 0, but just one kind of counterion for ζ > 0.

Fig. 1

Scaled electrophoretic mobility E _m = (3ηe / 2ε _r ε _o kT) μ of a positively charged spherical particle of radius a and zeta potential ζ > 0 in an aqueous mixed solution of KCl and MgCl₂ as functions of scaled zeta potential eζ / kT. Calculated with κa = 50 at 25 °C

Fig. 2

Same as Fig. 1 but for a negatively charged spherical particle with ζ < 0

Conclusion

We have derived simple approximate analytic expressions Eqs. (4) and (17) for the electrophoretic mobility μ of a charged spherical colloidal particle of radius a and zeta potential ζ in an mixed solution of 1:1 and 2:1 electrolytes with common anions by taking into account the relaxation effects. These expressions, which are obtained by neglecting terms of order 1 / κa in the general mobility expression and correct to the order of exp(ze∣ζ∣ / 2kT) / κa (where z is the valence of counterions), are applicable for all values of zeta potential at large κa (κa ≥ ca. 30).