Introduction

A number of theoretical studies have been made on the electrophoretic mobility of a charged colloidal particle in an electrolyte solution [1–25]. It is known that the magnitude of the electrophoretic mobility μ of a spherical colloidal particle with κa > 3 (κ = the Debye-Hückel parameter of the electrolyte solution and a = particle radius), when plotted as a function of the particle zeta potential ζ, exhibits a maximum μ max at a certain value of ζ, which we denote to be ζ max [6–11] (Fig. 1). The appearance of the mobility maximum is due to the relaxation effect. With further increase in zeta potential, the electrophoretic mobility tends to a non-zero limiting electrophoretic mobility, which was treated in detail in refs. [18] and [19]. The purpose of the present paper is to derive analytic expressions for μ max and ζ max. We employ approximate mobility expressions for a charged sphere derived in ref. [20], which are applicable for all values of zeta potentials and large particle radii a such that κa > 30 and have been applied to analyze the mobility of spherical particles with high zeta potentials [26–28].

Fig. 1
figure 1

Electrophoretic mobility μ of a spherical particle plotted as a function of the particle zeta potential ζ for the case where there is a maximum μ max of the magnitude of μ at ζ = ζ max and a non-zero limiting value μ ∞ at ζ → ∞

Full size image

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 symmetrical electrolyte solution of valence z. The electrophoretic mobility μ of the particle is defined by μ = U/E, where U = |U| and E = |E|. In a previous paper [20], 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 counter ions, 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 ≥ 30):

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

or its magnitude is given by

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

with

$$ F=\frac{2\left(1+3m\right)}{\kappa a}\left\{ \exp \left(\frac{ze\left|\zeta \right|}{2kT}\right)-1\right\} $$
(3)
$$ m=\frac{2{\varepsilon}_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}kT}{3\eta {z}^2{e}^2}\lambda =\frac{2{N}_{\mathrm{A}}{\varepsilon}_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}kT}{3\eta z{\varLambda}^{\mathrm{o}}} $$
(4)

where the minus and plus signs in Eq. (1) are used for positive and negative zeta potentials, respectively; λ is the drag coefficient of counterions; m is the scaled drag coefficient, which is further related to the limiting equivalent conductance Λ0 of counterions; 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. Note that F given by Eq. (3) corresponds to Dukhin’s number.

For high ζ, where the magnitude of μ reaches a maximum, Eq. (2) is approximated by

$$ \left|\mu \right|=\frac{\varepsilon_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}}{\eta}\left\{\frac{\left|\zeta \right|-\left(2kT/ze\right) \ln 2}{1+2\left(1+3m\right) \exp \left(ez\left|\zeta \right|/2kT\right)/\kappa a}+\left(\frac{2kT}{ze}\right) \ln (2)\right\} $$
(5)

The value of ζ at which μ reaches a maximum, which we denote by ζ max, is derived from the condition dμ/dζ = 0 with the result that

$$ \left\{\frac{ze\left|{\zeta}_{\max}\right|}{2kT}-1- \ln (2)\right\} \exp \left(\frac{ze\left|{\zeta}_{\max}\right|}{2kT}\right)=\frac{\kappa a}{2\left(1+3m\right)} $$
(6)

which gives

$$ {\zeta}_{\max }=\pm \left(\frac{2kT}{ze}\right)\left\{W\left(\frac{\kappa a}{4\left(1+3m\right) \exp (1)}\right)+1+ \ln (2)\right\} $$
(7)

where W(x) is the Lambert W function (or the product logarithm) and satisfies x = W(x)e W(x). By substituting Eq. (7) into Eq, (5), we have the following approximate expression for the maximum value μ max of the electrophoretic mobility μ:

$$ {\mu}_{\max }=\pm \frac{\varepsilon_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}}{\eta}\left(\frac{2kT}{ze}\right)\left\{W\left(\frac{\kappa a}{4\left(1+3m\right) \exp (1)}\right)+ \ln (2)\right\} $$
(8))

In Eqs. (7) and (8), the plus and minus signs are used for positive and negative zeta potentials, respectively. Equations (7) and (8) are the required expressions for μ max and ζ max.

Results and discussion

We have derived approximate analytic Eqs. (7) and (8) for the maximum μ max and its position ζ max of the magnitude of the electrophoretic mobility μ of a spherical particle of radius a and zeta potential ζ in a symmetrical electrolyte solution of valence z. These expressions are derived on the basis of Eq. (5) (or Eqs. (1) and (2)). Some examples of the calculation of the electrophoretic mobility μ for a negatively charged spherical particle for κa = 50 in an aqueous electrolyte solution with z = 1, 2, and 3 at 25 °C obtained from Eq. (1) are shown in Fig. 1. Figure 1 shows the scaled electrophoretic mobility E m  = (3ηe/2ε r ε o kT)μ plotted as a function of the scaled zeta potential eζ/kT. The results obtained from Eq. (5) are also plotted as dotted lines in Fig. 1, which agree with the results of Eq. (1) with negligible errors. As counterions, we have chosen K+, Mg2+, and La3+ ions for the cases of z = 1, 2, and 3, respectively. We use the following values of Λ0 and m: Λ0 = 73.5 × 10−4 m2 Ω−1 equiv−1 and m = 0.176 for K+ ions, Λ0 = 53.1 × 10−4 m2 Ω−1 equiv−1 and m = 0.122 for Mg2+ ions, and Λ0 = 69.6 × 10−4 m2 Ω−1 equiv−1 and m = 0.0618 for La3+ ions (Note that the values of Λ0 and m for Mg2+ ions given in ref. [25] are incorrect and should be replaced by the above values). The numerical values of the scaled mobility maximum defined by E max m  = (3ηe/2ε r ε o kT)μ max and the scaled zeta potential zeζ max/kT (at which μ = μ max) are (zeζ max/kT, E max m ) = (5.470, 5.249) for z = 1, (2.793, 2.714) for z = 2, and (1.913, 1.886) for z = 3, while the values obtained from Eqs. (7) and (8) are (zeζ max/kT, E max m ) = (5.490, 5.235) for z = 1, (2.803, 2.705) for z = 2, and (1.919, 1.879) for z = 3, showing that Eqs. (7) and (8) are excellent approximations with negligible errors (less than 0.4 %).

It follows from Eqs. (7) and (8) that ζ max and E max m are both inversely proportional to the valence z of counterions, that is,

$$ \left|{\zeta}_{\max}\right|\propto \frac{1}{z}\mathrm{and}\left|{E}_m^{\max}\right|\propto \frac{1}{z} $$
(9)

as is seen in Fig. 2. It also follows from Eqs. (7) and (8) that ζ max and E max m are both proportional to ln(κa) for large κa, that is

Fig. 2
figure 2

Scaled electrophoretic mobility E m  = (3ηe/2ε r ε o kT)μ of a negatively charged spherical particle of radius a and zeta potential ζ(ζ < 0) in an aqueous symmetrical electrolyte solution for three values of the valence of counterions z = 1, 2, and 3 as functions of scaled zeta potential eζ/kT. The values of the reduced ionic drag coefficients for Na+, Mg2+, and La3+ are used as those of counterions for z = 1, 2, and 3, respectively. Calculated with Eq. (1) at κa = 50 and 25 °C. The dotted lines are approximate results obtained from Eq. (5)

Full size image
$$ \left|{\zeta}_{\max}\right|\propto \ln \left(\kappa a\right)\; and\;\left|{E}_m^{\max}\right|\propto \ln \left(\kappa a\right) $$
(10)

as is actually seen in Figs. 3, 4, and 5. Figure 3 shows the dependence of the electrophoretic mobility μ upon κa, and Figs. 4 and 5 give the κa dependence of ζ max and E max m , respectively.

Fig. 3
figure 3

Scaled electrophoretic mobility E m  = (3ηe/2ε r ε o kT)μ of a negatively charged spherical particle of radius a and zeta potential ζ(ζ < 0) in an aqueous 1:1 symmetrical electrolyte solution for three values κa plotted as a function of scaled zeta potential eζ/kT. The value of the reduced ionic drag coefficient m (=0.176) for Na+ is used as that of counterions. Calculated with Eq. (1) at κa = 50, 100, and 200 at 25 °C

Full size image
Fig. 4
figure 4

eζ max/kT as a function of κa calculated with Eq. (7) for z = 1, 2, and 3 at 25 °C

Full size image
Fig. 5
figure 5

E m max as a function of κa calculated with Eq. (8) for z = 1, 2, and 3 at 25 °C

Full size image

Similar expressions for ζ max and E max m for the case of a spherical particle in 2:1 or 1:2 electrolyte solutions can be derived on the basis of the approximate mobility expressions given in ref. [20]. The results are given below.

  1. (i)

    For a positively charged particle (ζ > 0) in a 2:1 electrolyte solution, the electrophoretic mobility μ is given by

    $$ \mu =\frac{\varepsilon_r{\varepsilon}_o}{\eta}\left\{\zeta -\frac{2F}{1+F}\left(\frac{kT}{e}\right) \ln \left[\frac{1}{2}+\frac{1}{2}\sqrt{\frac{2}{3} \exp \left(\frac{e\zeta }{kT}\right)+\frac{1}{3}}\right]\right\} $$
    (11)

    with

    $$ F=\frac{\sqrt{6}}{\kappa a}\left(1+3{m}_{-}\right)\left\{ \exp \left(\frac{e\zeta }{2kT}\right)-1\right\},\kern1em {m}_{-}=\frac{2{\varepsilon}_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}kT}{3\eta {e}^2}{\lambda}_{-}=\frac{2{N}_{\mathrm{A}}{\varepsilon}_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}kT}{3\eta {\varLambda}_{-}^0} $$
    (12)

    where λ − and Λ 0− are, respectively, the drag coefficient and limiting equivalent conductance of counterions (anions of valence −1) and m − is the scaled drag coefficient. From Eq. (11), we obtain

    $$ {\zeta}_m=\left(\frac{2kT}{e}\right)\left\{W\left(\frac{\kappa a}{6\left(1+3{m}_{-}\right) \exp (1)}\right)+1+\frac{1}{2} \ln (6)\right\} $$
    (13)
    $$ {\mu}_{\max }=\frac{\varepsilon_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}}{\eta}\left(\frac{2kT}{e}\right)\left\{W\left(\frac{\kappa a}{6\left(1+3{m}_{-}\right) \exp (1)}\right)+\frac{1}{2} \ln (6)\right\} $$
    (14)
  2. (ii)

    For a negatively charged particle (ζ < 0) in a 2:1 electrolyte solution, the electrophoretic mobility μ is given by

    $$ \mu =\frac{\varepsilon_r{\varepsilon}_{\mathrm{o}}}{\eta}\left\{\zeta +\frac{2F}{1+F}\left(\frac{kT}{e}\right)\right. \ln \left[\frac{1}{2} \exp \left(\frac{e\left|\zeta \right|}{2kT}\right)+\frac{1}{2}\sqrt{\frac{1}{3} \exp \left(\frac{e\left|\zeta \right|}{kT}\right)+\frac{2}{3}}\right] $$
    (15)

    with

    $$ F=\frac{\sqrt{3}}{\kappa a}\left(1+3{m}_{+}\right)\left\{ \exp \left(\frac{e\left|\zeta \right|}{kT}\right)-1\right\},\kern1em {m}_{+}=\frac{\varepsilon_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}kT}{6\eta {e}^2}{\lambda}_{+}=\frac{N_{\mathrm{A}}{\varepsilon}_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}kT}{3\eta {\varLambda}_{+}^0} $$
    (16)

    where λ + and Λ 0+ are, respectively, the drag coefficient and limiting equivalent conductance of counterions (cations of valence +2) and m + is the scaled quantity. From Eq. (15), we obtain

    $$ {\zeta}_{\max }=-\left(\frac{kT}{e}\right)\left\{W\left(\frac{\kappa a\left(3+2\sqrt{3}\right)}{18\left(1+3{m}_{+}\right) \exp (1)}\right)+1+2 \ln \left(\frac{2}{1+1/\sqrt{3}}\right)\right\} $$
    (17)
    $$ {\mu}_{\max }=-\frac{\varepsilon_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}}{\eta}\left(\frac{kT}{e}\right)\left\{W\left(\frac{\kappa a\left(3+2\sqrt{3}\right)}{18\left(1+3{m}_{+}\right) \exp (1)}\right)+2 \ln \left(\frac{2}{1+1/\sqrt{3}}\right)\right\} $$
    (18)
  3. (iii)

    For a positively charged sphere (ζ > 0) in a 1:2 electrolyte solution, the electrophoretic mobility μ is given by

    $$ \mu =\frac{\varepsilon_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}}{\eta}\left\{\zeta -\frac{2F}{1+F}\left(\frac{kT}{e}\right)\right. \ln \left.\left[\frac{1}{2} \exp \left(\frac{e\zeta }{2kT}\right)+\frac{1}{2}\sqrt{\frac{1}{3} \exp \left(\frac{e\zeta }{kT}\right)+\frac{2}{3}}\right]\right\} $$
    (19)

    with

    $$ F=\frac{\sqrt{3}}{\kappa a}\left(1+3{m}_{-}\right)\left\{ \exp \left(\frac{e\zeta }{kT}\right)-1\right\},\kern1em {m}_{-}=\frac{\varepsilon_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}kT}{6\eta {e}^2}{\lambda}_{-}=\frac{N_{\mathrm{A}}{\varepsilon}_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}kT}{3\eta {\varLambda}_{-}^0} $$
    (20)

    where λ − and Λ 0− are, respectively, the drag coefficient and limiting equivalent conductance of counterions (anions of valence −2) and m − is the scaled quantity. From Eq. (19), we obtain

    $$ {\zeta}_{\max }=\left(\frac{kT}{e}\right)\left\{W\left(\frac{\kappa a\left(3+2\sqrt{3}\right)}{18\left(1+3{m}_{-}\right) \exp (1)}\right)+1+2 \ln \left(\frac{2}{1+1/\sqrt{3}}\right)\right\} $$
    (21)
    $$ {\mu}_{\max }=\frac{\varepsilon_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}}{\eta}\left(\frac{kT}{e}\right)\left\{W\left(\frac{\kappa a\left(3+2\sqrt{3}\right)}{18\left(1+3{m}_{-}\right) \exp (1)}\right)+2 \ln \left(\frac{2}{1+1/\sqrt{3}}\right)\right\} $$
    (22)
  4. (iv)

    For a negatively charged sphere (ζ < 0) in a 1:2 electrolyte solution

    $$ \mu =\frac{\varepsilon_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}}{\eta}\left\{\zeta +\frac{2F}{1+F}\left(\frac{kT}{e}\right) \ln \left[\frac{1}{2}+\frac{1}{2}\sqrt{\frac{2}{3} \exp \left(\frac{e\left|\zeta \right|}{kT}\right)+\frac{1}{3}}\right]\right\} $$
    (23)

    with

    $$ F=\frac{\sqrt{6}}{\kappa a}\left(1+3{m}_{+}\right)\left\{ \exp \left(\frac{e\left|\zeta \right|}{2kT}\right)-1\right\},\kern1em {m}_{+}=\frac{2{\varepsilon}_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}kT}{3\eta {e}^2}{\lambda}_{+}=\frac{2{N}_{\mathrm{A}}{\varepsilon}_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}kT}{3\eta {\varLambda}_{+}^0} $$
    (24)

    where λ + and Λ 0+ are, respectively, the drag coefficient and limiting equivalent conductance of counterions (cations of valence +1) and m + is the scaled quantity. From Eq. (23), we obtain

    $$ {\zeta}_{\max }=-\left(\frac{2kT}{e}\right)\left\{W\left(\frac{\kappa a}{6\left(1+3{m}_{+}\right) \exp (1)}\right)+1+\frac{1}{2} \ln (6)\right\} $$
    (25)
    $$ {\mu}_{\max }=-\frac{\varepsilon_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}}{\eta}\left(\frac{2kT}{e}\right)\left\{W\left(\frac{\kappa a}{6\left(1+3{m}_{+}\right) \exp (1)}\right)+\frac{1}{2} \ln (6)\right\} $$
    (26)

    Finally, it is to be noted that the electrophoretic mobility of an infinitely long cylindrical particle of radius a in an electric field when the particle axis is oriented perpendicular to the applied electric field coincides with that of a spherical particle of radius a provided that κa is large (κa ≥ 30) [29]. Thus, the expressions for ζ max and E max m obtained above hold good also for κa ≥ 30.

Conclusion

We have derived simple approximate analytic expressions for the maximum μ max of the magnitude of the electrophoretic mobility μ of a charged spherical colloidal particle of radius a and zeta potential ζ in an electrolyte solution and the zeta potential ζ max that gives μ = μ max. The obtained expressions are derived on the basis of approximate expressions of the electrophoretic mobility which take into account the relaxation effect. 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 ≥ 30). It is shown that ζ max∝ 1/z and μ max∝ 1/z, and it is also shown that ζ max∝ ln(κa) and μ max∝ ln(κa) for large κa. The corresponding expressions are also derived for a sphere in a 2:1 or 1:1 electrolyte solution. It is to be noted that the results for a cylindrical particle of radius a when the particle is oriented perpendicular to the applied electric field are the same as those for a spherical particle of radius a provided that κa is large (κa ≥ 30). Finally, it is to be mentioned that the existence of μ max at large κa becomes important also in micro-fluidic applications (see e.g., [30]).