Approximate Analytic Expression for the Electrophoretic Mobility of

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Approximate Analytic Expression for the Electrophoretic Mobility of Moderately Charged Cylindrical Colloidal Particles Hiroyuki Ohshima* Faculty of Pharmaceutical Sciences, Tokyo University of Science, 2641 Yamazaki Noda, Chiba 278-8510, Japan ABSTRACT: An approximate analytic expression for the electrophoretic mobility of an infinitely long cylindrical colloidal particle in a symmetrical electrolyte solution in a transverse electric field is obtained. This mobility expression, which is correct to the order of the third power of the zeta potential ζ of the particle, considerably improves Henry’s mobility formula correct to the order of the first power of ζ (Proc. R. Soc. London, Ser. A 1931, 133, 106). Comparison with the numerical calculations by Stigter (J. Phys. Chem. 1978, 82, 1417) shows that the obtained mobility formula is an excellent approximation for low-to-moderate zeta potential values at all values of κa (κ = Debye−Hückel parameter and a = cylinder radius).

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INTRODUCTION The zeta potential ζ of a charged colloidal particle in an electrolyte solution plays an essential role in determining the electrical behaviors of colloidal suspensions.1−7 Usually the zeta potential of a particle can be calculated from the measured value of the electrophoretic mobility μ of the particle under an applied electric field on the basis of a theoretical relationship between μ and ζ. A number of analytical or numerical theoretical studies have been made on the electrophoretic mobility of spherical particles8−36 and cylindrical particles.37−43 In biological systems, in particular, there are many biocolloidal particles of cylindrical shape such as DNA, viruses, and cells. Unlike the case of spherical particles, the electrophoretic mobility of an infinitely long cylindrical colloidal particle depends on the particle orientation relative to the applied electric filed. For a cylinder oriented parallel to an applied electric field, its electrophoretic mobility μ// is given by Smoluchowski’s equation,8 viz., μ =

εrεoζ η

kind. The following simple approximate expression to Henry’s eq 2 has been derived, which does not involve numerical integration:41 μ⊥ =

(3)

Henry’s equation for spheres is given also in ref 10, and its simple approximate equation has been derived.36 Note that eq 3 for μ⊥ approaches eq 1 for μ∥ in the limit of κa → ∞, that is, μ⊥ → μ =

εrεoζ ⎧ 4(κa)4 ⎨1 − K 0(κa) η ⎩

(1)

∞

∫κa

K 0(t ) t5

dt +

(κa)2 K 0(κa)

∞

∫κa

K 0(t ) t3

⎫ dt ⎬ ⎭

(2)

(4)

Received: August 10, 2015 Revised: November 25, 2015

where κ is the Debye−Hückel parameter (given later by eq 10), K0(t) is the zero-order modified Bessel function of the second © XXXX American Chemical Society

εrεoζ as κa → ∞ η

Note also that in the limit of κa → ∞, the electrophoretic mobility of a sphere and that of a cylinder both tend to Smoluchowski’s eq 1. Henry’s eq 3, which is correct to the first order of ζ, can only be applied to the case of low zeta potentials. For highly charged particles, the relaxation effect becomes appreciable. There are the following two different approaches to obtain mobility formulas which take into account the relaxation effect. In the first approach, one starts with the large κa limiting mobility formula (i.e., Smoluchowski’s eq 4) and obtain the next-order correction terms of the order of exp(ze|ζ|/2kT)/κa, where z is the valence of counterions of the electrolyte solution, e is the elementary electric charge, k is Boltzmann’s constant, and T is the absolute temperature. The factor exp(ze|ζ|/2kT)/ κa corresponds to Dukhin’s number. Several approximate analytic expressions have been derived for spherical or

where εr and η are, respectively, the relative permittivity and the viscosity of the electrolyte solution, and εo is the permittivity of a vacuum. If the cylinder is oriented perpendicular to the applied electric field, then the electrophoretic mobility μ⊥ does not follow eq 1 but depends on the cylinder radius a. For the case of low zeta potentials, μ⊥ is given by the following Henry equation:10 μ⊥ =

⎤ εrεoζ ⎡ 1 ⎢1 + ⎥ 2 2η ⎣ (1 + 2.55/[κa{1 + exp( −κa)}]) ⎦

A

DOI: 10.1021/acs.langmuir.5b02969 Langmuir XXXX, XXX, XXX−XXX

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Langmuir cylindrical particles. In particular, simple approximate mobility expressions for spherical particles derived in ref 30 are applicable for all values of zeta potentials with κa ≥ 30, so that they have been applied to analyze the mobility of spherical latex particles with high zeta potentials.44−46 The corresponding mobility expression for cylinders has also been recently derived.43 In the second approach, by starting with Henry’s mobility equation, one expresses the electrophoretic mobility in powers of zeta potential ζ and makes corrections to the third power of ζ in Henry’s mobility equation. This approach was first tried independently by Overbeek11 and Booth12 for the case of spherical particles. Their results, which are very complicated, are not convenient for practical mobility calculation. An alternative simple analytic expression for the electrophoretic mobility of spherical particles has been derived28 and employed to analyze the zeta potential of gold nanoparticles.47,48 For the case of cylinders, however, a simple mobility expression correct to the order of ζ3 has so far not been derived. The purpose of the present paper is to derive an analytic expression for the electrophoretic mobility μ⊥ of an infinitely long cylindrical particle correct to the of order ζ3 oriented perpendicular to the applied electric filed in a symmetrical electrolyte solution. The obtained mobility expression will be shown later to involve numerical integration. We further derive a simple interpolation formula without involving numerical integration. For a cylindrical particle oriented at an arbitrary angle between its axis and the applied electric field, its electrophoretic mobility μav averaged over a random distribution of orientation is given by the following:37−39 μav =

Figure 1. A cylindrical colloidal particle of radius a moving with a velocity U in a transverse field E.

equations, we obtain the following general expression for the electrophoretic mobility μ of an infinitely long cylindrical particle of radius a in a transverse electric field: μ⊥ =

3

G(r ) = −

(5)

⎛ r ⎞⎫⎤ ⎛ r ⎞2 ⎧ ⎢1 − ⎜ ⎟ ⎨1 − 2ln⎜ ⎟⎬⎥G(r ) dr ⎝ a ⎠⎭⎦ ⎝a⎠ ⎩ ⎣

∞⎡

(7)

⎛ zeψ (0) ⎞ ϕ ⎤ εr εoκ 2 dψ (0) ⎡ ⎛ zeψ (0) ⎞ ϕ+ ⎢exp⎜⎜− ⎟⎟ ⎟⎟ − ⎥ + exp⎜⎜+ 2η dr ⎢⎣ ⎝ kT ⎠ r ⎝ kT ⎠ r ⎥⎦ (8)

where ψ0)(r) is the equilibrium electric potential, and the functions ϕ±(r) are related to δμ±(r) by

■

δμ± (r ) = ∓zeϕ±(r )Ecos θ

THEORY Consider an infinitely long charged cylindrical colloidal particle of radius a moving with a velocity U in a symmetrical electrolyte solution of valence z and bulk (number) concentration n in an applied transverse electric field E. The drag coefficient of cations λ+ and that of anions λ− may be different. The drag coefficients λ± of cations and anions are related to their limiting equivalent conductances Λ0± by the following:

(9)

and E = |E|. For a symmetrical electrolyte of valence z and bulk concentration n, the Debye−Hückel parameter κ is given by the following:

■

⎛ 2nz 2e 2 ⎞1/2 κ=⎜ ⎟ ⎝ εrεokT ⎠

(10)

MOBILITY EXPRESSION CORRECT TO ORDER ζ3 We derive a mobility expression correct to order ζ3 in the following form:

NAe 2z Λ±0

∫a

with

μ + 2μ⊥

Stigter38 obtained numerical values for μav for an infinitely long cylinder. As will be seen later, a simple analytic mobility expression derived in the present paper (given later by eq 24) is found to be in excellent agreement with Stigter’s numerical results38 for low-to-moderate zeta potentials at all κa values.

λ± =

a2 8

(6)

where NA is Avogadro’s number. We employ the cylindrical coordinate system (r, θ, z) with its origin held fixed at the center of the particle. We treat the case where θ = 0 axis coincides with the applied electric field E so that E is perpendicular the cylinder axis, i.e., the z-axis (Figure 1). The fundamental electrokinetic equations can be expressed in terms of the liquid velocity u(r) at position r relative to the particle (u(r) → −U as r →∞, where r = |r|, and u(r) = 0 at r = a) and the deviations δμ+(r) and δμ−(r) of the electrochemical potentials μ+(r) and μ−(r) of cations and anions due to the applied electric field.41,43 By solving these electrokinetic

μ⊥ =

⎫⎤ ⎛ zeζ ⎞2 ⎧ 2εrεoζ ⎡ ⎛ m+ + m− ⎞ ⎟f (ka)⎬⎥ ⎟ ⎨f (ka) + ⎜ ⎢f1 (ka) − ⎜ 3 4 ⎝ ⎠ ⎝ ⎠ ⎭⎦ 3η ⎣ kT ⎩ 2 (11)

where m+ and m− are dimensionless ionic drag coefficient, defined by the following: m± =

2εrεokT

λ 2 2 ±

3ηz e

=

2NAεrεokT 3ηz Λ±0

(12)

and the first term on the right-hand side of eq 11 corresponds to Henry’s formula 2, that is, B

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Langmuir f1 (κa) =

4(κa)4 3⎧ ⎨1 − 2⎩ K 0(κa)

∞

∫κa

K 0(t ) t

5

(κa)2 K 0(κa)

dt +

∞

∫κa

K 0(t ) t

3

⎫ dt ⎬ ⎭

⎛ a2 ⎞ H1(x) = ⎜1 − 2 ⎟ x ⎠ ⎝

(13) −

which can be approximated well by eq 3, viz., ⎤ 3⎡ 1 ⎥ f1 (κa) = ⎢1 + 4⎣ (1 + 2.55/[κa{1 + exp( −κa)}])2 ⎦

⎡ ⎛ zeζ ⎞2 (0) ⎤ ⎟ ψ ψ (0)(r ) = ζ ⎢ψ1(0)(r ) + ⎜ ( r )⎥ ⎝ kT ⎠ 3 ⎣ ⎦

x

a 2 ⎞ dψ ⎜1 − 2 ⎟ 1 ⎝ t ⎠ dt

∞⎛

(0)

dt

(0) ⎛ t 2 ⎞⎛ a 2 ⎞ dψ ⎜1 + 2 ⎟⎜1 − 2 ⎟ 1 dt ⎝ x ⎠⎝ t ⎠ dt

H2(x) = −

3κ 2 16

∫a

+

3κ 2 16

∫a

(14)

In order to obtain approximate expressions for f 3(κa) and f4(κa), we employ an approximate expression for the equilibrium potential around a cylinder of radius a in a z−z type symmetrical electrolyte solution correct to order ζ3, which can be obtained as follows:

∫a

∫a

x

(20)

(0) ⎧ 2 ⎛ t ⎞⎫⎛ t4 a 2 ⎞ dψ ⎨x − 2 + 4t 2 ln⎜ ⎟⎬⎜1 + 2 ⎟ 1 dt ⎝ x ⎠⎭⎝ ⎩ x t ⎠ dt

⎛ ⎛ x ⎞⎫ a4 a2 ⎞ ⎨x 2 + 2 − 2a2 + 2⎜1 − 2 ⎟t 2 − 4t 2 ln⎜ ⎟⎬ ⎝ a ⎠⎭ x x ⎠ ⎝ ⎩

∞⎧

⎪

⎪

(0) ⎛ a 2 ⎞ dψ × ⎜1 + 2 ⎟ 1 dt ⎝ t ⎠ dt

(21)

In Figure 2, we plot f 3(κa) and f4(κa) as functions of κa. The values of f 3(κa) and f4(κa) for the case of spheres of radius a are also given in Figure 2 as dotted lines.28

(15)

with the following: ψ1(0)(r ) =

ψ3(0) =

K 0(κr ) K 0(κa)

(16)

⎡

⎧

⎫

I (κa) 1 ⎢K 0(κr )⎨ 0 ⎬ 6K 03(κa) ⎢⎣ ⎩ K 0(κa) ⎭ ∞

− I0(κr )

∫κr

∞

∫κa

K 04(t )t dt κr

∫κa

K 04(t )t dt − K 0(κr )

⎤ I0(t )K 03(t )t dt ⎥ ⎦

(17)

where I0(t) is the zero-order modified Bessel function of the first kind. Note that for symmetrical electrolytes, the terms proportional to ζ2 are absent in eqs 11 and 15, i.e., f 2(κa) = 0 and ψ2(0) = 0. Then we finally find the following: f3 (κa) =

3(κa)2 64

∫a

(0) ⎛ r ⎞⎫⎤ dψ r2 ⎧ ⎢1 − 2 ⎨1 − 2ln⎜ ⎟⎬⎥ 1 ⎝ ⎠ a ⎭⎦ d r ⎣ a ⎩

∞⎡

⎧ ⎪⎛ a2 ⎞ ×⎨ ⎜1 + 2 ⎟ ⎪ r ⎠ ⎩⎝ ×

∫a

∞

H1(x)

dψ1(0)

⎫ ⎪ 3(κa)2 ⎬ dr + dx ⎪ dx 64 ⎭

dψ1(0)

d(ψ1(0))2 dr

Figure 2. F3(κa) and F4(κa) as functions of the reduced cylinder radius κa (solid lines) in comparison with those for spherical particles of the reduced sphere radius κa (dotted lines).28

⎧ ⎪⎛ a2 ⎞ ⎨⎜1 + 2 ⎟ ×⎪ r ⎠ ⎩⎝

dx

∫a

∫a

dx −

∫a

r

■

INTERPOLATION FORMULA Equations 18 and (19) involve numerical integration of Bessel functions so that these equations are not convenient for practical mobility calculation. On the basis of eqs 18 and 19, we have obtained the following simple interpolation formulas for f 3(κa) and f4(κa) without involving numerical integration with relative errors less than 1% for 0.1 ≤ κa ≤ 100, in which region the relaxation effect becomes important:

⎛ x2 ⎞ ⎜1 − 2 ⎟H1(x) ⎝ r ⎠

∞⎡

⎛ r ⎞⎫⎤ r2 ⎧ ⎢1 − 2 ⎨1 − 2ln⎜ ⎟⎬⎥ ⎝ a ⎠⎭⎦ ⎣ a ⎩

a 2 ⎞ dψ ⎜1 − 2 ⎟ 1 dx ⎝ x ⎠ dx

∞⎛

(0)

(0) ⎫ ⎛ ⎪ x 2 ⎞⎛ a 2 ⎞ dψ ⎬ dr ⎜1 − 2 ⎟⎜1 − 2 ⎟ 1 dx ⎪ ⎝ r ⎠⎝ x ⎠ dx ⎭ ∞⎡ ⎛ r ⎞⎫⎤⎛ 3(κa)2 r2 ⎧ a2 ⎞ + ⎢1 − 2 ⎨1 − 2ln⎜ ⎟⎬⎥⎜1 + 2 ⎟ ⎝ a ⎠⎭⎦⎝ a ⎣ 16 a ⎩ r ⎠

−

∫a

r

f3 (ka) =

(22)

∫

(0) (0) 3 ⎫ ⎧ ⎪ dψ 1 d(ψ1 ) ⎪ 3 ⎬ dr ×⎨ + ⎪ ⎪ 6 dr ⎭ ⎩ dr

f4 (κa) =

3(κa)2 64

∫a

f4 (ka) =

⎛ r ⎞⎫⎤ dψ r2 ⎧ ⎢1 − 2 ⎨1 − 2ln⎜ ⎟⎬⎥ 1 ⎝ a ⎠⎭⎦ dr ⎣ a ⎩

⎧ ⎪⎛ a2 ⎞ ⎨⎜1 + 2 ⎟ ×⎪ r ⎠ ⎝ ⎩ ⎫ ⎪ dx ⎬ dr ⎪ dx ⎭

(0)

∫a

∞

H2(x)

dψ1(0) dx

dx −

∫a

r

9κa{κa + 0.361 exp( −0.475κa) + 0.0878} 8{(κa)3 + 10.8(κa)2 + 18.2κa + 0.0633}

(23)

By using eq 22 for f 3(κa) and eq 23 for f4(κa), and eq 14 for f1(κa), we find that eq 11 becomes the following:

(18)

∞⎡

κa(κa + 0.162) 2{(κa)3 + 9.94(κa)2 + 18.7κa + 0.147 exp(− 9.41κa)}

μ⊥ =

⎛ x2 ⎞ ⎜1 − 2 ⎟H2(x) ⎝ r ⎠

⎤ 2ε ε ζ εrεoζ ⎡ 1 ⎢1 + ⎥− ro 2η ⎣ 3η (1 + 2.55/[κa{1 + exp( − κa)}])2 ⎦ ×

⎛ zeζ ⎞2 ⎡ κa(κa + 0.162) ⎟ ⎢ ⎝ kT ⎠ ⎣ 2{(κa)3 + 9.94(κa)2 + 18.7κa + 0.147 exp(− 9.41κa)}

⎜

⎛ m + m− ⎞ 9κa{κa + 0.361 exp(− 0.475κa) + 0.0878} ⎤ ⎟ +⎜ + ⎥ ⎝ ⎠ 8{(κa)3 + 10.8(κa)2 + 18.2κa + 0.0633} ⎦ 2

dψ1(0)

(19)

(24)

Equation 24 is the required approximate analytic expression correct to the order of ζ3 for the electrophoretic mobility μ⊥ of

with C

DOI: 10.1021/acs.langmuir.5b02969 Langmuir XXXX, XXX, XXX−XXX

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Langmuir a cylindrical colloidal particle in a symmetrical electrolyte solution in a transvers electric field.

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RESULTS AND DISCUSSION In Figure 3, we plot the scaled electrophoretic mobility Em⊥ = (3ηe/2εrεokT)μ⊥ calculated via eq 24 for a cylindrical particle of

Figure 4. Scaled average electrophoretic mobility Em = (Em∥+ 2Em⊥)/3 of a cylindrical particle of radius a and zeta potential ζ in an aqueous electrolyte solution with m± = 0.184 (an appropriate value for KCl) as functions of scaled cylinder radius κa at three values of the reduced zeta potential e|ζ|/kT = 1, 2, and 3 (solid lines) in comparison with the results for a sphere of radius a (dotted lines).28 Numerical results38 are also given by symbol ● for cylinders and by ▲ for spheres. Figure 3. Scaled electrophoretic mobility Em⊥ = (3ηe/2εrεokT)μ⊥ of a cylindrical particle of radius a and zeta potential ζ oriented perpendicular to the applied electric field in an aqueous monovalent symmetrical electrolyte solution (z = 1) with m± = 0.184 (an appropriate value for KCl) as functions of scaled cylinder radius κa at three low-to-moderate values of the reduced zeta potential e|ζ|/kT = 1, 2, and 3. Solid lines are approximate results calculated with eq 24 for a cylinder and eq 25 for a sphere, and dotted lines are results calculated with Henry’s equation (eq 3).

Stigter’s numerical results38 are also given in Figure 4 by symbol ● for cylinders and by ▲ for spheres. It is found that agreement between eq 24 and the numerical results38 is excellent for low-to-moderate zeta potential values at all values of κa, including moderate values of κa, in which region the relaxation effect becomes appreciable. For the cylinder case, the maximum relative error is less than 1% for e|ζ |/kT = 1, about 1% for e|ζ |/kT = 2, and about 6% for e|ζ |/kT = 3. Thus, eq 24 is a considerable improvement of Henry’s formula 3. A similar tendency can be observed for the case of spheres. In the present paper, we have treated infinitely long cylindrical particles and neglected the end effects of the cylinder. For a cylinder of radius a and finite length L, its electrophoretic mobility should depend on the aspect ratio a/L. For a/L↔ 1, the relaxation effect due to the uneven charge distribution at both ends of the cylindrical particle should be taken into account.39 The results obtained in this paper can be applied to the case of a/L≪ 1 so that the end effects can be neglected.

radius a and zeta potential ζ oriented perpendicular to the applied electric field as functions of scaled cylinder radius κa at three low-to-moderate values of the reduced zeta potential e|ζ|/ kT = 1, 2, and 3 in an aqueous monovalent symmetrical electrolyte solution (z = 1) with m± = 0.184 (an appropriate value for KCl). Solid lines are results calculated with eq 24, and the dotted lines are results calculated with Henry’s eq 3. It is seen that for the low potential case (e|ζ|/kT = 1), where the relaxation effect is small, the difference between the results of eqs 24 and 3 is small, and the electrophoretic mobility Em⊥ agrees well with Henry’s eq 3. With increasing zeta potential, however, the relaxation effect becomes appreciable, and the contributions from terms proportional to ζ3 become appreciable, and the difference between Henry’s equation, and the present results become large. In Figure 4, we plot the scaled average electrophoretic mobility Em = (Em// + 2Em⊥)/3 of a cylindrical particle of radius a and zeta potential ζ as functions of scaled cylinder radius κa (solid lines), where Em⊥ = (3ηe/2εrεokT)μ⊥ and Em// = (3ηe/ 2εrεokT)μ∥ in comparison with the results for a sphere of radius a. The corresponding approximate expression for the electrophoretic mobility μ of a sphere is given by21 (dotted lines) the following: μ=

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CONCLUSIONS We have derived a simple analytic expression (eq 24) correct to the order of ζ3 for the electrophoretic mobility μ⊥ of an infinitely long cylindrical particle of radius a oriented perpendicular to an applied electric field in a symmetrical electrolyte solution. This mobility expression, which takes into account the relaxation effect, is a considerable improvement of Henry’s mobility eq 3. It is found that agreement with the numerical results38 is excellent for low-to-moderate zeta potentials (e|ζ |/kT ≤ 3) at all values of κa. Finally, it is noteworthy that the theory developed in the present paper is focused on the relaxation effect on the electrophoretic mobility of charged cylindrical particles, which becomes appreciable for moderate to high zeta potentials at moderate vales of κa. The present theory thus does not take into account the effects of viscosity and dielectric permittivity near a highly charged surface, which will deviate from their bulk values, and should affect the particle electrophoretic mobility.49,50 In particular, the theory of Bonthuis and Netz50 for the electro-osmotic velocity on a highly charged planar surface agrees well with experimental results given in ref 49. A planar

⎤ 2εrεoζ ⎡ 1 ⎢1 + ⎥ 3η ⎣ 2(1 + 2.5/[κa{1 + 2 exp( − κa)}])3 ⎦ −

2εrεoζ ⎛ zeζ ⎞2 ⎡ κa{κa + 1.3 exp(− 0.18κa) + 2.5} ⎜ ⎟ ⎢ 3η ⎝ kT ⎠ ⎣ 2{κa + 1.2 exp( −7.4κa) + 4.8}3

⎛ m + m− ⎞ 9κa{κa + 5.2 exp( −3.9κa) + 5.6} ⎤ ⎟ ⎥ +⎜ + ⎝ ⎠ 8{κa − 1.55 exp(− 0.32κa) + 6.02}3 ⎦ 2

(25) D

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surface considered in ref 50, however, corresponds to the limiting case of infinitely large κa, in which case the relaxation effect becomes negligible. As a future work, therefore, we need a theory that accounts for the relaxation effect and the effects of viscosity and dielectric permittivity near a curved charged surface of a particle with finite κa.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +81-4-7124-1501 ext. 6772. Fax: +81-3-6760-0891. Notes

The authors declare no competing financial interest.

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REFERENCES

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DOI: 10.1021/acs.langmuir.5b02969 Langmuir XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.langmuir.5b02969 Langmuir XXXX, XXX, XXX−XXX