pubs.acs.org/Langmuir © 2010 American Chemical Society
Electrophoretic Mobility of a Colloidal Particle with Constant Surface Charge Density Kimiko Makino†,‡,§ and Hiroyuki Ohshima*,†,‡ †
Faculty of Pharmaceutical Sciences, ‡Center for Colloid and Interface Science, Research Institute for Science and Technology, and §Center for Drug Delivery Research, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan Received September 6, 2010. Revised Manuscript Received October 6, 2010
When the electrophoretic mobility of a particle in an electrolyte solution is measured, the obtained electrophoretic mobility values are usually converted to the particle zeta potential with the help of a proper relationship between the electrophoretic mobility and the zeta potential. For a particle with constant surface charge density, however, the surface charge density should be a more characteristic quantity than the zeta potential because for such particles the zeta potential is not a constant quantity but depends on the electrolyte concentration. In this article, a systematic method that does not require numerical computer calculation is proposed to determine the surface charge density of a spherical colloidal particle on the basis of the particle electrophoretic mobility data. This method is based on two analytical equations, that is, the relationship between the electrophoretic mobility and zeta potential of the particle and the relationship between the zeta potential and surface charge density of the particle. The measured mobility values are analyzed with these two equations. As an example, the present method is applied to electrophoretic mobility data on gold nanoparticles (Agnihotri, S. M.; Ohshima, H.; Terada, H.; Tomoda, K.; Makino, K. Langmuir 2009, 25, 4804).
1. Introduction Electrophoretic mobility measurements are a powerful method of estimating the electrical properties of the surface of a charged colloidal particle in an electrolyte solution.1-15 Usually, the measured value of the electrophoretic mobility μ of a particle is converted to the particle zeta potential ζ (which is practically equal to the particle surface potential) with the help of an appropriate equation relating μ to ζ. This equation can be written as μ = μ(ζ, κa) for a spherical particle of radius a, where κ is the Debye-H€uckel parameter (given later in eq 3). Most colloidal particles, however, do not carry a constant surface potential (which depends on the electrolyte concentration) but a constant surface charge density σ. For such a particle, the surface charge density σ should be a more characteristic quantity than the zeta potential ζ, and thus the mobility μ should be rewritten as a function of σ and κa (i.e., μ = μ(σ, κa)). The purpose of this article is to present a systematic method of estimating the surface charge density σ of a spherical colloidal particle with a constant surface *Corresponding author. Tel and Fax: þ81-4-7121-3661. E-mail: ohshima@ rs.noda.tus.ac.jp. (1) Von Smoluchowski, M. In Handbuch der Elektrizitt und des Magnetismus; Greatz, L., Ed.; Barth: Leipzig, Germany, 1921; Vol. 2, p 366. (2) H€uckel, E. Phys. Z. 1924, 25, 204. (3) Henry, D. C. Proc. R. Soc. London, Ser. A 1931, 133, 106. (4) Overbeek, J. Th. G. Kolloid-Beih. 1943, 54, 287. (5) Booth, F. Proc. R. Soc. London, Ser. A 1950, 203, 514. (6) Wiersema, P. H.; Loeb, A. L.; Overbeek, J. Th. G. J. Colloid Interface Sci. 1966, 22, 78. (7) Dukhin, S. S.; Semenikhin, N. M. Kolloid Zh. 1970, 32, 360. (8) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607. (9) Ohshima, H.; Healy, T. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1983, 79, 1613. (10) Dukhin, S. S. Adv. Colloid Interface Sci. 1993, 44, 1. (11) Ohshima, H. J. Colloid Interface Sci. 1994, 168, 269. (12) Ohshima, H. J. Colloid Interface Sci. 2001, 239, 587. (13) Ohshima, H. Colloids Surf., A 2005, 267, 50. (14) Ohshima, H. Theory of Colloid and Interfacial Electric Phenomena; Elsevier: Amsterdam, 2006. (15) Ohshima, H. Biophysical Chemistry of Biointerfaces; John Wiley and Sons: Hoboken, NJ, 2010.
18016 DOI: 10.1021/la1035745
charge density from its electrophoretic mobility values in electrolyte solutions of various concentrations.
2. Theory Consider a charged spherical colloidal particle of radius a moving with a constant velocity in a symmetrical electrolyte solution of valence z and bulk (number) concentration n¥ in an applied electric field. The drag coefficient of cations λþ and that of anions λ- may be different. The drag coefficients of cations and anions λ( are related to their limiting conductances Λ0( by λ( ¼
NA e2 z Λ0(
ð1Þ
where NA is Avogadro’s number and e is the elementary electric charge. To calculate the zeta potential ζ of a colloidal particle in an electrolyte solution from the measured value of the electrophoretic mobility μ of the particle, one needs a relationship between μ and ζ. The most widely used relationship between the electrophoretic mobility μ of a spherical colloidal particle of radius a and its zeta potential ζ is Smoluchowski’s mobility equation1 μ ¼
εr εo ζ η
ð2Þ
where εr and η are, respectively, the relative permittivity and the viscosity of the electrolyte solution and εo is the permittivity of a vacuum. This equation is applicable when κa is sufficiently large (i.e., κa . 1), where κ is the Debye-H€uckel parameter defined by K ¼
2n¥ z2 e2 εr εo kT
!1=2 ð3Þ
k is the Boltzmann constant, and T is the absolute temperature. For arbitrary values of κa, the mobility becomes a function of κa.
Published on Web 11/03/2010
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Figure 3. Scaled electrophoretic mobility Em = 3ηeμ/2εrεokT of a
Figure 1. Scaled electrophoretic mobility Em = 3ηeμ/2εrεokT of a spherical colloidal particle of radius a in an aqueous KCl solution at 25 C (mþ = 0.176 and m- = 0.169) as a function of κa for several values of the scaled particle zeta potential eζ/kT.
spherical colloidal particle of radius a in an aqueous KCl solution at 25 C (mþ = 0.176 and m- = 0.169) as a function of κa for several values of the scaled surface charge density σ* = eaσ/εrεokT.
Figure 2. Scaled zeta potential eζ/kT of a particle of radius a as a unction of κa for several values of scaled surface charge density σ* = eaσ/εrεokT.
Figure 4. Electrophoretic mobility μ of gold nanoparticles of
For low ζ, the mobility is given by Henry’s equation εr εo ζ μ ¼ ½1 - eKa f5E7 ðKaÞ - 2E5 ðKaÞg η
3
ð4Þ
where En(κa) is the exponential integral of order n and is defined by Z ¥ -t Z ¥ - Kr e e n-1 En ðKaÞ ¼ ðKaÞn - 1 dt ¼ a dr ð5Þ n rn Ka t a 11
Ohshima showed that eq 4 can be approximated with negligible errors by ! 2εr εo ζ 1 μ ¼ ð6Þ 1þ 3η 2½1þ2:5=fKað1þ2e - Ka Þg3 Equations 2 and 4 (and 6) are correct to order ζ. Overbeek4 derived a mobility expression correct to order ζ3. Following Overbeek,4 we write a mobility expression correct to order ζ3 in the following form μ ¼
" 2 # 2εr εo ζ zeζ mþ þ mf3 ðkaÞ f2 ðkaÞ þ ð7Þ f1 ðkaÞ 2 3η kT
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radius 7.5 nm in an aqueous KCl solution as a function of ionic strength (M) in comparison with three theoretical predictions. Solid lines are the results for σ = -0.03/m2 calculated via eq 9 as combined with eq 10; the dotted line corresponds to Smoluchowski’s equation (eq 2), and the dashed line corresponds to Henry’s equation (eq 6). Experimental data are taken from ref 18.
where f1(κa)-f3(κa) are functions of κa and mþ and m- are dimensionless ionic drag coefficients defined by m( ¼
2εr εo kT λ( 3ηz2 e2
ð8Þ
Ohshima12 derived the following approximate mobility expression for eq 7 ! 2εr εo ζ 1 1þ μ ¼ 3η 2½1þ2:5=fKað1þ2e - Ka Þg3 " 2εr εo ζ zeζ 2 KafKa þ 1:3 expð - 0:18KaÞ þ 2:5g 3η kT 2fKaþ1:2 expð - 7:4KaÞþ4:8g3 # m þ þ m - 9KafKa þ 5:2 expð - 3:9KaÞ þ 5:6g þ ð9Þ 2 8fKa - 1:55 expð - 0:32KaÞþ6:02g3 DOI: 10.1021/la1035745
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Figure 5. Same as for Figure 4, but for a = 25 nm and σ = -0.02/m2.
Figure 6. Same as for Figure 4, but for a = 100 nm and σ =
-0.01/m2.
which is a good approximation to ze|ζ|/kT| e 4. An accurate expression for the electrophoretic mobility applicable for arbitrary ζ and κa g 10 is derived in ref 9. For the case of a particle in a general electrolyte solution, we refer the reader to ref 13. Equations 2, 4 (or 6), and 9 express the electrophoretic mobility μ as a function of ζ and κa (i.e., μ = μ(ζ, κa)). For a particle with a constant surface charge density σ independent of the electrolyte concentration, however, the particle surface potential, which is practically equal to the zeta potential ζ, becomes a function of the electrolyte concentration. For such a situation, in order to estimate the electrolyte concentration dependence of μ, eqs 2, 4 (or 6), and 9 should be rewritten as functions of σ and κa (i.e., μ = μ(σ, κa)) instead of functions of ζ and κa (i.e., μ = μ(ζ, κa)). To obtain the zeta potential ζ-surface charge density σ relationship, one must solve the spherical Poisson-Boltzmann equation for the electric potential around a spherical particle in an electrolyte solution. Numerical tables for the solution to the spherical Poisson-Boltzmann equation are given by Loeb et al.16 An acculturate approximate expression for the ζ-σ relationship was derived by Ohshima et al.,17 with the result being " 2εr εo KkT zeζ 1 2 1þ σ ¼ sinh ze 2kT Ka cosh2 ðzeζ=4kTÞ #1=2 8 ln½coshðzeζ=4kTÞ þ ðKaÞ2 sinh2 ðzeζ=2kTÞ 1
ð10Þ
which is an improvement of the following empirical equation derived by Loeb et al.:16 # " 2εr εo KkT zeζ 1 1 sinh 1þ σ ¼ ze 2kT Ka cosh2 ðzeζ=4kTÞ
ð11Þ
The maximum relative error in eq 11 is 21% at κa = 0.1, whereas that in eq 10 is reduced to 4% at κa = 0.1 and is less than 1% for κa g 1. Equation 9 combined with eq 10 gives μ as a function of σ and κa. (16) Loeb, A. L.; Overbeek, J.Th.G.; P. H. Wiersema, P. H. The Electrical Double Layer around a Spherical Colloid Particle; MIT Press: Cambridge, MA, 1961. (17) Ohshima, H.; Healy, T. W.; White, L. R. J. Colloid Interface Sci. 1982, 90, 17.
18018 DOI: 10.1021/la1035745
Results and Discussion Figure 1 shows the scaled electrophoretic mobility Em = 3ηem/ 2εrεokT of a spherical particle with radius a in a KCl solution at 25 C (mþ for Kþ ions = 0.176 and m- for Cl- ions = 0.169) plotted as a function of κa for four values of the scaled zeta potential eζ/kT. The κa dependence of μ represents the electrolyte concentration dependence of μ if a is considered to be constant. For a particle with a constant surface charge density, the zeta potential ζ becomes a function of κa, as is shown in Figure 2. Therefore, Figure 1 does not properly represent the electrolyte concentration dependence of the electrophoretic mobility μ of a spherical particle with a constant surface charge density of σ. We therefore plot the electrophoretic mobility μ as a function of κa for several values of the scaled surface charge density σ* = eaσ/ εrεokT in Figure 3. In view of the fact that most colloidal particles carry a constant surface charge density rather than a constant surface potential, one should analyze the measured values of electrophoretic mobility on the basis of eqs 9 and 10 (i.e., μ = μ(κa, σ) instead of μ = μ(κa, ζ)). In the following text, we analyze the electrolyte concentration dependence of the electrophoretic mobility of gold nanoparticles and determine the surface charge densities of gold nanoparticles of different sizes. Figures 4-6 exhibit electrophoretic mobility data for gold nanoparticles of different sizes, a = 7.5 nm (Figure 4), 25 nm (Figure 5), and 100 nm (Figure 6), as a function of ionic strength in an aqueous KCl solution at 25 C (taken from ref 18) in comparison with the results calculated with eq 9 as combined with eq 10 for σ = -0.03 C/m2 (Figure 4), -0.02 C/m2 (Figure 5), and -0.01 C/m2 (Figure 6). It is seen that eqs 9 and 10 with appropriate values of σ are in good agreement with the experimental results. In these Figures, the results calculated with Smoluchowski’s equation (eq 2) and Henry’s equation (eq 6), both combined with eq 10, are also given, showing poor agreement with the experimental results. It is also seen that deviation among these three equations (eqs 2, 6, and 9) becomes small for a large particle (Figure 6) but is large for a small particle (Figure 4). Thus, on the basis of the plot of the measured electrophoretic mobility values as a function of the electrolyte concentration in comparison with calculated results by using eqs 9 and 10, one can determine the surface charge density of a spherical particle by a curve-fitting procedure. (18) Agnihotri, S. M.; Ohshima, H.; Terada, H.; Tomoda, K.; Makino, K. Langmuir 2009, 25, 4804.
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Conclusions The surface charge density σ is a more characteristic quantity than the zeta potential ζ for a charged spherical particle with a constant surface charge density. We have shown that with the help of eqs 9 (the μ-ζ relationship) and 10 (the σ-ζ relationship) one (19) Lyklema, J. Fundamentals of Colloid and Interface Science; Academic Press: San Diego, CA, 1995; Vol. 2 Chapter 4. (20) Delgado, A. V.; Gonzalez-Caballero, E.; Hunter, R. J.; Koopal, L. K.; Lyklema, J. J. Colloid Interface Sci. 2007, 309, 194. (21) Dukhin, S. S.; Zimmerman, R.; Werner, C. Colloids Surf., A 2001, 195, 103.
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can determine the surface charge density σ of a spherical colloidal particle in an electrolyte solution from its electrophoretic mobility μ. The present method is based on two simple analytical equations (eqs 9 and 10) and thus does not require numerical computer calculation. Although the present electrophoresis theory (eq 9), which ignores the stagnant-layer effect,19-21 well explains the experimental results for the electrophoretic mobility of gold nanoparticles,18 this effect in general is important and should be taken into account to improve the present electrophoresis theory.
DOI: 10.1021/la1035745
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