Electrical Phenomena at Interfaces and Biointerfaces: Fundamentals and Applications in Nano-, Bio-, and Environmental Sciences

By Hiroyuki Ohshima

This book bridges three different fields: nanoscience, bioscience, and environmental sciences. It starts with fundamental electrostatics at interfaces and includes a detailed description of fundamental theories dealing with electrical double layers around a charged particle, electrokinetics, and electrical double layer interaction between charged particles. The stated fundamentals are provided as the underpinnings of sections two, three, and four, which address electrokinetic phenomena that occur in nanoscience, bioscience, and environmental science. Applications in nanomaterials, fuel cells, electronic materials, biomaterials, stems cells, microbiology, water purificiaion, and humic substances are discussed.

• Language: English
• Publisher: Wiley
• Release date: Jan 25, 2012
• ISBN: 9781118135426

Hiroyuki Ohshima

Hiroyuki Ohshima is Professor Emeritus at the Tokyo University of Science, Japan. He received the B.S. (1968), M.S. (1970), and Ph.D. (1974) degrees in physics from the University of Tokyo. He spent his post-doc study at University of Melbourne, Australia (1981-1983), State University of New York at Buffalo, USA (1983-1984), and University of Utah, USA (1984-1985). He has worked at the Tokyo University of Science, Japan since 1985. He was previously a Professor of Pharmaceutical Sciences at the Tokyo University of Science (1994-2012) and is now a Professor Emeritus and a Visiting Professor. He is the author of Theory of Colloid and Interfacial Electric Phenomena (Elsevier, 2006) and Biophysical Chemistry of Biointerfaces (Wiley, 2010). He edited Electrical Phenomena at Interfaces and Biointerfaces (Wiley, 2012) and Encyclopedia of Biocolloid and Biointerface Science (Wiley, 2016). He co-edited Electrical Phenomena at Interfaces. 2e (Marcel Dekker, 1998) and Colloid and Interface Science in Pharmaceutical Research and Development (Elsevier, 2014). He is also the author or co-author of over 400 book chapters and journal publications, reflecting his research interests in colloid and interface science. He was an Editor of the journal Colloids and Surfaces B: Biointerfaces (1994-2012, Elsevier) and currently he is an Associate Editor of Colloid and Polymer Science (2002-present, Springer). In 2016, Ohshima was awarded the 29th Khwarizmi International Award by the Iranian Research Organization for Science and Technology for his contribution to the theory of colloid and interfacial electric phenomena. In 2017, he was selected as one of Asia's top 100 scientists for 2017 (The Asian Scientist Magazine, June 2017). In 2022, he was awarded Fellow of Japan Oil Chemists' Society. On January 13, 2021, the layman's summary of Ohshima’s research article:?Gel Electrophoresis of a Soft Particle (Adv. Colloid Interface Sci., 271, 101977, 2019)?was published on the Atlas of Science website (https://atlasofscience.org/soft-particle-gel-electrophoresis/).

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PART I: FUNDAMENTALS

1

POTENTIAL AND CHARGE OF A HARD PARTICLE AND A SOFT PARTICLE

Hiroyuki Ohshima

1.1 INTRODUCTION

When a charged colloidal particle is immersed in an electrolyte solution, mobile electrolyte ions form an ionic cloud around the particle. As a result of electrostatic interaction between electrolyte ions and particle surface charges, in the ionic cloud the concentration of counterions (electrolyte ions with charges of the sign opposite to that of the particle surface charges) becomes very high, while that of coions (electrolyte ions with charges of the same sign as the particle surface charges) is very low. Figure 1.1 schematically shows the distribution of ions around a charged spherical particle of radius a. The ionic cloud together with the particle surface charge forms an electrical double layer. Such an electrical double layer is often called an electrical diffuse double layer since the distribution of electrolyte ions in the ionic cloud takes a diffusive structure due to the thermal motion of ions. The electrostatic interaction between colloidal particles and the motion of colloidal particles in an external field (e.g., electric field and gravitational field) depend strongly on the distributions of electrolyte ions and of the electric potential across the electrical double layer around the particle surface [1–5].

Figure 1.1. Electrical double layer of thickness 1/κ around a spherical charged particle of radius a.

c01f001

1.2 THE POISSON–BOLTZMANN EQUATION

Consider a uniformly charged particle immersed in a liquid containing N ionic species with valence zi and bulk concentration (number density) c01ue001 (i = 1, 2 … N) (in units of cubic meter). From the electroneutrality condition, we have

(1.1) c01e001

The electric potential ψ(r) at position r outside the particle, measured relative to the bulk solution phase, where ψ is set equal to zero, is related to the charge density ρel(r) at the same point by the Poisson equation, viz.,

(1.2) c01e002

where Δ is the Laplacian, εr is the relative permittivity of the electrolyte solution, and εo is the permittivity of a vacuum. We assume that the distribution of the electrolyte ions ni(r) obeys Boltzmann’s law, viz.,

(1.3) c01e003

where ni(r) is the concentration (number density) of the ith ionic species at position r, e is the elementary electric charge, k is Boltzmann’s constant, and T is the absolute temperature. The charge density ρel(r) at position r is thus given by

(1.4)

c01e004

Combining Equations 1.2 and 1.4 gives

(1.5) c01e005

This is the Poisson–Boltzmann equation for the potential distribution ψ(r), which is subject to the following boundary conditions:

(1.6) c01e006

and

(1.7) c01e007

If the internal electric fields inside the particle can be neglected, then the surface charge density σ of the particle is related to the potential derivative normal to the particle surface as

(1.8) c01e008

where n is the outward normal at the particle surface.

1.3 LOW POTENTIAL CASE

If the potential ψ is low, viz.,

(1.9) c01e009

then Equation 1.5 reduces to the following linearized Poisson–Boltzmann equation (Debye–Hückel equation):

(1.10) c01e010

with

(1.11) c01e011

where κ is called the Debye–Hückel parameter. The reciprocal of κ (i.e., 1/κ), which is called the Debye length, corresponds to the thickness of the double layer. Note that c01ue002 in Equations 1.5 and 1.10 is given in units of cubic meter. If one uses the units of M (mole per liter), then c01ue003 must be replaced by 1000 c01ue004 , NA being Avogadro’s number. Expressions for κ for various types of electrolytes are explicitly given in Table 1.1.

Linearized Equation 1.10 can be solved for particles of various shapes. Table 1.2 gives the potential distribution for a planar surface, a sphere of radius a, and a cylinder of radius a, which can be obtained by solving Equation 1.10 (with Δ = d²/dx² for a planar surface, Δ = d²/dr² + 2/r·d/dr for a sphere, and Δ = d²/dr² + 1/r·d/dr for a cylinder) subject to Equations 1.6 and 1.7, where x is the distance from the planar surface located at x = 0 and r is the distance from the sphere center or the cylinder axis. Table 1.2 also shows the surface potential ψo/surface charge density σ relationship, which can be obtained by substituting ψ into Equation 1.8.

TABLE 1.1. Debye–Hückel Parameter for Various Electrolytes

TABLE 1.2. Solution to the Linearized Poisson–Boltzmann Equation

Note: x (>0) is the distance from the planar surface and r (>a) is the distance from the center O of the sphere or that from the axis of the cylinder. Kn(z) is the modified Bessel function of the second kind of order n.

1.4 ARBITRARY POTENTIAL CASE

The nonlinear Poisson–Boltzmann Equation 1.5 (with Δ = d²/dx²) for a planar surface can be solved analytically. For a planar surface in contact with a z-z symmetrical electrolyte solution, a 2-1 electrolyte solution, or a mixed solution of 1-1 electrolyte of bulk concentration n1 and 2-1 electrolyte of bulk concentration n2, the potential distribution ψ(x) and the surface potential ψo/surface charge density σ relationship are given in Table 1.3.

TABLE 1.3. Potential Distribution ψ(x), Surface Potential ψo/Surface Charge Density σ Relationship, and Effective Surface Potential ψeff for a Planar Surface with Arbitral Surface Potential in an Electrolyte Solution

c01t00829qs

Note: c01ue030

c01ue031 , c01ue032 , c01ue033 , c01ue072 .

Consider the asymptotic behavior of the potential distribution at large distances, which will also be used for calculating the electrostatic interaction between two particles. When a planar surface is in contact with a z-z symmetrical electrolyte, the potential distribution ψ(x) around the surface (see Table 1.3) in the region far from the surface, that is, at large κx, takes the form

(1.12) c01e012

Comparing Equation 1.12 with the linearized form ψ(x) = ψoexp(−κx) (see Table 1.2), we find that the effective surface potential ψeff of the plate is given by

(1.13) c01e013

This result, together with those for other types of electrolytes, is given in Table 1.3.

For a sphere, the nonlinear Poisson–Boltzmann equation has not been solved analytically. Loeb et al. [6] tabulated numerical computer solutions to the nonlinear spherical Poisson–Boltzmann equation and approximate analytic solutions are given in References 7–9 (Table 1.4). For the case of an infinitely long cylindrical particle of radius a, approximate solutions are derived in References 7 and 10 (Table 1.5).

TABLE 1.4. Potential Distribution ψ(r) and Surface Potential ψo/Surface Charge Density σ Relationship for a Sphere of Radius a with Arbitrary Surface Potential

Note: c01ue039 , c01ue040 , c01ue041 , c01ue042 , c01ue043 , c01ue044 .

TABLE 1.5. Potential Distribution ψ(r) and Surface Potential ψo/Surface Charge Density σ Relationship for a Cylinder of Radius a with Arbitrary Surface Potential

Note: c01ue049 , c01ue050 , c01ue051 , c01ue052 , c01ue053 , c01ue073 .

1.5 SOFT PARTICLES

We consider the case where the particle core is covered by an ion-penetrable surface layer of polyelectrolytes, which we term a surface charge layer (or, simply, a surface layer). Polyelectrolyte-coated particles are often called soft particles (Fig. 1.2) [3–5]. Soft particles serve as a model for biocolloids such as cells. Figure 1.3 gives a schematic representation of ion and potential distributions around a hard surface (Fig. 1.3a) and a soft surface (Fig. 1.3b), which shows that the potential deep inside the surface layer is practicably equal to the Donnan potential ψDON, if the surface layer is much thicker than the Debye length 1/κ. Also we term ψo ≡ ψ(0) (which is the potential at the boundary between the surface layer and the surrounding electrolyte solution) the surface potential of the polyelectrolyte layer.

Figure 1.2. Soft particle (polyelectrolyte-coated particle).

c01f002

Figure 1.3. Ion and potential distribution around a hard surface (a) and a soft surface (b). When the surface layer is thick, the potential deep inside the surface layer becomes the Donnan potential.

c01f003

Consider a surface charge layer of thickness d coating a planar hard surface in a general electrolyte solution containing M ionic species with valence zi and bulk concentration (number density) c01ue005 (i = 1, 2, … , M). We treat the case where fully ionized groups of valence Z are distributed at a uniform density of N in the surface charge layer and the particle core is uncharged. We take an x-axis perpendicular to the surface charge layer with its origin x = 0 at the boundary between the surface charge layer and the surrounding electrolyte solution so that the surface charge layer corresponds to the region −d < x < 0 and the electrolyte solution to x > 0 (Fig. 1.3b). The Poisson–Boltzmann equations for the regions inside and outside the surface charge layer are given by

(1.14)

c01e014

and

(1.15)

c01e015

We have here assumed that the relative permittivity εr takes the same value in the regions inside and outside the surface charge layer. Note that the right-hand side of Equation 1.15 contains the contribution of the fixed charges of density ρfix = ZeN in the polyelectrolyte layer. The boundary conditions are

(1.16) c01e016

(1.17) c01e017

and

(1.18) c01e018

Equation 1.16 corresponds to the situation in which the particle core is uncharged.

Table 1.6 gives the potential distribution and the surface potential ψo/charge density N for the low potential case. Table 1.6 also gives the results for a soft sphere or a soft cylinder (i.e., a hard sphere or cylinder of radius a covered by a surface layer of thickness d = b − a). Table 1.7 gives the results for the case where a planar soft surface is in contact with a z-z symmetrical electrolyte solution and the thickness of the surface layer d is much greater than the Debye length 1/κ.

TABLE 1.6. Solution to the Linearized Poisson–Boltzmann Equation for Soft Particles with Low N

TABLE 1.7. Potential Distribution ψ(r), Surface Potential ψo/Charge Density N Relationship, and the Effective Surface Potential ψeff for a Planar Soft Surface with a Thick Surface Charge Layer Carrying Arbitrary N

Note: c01ue070

and κm is the Debye–Hückel parameter in the surface charge layer.

The potential distribution outside the surface charge layer of a soft particle with surface potential ψo is the same as the potential distribution around a hard particle with a surface potential ψo. The asymptotic behavior of the potential distribution around a soft particle and that around a hard particle are the same, and thus their effective surface potentials are also the same provided they have the same surface potential ψo (Table 1.7). It must be stressed here that for a hard plate, ψo is related to the surface charge density, σ, while for a soft plate, ψo is related to the volume charge density ρfix = ZeN.

REFERENCES

1 B. V. Derjaguin, L. Landau. Acta Physicochim. 14 (1941) 633.

2 E. J. W. Verwey, J. Th. G. Overbeek. Theory of the stability of lyophobic colloids. Elsevier, Amsterdam, 1948.

3 H. Ohshima, K. Furusawa (eds.), Electrical phenomena at interfaces, fundamentals, measurements, and applications, 2nd ed., revised and expanded. Dekker, New York, 1998.

4 H. Ohshima. Theory of colloid and interfacial electric phenomena. Elsevier/Academic Press, Amsterdam, 2006.

5 H. Ohshima. Biophysical chemistry of biointerfaces. John Wiley & Sons, Hoboken, NJ, 2010.

6 A. L. Loeb, J. Th. G. Overbeek, P. H. Wiersema. The electrical double layer around a spherical colloid particle. MIT Press, Cambridge, MA, 1961.

7 H. Ohshima, T. W. Healy, L. R. White. J. Colloid Interface Sci. 90 (1982) 17.

8 H. Ohshima. J. Colloid Interface Sci. 171 (1995) 525.

9 H. Ohshima. J. Colloid Interface Sci. 174 (1995) 45. Effective surface potential.

10 H. Ohshima. J. Colloid Interface Sci. 200 (1998) 291.

2

ELECTROSTATIC INTERACTION BETWEEN TWO COLLOIDAL PARTICLES

Hiroyuki Ohshima

2.1 INTRODUCTION

The Derjaguin–Landau–Verwey–Overbeek (DLVO) theory reveals that when two charged particles approach each other on an electrolyte solution, the electrical double layers around the particles overlap, resulting in an electrostatic interaction between them [1–9]. In this chapter, we derive expressions for the potential energy of the electrostatic interaction between charged particles for various cases.

2.2 INTERACTION BETWEEN TWO COLLOIDAL PARTICLES: LOW POTENTIAL CASE

Consider first two parallel dissimilar plates, plates 1 and 2, separated by a distance h immersed in a liquid containing N ionic species with valence zi and bulk concentration (number density) c02ue003 (i = 1, 2 … N). We take an x-axis perpendicular to the plates with its origin at the right surface of plate 1, as in Figure 2.1. We assume that the electric potential ψ(x) outside the plates (0 < x ≤ h) obeys the following one-dimensional planar Poisson–Boltzmann equation:

(2.1)

c02e001

Figure 2.1. Interaction between two parallel dissimilar plates, plates 1 and 2.

c02f001

The boundary conditions at the plate surface depends on the type of the double-layer interaction between plates 1 and 2. If the surface potentials of plates 1 and 2 remain constant at their unperturbed surface potentials ψo1 and ψo2 during interaction, respectively (interaction at constant surface potential), then

(2.2) c02e002

and

(2.3) c02e003

On the other hand, if the surface charge densities of plates 1 and 2 remain constant at σ1 and σ2 (electrostatic interaction at constant surface charge density), then

(2.4) c02e004

and

(2.5) c02e005

where we have assumed that the influence of the external fields within the plates can be neglected.

The interaction force P can be calculated by integrating the excess osmotic pressure and the Maxwell stress over an arbitrary closed surface, Σ, enclosing either one of the two interacting plates. As Σ, we choose two planes located at x = −∞ (in the bulk solution far from the plates) and x = x′ (0 < x′ < h) enclosing plate 1. Here x′ is an arbitrary point near the midpoint in the region 0 < x < h between plates 1 and 2. Thus, the force P(h) of the double-layer interaction per unit is given by [1–5]

(2.6)

c02e006

which, for the low potential case, can be linearized to

(2.7) c02e007

Here P(h) > 0 corresponds to repulsion and P(h) < 0 to attraction. The corresponding interaction energy Vpl(h) between two parallel plates per unit area can be obtained by integrating Ppl(h) with the result that

(2.8) c02e008

For the low potential case, Equation 2.1 can be linearized so that we can easily obtain expressions for the interaction energy between two parallel plates at separation h.

Once the interaction energy Vpl(h) between two parallel plates at separation h is obtained, one can derive the expression for the interaction energy Vsp(H) between two spheres of radii a1 and a2 at separation H between their surfaces and the interaction energy Vcy//(H) or Vcy⊥(H) between two parallel or crossed cylinders of radii a1 and a2 at separation H with the help of Derjaguin’s approximation [10–12], provided that κa1 » 1, κa2 » 1, and H « a1, a2. Figure 2.2 shows Derjaguin’s approximation to calculate the sphere–sphere interaction energy [10] and the cylinder–cylinder interaction energy [11, 12]. Figures 2.3 and 2.4 give expressions for Vpl(h), Vsp(H), Vcy//(H), and Vcy⊥(H) for the electrostatic interaction energy between two various hard particles. Figures 2.5 and 2.6 give expressions for Vpl(h), Vsp(H), Vcy//(H), and Vcy⊥(H) for the electrostatic interaction energy between two various soft particles carrying fixed charge densities ρfix1 and ρfix2.

Figure 2.2. Derjaguin’s approximation for sphere–sphere interaction [10] and cylinder–cylinder interaction [11, 12].

c02f002

Figure 2.3. Interaction energies between two parallel plates and between two hard spheres. The superscripts ψ, σ, and ψ–σ, respectively, corresponds to interactions at constant surface potential [13], constant surface charge density [14], and the mixed case in which particle 1 is at constant surface potential and particle 2 is at constant surface charge density [15]. ψo1 = σ1/εrεoκ and ψo2 = σ2/εrεoκ, respectively, are the unperturbed surface potentials of plates 1 and 2 or spheres 1 and 2 with κa1 » 1 and κa2 » 1.

c02f003

Figure 2.4. Interaction energies between two parallel or crossed hard cylinders [11, 12]. c02ue001 is the Lerch transcendent.

c02f004

Figure 2.5. Interaction energies between two parallel soft plates [16] and between two soft spheres [17]. ρfix1 and ρfix2, respectively, are the fixed charge densities in the surface layers on particles 1 and 2. b1 = a1 + d1 and b2 = a2 + d2.

c02f005

Figure 2.6. Interaction energies between two parallel soft cylinders 1 and 2 and between two crossed soft cylinders 1 and 2 [12]. ρfix1 and ρfix2, respectively, are the fixed charge densities in the surface layers on particles 1 and 2. b1 = a1 + d1 and b2 = a2 + d2. Lis(z) is the polylogarithm function, defined by c02ue002 .

c02f006

2.3 LINEAR SUPERPOSITION APPROXIMATION (LSA)

Although it is generally difficult to derive the interaction energy between two particles with an arbitrary surface potential, a simple approximation method (the LSA) is available to derive the interaction energy between particles at large particle separations, κh » 1 or κH » 1. Expressions for the interaction energy obtained with the help of the LSA, which do not depend on the type of the double-layer interaction, can be applied both to the constant surface potential and constant surface charge density cases as well as their mixed case. In addition, the results obtained on the basis of the LSA can be applied not only to hard particles but also to soft particles.

We now derive the interaction energy between two parallel plates at separation h with arbitrary (unperturbed) surface potentials (Fig. 2.7). We take an x-axis perpendicular to the plates with its origin at the surface of plate 1. The asymptotic forms of the unperturbed potentials ψ1(x) and ψ2(x) produced at large distances by plates 1 and 2, respectively, in the absence of interaction are given by Equation 1.12, viz.,

(2.9) c02e009

and

(2.10) c02e010

where ψeff1 and ψeff2, respectively, are the effective surface potentials of plates 1 and 2 (given in Chapter 1, Table 1.3). The potential distribution ψ(x) around the midpoint between plates 1 and 2 can thus be approximated by

(2.11) c02e011

Figure 2.7. Linear superposition approximation.

c02f007

By substituting Equation 2.11 into Equation 2.7, we obtain

(2.12) c02e012

The interaction energy V(h) per unit area of plates 1 and 2 is thus

(2.13) c02e013

From Equation 2.13 as combined with Derjaguin’s approximation, the interaction energy between two spheres or cylinders of radii a1 and a2 at large separations can be derived. The results are given in Figure 2.8. These results are also applicable to the interaction between soft spheres or cylinders consisting of hard spheres or cylinders of radii a1 and a2 covered with surface charge layers of thicknesses d1 and d2, in which case a1 and a2 must be replaced by b1 = a1 + d1 and a2 + d2.

Figure 2.8. Interaction energy Vpl(h) between two parallel plates, plates 1 and 2, with effective surface potentials ψeff1 and ψ eff2 at separation h. Vsp(H) and Vcy(H) are obtained with the help of Derjaguin’s approximation.

c02f008

REFERENCES

1 B. V. Derjaguin, L. Landau. Acta Physicochim. 14 (1941) 633.

2 E. J. W. Verwey, J. Th. G. Overbeek. Theory of the stability of lyophobic colloids. Elsevier, Amsterdam, 1948.

3 B. V. Derjaguin. Theory of stability of colloids and thin films. Consultants Bureau, New York, London, 1989.

4 J. N. Israelachvili. Intermolecular and surface forces, 2nd ed., Academic Press, New York, 1992.

5 J. Lyklema. Fundamentals of interface and colloid science, solid-liquid interfaces, vol. 2. Academic Press, New York, 1995.

6 T. F. Tadros (ed.), Colloid stability, the role of surface forces—Part 1. Wiley-VCH, Weinheim, 2007.

7 H. Ohshima, K. Furusawa (eds.), Electrical phenomena at interfaces, fundamentals, measurements, and applications, 2nd ed., revised and expanded. Dekker, New York, 1998.

8 H. Ohshima. Theory of colloid and interfacial electric phenomena. Elsevier/Academic Press, Amsterdam, 2006.

9 H. Ohshima. Biophysical chemistry of biointerfaces. John Wiley & Sons, Hoboken, 2010.

10 B. V. Derjaguin. Kolloid Z. 69 (1934) 155.

11 M. J. Sparnaay. Recueil 712 (1959) 6120.

12 H. Ohshima, A. Hyono. J. Colloid Interface Sci. 333 (2009) 202.

13 R. Hogg, T. W. Healy, D. W. Fuerstenau. Trans. Faraday Soc. 62 (1966) 1638.

14 G. R. Wiese, T. W. Healy. Trans. Faraday Soc. 66 (1970) 490.

15 G. Kar, S. Chander, T. S. Mika. J. Colloid Interface Sci. 44 (1973) 347.

16 H. Ohshima, K. Makino, T. Kondo. J. Colloid Interface Sci. 116 (1987) 196.

17 H. Ohshima. J. Colloid Interface Sci. 328 (2008) 3.

3

THE DERJAGUIN–LANDAU–VERWEY–OVERBEEK (DLVO) THEORY OF COLLOID STABILITY

Hiroyuki Ohshima

3.1 INTRODUCTION

The stability of colloidal systems consisting of charged particles can be essentially explained by the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory [1–8]. According to this theory, the stability of a suspension of colloidal particles is determined by the balance between the electrostatic interaction (Chapter 2) and the van der Waals interaction between particles. In this chapter, we first consider the van der Waals interaction between particles.

3.2 THE VAN DER WAALS INTERACTION BETWEEN MOLECULES

A neutral molecule can have a fluctuating instantaneous electric dipole originating from quantum mechanics. When two molecules are approaching each other, attractive intermolecular forces, which are called the van der Waals forces, are acting due to interactions between the fluctuating dipoles. The interaction energy u(r) between molecules 1 and 2 at separation r (Fig. 3.1) is given by

(3.1) c03e001

with

(3.2) c03e002

where C12 is called the London–van der Waals constant; αi and νi, respectively, are the electronic polarizability and the characteristic frequency of molecule i (i = 1, 2); and h is the Planck constant. Thus, hνi corresponds to the ionization energy of molecule i.

Figure 3.1. The van der Waals interaction energy u between molecules 1 and 2 at separation r and the interaction energy V between particles 1 and 2 of volumes V1 and V2 and molecular densities N1 and N2, respectively. C12 is the London–van der Waals constant.

c03f001

3.3 THE VAN DER WAALS INTERACTION BETWEEN PARTICLES

A most remarkable characteristic of the van der Waals interaction is that the additivity of interactions approximately holds. The interaction energy between two particles in a vacuum may thus be calculated approximately by a summation (or by an integration) of the interaction energies for all molecular pairs formed by two molecules belonging to different particles [9] (Fig. 3.1); that is, the interaction energy V between particles 1 and 2, containing N1 and N2 molecules per unit volume, respectively, can be obtained by the integration of the interaction energy u between two volume elements, dV1 and dV2, at separation r, containing N1dV1 and N2dV2 molecules, respectively, over volumes V1 and V2 of particles 1 and 2, viz.,

(3.3) c03e003

Expressions for the van der Waals interaction energy V between a molecule and a particle and between various particles have been derived [1–13] and are given in Figures 3.2–3.4.

Figure 3.2. The van der Waals interaction energy V between various particles and a molecule and between various particles. N, N1, and N2 are the line densities of molecules along rods and rings or the volume densities of molecules in a plate, a sphere, or a cylinder. C12 is the London–van der Waals constant. F(α, β, γ; x) is the hypergeometrical function defined by

c03ue001

, where Γ(z) is the gamma function.

c03f002

Figure 3.3. The van der Waals interaction energy V between two parallel plates, plates 1 and 2, of thicknesses d1 and d2, respectively, separated by distance h, and between spheres 1 and 2 of radii a1 and a2, respectively, separated by a distance R between their centers (or H between their closest surfaces). A is the Hamaker constant for the interaction between particles 1 and 2.

c03f003

Figure 3.4. The van der Waals interaction energy V between two parallel or crossed cylinders, cylinders 1 and 2, of radii a1 and a2, respectively, separated by distance R between their axes (or H between their closest surfaces). A is the Hamaker constant.

c03f004

Equation 3.2 is based on the assumption that the interacting particles are in a vacuum. If the medium between the particles is no longer a vacuum but a second substance (e.g., water), we must account for the fact that each particle replaces an equal volume of water. It follows from the assumption of the additivity of the van der Waals interaction energies that the Hamaker constant A132 for the interaction of two different particles of substances 1 and 2 immersed in a medium of substance 3 is

(3.4) c03e004

where A1 and A2, respectively, are referred to the interaction between two similar particles of substances 1 and 2 immersed in a vacuum. Thus, if two particles of substances 1 and 2 are interacting in medium 3, then A12 appearing in Figures 3.3 and 3.4 must be replaced by A132.

3.4 DLVO THEORY OF COLLOID STABILITY

A number of studies on colloid stability are based on the DLVO theory. As an example, we consider the interaction between two identical spherical particles of radius a carrying unperturbed surface potential ψo at separation H between their surfaces in a z-z symmetrical electrolyte solution of bulk concentration (number density) n (Fig. 3.5). The electrostatic interaction energy Vel(H) between the spheres is given by (from Fig. 2.8 in Chapter 2 and Table 1.3 in Chapter 1)

(3.5) c03e005

where γ = tanh(zeψo/kT) and κ is the Debye–Hückel parameter. The van der Waals interaction energy VvdW(H) between the spheres is given by (from Fig. 3.3)

(3.6) c03e006

Figure 3.5. Interaction between two identical hard spheres of radius a in an electrolyte solution.

c03f005

The total interaction energy Vt(H) = Vel(H) + VvdW(H) between the spheres is thus given by

(3.7) c03e007

Figures 3.6 and 3.7 show examples of the calculation of V(H) for the interaction between two spheres of radius a = 100 nm and A = 5 × 10−21 J, where the electrolyte concentration n (in units of cubic meter) is converted to C (in units of molar, M) by n = 1000 NAC, where NA = Avogadro’s constant. Figure 3.6 shows the interaction energy V at C = 0.01 M for various values of the particle surface potential ψo. Figure 3.7 plots the interaction energy V at ψo = 20 mV for various values of electrolyte concentration C. As seen from Figures 3.6 and 3.7, there can exist a maximum Vmax in the potential curves for particles with high surface potentials in low electrolyte concentrations. The DLVO theory predicts that a suspension of colloidal particles should be stable if the potential maximum Vmax is sufficiently higher than the thermal energy kT (which amounts to ca. 4 × 10−21 J at room temperature).

Figure 3.6. Interaction energy V between two identical hard spheres of radius a = 100 nm at C = 0.01 M for various values of the particle surface potential ψo. A = 5 × 10−21 J.

c03f006

Figure 3.7. Interaction energy V between two identical hard spheres of radius a = 100 nm at ψo = 20 mV for various values of electrolyte concentration C. A = 5 × 10−21 J.

c03f007

As shown in Figure 3.7, at a certain electrolyte concentration Ccr, the potential maximum becomes zero. Thus, at this concentration, which is called the critical aggregation concentration, the suspension becomes unstable, resulting in aggregation of the suspension. The critical coagulation concentration Ccr is given by the solution to the following coupled equations: V(H) = 0 and dV/dH = 0, with the solution

(3.8) c03e008

For particles with high surface potentials ψo, γ becomes unity. Thus, from Equation 3.8, we find that

(3.9) c03e009

which is in consistent with Schulze–Hardy’s empirical formula.

REFERENCES

1 B. V. Derjaguin, L. D. Landau. Acta Physicochim. 14 (1941) 633.

2 E. J. W. Verwey, J. Th. G. Overbeek. Theory of the stability of lyophobic colloids. Elsevier/Academic Press, Amsterdam, 1948.

3 B. V. Derjaguin. Theory of stability of colloids and thin films. Consultants Bureau, New York, London, 1989.

4 J. N. Israelachvili. Intermolecular and surface forces, 2nd ed. Academic Press, New York, 1992.

5 J. Lyklema. Fundamentals of interface and colloid science, solid-liquid interfaces, vol. 2. Academic Press, New York, 1995.

6 H. Ohshima, in H. Ohshima, K. Furusawa (eds.), Electrical phenomena at interfaces, fundamentals, measurements, and applications, 2nd ed., revised and expanded. Dekker, New York, 1998, Chapter 3.

7 T. F. Tadros (ed.), Colloid stabolity, the role of surface forces—Part 1. Wiley-VCH, Weinheim, 2007.

8 H. Ohshima. Biophysical chemistry of biointerfaces. John Wiley & Sons, Hoboken, 2010.

9 H. C. Hamaker. Physica 4 (1937) 1058.

10 M. J. Sparnaay. Recueil 78 (1959) 680.

11 D. Langbein. Theory of van der Waals attraction. Springer, Berlin, 1974.

12 V. A. Kirsch. Adv. Colloid Interface Sci. 104 (2003) 311.

13 H. Ohshima, A. Hyono. J. Colloid Interface Sci. 332 (2009) 251.

4

ELECTROPHORETIC MOBILITY OF CHARGED PARTICLES

Hiroyuki Ohshima

4.1 INTRODUCTION

When an external electric field is applied to a suspension of charged colloidal particles in a liquid, the particles in the stationary state move with a constant velocity as a result of the balance between the applied field and a viscous resistance exerted by the liquid on the particles. This phenomenon is called electrophoresis. The particle velocity, which is called electrophoretic velocity, depends strongly on the electrical diffuse double layer around the particles and the particle zeta potential ζ [1–27]. The zeta potential ζ is defined as the potential at the plane where the liquid velocity relative to the particle is zero. This plane is called the slipping plane or the shear plane. The slipping plane does not necessarily coincide with the particle surface. Only if the slipping plane is located at the particle surface, the zeta potential ζ becomes equal to the surface potential ψo. In this chapter, we treat the case where ζ = ψo and where particles are symmetrical (spherical or cylindrical). Usually, we treat the case where the magnitude of the applied electric field E is not very large. In such cases, the electrophoretic velocity U of the particles is proportional to E in magnitude. The ratio of the magnitude of the velocity U to that of the applied field E is called electrophoretic mobility, μ, which is defined by μ = U/E (where U = |U| and E = |E|). In this chapter, we derive equations for the electrophoretic mobilities of hard and soft colloidal particles.

4.2 GENERAL THEORY OF ELECTROPHORETIC MOBILITY OF HARD PARTICLES

Full electrokinetic equations determining electrophoretic mobility of spherical particles with arbitrary values of κa and ζ were derived independently by Overbeek [4] and Booth [5]. Wiersema, Loeb, and Overbeek [6] solved the equations numerically using an electronic computer. The computer calculation of the electrophoretic mobility of a spherical particle was considerably improved by O’Brien and White [9]. Approximate analytic mobility expressions, on the other hand, have been proposed by various authors [1–5, 7, 8, 10, 12, 17, 20–27].

Consider a spherical hard particle of radius a moving with a velocity U in a liquid containing a general electrolyte composed of N ionic species with valence zi and bulk concentration (number density) c04ue001 , and drag coefficient λi (i = 1, 2 … N). The origin of the spherical polar coordinate system (r, θ, ϕ) is held fixed at the center of the particle and the polar axis (θ = 0) is set parallel to E (Fig. 4.1). For a spherical particle, U takes the same direction as E.

Figure 4.1. Electrophoresis of a spherical particle of radius a with a velocity U in an applied electric field E.

c04f001

The main assumptions are as follows: (i) The Reynolds number of the liquid flow is small enough to ignore inertial terms in the Navier–Stokes equation and the liquid can be regarded as incompressible; (ii) the applied field E is weak so that the particle velocity U is proportional to E and terms of higher order in E may be neglected; (iii) the slipping plane (at which the liquid velocity relative to the particle becomes zero) is located on the particle surface (at r = a); (iv) no electrolyte ions can penetrate the particle surface.

Electrokinetic equations that govern the motion of charged colloidal particles in a liquid containing an electrolyte caused by an externally applied electric field consist of the Navier–Stokes equation for the liquid flow around the particle, continuity equations for the liquid flow and the flow of the ionic species, and the Poisson equation connecting the electric potential and the concentrations of the ionic species. The Navier–Stokes equation for the liquid flow u(r) around the particle is given by

(4.1) c04e001

where η is the viscosity of the electrolyte solution, p(r) is the pressure at position r, ρel(r) is the charge density resulting from the mobile charged ionic species, ψ(r) is the electric potential, and r = |r|.

Ohshima et al. [12] derived the general expression for the electrophoretic mobility of a sphere:

(4.2) c04e002

with

(4.3) c04e003

where ψ(0) is the equilibrium potential, y is the corresponding scaled quantity, and ϕi is related to the deviation δμi of the electrochemical potential of ith ionic species by

(4.4) c04e004

and

(4.5) c04e005

where the first term r on the right-hand side corresponds to the potential of the applied external field E·r, the second term a³/2r² to the deviation of the external field due to the presence of the particle core, and the third term fi(r) to the contribution of the relaxation effect of electrolyte ions.

4.3 SMOLUCHOWSKI’S, HÜCKEL’S, AND HENRY’S EQUATIONS

It can be shown that for the limiting case of κa → ∞, ϕi becomes

(4.6) c04e006

from which we obtain

(4.7) c04e007

This is Smoluchowski’s equation [1]. In the opposite limit of κa → ∞, ϕi becomes

(4.8) c04e008

so that we have

(4.9) c04e009

which is with Hückel’s equation [2]. For spherical particles with low ζ and arbitrary κa, ϕi becomes

(4.10) c04e010

which gives the following Henry equation [3]:

(4.11) c04e011

where f(κa) is called Henry’s function. Accurate approximate expressions for the mobilities of a sphere and a cylinder are derived in References 22 and 23. Figure 4.2 summarizes the aforementioned results. Similarly, we can derive the corresponding equations for the electrophoretic mobility of an infinitely long cylinders moving in a tangential or transverse electric field. The results are shown in Figure 4.3.

Figure 4.2. Mobility equations for a sphere of radius a. Smoluchowski’s equation for κa » 1, Hückel’s equation for κa « 1, and Henry’s equation for arbitrary κa and low ζ.

c04f002

Figure 4.3. Mobility equations for a cylinder of radius a. Smoluchowski’s equation for κa » 1, Hückel’s equation for κa « 1, and Henry’s equation for arbitrary κa and low ζ.

c04f003

4.4 MOBILITY EQUATIONS TAKING INTO ACCOUNT THE RELAXATION EFFECT

Henry’s mobility equation, Equation 4.10, assumes that the double layer potential distribution around a spherical particle remains unchanged during electrophoresis (Fig. 4.4). For high zeta potentials, however, the double layer is no longer spherically symmetrical. This effect is called the relaxation effect (the double layer polarization or the surface conduction) (Fig. 3.8). Henry’s equation, Equation 4.10, does not take into account the relaxation effect, and thus, this equation is correct to the first order of zeta potential ζ (Fig. 4.5). If the relaxation effect is taken into account, then the electrophoretic mobility μ becomes dependent on the ionic drag coefficient λi. We introduce the scaled ionic drag coefficient mi, defined by

(4.12) c04e012

Figure 4.4. Henry’s function f(κa) for a sphere [22] and a cylinder [23] of radius a as a function of scaled radius κa. The cylinder is oriented parallel (μ//) or perpendicular (μ⊥) to the applied electric field. For a cylindrical particle oriented at an arbitrary angle between its axis and the applied electric field, its electrophoretic mobility averaged over a random distribution of orientation is given by μav = (μ// + 2μ⊥)/3 [26, 27].

From Reference 20.

c04f004

Figure 4.5. Relaxation effect. The electrical double layer around a sphere with low ζ is spherically symmetrical (a) but loses its spherical symmetry for a sphere with high ζ (b).

c04f005

For an aqueous KCl solution at 25°C, m = 0.176 for K+ ions and m = 0.169 for Cl− ions. In Figure 3.9, we plot the electrophoretic mobility μ of a positively charged sphere in an aqueous KCl solution at 25°C (εr = 78.5 and η = 0.891 mPa/s) as a function of scaled zeta potential eζ/kT for several values of κa calculated by the method of O’Brien and White [9]. Here Em is the scaled electrophoretic mobility defined by (Fig. 4.6)

(4.13) c04e013

Figure 4.6. The scaled electrophoretic mobility Em of a positively charged spherical colloidal particle of radius a in an aqueous KCl solution at 25°C as a function of scaled zeta-potential eζ/kT for various values of κa. Calculated by the method of O’Brien and White [9]. Em is defined by Equation 4.13.

From Reference 20.

c04f006

The cases of κa = ∞ and κa = 0, which are both straight lines, correspond to Smoluchowski’s equation, Equation 4.7, and Hückel’s equation, Equation 4.9, respectively. It is seen that there is a mobility maximum due to the relaxation effect.

Accurate approximate mobility expressions of a sphere of radius a correct to order 1/κa applicable for κa ≥ 10 and arbitrary ζ [12] and that correct to order ζ³ [24] applicable for zeζ/kT ≤ 3 and arbitrary κa are given in Table 4.1. Approximate mobility expressions for a sphere with κa ≥ 30 and arbitrary ζ [25] is given in Table 4.2 (see also Fig. 4.7). In Figure 3.15, we show the range of validity of various mobility expressions: Smoluchowski’s equation (Eq. 4.7) [1], Hückel’s equation (Eq. 4.9) [2], Henry’s equation (Eq. 4.11) [3], Ohshima–Healy–White’s equation (Eq. (1) in Table 4.1) [12], and Ohshima’s equation (Eq. (2) in Table 4.1) [24].

TABLE 4.1. Accurate Mobility Expressions for a Hard Sphere of Radius a

Note: Equation (1) is correct to order 1/κa [12] and Equation (2) is correct to order ζ³ [24].

TABLE 4.2. Approximate Mobility Expressions for a Hard Sphere of Radius a with κa ≥ 30 [24]

c04t0432dnc

Figure 4.7. Range of validity of several mobility expressions. Smoluchowski’s equation (Eq. 4.7) [1], Hückel’s equation (Eq. 4.9) [10], Henry’s equation (Eq. 4.11) [12], Ohshima–Healy–White’s equation (Eq. (2) in Table 4.1) [19], and Ohshima’s equation (Eq. (2) in Table 4.1) [22]. Each area corresponds to the region with relative errors less than 1%. In the region labeled O’Brien–White, only a computer solution is available [16].

From Reference 20.

c04f007

4.5 ELECTROPHORETIC MOBILITY OF SOFT PARTICLES

Consider now a spherical soft particle, that is, a charged spherical particle covered with an ion-penetrable layer of polyelectrolytes moving with a velocity U in a symmetrical electrolyte solution of valence z and bulk concentration (number density) n in an applied electric field E [28–44]. We suppose that the uncharged particle core of radius a is coated with an ion-penetrable layer of polyelectrolytes of thickness d and that ionized groups of valence Z are distributed within the polyelectrolyte layer at a uniform density N so that the polyelectrolyte is later uniformly charged at a constant density, ρfix = ZeN. The polymer-coated particle has thus an inner radius a and an outer radius b = a + d (Fig. 4.8). Let the drag coefficient of coions be λ+ and that of counterions be λ–. We adopt the model of Debye–Bueche [45] wherein the polymer segments are regarded as resistance centers distributed uniformly in the polyelectrolyte layer, exerting frictional forces on the liquid flowing in the polyelectrolyte layer.

Figure 4.8. A spherical soft particle in an external applied electric field E. a = radius of the particle core and d = thickness of the polyelectrolyte layer coating the particle core. b = a + d.

c04f008

The fundamental electrokinetic equations for the liquid velocity u(r) at position r are [20, 21, 28, 32]

(4.14) c04e014

and

(4.15) c04e015

Equation 4.14 is the usual Navier–Stokes equation. The term γu on the left-hand side of Equation 4.15 represents the frictional forces exerted on the liquid flow by the polymer segments in the polyelectrolyte layer, and γ is the frictional coefficient. If it is assumed that each resistance center corresponds to a polymer segment, which in turn is regarded as a sphere of radius ap and the polymer segments are distributed at a uniform volume density of Np in the polyelectrolyte layer, then each polymer segment exerts the Stokes resistance 6πηapu on the liquid flow in the polyelectrolyte layer so that

(4.16) c04e016

The general mobility expression for a soft sphere is given by [32]

(4.17)

c04e017

where

(4.18) c04e018

(4.19)

c04e019

(4.20)

c04e020

(4.21) c04e021

and

(4.22)

c04e022

If we neglect the relaxation effect, we approximate ϕ±(r), where ϕ+(r) and ϕ–(r), respectively, are the ϕi function for coions and counterions in Equation 4.5 by

(4.23) c04e023

In order to take into account the relaxation effect outside the surface charge layer, we approximate ϕ±(r) by [44]

(4.24) c04e024

and

(4.25) c04e025

with

(4.26) c04e026

Here F corresponds to Dukhin’s number, which expresses the relaxation effect with respect to counterions and m- is the scaled drag coefficient of counterions (anions). By using Equations 4.23 or 4.24 and 4.25 for ϕ±(r) and equations in Table 1.7 (Chapter 1) for the potential distribution across the surface charge layer under the conditions κa » 1, κd » 1, λa » 1, and λd » 1, we finally obtain the electrophoretic mobility expressions given in Table 4.3.

TABLE 4.3. Approximate Mobility Expression for a Soft Sphere Consisting of the Particle Core of Radius a Coated with a Surface Layer of Thickness of d

Note: ψo and ψDON, respectively, are the surface potential and the Donnan potential of a soft particle, and κm is the Debye–Hückel parameter in the surface charge layer (see Table 1.7 in Chapter 1). c04ue034

REFERENCES

1 M. von Smoluchowski, in L. Greatz (ed.), Handbuch der Elektrizität und des Magnetismus, vol. 2. Barth, Leipzig, 1921, p. 366.

2 E. Hückel. Phys. Z. 25 (1924) 204.

3 D. C. Henry. Proc. R. Soc. Lond A 133 (1931) 106.

4 J. Th. G. Overbeek. Kolloid-Beihefte 54 (1943) 287.

5 F. Booth. Proc. R. Soc. Lond A 203 (1950) 514.

6 P. H. Wiersema, A. L. Loeb, J. Th. G. Overbeek. J. Colloid Interface Sci. 22 (1966) 78.

7 S. S. Dukhin, N. M. Semenikhin. Kolloid Zh. 32 (1970) 360.

8 S. S. Dukhin, B. V. Derjaguin, in E. Matievic (ed.), In surface and colloid science, vol. 7. Wiley, New York, 1974.

9 R. W. O’Brien, L. R. White. J. Chem. Soc. Faraday Trans. 2 74 (1978) 1607.

10 R. W. O’Brien, R. J. Hunter. Can. J. Chem. 59 (1981) 1878.

11 R. J. Hunter. Zeta potential in colloid science. Academic Press, New York, 1981.

12 H. Ohshima, T. W. Healy, L. R. White. J. Chem. Soc. Faraday Trans. 2 79 (1983) 1613.

13 T. G. M. van de Ven. Colloid hydrodynamics. Academic Press, New York, 1989.

14 R. J. Hunter. Foundations of colloid science, vol. 2. Clarendon Press University Press, Oxford, 1989, Chapter 13.

15 S. S. Dukhin. Adv. Colloid Interface Sci. 44 (1993) 1.

16 J. Lyklema. Fundamentals of interface and colloid science, solid-liquid interfaces, vol. 2. Academic Press, New York, 1995.

17 H. Ohshima, K. Furusawa. Electrical phenomena at interfaces, fundamentals, measurements, and applications, 2nd ed., revised and expanded. Dekker, New York, 1998.

18 A. V. Delgado (ed.), Electrokinetics and electrophoresis. Dekker, New York, 2000.

19 A. Spasic, J.-P. Hsu (eds.), Finely dispersed particles. Micro-. nano-, atto-engineering. CRC Press, Boca Raton, FL, 2005.

20 H. Ohshima. Theory of colloid and interfacial electric phenomena. Elsevier, Amsterdam, 2006.

21 H. Ohshima. Biophysical chemistry of biointerfaces. John Wiley & Sons, Hoboken, 2010.

22 H. Ohshima. J. Colloid Interface Sci. 168 (1994) 269.

23 H. Ohshima. J. Colloid Interface Sci. 180 (1996) 299.

24 H. Ohshima. J. Colloid Interface Sci. 239 (2001) 587.

25 H. Ohshima. Colloids Surf. A Physicochem. Eng. Aspects 267 (2005) 50.

26 W. P. J. T. van der Drift, A. de Keizer, J. Th. G. Overbeek. J. Colloid Interface Sci. 71 (1979) 67.

27 A. de Keizer, W. P. J. T. van der Drift, J. Th. G. Overbeek. Biophys. Chem. 3 (1975) 107.

28 H. Ohshima. J. Colloid Interface Sci. 163 (1994) 474.

29 H. Ohshima. Colloids Surf. A Physicochem. Eng. Asp. 103 (1995) 249.

30 H. Ohshima. Adv. Colloid Interface Sci. 62 (1995) 189.

31 H. Ohshima. Electrophoresis 16 (1995) 136.

32 H. Ohshima. J. Colloid Interface Sci. 228 (2000) 190.

33 J. J. López-García, C. Grosse, J. Horno. J. Colloid Interface Sci. 265 (2003) 327.

34 R. J. Hill, D. A. Saville. Colloids Surf. A Physicochem. Eng. Asp. 267 (2005) 31.

35 S. S. Dukhin, R. Zimmermann, C. Werner. Adv. Colloid Interface Sci. 122 (2006) 93.

36 J. F. L. Duval, H. Ohshima. Langmuir 22 (2006) 3533.

37 H. Ohshima. Electrophoresis 27 (2006) 526.

38 H. Ohshima. Colloid Polym. Sci. 285 (2006) 1411.

39 H. Ohshima. Sci. Technol. Adv. Mater. 10 (2009) 063001.

40 S. Ahualli, M. L. Jiménez, F. Carrique, A. V. Delgado. Langmuir 25 (2009) 1986.

41 C. Grosse, A. V. Dergado. Curr. Opin. Colloid Interface Sci. 15 (2010) 145.

42 J. F. L. Duval, F. Gaboriaud. Curr. Opin. Colloid Interface Sci. 15 (2010) 184.

43 H. Ohshima. J. Colloid Interface Sci. 349 (2010) 641.

44 H. Ohshima. Colloids Surf. A Physicochem. Eng. Asp. 376 (2011) 72.

45 P. Debye, A. Bueche. J. Chem. Phys. 16 (1948) 573.

5

ELECTROPHORETIC MOBILITY OF GOLD NANOPARTICLES

Kimiko Makino and Hiroyuki Ohshima

5.1 INTRODUCTION

When the electrophoretic mobility μ of a particle in an electrolyte solution is measured, the obtained electrophoretic mobility values are usually converted to the particle zeta potential ζ (which is practically equal to the particle surface potential) with the help of a proper relationship between μ and ζ, depending on the ratio of the particle size to the Debye–Hückel parameter κ and on the magnitude of the particle zeta potential [1–17]. For a particle with constant surface charge density, however, the surface charge density σ should be more of characteristic quantity than the zeta potential ζ, since for such particles, the zeta potential is not a constant quantity but depends on the electrolyte concentration. In this chapter, a systematic method is proposed to determine the surface charge density of a spherical colloidal particle on the basis of the particle electrophoretic mobility data [18]. This method is based on two analytic equations, that is, the relationship between the electrophoretic mobility and the zeta potential of the particle and the relationship between the zeta potential and the 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 [19].

5.2 ELECTROPHORETIC MOBILITY–ZETA POTENTIAL RELATIONSHIP

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 λ+ and that of anions λ− may be different. The drag coefficients λ± of cations and anions are related to their limiting conductances c05ue001 by

(5.1) c05e001

where NA is Avogadro’s number and e is the elementary electric charge.

As mentioned in the Introduction in Section 5.1, usually, the measured value of the electrophoretic mobility μ of a particle is converted to the particle zeta potential ζ 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, given by

(5.2) c05e002

where k is Boltzmann’s constant and T is the absolute temperature. 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 formula [1],

(5.3) c05e003

where εr and η, respectively, are the relative permittivity and viscosity of the electrolyte solution, and εo is the permittivity of a vacuum. This formula is applicable when κa is sufficiently large, that is, κa » 1. For arbitrary values of κa, the mobility becomes a function of κa. For low ζ, the mobility is given by Henry’s formula [3]:

(5.4) c05e004

where En(κa) is the exponential integral of order n and is defined by

(5.5) c05e005

Ohshima [11] showed that Equation 5.4 can be approximated with negligible errors by

(5.6) c05e006

In the limit of large κa, Henry’s formula (Eq. 5.4) tends to Smoluchowski’s formula (Eq. 5.3), while in the opposite limit (κa → 0), it becomes Hückel’s formula [2], given by

(5.7) c05e007

Equations 5.2, 5.4 (or Eq. 5.6), and 5.7 are correct to the order of ζ so that the relaxation effect is not taken into account. Overbeek [4] derived a mobility expression correct to the order of ζ³, which takes into account the relaxation effect. Following Overbeek [4], we write a mobility expression correct to the order of ζ³ in the following form:

(5.8)

c05e008

where f1(κa) − f3(κa) are some functions of κa, and m+ and m− are dimensionless ionic drag coefficients, defined by

(5.9) c05e009

Ohshima [12] derived the following approximate mobility expression for Equation 5.8:

(5.10)

c05e010

which is a good approximation for ze|ζ|/kT| ≤ 4. An accurate expression for the electrophoretic mobility applicable for arbitrary ζ and κa ≥ 10 is derived in Reference 9. In the case of a particle in a general electrolyte solution, the readers should be referred to Reference 13.

Equations 5.3, 5.4 (or Eq. 5.6), 5.7, and 5.10 express the electrophoretic mobility μ as a function of ζ and κa, that is, μ = μ (ζ, κa). Figure 5.1 shows the scaled electrophoretic mobility Em = 3ηeμ/2εrεokT of a spherical particle of radius a in a KCl solution at 25°C (m+ for K+ ions = 0.176 and m− for Cl− ions = 0.169) calculated via Equation 5.10 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 regarded as constant.

Figure 5.1. Scaled electrophoretic mobility Em = 3ηeμ/2εrεokT of a spherical colloidal particle of radius a in an aqueous KCl solution at 25°C (m+ = 0.176 and m− = 0.169) as a function of κa for several values of the scaled particle zeta potential eζ/kT.

From Reference 18.

c05f001

5.3 ZETA POTENTIAL–SURFACE CHARGE DENSITY RELATIONSHIP

Most colloidal particles, however, carry not 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 more characteristic quantity than the zeta potential ζ, which thus becomes a function of σ and κa. In order 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 that

(5.11)

c05e011

The maximum relative error of Equation 5.11 is 4% at κa = 0.1 and is less than 1% for κa ≥ 1. Figure 5.2 gives the scaled zeta potential eζ/kT of a sphere of radius a as a function of κa, calculated from Equation 5.11 with z = 1. Figure 5.2 shows the significant dependence of ζ upon κa (i.e., upon electrolyte concentration n). Thus, for a particle with a constant surface charge density, σ, independent of electrolyte concentration, the particle surface potential becomes a function of the electrolyte concentration. For such a situation, in order to estimate the electrolyte concentration dependence of μ, Equations 5.2, 5.4 (or Eq. 5.6), 5.7, and 5.10 should be rewritten as functions of σ and κa, that is, μ = μ (σ, κa), instead of as functions of ζ and κa, that is, μ = μ (ζ, κa).

Figure 5.2. Scaled zeta potential eζ/kT of a particle of radius a as a function of κa for several values of scaled surface charge density σ* = eaσ/εrεokT.

From Reference 18.

c05f002

5.4 ELECTROPHORETIC MOBILITY–SURFACE CHARGE DENSITY RELATIONSHIP

Therefore, Figure 5.1 does not properly represent the electrolyte concentration dependence of the electrophoretic mobility μ of a spherical particle with a constant surface charge density, σ. We plot the electrophoretic mobility μ as a function of κa (μ = μ (σ, κa)) for several values of the scaled surface charge density, σ* = eaσ/εrεokT, in Figure 5.3. 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 Equations 5.10 and 5.11, that is, μ = μ(κa, σ) instead of μ = μ(κa, ζ).

Figure 5.3. Scaled electrophoretic mobility Em = 3ηeμ/2εrεokT of a spherical colloidal particle of radius a in an aqueous KCl solution at 25°C (m+ = 0.176 and m− = 0.169) as a function of κa at several values of the scaled surface charge density σ* = eaσ/εrεokT for several values of the particle zeta potential eζ/kT.

From Reference 18.

c05f003

5.5 ANALYSIS OF ELECTROPHORETIC MOBILITY OF GOLD NANOPARTICLES

In this section, 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 [18]. Figures 5.4–5.6 exhibit the electrophoretic mobility data on gold nanoparticles of different sizes: a = 7.5 nm (Fig. 5.4), 25 nm (Fig. 5.5), and 100 nm (Fig. 5.6), as a function of electrolyte concentration n (molar, M) in an aqueous KCl solution at 25°C (experimental data are taken from Reference 19) in comparison with the results calculated with Equation 5.10 as combined with Equation 5.11 for σ = −0.03 C/m² (Fig. 5.4), σ = −0.02 C/m² (Fig. 5.5), and σ = −0.01 C/m² (Fig. 5.6) [18]. The values of κa are plotted on the upper abscissa of each of Figures 5.4–5.6 and are shown as functions of electrolyte concentration n in Figure 5.7. It is seen that Equations 5.10 and 5.11 with appropriate values of σ are in good agreement with the experimental results. In these figures, the results calculated with Smoluchowski’s formula (Eq. 5.3) and Henry’s formula (Eq. 5.6), both combined with Equation 5.10, are also given, showing only poor agreement with the experimental results. It is also seen that deviation among these three formulas (Eqs. 5.3, 5.6, and 5.10) becomes small for a large particle (Fig. 5.5) but is large for a small particle (Fig. 5.4). In Figure 5.8, the zeta potential ζ predicted for three different gold nanoparticles is given as a function of electrolyte concentration n. The significant dependence of ζ on the electrolyte concentration n is again seen. The observed poor agreement of Smoluchowski’s formula (Eq. 5.3) and Henry’s formula (Eq. 5.6) with the experimental data is due to the fact that the applicability of these formulas with respect to κa and ζ lies outside the region in which these formulas are applicable.

Figure 5.4. Electrophoretic mobility μ of gold nanoparticles of radius 7.5 nm in an aqueous KCl solution as a function of electrolyte concentration n (M) (or κa) in comparison with three theoretical predictions. Solid lines are results for σ = −0.03/m² calculated via Equation 5.9 as combined with Equation 5.10; a dotted line corresponds to Smoluchowski’s formula (Eq. 5.3) and a dashed line to Henry’s formula (Eq. 5.4 or 5.6).