Theory of electrostatics and electrokinetics of soft particles

Abstract

We investigate theoretically the electrostatics and electrokinetics of a soft particle, i.e. a hard particle covered with an ion-penetrable surface layer of polyelectrolytes. The electric properties of soft particles in an electrolyte solution, which differ from those of hard particles, are essentially determined by the Donnan potential in the surface layer. In particular, the Donnan potential plays an essential role in the electrostatics and electrokinetics of soft particles. Furthermore, the concept of zeta potential, which is important in the electrokinetics of hard particles, loses its physical meaning in the electrokinetics of soft particles. In this review, we discuss the potential distribution around a soft particle, the electrostatic interaction between two soft particles, and the motion of a soft particle in an electric field.

Introduction

The potential and charge of colloidal particles play fundamental roles in their interfacial electric phenomena, such as electrostatic interactions between colloidal particles and the motion of colloidal particles in an electric field [1–5]. When a charged colloidal particle is immersed in an electrolyte solution, mobile electrolyte ions, which are charged oppositely to the particle surface and which are called counter ions, tend to approach the particle surface to neutralize the particle surface charges, but the thermal motion of these ions prevents their accumulation so that an ionic cloud is formed around the particle. In the ionic cloud, the concentration of counter ions becomes very high, whereas that of coions (electrolyte ions with charges of the same sign as the particle surface charges) is very low. The ionic cloud, together with the particle surface charge, forms an electrical double layer. Such layer is often called an electrical diffuse double layer, since the distribution of electrolyte ions in the ionic cloud takes a diffusive structure owing to the thermal motion of ions. Here, we focus on 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 [4]. As seen in figure 1, a soft particle becomes a hard particle in the absence of the surface charge layer, while it tends to form a spherical polyelectrolyte (or a porous sphere) in the absence of the particle core. Soft particles serve as a model for biocolloids, such as cells. In such cases, the electrical double layer is formed not only outside but also inside the surface charge layer. Particles without surface structures or their surfaces, on the other hand, are called hard particles or hard surfaces.

Figure 1.

A soft particle becomes a hard particle in the absence of the surface charge layer, while it tends to form a spherical polyelectrolyte (or a porous sphere) in the absence of the particle core [4].

In this article, we start with the potential and charge of a soft particle in an electrolyte solution. We show that the Donnan potential plays an important role in determining the potential distribution across a surface charge layer. Then, we discuss the electrostatic interaction between soft particles. In conventional interaction models, it is assumed that the surface potential or the surface charge density remains constant during interaction. In the electrostatic interaction of soft particles, however, the potential deep inside the surface charge layer, which coats soft particles, remains constant at the Donnan potential. We thus call this type of interaction the Donnan-potential regulated interaction. Finally, we consider the motion of a soft particle in an external electric field. In the usual electrophoresis of hard particles, the particle zeta potential plays an essential role. In the electrophoresis of a soft particle, however, two parameters are involved, that is, the volume charge density of charged groups in the surface layer and the electrophoretic softness of the surface layer.

Charge and potential of a soft particle

Poisson–Boltzmann equation

Consider the simple case of a charged soft plate, that is, a hard plate covered with an ion-penetrable surface layer of polyelectrolytes. Imagine a surface charge layer of thickness d coating a planar hard surface that is immersed in a general electrolyte solution containing M ionic species with valence z_i and bulk concentration (number density in units of m⁻³) n_i^∞ (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 [4]. We assume that the relative permittivity ε_r takes the same value in the regions outside and inside the surface layer. See [6] for a detailed discussion on the equality of the relative permittivity outside and inside the surface charge layer. 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, as in figure 2.

Figure 2.

Schematic representation of ion distribution (upper) and potential distribution ψ(x) (lower) across an ion-penetrable surface charge layer [4].

The charge density ρ_el(x) resulting from the mobile charged ionic species is related to the electric potential ψ(x) by the Poisson equation,

where ε_r is the relative permittivity of the solution, ε_o is the permittivity of a vacuum, and e is the elementary electric charge. Note that the right-hand side of equation (2) contains the contribution of the fixed-charges of density ρ_fix=ZeN in the polyelectrolyte layer. We also assume that the distribution of the electrolyte ions n_i(x) obeys Boltzmann's law. Then, we have

and the charge density ρ_ex(x) at x is given by

The potential ψ(x) at position x in the regions x>0 and –d<x<0 thus satisfies the following Poisson–Boltzmann equations,

with

where y is the scaled potential and κ is the Debye–Hückel parameter of the solution. The boundary conditions are

Equation (9) corresponds to the situation in which the particle core is uncharged.

For the special case, where the electrolyte is symmetrical and has valence z and bulk concentration n, we have

or

with

where y and κ are, respectively, the scaled potential and the Debye–Hückel parameter of a symmetrical electrolyte solution. Note that the definition of y for symmetrical electrolytes contains z.

Donnan potential

If the thickness of the surface layer d is much greater than the Debye length 1/κ, then the potential deep inside the surface layer becomes the Donnan potential ψ_DON, which is obtained by setting the right-hand side of equation (15) or equation (17) to zero, viz.,

Equation (16) may be rewritten in terms of the Donnan potential ψ_DON as

When |ZN/zn|≪1, equation (21) gives the following linearized Donnan potential:

Moreover, we call ψ_o≡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.

For the simple case where ψ(x) is low, equations (14) and (15) can be linearized to

The solution to equations (23) and (24), subject to equations (9)–(13), is given by

and the surface potential ψ₀≡ψ(0) is given by

If we take the limit d→0 in equation (27), keeping the product Nd constant, i.e., keeping constant the total amount of fixed charges σ=ZeNd, then equation (27) becomes

where we have defined σ as

Equation (28) is the surface potential of a hard surface carrying a surface charge density σ.

Note that when κd≫ 1, the potential deep inside the surface layer approaches the linearized Donnan potential (equation (22)), that is,

and that the surface potential ψ_o (equation (27)) becomes half the Donnan potential

Arbitrary potential

The solution to the nonlinear differential equations (16) and (17), which is required for the arbitrary potential case, can be obtained as follows. Equation (16) subject to equations (12) and (13) can be integrated to give [4]

which is further integrated to give

where y_o≡y(0)=ze ψ(0)/kT is the scaled surface potential. Integration of equation (17) subject to equation (9) yields

Equation (34) can be further integrated to give

which determines y as a function of x. Evaluating equation (35) at x=−d, we obtain

On the other hand, evaluating equations (33) and (35) at x=0 and substituting the results into equation (11), we obtain

which can be rewritten as

Here equation (20) (i.e. sinh y_DON=zN/zn) has been used and y_DON≡ze ψ _DON/kT is the scaled Donnan potential. Equations (36) and (37) (or equation (38)) form coupled transcendental and integral equations for y_o and y(−d). By using the obtained values for y_o and y(−d), the potential y at an arbitrary point x can be calculated from equations (33) and (35).

Consider the case where the thickness of the surface layer d is much larger than the Debye length 1/κ. In this case, the electric field (dψ/dx) and its derivative (d²ψ/dx²) deep inside the surface layer become zero so that the potential deep inside the surface layer becomes the Donnan potential ψ_DON given by equation (20). Note that this is valid only for d≫ 1/κ. If the condition d≫1/κ does not hold, there is no region where the potential reaches the Donnan potential. When d≫1/κ, by replacing y(−d) by y_DON in equation (38), we obtain [4]

or

An approximate expression for the potential ψ(x) at an arbitrary point in the surface layer can be obtained by expanding y(x) in equation (35) around y_DON. Alternatively, it can be obtained in the following way. If we substitute ψ=ψ_DON+Δψ in equation (21) and linearize it with respect to Δψ, we obtain

with

where κ_m may be interpreted as the effective Debye–Hückel parameter of the surface charge layer that involves the contribution of the fixed-charges ZeN. Equation (41) is solved to give

We thus see that the potential distribution within the surface layer is characterized by ψ_DON and ψ_o, as schematically shown in figure 2.

When a thick soft plate is immersed in a general electrolyte solution, the Donnan potential y_DON is given by setting to zero the right-hand side of equation (6), in which −d is replaced by −∞. That is, y_DON is given as a root of the following transcendental equation

The relationship between y_DON and y_o is obtained in the following way. Integrating equations (5) and (6) gives

By evaluating equations (45) and (46) at x=0 and equating the results, we obtain

where equation (44) has been used. For a mixed solution of 1:1 electrolyte of bulk concentration n₁ and 2:1 electrolyte of bulk concentration n₂, equation (47) becomes

with

For the potential distribution around a spherical soft particle or a cylindrical soft particle, refer to [7, 8].

Electrostatic interaction between soft particles

Two parallel semi-infinite soft plates

Consider two parallel dissimilar soft plates 1 and 2 at a separation h between their surfaces immersed in an electrolyte solution [9]. We assume that each soft plate consists of a core and an ion-penetrable surface charge layer of polyelectrolytes covering the plate core and that there is no electric field within the plate core. We denote by d_l and d₂ the thicknesses of the surface charge layers of plates 1 and 2, respectively. The x-axis is taken perpendicular to the plates with the origin at the boundary between the surface charge layer of plate 1 and the solution, as shown in figure 3.

Figure 3.

Potential distribution y(s) across two parallel soft plates 1 and 2 [9].

We assume that each surface layer is uniformly charged. Let Z_l and N_l, respectively, be the valence and the density of fixed-charge groups contained in the surface layer of plate 1, and let Z₂ and N₂ be the corresponding quantities for plate 2. Thus, the charge densities ρ_fix1 and ρ_fix2 of the surface charge layers of plates 1 and 2 are, respectively, given by

The Poisson–Boltzmann equations for the present system are then

where ψ(x) is the electric potential at position x relative to that at a point in the bulk solution far from the plates (the plates are actually surrounded by the electrolyte solution). We assume that the relative permittivity ε_r in the surface layers takes the same value as that in the electrolyte solution.

We consider the case where N₁ and N₂ are low. The Poisson–Boltzmann equations (52)–(54) can be linearized to give

where κ is the Debye–Hückel parameter (equation (8)). Integration of equations (55)–(57) with appropriate boundary conditions gives

The electrostatic force acting between two plates can be obtained by integrating the osmotic pressure and Maxwell's stress over an arbitrary surface enclosing one of the plates [4]. We choose the surfaces x=−∞ (in the bulk electrolyte solution phase) and x=0 (at the boundary between plate 1 and the electrolyte solution) as a surface enclosing plate 1. When the potential is low, the electrostatic force P(h) between the two parallel plates 1 and 2 per unit area is then given by [9, 10]

The first term on the right-hand side of equation (61) corresponds to the osmotic pressure and the second term to Maxwell's stress, and P>0 means repulsion and P <0 attraction. The values of ψ(0) and dψ/dx at x=0⁺ can be obtained as functions of plate separation h from equation (59). Substituting the results into equation (61) gives the force P per unit area as a function of h. When the potential ψ(x) is low, the electrostatic force P_pl (h) between the two parallel plates 1 and 2 at separation h per unit area can be calculated from equation (61) with the result that

Integrating equation (62) with respect to h gives the potential energy V_pl(h) of the electrostatic interaction between the plates per unit area as a function of h:

For the special case of two similar soft plates carrying Z₁=Z₂=Z,N₁=N₂=N, and d₁=d₂=d so that ρ_fix1=ρ_fix2=ρ_fix, equations (62) and (63) reduce to

When the magnitude of ψ(x) is arbitrary, one must solve the original nonlinear Poisson–Boltzmann equations (52)–(54). Consider the case of two parallel similar plates in a symmetrical electrolyte with valence z and bulk concentration n. In this case, we need to consider only the region −d <x <h/2 so that the Poisson–Boltzmann equations to be solved are [4, 10]

Here, y=zeψ/kT, ψ_DON and y_DON=zeψ_DON/kT are, respectively, the scaled potential, Donnan potential given by equation (20) and scaled Donnan potential. The solution to equations (66) and (67), subject to appropriate boundary conditions, takes a complicated form, involving numerical integration. However, for κd>1 the value of y(h/2) can be calculated by solving the following coupled equations:

Here, dc is a Jacobian elliptic function with modulus 1/cosh(y(h/2)), and y_o≡y(0) is the scaled unperturbed potential at the front edge x=0 of the surface layer and is given by equation (40).

The interaction force between two parallel similar plates per unit area P_pl(h) is given by [4]

where ψ(h/2) is the potential at the midpoint between the plates. A simple approximate analytic expression for P_pl(h) can be obtained using the linear superposition approximation (LSA) [4]: ψ(h/2) in equation (70) is approximated by the sum of the asymptotic values of the two unperturbed potentials, which is produced by the respective plates in the absence of interaction. For two similar plates,

This approximation is correct in the limit of large κh. From equation (33), the value of the unperturbed potential of a single plate at x=h/2 is given by

where the unperturbed surface potential ψ_o is given by equation (39). For large κh , equation (72) becomes

Hence

For large κh, equation (70) asymptotes to

Substituting equation (74) into equation (75), we obtain

The potential energy V_pl(h) can be obtained by integrating equation (76) with the result

Interaction between two dissimilar soft spheres

Consider the electrostatic interaction between two dissimilar spherical soft spheres 1 and 2 (figure 4). We denote by d_l and d₂ the thicknesses of the surface charge layers of spheres 1 and 2, respectively. Let the radius of the core of soft sphere 1 be a₁ and that for sphere 2 be a₂. We imagine that each surface layer is uniformly charged. Let Z_l and N_l, respectively, be the valence and density of the fixed charge layer of sphere 1 and Z₂ and N₂ be those for sphere 2.

Figure 4.

Interaction between two soft spheres [9].

With the help of Derjaguin's approximation [11], viz.,

one can calculate the interaction energy V_sp(H) between two dissimilar soft spheres 1 and 2, separated by a distance H between their surfaces, via the corresponding interaction energy V_pl(h) between two parallel dissimilar plates. Substituting equation (77) into equation (78), we obtain [12]

For the special case of two similar soft spheres carrying Z₁=Z₂=Z, N₁=N₂=N, d₁=d₂=d so that ρ_fix1=ρ_fix2=ρ_fix, and a₁=a₂=a, equation (79) reduces to

We now introduce the quantities

which are, respectively, the amounts of fixed charges contained in the surface layers per unit area on spheres 1 and 2. If we take the limit d_l, d₂→0 and N₁, N₂→∞, keeping the products N₁d₁ and N₂d₂ constant, then σ₁ and σ₂ reduce to the surface charge densities of two interacting hard plates without surface charge layers. In this limit, equations (62), (63) and (79), respectively, reduce to

Equations (84) and (85), respectively, agree with the expression for the electrostatic interaction energy between two parallel hard plates at a constant surface charge density and that for two hard spheres at constant surface charge density [16]. When κd_l≫ 1 and κd₂≫ 1, which holds for practical cases, the potential deep inside the plates remains constant and equal to the Donnan potential for the respective surface charge layers, which are

Equations (86) and (87) are, respectively, also obtained by setting the right-hand sides of equations (55) and (57) to zero.

Consider the limiting case of κd₁≫ 1 and κd₂≫ 1. In this case, soft plates and soft spheres become planar polyelectrolytes and spherical polyelectrolytes, respectively, and equations (83), (84) and (85) reduce, respectively, to

Similarly, expressions for the force and potential energy of the electrostatic interaction between two parallel or crossed soft cylinders have been derived [14].

Electrophoretic mobility of soft particles

In this section, we deal with a general theory of electrophoresis of soft particles and approximate analytic expressions for the mobility of soft particles [4, 15–21]. This theory unites the electrophoresis theories of hard particles and of polyelectrolytes.

Consider a spherical soft particle, i.e., a charged spherical particle covered with an ion-penetrable layer of polyelectrolytes moving with a velocity U in a liquid containing a general electrolyte in an applied electric field E. We assume 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 layer is uniformly charged at a constant density ρ_fix=ZeN. The polymer-coated particle thus has an inner radius a and an outer radius b≡a+d (figure 5).

Figure 5.

A 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 [4].

The origin of the spherical polar coordinate system (r, θ, ϕ) is fixed at the center of the particle core and the polar axis (θ=0) is set parallel to E. Let the electrolyte be composed of N ionic mobile species of valence z_i, bulk concentration (number density) n_i^∞, and drag coefficient λ_i (i=1, 2, …, M). The drag coefficient λ_i of the ith ionic species is further related to the limiting conductance Λ_i^o of that ionic species by

where N_A is the Avogadro number. We adopt the model of Debye–Bueche [19], in which the polymer segments are regarded as resistance centers, which are uniformly distributed in the polyelectrolyte layer, exerting frictional forces on the liquid flowing in that layer. We give below the general theory of electrophoresis of soft particles [4, 15, 21].

The fundamental electrokinetic equations for the liquid velocity u(r) at position r relative to the particle (u(r)→−Uas r≡|r|→∞) and the velocity of the ith ionic species v_i are the same as those for rigid spheres, except that the Navier–Stokes equations for u(r) become different for the regions outside and inside the surface layer [4], viz.,

Here equation (92) is the usual Navier–Stokes equation. The term γu on the left-hand side of equation (93) 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 a_p and the polymer segments are distributed at a uniform volume density of N_p in the polyelectrolyte layer, then each polymer segment exerts the Stokes resistance 6πηa_pu on the liquid flow in the polyelectrolyte layer so that

Similarly, the Poisson equation relating the charge density ρ_el(r) resulting from the mobile charged ionic species and the electric potential ψ(r) are also different for the different regions,

For a weak applied field E, the electrokinetic equations (92) and (93) are linearized to give

and

where n_i⁽⁰⁾ (r) is the equilibrium concentration (number density) of the ith ionic species ( as r→∞), and δn_i⁽⁰⁾ (r) and δμ_i(r) are, respectively, the deviation of n_i⁽⁰⁾ (r) and that of the electrochemical potential μ_i(r) of the i th ionic species due to the applied field E.

Symmetry considerations permit us to write

where E=|E|. The fundamental electrokinetic equations can be transformed into equations for h(r) and φ_i(r) as

with

Here, y=eψ⁽⁰⁾/kT is the scaled equilibrium potential outside the particle core and the value 1/λ is called the electrophoretic softness.

The boundary conditions for ψ(r), u(r) and v_i(r) are as follows.

1. ψ(r) and −∇ψ(r) are continuous at r=b,

   

   

   where the continuity of −∇ψ(r) results from the assumption that the relative permittivity ε_r takes the same value both inside and outside the polyelectrolyte layer.
2. 
3. 
4. The normal and tangential components of u are continuous at r=b,

   

   

5. The normal and tangential components of the stress tensor , which is the sum of the hydrodynamic stress and the Maxwell stress , are continuous at r=b. From the continuity of −∇ψ(r), the normal and tangential components of are continuous at r=b. Therefore, the normal and tangential components of must be continuous at r=b so that the pressure p(r) is continuous at r=b.

6. The electrolyte ions cannot penetrate the particle core, viz.,

   

   where n is the unit normal outward from the surface of the particle core.
7. In the stationary state, the net force acting on the particle (the particle core plus the polyelectrolyte layer) or an arbitrary volume enclosing the particle must be zero. Consider a large sphere S of radius r containing the particle. The radius r is taken to be sufficiently large so that the net electric charge within S is zero. There is then no electric force acting on S and we need to consider only hydrodynamic force F_H, which must be zero, viz.,

   

   where the integration is carried out over the surface of the sphere S.

The electrophoretic mobility μ=U/E (where U=|U|) can be calculated from

with the result [4, 16]

with

We derive approximate mobility formulae for the simple but important case where the potential is arbitrary but the double layer potential still remains spherically symmetrical in the presence of the applied electric field (the relaxation effect is neglected). Moreover, we treat the case where the following conditions hold

which are satisfied for most practical cases. When the electrolyte is symmetrical with a valence z and bulk concentration n, and if κd≫1, then the potential inside the polyelectrolyte layer can be approximated using equation (43). Using this approximation, we obtain [4, 30]

with

In the limit of d≫a, in which case f(d/a)→2/3, equation (123) reduces to

For the low-potential case, equation (125) further reduces to

which agrees with Hermans–Fujita's equation for the electrophoretic mobility of a spherical polyelectrolyte (a soft particle with no particle core) [23].

In the opposite limit of d≪a, in which case f(d/a)→1, equation (123) reduces to

which, for low potentials, reduces to

Equation (117) covers a plate-like soft particle. Indeed, in the limiting case of a→∞, the general mobility expression (equation (117)) reduces to

which is the general mobility expression of a plate-like soft particle. For the case of κd≫1 and λd≫1, equation (129) becomes equation (127).

Equation (123) consists of two terms: the first term is a weighted average of the Donnan potential ψ_DON and the surface potential ψ_o. It should be stressed that only the first term is subject to the shielding effects of electrolytes, approaching zero as the electrolyte concentration n increases, while the second term does not depend on the electrolyte concentration. In the limit of high electrolyte concentrations, all the potentials vanish and only the second term of the mobility expression remains, viz.,

Equation (130) shows that as κ→∞, μ approaches a nonzero limiting value μ^∞. This is a characteristic of the electrokinetic behavior of soft particles, in contrast to the case of the electrophoretic mobility of hard particles, which should reduce to zero owing to the shielding effects, since the mobility expressions for rigid particles do not contain μ^∞. The term μ^∞ can be interpreted as resulting from the balance between the electric force acting on the fixed-charges (ZeN)E and the frictional force γu, viz.,

from which equation (130) follows.

To see this more clearly, in figures 6(a) and (b) we give a schematic representation of the liquid velocity u(x) (=|u(x)|) as well as potential distributions ψ(x) around a soft particle in comparison with those for a hard particle. There, x is given by x=r−a across the surface charge layer. Figure 6(c) shows the electrophoretic mobility μ as a function of electrolyte concentration n for a soft particle and a hard particle. It is seen that the liquid velocity u(x) in the middle region of the surface charge layer (the plateau region in figure 6(b)), where the potential is almost equal to the Donnan potential ψ_DON (figure 6 (a)), gives μ^∞. It is this term that yields the term independent of the electrolyte concentration. We also note that 1/λ can be interpreted as the distance between the slipping plane at the core surface at r=a (x=0) and the bulk phase of the surface layer (the plateau region in figure 6(b)), where u(x)/E is given by −μ^∞. Figure 6(a) shows that a small shift of the slipping plane has no effect on the mobility value. Therefore, the mobility of soft particles is insensitive to the position of the slipping plane and the zeta potential loses its significance.

Figure 6.

Schematic representation of liquid velocity distribution u(x) (a) as well as potential distribution ψ(x) (b) around a soft particle, and the electrophoretic mobility μ of a soft particle as a function of electrolyte concentration n (c) (left), in comparison with those for a hard particle (right) [4].

The above electrophoresis theory of soft particles has been applied to analyze the electrophoretic mobility data of biological colloids such as cells and their model particles [24–35]. Ohshima [36, 37] proposed a theory of electrophoresis of soft particles in concentrated suspensions. Keh and Liu presented theories of sedimentation [38] and electrical conductivity [39] of a dilute suspension of soft particles, and these theories were extended to the concentrated case [40, 41]. An approximate expression for the dynamic electrophoretic mobility of a spherical soft particle was derived in reference [39]. The electroosmotic velocity in an array of parallel soft cylinders was discussed in [40]. Dukhin et al [44] pointed out the importance of the degree of dissociation of charged groups in the polyelectrolyte layer. Duval and Ohshima [45] presented a new electrophoresis model of diffuse soft particles. In that model, a diffuse soft particle consists of an uncharged impenetrable core and a charged diffuse polyelectrolyte layer. The diffuse character of the polyelectrolyte layer is defined by a gradual distribution of the density of polymer segments in the interspatial region separating the core from the bulk electrolyte solution. Successful application of that model has been reported for humic acids [46], bacteria [47] and red blood cells [48]. For numerical calculations of the electrophoretic mobility based on more rigorous theories, the readers are referred to the studies of Hill et al [49–52] and Lopez-Garcia et al [53–55] as well as that of Duval and Ohshima [45].

Summary

In this article, we discussed the electrostatics and electrokinetics of soft particles in comparison with those of hard particles. It has been shown that the Donnan potential in the surface layer, which is related to the density of the fixed charge in the surface layer, plays an essential role in the potential distribution across the surface layer, electrostatic interactions between soft particles and the electrophoretic mobility of soft particle in an external electric field.