Electrophoresis of spherical polymer-coated colloidal particles

Electrophoresis of spherical polymer-coated colloidal particles

Journal of Colloid and Interface Science 258 (2003) 56–74 www.elsevier.com/locate/jcis Electrophoresis of spherical polymer-coated colloidal particle...

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Journal of Colloid and Interface Science 258 (2003) 56–74 www.elsevier.com/locate/jcis

Electrophoresis of spherical polymer-coated colloidal particles Reghan J. Hill, D.A. Saville,∗ and W.B. Russel Department of Chemical Engineering, Princeton University, Princeton, NJ 08542, USA Received 7 May 2002; accepted 22 October 2002

Abstract The motion of a spherical polymer-coated colloidal particle in a steady electric field is studied via “exact” numerical solutions of the electrokinetic equations. The hydrodynamic influence of the polymer is represented by a distribution of Stokes resistance centers. With neutral polymer, charge resides on the underlying bare particle, whereas for polyelectrolytes, the charge is distributed throughout the coating. The coatings may be brush-like or have long tails. As expected, neutral coatings lower the mobility because of increased drag and a decrease in the effective charge behind the shear surface. For polyelectrolyte coatings, the behavior is more complex. For example, the mobility becomes independent of the ionic strength and particle size when Donnan equilibrium prevails inside the coating and the coating is thick relative to the Brinkman screening length (square root of the coating permeability). In this limit, the mobility follows from a simple balance of forces within the coating and, therefore, becomes proportional to the fixed charge density and the coating permeability. If the permeability is sufficiently high, the mobility of a polyelectrolyte-coated particle may exceed that of its bare counterpart with the same net charge. In general, the effects of polarization and relaxation are as important for coated particles as they are for bare particles.  2003 Elsevier Science (USA). All rights reserved. Keywords: Electrophoresis; Electrophoretic mobility; Dynamic mobility; Polymer-coated colloidal particles; Soft particles; Fuzzy particles; Electrokinetics

1. Introduction The standard electrokinetic model, as originally set out by Overbeek [1] and Booth [2], accurately represents the behavior of bare particles in steady and oscillatory electric fields. Its success is due, in part, to the development of computational techniques that accurately solve the governing equations, taking into full account the deformation of the diffuse double layer by polarization and relaxation. This article describes an extension of the standard model to allow for the influence of an either neutral or charged polymer coating. After a general model for steady and oscillatory fields is formulated, the results of numerical calculations are presented for steady electrophoresis. Applications of the model to the dynamic mobility and dielectric response in oscillatory fields are treated in future work. Although the electrokinetic behavior of latex particles is complicated, experimental tests of the standard model, using (i) complementary measurements of electrophoretic mobility and complex conductivity at a given ionic strength [3], and (ii) measurements of mobility over a range of ionic * Corresponding author.

E-mail address: [email protected] (D.A. Saville).

strengths [4], attest to its validity for bare particles. Accordingly, the standard model provides a basis for understanding the behavior of polymer-coated particles, which are sometimes referred to as fuzzy or soft. The effects of a polymer coating manifest themselves in several ways. Given that the counterion density close to the underlying bare particle is much higher than in the bulk, even a thin polymer coating may enclose a significant portion of the mobile charge. Depending on the hydrodynamic permeability, an uncharged coating lowers the particle mobility by increasing the drag and decreasing the effective charge behind the shear surface [5]. For a charged coating, the situation is complicated because the fixed charge is distributed throughout the coating instead of being confined to a surface. For particles where the electrostatic potential is much higher than kB T /e (kB T is the thermal energy and e the fundamental charge), polarization and relaxation introduce additional complications. The standard model has been modified to mimic processes affecting the behavior of particles with thin coatings. These dynamic Stern-layer models [6–10] allow for dissociation of charged groups and altered ion migration rates within an infinitesimally thin surface layer. Although they capture the correct effects of a thin coating, bringing experiment and

0021-9797/03/$ – see front matter  2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0021-9797(02)00043-7

R.J. Hill et al. / Journal of Colloid and Interface Science 258 (2003) 56–74

theory into agreement, the “surface” parameters must be adjusted empirically [8]. The predictive capabilities of these models are therefore limited, and the situation with thicker coatings is less clear. Analytical solutions of the governing equations for a variety of so-called soft particles have been worked out—most notably by Ohshima [11]—under conditions where polarization and relaxation are absent. These models are useful for interpreting experimental results in terms of characteristics such as the amount of adsorbed polymer, its molecular weight, and the coating thickness. However, recent semianalytical solutions by Saville [12] show that polarization and relaxation are as important at high potentials for coated particles as for their bare counterparts. Clearly then, if theory for coated particles is to be compared with experiments over a range of double-layer and coating thicknesses, charge densities and particle sizes, numerical solutions of the governing equations are required. In this article, an electrokinetic model and a numerical method for solving the equations are described in Sections 2 and 3. Then, the electrophoretic mobility of particles with neutral coatings, various coating thicknesses, and surface charge densities is examined in Section 4. A similar analysis of the mobility of polyelectrolyte-coated particles, with various coating charge densities, is presented in Section 5, followed by a brief summary of all the results in Section 6. The dynamic mobility and dielectric response of dilute suspensions of polyelectrolyte-coated particles are addressed in forthcoming work.

2. The electrokinetic model for polymer-coated colloids 2.1. Field equations Consider a single spherical colloidal particle in an unbounded electrolyte to which a uniform electric field is applied with a steady or oscillating amplitude. Attached to the particle is a polymer coating with a radially varying segment density. The electrostatics are governed by the well-known Poisson–Boltzmann equation, ∇ 2ψ = −

N 

 f (j = 1, . . . , N), zj nj − nj

(1)

where the potential, ψ, is scaled with kB T /e, with kB T the thermal energy and e the fundamental charge, and zj denotes f the valence of the j th ion species. The densities nj and nj of the counterions and fixed ions, respectively, are scaled with twice the ionic strength, N 

zj2 n∞ j ,

opposite to their respective counterion in solution. Lengths are scaled with the double-layer thickness,  κ −1 = kB T s o /(2I e2 ), (3) where s is the dielectric constant of the solvent and o is the permittivity of free space. For an aqueous solvent at room temperature, the double-layer thickness is typically less than 100 nm. The ion transport equations are Pej (u − v j ) − zj ∇ψ − ∇ ln(nj ) = 0 (j = 1, . . . , N), (4) where the fluid and ion velocities, u and v j , are scaled with a characteristic velocity u∗ = s o (kB T /e)2 /(ηa).

(5)

Here, a is the bare particle radius and η is the solvent viscosity. For particles with a = 100 nm in an aqueous solvent, u∗ ∼ 10−3 m s−1 , and with ion diffusivities Dj ∼ 10−9 m 2 s−1 , the ion Péclet numbers, Pej = u∗ κ −1 /Dj ,

(6)

are O(10−1 /(κa)). Under steady conditions, the Péclet numbers indicate the extent to which convection deforms the equilibrium double layer. Note that the Dj = kB T /λj may be obtained from the respective ion mobilities, λ−1 j = Λj (e2 |zj |), where Λj are the limiting conductances for dilute electrolytes [13]. Note that fluid velocities may be significantly lower than suggested by the dimensional analysis leading to Eq. (5). For example, balancing an O(ηuc /a 2 ) viscous stress with an O(z1 eEn∞ 1 exp(−ζ z1 )) electrical stress, assuming κa ∼ 1 and ζ > 1, gives a characteristic electroosmotic velocity uc = u∗ exp(−ζ z1 )Eaez1 /(2kB T ). Thus, with an applied electric field E ∼ 102 V m−1 and a surface potential ζ ∼ 2, say, uc is O(10−2 ) smaller than u∗ . The ion conservation equations are ∂nj + Pej ∇ · (nj v j ) = 0 (j = 1, . . . , N), (7) ∂t where time, t, is scaled with the reciprocal angular frequency, ω−1 , of the applied electric field. Dimensionless frequencies

Ωj

Ωj = ωκ −2 /Dj

j =1

2I =

57

(2)

compare the time for ions to diffuse across the double layer with the reciprocal angular frequency. Thus, dynamic ion transport for particles with a = 100 nm in an aqueous solvent becomes important when Ωj = 1 at O(105 (κa)2 ) Hz frequencies. The fluid momentum and mass conservation equations are

j =1

where n∞ j denote ion densities in the bulk electrolyte. The valencies of the ions contributing to the fixed charge are

(8)

 ∂u = ∇ 2 u − ∇p − B(r)(u − V ) − κa nj zj ∇ψ ∂t N



j =1

(9)

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and ∇ · u = 0,

(10)

respectively, where the particle velocity, V , and the pressure, p, are scaled with u∗ and ηu∗ κ, respectively. Note that, in setting the third term on the right-hand side of Eq. (9) proportional to u − V , we have assumed that the underlying bare particle and the polymer coating move together as a rigid composite. The dimensionless frequency Ω = ωκ −2 /ν

(11)

is the square of the ratio of the double-layer thickness to the viscous penetration depth, (ν/ω)1/2 . In an aqueous solvent with kinematic viscosity ν ∼ 10−6 m2 s−1 , the viscous penetration depth for particles with a = 100 nm becomes comparable to the double-layer thickness at O(108 (κa)2 ) Hz frequencies. The dimensionless function 2  B(r) = 1 κ&B (r) (12) is the square of the ratio of the double-layer thickness to the local Brinkman screening length, &B , which characterizes the hydrodynamic permeability of the coating. The so-called Darcy drag accounts for the hydrodynamic drag force acting between the polymer segments and the interstitial fluid. The Brinkman screening length may be related to the structure of the polymer coating by expressing the body force as the product of the polymer segment density, ns (r), and the drag force on a single segment, fs = 6πηas (u − V )Fs . When the segments are approximated as spheres with a radius as , the dimensionless drag coefficient Fs depends on the volume fraction φs = ns (4/3)πas3 . Equating ns fs to the dimensional form of the Darcy drag term in Eq. (9), (η/&2B )(u − V ), gives &2B = 1/(ns 6πas Fs ) = 2as2/(9φs Fs ).

(13)

In this work, the local distribution of segments is approximated as a rigid “random” configuration of spheres, with a drag coefficient given by [14,15] Fs =

1 + 3(φs /2)1/2 + (135/64)φs ln φs + 16.456φs , (14) 1 + 0.681φs − 8.48φs2 + 8.16φs3

when the volume fraction is less than approximately 0.4, and the Carman correlation, Fs = 10φs /(1 − φs )3 ,

(15)

at volume fractions from 0.4 up to close-packing. For most of the coatings examined in this work, the segment volume fraction is much less than 0.1 and, hence, Fs is close to unity. The drag on each segment is therefore approximately equal to the well-known Stokes drag force. The permeability depends on the particular polymer coating under consideration. For polyelectrolyte brushes at low ionic strengths, the chains are stretched and the permeability is relatively high. However, increasing the ionic

strength causes the chains to “collapse,” because electrolyte ions help screen the fixed charge, thereby decreasing the permeability. Adsorbed neutral homopolymers are expected to have a low permeability close to the substrate surface, with the permeability increasing toward the outer region composed of loops and dangling tails. Estimates of the segment densities and coating length scales encountered in practice come from measurements of the electrophoretic mobility of human erythrocytes and polymer-grafted liposomes. Sharp and Brooks [16] interpreted the electrophoretic mobilities of human erythrocytes using a one-dimensional electrokinetic model that neglects double-layer deformation. Fitting their numerical model to data with Fs = 1, they find &B ≈ 1.5 nm with as ≈ 0.7 nm and ns ≈ 0.069 M (φs ≈ 0.06). Cohen and Khorosheva [17] interpreted the electrophoretic mobility of liposomes (2a = 3.5 µm) with neutral polymer chains terminally anchored to the surface. Using a one-dimensional model with the Debye–Hückel approximation, and relaxing continuity of the fluid velocity gradient at the edge of the coatings, they deduced the coating thickness to increase linearly from 2.6 to 13.2 nm as the molecular mass of the grafted polymer increased from 1 to 5 kg mol−1 . They correctly point out that this scaling corresponds to that for a polymer brush, but did not explain the varying permeability, with &B ranging from 1.6 to 4.7 nm, which should be independent of molecular mass if the coatings are really brush-like. Although it is unclear whether the inconsistency is due to the simplicity of the model or to the assumed grafting density and surface charge, their results give a basis for further interpretation using a more accurate model. With a = 100 nm and &B = 1 nm, B in Eq. (9) is O(104 /(κa)2), and the characteristic ratio of the Darcy drag force to the electrical body force inside the coating, B/κα, is therefore O(104 /(κa)3). Clearly, κa and, hence, the ionic strength significantly affect the electroosmotic flow inside a polymer coating. This motivates, in part, directing attention to how the mobility depends on κa. The effective solvent viscosity inside a porous medium is generally expected to depend on the local volume fraction [14,18–20]. However, in the absence of a well-tested model, we set it equal to that of the solvent, which is reasonable when φs 1. Note also that the Brinkman approximation in Eq. (9) is not strictly valid when velocity gradients are comparable to or greater than O(u∗ /as ). Assuming the largest velocity gradients within the polymer coating are O(u∗ /&B ) requires &B as . As suggested by Eq. (13), this requires φs 1, which is often the case for polymers in good solvents. Of course, the polymer segments are not spherical, and their density and effective size also depend on the solvent quality, ionic strength (for polyelectrolytes), and polymer architecture. Further details, including deformation of the polymer and the effects of rapid gradients in the segment density, are beyond the scope of this work—they do, however, deserve further attention.

R.J. Hill et al. / Journal of Colloid and Interface Science 258 (2003) 56–74

In this work, the segment density profiles are approximated by   ns = (1/2)Ns erfc (r − L − a)/δ , (16) where Ns is referred to as the nominal segment density, r is the (dimensional) radial position, L is the nominal coating thickness, and δ characterizes the distance over which the density decays. In this work, the hydrodynamic segment size as = 1 Å, so spatial variations in the coating permeability or Brinkman screening length result from only the varying segment density. While we will refer to the coating permeability based on Ns and as , it should be kept in mind that the Brinkman screening length and segment density vary according to Eqs. (13) and (16), respectively. For charged coatings, the fixed ion densities are assumed to be proportional to ns , giving   f f nj = (1/2)Nj erfc (r − L − a)/δ , (17) f

where Nj are referred to as nominal fixed ion densities. Note, because there is only one fixed ion species in this work, the subscript is omitted. With a z–z electrolyte, the nominal fixed charge density is −N f z1 e, where z1 is the counterion valence. For polyelectrolyte brushes, δ may be approximated by the blob size at the nominal edge of the coating [21], so δ ∼ d(a + L)/a, where d −2 is the surface density of polymers grafted to the bare particle. Polyelectrolyte chains typically have a short end group that anchors the chains, producing brush-like coatings with δ < L. For neutral homopolymers, the chains adsorb to the particle surface at multiple points along the chain, producing high segment densities close to the surface and a slowly decaying density further out. The slower decay of the segment density may be accounted for in our model by setting δ ∼ L in Eq. (16) or by adopting a segment density distribution   ns = Ns exp −(r − a)/δ , (18) for example. We will examine the merits of other density profiles in the future. For simplicity, most of the segment density profiles in this article are given by Eq. (16), with δ = 0.1L, and are therefore assumed to be brush-like.

and negative subscripts indicate the solvent and particle sides of the surface, respectively. Unless it is accompanied by dimensional units, σ will be in dimensionless form, with σ κa the charge density scaled with s o kB T /(ae). If the potential at the bare surface, ζ , is specified instead of the surface charge density, then (19) gives the surface charge density corresponding to the radial gradient of the potential at r = κa. In the far field, the gradient of the potential must approach the applied electric field; so ψ → −E · r

as r → ∞,

(20)

where the electric field strength is scaled with kB T κ/e. For an impenetrable bare particle, the ion flux normal to the surface vanishes, giving (Pej nj V − nj zj ∇ψ − ∇nj ) · n = 0 at r = κa.

(21)

In the far field, the ion densities approach those of the bulk electrolyte, so nj → n∞ j /(2I )

as r → ∞.

(22)

The fluid velocity at the bare particle surface satisfies the no-slip condition u=V

at r = κa,

(23)

and, in general, approaches a uniform velocity, U , as r → ∞. 2.3. Particle equation of motion In dimensionless form, the equation of motion for the bare particle and its coating, with the latter assumed to have negligible mass, is ∂V Ω(ρp /ρs )Vp ∂t   = Σ · nˆ dA + B(r)(u − V ) dV  − κa

2.2. Boundary conditions

59

σ ∇ψ · nˆ dA − κa

N  

f

zj nj ∇ψ dV ,

(24)

j =1

The boundary conditions are presented in terms of the dimensionless variables introduced for the field equations. Because the polymer segment density distribution and its radial derivatives are continuous, boundary conditions need to be applied only at the bare particle surface and in the far field. The surface charge density on the bare surface is assumed constant and uniform, imposing the boundary condition ∇ψ|+ · n − (p /s )∇ψ|− · n = −σ

at r = κa,

(19)

where p is the dielectric constant of the particle, the surface charge density, σ , is scaled with κo kB T /e, and the positive

where Vp is the volume of the bare particle scaled with κ −3 , ρp and ρs are the particle and solvent densities, respectively, Σ = −pI + ∇u + (∇u)T is the stress tensor scaled with ηu∗ κ, and nˆ is an outward unit normal to the particle surface; dA and dV denote the particle surface and the volume outside the particle, respectively. The first term on the right-hand side of Eq. (24) is from stresses acting on the underlying bare particle, the second term is the Darcy drag force on the (rigid) porous coating, and the third and fourth terms are the electrical forces on the bare surface and the polymer coating, respectively.

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3. Solution of the field equations

and

The solution of the field equations proceeds by first solving the nonlinear problem without an applied electric field, with the particle and surrounding fluid at rest. Then, the equations governing small-amplitude perturbations to the equilibrium base state are solved. This is achieved by eliminating the angular dependence of the dependent variables and decomposing the problem into two subproblems from which the full solution is obtained by linear superposition. The methodology is similar to that prescribed by O’Brien and White [22] and DeLacey and White [23], but the numerical method used to solve the resulting stiff system of linear ordinary differential equations is new. 3.1. Equilibrium base state

nj = n0j + nj

(32)

into the field equations and boundary conditions. Note that the fluid and particle velocities, u and V , are of the same order as the perturbations to the potential and ion densities. The ion velocities, v j , are eliminated by substituting the ion flux, v j nj , from the transport equation (4), into the ion conservation equation (7), and the pressure is eliminated by taking the curl of the momentum conservation equation (9). 3.3. Reduction to ordinary differential equations The fluid velocity may be expressed in terms of a scalar function h(r) via [24] u = ∇ × ∇ × h(r)X + U ,

In the absence of an applied electric field and buoyancy, the fluid and particle are stationary and the equilibrium ion densities, n0j , can be expressed in terms of the equilibrium potential, ψ 0 . Eliminating v j from the ion conservation equation (7) using the ion transport equation (4) gives n0j = nj (r → ∞) exp(−zj ψ 0 ).

(25)

Substituting these ion densities into the nonlinear Poisson– Boltzmann equation (1) gives an equation for the potential,  ∂ψ 0 ∂ 2ψ 0 f =− zj n0j − zj nj . + (2/r) 2 ∂r ∂r

(26)

j =1

The boundary conditions are either at r = κa

(27)

or ∂ψ 0 = −σ ∂r and ψ →0 0

where X ∈ {U , E}

(34)

and U is a far-field fluid velocity introduced to aid in the numerical solution of the equations. Note that the radial and tangential components of u are (U/X − 2hr /r)X cos θ

(U/X + hrr + hr /r)X sin θ,

(28)

(29)

Note that the fluid momentum equation (9) gives the (radial) hydrostatic pressure gradient,  ∂ψ 0 ∂p0 = −κa . n0j zj ∂r ∂r N

Equation (26) can be solved very accurately using the finitedifference method described briefly in Appendix B. 3.2. Solution of the perturbed field equations Linear equations for small-amplitude perturbations to the equilibrium base state are obtained by substituting perturbations of the form ψ = ψ0 − E · r + ψ

(31)

(38)

respectively, where rˆ is a radial unit vector. Under oscillatory conditions, time dependence is obtained by simply multiplying E, V (= −U ), and X by exp(−it). The linearized Poisson–Boltzmann, ion conservation, and momentum conservation equations become

(30)

j =1

(37)

and nj = nˆ j (r)X · rˆ ,

as r → ∞.

(36)

respectively, where the subscripts indicate differentiation with respect to r. Following O’Brien and White [22], the perturbations to the electrostatic potential and ion densities take the forms ˆ ψ  = ψ(r)X · rˆ

at r = κa,

(35)

and

N

ψ0 = ζ

(33)

L1 ψˆ = −

N 

zj nˆ j ,

(39)

j =1

∂n0j ∂n0j − Pej (U/X) −iΩj nˆ j = 2Pej (hr /r) ∂r ∂r

∂n0j ∂ ψˆ − E/X + zj ∂r ∂r ∂ nˆ j ∂ψ 0 ∂r ∂r + zj nˆ j L0 ψ 0 + L1 nˆ j (j = 1, . . . , N), + zj n0j L1 ψˆ + zj

(40)

R.J. Hill et al. / Journal of Colloid and Interface Science 258 (2003) 56–74

61

and

3.4. Decomposition and superposition

−iΩL1 hr = L2 hrrr − BL1 hr

∂B ∂hr − + hr /r + (V − U )/X ∂r ∂r N  ∂ψ 0 − (κa/r) zj nˆ j ∂r

While the linear ordinary differential equations for hr , ψˆ , and nˆ j may be solved directly, it is useful to obtain solutions when the particle is fixed at the origin (V = 0) and (i) a fluid velocity U is prescribed at r = ∞ in the absence of an electric field, and (ii) an electric field E is applied in the absence of a far-field flow. These are referred to as problems (U ) and (E), respectively, and the quantities obtained from their solution are denoted with appropriate superscripts. Because the equations are linear, the solution to the problem in which the particle is free to move (U = −V ) can be obtained from the superposition of problems (U ) and (E). In problems (U ) and (E), the net force on the particle may be obtained from the far-field velocity disturbances. Point forces, scaled with ηu∗ κ −1 , that produce the same farfield velocity disturbance as the coated particle are [25]

j =1

− zj

∂n0j ∂r

(ψˆ − rE/X) , (41)

respectively, where ∂2 ∂ + 2/r , 2 ∂r ∂r ∂2 ∂ L1 = 2 + 2/r − 2/r 2, ∂r ∂r ∂2 ∂ − 4/r 2. L2 = 2 + 4/r ∂r ∂r

L0 =

(42) (43) (44)

The boundary conditions for the perturbed potential give   ∂ ψˆ ˆ − E/X − (p /s ) ψ/(κa) − E/X = 0 ∂r

(45)

at r → ∞.

(46)

Note that the surface charge density is assumed constant, and the gradient of the potential on the internal side of the bare particle surface in (19) is given by the perturbed potential inside the bare particle, ˆ ψ  = X · r ψ(κa)/(κa).

(47)

The boundary conditions for the perturbed ion densities give

n0j zj



∂ nˆ j ∂ ψˆ − E/X + + nˆ j zj − Pej n0j V /X = 0 ∂r ∂r ∂r (48) ∂ψ 0

at r = κa, and nˆ j → 0 as r → ∞.

(49)

The radial and tangential components of the boundary conditions for the fluid velocity require hr = κa(U − V )/(2X)

(50)

and hrr = (U − V )/(2X)

(51)

and hrr → 0

r→∞

(53)

2 with C X = lim hX r r r→∞

(54)

under steady and oscillatory conditions, respectively. Note that these forces act on the fluid in a direction opposite to those acting on the particle. The far-field velocity disturbances include the contribution from the electrical force acting on the fluid, which is opposite to the electrical force on the particle. Under steady conditions, it follows that −f E equals the sum of the hydrodynamic (electroosmotic) and electrical forces acting on the particle. Therefore, the sum of the hydrodynamic and electrical forces on the particle vanishes, giving −f U − f E = −8πC U V + 8πC E E = 0.

(55)

The (dimensionless) electrophoretic mobility, defined as the ratio of the particle velocity to the electric field strength, is V /E = C E /C U .

(56)

Under oscillatory conditions, it is shown in Appendix A that the strength of a point force required to produce the same farfield velocity disturbance as a particle with finite size must be adjusted by adding a force equal to the acceleration of the fluid displaced by the particle, −iΩVp V . Equating the net force to the acceleration of the particle’s mass gives −iΩ(ρp /ρs )Vp V = 4πiΩC U V − 4πiΩC E E − iΩVp V (57) from which the dynamic mobility is   V /E = C E C U − (κa)3(ρp − ρs )/(3ρs ) . (58) Note that our results are presented in terms of another dimensionless mobility,

at r = κa, and hr → 0

with C X = lim hX r

and f X = 4πiΩC X X

at r = κa, and ψˆ → 0

f X = −8πC X X

as r → ∞.

(52)

M = 3V /(2Eκa),

(59)

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which is equal to the familiar electrophoretic mobility (the dimensional particle velocity divided by the dimensional electric field strength) scaled with 2s o kB T /(3ηe). To summarize, the field equations (39)–(41) are solved by setting V = 0 and either U = 1 and E = 0 or E = 1 and U = 0, in problems (U ) and (E), respectively. Then, the far-field decay of hr in (U ) and (E) gives the asymptotic coefficients C U and C E via Eq. (53) or (54). Finally, the mobility is obtained from the particle equation of motion, either Eq. (56) or (58), depending on whether the conditions are steady or oscillatory. Many single-particle and dilute suspension properties may be obtained from the asymptotic coefficients derived from problems (U ) and (E). From problem (U ) alone, one can determine the steady drag coefficient and polarizability, which lead to the Brownian diffusivity, sedimentation velocity, and sedimentation potential. Their dynamic analogs, the acoustic mobility and polarizability, give the particle velocity and polarization, or colloid vibration potential, resulting from a passing sound wave in a dilute suspension. From problems (U ) and (E), one obtains, in addition to the electrophoretic mobility, the dynamic mobility and the particle contributions to the effective conductivity and dielectric constant of dilute colloidal suspensions. In this work, we focus on particle motion due to steady electric fields. Equations (39)–(41) are deceptively difficult to solve under conditions where the parameters give rise to disparate length scales. To avoid some of the numerical difficulties already encountered by Mangelsdorf [26] in solving the equations for bare particles under oscillatory conditions (at high frequencies) and small κa, we devised a finite-difference method, described briefly in Appendix C, whereby the density of grid points is automatically increased in those regions of the domain where the functions undergo the most rapid changes. In addition to obtaining accurate results over a wide range of the parameter space, the method inherits the stability typically expected of implicit finite-difference methods.

4. Neutral coatings Our computational method was tested by comparing results for bare particles with those from the standard electrokinetic model using the software of White and coworkers [27]. Accuracy to within three significant figures is readily obtained over a wide range of surface potentials and double-layer thicknesses. Figure 1 shows the mobilities of particles with permeable and impermeable coatings, with mobilities obtained from the standard model for bare particles. Also shown are results from Saville’s [12] asymptotic theory for thin double layers (κa 1), thin coatings (L a), and high surface potentials (ζ 1). The asymptotic theory accounts for polarization and relaxation fairly accurately even though κL = 0.21 is not particularly small.

Fig. 1. Dimensionless electrophoretic mobility, M, of a particle with a neutral, thin and impermeable coating (open circles) as a function of the dimensionless surface potential, ζ : a = 200 nm; L = 0.01a; δ = 10−4 L; κa = 21 (aqueous KCl at 298 K). Also shown is Saville’s [12] thin-double-layer theory (long-dashed line) for κL 1. The filled circles show the mobility of the underlying bare particle interpolated with calculations from White and co-workers [27] (solid line). The dash-dotted lines are the thin-double-layer theories of Smoluchowski (straight) and Ohshima (curved) for bare and coated particles, respectively, both without polarization and relaxation. The short-dashed line comes from subtracting from the exact bare-particle mobility the difference between the thin-double-layer theories for bare and coated particles.

It is not surprising that polarization and relaxation are important for fuzzy particles, as demonstrated by comparing the mobilities from either the exact or asymptotic results with Ohshima’s [11] theory. Note that, for a hydrodynamically impermeable coating, Ohshima’s theory reduces to Smoluchowski’s well-known formula, M = (3/2)ζ,

(60)

with the “surface” potential evaluated at the outer edge of the coating. However, ion diffusion and electromigration inside the coating do affect the behavior of particles with a hydrodynamically impermeable coating. Consequently, the mobility maximum for the fuzzy particle shown in Fig. 1 is approximately 15% lower than for a bare particle whose surface potential is equal to the potential at the shear surface. Accordingly, using the standard model for bare particles to interpret the mobility of fuzzy particles under these conditions underestimates the charge. In the remainder of this section, the relationship between the mobility and the ionic strength is examined, first under conditions of constant surface potential and then at constant charge. Attention is focused on illustrating the effect of varying the coating thickness and the surface potential or charge, with a single nominal segment density or coating permeability.

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tency simply reflects the adopted representation of a segment and its corresponding hydrodynamic size. As mentioned previously, our calculations were mostly performed with as = 1 Å, which with Ns = 0.1 M gives a Brinkman screening length that is consistent with literature values, &B ≈ 2.96 nm. It is interesting to note that, if the monomer segments are represented by cylindrical particles, the radius of spherical particles with the same number density and giving the same Darcy permeability (Brinkman screening length) is much smaller than the cylinder length. To illustrate, consider the following semiempirical formula for the permeability of “random” fibrous porous media [29],   &2B = −3ac2 ln(φc ) + 0.931 (20φc ), (63)

Fig. 2. Dimensionless surface charge density, σ , of bare particles with neutral coatings as a function of κa (aqueous KCl at 298 K) for’various surface potentials, ζ = 1, 2, 4, and 8. The solid lines interpolate the coinciding computational results, and the dashed lines are the semiempirical formula (61).

4.1. Varying the double-layer thickness at constant surface potential Before examining the electrophoretic mobilities, it is helpful to recall that, as shown in Fig. 2, the surface charge density increases significantly with ionic strength at fixed surface potential. For a z–z electrolyte, the well-known formula [28] σ κaz = 2κa sinh(zζ /2) + 4 tanh(zζ /4)

(61)

accurately represents numerical solutions of the equilibrium problem. Note that σ κa is the dimensional surface charge density scaled with s o kB T /(ea). Note that a simple relation between the amount of adsorbed polymer and the polymer segment density is obtained for thin, uniform coatings. Then, the mass of absorbed polymer per unit area, Γ , is related to the segment density, Ns , by Γ ≈ Ns 103ms L,

(62)

where Ns is in mol L−1 and ms is the molecular mass of the segments. If the segments are taken to be monomers, then the degree of polymerization is mp /ms , where mp is the molecular mass of the polymer. For a PEO coating with mp = 100 kg mol−1 and ms = 44 g mol−1 adsorbed to a bare particle with a = 100 nm, Γ ≈ 0.7 mg m−2 and L ≈ 5 nm (see [5]), Ns ≈ 3.3 M and the number of absorbed polymers is approximately 530. This (monomer) segment density is an order of magnitude higher than the value used to obtain the results shown in Fig. 3, for example, where Ns = 0.1 M. However, the relevant parameter in the model is the permeability, which is, indeed, consistent with the available interpretations of electrokinetic experiments (see Section 2). This apparent inconsis-

where ac is the fiber (cylinder) radius and φc = ns πχ 2 lc2 is the volume fraction, with χ = ac / lc the aspect ratio. Equating the Brinkman screening lengths of “random” beds of fibers (Eq. (63)) and spheres (Eq. (14)) with the same particle density, ns , gives a sphere radius as = −lc 10/ (9Fs (ln(φc ) + 0.931)). Now, assuming φs and φc are 1 so Fs ∼ 1, as is much smaller than lc . Further, if the cylinder length is assumed to be comparable to the monomer length (≈ 0.44 nm [28, Table 6.1]) then as ∼ 0.4 Å, giving &B ∼ 0.8 nm. This is clearly of the required order of magnitude, suggesting that the segments may be better represented by cylindrical fibers, rather than spheres. Remaining differences are, of course, due to details of the polymer conformation, including the connectivity of the segments, the intrinsic flexibility of the chains, and, possibly, chain dynamics. Note also that attention should be given to the highly nonuniform structure of coatings composed of adsorbed homopolymers in good solvents, such as PEO in aqueous electrolytes. Then, the segment density is extremely high close to the surface, with an outer low-density region occupied by loops and tails. The hydrodynamic thickness of such coatings tends to be much larger than the thickness based on the first moment of the segment density distribution. Matters of this sort are beyond the scope of the present work, but may addressed using exponential distributions, as suggested by Eq. (18). Terminally anchored chains give much more uniform distributions, and these are clearly closer to those underlying the results presented here. As is well known and illustrated in Fig. 3, the relationship between the mobility and ionic strength for bare particles is sensitive to the surface potential. The electrical force on the particle is balanced by hydrodynamic drag from particle translation and electroosmotic drag arising from the electric field acting on charge in the diffuse double layer. At low potentials, polarization and relaxation are negligible, and so the mobility increases with κa because of the increasing charge. At higher potentials, the mobility first decreases with increasing κa because double-layer polarization weakens the electrical force on the particle. At all surface potentials, however, the mobility eventually approaches the Smoluchowski

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(a)

(b)

(c)

(d)

Fig. 3. Dimensionless electrophoretic mobility, M, of particles with neutral coatings as a function of κa (aqueous KCl at 298 K) for various coating thicknesses, L/a = 0, 0.0625, 0.125, 0.25, 0.5, and 1 (increasing downward), and surface potentials, ζ = 1 (a), 2 (b), 4 (c), and 8 (d): a = 100 nm; δ = 0.1L; &B (Ns ) ≈ 2.96 nm; Ns = 0.1 M. The solid lines interpolate the computational results, and the dash-dotted lines are Ohshima’s [11] flat-plate theory.

mobility, Eq. (60), because polarization decreases with decreasing double-layer thickness, and the electroosmotic drag force increases in such a way as to balance the effect of increasing charge. Clearly, the double-layer thickness where Smoluchowski’s result is accurate decreases with increasing surface potential. The relationship between the mobility and ionic strength for particles with various coating thicknesses, also shown in Fig. 3, can be understood as follows. When the double layer is much thicker than the coating, the mobility is similar to that of the underlying bare particle, since the electroosmotic flow is mostly unaffected by Darcy drag inside the coating. With increasing ionic strength, an increasing portion of the double layer resides inside the coating, so, when κL > 1,

the effect of increasing the charge is more than compensated for by the increasing Darcy drag. At higher potentials, polarization decreases the electrical force, so the mobility decreases with increasing κa until the charge is sufficiently high and the double layer sufficiently thin that the mobility begins to increase with κa, similarly to bare particles. Figure 3 also shows that the mobilities approach the mobility given by Ohshima’s [11] flat-plate theory for coated particles with κa 1 and L a. With thicker coatings, the mobility reaches a minimum at intermediate κa, as shown in Fig. 4 where the mobilities from Fig. 3a are replotted with logarithmically scaled axes. Higher ionic strengths are required to observe similar behavior with higher surface potentials. Similarly to bare par-

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Fig. 4. Electrophoretic mobility, replotted from Fig. 3a (ζ = 1) with logarithmically scaled axes to highlight the dependence on κa in the thin-double-layer limit. Ohshima’s flat-plate theory (dash-dotted lines) breaks down at high κa when the coating is insufficiently thin. (a)

Fig. 5. Drag coefficient, f U /(6π ηaU ), of particles with neutral coatings as a function of κa. The conditions are the same as in Fig. 3d (ζ = 8), where the corresponding electrophoretic mobilities are shown: L/a = 0.0625, 0.125, 0.25, 0.5, and 1 (increasing upward).

ticles, the mobility becomes independent of ionic strength as κa → ∞. The double-layer thickness where this limit is reached, however, decreases with increasing coating thickness and surface potential. The structure of the flow near a fuzzy particle is intricate. Before examining the underlying velocity fields, it is helpful to recall the decomposition used to compute solutions of the governing equations. The flow-induced velocity field— from translation of the particle in the absence of an applied electric field—is practically independent of the electrical body forces generated by the perturbed potential, as shown in Fig. 5. Note that, at smaller surface potentials, the constant

(b) Fig. 6. Streamlines of the field-induced flow for particles with neutral coatings: ζ = 8; L = a; (a) κa ≈ 5.88 and (b) κa ≈ 58.8. The inner and outer circles indicate the bare particle surface and nominal edge of the coating, respectively.

drag force is approximately equal to the Stokes drag force on a bare particle whose radius is a + L + δ. Figure 6 compares streamlines of field-induced velocity fields, for a particle with a thick neutral coating, at two values of κa. When the double-layer thickness is much smaller than the coating thickness, but greater than the Brinkman screening length, a high velocity is produced close to the bare particle surface (Fig. 6a). The strong electrical stress re-

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sulting from the high concentration of counterions within the (thin) double layer generates a pressure gradient sufficient to draw fluid into the coating from the upstream side and discharge it outward on the downstream side. Shear stresses inside the coating can be neglected because the length scale characterizing the velocity gradients (the double-layer thickness) is larger than the Brinkman screening length. Therefore, the electrical body force balances the pressure gradient and the Darcy drag. When the double-layer thickness is smaller than both the coating thickness and the Brinkman screening length, a recirculating flow appears inside the coating (Fig. 6b). This may be explained by considering the various stresses acting on fluid drawn into the double layer on the upstream side and discharged downstream. Within a Brinkman screening length of the bare-particle surface, shear stresses balance the electrical body force. However, as the fluid decelerates while being discharged downstream, the length scale over which the velocity changes increases, so the viscous stresses that previously balanced the electrical body force must instead be balanced by an adverse pressure gradient, which sustains a recirculating flow. Note that, on account of the low mobility, the fluid velocities are very low. 4.2. Varying the double-layer thickness at constant surface charge Conditions of constant surface charge are sometimes observed in experiments with latex particles when the ionic strength is varied [4], although at low ionic strengths the apparent charge can decrease with increasing ionic strength. To illustrate the range of surface potentials corresponding to representative charge densities over a wide range of doublelayer thicknesses, Fig. 7 shows how the surface potential depends on κa with dimensionless charge densities, σ κa, in the range 8–512; the corresponding dimensional charge densities are in the range 0.14–9.2 µC cm−2 . Note that κa > 1 for most colloidal particles, but smaller values may be achieved with nano-scale particles [30]. Accordingly, since potentials are often highest at low ionic strengths, the effects of polarization and relaxation tend to be most effective when κa is small. The relationship between the mobility and ionic strength at constant surface charge, for particles with various coating thicknesses and surface charge densities, is shown in Fig. 8. The overall decrease in the mobility with increasing κa is because decreasing the double-layer thickness increases the electroosmotic drag. Mobility maxima occur at high charge densities and intermediate κa because polarization decreases the electrical force. For bare particles with a low surface charge, the departure from Smoluchowski’s mobility is mainly because of surface curvature. As κa → ∞, polarization becomes negligible while the electroosmotic drag force continues to increase with the decreasing double-layer thickness. As Fig. 8 shows, an uncharged coating strongly modulates these effects. As much as the interaction between

Fig. 7. Dimensionless surface potential, ζ , of bare particles (with neutral coatings) as a function of κa (aqueous KCl at 298 K) for various dimensionless surface charge densities, σ κa = 8, 32, 128, and 512. The dashed lines are the semiempirical formula (61), and the solid line interpolates the coinciding computational results.

electroosmosis and polarization is weakened by Darcy drag within the coatings, the mobility maxima disappear if the coatings are sufficiently thick. Figure 8 also shows the mobilities of coated particles obtained by multiplying the “exact” mobilities of the bare particles by (κ&B )2 exp(−L/&B ) − exp(−κL) , (64) (κ&B )2 − 1 an approximate result derived by Cohen and Khorosheva [17] to correct the Smoluchowski mobility of bare particles with thin neutral coatings and low surface potentials. Adjusting the bare-particle mobilities in this way gives a reasonable approximation of the coated-particle mobilities over a wide range of coating and double-layer thicknesses. Because the effects of polarization are mostly accounted for in the calculation of the bare-particle mobilities, Eq. (64) also leads to reasonable approximations at low and high surface charge densities. Note that, similarly to Ohshima’s flat-plate theory (see Fig. 4), Eq. (64) gives a significant relative error in the mobility when κa is large and the coating is insufficiently thin.

5. Charged coatings For particles with charged coatings, the underlying equilibrium potential and ion concentrations differ fundamentally from those of bare particles. Since the solution of the perturbed electrokinetic equations depends significantly on the underlying equilibrium, it is helpful to examine the equilibrium potential for charged coatings with various fixed charge densities and electrolyte concentrations. In this work,

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67

(a)

(b)

(c)

(d)

Fig. 8. Dimensionless electrophoretic mobility, M, of particles with neutral coatings as a function of κa (aqueous KCl at 298 K) for various coating thicknesses, L/a = 0, 0.0625, 0.25, and 1 (increasing downward), and dimensionless surface charge densities, σ κa = 8 (a), 32 (b), 128 (c), and 512 (d): a = 100 nm; δ = 0.1L; &B (Ns ) ≈ 2.96 nm; Ns = 0.1 M. Solid lines interpolate the computational results, long-dashed lines are the mobility of the bare particles given by Smoluchowski’s formula, short-dashed lines are approximations to the coated-particle mobilities using Cohen and Khorosheva’s [17] formula (64) to adjust the exact bare-particle mobilities, and dash-dotted lines are Ohshima’s [11] flat-plate theory.

the bare particle is neutral, and the polymer segment and fixed charge density distributions are assumed to be independent of the ionic strength. The mobility of polyelectrolytecoated particles is examined after surveying the equilibrium base state and its relation to Donnan equilibrium. Note that, throughout this section, the mass of adsorbed polyelectrolyte per unit area, Γ , may be related to the nominal fixed ion density, N f , by Γ ∼ N f 103 ML/Z,

(65)

where N f is in mol L−1 , M is the polymer molecular mass, and Z is the polymer valence. For a coating with N f =

0.1 M and L = 100 nm (e.g., Fig. 9), Γ ∼ 1.88 mg m−2 when M = 150 kg mol−1 and Z = 800. Further, with a bareparticle radius a = 100 nm, the number of adsorbed chains is approximately 950. 5.1. Equilibrium electrostatic potential distributions The equilibrium distribution of charge and electrostatic potential inside a charged coating is closely related to the thermodynamic state known as Donnan equilibrium. This involves the distribution of ions and electrostatic potential between two semi-infinite, electrically neutral regions; one

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Fig. 9. Equilibrium potential, ψ 0 , scaled with the Donnan potential, ψD , as a function of radial distance, relative to the nominal edge of the coating, for a highly charged (N f = 10−1 M) polyelectrolyte-coated particle: a = 100 nm; L = a; δ = 0.1L. The ionic strength of the electrolytes (aqueous KCl at 298 K) I = 10−5 , 10−3 , and 10−1 M giving (κa, ψD ) ≈ (1.04, −9.21), (10.4, 4.61), and (104, −0.481), respectively. As shown more clearly in the inset, the potential of the coated particle with κa ≈ 104 decays on the length scale δ.

consists of electrolyte and a uniform distribution of fixed charges, and the other is electrolyte alone. The relation between the (dimensionless) Donnan potential, ψD , in a region with a fixed charge density N f zf e, in equilibrium ∞ with a z–z electrolyte with n∞ j = n , is given by sinh(zψD ) = zf N f /(2zn∞ ),

(66)

with (scaled) ion densities inside the coating n0j = (1/2)z−2 exp(−zj ψD )

(j = 1, 2).

(67)

The potential within the domain occupied by the fixed charge is proportional to N f /n∞ when this ratio is small, and grows logarithmically with N f /n∞ when the ratio is large. Equilibrium conditions inside a polyelectrolyte coating approach Donnan equilibrium when the electrostatic screening length is smaller than the characteristic distance over which the fixed charge density changes, i.e., when κδ 1 and L δ. Figure 9 shows potential distributions for a thick, brush-like coating with the nominal edge of the coating at r ≈ κa + κL. As the figure indicates, under these conditions the local electrostatic potential approaches the Donnan potential close to the underlying bare surface. However, this need not be the case with lower fixed charge densities. The figure also shows that the potential decays on the scale δ when κδ > 1, as is the case when κa ≈ 104. An appreciation for some of the effects of coating permeability can be gained by further inspection of the po-

Fig. 10. Dimensionless electrophoretic mobility, M, of polyelectrolyte-coated (L/a = 0.01) and bare (L/a = 0) particles, with κa = 50 and a = 150 nm, as a function of the (dimensionless) equivalent surface charge density, σ . The dashed line is Saville’s [12] thin-double-layer theory for a completely permeable coating.

tential distributions shown in Fig. 9. Recall that the charge in the double layer equals the fixed charge on the polymer and, furthermore, that the double layer occupies the region over which the potential decays. When κa ≈ 1.04, for example, the potential decays over the region −0.1 < (r − κa − κL)/(κa + κL) < 0.1. Suppose that the layer is completely permeable to flow. Then, the electrophoretic mobility reflects the net coating charge, subject to the effects of polarization and relaxation and viscous drag on the underlying bare surface. If the coating is impermeable to flow, the shear surface will be shifted outward to r ≈ κa + κL + κδ and the effective charge will be smaller in proportion to the amount of charge immobilized behind the new shear surface. This, of course, decreases the mobility. The analysis that follows quantifies this argument by comparing the behavior of bare and coated particles with the same net charge. 5.2. Thin charged coatings Contrary to expectations, the behavior of particles with thin coatings differs substantially from that of bare particles. According to Saville’s [12] asymptotic theory, the mobility of a particle with a thin polyelectrolyte coating can exceed the mobility of its bare counterpart with the same net charge. It is of interest to see whether full calculations reveal similar behavior for particles with arbitrary coating thicknesses. The electrophoretic mobilities of bare and polyelectrolyte-coated particles are shown in Fig. 10 as a function of the equivalent surface charge density. The completely permeable and almost impermeable polyelectrolyte coatings are thin compared to both the particle radius (L = 0.01a) and the double-layer thickness (κL = 0.5). To behave as an impermeable coating, the nominal Brinkman screening

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length &B (Ns ) ≈ 0.1 nm, giving (κ&B )−2 ≈ 900 as the coefficient of the Darcy drag term in Eq. (9). While an Angstrom-scale screening length is, perhaps, unrealistic, particularly when considering the segment densities and sizes of typical adsorbed polymers, such a low permeability might be more representative of a highly crosslinked gel. Saville’s theory for a completely permeable coating gives a much higher mobility than the full calculation because there is an insufficient separation of the double-layer and coating thicknesses. Nevertheless, even with κL = 0.5, the mobility of the bare particle lies between the completely permeable and impermeable limits for coated particles with the same charge, thereby bearing out inferences drawn from the asymptotic theory. 5.3. Electrophoretic mobility of particles with moderate coating thicknesses (a)

The relationship between the mobility and ionic strength for coated particles with thick (L = a) coatings and various fixed charge densities is shown in Fig. 11. The difference between Figs. 11a and 11b stems from the order of magnitude change in the nominal segment density. The nominal Brinkman screening length √ of the more permeable coatings in (a) are accordingly 10 times larger than in (b). For comparison, the mobilities of bare particles are shown with surface potentials set equal to the Donnan potentials of the coated particles. At low ionic strengths, when the double layer is well outside the coatings, the mobilities are nearly proportional to the Donnan potential, as might be expected by analogy with Hückel’s theory for bare particles, M = ζ , when ζ and κa are small. This behavior is difficult to observe experimentally because κa is typically greater than one with the particle sizes and ionic strengths achieved in practice. With nanoscale particles, however, κa ∼ 0.1 can be achieved in an aqueous electrolyte with a = 10 nm. With increasing ionic strength, the behavior departs significantly from that of a bare particle, and the mobilities become much more sensitive to the coating permeability. Whereas the mobility of a bare particle with a fixed surface charge vanishes at high ionic strength, because ions in the double layer are concentrated at the underlying bare surface, the mobility is finite when the fixed charge is distributed throughout a coating. In fact, the limiting mobilities in Fig. 11 are proportional to the fixed charge density and the coating permeability, and, as follows, can be predicted from a simple balance of forces inside the coatings. At high ionic strengths, Donnan equilibrium inside the coatings is characterized by a low potential and, because of electrical neutrality, the counterion density inside the coatings equals the fixed charge density. Therefore, the electrical body force on the fluid inside the coating in problem (E) is N f zf eE. Equating this body force to the Darcy drag force acting on the fluid, (η/&2B )Us , gives a slip velocity inside the coating Us = N f zf eE&2B /η. In problem

(b) Fig. 11. Dimensionless electrophoretic mobility, M, of polyelectrolyte-coated particles as a function of κa (aqueous KCl at 298 K) for various nominal fixed ion densities, N f = 0.01, 0.05, and 0.1 M: a = 100 nm; L = a; δ = 0.1L. Dashed lines show the mobilities of bare particles with radii and surface potentials set equal to a + L and ψD , respectively, of the coated particles with N f = 0.01 M (short) and 0.1 M (long).

(U ), the particle may be assumed bare and uncharged, with a radius a + L + δ if &B L, so the coating is impermeable to the external flow. In the thin-double-layer limit, the far-field fluid velocities in problems (U ) and (E) are, respectively, Stokeslet velocity disturbances proportional to U and Us . Balancing the forces on the particle gives V = Us , and hence the dimensional electrophoretic mobility is N f zf e&2B /η, or M = 3N f zf (e&B )2 /(2s o kB T ).

(68)

Thus, for a given fluid, the mobility depends only on the fixed charge density and the coating permeability. Equa-

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(a)

(b) Fig. 12. Streamlines of (a) the flow- and (b) field-induced contributions to the velocity field for electrophoresis of a polyelectrolyte-coated particle: a = 100 nm; L = a; δ = 0.1L, κa ≈ 58.8; N f = 0.01 M; &B (Ns ) ≈ 9.38 nm. The inner and outer circles indicate the bare particle surface and the nominal edge of the coating, respectively.

tion (68) accurately predicts the limiting mobilities shown in Fig. 11. While the coatings may be impermeable to an external flow (&B L), they permit an internal electroosmotic flow and a finite electrophoretic mobility, as pointed out by Ohshima [11]. Note that, if N f is proportional to &−2 B , as assumed here because N f is proportional to Ns , Eq. (68) does not require the segment or charge density to be uniform.

Fig. 13. Dimensionless electrophoretic mobility, M, of polyelectrolyte-coated particles as a function of κa (aqueous KCl at 298 K) for various nominal fixed ion densities, N f = 0.01, 0.05, and 0.1 M: a = 100 nm; L = a; δ = 1.0L; &B (Ns ) ≈ 2.91 nm; Ns = 0.1 M. Dashed lines show the mobilities of bare particles with radii and surface potentials set equal to a + L and ψD , respectively, of the coated particles with N f = 0.01 M (short) and 0.1 M (long).

To illustrate the flow structure when κa 1, Fig. 12 shows streamlines of the flow- and field-induced velocity fields for a moderately large value of κa. The streamlines in Fig. 12a show show that the coating is impermeable to the externally imposed flow at distances out to r ≈ κa + κL + κδ. The streamlines in Fig. 12b, however, show that the fieldinduced flow penetrates the coating, with the streamlines toward the edge of the coating being closely aligned with the applied electric field. Figure 13 shows the relationship between the mobility and ionic strength for particles with much more slowly decaying segment and charge density distributions. Recall, the distance over which the segment density decays is δ = L. Comparing these mobilities with those in Fig. 11a with δ = 0.1L shows that the mobility is lower when κa is small. This is because more of the mobile or diffuse charge is immobilized behind the shear plane. However, with increasing ionic strength, the double layer recedes further into the coating and the mobilities eventually approach values given by Eq. (68), also seen in Fig. 11a. Figure 14 shows the relationship between the mobility and ionic strength for particles with a much thinner coating L = 0.125a with δ = 0.1L. These mobilities may be compared with those in Fig. 11b with L = a and δ = 0.1L. With a thinner coating, vestiges of the bare-particle behavior are evident at intermediate values of κa. However, because the nominal Brinkman screening length is still smaller than the coating thickness (&B (Ns , as )/L ≈ 0.075), the same limiting mobilities given by Eq. (68) are approached when the double layer is sufficiently thin.

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wide range of conditions. In forthcoming work, we use the model developed here to examine the response of polymercoated particles to oscillatory fields, such as those used in dielectric spectroscopy and electrosonic amplitude (ESA) experiments. While the effects of charge density and ionic strength on the conformation of polyelectrolyte coatings is beyond the scope of this study, these too should be addressed.

Acknowledgment Supported by the Princeton Center for Complex Materials, a Materials Research Science and Engineering Center (DMR-9809483).

Fig. 14. Dimensionless electrophoretic mobility, M, of polyelectrolyte-coated particles as a function of κa (aqueous KCl at 298 K) for various nominal fixed ion densities, N f = 0.01, 0.05, and 0.1 M: a = 100 nm; L = 0.125a; δ = 1.0L; &B (Ns ) ≈ 0.938 nm; Ns = 1.0 M. Dashed lines show the mobilities of bare particles with radii and surface potentials set equal to a + L and ψD , respectively, of the coated particles with N f = 0.01 M (short) and 0.1 M (long).

6. Summary The electrophoretic mobility of polymer-coated colloidal particles has been studied via “exact” numerical solutions of the governing electrokinetic equations. The results for neutral and charged coatings cover a wide range of particle charges, and coating and double-layer thicknesses; the coating permeability and segment density distributions were also varied. In general, a polymer coating lowers the electrophoretic mobility according to how effectively the motion of free charge behind the shear surface is resisted by drag on the polymer. The mobility of a particle with a charged and highly permeable coating, however, may exceed that of its bare counterpart, even with the effects of polarization and relaxation. Some interesting effects transpire with charged coatings, since the polymer region is always occupied by mobile charge. At low ionic strengths, when the double layer is thicker than the coating, the mobility is similar to that of a bare particle with a surface potential comparable to the Donnan potential, which may be high. Under these conditions, polarization and relaxation are important, and the mobility may be obtained from the standard electrokinetic model for bare particles. At high ionic strengths, if the Brinkman screening length is smaller than the coating thickness, the electrophoretic mobility reflects only the polymer charge and segment densities. Under these conditions, the mobility is finite, even though the electrostatic potential, characterized by the Donnan potential, is low. Numerical solutions of the full electrokinetic equations permit comparisons between theory and experiment over a

Appendix A. Point-force representation of a particle with a charged porous coating Here we relate the net force on a spherical particle, with a charged or neutral porous coating, to the force on a point particle producing the same far-field velocity disturbance. The particles undergo oscillatory translational motion in an electrolyte that would otherwise be at rest. In dimensionless form, the velocity disturbances produced by a finite-sized coated particle and a point particle satisfy ∇ · Σ 1 = ∇ 2 u1 − ∇p1 = −iΩu1 + B(u1 − V ) + κa

N 

nj zj ∇ψ

(A.1)

j =1

and ∇ · Σ 2 = ∇ 2 u2 − ∇p2 = −iΩu2 + f δ(r),

(A.2)

respectively, where Σ = −pI + ∇u + (∇u)T is the fluid stress tensor and f is the force exerted on the point particle. The strength of the point force is obtained from the reciprocal identity [31]  (u1 · Σ 2 · nˆ − u2 · Σ 1 · n) ˆ dA  = (u1 · ∇ · Σ 2 − u2 · ∇ · Σ 1 ) dV , (A.3) where dV is the volume outside the underlying bare particle and dA is the surface (with outward unit normal n) ˆ composed of an inner surface coinciding with the bare surface and an outer spherical surface (centered at the particle) with a radius R → ∞. The integrals over the outer surface vanish for oscillatory flow because the far-field velocity disturbances decay faster than r −1 . Substituting the divergence of the stress tensors given by Eqs. (A.1) and (A.2) into (A.3) and applying the boundary

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condition u1 = V at the bare surface give   V · Σ 2 · nˆ dA − u2 · Σ 1 · nˆ dA = u1 · f    N  − u2 · B(u1 − V ) + κa nj zj ∇ψ dV .

(A.4)

j =1

The first integral on the left-hand side of (A.3) may be expressed in terms of an integral of ∇ · Σ 2 over the interior volume of the bare particle, Vp , giving   V · Σ 2 · nˆ dA = iΩV · u2 dV . (A.5) Now, u2 in the remaining volume integrals can be assumed constant, to leading order, when the point force is located far from the bare particle. This is because the porous medium and electrical body forces are nonzero only within a finite distance of the particle. Collecting all the terms and requiring the far-field velocity disturbances u1 and u2 to be equal give  iΩVp V − Σ 1 · nˆ dA    N  =f − B(u1 − V ) + κa (A.6) nj zj ∇ψ dV .

j =1

Note that − Σ 1 · nˆ dA is the force acting on the underlying bare surface, B(u1 − V ) dV is the hydrodynamic drag on  the porous coating, and κa N j =1 nj zj ∇ψ dV includes the electrical force acting on the particle and its charged coating. The sum of these forces must equal the acceleration of the particle’s mass, −iΩ(ρp /ρs )Vp V , so the strength of the point force is f = iΩVp V − iΩ(ρp /ρs )Vp V .

(A.7)

In other words, the acceleration of the mass of fluid displaced by a finite-sized particle, −iΩVp V , must be added to the force on a point particle producing the same far-field velocity disturbance, as shown by Mangelsdorf [32] for bare particles. Appendix B. Numerical solution of the equilibrium field equations Equation (26) is solved using a finite-difference method with a nonuniform distribution of grid points. The gridpoint allocation algorithm uses the approach described in Appendix C for solving the perturbed equations. The far-field boundary condition is implemented using the asymptotic form ψ 0 ∼ exp(−r)/r 2

as r → ∞,

(B.1)

which requires the ratio of the potentials at neighboring grid points at the boundary of the finite-difference mesh to be 0 2 ψi+1 /ψi0 = exp(ri − ri+1 )ri2 /ri+1 ,

(B.2)

where the subscripts i and i + 1 denote the ith and (i + 1)th grid points, respectively. The boundary condition at the particle surface is satisfied using standard finite-difference approximations, chosen to preserve the tridiagonal form of the resulting system of linearized finite-difference equations. This reduces the order of the approximations at the boundaries, relative to those in the interior of the domain, but the formal loss of accuracy is compensated for by the nonuniform grid. The nonlinear finite-difference equations are solved using Newton iteration, which converges rapidly if a reasonable initial guess of the solution is prescribed. Convergence is much slower when the gradient boundary condition (28) is implemented. In this case, the surface potential is adjusted— using Newton iterations, for example—to converge on the required surface potential gradient or surface charge.

Appendix C. Numerical solution of the perturbed field equations All lengths are scaled with κ −1 despite there being numerous other length scales (the distance over which the velocity field decays in problems (U ) and (E), a; the viscous penetration depth, (ν/ω)1/2 ; and the characteristic dimensions of the polymer coating, L and δ). The inner scales are resolved by a nonuniform grid-point allocation. To obtain accurate results, however, the dependent variables must satisfy their respective asymptotic forms in the far field. These are obtained from the solution of the governing equations by neglecting the terms with exponentially small coefficients— the terms with coefficients proportional to the equilibrium potential and the perturbations of the equilibrium ion densities from their respective bulk values. The resulting asymptotic forms for the perturbations can be obtained by examining the eigenfunctions of the matrix of linear differential operators [26]. The far-field perturbed potential takes the form ˆ · rˆ ∼ D X X · rˆ /r 2 ψ  = ψX

(C.1)

under both steady and oscillatory conditions, whereas the perturbed ion densities take the form nˆ j = nˆ j X · rˆ ∼ JjX X · rˆ /r 2

(C.2)

under steady conditions, but decay exponentially when the applied electric field is oscillatory, in which case nˆ j are set to zero at the outer boundary. Since all the dependent variables in the far field take the form of a function α(r), say, times an unknown asymptotic coefficient, the ratio of their values at the two outermost grid points gives αi+1 /αi = α(ri+1 )/α(ri ).

(C.3)

The asymptotic coefficients can then be determined from the values of the functions at the grid points where the boundary conditions are applied.

R.J. Hill et al. / Journal of Colloid and Interface Science 258 (2003) 56–74

Accuracy can be improved if higher-order approximations of the far-field asymptotic forms can be specified. While this approach was used successfully by Mangelsdorf [26], it is, unfortunately, not well suited for the finitedifference method used in this work. Nevertheless, it is still helpful to use the exact form of hr for the far-field (Stokeslet) velocity disturbances. Under steady conditions, hr may be written as hr = C X [3/a 2 − 1/r 2 ] as r → ∞,

(C.4)

whereas under oscillatory conditions, hr may be written as [24,25]     hr = (C X /c2 ) c1 exp(ikr) r − 1/(ik) + c2 r 2 as r → ∞

(C.5)

where c1 = −3a exp(−ika)/(2ik),   c2 = −(1/2)a 3 1 − 3/(ika) − 3/(ka)2 ,  1/2 . k = (1 + i) ω/(2ν)

(C.6)

with β ∈ {C, D, Jj }, or β/E = −(V /E)β U + β E .

(C.7)

To minimize the number of diagonal elements in the resulting banded matrix of finite-difference coefficients, it is convenient to reduce the third- and fourth-order derivatives of hr in the momentum equation to second and first order, respectively. Introducing an auxiliary variable, h =

∂ 2 hr , ∂r 2

and hr are then scaled with their maximum values, and their curvature scales, defined as 

(∂ 2 α/∂r 2 )/α

−1/2

ˆ nˆ j , hr }, with α ∈ {ψ,

(C.9)

are computed. From a suitable weighted average, a single scale is determined at each grid point. The grid points are then redistributed so their local density is inversely proportional to the local curvature scale. The coefficients of the finite-difference equations that depend on the equilibrium base state are obtained by interpolating the equilibrium base state variables onto the new grid. This procedure may be repeated until the relative change in the computed asymptotic coefficients reaches a specified tolerance. The number of iterations required to reach a given tolerance tends to increase with the equilibrium potential (or ion density in the double layer relative to the bulk) and as the ratio of the various characteristic length scales deviates from unity.

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Note that the complex constant C X /c2 allows both the phase and amplitude of the inner and outer velocity fields to match. After superposing the solutions to problems (U ) and (E) to determine V /E required to satisfy the particle equation of motion and the boundary conditions, the asymptotic coefficients for the complete problem are β = β U (U − V ) + β E E = −β U V + β E E,

73

(C.8)

increases the bandwidth of the matrix to be inverted, but avoids having to use cumbersome finite-difference approximations for third- and higher-order derivatives on the nonuniform grid. The boundary conditions for h are obtained from derivatives of the asymptotic form assumed for hr . These are implemented in the same way as for the other dependent variables. Beginning with the distribution of grid points converged on by solving the Poisson–Boltzmann equation for the equilibrium potential and ion densities, the finite-difference equations for the perturbed variables are solved using a standard algorithm that exploits the banded structure of the matrix of finite-difference coefficients. The functions ψˆ , nˆ j ,

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