## Abstract

The electric-field-induced response of an uncharged colloidal sphere embedded in a quenched polyelectrolyte hydrogel is calculated from a model where the polymer network is treated as an elastic, porous skeleton saturated with an aqueous electrolyte. We present exact analytical solutions for the steady response to a uniform electric field, as well as the steady susceptibility, defined as the ratio of the particle displacement to the strength of an optical or magnetic force. Even though the particle is uncharged, it attains a finite electric-field-induced displacement owing to hydrodynamic coupling with electroosmotic flow. The steady susceptibility decreases with increasing charge and decreasing electrolyte concentration; in general, charge imparts a small correction to the classical theory for an uncharged linearly elastic continuum.

## 1. Introduction

Hydrogels are water-saturated polymer networks that have widespread applications in drug delivery (Qiu & Park 2001) and tissue engineering (Brandl *et al.* 2007); they have also been identified as promising candidates for synthetic muscles (Calvert 2004; Bar-Cohen 2007). In the recent years, there has been much interest in micro-rheological characterization of hydrogels, particularly those of biological origin (Waigh 2005).

Micro-rheology probes viscoelastic properties of the micro-structure from the dynamics of embedded colloidal particles. The principal advantages of micro-rheology over macro-scale rheological methods include small sample size (MacKintosh & Schmidt 1999; Gardel *et al.* 2005), wide frequency range (Schnurr *et al.* 1997) and the ability to directly probe micro-scale characteristics of soft matter (Schnurr *et al.* 1997; MacKintosh & Schmidt 1999; Meyer *et al.* 2006).

While two-point micro-rheology probes non-local (bulk) characteristics (Crocker *et al.* 2000; Levine & Lubensky 2001*b*), single- and two-point methods are both sensitive to electrostatic, chemical and steric interactions between the particle and the matrix (McGrath *et al.* 2000; Valentine *et al.* 2004; Ehrenberg & McGrath 2005). Such interactions are generally undesirable in micro-rheology because they complicate an otherwise simple conversion of experimental data to bulk rheological characteristics. However, with an increasing interest in hydrogel–colloid composites (Schexnailder & Schmidt 2009), understanding how such interactions affect particle dynamics may furnish novel diagnostics, in a similar manner that micro-electrophoresis and electroacoustics, for example, have become routine for studying colloidal dispersions (O’Brien & White 1978; O’Brien 1990).

Micro-rheological techniques are generally classified as active or passive (Waigh 2005). In active methods, probe particles are driven by magnetic (Freundlich & Seifriz 1923; Ziemann *et al.* 1994) or optical (Valentine *et al.* 1996) forces, whereas the dynamics in passive experiments are entirely due to thermal fluctuations. Electrical forces have received much less attention because of complicating electrokinetic influences, such as diffuse double-layer dynamics and electroosmotic flow. Accordingly, few experiments have been reported (Mizuno *et al.*2000, 2001, 2004) and our understanding of the electric-field-induced displacement is poor.

Hill & Ostoja-Starzewski (2008) undertook the first theoretical study of electric-field-induced particle displacement. They calculated the steady electric-field-induced response of a charged, spherical colloid embedded in *incompressible*, uncharged polymer gels, showing that sub-nanometre displacements prevail under typical experimental conditions. Wang & Hill (2008) extended their model to *compressible*, but still uncharged, polymer skeletons, predicting displacements that are large enough to register with optical microscopy.

However, many hydrogels are charged, and even ideally uncharged gels (e.g. polyacrylamide) become weakly charged owing to chemical reactions, e.g. hydrolysis (Kizilay & Okay 2003). Fixed charge is well known to impact swelling and other responses to external stimuli (Skouri *et al.* 1995). Note also that the principal biological subjects of micro-rheology (e.g. F-actin networks) are charged.

In this paper, we take a first step towards quantifying the effect of polymer charge on the particle response to steady electrical and non-electrical forces. While our analysis is limited to uncharged inclusions in charged skeletons, it provides a simple physical and mathematical framework to furnish the exact analytical solutions. In principle, an experimental test of the theory could be undertaken using quenched polyelectrolyte hydrogels with an electrolyte whose pH is tuned to the isoelectric point of the immobilized inclusions. Interestingly, our theory predicts that particles with a dielectric constant that is much higher than water have a relatively strong response to electrical forcing when embedded in charged hydrogels.

We extend earlier theoretical analyses of spherical particles in charged viscoelastic matrices (Schnurr *et al.* 1997; Levine & Lubensky 2000), providing a more comprehensive interpretation of active and passive micro-rheology. Electrostatic interactions arising from changes in the density of the polymer skeleton upon deformation are expected to increase the effective rigidity and, therefore, attenuate the particle response. In previous theoretical treatments of active and passive micro-rheology, electrostatic influences have been implicitly lumped into the effective elastic constants, e.g. Poisson ratio and Young’s modulus (Levine & Lubensky 2000, 2001*a*). By separating the electrostatic penalty of compression from the intrinsic elastic energy, we seek to quantify how electrostatic screening owing to added electrolyte—and indeed from the charged matrix itself—attenuates the particle response.

Modern micro-rheological instruments, including contemporary electrophoresis devices (Minor *et al.* 1997), adopt oscillatory forcing. Nevertheless, small particles often respond in a quasi-steady manner. We pursue the quasi-steady response here to provide a sound understanding of the physics, in a similar manner to classical analyses of the Stokes hydrodynamic mobility and the Smoluchowski electrophoretic mobility, among other quasi-steady response functions. The frequency-dependent dynamic response may be more relevant to future interpretation of experiments, as amplitude attenuation and phase lag are expected due to draining and inertial influences. These are beyond the scope of this paper, but will be addressed elsewhere.

Note that a significant body of literature on the dynamics of polyelectrolyte hydrogels has emerged from the studies of articular cartilage (Lai *et al.* 1991; Gu *et al.* 1998). Lai *et al.* (1991) and Gu *et al.* (1998) derived the dynamical equations for charged, soft tissue based on multi-phase continuum theories that account for a charged porous solid, solvent (water) and added salt. Such theories are not suitable for the hydrogel–colloid composites considered in this paper, because they enforce local electroneutrality. Rather, these theories are appropriate on (macroscopic) scales larger than the Debye screening length, generally nm in aqueous electrolytes.

Li *et al.* (2004*a*,*b*, 2006) developed a so-called multi-physic, multi-effect model that adopts the Poisson equation to handle electrostatics. Similarly, Hill & Ostoja-Starzewski’s (2008) electrokinetic model for uncharged hydrogels with charged spherical inclusions extends the two-fluid model of Levine & Lubensky (2000, 2001*a*) by including electrokinetic influences. Our work extends these electrokinetic models to polyelectrolyte hydrogels by including the electrostatic forces owing to fixed charge on the polymer. In our parametric analysis, we approximate the polyelectrolyte as quenched, meaning that the fixed charge density is independent of pH, added salt and polymer concentration (Raphael 1990; Guo & Ballauff 2000, 2001). Nevertheless, such influences can be accounted for with models or experimental measurements of polyelectrolyte charge for a specific polymer architecture.

## 2. Theoretical model

Our continuum model for a polyelectrolyte hydrogel comprises three phases: a charged, soft, porous solid (polymer network), solvent (water) and ions (counterions and added salt) (Hill *et al.* 2003; Hill & Ostoja-Starzewski 2008). The porous medium is modelled as a compressible linear elastic solid with a continuous uniform distribution of fixed charge, and the solvent as an incompressible Newtonian fluid. The ionic charge is either mobile or fixed to the polymer skeleton. Mobile ions include *M* species of counterions of the fixed charge, and *N* species of ions from added salt. Electrostatics are governed by the Poisson equation
2.1
where *ψ*, *ϵ*_{°} and *ϵ*_{s} are, respectively, the electrostatic potential, vacuum permittivity and solvent dielectric constant. The mobile and fixed charge densities are and , where *e*, *n*_{j}, , *z*_{j} and are, respectively, the elementary charge, *j*th mobile ion number density, *j*th fixed charge number density, *j*th mobile ion valence and *j*th fixed charge valence. The flux of *j*th mobile ion is given by the well-known Nernst–Planck equation
2.2
where, at steady state, conservation demands
2.3
Here *D*_{j}, ** u**,

*k*

_{B}and

*T*are, respectively, the

*j*th ion diffusion coefficient, fluid velocity, Boltzmann constant and absolute temperature. Under steady conditions, the fluid velocity and ion fluxes are relative to a stationary porous skeleton.

Fluid momentum conservation is achieved via a linearized Navier–Stokes equation with electrical and Darcy drag forces
2.4
where **σ**^{h}=−*p*** I**+2

*η*

*e*^{h}is the Newtonian fluid stress tensor with

*e*^{h}=(1/2)[

**∇**

**+(**

*u***∇**

**)**

*u*^{T}] the (fluid) rate of strain tensor and

**the identity tensor. Here,**

*I**η*,

*p*and

*ℓ*are, respectively, the fluid shear viscosity, pressure and Brinkman screening length of the porous skeleton (Brinkman 1947). The Brinkman screening length is the square root of the Darcy permeability; in the polymer physics literature it is taken to be the mesh size of a polymer gel. The second and third terms on the right-hand side of equation (2.4) are, respectively, the hydrodynamic drag exerted by the polymer on the fluid, and the electrostatic body force acting on the fluid. Fluid mass conservation demands 2.5 because we assume the volume fraction of solvent approaches one (Lai

*et al.*1991; Levine & Lubensky 2001

*a*).

Static equilibrium of the poroelastic skeleton demands
2.6
with linear elastic stress tensor
2.7
Here, *e*^{e}=(1/2)[**∇**** v**+(

**∇**

**)**

*v*^{T}],

**,**

*v**ν*and are, respectively, the polymer elastic strain tensor, displacement, Poisson ratio and Young’s modulus. The second and third terms in equation (2.6) are the hydrodynamic drag of the fluid and electrostatic body force acting on the polymer, respectively. Substituting equation (2.7) into equation (2.6) gives an equation of static equilibrium: 2.8

The density of the polymer skeleton is expressed as (Landau & Lifshitz 1986) , where and *ρ*_{p} are the reference and deformed porous solid densities, respectively. Therefore, the fixed charge density under small-strain deformation is
2.9
where *ρ*^{f°} is the equilibrium fixed charge density of the polymer skeleton. Note that eqn (66) of Lai *et al.* (1991) approaches our equation (2.9) as the water volume fraction , which is often the case for the viscoelastic subjects of micro-rheology.

A spherical polar coordinate system (*r*, *θ*, *ϕ*) is adopted to solve the foregoing model equations. When an applied electric field ** E** is directed along the polar axis

*e*_{z}, with the origin centred on the particle, the boundary conditions for the electrostatic potential are at

*r*=

*a*,

*ψ*continuous at

*r*=

*a*, as , and

*ψ*finite at

*r*=0. Here, is an outward unit normal to the particle surface (for a spherical particle ); subscripts < and >, respectively, distinguish the particle and the solvent sides of the interface. Boundary conditions for the polymer displacement are

**=**

*v***at**

*Z**r*=

*a*and as . Other boundary conditions (see Hill & Ostoja-Starzewski 2008) are

**=**

*u***0**at

*r*=

*a*, at

*r*=

*a*, at , as , as . Note that

*ρ*

^{m°}is the equilibrium density of mobile charge, and bulk electroneutrality demands

*ρ*

^{m°}=−

*ρ*

^{f°}.

To determine the particle displacement under the influence of a weak electric field, a perturbation methodology is adopted (O’Brien & White 1978; Hill *et al.* 2003). The fields are calculated for equilibrium conditions, i.e. in the absence of an electric field and external force. For this equilibrium base state, the solution of the governing equations is simply *ρ*^{f}=*ρ*^{f°}=−*ρ*^{m}=−*ρ*^{m°}=constant, *ψ*=*ψ*^{°}=constant, *p*=*p*^{°}=constant, ** u**=

*u*^{°}=

**0**, and

**=**

*v*

*v*^{°}=

**0**, where the superscript ° denotes the equilibrium base state. Then, in the presence of an electric field, the model equations are linearized with

*ρ*

^{m}=

*ρ*

^{m°}+

*ρ*

^{m}′,

*…*,

*ψ*=

*ψ*

^{°}+

*ψ*′,

*…*,

**=**

*v***′, where primed quantities denote the perturbations from equilibrium. Linearized equations for the perturbations, which are generally valid for weak electric fields**

*v**E*=|

**|≪**

*E**κ*

*k*

_{B}

*T*/

*e*and small particle displacements

*Z*=|

**|≪**

*Z**a*, are 2.10 2.11 2.12 2.13 2.14 where . The foregoing

*M*+

*N*+8 independent scalar equations can be solved analytically to ascertain

*ψ*′,

*p*′,

**′,**

*u***′ and**

*v**n*

_{j}′. From the resulting force on the inclusion, the superposition methodology detailed in the next section gives the particle displacement in response to an applied force.

### (a) Particle displacement

The general solution for the perturbations is obtained from two independent sub-problems. In the so-called *Z*-problem, the particle is displaced a distance ** Z** in the absence of an applied electric field (

**=**

*E***0**). This is equivalent to a uniform translation of the far field with the particle fixed at the origin, giving boundary conditions

**′=**

*v***0**at

*r*=

*a*and as In the so-called

*E*-problem, the particle is fixed at the origin (

**=**

*Z***0**) in the presence of an external electric field

**. The boundary conditions for the polymer displacement are then**

*E***′=**

*v***0**at

*r*=

*a*and as .

The total force on the particle ** F** is the sum of the forces

*F*^{Z}and

*F*^{E}from the foregoing

*Z*- and

*E*-problems. To satisfy the particle equation of motion at steady state,

*F*^{Z}=−

*F*^{E}. Because the perturbed problem is linear, the forces can be written (O’Brien & White 1978) where

*f*

^{E}and

*f*

^{Z}are independent of

*E*and

*Z*. Accordingly, the electrical response is defined as where

*f*

^{E}and

*f*

^{Z}are from equations (2.10)–(2.14) and their boundary conditions. It is expedient to separate the forces acting on the colloidal particle into electrical, hydrodynamic, and elastic (or mechanical-contact) contributions. However, there is no electrical force, so the hydrodynamic and elastic forces are furnished by integrals of the hydrodynamic and elastic stress.

#### (i) *Z*-problem

In this problem, the fluid is at rest, so there is no force from the deviatoric fluid stress. However, in contrast to uncharged hydrogels (Hill & Ostoja-Starzewski 2008; Wang & Hill 2008), there exists a hydrostatic force owing to the gradients of osmotic pressure. The mechanical-contact force is due to polymer displacement where, in contrast to uncharged hydrogels, the displacement is coupled to the electrostatic potential.

It is well known that a vector field can be decomposed into irrotational and solenoidal parts, so
2.15
where ** A** (with

**∇**⋅

**=0) and**

*A***∇**

*ϕ*(with

**∇**×

**∇**

*ϕ*=0) decay as . Linearity and symmetry require (Landau & Lifshitz 1987) 2.16 so equation (2.11) becomes 2.17 Multiplying both sides of equation (2.17) by

*z*

_{j}

*e*, and summing over the mobile ions (

*j*=1,

*…*,

*M*+

*N*) gives 2.18 where is the ionic strength. The sum includes counterions of the fixed charge, so the ionic strength is non-zero in the absence of added salt.

Substituting equation (2.15) into equations (2.10) and (2.13), taking the divergence and curl of the resulting equation (2.13), and noting that ** u**′=

**0**, gives 2.19 2.20 2.21 2.22 From an exact analytical solution of these equations, the hydrodynamic (osmotic pressure) and mechanical-contact forces are 2.23 and 2.24 respectively, where 2.25 with

*κ*

^{2}=2

*I*

*e*

^{2}/(

*k*

_{B}

*T*

*ϵ*

_{°}

*ϵ*

_{s}) and . Note that

*κ*

^{−1}is the well-known Debye screening length, and

*β*

^{−1}is a new length scale whose physical significance will be discussed below. Recall,

*I*includes contributions from the added salt and counterions of the fixed charge. Therefore, if the polymer bears a finite charge,

*κ*

^{−1}and

*D*

^{−1}are finite in the absence of added salt.

Finally, summing the forces above gives
2.26
As the fixed charge density vanishes, and it is readily verified that we recover the same formula as Hill & Ostoja-Starzewski (2008) and Lin *et al.* (2005) for uncharged hydrogels.

#### (ii) *E*-problem

In this problem, the polymer displacement, fluid velocity and pressure are all non-zero. As the fluid velocity is divergence-free,
2.27
where ** B**=

**∇**

*g*(

*r*)×

**. Similarly to the**

*E**Z*-problem, the polymer displacement takes the form 2.28 where

**=**

*C***∇**

*k*(

*r*)×

**, and the vector fields**

*E***and**

*C***∇**

*φ*vanish as .

We now obtain 2.29 2.30 2.31 2.32 2.33 2.34 From an exact analytical solution of these equations for the polymer displacement, and fluid velocity and pressure, the hydrodynamic and mechanical-contact forces are 2.35 and 2.36 where 2.37 The hydrodynamic force comprises viscous, dynamic- and osmotic-pressure terms. Summing the forces above gives 2.38

### (b) Dielectric contrast

Surprisingly, the electrical response from equations (2.26) and (2.38) depends on the particle dielectric constant *ϵ*_{p}. This is contradictory to the expectations from O’Brien & White’s (1978) well-known analysis of the electrostatic boundary conditions for electrophoresis of colloidal particles in Newtonian electrolytes. They proved that the electrophoretic mobility of a charged colloidal particle is independent of the particle dielectric constant. Moreover, Hill & Ostoja-Starzewski (2008) identified a close connection between the steady electrical displacement of charged inclusions embedded in uncharged, incompressible hydrogels. Accordingly, for uncharged, compressible hydrogels, the particle displacement is also independent of the dielectric constant (Wang & Hill 2008).

To help verify the distinctly different behaviour with a charged polymer skeleton, we follow O’Brien & White (1978) and introduce *ψ*′^{1}, *p*′^{1}, ** u**′

^{1}and

**′**

*v*^{1}as solutions of the

*X*-problem (with

*X*=

*Z*or

*E*) with particle dielectric constant . The electrostatic boundary conditions at the particle surface

*r*=

*a*are with continuous

*ψ*′

^{1}. Here, subscripts < and > distinguish, respectively, the particle and the solvent sides of the interface. Similarly,

*ψ*′

^{2},

*p*′

^{2},

**′**

*u*^{2}and

**′**

*v*^{2}denote solutions of the same problem, but with a particle dielectric constant and boundary conditions and continuous

*ψ*′

^{2}.

As the equations governing the perturbations are linear, differences (denoted with the symbol Δ below) owing to changing *ϵ*_{p} are the solutions of the same equations, but with electrostatic boundary conditions and continuous Δ*ψ*′, where . All other differences at the particle surface vanish, and all the differences vanish as . Note also that the governing equations are independent of whether *X*=*Z* or *E*. Our solutions for *ψ*′ yield
2.39
where *θ* is the azimuthal angle with ** X** directed along the polar axis. Accordingly, the differences are the perturbations that arise from endowing an originally uncharged particle embedded in a uniform, unperturbed hydrogel with the non-uniform surface charge density given by equation (2.39).

With an uncharged polymer skeleton, the surface charge *q* cannot give rise to a force, because excess negative charge on one side is compensated for by an equal excess of positive charge on the other side. Overall, the osmotic pressure retains fore–aft symmetry, so there is no net force on the particle. In a charged polymer skeleton, however, negative surface charge on one side of the particle repels (attracts) a negatively (positively) charged skeleton, while positive charge on the other side attracts (repels) a negatively (positively) charged skeleton. Thus, the particle experiences a net elastic (mechanical-contact) force whose magnitude is expected to increase with the (dipole) strength of the surface charge density given by equation (2.39).

The force on the particle in the *X*-problem can be calculated from the far-field decay of the pressure *p*, velocity ** u** and displacement

**. In the electrophoretic mobility problem, as O’Brien & White have shown, the far-field decays of velocity and pressure are independent of**

*v**ψ*. Therefore, the force is independent of

*ϵ*

_{p}. However, for the problem addressed in this work,

*p*and

**are coupled to**

*v**ψ*in the far field.

By introducing a function *Φ*_{j} defined by
2.40
O’Brien & White show that the far-field decays of *p* and ** u** in the electrophoretic mobility problem are independent of the potential

*ψ*and ion densities

*n*

_{j}. We have 2.41 2.42 2.43 2.44 2.45 where the boundary conditions for

*Φ*

_{j}are at

*r*=

*a*and as .

Equation (2.45) with the curl of equation (2.42) shows that ** u** is independent of

*ψ*. Thus, with a uniformly charged polymer skeleton at equilibrium, the fluid velocity disturbance is the same as for pressure-driven flow past a spherical inclusion in a uniform Brinkman medium (Brinkman 1947). However, the far-field fluid velocity rather than −

*η*

^{−1}

*ℓ*

^{2}

**∇**

*p*. Note that the far-field decay of

*p*must be obtained from equation (2.42) with the knowledge of

*ψ*. Similarly,

*ψ*influences

**through equations (2.41) and (2.44). These couplings arise from terms involving the fixed charge**

*v**ρ*

^{f°}, so the force on the particle and, hence, its displacement depend on

*ϵ*

_{p}through the far-field decay of

*ψ*. Quantitative influences are examined below.

## 3. Electrical response

We examine the general features of the perturbed fields with the representative parameters in table 1. As noted previously, the ionic strength includes counterions of the fixed charge and ions from added salt. Accordingly, the Debye length *κ*^{−1} involves a sum over all mobile ions. The perturbations to the electrostatic potential, pressure, ion concentrations, fluid velocity and the particle and polymer displacement are proportional to the electric-field. This is the only way the electric-field strength enters the problem. Note that only four of the five identified dimensionless parameters are independent, because *D* is related to *κ* and *β* by equation (2.25). Generally, *κ* can be considered a measure of the mobile ion concentration (counterions and added electrolyte), with *β* a measure of the fixed charge concentration, and *D* a measure of the total ion concentration. More detailed, quantitative parametric studies—based on the five independent dimensionless variables—are undertaken below.

Streamlines of the fluid velocity (from right to left), and isocontours of the electrostatic potential (proportional to the free charge density) are shown in figure 1*a*. The polymer displacement and isocontours of the pressure, which, recall, has osmotic and hydrodynamic contributions, are shown in figure 1*b*. Note that the fixed charge on the polymer is positive, and the electric field is directed from left to right. Although the particle has zero charge, it is displaced (right to left) in the direction of the undisturbed electroosmotic flow.

An accumulation and depletion of free charge, respectively, is evident at the front of and behind the particle. Compression of the polymer skeleton at the front increases the electrostatic energy of the (positive) fixed charge, thereby increasing the effective elastic restoring force. Accordingly, the apparent elastic modulus is larger than the intrinsic value for an uncharged skeleton. These observations are consistent with the modelling of articular cartilage by Sun *et al.* (2004), which revealed a higher apparent Young’s modulus in unconfined compression tests than under shear.

The particle displacement in figure 1 is co-linear with the undisturbed electroosmotic flow. Thus, even though the polymer experiences an electrical force (left to right), the polymer displacement in close proximity to the particle reflects the hydrodynamic drag exerted by the fluid on the particle and polymer. Alternatively, the particle can be considered as responding to the electric field as if it bears the same signed charge as the counterions of the fixed charge. Clearly, the hydrodynamic drag of the polymer is expected to play an important role in transferring the electrical charge on the counterions to the particle. Note also that the electroosmotic flow exerts a force on the particle whose magnitude is proportional to the mobile charge density. The pressure isocontours in figure 1*b* are similar to those of the perturbed fixed charge density, but with opposite sign. This reflects the *O*(*ρ*^{f°}*E**a*) hydrodynamic pressure dominating the *O*(*ρ*^{f°}*k*_{B}*T*/*e*) osmotic contribution.

In the following parametric studies, Young’s modulus, Poisson ratio and the Brinkman screening length (hydrodynamic permeability) are implicitly specified as independent of the fixed charge density and ionic strength. We also neglect annealing influences on the fixed charge, i.e. we assume that the fixed charge is quenched and, thus, independent of pH, electrolyte concentration and electrostatic potential.

Note that counterion condensation places a practical upper limit on the effective fixed charge density. According to Manning’s well-known theory, the fixed charge density *ρ*^{f°} is limited to values with less than one elementary charge per Bjerrum length of polymer contour (Manning 1969). For a representative hydrogel comprising 5 per cent polymer with monomer molecular weight 100 g mol^{−1}, and 10 per cent charged monomers (Tong & Liu 1993; Okay & Durmaz 2002), the maximum fixed charge density *ρ*^{f°}∼10^{7} C m^{−3} (equivalent to ≈ 0.062*e* nm^{−3}). This is consistent with values reported for articular cartilage at physiological pH, e.g. 2×10^{7} C m^{−3} from Lai *et al.* (1991). With Young’s modulus kPa and particle diameter ∼1 μm, .

We term the ratio of the particle displacement to the electric-field strength the *electrical response function*
3.1
where is a dimensionless function—given explicitly by the ratio of equations (2.26) and (2.38)—of the five indicated dimensionless parameters. The dimensional prefactor in equation (3.1) is the scaling of *Z*/*E* that prevails for incompressible skeletons (*ν*=0.5).

Independent calculations with *ν*=0.5 show that *Z**_{E}=3 for incompressible skeletons, i.e.
3.2
This formula can be derived by summing the fluid and polymer equations of motion (with **∇**⋅** v**′=0) giving

**0**=−

**∇**

*p*′+∇

^{2}

**and**

*w***∇**⋅

**=0, where −**

*w**p*′

**and**

*I***∇**

**+(**

*w***∇**

**)**

*w*^{T}are the isotropic and deviatoric stresses, respectively (Hill & Ostoja-Starzewski 2008). Here,

**=**

*w**μ*

**+**

*v**η*

**with the polymer shear modulus when**

*u**ν*=0.5. The boundary conditions for the

*E*-problem are

**=**

*w***0**at

*r*=

*a*and as . Therefore, by analogy with the well-known problem of Stokes flow past an impenetrable sphere, the solution yields a force

*F*^{E}=−6π

*a*

*ℓ*

^{2}

*ρ*

^{f°}

**. Balancing this with the elastic restoring force in the**

*E**Z*-problem gives equation (3.2).

Let us now consider practically relevant situations where *ν*<0.5. Following Wang & Hill (2008), we adopt *ν*=0.2 as a representative value for charged and uncharged hydrogels, e.g. *ν*≈0.15 for agarose (Freeman *et al.* 1994) and *ν*≈0.2 for articular cartilage (Jurvelin *et al.* 1997). In addition, we fix the ratio of dielectric constants *ϵ*_{p}/*ϵ*_{s}=0.02, which is representative of a wide variety of inclusions in aqueous electrolytes. Accordingly, figure 2 shows how *Z**_{E} varies with *κ**a* for various *β**a* and several values of *a*/*ℓ*. These plots reveal three physically distinct regions of the parameter space, each of which is examined below.

Firstly, when *κ**a*≪1 and *β*≪*κ*, the scaled displacement *Z**_{E} plateaus to a larger value than in the high *κ**a* limit where electrostatic interactions are screened by the added electrolyte. Thus, the skeleton of very weakly charged polymers (vanishing *β*) is softened in the absence of added salt. For example, this yields a decreasing particle displacement with increasing concentration of added electrolyte (increasing *κ*). This unexpected result may be due to the gradient of fixed charge density that accompanies dilation. This would induce an accompanying electrostatic dipole moment that enhances the local electric-field, which, in turn, enhances electroosmotic flow. As discussed above, the particle displacement is generally attributed to viscous drag on the particle. Thus, in striking contrast to charged inclusions dispersed in uncharged media, electrical polarization increases the particle displacement. This is only possible when the Debye length *κ*^{−1} is much larger than the characteristic length scale for dilation, i.e. when *κ**a*≪1. Because this mechanism depends on the intrinsic elasticity of the gel ( and *ν*), the softening effect vanishes as .

Next, when *κ**a*≪1 and *β*≫*κ*, the scaled displacement *Z**_{E} plateaus to a smaller value than in the high *κ**a* limit. For example, increasing the added salt concentration increases the particle displacement. Here, the polymer skeleton is electrostatically stiffened by the fixed charge, and this stiffening is evidently more influential than the accompanying enhancement of electroosmotic flow. Again, this influence vanishes as , because, under these conditions, the skeleton incompressibility is independent of polymer charge.

Finally, when *κ**a*≫1 and *κ*≫*β*, the scaled displacement *Z**_{E} plateaus to a value that depends only on *a*/*ℓ* when *ν* and *ϵ*_{p}/*ϵ*_{s} are fixed. In this regime, viscous shear stresses on the inclusion scale as τ∼*η**U*/*ℓ* when *a*/*ℓ*≫1, where the characteristic fluid velocity beyond a Brinkman screening length *ℓ* of the particle surface is the velocity of the undisturbed electroosmotic flow, *U*=−*ℓ*^{2}*η*^{−1}*ρ*^{f°}*E*. Thus, the hydrodynamic drag force on the particle *F*∼τ*a*^{2}∼−*ρ*^{f°}*E**ℓ**a*^{2}. Next, balancing this force with the intrinsic elastic restoring force of the hydrogel, which is when the particle is displaced a distance *Z*, gives
3.3
Accordingly, in contrast to intrinsically incompressible skeletons, we find *Z**_{E}∼*a*/*ℓ*. This scaling is highlighted in figure 3*a* where *Z**_{E} is plotted as a function of *a*/*ℓ* for several values of *β**a* with *κ**a*=100. As expected from the preceding analysis for incompressible skeletons, the foregoing scaling vanishes when *ν*=0.5 (figure 4). Similarly to the earlier studies of charged inclusions in uncharged skeletons (Hill & Ostoja-Starzewski 2008; Wang & Hill 2008), the particle displacement is independent of particle size when *ν*=0.5, but otherwise increases in proportion to the particle radius *a*. Note, however, that while the absolute displacement increases with *a*, the displacement remains a small fraction of *a*. This fraction increases with charge density *ρ*^{f°}, hydrodynamic permeability *ℓ* and compliance . As noted above, these parameters are generally not independent (see Sasaki *et al.* 1995; Sasaki 2006, and the references therein), but are coupled according to the polymer architecture and gel synthesis.

The scaled particle displacements *Z**_{E} in figure 2 partially obscure how the dimensional displacement depends explicitly on hydrogel charge. Therefore, figure 3*b* shows a representative plot of the dimensional electrical response *Z*/*E* as a function of the fixed charge density *ρ*^{f°} for several values of Young’s modulus . With kPa, for example, *Z*≈1.08 nm with *E*=100 V cm^{−1} and *ρ*^{f°}=10^{5} C m^{−3}. Obviously, more compliant gels with a higher charge density yield larger particle displacements. Nevertheless, while these displacements are within the range of detection using optical tweezers with back-focal-plane interferometry (Gittes & Schmidt 1998), the particle displacements are generally much smaller than for charged colloidal inclusions (with typical surface charge densities) in uncharged hydrogels with a comparable intrinsic Young’s modulus. Thus, if the response of a charged particle in an uncharged gel (Wang & Hill 2008) were naively superposed with the response of an uncharged particle in a charged gel, the displacement would tend to reflect the particle charge. Clearly, this important problem deserves future attention, as the most general problem of practical significance involves charged particles in charged hydrogels.

Poisson ratios of hydrogels are almost exclusively in the range 0≤*ν*≤0.5. Accordingly, figure 4*a* shows how varies with *ν* in this range for several representative values of *β**a*. Generally, there is a rapid change in as , but this sensitivity vanishes as . is insensitive to *ν* when for all *β**a*; when *β**a* is small, however, there exists a maximum in when *ν*≈0, but this vanishes with increasing *β**a*. In the thermodynamic limit , the strain tensor of a linearly elastic medium is symmetric with vanishing deviatoric terms (Landau & Lifshitz 1986). Thus, any stress is accompanied by dilation in the absence of shear/extension. Similarly to uncharged hydrogels (Wang & Hill 2008), the particle displacement vanishes as . Figure 4*b* shows the results for several values of *β**a* with −1≤*ν*≤−0.8.

Finally, figure 5 shows how the scaled displacement *Z**_{E} depends on the dielectric constants of the particle and electrolyte. Recall, this dependence is absent for particles in uncharged media, i.e. electrophoresis, and charged inclusions in uncharged hydrogels. However, the scaled particle displacement with polyelectrolyte gels is particularly sensitive to the particle dielectric constant when *κ**a* and *β**a* are small. Moreover, the forces and particle displacement increase with particle dielectric constant, approaching finite limits as . Accordingly, for uncharged metallic inclusions (), we have
3.4
and
3.5
with
3.6

## 4. Micro-rheological response (steady susceptibility)

Using mode-coupling theory (MCT), Nägele (2003) identified a breakdown of the widely adopted generalized Stokes–Einstein relation (GSER) (Mason & Weitz 1995; Mason *et al.* 1997) for a charged-colloidal sphere in a dispersion of charged-stabilized colloidal particles. Earlier theoretical studies of the susceptibility have not explicitly considered the influence of charge. Rather, such influences have been lumped into the effective shear and bulk moduli for a linearly elastic continuum. Continuum theories include the GSER, which is exact for incompressible elastic skeletons; Levine and Lubensky’s approximate solution of a two-fluid continuum model (Schnurr *et al.* 1997; Levine & Lubensky 2000, 2001*a*)—recently solved exactly in the course of studying electrical influences (Wang & Hill in press); and the theory of Fu *et al.* (2008) accounting for slip at the particle surface. Explicit neglect of charge is generally justified by compressibility, i.e. a Poisson ratio *ν*<0.5, increasing particle displacements by an amount that is less than the experimental uncertainty. Nevertheless, in experiments where small *changes* in susceptibility can be accurately measured, it will be invaluable to interpret such changes in terms of the accompanying changes in charge density rather than adopting effective properties.

It is customary to write the particle displacement as
4.1
where ** Z**,

**and**

*F**α*are, respectively, the particle displacement, applied force and steady susceptibility. For uncharged, elastic, compressible matrices (Schnurr

*et al.*1997), 4.2 where is the shear modulus, often reported as the zero-frequency storage modulus

*G*′. In the thermodynamically accessible range of Poisson’s ratio (−1≤

*ν*≤0.5), the factor 4(1−

*ν*)/(5−6

*ν*) in equation (4.2) has a maximum value of one when

*ν*=0.5, and decreases monotonically to 8/11 when

*ν*=−1.

Equation (4.2) motivates writing
4.3
where the dimensional prefactor is the scaling that prevails for uncharged, incompressible skeletons (*ν*=0.5) with shear modulus . We will refer to this limit as the GSER. Note also that obtained directly from equation (2.26) recovers equation (4.2) when .

The polymer displacement and isocontours of the electrostatic potential are shown in figure 6*a* with the parameters listed in table 1. Isocontours of the pressure (not shown) are qualitatively the same as those of the electrostatic potential (see equation (2.20)), and the perturbed fixed charge density (not shown) is similar to that in figure 1. The electrostatic potential is clearly perturbed at the front and rear of the particle, and the accompanying increase in electrostatic energy with dilation increases the skeleton’s resistance to deformation, thereby decreasing the particle susceptibility.

Figure 6*b*–*d* shows the effect of various parameters on the response of an uncharged particle in a charged hydrogel. To distinguish the influences of the added salt from the polymer counterions, we consider the ionic strength of the added salt *I*_{s} and *β**a* as independent variables rather than *κ**a* and *β**a*.

In figure 6*b*, the influence of *β**a* (scaled fixed charge density) is shown for several values of *I*_{s}; our results (solid lines) are compared with the GSER and equation (4.2) (Schnurr *et al.* 1997). At high ionic strength, the displacement plateaus to the value expected for uncharged, compressible polymer networks (equation (4.2)). More precisely, when *κ**a*≫*β**a*, the screening of electrostatic interactions by the added electrolyte eliminates electrostatic resistance to deformation. However, at low ionic strength, the displacement plateaus to the value for uncharged, incompressible skeletons (GSER). When *κ**a*≪*β**a*, electrostatic stiffening yields an incompressible skeleton without affecting the shear modulus. Clearly, the transition from the low to the high electrolyte concentration regimes depends on the fixed charge density and intrinsic stiffness of the uncharged polymer skeleton. These findings are consistent with our discussion of the electrical response and with the independent studies of articular cartilage (Sun *et al.* 2004).

Figure 6*c* shows how varies with *I*_{s} for the various values of *β**a*. The response increases with the accompanying change in *κ**a*, and plateaus to the value for uncharged, compressible skeletons. Again, counterions screen the fixed charge, thereby increasing the effective compressibility.

Figure 6*d* shows the effect of Poisson ratio for the various values of *β**a*. With increasing *β**a*, asymptotes to the value for uncharged, incompressible skeletons. On the other hand, with decreasing *β**a*, plateaus to the value for compressible hydrogels. Note that, in contrast to the electrical susceptibility, is practically independent of the particle dielectric constant *ϵ*_{p}. Accordingly, to an excellent approximation,
4.4

From figure 6, the maximum variation in the response with *κ**a* or *β**a* is approximately 15 per cent. Moreover, with changes in *β**a* and *κ**a*, the response is bounded by the limits for uncharged skeletons. The resulting absolute change in displacement is rather small under the forces typically used in micro-rheology, but within the typical limits of detection. For example, the displacement resulting from a 1 pN force, which is representative of active micro-rheology (Ziemann *et al.* 1994; Valentine *et al.* 1996), yields a displacement in the range 0.26–0.30 nm with the parameters adopted in figure 6. Such displacements are consistent with the experiments of Di Cola *et al.* (2007), who established consistency of micro- and macro-rheology for highly charged linear polyelectrolytes. However, with recent technological advances, *F*∼1 nN forces can be achieved (Uhde *et al.*2005*a*,*b*; Kollmannsberger & Fabry 2007). For example, a *F*∼1 nN force produces a *Z*∼50 nm displacement, which is well within the range of digital-imaging optical microscopy.

## 5. Summary

The electric-field-induced response of an uncharged spherical particle embedded in a charged hydrogel was studied theoretically. A three-phase electrokinetic model (solvent, mobile ions and charged polymer) for the quenched polyelectrolyte hydrogel was presented as an extension of Hill & Ostoja-Starzewski’s (2008) model for uncharged skeletons. Linear perturbation and superposition were used to derive the exact analytical solutions for the steady response to a steady electric field and external force.

Noteworthy is that the uncharged particles are displaced by an electric field. This is primarily due to electroosmotic flow, with secondary influences attributed to the polarization of the diffuse double layer and electrostatic stiffening, the latter of which is apparent when the underlying uncharged polymer skeleton is compressible. Accordingly, the electrical response is sensitive to the fixed charge density and ionic strength. Overall, increasing the fixed charge density increases the particle displacement because of the enhanced electroosmotic flow. Moreover, the response generally increases with increasing ionic strength, owing to increasing compressibility from the screening of electrostatic repulsion among fixed charges.

In contrast to the electrophoretic mobility of colloidal particles dispersed in Newtonian electrolytes, the electrical particle displacement depends on the dielectric constant of the inclusion. Increasing the particle dielectric constant increases the electric-field-induced displacement.

Finally, our theory captures the influence of charge on the static susceptibility widely used to interpret active and passive micro-rheology experiments. We quantified the roles of fixed charge and ionic strength, showing that the response is bounded by the compressible (upper) and incompressible (lower) limits for the uncharged polymer skeleton. While the influences of charge are most significant for the electrical response, which involves electroosmotic flow and electrical polarization of diffuse double layers, we demonstrated that charge is unlikely to significantly impact the present interpretations of classical micro-rheology.

## Acknowledgements

R.J.H. gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada and the Canada Research Chairs Program; and A.M. thanks the McGill Faculty of Engineering for the generous financial support through a McGill Engineering Doctoral Award (the Hatch Graduate Fellowships in Engineering), and M. Wang for helpful discussions.

## Footnotes

- Received May 27, 2009.
- Accepted September 11, 2009.

- © 2009 The Royal Society