## Abstract

Electrophoresis of a diffuse soft particle with a charged hydrophobic core is considered under the weak field and low charge density assumptions. The hydrophobic surface of the core is coated with a diffuse polyelectrolyte layer (PEL) in which a gradual transition of the polymer segment distribution from the impenetrable core to the surrounding electrolyte medium is considered. A mathematical model is adopted to analyse the impact of the core hydrophobicity on the diffuse soft particle electrophoresis. The mobility based on the present model for the limiting cases such as bare colloids with hydrophobic core and soft particles with no-slip rigid cores are in good agreement with the existing results. The presence of PEL charges produces the impact of the core hydrophobicity on the soft particle mobility different from the corresponding bare colloid with hydrophobic surface in an electrolyte medium. The impact of the core hydrophobicity is subtle when the hydrodynamic screening length of the PEL is low. Reversal in mobility can be achieved by tuning the core hydrophobicity for an oppositely charged core and PEL.

## 1. Introduction

A wetted surface favours a no-slip condition, whereas partially wetting or non-wetting (hydrophobic) surfaces exhibit a velocity slip [1]. The slippage is considered to be linear, i.e. the tangential velocity at the boundary is proportional to the shear stress. The degree of the boundary slip is represented by the slip length which can be interpreted as the fictitious distance below the surface at which the velocity would be equal to zero if extrapolated linearly. An infinite slip length corresponds to zero shear stress at the boundary and the surface is termed as superhydrophobic surface. A zero slip length, a hydrophilic surface, implies a no-slip velocity condition. The slip creates a reduction in shear stress and enhanced transport results. In recent years, there has been interest to consider transport near hydrophobic and superhydrophobic surfaces in the context of microfluidics and nanofluidics devices, where the surface area is larger than the volume, aiming to reduce the friction loss at the boundary for flows at low Reynolds number.

Studies of electrokinetics involving a slipping surface have drawn interest in recent years with an intention to achieve enhanced transport in microfluidics. The electrophoresis of colloids whose surface exhibit a hydrodynamic slip was studied by Khair & Squires [2]. Their results based on the first-order perturbation under a weak-field condition, as adopted by O’Brien & White [3], show that the velocity slip condition leads to an enhancement in mobility by a factor proportional to the ratio of the effective slip length to the electric double layer (EDL) thickness when the particle zeta-potential is lower than the thermal potential. A slip-induced amplification in mobility has been observed experimentally by Churaev *et al.* [4] and Bouzigues *et al.* [5]. However, the theoretical analysis due to Khair & Squires [2] shows that such enhancement in mobility at high zeta-potential (i.e. much larger than the thermal potential 25 mV) is subtle due to non-uniform surface conduction and concentration polarization. Park [6] has provided an analytic formula to determine the electrophoretic mobility of a particle with a hydrophobic surface in terms of slip length and zeta-potential. The dipole coefficient of a charged dielectric particle with a hydrophobic surface under an alternating electric field was obtained numerically by Zhao [7]. They found that the hydrodynamic slip increases the dipole moment at small and moderate zeta-potentials, whereas at large zeta-potentials, the advantage of the slip is lost where the dipole moment is independent of the slip length.

The soft particle, comprising a hard inner core with an ion and solvent permeable polymer layer of finite thickness around the core, mimics several micro-organisms, humic substances, biocolloids and have drawn wide spread interest (e.g. [8–13]). The polymer layers with ionizable groups that carry charge dissociate in a polar solvent to create charges on polymer chains. Such ionic polymer (or polyelectrolyte) layers have a fixed surface charge density and can be modelled as porous medium. A diffuse representation of the outer soft layer is found to be more appropriate in a practical context [14–16]. Unlike the step-like polyelectrolyte layer (PEL), the diffuse representation of the PEL considers the gradual transition of the polymer layer properties to the corresponding electrolyte medium. Recently, Gopmandal *et al.* [17] have studied the electrophoresis of a diffuse soft particle with a no-slip rigid core which possesses volumetric core charge.

In this paper, we have considered the electrophoresis of a diffuse soft particle with a hydrophobic rigid core coated with an ion and solvent permeable PEL. The velocity slip at a polymer–solid interface has been reported by Brochard & de Gannes [18]. Velocity slip may occur at the interface between two immiscible polymer layers [6]. The bacterial cell surface is hydrophobic with negatively charged groups [19]. The experimental analysis by Rochette *et al.* [20] on natural rubber particles reveals the presence of hydrophobic inner layers in the core–shell structure of those particles. Tribet *et al.* [21] made an experimental study to develop hydrosoluble polymeric agents that can stick to hydrophobic surfaces in the context of stabilization of aqueous dispersion of highly hydrophobic colloidal particles. These suggest that studying the electrokinetics of a soft particle in which a velocity slip occurs at the core–shell interface has several practical relevances. The electrophoresis of a soft particle consisting of a hydrophobic core coated with a charged PEL has not been addressed before. The existing electrokinetics theory on soft particles corresponds to a hydrophilic core with a no-slip surface. Owing to the presence of charged PEL, the electrohydrodynamics in the vicinity of the hydrophobic surface will be different from that of a bare colloid suspended in an electrolyte medium. The fluid convection near a hydrophobic surface at a thin Debye length may become independent of the slip length as the loss of shear stress due to the presence of a slipping plane is compensated by the reduced electric force for the electrically neutral fluid. However, for the soft particle, the charge density in PEL attracts counterions and thus the fluid in the vicinity of the hydrophobic core is electrically non-neutral even at a thin Debye length.

We have extended the existing mathematical model due to Ohshima [8–10] to analyse the electrophoresis of a diffuse soft particle with hydrophobic core under a weak applied electric field and low charge density assumption. A weak-field implies that the strength of the applied field is much lower than the field induced by the Debye layer. The impact of the hydrophobic core on the electrophoretic mobility of a diffuse soft particle is analysed for a wide range of Debye length. The impact of the velocity slip at the core–shell interface reduces with the reduction of PEL hydrodynamic screening length and enhances with the increase of PEL decay length.

The paper is organized as follows. The electrokinetic model and the methodology are outlined in §2. The closed form expression for electrophoretic mobility for a bare colloid with a hydrophobic surface and a step-like soft particle with a hydrophilic rigid core are discussed in §3a,b, respectively. Results and discussions on electrophoresis of a diffuse soft particle with a hydrophobic core is made in §3c followed by a brief summary of all the results in §4.

## 2. Mathematical model

We consider a charged diffuse soft particle moving with electrophoretic velocity *U*, is unknown *a priori*, in a liquid containing a binary symmetric electrolyte under a weak electric field *E*_{0} (figure 1). A spherical polar coordinate system with its origin at the centre of the particle and initial line along the direction of the applied electric filed (*E*_{0}) is considered. The diffuse particle consists of a non-conducting charged rigid core of radius *a* possessing a surface charge density *σ* distributed along the core surface. The ionic diffusion coefficient within the PEL is considered to be constant, and equal to that in the bulk liquid. This is a valid assumption for a PEL with sufficiently high water content [16,22]. The surface of the core is considered to be hydrophobic with finite slip length *Λ*. The core is coated with a concentric layer of polyelectrolyte, modelled as a porous medium saturated with an aqueous electrolyte of nominal thickness *δ* and bears a constant volume charge density *ρ*_{fix}, which is entrapped within the PEL. If dissociated groups of valence *z* and molar concentration *N* are distributed within the diffuse PEL, then we have *ρ*_{fix}=*zFN*, where *F* is the Faraday constant.

The equations describing the steady-state motion of the ionized incompressible fluid in and around the diffuse soft particle are [12,16]
*η*ℓ^{2}**u**. The first and second terms on the left-hand side of the equation (2.1) represent the contribution of viscous forces and pressure, respectively. Equation (2.2) is the equation of continuity for the incompressible fluid. Here, **u** is the velocity vector, *η* is the viscosity of the Newtonian electrolyte, *p* is the pressure, *e* is the elementary charge and *z*_{i} (=±1 for *i*=1,2) are the valence of the ionic species present in the system. The electric potential *Φ* comprises the the potential due to the applied electric field, equilibrium double layer potential along with the the polarization of the particle and the double layer due to the applied electric field.

The rigid core of the soft particle is considered to be hydrophobic on which a Navier-slip boundary condition, i.e. the slip velocity is proportional to the shear strain rate of the fluid, is imposed. The proportionality constant is referred as the slip length (*Λ*). The slip boundary condition at the interface between the core surface and soft layer can be expressed as follows [1,2]:
*u*_{r} and *u*_{θ} are radial and cross-radial velocity components, respectively. It may be noted that for the hydrophilic (non-slip) inner core, *Λ* is considered to be zero and for the superhydrophobic case

The far-field liquid velocity **u** relative to the particle satisfies the following condition

For a diffuse soft particle the position dependent hydrodynamic softness of the PEL, ℓ(*r*), can be expressed as [16]
*α*. Here *α*, the thickness of the PEL extends beyond the nominal thickness *δ* and an estimate for the PEL thickness is *δ*+2.3*α* [16]. The dimensionless quantity *ω* appearing in the equation (2.5) ensures the constant values of total number of polymer segments and hence, total charges enclosed by the diffuse PEL upon variation of diffuse property of the PEL and is given by [16]
*b*=*a*+*δ*, the nominal radius of the soft particle as the length scale with scaled radial position *r**=*r*/*b*. The dimensionless form of the position dependent polymer segment distribution function is
*α*_{1}=(1−*γ*)*α*/*δ*. The asterisk from the radial coordinate is dropped for the sake of simplicity.

The distribution of the mobile ions follows the Boltzmann distribution under a small potential limit. Based on the Gauss Law, the electric potential can be related to the charge density by the Poisson equation. The equilibrium double layer potential *ϕ*(*r*), scaled by *ϕ*_{0}=*k*_{B}*T*/*e*, i.e. the scaled potential *y*(*r*)=*ϕ*(*r*)/*ϕ*_{0}, where *k*_{B} is the Boltzmann constant and *T* is the absolute temperature, is governed by [12,16]
*ρ*_{e} is the scaled charge density, i.e. *n*_{i} (*i*=1,2) is the scaled ionic concentration of the *i*th ionic species, scaled by the bulk number density *n*_{0}. The non-dimensional parameter *γ*=*a*/*b*. It may be noted that *FC*=*en*_{0}, where *C* is the bulk molar concentration of the electrolyte. The inverse of the EDL thickness is given by *ϵ*_{f} is the permittivity within the shell and the electrolyte medium. The scaled charge density of the PEL is given by *Q*_{fix}=*FNb*^{2}/*ϵ*_{f}*ϕ*_{0}.

Suppose that the surface of the inner rigid core of the soft diffuse particle is non-conductive, ion-impenetrable and maintained at a constant charge density *σ*, and the equilibrium double layer potential far away from the particle reaches the bulk value. It leads to the following boundary conditions for the equation (2.7)
*σ*_{s} is the scaled surface charge density of the core surface with *σ*_{s}=*σb*/*ϵ*_{f}*ϕ*_{0}. Modelling the electrokinetics of a soft particle by prescribing the surface charge density of the core instead of fixed *ζ*-potential is physically more appropriate [20]. For a soft particle with step-like coating (for *α*=0), we need to consider the continuity of potential and electric displacement at the interface of the PEL-electrolyte interface as follows:

Under a spherical symmetry consideration, the scaled fluid velocity, scaled by *ϵ*_{f}*E*_{0}*ϕ*_{0}/*η*, can be expressed as [23]
*h*(*r*), as introduced above, can be related to the stream function *ψ*(*r*,*θ*) as
*μ*_{E}, scaled by *ε*_{f}*ϕ*_{0}/*η*, can be obtained as
*h*(*r*) can be obtained as
*β*=ℓ_{0}*b* measures the scaled softness of the PEL. Here the operator *L* is defined as
*G*(*r*) present in right-hand side of the equation (2.12) is given by
*ϕ*_{i} satisfies the following equation
*N*_{A} is the Avogadro number and *i*=1,2) are the limiting conductance of the ionic species.

The boundary condition for *ϕ*_{i}(*i*=1,2) and *h*(*r*) on the inner core surface results from the condition that the ions cannot penetrate the rigid core and the surface of the particle core exhibits slip, as described in (2.3), with scaled slip length *Λ*_{s}=*Λ*/*b*, i.e.
*ϕ*_{i}(*i*=1,2) are governed by the insignificant perturbations of the local electric potential and local ion concentrations caused by the applied field. In addition, for the far field condition of *h*(*r*), we consider a sphere *S* of radius ℜ(≫*b*) that is sufficiently large so that the net force acting on *S* is zero. Thus, the far-field conditions can be expressed as [16,17]

The governing equation (2.12) for *h*(*r*) explicitly involves the double layer potential *y*(*r*) and *ϕ*_{i}(*r*). The equation (2.7) for double layer potential subject to the boundary conditions (2.8) is first solved iteratively through a second-order central difference scheme. With the known *y*(*r*), the set of boundary-value problems (2.12) and (2.13) with boundary conditions (2.14) and (2.15) is solved simultaneously through a numerical method as illustrated in Gopmandal *et al.* [17].

The electric permittivity of the solvent is taken to be the same as that of water. The bulk electrolyte concentration is varied to span the Debye–Huckel parameter from *κb*∼*O*(1) (thick Debye layer) to *κb*≫1 (thin Debye layer). This implies that for a fixed *b* (=15 nm, say), the variation of *κb* from 1.54 to 100 corresponds to the bulk molar concentration to vary between 1 mM and 10^{4} mM. We have considered positively charged PEL ions, i.e. PEL ions have valence *z*=1 and the core is negatively charged with surface charge density *σ* and the soft layer is concentric to the core with a fixed dimension. We restrict our analysis to low molar concentration of PEL ions and low surface charge density. The molar concentration of PEL ion is taken to be *N*∼*O*(50) mM and the surface charge density of the inner rigid core is considered to be ∼*O*(50) mC m^{−2}. This type of surface charge conditions can be considered for the biological particles [24] with reactive functional groups along the core’s surface with a slow rate of reaction [25]. It may be noted that in the absence of the PEL (i.e. *α*=0 and *δ*=0), the model corresponds to that of a bare colloid, i.e. *γ*=1 and the soft particle with step-like PEL corresponds the case when *α*=0.

## 3. Results and discussion

We have compared our solutions for mobility based on the present model with the results available in the literature for the case of hydrophobic bare colloid and step-like soft particle with a hydrophilic core (figures 2 and 3). An analytic expression for mobility of a bare colloid with hydrophobic core and soft particle with hydrophilic core is derived in §3a,b, respectively. In §3c, the effect of Debye length, slip length, electrokinetic parameters of PEL on electrophoresis of the diffuse soft particle and mobility reversal are analysed.

### (a) Electrophoresis of a bare colloid (*γ*=1) with hydrophobic surface

An expression for the electrophoretic mobility of a bare colloid (*γ*=1) with a hydrophobic surface possessing constant surface charge density is derived through the closed form solution of the electric potential under the Debye–Huckel approximation. Recently, Park [6] provided mobility for a hydrophobic colloid with uniform surface potential (*ζ*-potential) under the Debye–Huckel approximation with a balance of force condition. Following the same strategy as adopted by Park [6], in which the velocity slip conditions are incorporated in the solutions as provided by Henry [26], an analytical expression for the scaled mobility, scaled by *ϵ*_{f}*ϕ*_{0}/*η*, for the hydrophobic bare colloid with uniform surface charge density (*σ*_{s}), scaled by *ϵ*_{f}*ϕ*_{0}/*b*, can be expressed as
*E*_{n}(*κb*) is the *n*th order exponential integral, defined by

By using the recurrence relation for exponential integrals
*ζ*-potential condition. This suggest that (3.2) is valid for either low *σ*_{s} or high *κb* so that the *ζ*-potential becomes lower than the thermal potential (*ϕ*_{0}). Note that the first term on the right-hand side of equation (3.2) agrees with Henry’s formula [26]. In figure 2, the analytic solution for mobility (3.2) is compared with the computed mobility based on the present model for a soft particle by setting *a*=*b* (i.e. *γ*=1) and *g*(*r*)=0 for *r*>1 and found them in good agreement. Figure 2 shows that the mobility for the rigid colloid with hydrophobic surface increases monotonically with the increase of the slip length. A similar observation has already been made by several previous authors namely, Khair & Squires [2] and Jolly *et al.* [27].

### (b) Electrophoresis of a soft particle (*α*=0) with hydrophilic core (*Λ*_{s}=0)

We consider the electrophoresis of a soft particle with a step-like coating (*α*=0) possessing a hydrophilic (*Λ*_{s}=0) rigid core bearing constant surface charge density *σ*_{s}. For a soft particle with step like coating, the charge distribution within the PEL becomes homogeneous, i.e. *g*(*r*)=1 inside the PEL and *g*(*r*)=0 within the electrolyte. For this type of soft particles under the Debye–Huckel approximation, the closed form solution of equation (2.7) for the double layer potential subject to the boundary conditions (2.8) and (2.9) can be obtained as [28]
*F*_{1}=((1+*κb*)/*κb*)((1−*κbγ*)/(1+*κbγ*))(*Q*_{fix}/2(*κb*)^{2}) *e*^{κb(γ−1)}+*σ*_{s}*γ*^{2}/(1+*κbγ*), *F*_{2}=−(*Q*_{fix}/2(*κb*)^{2})((1+*κb*)/*κb*) and *F*_{3}=*F*_{1}−((1−*κb*)/*κb*)(*Q*_{fix}/2(*κb*)^{2}) *e*^{κb(1−γ)}.

In order to validate our computed solution, we set the value of *α*/*δ* in the model described before sufficiently small (less than or equal to 10^{−3}) so that the effect of decay length becomes negligible and the diffuse PEL behaves like a step-like coating. In figure 3*a*, we have shown the comparison of the computed double layer potential with the corresponding analytical solution given in (3.3). In addition, we have also compared (figure 3*b*) the computed mobility with the existing results for mobility due to Ohshima [9,10]. The softness parameter *β*, as defined before, measures the ratio between the radius of the particle to the Brinkman screening length of the PEL. It provides a measure of the permeability of the PEL. A higher value of *β* implies a lower permeability of the shell, which produces an enhancement in the hydrodynamic frictional coefficient of the soft layer. For this, figure 3*b* shows that the mobility reduces as *β* increases.

### (c) Diffuse soft particle with hydrophobic core

#### (i) Effect of Debye layer thickness and slip length

We now consider the effect of Debye layer thickness on the mobility of a diffuse soft particle at different slip length of the hydrophobic core with a fixed surface charge density in figure 4*a*,*b*. The results for a no-slip case (*Λ*_{s}=0) and the corresponding results due to Ohshima [10] and Ohshima *et al.* [29] for step-like coating (*α*/*δ*=0) are also included in figure 4*a*. Here the core is considered to be negatively charged while the PEL posses a positive volume charge density. Figure 3*b* shows that the variation of mobility with the Debye length for the hydrophobic core is similar to the case of a soft particle with hydrophilic core. For a lower range of *κb* (i.e. *κb*∼*O*(1)), the negatively charged core have an influence on the particle mobility as the Debye layer induced by the core can influence the ion distribution outside the PEL layer. For a moderate range of the Debye length, the shielding effect created by the core charge density as well as the counterion condensation of the PEL charges grows with the rise of the electrolyte concentration (i.e. increase of *κb*), which causes a gradual reduction in mobility with the increase of the Debye length in the range for which *κb* is *O*(1). Further reduction of the Debye length diminishes the contribution of the core charge density and the mobility gradually approaches a constant value. The hydrophobicity of the core surface attenuates the counterion condensation effect due to the enhanced convective transport of ions. For this, an increase in slip length parameter produces a monotonic increment in the particle mobility for low to moderate range of the Debye length. Our results show that the impact of the slip length parameter is enhanced when a thin Debye length is considered. The variation of *μ*_{E} with *κb* at different slip length of a diffuse soft particle and soft particle with step-like coating exhibits a similar pattern.

The hydrophobicity of the core increases its effective surface potential and the impact of the slip length grows as the Debye layer becomes thinner. By contrast, the effect of the core with a no-slip interface diminishes as the Debye length reduces. For this, a large deviation of the mobility of a soft particle with a hydrophobic core from the corresponding no-slip case occurs when the Debye length becomes thinner. Figure 4*c* shows the occurrence of mobility reversal for a soft particle with hydrophobic core when the Debye length is varied at a fixed value of the core *ζ*-potential. When the PEL is considered to be uncharged with *ζ*=−1 on the core, the dependence of mobility on *Λ*_{s} becomes significantly larger as the Debye layer becomes thinner. In this case, the dependence of *μ*_{E} with *Λ*_{s} at different *κb* resemble to that of a bare colloid.

We now consider the slip length of the negatively charged core surface to vary from 0 (no-slip) to *O*(1+*Λκ*). The results as presented in figure 5*a* show that the mobility for the present diffuse soft particle does not follow the above prediction. The variation of the soft particle mobility with slip length depends on the charge density of the PEL. When the mobility is dominated by the PEL charge density i.e. mobility is positive, it increases with the increase of the slip length. An increment of 50% in mobility compared with a no-slip core (*Λ*_{s}=0) is found for *N*=50 mM. With the increase of slip length, the effect due to fluid convection on ions increase. For a core with non-slip surface the counterion condensation effects become stronger at a thinner Debye length and the effective charge density of the PEL becomes low. However, the stronger fluid convection induced by the core with a slip surface attenuates the counterion condensation effects. For this, we find that the slip length produces a larger rate of increment in particle mobility for a higher range of PEL-fixed charge density. The impact of the slip length for lower range of PEL nominal fixed charge density is relatively low. The hydrophobicity of the core reduces the frictional force as well as increases the effective charge density of the PEL. These two effects counteract to create the dependence of soft particle mobility on the core slip length different from the mobility dependence on slip length for a bare colloid. We find from figure 5 that the mobility approaches a saturation limit, which is equal to the corresponding superhydrophobic core, for a large value of *Λ*_{s}.

We now investigate the effect of the surface charge density of the hydrophobic core at a fixed value of the PEL volume charge density at different values of the Debye length (figure 5*b*). We have also included the case of an uncharged hydrophobic core, i.e. *σ*=0. Previous investigations [30] have shown that the electroosmotic flow at a thin Debye length is unaffected by the presence of an uncharged hydrophobic surface. In case of EOF near an uncharged hydrophobic surface, the low hydrodynamic friction is compensated by the low electric body force as the fluid is electrically neutral at a thin Debye length. For this, the electrokinetics is reported to be unperturbed due to the presence of an uncharged hydrophobic surface for a thin Debye layer. The electrophoresis of the soft particle with an uncharged hydrophobic core is found to be influenced by the velocity slip length even at *κb*=50. We find an increment of about 60% of the mobility for *σ*=0 is considered with *κb*=50(≫1). The presence of mobile counterions (negative ions) within the positively charged PEL creates the fluid to experience an electric force in the vicinity of the uncharged hydrophobic core. The electrophoretic mobility of the soft particle diminishes with the rise of negative charge density of the core, which follows the trend as derived by earlier studies [8–11]. The increase in slip length creates a reduction in positive value of the mobility when the core is highly charged. The slip condition at the core surface appears to enhance the core surface charge density. The particle electrophoretic velocity switches its direction from positive to negative for larger values of the slip length. This mobility reversal shows that at large slip length, the electrophoresis is dominated by the core charge density even at this low Debye length.

In figure 6, we present the mobility for a fixed PEL charge while surface charge density of the hydrophobic core is varied. Results are presented for different values of the slip length parameter at a fixed Debye length. It is evident from these results that the impact of the slip length is enhanced at higher range of the surface charge density of the core when Debye length is lower than the particle size, i.e. *κb*>1. Results show that the mobility curves for different *Λ*_{s} crosses over as the surface charge density is increased at a given value of the Debye length. For lower range of *σ*, for which soft particle electrophoresis is dominated by the PEL charge density (i.e. *μ*_{E}>0), an increase in *Λ*_{s} reduces the counterion condensation effect and the *μ*_{E} increases with the increase of *Λ*_{s}. However, as *σ* is increased, the mobility reduces and the hydrophobicity of the core creates further reduction in *μ*_{E}. This leads to a mobility reversal as the slip length is increased.

#### (ii) Effect of polyelectrolyte layer thickness and decay length

The dependence of mobility on the slip length for different choice of the core size *γ*(=*a*/*b*) is shown in figure 7 when *κb*=1.54, 10, 50. Increase in slip length creates an enhancement in the magnitude of the mobility and it then approaches a saturation at large *Λ*_{s}, i.e. the core can be considered as superhydrophobic. As the core size at a fixed *b* is increased, the impact of the core enhances and, consequently, the influence of the slip length *Λ*_{s} grows at larger values of *γ*. However, the impact of the core reduces as the Debye layer becomes thinner. These results show that at thinner Debye layer, unlike a bare colloid, a large amplification in mobility with the increase of the slip length does not occur. The mobility of a soft particle with thin PEL (*γ*=0.9) is enhanced by 70% for a superhydrophobic core compared with the no-slip core when *κb*=1.54, it is 200% for *κb*=10 and 630% for *κb*=50.

We now analyse the impact of the diffuse character of the PEL on the mobility with a hydrophobic inner core. The results presented in figure 8*a*,*b* are for different values of the decay length *α*/*δ* of the PEL at different choice of the nominal screening length *β*. We have increased the parameter *α*/*δ* by increasing the decay length (*α*) of the diffuse layer with fixed nominal thickness *δ*=*b*−*a*. We found a monotonic increment of the mobility *μ*_{E} with *α*/*δ* for smaller values of *β*(=1), i.e. larger hydrodynamic penetration length, while it decreases with *α*/*δ* for higher values of *β* (i.e. *β*=3), lower hydrodynamic penetration length. With an increment of the decay length, the swelling of the diffuse layer increases and it results in a local increase in frictional force [16,31]. In addition, an increase in *α*/*δ* leads to an increase in overall effective permeability of the diffuse PEL. The electrostatic effects at *r*>*γ*+*δ* become stronger for increasing values of *α*/*δ*, which results in a significant enhancement of the electric field-induced driving force overwhelming the hydrodynamic frictional force for smaller values of *β*. We find that for a highly permeable PEL, the mobility rises with the increase of *α*/*δ*. Mobility also increases with the increase of slip length for the lower value of core charge density in which PEL charge density dominates the electrophoresis. Increase of *Λ*_{s} or *α*/*δ* causes the fluid convection within the PEL to become stronger. On the other hand, for larger *β*, increasing hydrodynamic friction is responsible for the decrease in mobility. From figure 8*a*,*b*, it is clear that the impact of the slip length diminishes as the PEL becomes denser (increase of *β*). The fluid convection in PEL reduces as the permeability of the soft layer decreases and, hence, the impact of the slip length also diminishes.

#### (iii) Parameters range for zero mobility and mobility reversal

The reversal in soft particle mobility for the case of an oppositely charged core and PEL arises when the electrophoresis switches over from the core (or PEL) dominated situation to the PEL (or core) dominated case. Previously, Raafatnia *et al.* [32] and De *et al.* [33] have shown the occurrence of mobility reversal of a soft particle by varying the Debye length. Figure 9*a* provides an estimation of the core and PEL charge density at a different Debye length beyond which a reversal in electrophoretic propulsion direction occurs. Results show that critical charge density of the core for a mobility reversal grows as the Debye length reduces. For a soft particle with hydrophobic core, the impact of the core enhances as the slip length becomes large. For this, the critical core charge density reduces as the slip length is increased and the reduction in critical value of −*σ*_{s} with increment of *Λ*_{s} occurs at a faster rate as the Debye length becomes thinner (larger *κb*). For a superhydrophobic core, a linear variation of the critical −*σ*_{s} with the critical PEL charge density *N* is found for the thinner Debye length (figure 9*b*). Figure 9*c* shows that the critical value of *σ*_{s} rises as *α*/*δ* of the PEL rises, which shows that the impact of the PEL grows as the decay length increases. The impact of the slip length on the critical charge density of the core is enhanced at large values of the PEL decay length. Increase in *α*/*δ* creates a swelling of the PEL along with an increment in its hydrodynamic permeability. For this, the impact of the slip length grows with the increase of *α*/*δ*.

## 4. Conclusion

The impact of the hydrophobic core on the electrophoresis of a diffuse soft particle is analysed. The core and PEL is considered to be oppositely charged. A mathematical model to analyse the electrophoresis under a weak applied field and low charge density assumptions is presented. The reduced nonlinear set of boundary value problems are solved numerically to determine the particle mobility. An analytic expression for the mobility of a bare colloid with hydrophobic core and fixed surface charge density is obtained. Computed results for mobility based on the present model agrees well with this analytic solution as well as the existing results for a soft particle with hydrophilic rigid core. Unlike a bare colloid, the hydrophobic core does not produce an enhanced particle propulsion for the entire range of Debye length. Our analysis shows that the hydrophobicity of the core when core and PEL are oppositely charged enhances the mobility at a thinner Debye length for which the electrophoresis is dominated by the PEL charge density, it reduces the mobility for low to moderate range of Debye length in which the core influences the electrophoresis. This may lead to a zero mobility and, subsequently, a mobility reversal. The impact of the core hydrophobicity depends also on the Brinkman screening length of the PEL and its decay length. The slip length of the core has no significant influence for a dense PEL.

## Authors' contributions

All the authors conceived the idea for the present research. P.P.G. performed the numerical simulations. P.P.G. and S.B. analysed the results. All the authors wrote the manuscript.

## Competing interests

We declare we have no competing interests.

## Funding

There is no external funding source and support to carry out our work.

## Acknowledgements

The authors thank the board member and two anonymous referees for carefully reading the manuscript and offering helpful suggestions.

- Received December 27, 2016.
- Accepted February 24, 2017.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.