Gel Diffusiophoresis of a Spherical Colloidal Particle

Abstract

A theoretical framework is established for the gel diffusiophoresis of a spherical colloidal particle moving through an uncharged dilute porous polymer gel medium when an electrolyte concentration gradient field is applied. The network of cross-linked polymer segments is treated as a porous skeleton containing an electrolyte solution using the Brinkman–Debye–Bueche model. We derive a general expression for the gel-diffusiophoretic mobility of a charged spherical colloidal particle. Based on this general mobility expression, we farther derive a closed-form approximate expression for the gel-diffusiophoretic mobility of a weakly charged spherical particle correct to the second order of the particle’s zeta potential. The obtained mobility expression depends on the Debye–Hückel parameter and the Brinkmann parameter.

1. Introduction

The zeta potential of colloidal particles plays an essential role in the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory concerning the stability of colloidal suspensions [1,2,3,4,5]. The particle’s zeta potential can be estimated via electrokinetic measurements, including electrophoresis and diffusiophoresis. Diffusiophoresis, alongside electrophoresis, is a fundamental electrokinetics describing the motion of charged particles in a liquid medium. While electrophoresis involves the movement of charged particles under an applied electric field, diffusiophoresis refers to the movement of charged particles in response to a concentration gradient of electrolytes. This process is driven by the natural variation in ion concentration, rather than an external electric field, and plays a crucial role in areas such as drug delivery and environmental sciences.

Diffusiophoresis has two components, that is, the chemiphoresis component and the electrophoresis component. By the chemiphoresis component, the particles move in the direction of higher electrolyte concentration. The electrophoresis component is related to an induced electric filed, which is called the diffusion field. While no external electric field is applied to a suspension of colloidal particles under the electrolyte concentration gradient, if the mobilities of electrolyte cations and inions are different, then a macroscopic average electric field can be induced so as to act to accelerate the slower electrolyte ions and to retard the faster ions, nullifying the electric current through the suspension at steady state. Understanding diffusiophoresis is essential as it complements the knowledge of electrophoresis, providing a more comprehensive view of particle electrokinetics in various scientific and practical applications.

There have been a lot of theoretical studies on diffusiophoresis of colloidal particles [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. The theories of diffusiophoresis are quite similar to those of electrophoresis. Indeed, the governing equations for diffusiophoresis can be derived from those for electrophoresis by replacing the applied electric field with the applied electrolyte concentration gradient field and modifying the far-field boundary conditions for the ionic electrochemical potentials.

The electrokinetics of colloidal particles described above typically refer to the motion of particles in a free electrolyte solution, which is known as free-solution electrokinetics. In addition to this, there is electrophoresis in a porous gel medium, which is important both practically and theoretically [22,23,24]. In the case of electrophoresis, this is referred to as gel electrophoresis. When a particle migrates through the pores of a gel matrix, two primary types of interactions come into play: (i) short-range steric interactions due to particle–gel friction and (ii) long-range hydrodynamic interactions. In dense gels, where the particle size exceeds the pore size, the steric effect prevails, allowing the application of reptation theory. Conversely, in dilute gels, where the particle is much smaller than the gel pores, long-range hydrodynamic interactions between the particle and the gel medium become more significant. In common gels like polyacrylamide and agarose, the mesh size typically ranges from 2 to 200 nm. Therefore, under dilute gel conditions, the mesh size is expected to be significantly larger than that of small colloidal particles, which are typically around 10–50 nm in size (e.g., several to tens of times larger), thereby making long-range hydrodynamic interactions more pronounced. The cross-linked polymer network can be modeled as a porous skeleton filled with an electrolyte solution, following the Brinkman–Debye–Bueche framework [25,26]. This model views the polymer segments as resistance centers that impose frictional forces on the fluid within the gel [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46]. Allison et al. [27] particularly examined particle gel electrophoresis in uncharged gels using this model, deriving expressions for electrophoretic mobility. Li and Hill [32] and Li et al. [33] expanded Allison et al.’s theory to encompass electrophoresis in charged gels. In a previous study [38], we formulated a general theory for the gel electrophoresis of a spherical hard particle, presenting approximate expressions for the electrophoretic mobility of a charged spherical colloidal particle in both uncharged and charged polymer gels. Additionally, there are theoretical investigations on the diffusiophoresis of liquid droplets [39] and soft particles [40,41,42,43,44,45,46]. Note that Trabzon et al. [47] discussed the formation of liposomes through the collision of smaller particles based on electro hydrodynamics. This process is similar to gel electrophoresis in that both involve interactions between particles. In the particle formation process, new particles are created as a result of collisions between smaller particles. In gel electrophoresis, particles interact with the polymer matrix within the gel.

In the present paper, we consider a third type of electrokinetics, which is neither free-solution electrophoresis, free-solution diffusiophoresis, nor gel electrophoresis. Namely, we deal with the motion of spherical colloidal particles under an electrolyte concentration gradient in a porous polymer gel medium, which we term gel diffusiophoresis. The theory of the diffusiophoresis of charged spherical particles in a porous gel medium was proposed by Sambamoorthy and Chu [48] and Bhaskar and Bhattacharyya [49]. In their seminal work [48,49], they derived a comprehensive set of governing equations that describe the gel-diffusiophoretic mobility of these particles. Additionally, they provided a detailed methodology for evaluating the particle gel-diffusiophoretic mobility, which involves numerically solving the system of simultaneous equations that they developed. In the present study, we build upon these foundational theories by further advancing their concepts and deriving a more general expression that captures the particle’s diffusiophoretic mobility within a gel medium. Moreover, utilizing this general expression, we present an approximate closed-form formula specifically designed to estimate the gel-diffusiophoretic mobility of weakly charged spherical particles. This refinement aims to provide a more practical tool for analyzing the behavior of such particles in gel diffusiophoresis experiments. They derived a set of governing equations for diffusiophoretic mobility and provided a method to evaluate the particle velocity by numerically solving this set of simultaneous equations. In this paper, we further develop the above theories and derive a general expression for the particle’s diffusiophoretic mobility in a gel medium. Furthermore, based on this general expression, we derive an approximate closed-form expression for the diffusiophoretic mobility of weakly charged spherical particles. In the present paper, we treat the gel diffusiophoresis of a spherical colloidal particle. The gel electrophoresis of colloidal particles in dilute gels is not limited to spherical particles; particles with different shapes can also be subjects of study. However, spherical particles are commonly used due to the ease of analysis and experimentation.

2. Theory

Let us consider a spherical colloidal particle with a radius a and zeta potential ζ moving with a diffusiophoretic velocity U within an uncharged porous polymer gel medium of infinite volume containing a symmetrical electrolyte solution with viscosity η, relative permittivity ε_r and bulk electrolyte concentration (number density) n^∞. The electrolyte is of the symmetrical type with valence z, though it may have different ionic drag coefficients λ₊ and λ₋ for cations and anions, respectively. Let n₊(r) and n₋(r) denote the concentrations (number densities) of the electrolyte’s cations and anions, respectively, at position r, and n^∞ represent their concentrations outside the electrical double layer surrounding them. We now apply a constant gradient of electrolyte concentration ∇n^∞ and introduce a constant vector α proportional to ∇n^∞, viz.,

$$\mathit{\alpha }={\displaystyle \frac{kT}{ze}}\nabla \mathrm{l}\mathrm{n}(n^{\infty })$$

(1)

where k represents the Boltzmann constant, T is the absolute temperature, and e denotes the elementary electric charge.

For the gel medium, we utilize a Brinkman–Debye–Bueche continuum model [25,26], where polymer segments are considered as resistance centers uniformly distributed throughout the gel, exerting frictional forces γu on the liquid flowing through the gel medium. Here, γ denotes the frictional coefficient, and u represents the fluid velocity. The gel medium is treated as a uniform continuum. We adopt a reference frame fixed at the center of the particle. That is, the origin of the coordinate system (r, θ, ϕ) is anchored at the particle’s center, and the polar axis (θ = 0) is aligned parallel to α (αcosθ, −αsinθ, 0), α being the magnitude of α (Figure 1).

For a spherical particle, U aligns with the direction as α. We assume that in the absence of the applied electrolyte concentration gradient filed α, the particle has a uniform zeta potential ζ at r = a, where r = |r|, r being the position vector.

The primary assumptions are as follows. (i) The Reynolds number for the liquid flow within the gel medium is sufficiently small to neglect inertial terms in the Navier–Stokes equation, allowing the liquid to be considered incompressible. (ii) The applied field α is weak enough that U varies linearly with α, where U and α are the magnitudes of U and α, respectively. (iii) The slipping plane, where the liquid velocity relative to the particle becomes zero, is situated on the particle surface (at r = a). (iv) Electrolyte ions cannot penetrate the particle surface.

The fundamental electrokinetic equations for the flow velocity u(r) = (u_r(r), u_θ(r), 0) of the liquid at position r(r, θ, ϕ) and those of the electrolyte cations and anions v_±(r) = (v_±r(r), v_±θ(r), 0) are

$$\eta \nabla \times \nabla \times \mathit{u}\left(\mathit{r}\right)+\nabla p\left(\mathit{r}\right)+{\rho }_{\mathrm{e}\mathrm{l}}\left(\mathit{r}\right)\nabla \psi \left(\mathit{r}\right)+\gamma (\mathit{u}\left(\mathit{r}\right)+\mathit{U})=0$$

(2)

$$\nabla ·\mathit{u}\left(\mathit{r}\right)=0$$

(3)

$${\mathit{v}}_{\pm }(\mathit{r})=\mathit{u}(\mathit{r})-{\displaystyle \frac{1}{{\lambda }_{\pm }}}\nabla {\mu }_{\pm }(\mathit{r})$$

(4)

$$\nabla ·{(n}_{\pm }{\mathit{v}}_{\pm }\left(\mathit{r}\right))=0$$

(5)

$${\rho }_{\mathrm{e}\mathrm{l}}\left(\mathit{r}\right)=ze\left\{n_{+}\left(\mathit{r}\right)-n_{-}\left(\mathit{r}\right)\right\}$$

(6)

$${\mu }_{\pm }\left(\mathit{r}\right)={\mu }_{\mathrm{o}}^{\pm }\pm ze\psi \left(\mathit{r}\right)+kT\mathrm{l}\mathrm{n}{n}_{\pm }\left(\mathit{r}\right)$$

(7)

Here, p(r) denotes the pressure, ρ_el(r) represents the charge density as defined by Equation (6), ψ(r) is the electric potential, μ₊(r) and μ₋(r) are the electrochemical potentials of cations and anions, respectively, as given by Equation (7), and ${\mu }_{\mathrm{o}}^{\pm }$ are constant terms of μ_±(r). Equations (2) and (3) represent the Navier–Stokes equation for a steady incompressible flow at low Reynolds numbers and the continuity equation, respectively. Equation (4) specifies that the ionic flow v_±(r) is driven by the liquid flow u(r) and the gradient of the electrochemical potential μ_±(r). Equation (5) is the continuity equation for ions. Equations (2) and (3) are the Navier–Stokes equation and the equation of continuity for an incompressible flow (condition (i)). The term involving the particle velocity U in Equation (2) reflects the fact that the particle has been chosen as the reference frame for the coordinate system. Equation (4) indicates that the flows v_±(r, t) of cations and anions are driven by the liquid flow u(r, t) and the gradient of their electrochemical potentials μ_±(r, t). Equation (5) serves as the continuity equation for cations and anions.

The following boundary conditions for u(r) and v_i(r) must be satisfied:

$$\mathit{u}\left(\mathit{r}\right)=\mathbf{0} \mathrm{a}\mathrm{t} r=a$$

(8)

$$\mathit{u}\left(\mathit{r}\right)\to -\mathit{U} \mathrm{a}\mathrm{s} r\to \mathrm{\infty }$$

(9)

$${\mathit{v}}_{\pm }\left(\mathit{r}\right)·\widehat{\mathit{n}}=0 \mathrm{a}\mathrm{t} r=a$$

(10)

where $\widehat{\mathit{r}}$ = r/r (r = |r|) and $\widehat{\mathit{n}}$ is the unit normal outward from the particle surface. Equation (8) states that slipping plane (at which u(r) = 0) is located on the particle surface. Equation (10) follows from assumption (iv) that electrolyte ions cannot penetrate the particle surface.

Along with the aforementioned boundary conditions, it is necessary to satisfy the constraint that, in the stationary state the net force acting on the particle or an arbitrary volume enclosing the particle must be zero. Consider a large sphere S with radius r containing the particle at its center. The radius r of S is chosen to be sufficiently large so that the net electric charge within S is zero. Consequently, there is no net electric force acting on S, and we only need to consider the hydrodynamic force, which must be zero. The force-free condition is thus given by

$${\mathit{F}}_{\mathrm{H}}=0$$

(11)

where F_H acting on S is obtained from the asymptotic form of the liquid velocity u(r) far from the particle (beyond the electrical double layer) as follows.

$${\mathit{F}}_{\mathrm{H}}=\underset{r\to \mathrm{\infty }}{\lim }{\int }_0^{\pi }\left[\left(-p+2\eta {\displaystyle \frac{\partial u_r}{\partial r}}\right)\mathrm{c}\mathrm{o}\mathrm{s}\theta -\eta \left({\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial u_r}{\partial \theta }}+{\displaystyle \frac{\partial u_{\theta }}{\partial r}}-{\displaystyle \frac{u_{\theta }}{r}}\right)\mathrm{s}\mathrm{i}\mathrm{n}\theta \right]2\pi r^2\mathrm{s}\mathrm{i}\mathrm{n}\theta d\theta {\displaystyle \frac{\mathit{\alpha }}{\alpha }}$$

(12)

When the concentration gradient field α is weak, the deviations δn_±(r), δψ(r), δμ_±(r) and δρ_el(r) of n_±(r), ψ(r), μ_±(r) and ρ_el(r) from their equilibrium values are small so that we may write

$$n_{\pm }\left(\mathit{r}\right)=n_{\pm }^{\left(0\right)}\left(r\right)+\delta n_{\pm }\left(\mathit{r}\right)$$

(13)

$$\psi \left(\mathit{r}\right)={\psi }^{\left(0\right)}\left(r\right)+\delta \psi \left(\mathit{r}\right)$$

(14)

$${\mu }_{\pm }\left(\mathit{r}\right)={\mu }_{\pm }^{\left(0\right)}+\delta {\mu }_{\pm }\left(\mathit{r}\right)$$

(15)

$${\rho }_{\mathrm{e}\mathrm{l}}\left(\mathit{r}\right)={\rho }_{\mathrm{e}\mathrm{l}}^{\left(0\right)}\left(r\right)+\delta {\rho }_{\mathrm{e}\mathrm{l}}\left(\mathit{r}\right)$$

(16)

where quantities with the superscript (0) denote equilibrium values, i.e., in the absence of α and ${\mu }_{\pm }^{\left(0\right)}$ which are constants independent of r. It is assumed that the equilibrium concentrations $n_{+}^{\left(0\right)}\left(r\right)$ and $n_{-}^{\left(0\right)}\left(r\right)$ of cations and anions, respectively, which depend r only, follow the Boltzmann distribution and that the equilibrium electric potential ψ⁽⁰⁾(r) around the particle satisfies the spherical Poisson–Boltzmann equation, viz.,

$$n_{\pm }^{\left(0\right)}\left(r\right)=n^{\infty }{e}^{\mp y\left(r\right)}$$

(17)

$$∆y\left(r\right)={\kappa }^2\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}y\left(r\right)$$

(18)

with

$$y(r)={\displaystyle \frac{ze{\psi }^{\left(0\right)}\left(r\right)}{kT}}$$

(19)

$$\kappa =\sqrt{{\displaystyle \frac{2z^2{e}^2{n}^{\infty }}{{\epsilon }_{\mathrm{r}}{\epsilon }_{0}kT}}}$$

(20)

where y(r) is the scaled equilibrium electric potential, κ is the Debye–Hückel parameter, and ε₀ is the permittivity of a vacuum. The boundary conditions for n_±⁽⁰⁾(r) and ψ⁽⁰⁾(r) are given by

$$n_{\pm }^{\left(0\right)}\left(r\right)\to n^{\infty } \mathrm{a}\mathrm{s} r\to \mathrm{\infty }$$

(21)

$${\psi }^{\left(0\right)}\left(r\right)\to 0 \mathrm{a}\mathrm{s} r\to \mathrm{\infty }$$

(22)

$${\psi }^{\left(0\right)}\left(a\right)=\zeta$$

(23)

The boundary condition for δn_±(r) far from the particle when α is applied is

$$\delta n_{\pm }\left(\mathit{r}\right)\to \left|\nabla n\right|r\mathrm{c}\mathrm{o}\mathrm{s}\theta ={\displaystyle \frac{ze{n}^{\infty }}{kT}}\alpha r\mathrm{c}\mathrm{o}\mathrm{s}\theta \mathrm{a}\mathrm{s} r\to \mathrm{\infty }$$

(24)

We determine the boundary condition for δψ(r) at large distances from the particle. A macroscopic electric field—i.e., the diffusion potential field—is generated by the ionic flows v_±(r) induced by α. This field counteracts the net electric current, meaning that δψ(r) does not approach zero as r → ∞. The electric current density i(r) at position r is given by

$$\mathit{i}\left(\mathit{r}\right)=ze\left\{n_{+}\left(\mathit{r}\right){\mathit{v}}_{+}\left(\mathit{r}\right)-n_{-}\left(\mathit{r}\right){\mathit{v}}_{-}\left(\mathit{r}\right)\right\}$$

(25)

By substituting Equations (4), (13) and (15) into Equation (25) and neglecting the products of the small quantities u(r), δn_±(r) and δμ_±(r), we obtain

$$\mathit{i}\left(\mathit{r}\right)={\rho }_{\mathrm{e}\mathrm{l}}^{\left(0\right)}\left(r\right)\mathit{u}\left(\mathit{r}\right)-ze\left\{{\displaystyle \frac{n_{+}^{\left(0\right)}\left(r\right)}{{\lambda }_{+}}}\nabla {\mathsf{\delta }\mu }_{+}\left(\mathit{r}\right)-{\displaystyle \frac{n_{-}^{\left(0\right)}\left(r\right)}{{\lambda }_{-}}}\nabla \mathsf{\delta }{\mu }_{-}\left(\mathit{r}\right)\right\}$$

(26)

The right-hand side of Equation (26) must be zero beyond the electrical double layer around the particle. We thus find that

$$\mathsf{\delta }\psi \left(\mathit{r}\right)\to -\beta \alpha r\mathrm{c}\mathrm{o}\mathrm{s}\theta \mathrm{a}\mathrm{s} r\to \mathrm{\infty }$$

(27)

where

$$\beta ={\displaystyle \frac{{1/\lambda }_{+}-{1/\lambda }_{-}}{1/{\lambda }_{+}+1/{\lambda }_{-}}}=-{\displaystyle \frac{{\lambda }_{+}-{\lambda }_{-}}{{\lambda }_{+}+{\lambda }_{-}}}$$

(28)

Equations (24) and (27) can be combined to give the following boundary condition for δμ_±(r) far from the particle:

$${\delta \mu }_{\pm }\left(\mathit{r}\right)=\pm ze\mathsf{\delta }\psi \left(\mathit{r}\right)+kT{\displaystyle \frac{{\delta n}_{\pm }\left(\mathit{r}\right)}{n_{\pm }^{\left(0\right)}\left(r\right)}}\to ze\left(1\mp \beta \right)\alpha r\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\theta } \mathrm{a}\mathrm{s} r\to \mathrm{\infty }$$

(29)

By symmetry, we may write

$${\delta \mu }_{\pm }\left(\mathit{r}\right)=\mp ze\alpha {\varphi }_{\pm }\left(r\right)\mathrm{c}\mathrm{o}\mathrm{s}\theta$$

(30)

and

$$\mathit{u}\left(\mathit{r}\right)=\left(-{\displaystyle \frac{2}{r}}h\left(r\right)\mathsf{\alpha }\mathrm{c}\mathrm{o}\mathrm{s}\theta , {\displaystyle \frac{1}{r}}{\displaystyle \frac{d}{dr}}\left[rh\left(r\right)\right]\mathsf{\alpha }\mathrm{s}\mathrm{i}\mathrm{n}\theta , 0\right)$$

(31)

where ϕ_±(r) and h(r) are functions of r. By substituting Equations (30) and (31) into Equations (2)–(4), we obtain the following equations for ϕ_±(r) and h(r):

$$L{\varphi }_{\pm }\left(r\right)=g_{\pm }\left(r\right)$$

(32)

$$L\left(Lh\left(r\right)\right)=G\left(r\right)$$

(33)

with

$$g_{\pm }\left(r\right)=\pm {\displaystyle \frac{dy}{dr}}\left({\displaystyle \frac{d{\varphi }_{\pm }}{dr}}\mp {\displaystyle \frac{2{\lambda }_{\pm }}{ze}}{\displaystyle \frac{h}{r}}\right)$$

(34)

$$G\left(r\right)=-{\displaystyle \frac{ze{n}^{\infty }}{\eta r}}{\displaystyle \frac{dy}{dr}}\left(e^{-y}{\varphi }_{+}+e^y{\varphi }_{-}\right)$$

(35)

where

$$L={\displaystyle \frac{d}{dr}}{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{d}{dr}}{r}^2={\displaystyle \frac{d^2}{d{r}^2}}+{\displaystyle \frac{2}{r}}{\displaystyle \frac{d}{dr}}+{\displaystyle \frac{2}{r^2}}$$

(36)

is a differential operator. The boundary conditions for ϕ_±(r) and h(r) are

$${\displaystyle \frac{d{\varphi }_{\pm }}{dr}}=0 \mathrm{a}\mathrm{t} r=a$$

(37)

$${\varphi }_{\pm }(r)\to \left(\mp 1+\beta \right)r \mathrm{a}\mathrm{s} r\to \mathrm{\infty }$$

(38)

$$h={\displaystyle \frac{dh}{dr}}=0 \mathrm{a}\mathrm{t} r=a$$

(39)

$$h\left(r\right)\to {\displaystyle \frac{U}{2\alpha }}r+O\left({\displaystyle \frac{1}{r}}\right) \mathrm{a}\mathrm{s} r\to \mathrm{\infty }$$

(40)

where the term O(1/r) in Equation (40) is necessary to ensure the constraint that the net force acting on the particle must be zero in the stationary state (Equation (11)). Equations (32) and (33) are similar to the equations that give the diffusiophoretic mobility [18] and the gel-electrophoretic mobility [38] of colloidal particles. Readers should refer to references [18] and [38] to solve these equations.

By solving Equations (32) and (33) subject to the boundary conditions (37)–(40), we finally obtain

$${\varphi }_{\pm }\left(r\right)=\left(\mp 1+\beta \right)\left(r+{\displaystyle \frac{a^3}{2r^3}}\right)-{\displaystyle \frac{1}{3}}\left(r+{\displaystyle \frac{a^3}{2r^3}}\right){\int }_a^{\infty }{g}_{\pm }\left(r\right)dr+{\displaystyle \frac{1}{3}}{\int }_a^{\infty }\left(r-{\displaystyle \frac{x^3}{r^2}}\right)g_{\pm }\left(x\right)dx$$

(41)

and

$$h\left(r\right)={\displaystyle \frac{r}{3{\lambda }^2}}{\int }_r^{\infty }\left(1-{\displaystyle \frac{x^3}{r^3}}\right)G\left(x\right)dx-{\displaystyle \frac{\left(1+\lambda r\right)a^3}{3{{\lambda }^2\left(1+\lambda a\right)r}^2}}{e}^{-\lambda \left(r-a\right)}{\int }_a^{\infty }\left(r-{\displaystyle \frac{x^3}{a^3}}\right)\mathrm{G}\left(x\right)dx$$

$$-{\displaystyle \frac{1}{3{\lambda }^2{\mathsf{\Omega }}_0}}\left[r-{\displaystyle \frac{2a^3}{3r^2}}\left\{1+{\displaystyle \frac{\left(1+\lambda r\right)}{2\left(1+\lambda a\right)}}{e}^{-\lambda \left(r-a\right)}\right\}\right]{\int }_a^{\infty }\left\{\left(1+\lambda a+{\displaystyle \frac{{\lambda }^2{a}^2}{3}}\right)-\left(1+\lambda x\right)e^{-\lambda \left(x-a\right)}-{\displaystyle \frac{{\lambda }^2{x}^3}{3a}}\right\}G(x)dx$$

$$+{\displaystyle \frac{1}{{\lambda }^3}}{\int }_r^{\infty }\left\{\left({\displaystyle \frac{x}{r}}-{\displaystyle \frac{1}{{\lambda }^2{r}^2}}\right)\sinh \left[\lambda \left(x-r\right)\right]+\left({\displaystyle \frac{x}{\lambda r^2}}-{\displaystyle \frac{1}{\lambda r}}\right)\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left[\lambda \left(x-r\right)\right]\right\}G(x)dx$$

$$-{\displaystyle \frac{\left(1+\lambda r\right)a^2}{{{\lambda }^3\left(1+\lambda a\right)r}^2}}{e}^{-\lambda \left(r-a\right)}{\int }_a^{\infty }\left\{\left({\displaystyle \frac{x}{a}}-{\displaystyle \frac{1}{{\lambda }^2{a}^2}}\right)\sinh \left[\lambda \left(x-a\right)\right]+\left({\displaystyle \frac{x}{\lambda a^2}}-{\displaystyle \frac{1}{\lambda a}}\right)\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left[\lambda \left(x-a\right)\right]\right\}G\left(x\right)dx$$

(42)

where

$$\lambda =\sqrt{{\displaystyle \frac{\nu }{\eta }}}$$

(43)

is the Brinkmann parameter and 1/λ is the Brinkmann screening length and

$${\mathsf{\Omega }}_0=1+\lambda a+{\displaystyle \frac{{\lambda }^2{a}^2}{9}}$$

(44)

It should be noted that the Stokes drag 6πηaU acting on a sphere of radius a moving with velocity U in a liquid of viscosity η becomes 6πηaUΩ₀ when the sphere moves through a polymer gel medium [50].

It follows from Equation (40) that the magnitude U of the diffusiophoretic velocity U is given by

$$U=2\alpha \underset{r\to \infty }{\lim }{\displaystyle \frac{h\left(r\right)}{r}}$$

(45)

From Equations (42) and (45), we find that U is given by

$$\mathit{U}={\displaystyle \frac{2}{3{\lambda }^2{\mathsf{\Omega }}_0}}{\int }_a^{\infty }\left\{-\left(1+\lambda a+{\displaystyle \frac{{\lambda }^2{a}^2}{3}}\right)+\left(1+\lambda r\right)e^{-\lambda \left(r-a\right)}+{\displaystyle \frac{{\lambda }^2{r}^3}{3a}}\right\}G(r)dr\mathit{\alpha }$$

(46)

It is noteworthy that one can derive Equation (46) for the gel-diffusiophoretic velocity U from the expression for the gel-electrophoretic velocity U_E of a spherical particle with radius a and zeta potential ζ moving in an applied electric field E by replacing E with α. This is because the fundamental electrokinetic equations governing these two velocities are identical. The only distinction lies in the far-field boundary condition for ϕ_±(r) for r → ∞: for the gel-diffusiophoresis problem, it is specified by Equation (38), whereas for the gel electrophoresis problem, the boundary condition is ϕ_±(r) → r as r → ∞.

3. Results and Discussion

The general expression for the gel-diffusiophoretic velocity of a spherical particle of radius a in an electrolyte concentration gradient field α (Equation (1)) is given by Equation (46). Let us now derive an approximate expression for U for the case where the zeta potential is low, in which case, Equation (18) for the scaled equilibrium electric potential ψ(r) gives

$$y\left(r\right)=\tilde{\zeta }{\displaystyle \frac{a}{r}}{e}^{-\kappa (r-a)}$$

(47)

where

$$\tilde{\zeta }={\displaystyle \frac{ze}{kT}}\zeta$$

(48)

is the scaled zeta potential, and Equations (34) and (41) tend to

$$g_{\pm }\left(r\right)=\pm {\displaystyle \frac{dy}{dr}}{\displaystyle \frac{d{\varphi }_{\pm }}{dr}}=\left(-1\pm \beta \right){\displaystyle \frac{dy}{dr}}\left(1-{\displaystyle \frac{a^3}{r^3}}\right)$$

(49)

and

$${\varphi }_{\pm }\left(r\right)=\left(\mp 1+\beta \right)\left(r+{\displaystyle \frac{a^3}{2r^2}}\right)-{\displaystyle \frac{1}{3}}\left(-1\pm \beta \right)\left(r+{\displaystyle \frac{a^3}{2r^2}}\right){\int }_a^{\infty }{\displaystyle \frac{dy}{dr}}\left(1-{\displaystyle \frac{a^3}{r^3}}\right)dr+{\displaystyle \frac{1}{3}}\left(-1\pm \beta \right){\int }_a^r\left(r-{\displaystyle \frac{x^3}{r^2}}\right){\displaystyle \frac{dy}{dx}}\left(1-{\displaystyle \frac{a^3}{x^3}}\right)dx$$

(50)

from which we obtain

$$G\left(r\right)=-{\displaystyle \frac{2ze{n}^{\infty }}{\eta }}{\displaystyle \frac{dy}{dr}}\left(1+{\displaystyle \frac{a^3}{2r^3}}\right)\left\{\beta +y+{\displaystyle \frac{1}{3}}{\int }_a^{\infty }{\displaystyle \frac{dy}{dr}}\left(1-{\displaystyle \frac{a^3}{r^3}}\right)dr\right\}+{\displaystyle \frac{2ze{n}^{\infty }}{3\eta }}{\displaystyle \frac{dy}{dr}}{\int }_a^r{\displaystyle \frac{dy}{dx}}\left(1-{\displaystyle \frac{x^3}{r^3}}\right)\left(1-{\displaystyle \frac{a^3}{x^3}}\right)dx$$

(51)

Equation (46) as combined with Equations (47) and (51) for y(r) and G(r) gives an approximate expression for U correct to the order of ζ². We now introduce the scaled diffusiophoretic mobility U* defined by

$$\mathit{U}={\displaystyle \frac{2{\epsilon }_{\mathrm{r}}{\epsilon }_{0}kT}{3\eta ze}}{U}^{*}\mathit{\alpha }={\displaystyle \frac{4ze{n}^{\infty }}{3\eta {\kappa }^2}}{U}^{*}\mathit{\alpha }$$

(52)

We find that U* is given by

$$U^{*}=\beta \tilde{\zeta }{\displaystyle \frac{F_1\left(\kappa a,\lambda a\right)}{{\mathsf{\Omega }}_0}}+{\tilde{\zeta }}^2{\displaystyle \frac{F_2\left(\kappa a,\lambda a\right)}{{\mathsf{\Omega }}_0}}$$

(53)

with

$$F_1\left(\kappa a,\lambda a\right)=1+{\displaystyle \frac{\kappa \lambda a}{\kappa +\lambda }}+{\displaystyle \frac{3{\kappa }^2}{2{\lambda }^2}}{\mathsf{\Omega }}_1{f}_5\left(\kappa a\right)-{\displaystyle \frac{3{\kappa }^2}{2{\lambda }^2}}\left\{f_5\left(\left(\kappa +\lambda \right)a\right)+\lambda a{f}_4\left(\left(\kappa +\lambda \right)a\right)+{\displaystyle \frac{{\lambda }^2{a}^2}{3}}{f}_3\left(\left(\kappa +\lambda \right)a\right)\right\}$$

(54)

and

$$F_2\left(\kappa a,\lambda a\right)={\displaystyle \frac{\kappa a(1+\kappa a)}{4}}{f}_4\left(\kappa a\right)-\left\{F_1\left(\kappa a,\lambda a\right)+{\displaystyle \frac{3(1+\kappa a)(1+\lambda a)}{{\lambda }^2{a}^2}}\right\}{f}_5\left(\kappa a\right)+{\displaystyle \frac{3{\mathsf{\Omega }}_1}{8{\lambda }^2{a}^2}}\left\{\kappa a+3-7f_8\left(2\kappa a\right)\right\}$$

$$+{\displaystyle \frac{3(1+\kappa a)}{{\lambda }^2{a}^2}}\left\{f_5\left(\left(\kappa +\lambda \right)a\right)+\lambda a{f}_4\left(\left(\kappa +\lambda \right)a\right)+{\displaystyle \frac{{\lambda }^2{a}^2}{3}}{f}_3\left(\left(\kappa +\lambda \right)a\right)\right\}-{\displaystyle \frac{{\kappa }^2{a}^2}{2}}{f}_1\left(\left(2\kappa +\lambda \right)a\right)$$

$$-{\displaystyle \frac{\kappa \left(\kappa +\lambda \right)a}{\lambda }}{f}_2\left(\left(2\kappa +\lambda \right)a\right)-\left({\displaystyle \frac{{\kappa }^2+{\lambda }^2+3\kappa \lambda }{{\lambda }^2}}\right)f_3\left(\left(2\kappa +\lambda \right)a\right)-\left\{{\displaystyle \frac{{\kappa }^2{a}^2}{4}}+{\displaystyle \frac{3\left(\kappa +\lambda \right)}{{\lambda }^{2}a}}\right\}{f}_4\left(\left(2\kappa +\lambda \right)a\right)$$

$$-\left\{{\displaystyle \frac{3}{{\lambda }^2{a}^2}}-{\displaystyle \frac{{\kappa }^{3}a}{4\left(\kappa +\lambda \right)\lambda }}+{\displaystyle \frac{3{\kappa }^{2}a}{4\left(\kappa +\lambda \right)}}\right\}{f}_5\left(\left(2\kappa +\lambda \right)a\right)+{\displaystyle \frac{{\kappa }^2}{4{\lambda }^2}}{f}_6\left(\left(2\kappa +\lambda \right)a\right)$$

(55)

with

$${\mathsf{\Omega }}_1=1+\lambda a+{\displaystyle \frac{{\lambda }^2{a}^2}{3}}$$

(56)

$$f_n\left(z\right)=e^z{E}_n\left(z\right)=e^z{z}^{n-1}{\int }_z^{\infty }{\displaystyle \frac{e^{-t}}{t^n}}dt$$

(57)

where E_n(z) is the exponential integral of order n. The first and second terms on the right-hand side of Equation (53) are, respectively, the electrophoresis and chemiphoresis components of the diffusiophoretic mobility U*.

Let us consider some limiting cases. In the limit of λa → 0, Equation (53) reduces to

$$U^{*}={\displaystyle \frac{3}{2}}\beta \tilde{\zeta }\left\{1+2f_5\left(\kappa a\right)-5f_7\left(\kappa a\right)\right\}+{\displaystyle \frac{3}{16}}{\tilde{\zeta }}^2\left[1-{\displaystyle \frac{8}{3}}{f}_3\left(\kappa a\right)\right.+8f_4\left(\kappa a\right)+{\displaystyle \frac{8}{3}}{f}_5\left(\kappa a\right)$$

$$-8\left\{1+2f_5\left(\kappa a\right)-5f_7\left(\kappa a\right)\right\}{f}_5\left(\kappa a\right)-{\displaystyle \frac{40}{3}}{f}_6\left(\kappa a\right)+\left.{\displaystyle \frac{10}{3}}{f}_6\left(2\kappa a\right)+{\displaystyle \frac{7}{3}}{f}_8\left(2\kappa a\right)\right]$$

(58)

which agrees with the diffusiophoretic mobility of a sphere of radius a carrying zeta potential ζ in a free electrolyte solution [11].

Next consider the limiting case of κa → ∞ (Smoluchowski limit). In this case, Equation (53) tends to

$$U^{*}={\displaystyle \frac{3(1+\lambda a)}{{2\mathsf{\Omega }}_0}}\left(\beta \tilde{\zeta }+{\displaystyle \frac{1}{8}}{\tilde{\zeta }}^2\right)$$

(59)

In the opposite limit κa → 0 (Hückel limit), Equation (53) becomes

$$U^{*}={\displaystyle \frac{\beta \tilde{\zeta }}{{\mathsf{\Omega }}_0}}$$

(60)

By using Equation (53), one can easily calculate the gel-diffusiophoretic mobility U* of a weakly charged spherical colloidal particle of radius a carrying zeta potential ζ as a function of λa and κa. Figure 2 shows some examples of the calculation of U* of a spherical colloidal particle of radius a as a function of the scaled zeta potential $\tilde{\zeta }$ at 25 °C calculated for several values of λa at κa = 10 for KCl (m₊ = 0.176, m₋ = 0.169, β = −0.02) (a) and NaCl (m₊ = 0.258, m₋ = 0.169, β = −0.2) (b) and those at κa = 50 for KCl (c) and NaCl (d). These figures illustrate how the gel-diffusiophoretic mobility U* depends on ζ, λa and κa.

In Figure 2, U* > 0 corresponds to the situation where the particle moves in the direction of higher electrolyte concentration, while U* < 0 corresponds to the situation where the particle moves in the direction of lower electrolyte concentration. Figure 2 demonstrate that in the case of KCl with a small β (β = −0.02), the mobilities of electrolyte cations and anions are similar, making the chemiphoresis component dominant. Therefore, the particle moves in the direction of higher electrolyte concentration (U^∗ > 0). In the case of NaCl with a large β (β = −0.2), on the other hand, the mobility of Cl⁻ ions is much higher than that of Na⁺ ions, causing the diffusion field to accelerate the slower ions (Na⁺) and retard the faster ions (Cl⁻). As a result, the particle moves in the direction of lower electrolyte concentration (U^∗ < 0) when ζ > 0 in the range shown in Figure 2. Additionally, Figure 2 illustrates that the gel-diffusiophoretic mobility U* for KCl as a function of the zeta potential ζ is almost symmetrical with respect to the ordinate. This is because in this case, U* is practically equal to its chemiphoresis component (which is proportional to ζ²). On the other hand, in the case of NaCl, where the electrophoresis component (which is proportional to ζ) becomes appreciable, the U*-zeta potential ζ curve is no longer symmetrical with respect to the ordinate. Figure 2 shows how U* decreases in magnitude with increasing λa, tending to zero as λa → ∞. In accordance with this, the dependence of U* upon the zeta potential ζ decreases with increasing λa. This is because since the dependency of the gel-diffusiophoretic mobility U* is on the square of the zeta potential ζ (see Equation (53)), the change in sensitivity of U* upon ζ; that is, ∂U*/∂ζ is linear with respect to ζ. As a result, the sensitivity to ζ decreases. In the opposite limit of λa → 0, U* tends to the result for the scaled diffusiophoretic mobility in a free electrolyte solution.

4. Conclusions

We have developed a theory of gel diffusiophoresis of a spherical colloidal particle moving in an uncharged dilute porous polymer gel medium under an application of an electrolyte concentration gradient field α. We have derived a general expression (Equation (46)) for the gel-diffusiophoretic mobility of a charged spherical particle. Based on this general mobility expression, we farther derived a closed-form approximate expression (Equation (53)) for the scaled gel-diffusiophoretic mobility U* of a weakly charged spherical particle correct to the second order of the particle’s zeta potential ζ, which is applicable for arbitrary values of κa and λa. With this closed-form approximate expression for U*, one can easily calculate U* as a function of ζ, κ and λ.