Diffusiophoresis of a Charged Dielectric Fluid Droplet in a Cylindrical Pore in the Presence of Diffusion Potential

Abstract

We conducted a theoretical analysis on the diffusiophoretic motion of a dielectric droplet in a cylindrical pore in the presence of an induced diffusion potential, such as that in a NaCl electrolyte solution. The fundamental electrokinetic governing equations are solved using a patched pseudo-spectral method based on Chebyshev polynomials, coupled with a geometric mapping scheme to handle the irregular solution domain. The impact of the boundary confinement effect on droplet mobility is examined in detail. Interesting electrokinetic phenomena are found in this work, such as mobility reversal in narrow cylindrical pores with the droplet moving against the direction expected based on the classical Coulomb electrostatic law due to the strong boundary confinement effect. Moreover, “solidification phenomenon” is also found at some specific pore radius where the droplets move as rigid particles with no interior recirculating vortex flows regardless of the droplet viscosities. Corresponding critical points of ${\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$, the ratio of droplet radius to the cylindrical radius are found where the spinning orientation on the droplet surface changes each time as it passes them. The profound boundary confinement effect, both electrostatically and hydrodynamically, is responsible for these peculiar phenomena. The results presented here have direct applications in microfluidic and nanofluidic operations as well as drug delivery applications.

1. Introduction

Droplets have many merits which make them very attractive in practical applications [1,2,3,4,5,6,7,8]. For instance, its tiny size provides excellent mass and thermal transport properties within it, which are further enhanced by the interior recirculating vortex flow typically generated when it is in motion [2,3,9,10,11,12]. As a result, it finds enormous applications in drug delivery with the therapeutics dissolved in it to form the drug-carrying liposome [13,14,15,16,17,18,19]. The droplet liposome migrates along with the blood and other biological fluids circulating in the human body and eventually reaches the region needing therapy, which often releases some specific chemicals in its neighborhood, for instance, Ca²⁺, OH⁻, and H₂PO₄⁻, which are often released by a fractured bone tissue [20]. As a result, a concentration gradient is established in the vicinity of this region needing therapy. In this area, it is highly desirable to maximize the concentration of the drug-carrying liposomes to optimize therapeutic performance. And diffusiophoresis, the motion of a colloidal entity in response to a solute concentration gradient in the solution, cuts in and provides the vital extra driving force needed to reach this goal. Moreover, it serves as a self-guiding mechanism to guide the liposome toward its destiny. In particular, diffusiophoresis has no or negligible Joule heating effect as no external electric field is applied. This makes it extremely suitable for biomedical applications such as drug delivery, as a temperature rise over 40 °C is fatal in mammalian cells.

Corresponding studies on the droplet diffusiophoresis have been carried out by Lee and his coworkers [21,22,23]. In particular, the diffusiophoresis of a dielectric droplet in a cylindrical pore was reported recently, focusing on the boundary confinement effect due to the presence of a nearby wall [11]. Interesting and unique features were discovered there, such as mobility reversal, which is observed in very narrow channels. This certainly has a profound impact on drug delivery applications as well as conventional microfluidic operations [24,25], for instance, as it implies that undesirable migration direction might happen, and one should find ways to prevent it from happening or minimize the negative impact at least, if possible. The findings there are applicable in fundamental electrokinetics involving a fluid droplet in a cylindrical pore in general, which is a classic geometric configuration in itself. The narrower the cylindrical pore is, the more profound the boundary confinement effect is in general, both electrostatically and hydrodynamically. The net impact is often motion-deterrent in terms of droplet mobility. It should be noted that drug delivery is chosen only as a convenient demonstrative example of its potential in various possible practical applications. The cylindrical pore certainly is an excellent geometric model of a human blood vessel. The diameter of a micro-blood vessel typically ranges from 1000 to 2500 nanometers. Compared with the typical size of a liposome, which is in the range of 50 to 500 nanometers, the migration of a liposome droplet in a very narrow channel is very likely to happen during drug delivery in the human body. However, there are some limitations, of course. For instance, the droplet is assumed to maintain a perfect spherical shape throughout the entire process, without considering real behaviors such as squeezing deformation in narrow pores, uneven interfacial tension, and droplet breakup. This assumption is necessary to make the rigorous theoretical analysis possible. This study focuses on the fundamental electrokinetic response to a droplet conducting diffusiophoretic motion in a cylindrical pore. Moreover, a cylindrical pore is often adopted to mimic a porous medium, with the pore radius representing the porosity of the medium [26]. For instance, in confined interstitial spaces in biological tissues, the extracellular matrix (ECM) forms a complex porous environment for molecular and colloidal transport.

The previous paper treated the diffusiophoretic motion of a droplet suspended in KCl solution, where the nearly equal diffusivities of K⁺ and Cl⁻ make the diffusion potential negligible [11]. In general, if the cations and the anions in the suspending electrolyte solution are distinctive, there will be an inner potential established within the solution to speed up the slowly moving ions and slow down the faster ions simultaneously so that there will be no net electric current generated across any cross-sectional plane perpendicular to the migration of ions due to the macro concentration gradient of total ions [27,28]. The resulting electric potential is referred to as the diffusion potential in diffusiophoresis [27,28]. Basically, the presence of the diffusion potential is just like an “inner battery” as compared with the external battery in conventional electrophoresis. In other words, the charged droplet or a colloidal entity in general will respond to this diffusion potential based on the fundamental Coulomb electrostatic law [29]. Note that in response to this extra diffusion potential, a corresponding extra driving force upon the droplet motion results, which is referred to as the “electrophoresis component,” whereas the one investigated in the previous paper in the absence of the diffusion potential is referred to as the “chemiphoresis component” [30,31]. They are coupled in general if both are present. In the human blood and other biological fluids in the human body, there are many different electrolyte ions, with Na⁺ and ${\mathrm{C}\mathrm{l}}^{-}$ as the two major species [32,33,34]. Hence, there will be a diffusion potential established simultaneously if somehow a concentration gradient of these ions is established and ion migration takes place, accordingly, due to the diffusion mechanism [30]. Hence, we propose here to investigate the diffusiophoretic motion of a dielectric droplet within a cylindrical pore in the presence of the diffusion potential, such as in the NaCl solution. Moreover, we will explore the highly charged droplet situation as well in this study to provide a complete and comprehensive understanding of the diffusiophoretic motion of a dielectric droplet, both weakly and highly charged, in a cylindrical pore with or without the presence of the diffusion potential. In particular, the electrokinetic flow field will be examined in detail to explore its impact on the droplet motion.

2. Theory

As shown in Figure 1, we consider the diffusiophoretic motion of a Newtonian dielectric fluid droplet in response to an ion concentration gradient in the bulk solution along the axis of a cylindrical pore. In particular, we focus on the NaCl solution to examine the impact of the induced diffusion potential. The geometric system and corresponding basic assumptions are essentially the same as the ones adopted in the previous paper, where a KCl solution is investigated [11]. Briefly speaking, the droplet is assumed to be composed of a dielectric fluid, and it carries uniformly distributed surface charges, with the interior fluid containing no electrolyte ions. In other words, the surface charge density is assumed to be uniform and remains constant with a varying electrostatic environment. The droplet is assumed to preserve its spherical shape throughout the motion, justified by the trivial hydrodynamic Weber number of the order of 10⁻⁷ [22]. As indicated in the Introduction, there is no consideration of other possible deviations from the above assumption, such as squeezing deformation in narrow pores, uneven interfacial tension, and droplet breakup.

For the fluid domain inside the droplet, a spherical coordinate system (r, θ, φ) is adopted with the origin located at the center of the moving droplet. On the other hand, cylindrical coordinates (R, Θ, Z) are adopted for the domain between the droplet surface and the cylindrical pore. The droplet moves along the axis of the cylindrical pore with a constant velocity U.

2.1. Governing Electrokinetic Equations

The governing electrokinetic equations of the system considered here consist of equations for the electric potential, the ion distribution, and the momentum of the fluid, which are essentially the same as those adopted in the previous paper [35]:

$${\mathbf{\nabla }}^2\mathsf{\varphi }=-{\displaystyle \frac{\mathsf{\rho }}{{\mathsf{\epsilon }}_{\mathrm{m}}}},$$

(1)

$${\mathbf{f}}_{\mathbf{j}}=-{\mathrm{D}}_{\mathrm{j}}\left(\nabla {\mathrm{n}}_{\mathrm{j}}+{\displaystyle \frac{{\mathrm{z}}_{\mathrm{j}}\mathrm{e}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}}{\mathrm{n}}_{\mathrm{j}}\nabla \mathsf{\varphi }\right)+{\mathrm{n}}_{\mathrm{j}}\mathbf{v},$$

(2)

$$\nabla \cdot {\mathbf{f}}_{\mathbf{j}}=0,$$

(3)

$${\mathsf{\eta }}_{\mathrm{m}}{\nabla }^2\mathbf{v}-\nabla \mathrm{P}-\mathsf{\rho }\nabla \mathsf{\varphi }=0,$$

(4)

$${{\mathsf{\eta }}_{\mathrm{D}}\nabla }^2{\mathbf{v}}_{\mathrm{I}}-\nabla {\mathrm{P}}_{\mathrm{I}}=0,$$

(5)

$$\nabla \cdot \mathbf{v}=0,$$

(6)

$$\nabla \cdot {\mathbf{v}}_{\mathrm{I}}=0,$$

(7)

where Equation (1) is the Poisson equation based on Gauss’s divergence theorem, ϕ is the electric potential and $\mathsf{\rho }={\displaystyle {\sum }_{\mathrm{j}=1}^{\mathrm{N}}}{\mathrm{z}}_{\mathrm{j}}\mathrm{e}{\mathrm{n}}_{\mathrm{j}}$ is the space charge density, the total number of electric charges per unit volume of electrolyte solution. N is the number of the ion species in the fluid. Moreover, n_j refers to the number concentration of ion species j, z_j its valence number, and D_j its diffusivity coefficient. In addition, ε_m is the electric permittivity of the ambient electrolyte solution. The definitions of the rest of the symbols can be found in the List of Symbols in the Supplementary Material, which is essentially the same as the ones used in the previous paper [11], except for the inclusion of symbols relating to the diffusion potential.

Corresponding mathematical treatment can be found elsewhere for details [36]. After the standard non-dimensionalization procedure is applied, we end up with the complete governing equations in dimensionless form as follows:

$${\nabla }^{\mathrm{*}2}{\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}+{\displaystyle \frac{{(\mathsf{\kappa }\mathrm{a})}^2}{1+\mathsf{\alpha }}}[\exp \left(-{\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}\right)-\exp \left({\mathsf{\alpha }\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}\right)]=0,1\le {\mathrm{r}}^{\mathrm{*}}<{\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$$

(8)

$${\nabla }^{\mathrm{*}2}{\mathsf{\varphi }}_{{\mathrm{e}}_{\mathrm{I}}}^{\mathrm{*}}=0,1\le {\mathrm{r}}^{\mathrm{*}}<{\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$$

(9)

$$\begin{gathered} {{\nabla }^{\mathrm{*}}}^2\mathsf{\delta }{\mathsf{\varphi }}^{\mathrm{*}}-{\displaystyle \frac{(\mathsf{\kappa }\mathrm{a}{)}^2}{1+\mathsf{\alpha }}}\left[\exp (-{\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}})+\mathsf{\alpha exp}(\mathsf{\alpha }{\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}})\right]\mathsf{\delta }{\mathsf{\varphi }}^{\mathrm{*}} \\ ={\displaystyle \frac{(\mathsf{\kappa }\mathrm{a}{)}^2}{1+\mathsf{\alpha }}}\left[\exp (-{\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}){\mathrm{g}}_1^{\mathrm{*}}+\mathsf{\alpha }\exp (\mathsf{\alpha }{\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}){\mathrm{g}}_2^{\mathrm{*}}\right],1\le {\mathrm{r}}^{\mathrm{*}}<{\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}} \end{gathered}$$

(10)

$${\nabla }^{\mathrm{*}2}\mathsf{\delta }{\mathsf{\varphi }}_{\mathrm{I}}^{\mathrm{*}}=0,0\le {\mathrm{r}}^{\mathrm{*}}<1$$

(11)

$$\begin{gathered} {{\nabla }^{\mathrm{*}}}^2{\mathrm{g}}_1^{\mathrm{*}}-({\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}}}{\displaystyle \frac{\partial {\mathrm{g}}_1^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}}}+{\displaystyle \frac{1}{{{\mathrm{r}}^{\mathrm{*}}}^2}}{\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial \mathsf{\theta }}}{\displaystyle \frac{\partial {\mathrm{g}}_1^{\mathrm{*}}}{\partial \mathsf{\theta }}}) \\ ={\displaystyle \frac{{\mathrm{P}\mathrm{e}}_1}{{{\mathrm{r}}^{\mathrm{*}}}^2\sin \mathsf{\theta }}}({\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial \mathsf{\theta }}}{\displaystyle \frac{\partial {\mathsf{\psi }}^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}}}-{\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}}}{\displaystyle \frac{\partial {\mathsf{\psi }}^{\mathrm{*}}}{\partial \mathsf{\theta }}}),1\le {\mathrm{r}}^{\mathrm{*}}<{\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}} \end{gathered}$$

(12)

$$\begin{gathered} {{\nabla }^{\mathrm{*}}}^2{\mathrm{g}}_2^{\mathrm{*}}+\mathsf{\alpha }({\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}}}{\displaystyle \frac{\partial {\mathrm{g}}_2^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}}}+{\displaystyle \frac{1}{{{\mathrm{r}}^{\mathrm{*}}}^2}}{\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial \mathsf{\theta }}}{\displaystyle \frac{\partial {\mathrm{g}}_2^{\mathrm{*}}}{\partial \mathsf{\theta }}}) \\ ={\displaystyle \frac{{\mathrm{P}\mathrm{e}}_2}{{{\mathrm{r}}^{\mathrm{*}}}^2\sin \mathsf{\theta }}}({\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial \mathsf{\theta }}}{\displaystyle \frac{\partial {\mathsf{\psi }}^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}}}-{\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}}}{\displaystyle \frac{\partial {\mathsf{\psi }}^{\mathrm{*}}}{\partial \mathsf{\theta }}}),1\le {\mathrm{r}}^{\mathrm{*}}<{\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}} \end{gathered}$$

(13)

$${\mathrm{E}}^4{\mathsf{\psi }}^{\mathrm{*}}=-{\displaystyle \frac{(\mathsf{\kappa }\mathrm{a}{)}^2}{1+\mathsf{\alpha }}}\left\{\begin{array}{c}\left[{\displaystyle \frac{\partial {\mathrm{g}}_1^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}}}\exp (-{\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}})+{\displaystyle \frac{\partial {\mathrm{g}}_2^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}}}\mathsf{\alpha }\exp (\mathsf{\alpha }{\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}})\right]{\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial \mathsf{\theta }}}\\ -\left[{\displaystyle \frac{\partial {\mathrm{g}}_1^{\mathrm{*}}}{\partial \mathsf{\theta }}}\exp (-{\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}})+{\displaystyle \frac{\partial {\mathrm{g}}_2^{\mathrm{*}}}{\partial \mathsf{\theta }}}\mathsf{\alpha }\exp (\mathsf{\alpha }{\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}})\right]{\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}}}\end{array}\right\}\sin \mathsf{\theta },1\le {\mathrm{r}}^{\mathrm{*}}<{\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$$

(14)

$${\mathrm{E}}^4{\mathsf{\psi }}_{\mathrm{I}}^{\mathrm{*}}=0,0\le {\mathrm{r}}^{\mathrm{*}}<1$$

(15)

where ${\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$ is the ratio of the pore radius to the droplet radius. α is defined as $\mathsf{\alpha } = - {\displaystyle \frac{{\mathrm{z}}_{1 }}{{\mathrm{z}}_2}}$, where ${\mathrm{z}}_{1 }$ and ${\mathrm{z}}_2$ are the valences of cations and anions, respectively. For NaCl considered here, α = 1. The parameter κ is the Debye–Hückel parameter, where κ⁻¹ is the Debye length, a measurement of the double-layer thickness, and a is the droplet radius. κa is often viewed as an overall index of the electrostatic environment a colloidal entity is facing. In addition, ${\mathrm{g}}_1^{\mathrm{*}}$ and ${\mathrm{g}}_2^{\mathrm{*}}$ represent the perturbation functions of the anion and cation concentration fields, respectively. They represent the concentration flux in a sense. And $\mathsf{\delta }{\mathsf{\varphi }}^{\mathrm{*}}$ denotes the perturbation of the equilibrium electric potential.

The resulting final boundary conditions in dimensionless form are as follows:

$${\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial {\mathrm{R}}^{\mathrm{*}}}}=-{\mathsf{\sigma }}^{\mathrm{*}}{\mathrm{R}}^{\mathrm{*}}=1$$

(16)

$${\left.{\displaystyle \frac{\partial \mathsf{\delta }{\mathsf{\varphi }}^{\mathrm{*}}}{\partial {\mathrm{R}}^{\mathrm{*}}}}\right|}_{{\mathrm{R}}^{*}=1}\mathrm{ }=0\mathrm{ }\mathrm{ }\mathrm{ }{\mathrm{R}}^{\mathrm{*}}=1$$

(17)

$${\displaystyle \frac{\partial {\mathrm{g}}_1^{\mathrm{*}}}{\partial {\mathrm{R}}^{\mathrm{*}}}}=0{\mathrm{R}}^{\mathrm{*}}=1$$

(18)

$${\displaystyle \frac{\partial {\mathrm{g}}_2^{\mathrm{*}}}{\partial {\mathrm{R}}^{\mathrm{*}}}}=0{\mathrm{R}}^{\mathrm{*}}=1$$

(19)

$${\left.{\displaystyle \frac{\partial {\mathsf{\Psi }}^{\mathrm{*}}}{\partial {\mathrm{R}}^{\mathrm{*}}}}\right|}_{{\mathrm{r}}^{\mathrm{*}}=1^{+}}={\left.{\displaystyle \frac{\partial {\mathsf{\Psi }}_{\mathrm{I}}^{\mathrm{*}}}{\partial {\mathrm{R}}^{\mathrm{*}}}}\right|}_{{\mathrm{r}}^{\mathrm{*}}=1^{-}}{\mathrm{R}}^{\mathrm{*}}=1$$

(20)

$${{(\mathsf{\tau }}^{\mathrm{*}}}_{\mathrm{r}\mathsf{\theta }}^{\mathbf{N}}+{{\mathsf{\tau }}^{\mathrm{*}}}_{\mathrm{r}\mathsf{\theta }}^{\mathbf{M}}{)|}_{{\mathrm{r}}^{\mathrm{*}}=1^{+}}={({\mathsf{\tau }}^{\mathrm{*}}}_{\mathrm{r}\mathsf{\theta }\mathrm{I}}^{\mathbf{N}}+{{\mathsf{\tau }}^{\mathrm{*}}}_{\mathrm{r}\mathsf{\theta }\mathrm{I}}^{\mathbf{M}}){|}_{{\mathrm{r}}^{\mathrm{*}}=1^{-}}{\mathrm{R}}^{\mathrm{*}}=1$$

(21)

$${\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}=\mathrm{ }{\mathsf{\varphi }}_{\mathrm{w}}^{\mathrm{*}}=0{\mathrm{R}}^{\mathrm{*}}\mathrm{ }=\mathrm{ }{\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$$

(22)

$${\displaystyle \frac{\partial \mathsf{\delta }{\mathsf{\varphi }}^{\mathrm{*}}}{\partial {\mathrm{R}}^{\mathrm{*}}}}=0\mathrm{ }\mathrm{ }{\mathrm{R}}^{\mathrm{*}}\mathrm{ }=\mathrm{ }{\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$$

(23)

$${\displaystyle \frac{\partial {\mathrm{g}}_{\mathbf{j}}^{\mathrm{*}}}{\partial {\mathrm{R}}^{\mathrm{*}}}}=0{\mathrm{R}}^{\mathrm{*}}\mathrm{ }=\mathrm{ }{\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$$

(24)

$${\mathsf{\psi }}^{\mathrm{*}}=\mathrm{ }0{\mathrm{R}}^{\mathrm{*}}\mathrm{ }=\mathrm{ }{\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$$

(25)

$${\displaystyle \frac{\partial {\mathsf{\psi }}^{\mathrm{*}}}{\partial {\mathrm{R}}^{\mathrm{*}}}}=0{\mathrm{R}}^{\mathrm{*}}\mathrm{ }=\mathrm{ }{\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$$

(26)

$${\mathsf{\psi }}^{\mathrm{*}}=\mathrm{ }{\displaystyle \frac{1}{2}}{\mathrm{U}}^{\mathrm{*}}{\mathrm{R}}^{\mathrm{*}2}+\mathrm{ }{\mathsf{\psi }}_{\mathrm{ }\mathrm{\infty }}^{\mathrm{*}}\left({\mathrm{R}}^{\mathrm{*}}\right)\mathrm{ }\mathrm{ }\mathrm{ },\mathrm{ }\mathrm{ }\mathrm{ }{\displaystyle \frac{\partial {\mathsf{\psi }}^{\mathrm{*}}}{\partial {\mathrm{Z}}^{\mathrm{*}}}}=0{\mathrm{Z}}^{\mathrm{*}}=\mathrm{ }\pm {\mathrm{L}}^{\mathrm{*}}$$

(27)

$${\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}=\mathrm{ }{\mathsf{\varphi }}_{\mathrm{e}\mathrm{ },\mathrm{ }\mathrm{\infty }}^{\mathrm{*}}\left({\mathrm{R}}^{\mathrm{*}}\right)=0\mathrm{ }\mathrm{ }{\mathrm{Z}}^{\mathrm{*}}=\mathrm{ }\pm {\mathrm{L}}^{\mathrm{*}}$$

(28)

$${\displaystyle \frac{\partial \mathsf{\delta }{\mathsf{\varphi }}^{\mathrm{*}}}{\partial {\mathrm{Z}}^{\mathrm{*}}}}=-\mathsf{\beta }{\nabla }^{\mathrm{*}}{\mathrm{C}}^{\mathrm{*}}{\mathrm{Z}}^{\mathrm{*}}=\mathrm{ }\pm {\mathrm{L}}^{\mathrm{*}}$$

(29)

$${\mathrm{g}}_{\mathbf{j}}^{\mathrm{*}}=\left(\mathsf{\beta }-1\right){\nabla }^{\mathrm{*}}{\mathrm{C}}^{\mathrm{*}}{\mathrm{R}}^{\mathrm{*}}\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\theta }{\mathrm{Z}}^{\mathrm{*}}=\mathrm{ }\pm {\mathrm{L}}^{\mathrm{*}}$$

(30)

where σ^∗ stands for the dimensionless surface charge density and σ_H the viscosity ratio of the droplet to that of the ambient solution. Note that these two symbols are not related to each other in physical essence. Equation (16) is obtained based on the two-dimensional Gauss divergence theorem across the droplet surface: ${\mathsf{\epsilon }}_{\mathrm{M}}{\left[{\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}}{\partial \mathrm{r}}}\right]}_{\mathrm{r}=\mathrm{a}}-{\mathsf{\epsilon }}_{\mathrm{D}}{\left[{\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}}{\partial \mathrm{r}}}\right]}_{\mathrm{r}=\mathrm{a}}=\mathsf{\sigma }$, where ${\mathsf{\epsilon }}_{\mathrm{M}}$ and ${\mathsf{\epsilon }}_{\mathrm{D}}$ are the electric permittivity of the ambient solution and the droplet, respectively. σ is the dimensional surface charge density. Since the droplet interior is assumed chargeless or containing no electrolyte ions, the electric potential within the droplet is thus in an equal-potential situation. This yields ${\left[{\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}}{\partial \mathrm{r}}}\right]}_{\mathrm{r}=\mathrm{a}}$ = 0 in the above equation and leads to the boundary condition for the constant surface charge density on the droplet interface. Equation (16) is after a proper non-dimensionalization procedure. In addition, since the droplet interior is assumed chargeless, there is no need to solve for the electric field within the droplet since it assumes the same value as the droplet surface potential. It is thus decoupled from the exterior electric field. Only the flow fields are coupled. There is an analytical solution for the chargeless fluid inside a droplet [37,38], which greatly simplifies the subsequent mathematical treatment. Equations (27) to (30) are the boundary conditions far away from the droplet. The dimensionless index β is a measurement of the strength of the induced diffusion potential, defined as β $= {\displaystyle \frac{{\mathrm{D}}_{1 }-{ \mathrm{D}}_2}{{\mathrm{D}}_1 +{ \mathrm{D}}_2}}$ in a binary electrolyte solution, it appears in Equation (29), where D₁ is the diffusivity of the cations, and D₂ is the diffusivity of the anions [39]. Taking NaCl as an example, at 25 °C the diffusivity of Na⁺ in water is approximately 1.334 × 10⁻⁹ m²/s, whereas that of Cl⁻ is 2.032 × 10⁻⁹ m²/s, giving β = −0.208. Equation (30) is derived based on the electro-neutrality constraint in diffusiophoresis [22]. The details can be found in [31].

2.2. Evaluation of Droplet Mobility

Similar to the treatment of our previous paper [11], a patched pseudo-spectral method based on Chebyshev polynomials is applied to solve for the coupled electric and flow fields. The details can be found in [31,40]. The resulting dimensionless hydrodynamic drag force (F_Dz^*) as well as the corresponding electric driving force (F_Ez^*) acting on the droplet surface can thus be evaluated as follows once the electric and flow fields are obtained:

$$\begin{array}{c}{\mathrm{F}}_{\mathbf{D}\mathbf{z}}^{\mathrm{*}}=\mathsf{\pi }{\int }_0^{\mathsf{\pi }}{\left|{\mathrm{r}}^{\mathrm{*}4}{\sin }^3\mathsf{\theta }{\displaystyle \frac{\partial }{\partial {\mathrm{r}}^{\mathrm{*}}}}\left({\displaystyle \frac{{\mathrm{E}}^{\mathrm{*}2}{\mathsf{\psi }}^{\mathrm{*}}}{{\mathrm{r}}^{\mathrm{*}2}{\sin }^2\mathsf{\theta }}}\right)\right|}_{{\mathrm{r}}^{\mathrm{*}}=1}\mathrm{d}\mathsf{\theta }-\mathsf{\pi }{\int }_0^{\mathsf{\pi }}|{\mathrm{r}}^{\mathrm{*}2}{\sin }^2\mathsf{\theta }\{{\displaystyle \frac{(\mathsf{\kappa }\mathrm{a}{)}^2}{1+\mathsf{\alpha }}}[\exp (-{\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}})\left(1-\mathsf{\delta }{\mathsf{\varphi }}^{\mathrm{*}}-{\mathrm{g}}_1^{\mathrm{*}}\right)\\ -\exp (\mathsf{\alpha }{\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}})(1+\mathsf{\alpha }(\mathsf{\delta }{\mathsf{\varphi }}^{\mathrm{*}}+{\mathrm{g}}_2^{\mathrm{*}}))]{\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial \mathsf{\theta }}}+\left[\exp (-{\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}})-\exp (\mathsf{\alpha }{\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}})\right]{\displaystyle \frac{\partial \mathsf{\delta }{\mathsf{\varphi }}^{\mathrm{*}}}{\partial \mathsf{\theta }}}\}{|}_{{\mathrm{r}}^{\mathrm{*}}=1}\mathrm{d}\mathsf{\theta },\end{array}$$

(31)

$$\begin{gathered} {\mathrm{F}}_{\mathbf{E}\mathbf{z}}^{\mathrm{*}}=\mathsf{\pi }{\int }_0^{\mathsf{\pi }}{\left|\left\{{\mathrm{r}}^{\mathrm{*}2}\sin \mathsf{\theta }\cos \mathsf{\theta }\left[{\left({\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}}}\right)}^2+2{\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}}}{\displaystyle \frac{\partial \mathsf{\delta }{\mathsf{\varphi }}^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}}}\right] \\ +{\mathrm{r}}^{\mathrm{*}2}{\sin }^2\mathsf{\theta }\left({\displaystyle \frac{{\partial }^2{\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}\partial \mathsf{\theta }}}{\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}}}+{\displaystyle \frac{{\partial }^2{\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}\partial \mathsf{\theta }}}{\displaystyle \frac{\partial \mathsf{\delta }{\mathsf{\varphi }}^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}}}+{\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}}}{\displaystyle \frac{{\partial }^2\mathsf{\delta }{\mathsf{\varphi }}^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}\partial \mathsf{\theta }}}+{\displaystyle \frac{1}{{\mathrm{r}}^{\mathrm{*}2}}}{\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial \mathsf{\theta }}}{\displaystyle \frac{{\partial }^2{\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial {\mathsf{\theta }}^2}} \\ +{\displaystyle \frac{1}{{\mathrm{r}}^{\mathrm{*}2}}}{\displaystyle \frac{{\partial }^2{\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial {\mathsf{\theta }}^2}}{\displaystyle \frac{\partial \mathsf{\delta }{\mathsf{\varphi }}^{\mathrm{*}}}{\partial \mathsf{\theta }}}+{\displaystyle \frac{1}{{\mathrm{r}}^{\mathrm{*}2}}}{\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial \mathsf{\theta }}}{\displaystyle \frac{{\partial }^2\mathsf{\delta }{\mathsf{\varphi }}^{\mathrm{*}}}{\partial {\mathsf{\theta }}^2}}\right) \\ -2{\mathrm{r}}^{\mathrm{*}}{\sin }^2\mathsf{\theta }\left({\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}}}{\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial \mathsf{\theta }}}+{\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}}}{\displaystyle \frac{\partial \mathsf{\delta }{\mathsf{\varphi }}^{\mathrm{*}}}{\partial \mathsf{\theta }}}+{\displaystyle \frac{\partial \mathsf{\delta }{\mathsf{\varphi }}^{\mathrm{*}}}{\partial {\mathrm{r}}^{\mathrm{*}}}}{\displaystyle \frac{\partial {\mathsf{\varphi }}_{\mathrm{e}}^{\mathrm{*}}}{\partial \mathsf{\theta }}}\right)\right\}\right|}_{{\mathrm{r}}^{\mathrm{*}}=1}\mathrm{d}\mathsf{\theta } \end{gathered}$$

(32)

The droplet diffusiophoretic mobility, defined as droplet velocity divided by the magnitude of the concentration gradient imposed, is evaluated as follows:

$${\mathsf{\mu }}^{\mathbf{*}}={\displaystyle \frac{{\mathbf{U}}^{\mathrm{*}}}{{\nabla }^{\mathrm{*}}{\mathrm{C}}^{\mathrm{*}}}}=-{\displaystyle \frac{{\mathrm{F}}_{\mathrm{E}}^{\mathrm{*}}}{{\mathrm{F}}_{\mathrm{D}}^{\mathrm{*}}}}$$

(33)

3. Results and Discussion

We first check the mesh convergence, ensuring that the calculation results do not change with further mesh refinement. It turns out that 51 grid points in the θ-direction plus 100 grid points in the r-direction are sufficient. The details can be found in the Supplementary Material, where the mesh convergence is clearly demonstrated. As a result, this mesh scheme is used throughout the calculation in this study.

We then check the asymptotic behavior of the droplet mobility with increasing radius of the cylindrical pore. Indeed, they all approach the corresponding single droplet mobilities asymptotically in the absence of the cylindrical pore, shown as the color dots in Figure 2, where the droplet mobilities are expressed as functions of the ratio of the droplet radius to the radius of the cylindrical pore, for several representative viscosities of the droplet interior fluids. These selected representative droplets are the benchmark cases: the rigid particle with σ_H = 100, the corn oil droplet with σ_H = 0.5, and a gas bubble with σ_H = 0.01. We thus conclude that the numerical scheme is reliable and the calculation results are accurate. We thus investigate the droplet motion based on it.

It is interesting to note that, as shown in Figure 2, mobility reversals are observed for each and every fluid droplet under consideration, which indicates that the moving direction of a droplet in narrow channels may be opposite to that of a droplet in wider channels. Moreover, comparison with the corresponding KCl solution is also shown in Figure 2 to reveal the difference when the diffusion potential is present, for instance, in the NaCl solution. Qualitatively speaking, they are similar in that both the mobility reversal and the solidification phenomenon are observed in either electrolyte solution. The involvement of diffusion potential, however, significantly increases the magnitude of the mobility for a positively charged droplet under consideration.

More importantly, the magnitudes of the droplet mobilities of different droplets in NaCl solution are comparable to one another when ${\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$ is relatively large, say ${\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$> 5. All the positively charged droplets move down to the region with lower electrolyte concentrations, as the diffusion potential yields a negative corresponding electric field. This is indicated by the dimensionless β parameter defined for a binary symmetric electrolyte solution like NaCl as β $= {\displaystyle \frac{{\mathrm{D}}^{+}-{\mathrm{D}}^{-}}{{\mathrm{D}}^{+} +{\mathrm{D}}^{-}}}$, where ${\mathrm{D}}^{+}$ refers to the diffusivity of the cations, and ${\mathrm{D}}^{-}$ that of anions. Here the ${\mathrm{C}\mathrm{l}}^{-}$ migrates faster than Na⁺ in the bulk electrolyte solution, hence a downward electric field is generated to speed up the downward motion of the Na⁺ in the bulk electrolyte solution and slow down the speed of ${\mathrm{C}\mathrm{l}}^{-}$ simultaneously. This is indicated by β $=-0.208$ for the NaCl solution considered here. Apparently, the boundary confinement effect upon the droplet is negligible in general; thus, the droplet mobilities are about the same as their corresponding mobilities in an infinitely large medium of electrolyte solution, indicated by the colorful dots at ${\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$ about 25.

At ${\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$ somewhere between four and five, however, all the mobility profiles intersect at the same point. In other words, all the droplets move at the same speed regardless of their viscosities. This implies they move like rigid particles without the normally observed recirculating axisymmetric vortex flow within the droplet, as shown in Figure 3A. As a result, it is referred to as the “solidification phenomenon” [11]. This is due to the deadlock between the spinning Maxwell traction driven by electrostatic force and the spinning hydrodynamic drag driven by the diffusioosmosis flow relative to the moving droplet [11,12,22]. The intriguing part of it is that the droplet surface shear stress is zero in this situation. As the liposome bilayer membrane is very vulnerable to strong shear stress, which may lead to breakup of the droplet surface and early release of medicines inside before it reaches the intended region needing therapy, the critical κa value where the surface shear stress is zero provides the ideal liposome size in its fabrication stage to minimize the potential damage of the shear stress upon the droplet surface when it is enroute to its intended region in the human blood vessels, for instance.

Moreover, when these two spinning forces, the spinning Maxwell traction and the spinning hydrodynamic drag force, are moving opposite to each other, an exterior axisymmetric vortex flow is often generated to reconcile the infinite hydrodynamic stress paradox when the flows driven by them meet each other in the electrolyte solution near the droplet exterior region, as shown in Figure 4, which has been reported before [22]. This phenomenon happens for a highly charged single droplet immersed in an infinite medium of electrolyte solution. It is found here that it can happen for a weakly charged droplet as well in a cylindrical pore. When the ${\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$ is beyond this critical point, the hydrodynamic spinning force dominates and turns back the spinning Maxwell traction on the droplet surface. The droplet is pushed downward with a negative droplet mobility. No exterior axisymmetric vortex flow is observed, as shown in Figure 3B. Once ${\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$ becomes smaller than this critical point (${\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$= 4.577), the situation becomes the other way around, and an exterior axisymmetric flow vortex appears, as shown in Figure 3C. The boundary confinement effect, both hydrodynamically and electrostatically, makes it happen. The droplet tends to move this way and become slower with further decrease of ${\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$. Eventually, it reaches a point where the droplet becomes stationary, and mobility reversal is observed with an even further decrease of ${\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$. Interestingly enough, the exterior vortex flow disappears completely once passing this critical point, as shown in Figure 3D. The entire flow field reduces to the original situation before the onset of the solidification phenomenon. The droplet moves upward toward the region with a higher concentration of electrolyte ions. This is against the intuitive prediction: A positively charged droplet moves against the electric field. The electrophoresis component is not the single dominant force in a cylindrical pore. The profound boundary confinement effect in a narrow cylindrical pore changes the entire electrokinetic scenario, as we observed above.

Corresponding mobility profiles for a negatively charged droplet are also shown in Figure 5. Again, contradictory polarity-dependence against the intuitive prediction is observed in a narrow cylindrical pore. This tells us that it is very dangerous to apply the electrokinetic knowledge of a single droplet in an infinite medium to a droplet in a cylindrical pore. The severe boundary confinement effect in a narrow channel changes everything. For a highly charged droplet, the fundamental phenomena discussed earlier hold nonetheless, except that the mobility magnitude is somewhat different, as might be expected. Corresponding mobility profiles for a benchmark highly charged droplet as a function of ${\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$ are shown in Figure 6 and Figure 7. Moreover, corresponding mobility profiles for a droplet with other surface charges in both NaCl and KCl electrolyte solutions are also presented in the Supplementary Material for completeness. Together, they shed light on a comprehensive understanding of the diffusiophoretic motion of a dielectric droplet, weakly or highly charged, in an electrolyte solution, with or without the presence of the diffusion potential.

4. Conclusions

Diffusiophoresis of a charged dielectric fluid droplet in a cylindrical pore is theoretically investigated, focusing on the impact of the diffusion potential, such as in the NaCl electrolyte solution. Interesting phenomena are found. Mobility reversal is observed in narrow channels. Moreover, the resulting droplet moving direction is against the intuitive prediction that one might have. The electrophoresis component resulting from the diffusion potential turns out not to be the single dominant factor in determining the droplet’s moving direction, unlike what is normally observed in the corresponding situation where a single droplet is in an infinite medium of electrolyte solution. The profound boundary confinement effect, both electrostatically and hydrodynamically, is found to be responsible for it. In addition, two critical points are found where the orientation of the spinning droplet surface changes each time passing these critical points. Furthermore, “solidification phenomenon” is also observed in some specific situations, where the droplet moves like a rigid particle without interior recirculating vortex flow. All the droplets move with identical speeds regardless of their viscosities, indicating a zero shear rate and shear stress on the surfaces of the droplets. The characteristic spinning motion on the droplet surface disappears completely when this “solidification phenomenon” happens.

The results presented here have potential applications in microfluidic and nanofluidic operations as well as drug delivery. However, the present study focuses on the fundamental electrokinetic response of a spherical non-deforming dielectric droplet with fixed surface charge and idealized cylindrical confinement. Direct applications to practical operations like drug delivery should be cautious as real liposomes and biological pores may involve membrane deformation, charge regulation, complex electrolytes, surface adsorption, background flow, and nonuniform geometries. Moreover, as a possible future direction, the present framework could be naturally extended beyond binary, charge-symmetric electrolytes such as the NaCl treated here. Since the diffusion potential is central to the present study, electrolytes with unequal valences, such as BaCl2 or LaCl3, and multispecies electrolyte solutions may lead to richer droplet-migration behavior, as indicated in related research papers in the literature in recent years [41,42].