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

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 property 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 application 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 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^- is often released by a fractured bone tissue [20]. As a result, a concentration gradient is established in the vicinity of this region needing therapy, where it is highly desirable to increase the concentration of the drug-carrying droplet liposomes as high as possible in order to maximize the therapy 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 in biomedical applications such as drug delivery, as a temperature raise over 40℃ 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 is observed in very narrow channels. This certainly has a profound impact in drug delivery application 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. Moreover, cylindrical pore is often adopted to mimic a porous medium with the pore radius representing the porosity of the medium [26]. For instance, for the confined interstitial spaces in biological tissues, where the extracellular matrix (ECM) forms a complex porous environment for molecular and colloidal transport. The diameter of a micro-blood vessel typically ranges from 1000 to 2500 nanometers. This is the region beyond the capillary all the way to the tissues. 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 in drug delivery in the human body.

The previous paper treated the droplet migration in the absence of diffusion potential, as the diffusivity of ${\mathrm{K}}^{+}$ and ${\mathrm{C}\mathrm{l}}^{-}$in the KCL electrolyte solution where the droplet is assumed to be suspended are approximately equal [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 ion 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]. This is based on the fundamental Coulomb's electrostatic law [29]. And the resulted electric potential is referred to as the diffusion potential in diffusiophoresis [27,28]. Note that in response to this extra diffusion potential, a corresponding driving force upon the droplet motion is resulted, 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." They are coupled in general if both are present. In the human blood and other biological fluids in the human body, there are many electrolyte ions with Na⁺ and ${\mathrm{C}\mathrm{l}}^{-}$as the two major species [30,31,32]. Hence, there will be a diffusion potential established simultaneously if somehow a concentration gradient of these ions is established and ions migration takes place driven by the diffusion mechanism [33]. 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 charged 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 a solute concentration gradient in the bulk solution along the axis of a cylindrical pore. In particular, have we focus on the NaCl solution to examine the impact of the induced diffusion potential in particular. The geometric system and corresponding basic assumptions are essentially the same as the ones adopted in the previous paper where KCl solution is investigated [11]. Briefly speaking, the droplet is composed of a dielectric fluid and carries a uniformly distributed surface charge, and the interior fluid contains no electrolyte ions. In the present model, the surface charge density is assumed uniform and remain constraint. 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].

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. In addition, 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.

Governing Electrokinetic Equations

The governing electrokinetic equations of the system considered here consist of equations for the electric potential, the ions distribution, and the momentum of the fluid, which are essentially the same as adopted in the previous paper [34] :

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

(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 eqns. (1) is the Poisson equations based on Gauss's divergence theorem, ϕ is the electric potential and $\mathsf{\rho }={\displaystyle {\sum }_{\mathrm{j}=1}^{\mathrm{N}}}{\mathrm{z}}_{\mathrm{j}}\mathrm{e}{\mathrm{n}}_{\mathrm{j}}$ the space charge density, the total number of electric charges per unit volume. 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 is 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 that the inclusion of symbols relating to the diffusion potential.

Corresponding mathematical treatment can be found elsewhere for detail [35]. After the standard non-dimensionlization procedure is applied, we end up with the complete governing equations in dimensionless form are as follows:

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

(8)

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

(9)

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

(10)

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

(11)

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

(12)

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

(13)

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

(14)

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

(15)

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

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

(16)

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

(17)

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

(18)

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

(19)

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

(20)

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

(21)

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

(22)

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

(23)

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

(24)

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

(25)

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

(26)

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

(27)

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

(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)

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

(30)

Eqns. (27) to (30) are the boundary conditions far away from the droplet. A 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 appears in Eqn. (29), where D₁ is the diffusivity of the cations and D₂ is the diffusivity of the anions[36,37]. Taking NaCl as an example, at 25 °C the diffusivity of Na⁺ in water is approximately1.334×10⁻⁹ m²/s, whereas that of Cl⁻ is 2.032×10⁻⁹ m²/s, giving β = −0.208. Eqn. (30) indicates the impact of the diffusion potential upon the droplet motion, which is derived based on the electro-neutrality constraint in diffusiophoresis [22]. The details can be found elsewhere [38].

Evaluation of Droplet Mobility

As the droplet interior fluid is assumed chargeless, there is no need to solve for the electric field there. Since there is an analytical formula for the purely hydrodynamic fluid field inside a droplet, which can be incorporated into the corresponding exterior fluid field as a boundary condition, only the electric and fluid flow fields in the exterior region need to be solved. A patched pseudo-spectral method based on Chebyshev polynomials is then applied to accomplish this. The details can be found elsewhere [38,39]. The resulting dimensionless hydrodynamic drag force (F_Dz^*) and electric driving force (F_Ez^*) acting on the droplet surface can be evaluated as follows:

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

(31)

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

(32)

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

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

(33)

3. Results and Discussion

We first check the convergence of the numerical scheme in that the calculation results do not change any more 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. As a result, this mesh scheme is used throughout the calculation in this study. We also check the convergence behavior of the droplet mobility with increasing radius of the cylindrical pore. Indeed, they all approach asymptotically the corresponding single droplet situations in the absence of the cylindrical pore, as shown in Figure 1, where the droplet mobilities as functions of the ratio of the droplet radius to the radius of the cylindrical pore are for several viscosities of the droplet interior fluids. The representative droplets are chosen to be the benchmark case: 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, where σ_H is the ratio of the droplet viscosity to that of the ambient aqueous electrolyte solution. We thus conclude that the numerical scheme is reliable and the calculation results are accurate. We then go on to investigate the droplet motion based on it. 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 a narrow channel may be opposite to a droplet in a wider channel.

More importantly, the magnitude of the droplet mobility is 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 (NaCl) 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 ${\mathrm{D}}^{+}$ refer to the diffusivity of the cations, and ${\mathrm{D}}^{-}$ that of anions. Here the ${\mathrm{C}\mathrm{l}}^{-}$ (${\mathrm{D}}^{-}$= 2.032×10⁻⁹ m²/s) migrates faster than Na⁺ (${\mathrm{D}}^{+}$=1.334×10⁻⁹ m²/s) 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 slowdown that of ${\mathrm{C}\mathrm{l}}^{-}$ simultaneously. This is indicated by β $=-0.208$ for the NaCl solution considered here. Note that the generation of the above electric field by the coulomb electrostatic law can also be viewed as the presence of an induced diffusion potential in the bulk electrolyte solution, where an osmosis flow is driven by these downward-moving ions, which is often referred to as the diffusioosmosis flow. The driving force upon the droplet considered here, or any colloidal entities, is referred to as the “electrophoresis component” in Diffusiophoresis. Obviously, the boundary confinement upon the droplet is negligible, thus the droplet mobilities are about the same as their corresponding mobilities in an infinite large medium of electrolyte solution, indicated by the colorful dots at ${\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$ about 25.

Moreover, at ${\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$ between 4 and 5, 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 axi-symmetric vortex flow within the droplet, as shown in Figure 3. 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 [11,12,22]. A’’solidification phenomenon’’ takes place at a critical value of κa, where κ is the electrolyte strength and a is the droplet radius. At this critical value, all the droplet moves with identical speeds regardless of their droplet viscosities. The recirculating interior vortex flow typically observed within a moving droplet disappears, and the droplet moves like a rigid solid particle. That is the reason it is referred to as the’’solidification phenomenon’’. The intriguing part of it is that the surface shear stress is zero at this situation. As the bi-layer of a liposome 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 sizes in its fabrication stage to minimize the damage of shear stress upon the potential droplet surface when it is enroute to the intended region in the human blood vessels, for instance.

When these two spinning forces are moving opposite to each other, an exterior axi-symmetric 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. We demonstrate 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 turn back the spinning Maxwell traction on the droplet surface. The droplet is pushed downward with a negative droplet mobility. No exterior axi-symmetric flow is observed, as shown in Figure 5. 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 axi-symmetric flow is observed, as shown in Figure 6. The boundary confinement effect, both hydrodynamically and electrostatically, make it happen.

The droplet still moves and becomes slower with decreasing ${\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$. A finally it reaches a point where the droplet becomes stationery and mobility reversal is observed with 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 7! The entire flow field reduces to the original situation before the onset of the solidification phenomenon. The droplet moves upward toward the region with higher concentration of electrolyte ions. This is against the intuitive prediction law: A positively charged droplet moves against the electric field! The electrophoresis component is not a dominant force alone in a cylindrical pore. The profound boundary confinement effect in a narrow cylindrical power changes the entire electrokinetic scenario.

Corresponding mobility profiles for a negatively charged droplet is shown in Figure 8. Again, contradictory polarity-dependence against the prediction by Coulomb’s electrostatic law 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 change everything. For a highly charged droplet, the fundamental phenomena discussed above hold nonetheless, except that the mobility magnitude is different, as might be expected. Corresponding mobility profiles for a benchmark highly charged droplet a function of ${\mathrm{R}}_{\mathrm{w}}^{\mathrm{*}}$ ia shown in Figure 9 and Figure 10. Moreover, corresponding mobility profiles for a droplet with other surface charges in both NaCl and KCl solutions in a cylindrical pore is 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.

4. Conclusions

Diffusiophoresis of a charged dielectric fluid droplet in a cylindrical pore is investigated, focusing on the impact of the diffusion potential, such as in the electrolyte solution. Interesting phenomena are found. Mobility reversal is observed in narrow channels. Moreover, the resulting droplet moving direction is against the intuitive prediction based on the Coulomb electrostatic law. In other words, the electrophoresis component resulted from the inner diffusion potential turn out to be not the dominant factor in determining the droplet moving direction as normally observed in the corresponding situation where a single droplet is in an infinite medium if electrolyte solution. The profound boundary confinement effect, both electrostatically and hydrodynamically, is found to be responsible for it. In addition, two critical points of the spinning droplet surface changes each time passing these critical points. Furthermore, ’’solidification phenomenon’’ 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 speed 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.