Diffusiophoresis of a perfectly conducting droplet-like liquid metal in electrolyte solutions is investigated theoretically, focusing on the chemiphoresis component, the very heart of diffusiophoresis, where the droplet motion is induced solely by the chemical gradient. The resulting electrokinetic equations are solved with a pseudo-spectral method based on Chebyshev polynomials. For the isothermal electrokinetic system of a perfectly conducting droplet considered here, there is no Marangoni effect, which is a motion-inducing effect due to the variation of interfacial tension along the droplet surface. No Maxwell traction is present as well. The droplet motion is full of hydrodynamic nature. It is found, among other things, that contrary to a dielectric droplet, a conducting droplet always moves up the chemical gradient toward the region with a higher concentration of ions in chemiphoresis. This implies that a perfectly conducting droplet like a gallium or its alloy droplet is superior to the commonly utilized dielectric droplet like a liposome in drug delivery in terms of self-guarding itself toward the desired destination of injured or infected area in the human body, as specific ionic chemicals are often released there. Optimum droplet size yielding the fastest migration rate is predicted.

Diffusiophoresis of a highly charged conducting fluid droplet

The present article deals with the theoretical study on electrophoresis of hydrophobic and dielectric spherical fluid droplets possessing uniform surface charge density. Unlike the ideally polarizable liquid droplet bearing constant surface ζ-potential, the tangential component of the Maxwell stress is nonzero for dielectric fluid droplets with uniform surface charge density. We consider the continuity of the tangential component of total stress (sum of the hydrodynamic and Maxwell stresses) and jump in dielectric displacement along the droplet-to-electrolyte interface. The typical situation is considered here for which the interfacial tension of the fluid droplet is sufficiently high so that the droplet retains its spherical shape during its motion. The present theory can be applied to nanoemulsions, hydrophobic oil droplets, gas bubbles, droplets of immiscible liquid suspended in aqueous medium, etc. Based on weak field and low charge assumptions and neglecting the Marangoni effect, the resultant electrokinetic equations are solved using linear perturbation analysis to derive the closed form expression for electrophoretic mobility applicable for the entire range of Debye-Hückel parameter. We further deduced an alternate approximate expression for electrophoretic mobility without involving exponential integrals. Besides, we have derived analytical results for mobility pertaining to various limiting cases. The results are further illustrated to show the impact of pertinent parameters on the overall electrophoretic mobility.

Electrophoresis of Dielectric and Hydrophobic Spherical Fluid Droplets Possessing Uniform Surface Charge Density.

The mixing process of two liquids inside an open minichannel was experimentally studied in the presence of liquid metal and an electric field. The Y-type mixers under study were made of Plexiglass and two liquid metal-enabled pumping systems (based on electrically induced surface tension gradients) were placed at the inlets of the mixer instead of conventional syringe pumps. The effects of the mixing angle, the voltage applied to the liquid metals, and the Reynolds number on the mixing process were investigated. To accurately determine the mixing index, the image processing toolbox of MATLAB software was employed. The results showed that the mixing intensity increased as the applied voltage signal increased, thereby creating a chaotic advection in the minichannel. Furthermore, although the Reynolds number of induced flow and the applied voltages were directly proportional, however, the input angle plays an important role in the mixing. Among the considered models, in the constant voltage, the 30 and 90 degrees had the best and the worst mixing, respectively. The maximum mixing intensity of 94% was obtained at an input angle of 30 degrees and voltage of 14 volts, where, in the absence of an electric field, the maximum mixing intensity was 55%.

Experimental study on the performance of a mini-scale Y-type mixer with two liquid metal enabled pumps

The present article deals with the diffusiophoresis of hydrophobic rigid colloids bearing arbitrary $\zeta$-potential. We derived the generic expression for the diffusiophoretic velocity of such a colloid exposed in an externally applied concentration gradients of general electrolyte solution. The derived expression takes into account the relaxation effect and is applicable for all values of surface $\zeta$-potential and hydrodynamic slip length at large $\kappa a$ ($\kappa a \ge ca. 50)$, where $\kappa^{-1}$is the thickness of the electric double layer and $a$ is the particle radius. We further derived several closed form expressions for particle velocity derived under several electrostatic and hydrodynamic conditions when the particle is exposed in an applied concentration gradient of binary symmetric (e.g., ), asymmetric (1:2, 2:1, 3:1, 1:3) and a mixed electrolyte (mixture of 1:1 and 2:1 electrolytes). The results for diffusiophoretic velocity are further illustrated graphically to indicate the mutual interaction of chemiphoresis, induced electrophoresis due to unequal mobilities of cations and anions of the electrolyte and the mechanism by which the sufficiently charged particle migrates opposite to the applied concentration gradient direction. Impact of hydrophobicity is further discussed.

Diffusiophoresis of hydrophobic spherical particles in a solution of general electrolyte

This study aims to quantify the impact of the dielectric permittivity of a droplet on its diffusiophoresis in different types of electrolytes. The dielectric droplet polarizes by the diffusion field along with the local electric field created by the interactions of the doublelayer with the imposed ionic concentration gradient, which generates an induced surface charge density anti-symmetrically distributed on the droplet surface. This induced surface charge influences both electrophoresis and chemiphoresis parts. Based on a low imposed concentration gradient, a simplified model is derived through a first-order perturbation technique. Dielectric polarization of the droplet attenuates the spinning force at the interface. This creates the mobility of a droplet of higher dielectric permittivity in presence of a stronger diffusion field significantly higher than that of a perfectly dielectric droplet and its value depends on the polarity of the droplet surface charge. In absence of the diffusion field, the mobility of a conducting droplet remains positive immaterial of the polarity of its surface charge density. We find that the impact of the dielectric polarization becomes significant as the surface charge density increases and attenuates with the increase of droplet viscosity. For a dielectric droplet at a thinner Debye length, a step-jump in mobility occurs at a higher value of the surface charge density. Such type of step-jump in mobility does not appear for the conducting droplet due to the absence of the Maxwell stress at the interface.

https://aip.scitation.org/doi/pdf/10.1063/5.0142875

A simplified model for the impact of dielectric polarization of a charged droplet on its diusiophoresis

Diffusiophoresis of a dielectric fluid droplet in electrolyte solutions is investigated theoretically, focusing on the electrophoresis component resulting from the induced diffusion potential in the electrolyte solution when the diffusivities of cations and anions there are different. The resultant electrokinetic equations are solved with a pseudo-spectral method based on the Chebyshev polynomials. We found, among other things, that the electrophoresis component dominates at a larger Debye length, whereas the chemiphoresis component at a smaller Debye length for a dielectric droplet of a constant surface charge density. The two components are of comparable magnitudes in the NaCl solution. The dual between the spinning electric driving force tangent to the droplet surface and the hydrodynamic drag force reinforced by the motion-deterring electrokinetic Maxwell traction from the surrounding exterior osmosis flow is crucial in the determination of the ultimate droplet motion. Unlike the chemiphoresis component, which is independent of the sign of charges carried by the droplet, the droplet moving direction as well as its magnitude in the electrophoresis component depends on the sign of charges carried by the droplet as well as the direction of the electric field induced by the diffusion potential. This gives the electrophoresis component excellent maneuverability in practical applications like drug delivery and enhanced oil recovery, where migration of droplets toward regions of higher solute concentrations is often desired.

Diffusiophoresis of a highly charged dielectric fluid droplet induced by diffusion potential

An analytical formula is presented here for the electrophoresis of a dielectric or perfectly conducting fluid droplet with arbitrary surface potentials suspended in a very dilute electrolyte solution. In other words, when the Debye length (κ−1) is very large, or κa ≪$\ll $ 1, where κ is the electrolyte strength and a stands for the droplet radius. This formula can be regarded as an extension of the famous Hückel solution valid for weakly charged rigid particles to arbitrarily charged fluid droplets. The formula reduces successfully to the ones obtained by Booth for a dielectric droplet, and Ohshima for a perfectly conducting droplet, both under Debye–Hückel approximation valid for weakly charged droplets. Moreover, the formula is valid for a gas bubble and a rigid solid particle as well. Classic results obtained by Hückel for a rigid particle are reproduced as well. We found that for a dielectric droplet, the more viscous the droplet is, the faster it moves regardless of its surface potential, contrary to the intuition based on the purely hydrodynamic consideration. For a perfectly conducting liquid droplet, on the other hand, the situation is reversed: The less viscous the droplet is, the faster it moves. The presence or absence of the spinning electric driving force tangent to the droplet surface is found to be responsible for it. As a result, an axisymmetric exterior vortex flow surrounding the droplet is always present for a dielectric liquid droplet, and never there for a conducting liquid droplet.

Electrophoresis of a highly charged fluid droplet in dilute electrolyte solutions: Analytical Hückel‐type solution

Diffusiophoresis of a weakly charged liquid metal droplet (LMD) is investigated theoretically, motivated by its potential application in drug delivery. A general analytical formula valid for weakly charged condition is adopted to explore the droplet phoretic behavior. We determined that a liquid metal droplet, which is a special category of the conducting droplet in general, always moves up along the chemical gradient in sole chemiphoresis, contrary to a dielectric droplet where the droplet tends to move down the chemical gradient most of the time. This suggests a therapeutic nanomedicine such as a gallium LMD is inherently superior to a corresponding dielectric liposome droplet in drug delivery in terms of self-guiding to its desired destination. The droplet moving direction can still be manipulated via the polarity dependence; however, there should be an induced diffusion potential present in the electrolyte solution under consideration, which spontaneously generates an extra electrophoresis component. Moreover, the smaller the conducting liquid metal droplet is, the faster it moves in general, which means a smaller LMD nanomedicine is preferred. These findings demonstrate the superior features of an LMD nanomedicine in drug delivery.

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Diffusiophoresis of a Weakly Charged Liquid Metal Droplet

Diffusiophoresis phenomenon of aoft particles suspended in binary electrolyte solutions is explored theoretically in this study based on the spherical cell model, focusing on the chemiphoresis component in absence of diffusion potential. Both the electrostatic and hydrodynamic aspects of the boundary confinement, or steric effect, due to the presence of neighboring particles are examined extensively under various electrokinetic conditions. Significant local extrema are found in mobility profiles expressed as functions of the Debye length in general, synchronized with the strength of the motion‐inducing double layer polarization. Moreover, a seemingly peculiar phenomenon is observed that the soft particles may move faster in more concentrated suspensions. The competition between the simultaneous enhancement of the motion‐inducing electric driving force and the motion‐retarding hydrodynamic drag force from the boundary confinement effect of the neighboring particles is found to be responsible for it. The above findings are also demonstrated experimentally in a very recent study on the diffusiophoretic motion of soft particles through porous collagen hydrogels. The results presented here are useful in various practical applications of soft particles like drug delivery.

Eric Lee

Papers

Diffusiophoresis of a highly charged dielectric fluid droplet induced by diffusion potential

Diffusiophoresis of a highly charged conducting fluid droplet

Electrophoresis of a highly charged fluid droplet in dilute electrolyte solutions: Analytical Hückel‐type solution

Diffusiophoresis of a Weakly Charged Liquid Metal Droplet

Diffusiophoresis in suspensions of highly charged soft particles