Unconventional hydrodynamics of hybrid fluid made of liquid metals and aqueous solution under applied fields

Abstract

The hydrodynamic characteristics of hybrid fluid made of liquid metal/aqueous solution are elementary in the design and operation of conductive flow in a variety of newly emerging areas such as chip cooling, soft robot, and biomedical practices. In terms of physical and chemical properties, such as density, thermal conductivity and electrical conductivity, their huge differences between the two fluidic phases remain a big challenge for analyzing the hybrid flow behaviors. Besides, the liquid metal immersed in the solution can move and deform when administrated with non-contact electromagnetic force, or even induced by redox reaction, which is entirely different from the cases of conventional contact force. Owing to its remarkable capability in flow and deformation, liquid metal immersed in the solution is apt to deform on an extremely large scale, resulting in marked changes on its boundary and interface. However, the working mecha- nisms of the movement and deformation of liquid metal lack appropriate models to describe such scientific issues via a set of well-established unified equations. To promote investigations in this important area, the present paper is dedicated to summarizing this unconventional hydrodynamics from experiment, theory, and simulation. Typical experimental phenomena and basic working mechanisms are illustrated, followed by the movement and deformation theories to explain these phenomena. Several representative simulation methods are then proposed to tackle the governing functions of the electrohydrodynamics. Finally, prospects and challenges are raised, offering an insight into the new physics of the hybrid fluid under applied fields.

Keywords

liquid metal / hybrid fluid / hydrodynamics / surface tension / applied fields / self-actuation

Introduction

The hybrid fluid hydrodynamics can be applied to various industrial fields such as oil extraction [¹–³], electrostatic spraying [⁴], inkjet printing [⁵], and metal recycling [⁶]. The liquid metal drops immersed in immiscible solution under electric field will present intriguing phenomena such as movement [⁷–¹⁰], deformation [¹⁰], rotation [¹¹], breakup [¹⁰,¹²], oscillating [¹³], collision and coalescence [¹⁴,¹⁵]. Precise description of the electrohydrodynamics of liquid metal drops remains unclear, due to the mutual coupling of the electric field and the flow field.

The room temperature liquid metal(RTLM) is an attractive material option owing to its combination of advantages of metals such as high electrical conductivity, thermal conductivity and reflectivity, with the inherently dynamic nature of fluids [¹⁶]. Though mercury is the most well-known RTLM, its further applications are limited by the hyper toxicity involved. In contrast to mercury, gallium-based alloys have little toxicity and a low vapor pressure at a high temperature, on the order of 10⁻⁹ atm at 538°C and nearly zero at room temperature [¹⁷]. Recently, it has been found that liquid metal immersed in immiscible solutions under applied fields displays rather unconventional hydrodynamic characteristics, including planar locomotion [¹⁸], rotation [¹⁸], self-propelled [¹⁹], oscillation [²⁰], climbing [²¹], breathing [²²], deformation [¹⁶,²¹], and surfing [²³]. The potential of fluidic metallic liquid metal has been verified in the fields of chip cooling [²⁴–²⁶], heat dissipation [²⁷,²⁸], energy harvest [²⁹,³⁰], drug delivery [³¹], liquid pump [³²,³³], soft robot manufacture [³⁴], and printed electronics [³⁵,³⁶], etc.(Fig. 1)

Distinctive from the classical hybrid fluid in nature such as oil/water hybrid fluid and debris flow, the liquid metal/aqueous solution hybrid flow is unique in the following aspects: liquid metal immersed in the solution can move and deform when exerted with non-contact electromagnetic force, or even the force induced by redox reaction, which is significantly different from the conventional contact force. Owing to its excellent characteristics of flow and deformation, liquid metal is apt to deform on a large scale, resulting in marked changes on the boundary between the liquid metal and the solution. Because of the strong disparity in density between liquid metal (5907 kg/m³) [³⁷] and water (1000 kg/m³)), liquid metal invariably sinks into the bottom of the container. The surface tension of liquid metal (700 mN/m) [³⁸], another leading property in hybrid flow study, is almost ten times larger than that of water (72 mN/m, 25°C) [³⁹]. Besides, the electrical conductivity of liquid metal is 7 orders of magnitude higher than that of water. Chemical reactions such as the transformation between gallium and its oxide have an important effect on the movement and deformation of the liquid metal. In light of the uniqueness of liquid metal/aqueous solution, the working mechanism of the movement and deformation of liquid metal is not yet clear enough, with nosuitable model established to describe it with a set of unified equations.

This paper mainly reviewed the unconventional fluid mechanics of the movement and deformation of liquid metal in aqueous solutions. Former experiment phenomena of liquid metal immersed in the aqueous solution under applied fields are introduced. Foundations of electrohydrodynamics are presented, including the principles of electric double layer, the physical model of liquid metal immersed in aqueous solutions under the applied fields and the different deformation mechanism between the liquid metal and the perfectly conducting drop. Several classical simulation methods in the history are then summarized. Finally, in the light of the characteristics, mechanism and applications of liquid metals, challenges and prospects of future researches in liquid metal are proposed.

Experiments

Liquid metal possesses proprieties of both liquid and metal. Apart from the ability to move and deform under the pressure field, many intriguing phenomena occur under electric field [¹⁸,²⁷,⁴⁰,⁴¹], magnetic field [⁴²,⁴³], and chemical field [¹⁹,²¹,⁴⁴–⁴⁶] as well.

Electric field

It was observed that when the spherical liquid metal was exposed to the electric field after the electrodes were rearranged, self-rotation of the metal accompanied by two vortexes of water could occur (Fig. 2). Meanwhile, planar locomotion of the sphere occurred as a result of the driving forces applied by the electric field. Both the self-rotation and the planar locomotion could be used to control the movement of liquid objects [¹⁸].

Figure 3 shows an electrohydrodynamic shooting phenomenon of liquid metal stream. A low voltage direct current electric field can induce the ejection of liquid metal inside capillary tube, which ultimately shoot into sodium hydroxide solution and form discrete droplets [⁴⁷].

Tang et al. [³³] demonstrated a system called the liquid metal enabled pump capable of driving liquid with modest electric field, abandoning traditional mechanical moving parts. This pump incorporates a droplet of liquid metal, which induces liquid flow at high flow rates, yet with exceptionally low power consumption by electrowetting/deelectrowetting at the metal surface. Figure 4 describes the continuous pumping effect with the liquid metal droplet ceased by the neck of the chamber under the effect of the external electric field.

Although liquid metal can be actuated via numerous ways, it remains a huge challenge to flexibly control the shape of a liquid metal drop due to its extremely high surface tension. In terms of this issue, Zhang et al. [⁴⁸] introduced a SCHEME (synthetically chemical-electrical mechanism) to regulate the surface tension. It was found that when liquid metal, which had been previously immersed in NaOH solution, touched the anode, gallium oxide with smaller surface tension would be formed. Liquid metal spread out and its surface area increased by 5 times. When the electrode was removed, the liquid metal recovered to spherical shape as the oxide layer gradually dissolved and the surface tension increased (Fig. 5).

The non-coalescence phenomenon between a droplet and the same-component liquid surface has been revealed recently [²³]. The cathode and the anode were inserted respectively in the liquid metal bulk and the electrolyte. After applying voltage between the electrodes, a bearing electrolyte film filled the gap between the dripped liquid metal droplets and the underneath liquid metal bath, which maintained non-coalescent so long as the electric field existed, as if the droplets were surfing on the interface (Fig. 6).

Self-driven motion

Eutectic Ga-In alloy droplets can shift spontaneously in NaOH solution by only eating a small piece of aluminum. A milli-/centimeter scaled Ga-In-Al drop is capable of navigating itself to fit in different geometrical spaces where it voyaged at a considerably large velocity of 5 cm/s for more than one hour (Fig. 7). The soft machine works just like a biomimetic mollusk which eats Al as food [¹⁹], offering inspirations on the future development of bio-robots.

More interestingly, a self-powered copper wire oscillator was also disclosed, as seen in Fig. 8 [²⁰]. When contacting a copper wire with the liquid GaIn₁₀ alloy pre-fueled with a small piece of aluminum, the copper wire was wetted and then swallowed into the liquid metal body. After a while, it began oscillating across the liquid metal horizontally just like a violin bow.

A fundamental scientific finding was reported that a bouncing bright liquid-metal droplet in an alkaline electrolyte could be transformed to a flat and dull puddle when placed on a graphite surface without the use of electric field. Through the intrinsic interactions between the liquid metal and the graphite, the liquid-metal puddle on the graphite could be manipulated as desired into various stable shapes with sharp angles in a semi-open space via a simple and highly feasible method (Fig. 9) [²¹].

Unusual biomimetic amoeba-like behaviors of liquid metal were discovered when amalgamated with Al, with the multi-material system placed on a graphite substrate immersed in the alkaline electrolyte [⁴⁹]. Systems with varied Al content presented distinctive behaviors, as the less Al involved, the more resembling to the amoeba movement (Fig. 10). The underlying mechanism is the surface tension gradient on account of the reductive effect by Al reaction and the oxidative effect by graphite.

Coupled-fields

Tan et al. [⁴³] reported a phenomenon that the magnetic field could make up a boundary to restrict motion of the aluminum powered liquid metal motor. The distribution of self-propelled Al-Ga-In tiny motors which was achieved via the hydrogen gas generated from the galvanic cell reaction among alloy, aluminum and surrounding electrolyte, is given in Fig. 11. Most of the motors were trapped in the interior zone under the effect of magnetic field.

A liquid metal galinstan sphere, along with a NaOH solution, was stimulated to rotate centrifugally around the central electrode, while the rotating speed was increasing with the voltage. The apparatus and experiment phenomena are illustrated in Fig. 12. Liquid metal was actuated through the coupled electric field produced by previously arranged electrodes and vertically magnetic field which was produced by magnet underneath [⁵⁰].

It has been discovered that adding aluminum to liquid metal droplets will significantly magnify its electric controlling capability, which provides dozens of times of driving force compared to pure GaIn₁₀ droplets. After switching on the electrical field, the aluminum powered liquid metal droplet would accelerate to an extremely high velocity [⁴⁰], as compared in Figs. 13(a) and 13(b). Therefore, the Al-Ga-In motors in a Petri dish filled with NaOH solution could be easily controlled to a specific speed and direction, shown in Fig. 13(c), as desired by an external electric filed.

In conclusion, these special phenomena indicate that the liquid metal can be actuated by potential fields, which avoids direct contact. The present findings have both fundamental and practical significance, e.g., serving as the premise of a generalized way of making a soft machine, collecting discrete metal fluids, as well as flexibly manipulating liquid metal objects.

Theory

Physical and chemical properties of gallium

Liquid metal is an attractive material choice for researchers wishing to combine the advantages of metals, such as high electrical conductivity, thermal conductivity, and reflectivity, with the inherently dynamic nature of fluids [¹]. Some physical parameters of liquid metals are listed in Table 1. Compared to other liquid metals, gallium and gallium-based alloys are more commonly used in the laboratory, owing to their extremely little toxicity and high stability.

Liquid gallium is easily oxidized in the air by the chemical formula [⁵⁵]

(1)\mathrm{4}\mathrm{G}\mathrm{a}+\mathrm{3}{{\displaystyle \mathrm{O}}}_{\mathrm{2}}\to {{\displaystyle \mathrm{G}\mathrm{a}}}_{\mathrm{2}}{{\displaystyle \mathrm{O}}}_{\mathrm{3}},

which results in a thin layer of gallium oxide on the surface which prevents the continuation of oxidation reaction. Owing to the greater surface tension of gallium than that of gallium oxide which invariably exists on the surface, gallium presents paper sheet in the air, as shown in Fig. 14(a).

While the liquid metal is immersed in the NaOH solution, its surface oxide is removed by the electrolyte and the chemical formula can be expressed as [⁵⁶]

(2){{\displaystyle \mathrm{G}\mathrm{a}}}_{\mathrm{2}}{{\displaystyle \mathrm{O}}}_{\mathrm{3}}+\mathrm{N}\mathrm{a}\mathrm{O}\mathrm{H}\to \mathrm{N}\mathrm{a}\mathrm{G}\mathrm{a}{{\displaystyle \mathrm{O}}}_{\mathrm{2}}+{{\displaystyle \mathrm{H}}}_{\mathrm{2}}\mathrm{O}.

Therefore, the gallium placed on the glass substrate maintains an ellipsoidal shape in the alkali solution (Fig. 14(b)). Different from gallium in the NaOH solution on the glass substrate, a bouncing bright gallium droplet in NaOH electrolyte can be transformed to a flat and dull puddle when placed on a graphite surface. The special EDL in the surface of gallium and graphite makes the inhomogeneous distribution of the surface tension around the gallium, which contributes to the transformation (Fig. 14(c)).

Gallium is an amphoteric metal, which is soluble not only in acid generating Ga³⁺, but also in alkali, generating [Ga(OH)₄]⁻

(3)\mathrm{G}\mathrm{a}+{\mathrm{O}\mathrm{H}}^{-}\to {[\mathrm{G}\mathrm{a}{{\displaystyle (\mathrm{O}\mathrm{H})}}_{\mathrm{4}}]}^{-},

which makes liquid metal negatively charged. Therefore, the positive ions in the solution are electrostatically attracted by these anions and an electric double layer is generated at the liquidmetal/electrolyte interface, as shown in Fig. 15(a). According to the Stern’s electric double layer (EDL) theory [⁵⁷,⁵⁸], as sketched in Fig. 15(b), there exist two ion layers, contact layer and diffuse layer, with opposite charges and equal electrical quantity on the interface of two different liquid phases.

In addition, owing to the larger density of liquid metal, when a droplet of liquid metal is placed in NaOH solution, it presents ellipsoid with an evident boundary. The force of the kinetic friction between liquid metal and the substrate is small because of the few contact areas.

Movement theory

It is discussed in Sub-section 2.1 that when a conducting drop (e.g. gallium) suspended in unbounded electrolyte is exposed to a uniform electric field, it migrates along the field direction and deforms to achieve equilibrium. The electrohydrodynamic effect on the drop is commonly associated with the electrical double layer, which plays a significant role in the interaction between the drop and the electrolyte. Differed by the conductivity, the physical models of drop can be divided into the perfect dielectrics model, the leaky dielectrics model, and the conducting drop model [⁵⁹].

The development of conducting drop model is summarized in Table 2. von Smoluchowski [⁶⁰] made a tremendous impetus on the theory of electroosmosis and electrophoresis by presenting the formula of the electrophoretic velocity of solid-particle electrophoresis and electrolyte drops. Likewise, Frumkin [⁶¹] conducted the first analysis of liquid metal drops in weak applied fields. Booth [⁶²] then studied spherical fluid drops and concluded a same electrophoretic velocity for all dielectrics. In 1962, Levich and Rice [⁶³] modified the velocity for highly conducted liquid drops, multiplying the Smoluchowski’s scale by δ^–1, and confirmed the Frumkin-Levich velocity scaling experimentally.

To acquire the theory solution, O’Brien and White [⁸] put forward the weak-field linearization when studying electrophoretic mobility of colloidal particles, which was applied by Ohshima et al. [⁶⁴], who made the first systematic analysis of charged liquid metal, independent of Debye thickness and zeta potential. To overcome the limit of weak applied field, Schnitzer et al. [⁹] and Schnitzer and Yariv [⁶⁵] simulated the conducting drop using the nonlinear macroscopic model, which was verified in the weak applied field condition.

An uncharged liquid metal drop in unbounded electrolyte in the presence of applied electric field is an equipotential body since electric field is unable to penetrate the drop, resulting in no tangential Maxwell stress exerting on it. Therefore, viscous force arising from the electrolyte is the propulsive force for the drop.

Liquid metal drop suspended in the electrolyte in the presence of an electric field has the following characteristics: The Reynolds numbers of the liquid flows outside and inside the drop are small enough to allow ignorance of inertial terms in the Navier-Stokes equation while the liquids can be regarded as incompressible. Considering the highly polarization of liquid metal drops, it is reasonable to assume that electrostatic charge, field or current do not exist inside the drop and that the drop surface is always equipotential. The drop remains spherical throughout the entire process of motion under the field E. This assumption is satisfied when the surface tension is sufficiently large.

In the study of fluid mechanics, it is crucial to select the proper dimensionless number. Unlike the traditional capillary number used to describe the relationship between the viscosity force and the surface tension, the number of electric capillary number {{\displaystyle Ca}}_E is put forward to describe the relationship between the electric field stress and the surface tension [⁶⁶–⁶⁸], i.e.

(4){{\displaystyle Ca}}_E={\displaystyle \frac{a{{\displaystyle \epsilon }}_{\mathrm{0}}{{\displaystyle E}}_{\infty }^{\mathrm{2}}}{\sigma }},

where E_∞ is applied electric field, σ is the surface tension of gallium, ε₀ is the dielectric constant of gallium in vacuum, a is the diameter of the drop, and {{\displaystyle Ca}}_E serves as a criterion to predict the possibility of deformation and fracture of dispersed liquid droplets.

The ratio of the density difference between the two phase liquids and the interfacial tension is represented by a non-dimensional parameter Bond number [⁶⁹–⁷¹], which represents the effect of relative gravity of the drop on drop deformation. Bo can be expressed as

(5)Bo={\displaystyle \frac{\Delta \rho g{R}^{\mathrm{2}}}{\sigma }},

where Δρ is the density difference between the two phases, g is acceleration of gravity, and σ represents the interfacial tension coefficient of the two-phase liquid. When the drop radius is smaller than the capillary length, (Bo<1) would be flattened by gravity and form puddles.

Ohnesorge number Oh is used to measure the relationship between viscosity force, inertial force and surface tension [⁷¹–⁷³]. The expression of Oh is

(6)Oh={\displaystyle \frac{\mu }{\sqrt{\rho \sigma R}}},

where σ is surface tension, μ is viscosity, ρ is the density of the ball, and R is the diameter of the ball.

The liquid metal drop and the electrolyte inside the electric double layer are considered separately at first and integrated with boundary conditions belatedly. Considering the fact that the liquid metal drop is scaled by millimeters or centimeters and the velocity is less than 1m/s, the Reynolds numbers of the liquid flows outside and inside the drop are small enough to allow ignorance of inertial terms.

For liquid metal drop, the steady dynamics equations are expressed as

(7)\nabla \cdot \mathit{\mu }=\mathrm{0},

(8)\nabla \mathit{p}=\mu {\nabla }^{\mathrm{2}}\mathit{\mu }+\gamma ,

(9)\gamma ={{\displaystyle \gamma }}_{\mathrm{0}}-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{CU}^{\mathrm{2}},

where u is the velocity field of liquid metal, p is drop pressure field, {{\displaystyle \gamma }}_{\mathrm{0}} is maximum surface tension value, \gamma is the surface tension of the liquid metal under the electric voltage U applied on the liquid metal/solution interface, and C is the capacitance per unit area of the EDL.

For electrolyte, the steady dynamics equations are expressed as

(10)\nabla \cdot \mathit{\mu }=\mathrm{0},

(11)\nabla \mathit{p}={\nabla }^{\mathrm{2}}\mathit{\omega }+{\nabla }^{\mathrm{2}}\phi \nabla \phi ,

where \phi is electric potential, u is electrolyte velocity field, and p is electrolyte pressure field.

To acquire the electric potential distribution, the electric field equation sets need to be solved:

(12)j^{\pm }=-\nabla c^{\pm }\pm c^{\pm }\nabla \phi ,

(13)\nabla \cdot j^{\pm }+\alpha \mu \cdot \nabla c^{\pm }=\mathrm{0},

(14)-\mathrm{2}\delta {\nabla }^{\mathrm{2}}\phi =c^{+}-c^{-},

in which j^{\pm } are molecular fluxes, c^{\pm } are ionic concentrations, \phi is electric potential, \alpha is dimensionless drag coefficient, and δ is dimensionless Debye thickness.

Boundary conditions are also derived to combine the equation sets for liquid metal and electrolyte together. First, the normal ionic fluxes vanish at the interface. Second, the electric potential of liquid metal drop is uniform and the total charge of the drop is conservative. Third, tangential velocities of both liquid medium are equal and radial velocity vanishes to zero. Fourth, tangential-stress is balanced at the interface. Fifth, radial force exerted on the drop vanishes along the surface of liquid metal drop. Finally, at the remote place from the drop, ionic concentration, electric field and electrolyte velocity equal the initial values.

Deformation theory

Drop deformation D is defined as

(15)D={\displaystyle \frac{L-B}{L+B}},

where L represents the length of deformation and B represents the breadth. The first and second order in Ca_E can describe the small deformation of a spherical drop, put forward by Taylor [⁷⁴] and later by Ajayi [⁷⁵]. Taylor’s discriminating function F_d can be used to decide whether the drop will deform into a prolate (F_d >0) or an oblate (F_d<0)one.

(16){{\displaystyle F}}_{\mathrm{d}}(R,Q,\lambda )={(\mathrm{1}-R)}^{\mathrm{2}}+R(\mathrm{1}-RQ)\left[\mathrm{2}+{\displaystyle \frac{\mathrm{3}}{\mathrm{5}}}{\displaystyle \frac{\mathrm{2}+\mathrm{3}\lambda }{\mathrm{1}+\lambda }}\right].

In the electrohydrodynamic analysis of the perfect conducting drop, researchers usually hypothesize that the perfect conducting drop is chemically inertness, which does not react with air or the solution. Another ideal assumption is that the perfect conducting drop is suspended in the solution, ignoring the gravity. The surface tension of the liquid metal is generally considered constant to simplify the analysis. Hence, the capillary number (Eq. (4)) which is used to identify deformation is small. Therefore, the liquid metal drop remains spherical in researches as mentioned above, ignoring the fact that surface tension varies with the transformation of the charge distribution on the surface. However, surface tension (Eq. (9)) remains a significantly important parameter to decide the deformation pattern of the liquid metal. The pressure difference between the electrolyte and the liquid metal droplet p (pressure of Galinstan is higher) can be obtained from the Young-Laplace equation

(17)p=\mathrm{2}\times {\displaystyle \frac{\gamma }{r}},

where r is the radius of the liquid metal droplet.

Under applied external electric field, the surface tension changes with the charge redistribution on the surface and subsequently alters the force acting on the liquid metal (Fig. 16). The imbalance of the surface tension g induces a pressure difference Δp between the downstream and upstream hemispheres of the droplet. The unbalanced surface tension will decrease with the increasing voltage, rapidly increasing the dimensionless number Ca to enable evident deformation.

In fact, gallium-based liquid metal is reactive with oxygen, generating a thin layer of gallium oxide with a smaller surface tension and a higher viscosity. Gallium-based liquid metal has a density six times larger than the aqueous solution, allowing it to sink to the bottom of the container. Therefore, huge discrepancies have been found on the deformation of gallium-based liquid metal and the ideal conducting drop.

When the liquid metal contacts with an anode in NaOH solution, electrochemical oxidation occurs and immediately a layer of oxide is formed. The oxide film decreases the surface tension and then alters the contact angle at the interface between liquid metal and electrolyte at once, as Fig. 17 depicts [⁷⁶]. Liquid metal sphere spreads out to an asymmetric layer, as the surface area becomes almost five times that of the original one. The explanation lies in the fact that gallium around the anode loses electrons and reacts with oxygen, generating gallium oxide. Therefore, the surface tension is weaker and the gravity is thus relatively dominant. To retain balance, the liquid metal deforms itself to decrease the effect of gravity. Figure 18 suggested that the liquid metal immersed in the alkaline solution placed on the graphite substrate should display a flat pancake without the electric field [²¹]. The predominant core is the transference of electrons from liquid metal to graphite, leaving liquid metal positively charged and easily oxidized. Sequentially, an oxide film is formed on the surface of the liquid metal, which highly resembles mud.

Although the movement of the gallium-based liquid metal is similar to the conducting drop model, the deformation theory is actually different.

First, the deformation mechanism is different. As shown in Fig. 19, the deformation of the gallium-based liquid metal is mainly resulted from the chemical reaction which evidently decreases the surface tension. Once the surface tension is reduced, liquid metal will expand itself to make more contact with the container. However, the deformation of the ideal conducting drop is mainly ascribed to the inhomogeneous surface charge distribution which is induced by the electric field.

Next, the difficulty of deformation is different. The deformation of gallium-based liquid metal occurs quickly requiring a lower voltage, ascribable to the active chemical oxidation reaction. However, to achieve visible deformation, a high electric field is required to make the Coulomb force at both ends of the ideal conducting drop big enough.

Finally, the deformation scale is different. Liquid metal can transform itself from an ellipsoid to a flat surface under proper electric field, whose surface expands over 1000 times. In contrast, the ideal conducting drop only deforms from sphere to ellipsoid.

Simulation

With applied electric field, the fluidic movement contributes to the transport and redistribution of internal charges, which, in turn, affects the electric field distribution in the flow field. For the discrepancy in physical properties of the two-phase fluid, the electric field gradient on the interface is discontinuous under external field. Conductive droplets induce spontaneous electric field, which can be coupled with the original electric field, generating tangential electric force on the interface while inducing fluid motion along the boundary. Under different combinations of the electric field, the net charges and fluid properties, prolate or oblate-shaped deformations as well as circulation flows will be induced. Taylor [⁷⁴] proposed the well-known leaky-dielectric model, whose accuracy was validated for small deformations. He put forward a parameter \psi to determine the extent of drop deformation

(18a)

(18b)\psi =\mathrm{0},\mathrm{spherical},

(18c)\psi >\mathrm{0},\mathrm{prolate},

Where \psi is defined as

(18d)\psi (R,S,k)={\displaystyle \frac{S}{{(R+\mathrm{2})}^{\mathrm{2}}}}\cdot \left[{\displaystyle \frac{R^{\mathrm{2}}+\mathrm{1}}{S}}-\mathrm{2}{\displaystyle \frac{\mathrm{3}\cdot \left({\displaystyle \frac{R}{S}}-\mathrm{1}\right)(\mathrm{3}k+\mathrm{2})}{\mathrm{5}k+\mathrm{5}}}\right],

where R is conductivity ratio, S is permittivity ratio, and k is the viscosity ratio of the drop to the surrounding fluid. He also suggested the criteria for the flow direction on the interface

(19a)

(19b)\mathrm{R\; >\; S,\; from\; the\; equator\; to\; the\; pole}.

Torza et al. [⁷⁷] gave the first quantitative experimental results on a single drop deformation, and extending Taylor’s theory to conditions of oscillatory (up to 60 Hz) fields. Alternating fields endowed the deformation an oscillatory part, apart from the steady part. However, disappointing quantitative agreement with Taylor’s theory was also reported when large deformation occurred, which boiled down to the electrokinetic effects, demanding more accurate dielectric constants and other properties.

In this paper, the deformation and movement of conductive droplets are focused on, which are dispersed in immiscible surrounding medium under uniform electric field. Numerous efforts on theoretical studies have been made to simulate the electrohydrodynamic phenomena of two-phase flow and different numerical simulation methods have been developed to mimic the liquid metal/aqueous solution interface, such as the VOF method, the LBM method, the boundary integral method, the finite-element method, and the front-tracking method.

VOF method

The volume-of-fluid (VOF) method uses a marker function F to track the fluid volume changes in a computational cell, anywhere between zero and one indicating a free boundary while at either ends representing a single phase. F satisfies [⁷⁸]

(20){\displaystyle \frac{\partial F}{\partial t}}+\nabla \cdot (UF)=\mathrm{0.}

The accuracy and the versatility of the method were verified by Tomar et al. [⁷⁹], who proposed the coupled level set and volume-of-fluid (CLSVOF) algorithm to simulate the electrohydrodynamic behaviors of two-phase flows, combining the virtues of both schemes. The electric field equations for perfectly dielectric liquids and conducting liquids are

(21)\nabla \cdot \mathit{J=}\nabla \cdot (\sigma \mathit{E}\mathrm{)=0,}

(22)\nabla \cdot \mathit{v}=\mathrm{0.}

respectively, while the hydrodynamic governing equations are

(23)\rho ({{\displaystyle H}}_{\delta })\left({\displaystyle \frac{\partial \mathit{v}}{\partial t}}+\mathit{v}\cdot \nabla \mathit{v}\right)=-\nabla p+\nabla \cdot [\mu ({{\displaystyle H}}_{\delta })(\nabla \mathit{v}+{\nabla \mathit{v}}^T)]+\rho ({{\displaystyle H}}_{\delta })g+{{\displaystyle f}}_v^{\gamma }+{{\displaystyle f}}_v^E,

where J = σE is current density, \mathit{v}=(\mu ,v) is velocity vector, t is time, p is pressure, g is gravitational acceleration, {{\displaystyle f}}_v^{\gamma } is the surface tension force which equals zero except in interfacial transition area, {{\displaystyle f}}_v^E is electric field force, ρ(H_δ) is density, and μ(H_δ) is dynamic viscosity,

(24)\rho ({{\displaystyle H}}_{\delta })={{\displaystyle \rho }}_{\mathrm{1}}{{\displaystyle H}}_{\delta }+(\mathrm{1}-{{\displaystyle H}}_{\delta }){{\displaystyle \rho }}_{\mathrm{2}},

(25)\mu ({{\displaystyle H}}_{\delta })={{\displaystyle \mu }}_{\mathrm{1}}{{\displaystyle H}}_{\delta }+(\mathrm{1}-{{\displaystyle H}}_{\delta }){{\displaystyle \mu }}_{\mathrm{2}},

To solve the governing equations, ρ(H_δ) and μ(H_δ) are interpolated using a smoothed Heaviside function which indicates different fluids at value 1 or 0 and varies smoothly in the transition region

(26)

where 2δ is transition region thickness and ϕ is a distance function which equals 0 at the interface.

The research shows that under the effect of electric field, the differences of the physical properties between the conducting drop and the surrounding fluid lead to the surface deformation and different forms of free charge redistribution on the interface. As exhibited in Fig. 20, for small deformations, the predicted deformation agrees well with the simulation results.

LBM method

Shan and Chen [⁸⁰]developed the Lattice Boltzmann method to simulate flows containing multiple phases and components, which was widely employed in further multiphase/multicomponent or interfacial studies. The Lattice Boltzmann method is generally adopted to model complicated boundary conditions and multiphase/multicomponent interfaces, with no need for special treatment on the interface.

Zhang and Kwok [⁸¹] applied the 2-dimensional, 9-velocities Lattice Boltzmann method to simulate the drop deformation under electric field with the leaky dielectric theory. They solved the following equation

(27)\rho {{\displaystyle f}}_i^{(k)}(\mathit{x}+{{\displaystyle \mathit{e}}}_i,t+\mathrm{1})-{{\displaystyle f}}_i^{(k)}(\mathit{x},t)=-{\displaystyle \frac{{{\displaystyle f}}_i^{(k)}(\mathit{x},t)-{{\displaystyle \overline{f}}}_i^{(k)}(\mathit{x},t)}{{\tau }^{(k)}}},

where {{\displaystyle f}}_i^{(k)}(\mathit{x},t) is the number density distribution of the kth component in the ith lattice velocity direction {{\displaystyle \mathit{e}}}_i at position x and time t, {\tau }^{(k)} is the relaxation time, {{\displaystyle \overline{f}}}_i^{(k)}(\mathit{x},t) is the corresponding equilibrium distribution. The lattice velocities are given by

(28){{\displaystyle e}}_{\mathrm{0}}=\mathrm{0},

(29){{\displaystyle e}}_i=\left(\cos {\displaystyle \frac{i-\mathrm{1}}{\mathrm{2}}}\pi ,\sin {\displaystyle \frac{i-\mathrm{1}}{\mathrm{2}}}\pi \right)(i=\mathrm{1}-\mathrm{4}),

(30){{\displaystyle e}}_i=\sqrt{\mathrm{2}}\left(\cos {\displaystyle \frac{\mathrm{2}i-\mathrm{9}}{\mathrm{4}}}\pi ,\sin {\displaystyle \frac{\mathrm{2}i-\mathrm{9}}{\mathrm{4}}}\pi \right)(i=\mathrm{5}-\mathrm{8}).

To implement LBM to electrohydrodynamics, it is assumed that the mixed dielectric properties are

(31)ϵ=ϵ{}^{(\mathrm{1})}{\rho }^{(\mathrm{1})}+{ϵ}^{(\mathrm{2})}{\rho }^{(\mathrm{2})},

(32)\sigma ={\sigma }^{(\mathrm{1})}{\rho }^{(\mathrm{1})}+{\sigma }^{(\mathrm{2})}{\rho }^{(\mathrm{2})},

where ϵ⁽ⁱ⁾, σ⁽ⁱ⁾ and ρ⁽ⁱ⁾ are the permittivity, conductivity, and density of different phases. The motion characteristics of the two-phase flow depend on R= σ_in/σ_out and S = ϵ_in/ϵ_out, among which the flow direction is determined only by the relative quantities of R and S, not directly related to drop deformation. The relations of drop deformation, property ratios, and velocity vectors are displayed in Figs. 21 and 22.

Boundary integral method

Miksis [⁸²] developed a boundary integral method to calculate the shape of a suspended dielectric drop immersed in an unbounded medium under the influence of a steady electric field. Similar approach was applied by Sherwood [⁸³] to study deformation and breakup of a single drop under the effect of electric fields, who solved both Laplace’s equation for the applied electric field and the Navier-Stokes equations for the fluid motion. Tip-streaming from drops with pointed ends and division of the drop into two blobs connected by a thin thread were observed and predicted by using numerical methods. Baygents et al. [⁸⁴]employed the same technique to study the motion of two drops with no native charge in a uniform electric field using the leaky dielectric model, giving a boundary integral formulation of the electric field, which is represented as a surface distribution of dipoles.

Applying the boundary integral method to calculate the electric and velocity field, Lac and Homsy [⁸⁵] studied the deformation and stability of a neutrally buoyant and initially uncharged drop in another solution under the effect of a uniform electric field. To solve hydrodynamic equations for the calculation of the interfacial velocity, stress distribution needs to be determined beforehand, which requires knowledge of the external electric field on the surface of the drop S. Surface distribution of dipoles are extensively used to represent the electric field in the boundary integral method [⁸⁶]

(33)E^{\infty }+{\displaystyle {\propto }_s}{\displaystyle \frac{r}{\mathrm{4}\pi r^{\mathrm{3}}}}\left[{{\displaystyle E}}_n(y)-{{\displaystyle \overline{E}}}_n(y)\right]dS(y)=\{\begin{array}{l}\overline{E}(x)\\ \mathrm{0.5}\left[E(x)+\overline{E}(x)\right]\\ E(x)\end{array}\begin{array}{l}\mathrm{i}\mathrm{f}\hspace{0.17em}x\in {{\displaystyle V}}_{\mathrm{1}},\\ \mathrm{i}\mathrm{f}\hspace{0.17em}x\in \mathit{S},\\ \mathrm{i}\mathrm{f}\hspace{0.17em}x\in {{\displaystyle V}}_{\mathrm{2}}.\end{array}

Bring in the boundary condition, an integral equation for E_n can be written as

(34)E^{\infty }\cdot \mathit{n}(x)+{\displaystyle \frac{\mathrm{1}-R}{\mathrm{4}\pi }}{\displaystyle {\propto }_{\phi }}{\displaystyle \frac{\mathit{r}\cdot \mathit{n}(x)}{r^{\mathrm{3}}}}{{\displaystyle E}}_n(y)\mathrm{d}S(y)={\displaystyle \frac{\mathrm{1}+R}{\mathrm{2}}}{{\displaystyle E}}_n(x)

and E_t as

(35){{\displaystyle E}}_t={\displaystyle \frac{\mathit{E}+\overline{\mathit{E}}}{\mathrm{2}}}-{\displaystyle \frac{\mathrm{1}+R}{\mathrm{2}}}{{\displaystyle E}}_n\mathit{n}.

Finite-element method

Tsukada et al. [⁸⁷] conducted simulations of the induced circulations inside and outside a stationary suspended deformable dielectric drop with the Galerkin finite-element scheme [⁸⁸], which subdivided the axisymmetric problem domain into quadrilateral elements (Fig. 23). It is found that when Ψ = 0, the suspended drop remains spherical even when the dimensionless uniform electric field E₀* = 5. For a silicone oil drop immersed in vegetable-oil mixture, where Ψ<0 and the drop shape is oblate, the inner potential gradient of the drop is larger than the outside potential gradient of the drop. The induced circulation flows from the pole to the equator. For an opposite situation, where Ψ>0 and the drop shape is prolate, the process goes into reverse. The simulation results showed good agreement with the theory of Taylor’s [⁷⁴] for small drop deformations. Conducting the same simulating scheme, Feng and Scott [⁸⁹]considered a deformable leaky dielectric fluid drop suspended in an immiscible liquid subjected to an electric field and illustrated the deformation phenomena of the drop at finite Reynolds number and different viscosities. The work extended Taylor’s linear asymptotic results by introducing nonlinearities which are ascribable to large deformation.

Front-tracking method

Front-tracking method is a numerical technique to solve the Stokes equations for the fluid flows, using connected marker points to track the boundary of the fluids, or the front, where generally the deformation and charge redistribution take place. Fernández et al. [⁹⁰] studied the effects of an electrostatic field on the distribution of drops in a channel flow using the front tracking method Tryggvason et al. [⁹¹] combined the front-capturing and front-tracking schemes, applying one set of governing equations while treating different phases as one fluid with different physical properties. It turns out that when conducting surface tension computations, despite its complexity, the front explicitly tracked by marker points is more direct and clearer than reconstructing it from a marker function.

Unverdi and Tryggvason [⁹²] developed the finite-volume/front tracking method for incompressive flows of multi-phases fluids to explicitly track the interface while taking into consideration the surface tension and keeping the density and viscosity field sharp. The identical scheme was conducted by Hua et al. [⁹³] to simulate the deformation and motion of the two-phase drop suspension system. The leaky dielectric model, the perfect dielectric model, and the constant surface charge model were applied, for the comprehensive consideration of the electric field, the net charges, and the properties of the two phases. For leaky dielectric drops, the ratio of electrical conductivity to permittivity decides whether the drops will deform into a prolate or oblate one. The intensity of the electric field-induced circulation flow inside the drop increases with the extent of the drop deformation. However, circulation flows are not obvious in a perfect dielectric drop, which generally deforms into a prolate shape. Drops with constant net charge, which provide resistance to deformation, usually move along the electric field in a prolate shape. (Fig. 24)

Challenges and prospects

Although several breakthroughs have been made in modeling the movement and deformation mechanism of the liquid metal immersed in solution, there are still many challenges because of the unique characteristics of liquid metal and the complexity of applied fields.

(1) Non-uniform potential field. Concerning liquid metal under electric field, a majority of former researches focus on the non-uniform electric field and non-spherical liquid metal, which does not perfectly satisfy existing physical models, resulting in the difficulty of predicting the behaviors. The precise equations of the large-scale deformation, rotation, directional movement and merging of liquid metal in the immiscible fluid remain unknown. To settle these complex fluidic problems, a promising approach is to use computational fluid mechanics to calculate the hydrodynamics of liquid metal according to its characteristics.

(2) Charged liquid metal. In the experiment mentioned previously, liquid metal has been charged by contacting with the electrode, resulting in partly electrophoresis movement, which is different from the ideal conducting drop model. Meanwhile, the change of the electrical potential distribution on the surface of the charged liquid metal has an indescribable effect on the deformation as well. Therefore, the movement and deformation mechanism of the charged liquid metal needs to be further investigated.

(3) Superpositional potential fields. The theoretical researches on hydrodynamics of liquid metal in aqueous solution have been mainly focusing on situations with electric field exerted. Although the movement and deformation of liquid metal can be calculated by simulating the physical model, it is still tough to quantitate the deformation of the liquid metal, which combines numerous factors, e.g., the electric field, flow field, and chemical field. The specific mechanisms of acoustic, optic, magnetic, thermal and other potential fields on hydrodynamics of liquid metal still lack sufficient researches. It is highly expected to find a new method to control liquid metal in the aqueous solution to achieve more complex applications.

(4) Internal flow field image. The observation of the liquid metal internal flow field is significantly important on constructing the movement and deformation mechanism. However, the internal flow field image has not yet been acquired because of the opaque property while only surface changes are visually.

The theoretical breakthroughs of the hydrodynamics of liquid metal will bring a significant impact on building deformable liquid machines and further on promoting soft robots. Liquid metal is expected to show extraordinary potential in fields of enhanced heat dissipation, soft robot, biomedicine, and ink-jet printing.

Conclusions

Liquid metal, endowed with both the outstanding fluidity of fluids and the metallic characteristics, has been proven promising in various fields, including both industrial manufacture and scientific studies. This review summarizes the unconventional hydrodynamics of the liquid metal immersed in immiscible solution under applied fields. Interesting experiment results are introduced to describe the hydrodynamic phenomena of the liquid metal. Physical models are then provided to qualitatively explain the movement and deformation behavior in the electric field. Finally, several simulation methods are suggested to solve the physical model. However, issues related to non-uniform potential fields, charged drops, coupled fields, internal streamline, etc., remain unsolved. Tremendous efforts ought to be made to precisely predict the movement and deformation of liquid metal immersed in solution, which is the premise of manufacturing and controlling a liquid metal soft robot.

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