Liquid metal immersed in an electrolyte develops a net surface charge as a result of ion transfer across the interfacial boundary. This charge attracts opposite ions from the surrounding electrolyte, resulting in the formation of an electrical double layer (EDL) [14]. The surface tension *γ* of the liquid metal is a function of the EDL, as described by the Young-Lippman equation:$\gamma ={\gamma}_{o}-\frac{1}{2}{\mathit{CV}}_{EDL}^{2}\text{,}$1

where *γ*_{o} is the surface tension minus electrical influence, *C* is the capacitance per unit area of the EDL, and *V*_{EDL} is the voltage across the EDL [14]. This phenomenon, where surface tension is influenced by electrical conditions at the liquid-liquid boundary, is known as electrocapillarity [15].

The electrical control of surface tension can also adjust the pressure at the interfacial boundary. A pressure differential Δ*p* naturally exists between the liquid metal and electrolyte, and is related to the surface tension *γ* by the Young-Laplace equation:$\Delta p=\gamma \left({\scriptscriptstyle \frac{1}{{R}_{1}}}+{\scriptscriptstyle \frac{1}{{R}_{2}}}\right)\text{,}$2

where *R*_{1} and *R*_{2} are the principal curvature radii [16]. Liquid metal in an electrolyte-filled channel has a near-180° contact angle with the channel walls [14], so the pressure is higher within the liquid metal relative to the electrolyte. CEW actuation takes advantage of this relationship by creating a surface-tension gradient across the length of a liquid-metal slug, resulting in a pressure imbalance from one end of the slug to the other and inducing motion [14], [17].

In ECA, the EDL voltage is altered by directly applying a DC bias between the liquid metal and electrolyte. This has the same low-voltage and low-power advantages as CEW, but allows for the liquid metal to be deformed and shaped instead of simply moved from point to point. Previous attempts at utilizing this technique resulted in only minor deformation [10] due to the low surface-area-to-volume ratio of the liquid metal. Here we show how these deformations can be greatly increased by maximizing this ratio.

In the absence of external forces, the surface tension of a liquid droplet works to minimize the surface area, and thus the free surface energy, of the liquid by forming a sphere [16]. Any further increase in this surface area *dS* requires external work equaling *αdS*, where *α* is the surface tension of the liquid [16]. If the deformation of a spherical liquid-metal droplet under electrical manipulation is approximated as a transformation of the sphere into a prolate spheroid, the new liquid-metal surface area *S* can be calculated by${S}_{\mathit{spheroid}}=2\pi {b}^{2}\left(1+\frac{a}{\mathit{be}}{sin}^{-1}e\right)\text{,}$3where *a* and *b* are the semi-major and semi-minor axes respectively, and *e* is the eccentricity of the spheroid, defined as (ref. [18])$e=\sqrt{1-{\left(\frac{b}{a}\right)}^{2}}\text{.}$4

The axes *a* and *b* are further related by the finite volume *V* of the liquid metal,$b=\sqrt{\frac{3V}{4\pi a}}\text{.}$5

Using this prolate spheroid approximation, the deformation of the liquid metal can be characterized as a change to the length of the semi-major axis *a*, and the resulting increase in the surface area of the spheroid can be calculated.

Now, consider the same volume *V* of liquid metal, trapped between two plates separated by height *h*, where *h* is much smaller than the original spherical radius *r*. The surface tension minimizes the free surface energy of the liquid metal as before, but with the new quasi-planar boundary conditions imposed by the plates, the optimum shape is now a flattened cylinder with height *h* and facial radii *R* (the curvature of the walls of the cylinder is assumed to be negligible compared to *R*). The surface area of this cylinder can be expressed generally as${S}_{\mathit{cylinder}}=2\pi {R}^{2}+Ph\text{,}$6where the general term *P* is used for the perimeter. In the minimum-energy case *P* is the circumference of the cylinder, *2πR*.

The majority of the surface area of the cylinder is comprised of the circular faces at each of the two planar boundaries. Furthermore, if *h* is held constant and the deformation of the liquid metal is limited such that the equivalent semi-minor spheroid axis *b* is always much greater than *h*, then the area of these faces remains constant. That is, deformation of the liquid metal transforms these circles into ellipses and increases the perimeter length *P*, but does not alter the area of the liquid metal in contact with each plate.

This is significant because the product *P* × *h* is a small percentage of the overall surface area of the cylinder, so even dramatic liquid-metal deformations require only a very small increase in surface energy. This effect is illustrated in Figure , which plots the extent that the semi-major axis of the liquid metal can be elongated as a function of the required change in normalized surface energy. The smaller the height *h* between the planar boundaries becomes, the smaller the excess surface energy that is required for deformation. This in turn makes the liquid metal much more responsive to perturbation: for example, given a constant change in surface energy, a volume of liquid metal flattened such that its cylindrical height *h* becomes 10% of the original spherical radius will elongate approximately 40 times further than it would if the same surface energy change was applied to the original sphere.

Figure 1 **Effects of confinement on surface energy increase required for deformation. (a)** A finite volume of liquid metal (or any liquid) minimizes its surface energy by contracting into a sphere with radius *r*. Flattening this sphere leads to a cylinder with facial radius *R* and perimeter *P*. **(b)** Elongating the liquid sphere can be approximated as transforming it into a prolate spheroid whose semi-major axis *a* is longer than both the original radius *r* and the semi-minor axis *b*. This action necessarily involves increasing its surface area, and thus its free surface energy. Similarly distorting the cylinder will transform it into an ellipse, but its semi-major axis will lengthen by a much larger amount for the same change in overall surface energy. **(c)** Calculated maximum deformation of a liquid-metal spherical droplet with a 1-mm radius as it is compressed into a cylinder of varying heights. As the liquid metal is flattened it is capable of much greater deformation for a given increase in its surface energy.

Figure shows a top-view illustration of a liquid-metal droplet trapped in a quasi-planar (*R > > h*) reservoir filled with an alkaline electrolyte. The reservoir is connected to a long, narrow channel of the same height, which is also filled with the electrolyte. A smaller pressure-relief channel extends from the reservoir in the opposite direction, which allows the liquid metal to flow into the larger channel without creating negative pressure at its trailing edge. In ambient conditions the liquid metal is trapped in the reservoir, as opposing capillary forces block flow into either channel.

Figure 2 **Electrocapillary actuation of liquid metal. (a)** Liquid metal resting in a quasi-planar reservoir exerts no lateral pressure. **(b)** A positive DC voltage creates a potential gradient on the liquid-metal surface near the primary channel entrance, resulting in a corresponding surface-tension gradient and generating Marangoni forces. The liquid metal now exerts a positive pressure on the channel entrance that is opposed by the channel’s capillary pressure. **(c)** Above the critical voltage threshold, the pressure exerted by the liquid metal exceeds the capillary pressure, and the liquid metal enters the channel. Flow is facilitated by the pressure-relief channel at the opposite end of the reservoir, which prevents a negative pressure build-up at the trailing edge of the liquid metal.

At the end of the primary channel opposite the reservoir, a positive voltage is applied to the electrolyte relative to the liquid metal. As the electrolyte is semi-conductive, a potential gradient is established within the channel and along the interface between the electrolyte and liquid metal. This creates a surface-tension gradient along the interfacial boundary as described by (1), with the lowest surface tension at the channel entrance. Marangoni forces (*F*_{Mar}) that develop along this interface generate a flow of the electrolyte away from the channel entrance, and exert negative displacement pressure on the liquid metal, pulling it flush across the channel entrance. Entrance of the liquid metal into the channel is opposed by the capillary pressure of the channel; this additional pressure threshold must be overcome if flow is to be initiated.

From (2), the pressure discontinuity *p* of the liquid metal relative to the surrounding electrolyte within the reservoir can be described as:${p}_{\mathit{reservoir}}={\gamma}_{t=0}\left(\frac{1}{R}+\frac{2}{h}\right)\text{,}$7where *γ* is the initial surface tension prior to electrical actuation, and *R* and *h* are the radius and height of the reservoir, respectively. The vertical curvature radius can be approximated as half the reservoir height because the contact angle at the liquid-liquid–solid interface is approximately 180°.

To induce flow into the channel, the pressure exerted by the liquid metal must exceed the capillary resistance of the channel. The pressure differential of the liquid metal within the channel once this occurs is:${p}_{\mathit{channel}}=2{\gamma}_{1}\left(\frac{1}{w}+\frac{1}{h}\right)\text{,}$8where *γ*_{1} is the reduced surface tension resulting from the applied bias potential and *w* and *h* are the width and height of the channel, respectively. At the threshold point, these pressures are equal, and so by combining (7) and (8) we can determine the normalized surface tension at which flow is induced:$\widehat{\gamma}=\frac{{\gamma}_{1}}{{\gamma}_{t=0}}=\frac{w\left(h+2R\right)}{2R\left(h+w\right)}\text{.}$9

This equation can be re-written in terms of the channel aspect ratio *h/w*, and approximated for conditions when *h/2R* is negligibly small:$\widehat{\gamma}=\frac{1}{\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$w$}\right.+1}\cdot \left(1+\frac{h}{2R}\right)\approx \frac{1}{\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$w$}\right.+1}\text{.}$10

Figure shows this normalized surface tension plotted as a function of channel geometry for a reservoir radius of 5 mm. For channels that have a small aspect ratio (*h/w*), very little reduction in surface tension is required to induce flow: for a channel with a width of 3 mm and a height of 0.1 mm, a normalized surface tension of approximately 97% will result in liquid-metal deformation due to electrocapillary actuation. As the channel aspect ratio is increased, a greater reduction in surface tension is required, which in turn requires higher actuation voltages. The relationship between normalized surface tension and channel aspect ratio, approximated for cases in which *h/2R* is negligibly small, is plotted as an inset in Figure .

Figure 3 **Normalized surface tension required for flow of liquid metal.** Minimum normalized surface tension required to initiate flow for liquid-metal in a cylindrical reservoir into a channel of a given height and width. The assumed reservoir radius is 5 mm. Inset: approximate normalized surface tension required for flow as a function of channel aspect ratio *h/w*.