Surface tension of liquid metal: role, mechanism and application

Abstract

Surface tension plays a core role in dominating various surface and interface phenomena. For liquid metals with high melting temperature, a profound understanding of the behaviors of surface tension is crucial in industrial processes such as casting, welding, and solidification, etc. Recently, the room temperature liquid metal (RTLM) mainly composed of gallium-based alloys has caused widespread concerns due to its increasingly realized unique virtues. The surface properties of such materials are rather vital in nearly all applications involved from chip cooling, thermal energy harvesting, hydrogen generation, shape changeable soft machines, printed electronics to 3D fabrication, etc. owing to its pretty large surface tension of approximately 700 mN/m. In order to promote the research of surface tension of RTLM, this paper is dedicated to present an overview on the roles and mechanisms of surface tension of liquid metal and summarize the latest progresses on the understanding of the basic knowledge, theories, influencing factors and experimental measurement methods clarified so far. As a practical technique to regulate the surface tension of RTLM, the fundamental principles and applications of electrowetting are also interpreted. Moreover, the unique phenomena of RTLM surface tension issues such as surface tension driven self-actuation, modified wettability on various substrates and the functions of oxides are discussed to give an insight into the acting mechanism of surface tension. Furthermore, future directions worthy of pursuing are pointed out.

Graphical abstract

Keywords

surface tension / liquid metal / soft machine / printed electronics / electrowetting / self-actuation

1 Introduction

Surface tension is an inherent characteristic of surfaces and interfaces of materials. The interfacial motion of fluid induced by surface tension plays a fundamental role in many natural and industrial phenomena [1].The surface and interfacial properties of liquid metals are of great importance in metallurgical industry for controlling the processes of casting, welding and solidification [2–6]. For example, the interfacial properties are key parameters to understand slag entrainment and determine the complexity to separate impurities from liquid steel [7]. To optimize these processes, a deep understanding of surface properties is crucial, especially surface tension which dominates the mass transport procedure of fluids. Surface tension is an energy source required to change the surface area of a material [8]. It decides the shape and other characters of the liquid steel on a free surface or an interface where the liquid phase is in contact with the atmosphere or other condensed phase such as inclusion, slag, and refractory, etc [9]. As a consequence, it is practically essential to probe into the behaviors of surface tension of liquid metals.

The surface tension of a liquid metal is not only technologically important but also a scientifically interesting parameter. Distinctive from the common liquid metal whose melting temperature is extremely high, the room temperature liquid metal (RTLM) typically consisted of gallium-based alloys possesses a very low melting point thus could maintain a liquid phase at room temperature. This special material combines the virtues of liquid, such as fluidity, flexibility, and deformability with the advantages of metals, such as high electrical conductivity, heat conductivity, etc [10]. Though mercury is the most well-known RTLM, its further applications are limited by its serious toxicity. In comparison with mercury, gallium-based alloys have an extremely low toxicity and all components have a very low vapor pressure at elevated temperature, on the order of 10⁻⁹ atm at 538°C and nearly zero pressure at room temperature [11]. RTLM is a chemically benign metal alloy [12] and is, meanwhile, compatible with most metals and plastics. Owing to these outstanding physical merits, RTLM is believed to be an alternative material to realize numerous functions [13]. RTLM has been applied in a wide variety of significant areas, as depicted in Fig. 1, including chip cooling [14–19], optical switches [20–22], flexible sensors [23–27], stretchable antennas [28–31], micro-pump [32,33], bio-medical technology [34–36], drug delivery [37,38], flexible electronics [39–41], and 3D printing [42,43], etc.

Recently, liquid Ga-In alloy has been found to shift automatically in solution like a mollusk without any external energy support only by eating a piece of aluminum [44]. The driving mechanism is related to the imbalance of the surface tension induced by the asymmetric chemical reaction between Al and the solution as well as the electrochemical interaction of the eutectic Ga-In alloy and the solution. This phenomenon paves the way for self-driving of liquid metal machine and immediately attracts the attention of researchers to seek after surface tension-driven motors. So far, some external fields such as electrical field [45], magnetic field [46], illumination [47], ionic concentration [48], etc. have been found to effectively affect the propelling behavior of RTLM through intentionally altering its surface tension. These factors present rather promising capability to flexibly manipulate RTLM objects or machines. Therefore, their effects on the surface tension will be mainly discussed in the present paper. In fact, surface and interface properties are crucial in nearly all the applications of RTLM since RTLM has a pretty large surface tension of around 700 mN/m which is nearly ten times that of water [49]. In addition to the RTLM/electrolyte interface, the wettability of RTLM on solid is also one of the core factors that determine the deposition quality in 3D printing and electronics printing when utilizing RTLM as the ink [50]. Therefore the RTLM/substrate interface characters also deserve exploration to obtain satisfied printing quality. Naturally, Ga-based RTLM is quite easy to be oxidized even in a trace amount of oxygen by forming Ga₂O₃ oxides on its surface (~1 nm in dry air) [51–53]. To a large extent, this oxide film restricts the fluidity and flexibility of RTLM which is disfavored for flow task. To solve such an issue, the liquid metal can be manipulated in certain solutions to remove its oxides [54,55] or in a protected atmosphere [56,57] to prevent formation of the oxides. But, on the other hand, surface oxides can change the viscosity as well as the surface tension apparently which can be controlled and reversed via electrical manipulation. Meanwhile, oxidization of RTLM has been identified as an important tool to significantly regulate its surface properties, especially wetting behavior which can then help fulfill various desirable functions in a rather easy going way [58,59]. By this token, the wetting properties of surface oxides of RTLM are also discussed in this paper.

For a precise understanding of these specific performances, a fundamental probe into the surface tension is indispensable. This paper is dedicated to summarizing some main aspects of surface tension including basic theories, influencing factors, and measuring methods. Additionally, it discusses useful tools to modify the surface tension such as electrical technique. Furthermore, it interprets RTLM which is a novel and functional material, involving surface tension issues, to provide an insight into the intrinsic mechanism of numerous miraculous phenomena.

2 Fundamental theory

2.1 Thermodynamic theory

From the perspective of thermodynamics, surface tension correlates closely with Gibbs free energy. For liquid, the required work to extend the surface, which is equal to the increment of Gibbs free energy, is positively related to the increased surface area under certain temperature and pressure. Free surface energy is the free energy per unit area of surface, and can be defined as the work necessary to increase the surface by unit area [60].

The surface tension of liquid decreases with increasing temperature [61], i.e.

(1){\mathit{\gamma }}_T={\mathit{\gamma }}_{\mathrm{m}}+{\displaystyle \frac{\mathrm{d}\mathit{\gamma }}{\mathrm{d}T}}\left(T-T_{\mathrm{m}}\right),

where γ_T is the surface tension at temperature T, T_m is the melting temperature, and γ_m is the surface tension at the melting point, .

According to Gibbs’s thermodynamics theory [62,63], for the unitary liquid metal separated from the vapor phase, the surface tension is given by

(2)\mathit{\gamma }=-{\kappa }_{\mathrm{B}}T{\left({\displaystyle \frac{\partial \ln Z_N}{\partial A}}\right)}_{N,V,T},

where {\kappa }_B is Boltzmann constant, Z_N is the partition function including the surface effect, A is the surface area, N is the particle number, V is the volume, and T is the temperature.

Considering a curved interface with different pressures on each side, the free energy G can be written as

(3)G=\mathit{\gamma }{\int }^{}\mathrm{d}A-p{\int }^{}\mathrm{d}V,

where dA is the surface area element, dV is the volume element, and p is the pressure.

Obtained from the partial differential of Eq. (3), the surface tension reads as

(4)\mathit{\gamma }={\displaystyle \frac{\partial G}{\partial A}}.

According to the Ginzberg-Laudon theory [63], the Gibb’s free energy G is written as

(5)G\left[\varphi \left(\overrightarrow{r}\right)\right]={\int }^{}\left[-{\displaystyle \frac{\epsilon }{\mathrm{2}}}{\varphi }^{\mathrm{2}}\left(\overrightarrow{r}\right)+{\displaystyle \frac{c}{\mathrm{4}}}{\varphi }^{\mathrm{4}}\left(\overrightarrow{r}\right)+{\displaystyle \frac{B}{\mathrm{2}}}{\left|\overrightarrow{\nabla }\varphi \left(\overrightarrow{r}\right)\right|}^{\mathrm{2}}\right]\mathrm{d}\overrightarrow{r},

where \overrightarrow{r} represents the particle position, \varphi \left(\overrightarrow{r}\right) is a state function which quantifies the difference to the average volume fraction at the specific position \overrightarrow{r}, \epsilon =\left(\mathrm{4}{T}_{\mathrm{c}}-\mathrm{4}T\right)/a^{\mathrm{3}},c=\mathrm{16}T/\left(\mathrm{3}{a}^{\mathrm{3}}\right), B=\mathrm{2}{T}_{\mathrm{c}}/a, T_c is the critical temperature, and a is known as the virial coefficient.

To minimize the approximate free energy, the variation of G with respect to \varphi \left(\overrightarrow{r}\right) is calculated. According to Eq. (5), the Gibbs free energy can be expressed as

(6)G={\displaystyle \int g^{\prime }\left[\varphi \right(\overrightarrow{r}),\nabla \varphi ]}\mathrm{d}\overrightarrow{r}.

Therefore, the Lagrange equation of G is

(7){\displaystyle \frac{\delta G}{\delta \varphi \left(\overrightarrow{r}\right)}}={\displaystyle \frac{\partial g^{\prime }}{\partial \varphi }}-{\displaystyle \frac{\partial }{\partial r_i}}{\displaystyle \frac{\partial g^{\prime }}{\partial {\varphi }_{r_i}}}=\mathrm{0},

where {\varphi }_{r_i}=\partial \varphi /\partial r_i, and i represents different dimensions.

With the boundary condition that the system is uniform away from the interface, there is

(8)-\epsilon \varphi +c{\varphi }^{\mathrm{3}}-B{\nabla }^{\mathrm{2}}\varphi =\mathrm{0.}

The surface tension can be finally presented as

(9)\mathit{\gamma }=B{\displaystyle {\int }_{-\infty }^{+\infty }{{\displaystyle \varphi }}_z^{\mathrm{2}}}\mathrm{d}z={\displaystyle \frac{\mathrm{2}\epsilon }{\mathrm{3}c}}\sqrt{\mathrm{2}B\epsilon }.

2.2 Statistics theory

Surface tension indicates the force acting on the liquid surface vertically by unit length, causing the surface area to shrink. In term of mechanics, for the molecules located on the gas-liquid interface, the acting force from the gas phase is much weaker than that from the liquid phase. Such unbalance is counterbalanced by surface tension. Until now, several respectable models have been proposed to calculate interface tension [64–68], where the interface is simplified as a plane without thickness or a monomolecular layer, which is not corresponding to the reality that the thickness of interface is one or several times the molecular diameter.

Functional expansion becomes an effective tool to determine the interface tension. Based on the density functional theory (DFT), the systemic space density ρ(r) is a fundamental variable, and grand thermodynamic potential can be obtained via variation principle [69–72]. Sequentially, other systemic properties can be approached. The primary step of DFT is to construct the grand potential functional Ω[ρ(r)], i.e.,

(10)\mathit{\Omega }\left[\rho \left(r\right)\right]=F\left[\rho \left(r\right)\right]+{\displaystyle \sum }_{i=\mathrm{1}}^{\mathrm{2}}{\int }^{}\rho \left(r\right)\left[E_{\mathrm{ext}}\left(r\right)-{\mu }_i\right]\mathrm{d}r,

where E_ext(r) is the external potential energy, μ_i is the chemical potential of component i, dr is the volume differential, ρ(r) is the density of molecule number, and F[ρ(r)] is the Helmholtz free energy.

If the interface is divided into numbers of ultrathin micro layers, infinitesimal grand potential can be expressed as

(11)\mathrm{d}\mathit{\Omega }=\mathit{\gamma }\mathrm{d}A-p\mathrm{d}V

where dA, and dV respectively represent the area and volume of the interface micro layer, and p is the normal pressure tensor.

Thereby, the surface tension in layer i is γ_i, i.e.,

(12){\mathit{\gamma }}_i={\displaystyle \frac{\mathrm{d}\mathit{\Omega }\left[{\rho }_{\mathrm{0}}\left(r\right)\right]+p\mathrm{d}V}{\mathrm{d}A}}={\displaystyle \frac{\mathrm{d}\mathit{\Omega }\left[{\rho }_{\mathrm{0}}\left(r\right)\right]+p\mathrm{d}V}{\mathrm{d}V}}\mathrm{d}z,

where ρ₀(r) is the density of molecule number in balance and z is the thickness of the selected layer.

The total surface tension is equal to the sum of surface tension in each layer:

(13)\mathit{\gamma }={\displaystyle \sum }_{i=\mathrm{1}}^n\left\{{\displaystyle \frac{\mathrm{\Delta }\mathit{\Omega }\left[{\rho }_{\mathrm{0}}\left(z\right)\right]}{\mathrm{\Delta }V}}+p\right\}\mathrm{\Delta }z\approx {\displaystyle {\int }_{-\infty }^{\infty }\left\{{\displaystyle \frac{\mathrm{d}\mathit{\Omega }\left[{\rho }_{\mathrm{0}}\left(z\right)\right]}{\mathrm{d}V}}+p\right\}\mathrm{d}z}.

In view of thermodynamic formula on the interface, there is

(14)p\left[{\rho }_{\mathrm{0}}\left(z\right)\right]=-{\displaystyle \frac{\mathrm{d}\mathit{\Omega }\left[{\rho }_{\mathrm{0}}\left(z\right)\right]}{\mathrm{d}V}}.

The surface tension can be exhibited as

(15)\mathit{\gamma }=\underset{}{\overset{}{{\displaystyle {\displaystyle {\int }_{-\infty }^{\infty }\left\{p-p\left[{\rho }_{\mathrm{0}}\left(z\right)\right]\right\}\mathrm{d}z}}}},

where p[ρ₀(z)] is the tangential stress tensor in every layer of the interface.

The aforementioned theory is based on local density approximation, which brings about inevitable errors for the fluid with the short-range correlation property. Another more compatible approach is weighted density approximation and the systemic excess free energy F_ex is given by Eq. (16) [73].

(16)F_{\mathrm{ex}}\left[\rho \left(r\right)\right]={\int }^{}\rho \left(r^{\prime }\right)f\left[\overline{\rho }\left(r^{\prime }\right)\right]\mathrm{d}{r}^{\prime }.

2.3 Electrodynamic theory

Liquid metal droplets move towards the anode direction in an electrical field. When some of the momentum of a moving electron is transferred to a nearby activated ion, electromigration occurs [74,75]. Placing Ga-based RTLM in the alkali solution, the gallium reacts with alkali, producing gallate like {\left[\mathrm{Ga}{\left(\mathrm{OH}\right)}_{\mathrm{4}}\right]}^{-}, which causes a negative charge aggregation at the liquid metal droplet surface. Then, the negatively charged surface attracts positive ions of the solution to form a diffuse layer—electrical double layer (EDL) [76]. According to the Stern’s EDL theory [77,78], at the interface between solid and liquid, there exist two ion layers with opposite charges and equal electrical quantity, as sketched in Fig. 2.

Based on the Lippmann electrical theory [79,80], a phase interface is formed when two different phases contact. If the solid phase is a nonionic crystal, the adsorption of the ionic one is corresponding to Lippmann’s equation, which is deduced from the Gibbs adsorption equation (Eq. (17)) [81].

(17)-\frac{\mathrm{d}\mathit{\gamma }}{\mathrm{d}E}=e({\Gamma }_2^{+}-{\Gamma }_2^{-})\mathrm{.}

In the experiment, by testing the effect of the external potential difference E on the surface tension γ, the capillary curve, namely the γ-E relation curve, can be acquired. Here, e is the electron charge, Г₂ is the adsorptive capacity of one composition on the unit interface, and the plus-minus superscript represents the adsorption capacity after and before addition of the electrical field respectively.

The double electrical theory of metal surface tension presents that the EDL composed by positrons and negative electrons produces a potential barrier, and pressure formed by the action force between positive and negative ions shrinks the surface tension. The relationship between surface tension, electron charge, atom distance can be expressed as

(18)\mathit{\gamma }={\displaystyle \frac{\mathrm{4}\mathrm{\pi }Ze}{R^{\mathrm{3}}}},

where Z is the valence electron count, e is the electron charge, and R is the distance between atoms.

The EDL on the surface affects the surface tension which can be implicated by Lippmann’s equation (Eq. (19)) [82,83].

(19)\mathit{\gamma }={\mathit{\gamma }}_{\mathrm{0}}-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}C{U}^{\mathrm{2}},

where γ₀ is the value of maximum surface tension, C is the capacitance per unit area of the EDL, and U is the voltage applied on the liquid metal-solution interface.

When applying an external electrical field, the surface tension changes due to the transformation of the charge distribution on the surface, which further alters the force acting on the liquid metal. The unbalanced surface tension can impel a number of peculiar phenomena which will be described in detail in Section 6.

3 Typical effects on surface tension

3.1 Temperature

Surface tension energy results from the mutual effect of molecules. According to Nogi et al. [84] and Keene [85], the surface tension of metal tightly correlates to the temperature. So when the surface tension is mentioned, the corresponding temperature must be stated. There are no theoretical functions to describe the relationship between surface tension and temperature, except some empirical equations available.

The Eötvös rule [86], named after the Hungarian physicist Loránd Eötvös, makes it possible to predict the surface tension of an arbitrary liquid pure substance at all temperatures. At the critical point, the surface tension is zero. Then, there is

(20)\mathit{\gamma }{V}^{\frac{\mathrm{2}}{\mathrm{3}}}=k_{\mathrm{0}}\left(T_{\mathrm{c}}-T\right),

where V is the molar volume of a substance, T_c is the critical temperature, and k₀ is a constant valid for almost all substances.

Guggenheim [86] has given a variant equation as

(21)\mathit{\gamma }={\mathit{\gamma }}_{\mathrm{0}}{\left(\mathrm{1}-{\displaystyle \frac{T_{\mathrm{c}}}{T}}\right)}^n,

where γ₀ is a constant, and n is an empirical factor.

Van der Waals [86] also proposed this equation and further pointed out that γ₀ could be calculated as

(22){{\displaystyle \mathit{\gamma }}}_{\mathrm{0}}={{\displaystyle K}}_{\mathrm{2}}{{\displaystyle T}}_{\mathrm{c}}^{\frac{\mathrm{1}}{\mathrm{3}}}{{\displaystyle P}}_{\mathrm{c}}^{\frac{\mathrm{2}}{\mathrm{3}}},

where K₂ is nearly constant for all liquids and P_c is the critical pressure of the liquid.

Lu and Jiang [87] have proposed formulas to calculate the surface tension of liquid metals γ and its temperature coefficient {\mathit{\gamma }}^{\prime }\left(T\right), i.e.,

(23)\mathit{\gamma }\left(T\right)=\left({\displaystyle \frac{m^{\prime }{{\displaystyle H}}_{\mathrm{v}}-TS}{w{{\displaystyle N}}_{\mathrm{a}}^{\frac{\mathrm{1}}{\mathrm{3}}}}}\right){{\displaystyle \left[{\displaystyle \frac{{{\displaystyle \rho }}_{\mathrm{L}}\left(T\right)}{M}}\right]}}^{\mathrm{2}/\mathrm{3}},

(24)-{\mathit{\gamma }}^{\prime }\left(T\right)={\displaystyle \frac{\mathit{\gamma }\left(T\right)}{T}}\left[{\displaystyle \frac{\mathrm{1}}{{\displaystyle \frac{m^{\prime }{H}_{\mathrm{v}}}{TS}}-\mathrm{1}}}-{\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}{\displaystyle \frac{T}{{\rho }_{\mathrm{L}}\left(T\right)}}{\displaystyle \frac{\mathrm{d}{\rho }_{\mathrm{L}}}{\mathrm{d}T}}\right],

where m^{\prime }=\left(\mathrm{2}-k_{\mathrm{1}}-\sqrt{k_{\mathrm{1}}}\right)/\mathrm{2}, k₁ is the ratio of the coordination number between surface atoms, H_v is the heat of evaporation, S is surface entropy, w=\left({\displaystyle \frac{\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{3}}}\right)\times {\left({\displaystyle \frac{\mathrm{6}\eta }{\mathrm{\pi }}}\right)}^{\frac{\mathrm{2}}{\mathrm{3}}}, η is the packing density, N_a is Avogadro’s number, ρ_L(t) is the density at temperature T, and M is the atomic weight.

The surface tension of a specific liquid metal can be reflected by its melting temperature to some extent. As is known that as the melting point increases, the atomic binding force increases. Consequently, the surface tension increases with respect to the melting point, and decreases with respect to the atom volume. In the antecedent research, many empirical formulas have been concluded to describe the relationship between surface tension and the melting point.

Dayal [88] has described the surface tension at the melting point as

(25){\mathit{\gamma }}_{\mathrm{m}}=\mathrm{3}{\left(N/V\right)}^{\mathrm{2}/\mathrm{3}}{\kappa }_{\mathrm{B}}{T}_{\mathrm{m}},

where V is the molar volume of the metal droplet, {\left(N/V\right)}^{\mathrm{2}/\mathrm{3}} is the number of molecules in a unit area of the surface, {\kappa }_{\mathrm{B}} is the Boltzmann constant, and T_m is the melting point.

Considering the negative relationship between surface tension and surface area, Xiao [89] has simplified the coefficient and obtained Eq. (26),

(26){\mathit{\gamma }}_{\mathrm{m}}=\beta {\displaystyle \frac{T_{\mathrm{m}}}{V^{\mathrm{2}/\mathrm{3}}}},

where β is a constant between 3 and 4.

Ceotto [90] has introduced an empirical equation for predicting the surface tension of some liquid metals at their melting point:

(27){\mathit{\gamma }}_{\mathrm{m}}=\sigma {\displaystyle \frac{Q_{\mathrm{L}}}{A_{\mathrm{mol}}}},

where σ is a non-dimensionless factor, equal to 0.2372×10⁻³, Q_L is the latent heat of the metal whose unit is J/mol, and A_mol is the surface area of a spherical drop of one mole of liquid metal.

The knowledge on temperature-dependent surface tension is crucial because it drives Marangoni convection [91,92]. Marangoni convection is the mass transfer along an interface between two fluids caused by the surface tension gradient [93]. The Marangoni effect plays a dominant role in many metallurgical processes such as casting, welding and solidification which involve the fluid phase and a free surface [94,95]. Since a liquid with a high surface tension pulls more strongly on the surrounding liquid with a low surface tension, the presence of a gradient in surface tension naturally drives the liquid to flow from regions of low surface tension to those of high surface tension. The surface tension gradient can be caused by concentration or temperature gradients. If the gradient is caused by temperature, it is called the thermocapillary convection [96,97] which can be expressed by the dimensionless Marangoni number, Ma, which is named after the Italian scientist Carlo Marangoni and defined as [98]

(28)Ma=-{\displaystyle \frac{\mathrm{d}\mathit{\gamma }}{\mathrm{d}T}}{\displaystyle \frac{L\mathrm{\Delta }T}{\alpha \mu }},

where L is a characteristic length, dγ is the difference in surface tension along L, dT is the temperature difference, α is the thermal diffusivity, and μ is the dynamic viscosity.

The thermophysical parameters α and μ are precisely given in general cases. Therefore, the temperature coefficient of the surface tension dγ/ dT is the key problem to decide the Ma number which represents the flow pattern. dγ/ dT is the relationship of the surface tension with respect to temperature. In general, surface tension decreases as temperature rises, reaching 0 mN/m at a critical temperature. A lot of researchers have proven that this influence pattern in metals such as Ga [99], In [100], Hg [101], Fe [102], Ni [103], Bi [104], Au [105], Ag [106], Li [107], Sn [108], etc. Figure 3 presents the dependence of surface tension on temperature for tin [109].

But for binary and ternary alloys, the temperature coefficient may be positive in some compositions. A positive dγ/ dT for Sn-Ag alloy of 0.5 mole fractions of Ag has been obtained in Ref. [110]. Similarly, a positive dγ/ dT has also been observed in Ref. [111] for Sn-Cu alloy of 0.5 mole fractions of Cu. The positive temperature coefficient is, perhaps, caused by the presence of the surface-active elements in materials or oxygen from protective atmosphere. However, the results in Ref. [112] have revealed that the surface tension of Ga-Bi alloys increases with increasing temperature for all Bi concentrations, as shown in Fig. 4. This unusual behavior can be explained by the reduction of surface enrichment of Bi component, which has a lower surface tension than Ga with increasing temperature.

3.2 Oxygen pressure

Surface tension is hard to be measured accurately because even a small quantity of surfactants or impurities can significantly reduce the value [113]. Oxygen pressure is one of these surface-active factors which can lead to oxidation of pure metals and then make the surface tension data unreliable. Moreover, the temperature coefficient also changes at different oxygen pressures. Measurements of some pure liquid metals surface tension as a function of oxygen content show an anomalous trend of the temperature coefficient that surface tension increases with temperature [114–117]. This effect is attributed to the different actual availabilities of oxygen near the surface as a function of temperature [118,119]. Additionally, gas flow rate is also an influencing factor [120,121]. It is understood that the wettability and adhesion are influenced by the affinity of the liquid phase metal to the reactive gaseous species [118,122]. Many researchers have proven that oxygen can strongly affect the surface tension of pure metals such as Ag [114], Bi [123], Cu [124], Fe [125,126], Ni [127], and Sn [117,128].

What presented in Fig. 5 is a typical “sigmoid” curve obtained in Ref. [129]. In the range from none oxygen content (X_o=0) to the saturation point (X_o=X_SAT) where the temperature is constant, the effect of oxygen can be clearly observed from the corresponding change of the surface tension. The curve can be subdivided into three regions according to the different adsorption velocity under various oxygen contents. In the first region (X_o≈0), the surface tension is almost equal to the value of pure metal γ≈ γ₀ since the oxygen content is very low. In the second region 0≤X_o≤X_Γmax, where X_Γmax is the inflection point of the curve where the adsorption velocity reaches the maximum Γ_max, the surface tension begins to decrease since the bulk gradually gets oxidized. In the third region X_o>X_Γmax, the surface tension decreases slowly and finally reaches a constant value of γ_SAT corresponding to the saturation value of X_SAT.

3.3 Particle size

With the development of nanotechnology, research objects enter into the nanoscale level where the effect of surface property strengthens on account of the increased specific surface area. Some phenomena in nanoscale are quite different from those in macroscale such as the size dependence of surface tension which only becomes significant for extremely tiny droplets. The interaction of size dependence and surface tension has been investigated in recent years. The Tolman length δ is a characteristic parameter to describe the size effect, which is defined as [130]

(29)\delta =\underset{r\to \infty }{\lim }\left(r_{\mathrm{e}}-r_{\mathrm{s}}\right),

where r_e is the radius of the equimolecular dividing surface, r_s is the radius of the Gibbs tension surface, and δ is the limiting distance between these two surfaces for a flat interface.

The comparison of the surface tension of a flat surface γ_f and the surface tension of a spherical droplet γ whose radius is r is given by [131]

(30)\log {\displaystyle \frac{\mathit{\gamma }}{{\mathit{\gamma }}_{\mathrm{f}}}}={\int }_{\infty }^r{\displaystyle \frac{\left[\mathrm{2}\delta /r^{\mathrm{2}}\right]\left[\mathrm{1}+\left(\delta /r\right)+{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}({\delta }^{\mathrm{2}}/r^{\mathrm{2}})\right]}{\mathrm{1}+\left[\mathrm{2}\delta /r\right]\left[\mathrm{1}+\left(\delta /r\right)+{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}({\delta }^{\mathrm{2}}/r^{\mathrm{2}})\right]}}\mathrm{d}r.

Neglecting higher order terms and treating the Tolman parameter as a constant, a simplified form has been obtained [131] and expressed as

(31){\displaystyle \frac{\mathit{\gamma }}{{\mathit{\gamma }}_{\mathrm{f}}}}={\displaystyle \frac{\mathrm{1}}{\mathrm{1}+\mathrm{2}\delta /r}}.

It is confirmed from Eq. (31) that the surface tension decreases with a reduction in the radius of the metal [132]. Calculations in Ref. [133] have shown that the surface tension of liquid metal K, Rb, Au, Pt, and Mo monotonically decreases with a reduction in the radius of the particle size, especially in case of extremely curved boundaries. Table 1 presents the similar results published in Ref. [130].

4 Experimental measurements

There are various methods available to measure the surface tension of liquid such as the capillary rise method [134,135], the Wilhelmy plate method [136,137], the Du Noüy ring method [138,139], the maximum bubble pressure method [140,141], the sessile [142] and pendant drop method [143,144], the levitated drop method [145] and the capillary wave scattering method [146,147], etc. But only a few of them can be applied to high temperature melts such as liquid metal which has a relatively high chemical activity [148]. Of these methods, the drop shape method is the most primary type, in which the sessile drop method and pendant drop method are especially widely adopted in the early work of Bashforth and Adams [149] and Adreas et al. [150].

Due to the strong chemical activity of high temperature melts, the results of surface tension may be affected by plenty of factors. Besides, the substrate may react with liquid metal specimen which reduces the accuracy. The drop shape methods face such problems because of the substrates utilized to form a liquid metal drop. The main difficulty of the drop shape methods is to find a suitable substrate which can keep stable and chemically compatible with the specimen at a high temperature. To solve this problem, the containerless methods come into existence. The prominent advantage of the containerless methods lies in the fact that liquid melt specimens does not need to contact with any containers so that the contamination can be avoided. Of the containerless methods, electromagnetic levitation is the most applicable one for metallic melts. It is easy to operate since it can provide levitating and oscillating simultaneously. The surface tension of liquid metal specimen can be calculated from the oscillating frequency of levitated liquid drop [151]. This method is widely utilized at a high temperature of up to 2000°C thanks to its extreme stability. Additionally, capillary wave scattering is another universal containerless method to acquire surface tension since capillary wave is ubiquitous for liquids.

4.1 Sessile drop and pendant drop method

The drop shape method is to measure the surface tension by combing the shape of a liquid drop with the Laplace equation [152]. The sessile drop method and the pendant drop method are two basic drop shape methods. A liquid drop is placed on a substrate in the sessile drop method while the pendant drop method uses a capillary to form a liquid drop. Surface tension measurement by the typical drop shape methods is presented in Fig. 6. Figure 6(a), (b) denotes the experimental setup of the sessile and the pendant drop methods, respectively. Besides the two fundamental methods, a variant of the sessile drop method which is called the constrained drop method is also practical. This method is based on the concept of large drop [153]. Different from the traditional sessile drop method which uses a substrate to support liquid drop specimen, the constrained drop method adopts a special circular crucible with sharp edges [154]. The constrained drop method can obtain the axial symmetry of drops, which makes the measurements more accurate [155]. Figure 6(c) displays three experimental images of the sessile drop method, the pendant drop method, and the constrained drop method. In these three methods, surface tension is calculated by numerically fitting the Laplace equation to the drop shape image acquired by the camera.

Due to the rapid development of computer-aided imaging technique as well as computerized numerical calculation techniques, the precision of the drop shape method has been improved a lot. Bashforth and Adams [149] have investigated the relationship between drop shape parameters and surface tension. Moreover, dedicated software packages have been developed to obtain surface tension data simultaneously with other parameters, such as drop shape and size, surface area, contact angle, etc [157]. The surface tension is calculated using an improved version of the Maze and Burnet algorithm [158].

Table 2 lists some surface tension values of pure elements, binary and ternary alloys measured by using the drop shape methods. In Table 1, the surface tension of Al-Ni [159], Al-Ti [160], Cu-Ag [161], Sn-Sb [162], Fe-Mn [163], Al-Cu-Ag [164], Cu-Fe-Sb [165], Cu-In-Sn [166], and Au-Bi-Sn [167] has also been studied. The surface tension is generally measured as a function of temperature and composition. The confidence error of the surface tension of liquid gallium and indium as a function of temperature studied in Ref. [168] is 1%.

4.2 Pendant/Sessile drop combined method

Even though the two fundamental drop shape methods are easy to operate to measure surface tension, there come many problems when they are used in high temperature situations [170]. To overcome the shortcomings of the sessile and pendant drop method, a new methodology integrating both former methods emerges. This procedure consists of two main steps corresponding to the two fundamental drop methods in one test. Figure 7 demonstrates the typical combined methods to measure surface tension of liquid drop. Figure 7(a) shows the schematic of the pendant/sessile drop combined method [156].

A general process of the pendant/sessile drop combined method is presented here. At the beginning, the experimental solid metal is placed in a capillary, then melted at a specific temperature and squeezed from the capillary. The drop shape images are recorded by high speed cameras. The above is the pendant drop method procedure. After that, by moving the capillary or substrate to make the droplet in contact with the substrate and then separating them in diverse directions, the droplet breaks and deposits on the substrate. Finally, the support slightly rotates in order to better position the drop and to ensure the drop symmetry [170]. Figure 7 (b) shows the drop deposition procedure. An image of two coexisting pendant/sessile drops can be observed in Fig. 7(c).

Compared to single pendant or sessile drop method, the combined method has particular advantages. First of all, though there is contact between the liquid metal and the capillary or the substrate, this contact time is very short, just a few seconds [173]. Besides, due to the contrary shift of the capillary and substrate, the liquid metal drop extends completely, which results in a highly symmetric graph. A large drop method has the same character. Finally, the liquid metal drop squeezed from the capillary is free of oxide film, which is very hard to remove in any other ways, thus improving the measurement accuracy.

The pendant/sessile drop combined method has been used for the first time to measure the surface tension of high melting Al-Ti based alloys [171]. The results obtained in Ref. [171] demonstrated the validity of such a method. This particular method can be widely used to measure metals which have high melting temperatures and strongly chemically-active elements. The surface tension of pure Cu, Ni, Al, and Fe using the pendant/sessile drop combined method investigated by Ricci satisfactorily agrees with the data from literature with a difference less than 1.5% [172].

4.3 Electromagnetic levitation-oscillating drop (EML-OD) method

Though the drop shape methods have been successfully used to measure surface tension, there is a fatal disadvantage. All the experimental liquid metal specimens are in contact with the containers. To overcome this problem, the containerless methods appear. For conductive metals, the EML-OD method is the best applicable containerless method to measure surface tension. First introduced by Okress et al. [174], this method has been explicitly studied by Lu et al. [175–177]. The basic principle is to adopt an alternating inhomogeneous electromagnetic field to produce a lifting force which counteracts the gravitational force to keep the drop levitating around its equilibrium position. The lifting force which compensates the gravitational force is the Lorentz force caused by the interaction between the electromagnetic field and a current in the specimen induced by the changeable electromagnetic field. Due to the restoring force of surface tension, the surface of a liquid drop always oscillates around its equilibrium shape with a critical frequency [178,179]. Therefore, the surface tension can be calculated by this frequency. The deformation of the drop shape due to surface oscillations can be described by spherical harmonics, Y_n^m. Particularly, when n=2, the fundamental frequency is called the Rayleigh frequency, ω_R. For a non-rotating, spherical and force-free sample, the Rayleigh frequency is given by [180]

(32){\omega }_{\mathrm{R}}^{\mathrm{2}}={\displaystyle \frac{\mathrm{32}\mathrm{\pi }}{\mathrm{3}}}{\displaystyle \frac{\mathit{\gamma }}{M}},

where γ is the surface tension and M is the mass of the specimens.

But the experimental specimen is not force-free due to gravity which results in a non-spherical shape. Consequently, the Rayleigh frequency, ω_R, splits into three unequally peaks (m=0, |m|=1, and |m|=2) corresponding to different surface oscillation modes of the liquid metal specimens. Cummings and Blackburn have proposed a summarized formula to calculate the Rayleigh frequency under this situation [181], as expressed in Eq. (33).

(33){\omega }_{\mathrm{R}}^{\mathrm{2}}={\displaystyle \frac{\mathrm{1}}{\mathrm{5}}}\sum _m{\omega }_{\mathrm{2},m}^{\mathrm{2}}-\mathrm{1.9}{\overline{\mathit{\Omega }}}_{\mathrm{tr}}^{\mathrm{2}}-\mathrm{0.3}{\left({\overline{\mathit{\Omega }}}_{\mathrm{tr}}^{\mathrm{2}}\right)}^{-\mathrm{1}}{\left({\displaystyle \frac{g}{R_{\mathrm{0}}}}\right)}^{\mathrm{2}},

where ω_2,m are the separated surface oscillation frequencies for n=2, {\overline{\mathit{\Omega }}}_{\mathrm{tr}}^{\mathrm{2}} is the mean squared translation frequency, g is the gravitational acceleration, and R₀ is the radius of the specimen.

The latter two terms are a correction of the original Rayleigh frequency. Generally, these two terms are approximately 5%–10% of the original value. By applying the Cummings correction, the surface tensions obtained by using the EML-OD method agree closely with the real surface tensions. An explicit error analysis of the EML-OD method in Ref. [182] indicates that the error of the surface tension is approximately 2%–5% while the error of temperature coefficient is approximately 10%–20%. Because of the extreme stability and accuracy of the EML-OD method, plenty of metals have been studied. Besides pure metals such as Fe [183], Ni [184], and Cu [185], the surface tension of considerable binary and ternary alloys have also been investigated as a function of temperature or composition including Al-Au [186], Al-Ag [187], Ag-Cu [188], Ni-Al [189], Cu-Ti [190], Al-Cu-Ag [164], Ti-Al-Nb [191], and Ni-Cu-Fe [192], etc.

The outcomes of the EML-OD method can be improved a lot by performing the experiments in a microgravity environment. Compared to experiments on the earth, there are additional advantages in microgravity. Due to the absence of gravitational acceleration, the lifting force to counteract the gravity is not required anymore so that the electromagnetic field intensity to position the samples is greatly reduced [193]. Consequently, its effect is negligible and there is no deformation of liquid drop or frequency splitting [194]. Therefore, Rayleigh’s formula can be used directly without using Cummings correction [195].

4.4 Capillary wave scattering method

Thermal excitation capillary waves propagate everywhere on a liquid surface [196]. The order of wavelength can range from millimeter (capillary wave) to kilometer (flood wave), and the amplitude can range from a fraction of a millimeter to dozens of meters. The properties of the surface wave are affected by the surface tension and gravity. When the wavelength of surface wave is large λ>10 mm, gravity plays a main role compared to surface tension. Otherwise, surface tension dominates. As is known from fluid mechanics, the surface wave dispersion of the propagating mode is expressed as [197]

(34){\omega }^{\mathrm{2}}=gk+{\displaystyle \frac{\mathit{\gamma }}{\rho }}{k}^{\mathrm{3}},

where ω=2πf_s is the surface wave circular frequency and f_s is the frequency, g is the gravitational acceleration, k=2π/λ is the wave number where λ is the wavelength, and γ and ρ are the surface tension and density of liquid, respectively.

For the capillary wave whose wavelength is relatively small, the effect of gravity can be ignored. Ignoring the first term gk and substituting ω and k into Eq. (34), the expression of surface tension can be derived as

(35)\mathit{\gamma }={\displaystyle \frac{{{\displaystyle f}}_{\mathrm{s}}^{\mathrm{2}}{\lambda }^{\mathrm{3}}\rho }{\mathrm{2}\mathrm{\pi }}}.

Equation (35) shows that the surface tension of the liquid whose density is given can be calculated if the frequency and wave length of the capillary wave are measured. Consequently, there are two methods to acquire surface tension. One is by measuring the capillary wave wavelength when the capillary wave frequency is fixed. The other is by measuring the capillary wave frequency when the capillary wave wavelength is fixed. The latter is more universally used due to its unique advantages. In this method, non-contact measurement can be achieved thus avoiding disturbance on the specimen [198]. Besides, this method can be applied to measure the surface tension in real time to study physical dynamics of the liquid. The basic principle is explained by Fig. 8. Laser can produce Brillouin scattering on a liquid surface. The frequency of the scattered light changes due to the thermal excitation capillary wave. This change is equal to the capillary wave frequency. Mixing the scattered light with the reflected light, the capillary wave frequency can be obtained by using difference frequency detection methods [199]. Additionally, the wavelength of the capillary wave can be calculated through the angle of the incident light, thus the surface tension of the liquid can be obtained by Eq. (35). The capillary wave scattering method is widely applicable to the measurement of surface tension of a broad range of materials, such as pure liquid Ga [200], liquid Ga-Bi alloy [201], liquid Sn-Bi alloy [202] and others [203].

5 Electrowetting

5.1 Theory

Electrowetting is to modify the wetting properties of a surface by applying electric field. It can change the contact angle of electrolyte-conductor interface or the interface between two immiscible electrolytic solutions due to the additional force resulted from the applied electric field [204–206]. Electrowetting can be comprehended in terms of the thermodynamic perspective. Surface tension is defined as the Gibbs free energy required to develop a unit surface area. The energy consists of both chemical and electric parts. The chemical part is just the natural surface tension without electric field while the electric part depends on the electric environment [207].

Once an electric field is implemented on an interface, the charge distribution transforms to a different situation, thereby affecting interfacial tension dramatically. The contrary electric charges in the electrolyte and the substrate or the two immiscible electrolytic solutions form an interface capacitor which stores electric energy. The relationship between the interfacial tension and the voltage difference across the interface can be expressed by Lippmann’s equation (Eq. (19)) which implies that surface tension decreases with increment of voltage. Accordingly, the contact angle is also altered by applying electric field. This is the mechanism of electrowetting which is accepted by the majority of investigators [208,209]. It is worthy to note that liquids universally exhibit a saturation phenomenon. Till reaching a certain voltage which is defined as the saturation voltage, further increase of the voltage hardly affects the contact angle anymore. The interface only becomes instable at an extreme voltage [210]. Numerous experiments agree well with this varying pattern for both liquid-solid and liquid-liquid interfaces [211–213]. This behavior is not well clarified, and there still remain diverse controversies about its origin [214–218].

A pressure gap across the interfacial boundary Δp as a consequence of surface tension is depicted by the Young-Laplace equation [219]

(36)\mathrm{\Delta }p=\mathrm{2}\mathit{\gamma }/R,

where R represents the curvature radius of the liquid metal drop.

Figure 9 shows the liquid metal droplet inside an electrolyte. Without electric field, the pressure is constant everywhere around the interfacial boundary of a spherical droplet as depicted in Fig. 9(a), (c). For a case with electric field, the electric field causes uneven charge distribution, thus leading to surface tension gradient. Take Fig. 9(b), (d) as an example, the unbalanced surface tension makes the droplet move right. Continuous electrowetting (CEW) actuation is based on this mechanism. Such driving mode is particularly stable and convenient with no moving parts. In addition, the power consumption to drive the liquid is considerably small compared to other mechanical methods.

5.2 Applications

Surface tension is relatively weak for macroscopic systems. But it plays a dominating role in microscale systems since the functions of other forces weaken as the size reduces [221]. The electrowetting and electrocapillary phenomena are active means to control surface tension in microscale systems to manipulate the behavior of liquid droplets [222]. Unlike mechanical ways, electrowetting is a convenient and smart approach to realize controllability of liquid movements by consuming a fairly low power. In recent years, electrowetting has attracted many researchers to investigate its potential applications in microfluidic devices. Lee and Kim have described a microelectromechanical systems (MEMS) demonstration device by utilizing the surface tension as the driving force based on continuous electrowetting (CEW) actuation. A liquid mercury micromotor travelling along a 2 mm-diameter channel loop has a maximum speed of 420 r/min. It is notable that the operating voltage is only 2.8 V and the average current is 10 μA [223]. By applying the CEW technique, a surface tension-driven micro-pump has also been fabricated to transfer liquids with very few energy inputs [224,225]. This direct electrical manipulation of liquid drops enables integrated microfluidic systems to be more flexible, efficient, smart and durable without the conventional pumps, valves and channels [226]. Based on the traditional structure of electrowetting on dielectric (EWOD) chip which uses two parallel plates, Yi and Kim have characterized a single-side coplanar electrodes without a cover plate by arranging the driving and reference electrodes necessary for EWOD actuation on one plate [227]. This simplified configuration makes it possible to integrate additional functions on the chip without any side effects on EWOD actuation, thus redoubling the flexibility of chip systems [228].

Another effect called reverse electrowetting (REWOD) is regarded as a new approach to high-power energy harvesting. The electric energy can be generated through the interaction of arrays of moving microscopic liquid droplets with nanometer-thick multilayer dielectric films [229]. This novel method has extraordinary advantages including high power densities of up to10³ W/m²; a broad range of output, from a few volts to dozens of volts; and most importantly, the ability to use all kinds of mechanical forces and displacements [230]. A ski or snowboard incorporated with a microfluidic device has already been designed to generate electrical energy through REWOD [231]. Furthermore, there are many other applications including the liquid lens [232,233], electronic displays [234–236], lab-on-chip devices [237], and microfluidic devices [238].

6 Room temperature liquid metal involved surface tension

6.1 Self-actuated RTLM

RTLM has been found to have such an unexpected phenomenon: eutectic Ga-In alloy droplets could shift spontaneously in NaOH solution by only eating a small piece of aluminum [44]. In a former study, Yu et al. have found that injecting liquid metal into electrolyte would quickly generate a large number of metal droplets [239]. If such liquid metals are fueled with aluminum in advance, fantastic phenomena would be observed according to the researches by Sheng et al. [240]. By injecting the liquid Ga-In-Al alloy into the Petri dish through a syringe, tremendously tiny liquid metal motors are quickly formed. These tiny liquid metal motors either move independently at a high speed or gather back to an original large size motor. During the process, every tiny motor exhibits running, oscillation, collision, bounce states randomly before the swarm of tiny motors finally assembles to a large one, as displayed in Fig. 10. The single large drop could still move automatically and disperse reversibly to create tiny motors through sucking and injecting control [240]. This peculiar behavior opens a new conceptual soft machine, the transient state machine, which can work autonomously or integrally in a reversible way.

The above experiments are basically originated from investigating the behavior of a single Ga-In-Al soft motor in NaOH solution [44]. Figure 11 shows the behavior of a liquid metal mollusk moving by itself inside the electrolyte. A milli-/centimeter scaled Ga-In-Al drop is capable of navigating itself to match up different geometrical spaces where it voyages in at a pretty large velocity of 5 cm/s for more than an hour, as shown in Fig. 11(a–d). The soft machine works just like a biomimetic mollusk which eats Al as food, as illustrated in Fig. 11(e). In general, to actuate a macroscopic object for a long time, external auxiliaries such as electrical field, electromagnetic field, thermal sources, mechanical devices must be applied. But these methods are, to some extent, cumbersome, thereby making the whole system complicated and hard to handle. Once the external energy is unavailable, the whole system cannot operate anymore. Moreover, the energy consumption is considerably great for a milli-centimeter object according to the equation F=6πμrv which characterizes the force required to drive a spherical object in liquid medium, where μ, r, and v are the dynamic viscosity, radius, and velocity of the object, respectively [241]. Consequently, it is intriguing that a milli-centimeter eutectic liquid Ga-In-Al alloy moves at a velocity of several centimeters per second for more than half an hour without external energy inputs.

In addition to self-actuated motors, a self-powered copper wire oscillator has also been disclosed recently [242]. When contacting a copper wire with the liquid GaIn₁₀ alloy pre-fueled with a small piece of aluminum, the copper wire is wetted and then swallowed into the liquid metal body. After a while, it begins to oscillate across the liquid metal horizontally just like a violin bow, as shown in Fig. 12. The whole procedure takes place in the 0.5 mol/L NaOH solution and lasts for half an hour without any supplementary energy. The base of this self-powered oscillator is totally soft, without any rigid part, therefore, setting up a platform for fabrication of future soft machines.

6.1.1 Mechanism of self-actuated RTLM

The basic mechanism of the self-actuation phenomenon comes from the galvanic interaction among RTLM, aluminum, and electrolyte. Figure 13 illustrates the force of self-driving motions generated from the liquid metal. When immersing it in electrolytes such as sodium hydroxide, Ga-In liquid metal can react spontaneously with the solution, producing [Ga(OH)₄]- gallate and subsequently leaving the surface with negative charges. These negative charges then attract the positive ions in the solution and produce a uniform back-to-back EDL (Fig. 13(a1)) [243,244]. The EDL is like a capacitance, hence resulting in a voltage difference [245]. According to Lippmann’s equation, the surface tension between the liquid metal and the solution depends on the capacitance of the EDL and the voltage difference across the interface [246]. The pressure jump across the EDL interface Δp can be calculated by the Young-Laplace equation, Δp=2γ/R. In addition to the electrochemical interaction between liquid Ga and the solution, the added Al in Ga-In alloy reacts with NaOH, expressed as 2Al+ 2NaOH+ 2H₂O= 2NaAlO₂ + 3H₂↑, leading to a charge redistribution upon the surface. Since this reaction is not uniformly distributed on the alloy surface (mainly on the interface between the droplet and substrate due to the high roughness), the symmetry of the surface tension is disrupted, which further changes the pressure jump between eutectic Ga-In-Al alloy and the solution. This unbalanced surface tension provides a driving force to actuate the liquid metal drop until the aluminum is used up (Fig. 13(a2)) [44].

The driving force for the self-powered copper wire oscillator comes from the dynamic unbalanced wetting force on two edges of the contacting area between the wire and liquid metal droplet. The copper wire carries a fraction of NaOH solution into the liquid metal body, assuming that it is moving from left to right. The inner solution on the right side reacts with aluminum instantly to produce hydrogen bubbles, which reduces the contact area and further lowers the adhesion force of the wire on the right side. Consequently, the higher wetting force on the left side drags the copper wire to left, and the above procedure repeats to produce the oscillating behavior (Fig. 13(b)) [242].

6.1.2 Manipulation of RTLM motion

Self-actuated RTLM motors suggest a new concept to develop future soft robots. However, the moving pattern of these self-propelled liquid metal motors is random and unordered like the Brownian motion [247]. In order to control the movement of such soft liquid metal motors including direction and velocity, thereby taking full advantage of these motors, various methods have been developed.

Since the driving force of eutectic Ga-In-Al alloy RTLM comes from the unbalanced surface tension induced by asymmetric charge distribution, the most important approach is to control the charge distribution by applying an electric field. Tan et al. [45] have utilized an electric field to control the motion of self-fueled liquid Ga-In-Al tiny motors in a Petri dish filled with NaOH solution (0.5 mol/L). Before applying the electric field, the tiny motors just move in a random way. Once switching on the electric field, all of the tiny motors run towards the anode at an average speed of 15 cm/s. When focusing on one tiny droplet with a diameter of 1.8 mm, the velocity in equilibrium state could even reach a surprisingly high magnitude of 43 cm/s when the voltage input is 20 V. Compared to the experiments in Ref. [44] which shows an average velocity of 5 cm/s without electrical actuation, the results reflect the powerful ability to control the motors by electrical technique. Besides, it is found that the trajectory of liquid metal motors approximately reflected the electric lines in solution, which is hard to visualize by other ways. Even though there is no addition of aluminum, the electrical method is found to be competent to handle the behavior of RTLM. In addition, the electrical aid enables fast fabrication of RTLM droplets in large quantity with relatively less trivial. A shooting of liquid Ga₆₇In_20.5Sn_12.5 stream through a capillary nozzle breaks into tremendous discrete drops due to its large surface tension with an applied voltage of 2.5 V [248]. Furthermore, a liquid metal enabled-micro-pump based on the CEW technique has also been demonstrated and fabricated to transport a variety of liquids with exceptionally low power consumption. This simple pump is capable of efficiently actuating liquids in micro-systems including MEMS, microfluidics, and micro-coolers [32,33]. Tang et al. have adopted the CEW technique to drive the flow of liquid Ga₇₅In₂₅ metal in complex micro-channels [249]. A cooling device utilizing hybrid liquid GaIn₂₀ alloy and water coolants has a great cooling efficiency, and the ratio between heat transfer and energy consumption is almost 10:1 [220]. It is noteworthy that the driving force comes just from the electrowetting under a low voltage below 30 V. Therefore, the whole system could be simplified to support more systems. Apart from the bare liquid metal droplets, nanoparticle-coated liquid metal marbles could move faster under the same electrical condition due to the extra bipolar electrochemical actuation [250]. Electrical technique can not only control the direction and velocity of liquid metal drops, but also flexibly regulate the shape of liquid metal in large-scale [251]. It is reported for the first time that the liquid metal such as Ga₆₇In_24.5Sn_12.5 immersed in electrolyte (water or NaOH solution) is able to display extraordinary transformations between different morphologies and configurations through applying the electrical field. When attaching the cathode to the surface of a large liquid metal film, the liquid metal immediately shrinks into a sphere, which changes the surface area over one thousand times [252]. Combined with the magnetic field, the liquid metal can exhibit more complex formations. Wang et al. have disclosed a series of surface folding patterns of liquid metal including gear pattern, fan blade pattern, glitch pattern, high dense folding pattern, and so on [253]. Figure 14 shows the schematic diagram of the experimental configuration and some snapshots of different folding patterns. These various folding patterns reflect the outcomes of the electric capillary force induced by surface tension gradients and the Lorentz force. The electric capillary force results in a destabilizing effect to induce the surface folding patterns while the Lorentz force is an inhibition factor. The interaction of these two mechanisms is found to be responsible for this magical phenomenon.

Apart from electrical and magnetic techniques, a straightforward way to realize self-actuated motion of liquid metals without addition of aluminum is to modify the ionic environment, since the movement mechanism directly comes from galvanic interaction. According to Lippmann’s equation (Eq. (19)), surface tension difference results from the properties of EDL, which is closely linked to the ionic environment. Below the saturation point, as the ions in the solution increase, the chemical reaction between the solution and liquid metal intensifies. As a result, a discrepant ionic concentration environment can give rise to an asymmetric EDL, subsequently leading to imbalanced surface tension to drive the liquid metal. Figure 15 presents the ionic gradient driven flow of liquid metal. By arranging the two sides of an eutectic Ga_68.5In_21.5Sn₁₀ alloy droplet with 1.2 mol/L HCl \left(\mathrm{p}\mathrm{H}\approx -\mathrm{0.1}\right) and 0.6 mol/L NaOH \left(\mathrm{p}\mathrm{H}\approx \mathrm{13.8}\right) solution respectively, a liquid metal droplet could travel continuously along the channel at a maximum speed of 25 mm/s, as shown in Fig. 15(a). Besides, the switching capability and pumping effect of these self-actuated liquid metal droplets induced by the ionic difference on either side of the droplet are also tested, as presented in Fig. 15(b), (c) [48]. This new technique makes it possible to conduct the self-actuated behavior of liquid metals completely through fluid without any hard components, thereby yielding an entire soft system.

Inspired by the intrinsic character of Ga to penetrate into a variety of metals which further alters the wetting properties [44], more extended ways can be developed to realize the self-actuated behavior of RTLM. For example, the liquid metal droplet could gain asymmetric driving force by wetting, delaminating and dissolving metal film while accelerating itself [254]. The driving force resulted from imbalanced wettability of the leading edge and lagging edge of the liquid metal droplet is responsible for this self-running motion. It is this asymmetric surface tension that propels the liquid metal drop to move spontaneously.

6.2 Printing devices utilizing RTLM ink

RTLM is a dream ink for direct writing electronics, printed flexible circuit, 3D printing and biomedical electrodes in virtue of its unique electrical conductivity as well as its inherent flexibility and fluidity at room temperature. The wetting properties of liquid metal on substrate are core factors to determine the print quality and accuracy. However, due to the high surface tension of RTLM, it is often a tough problem to obtain a good wettability [255]. In order to identify further applications of liquid metal ink, it is very necessary to conduct studies on different substrates.

Current printing methods lie in dispensing a stream of inks through the nozzle to the substrate. However, for a liquid metal which has a relatively low viscosity and a high surface tension, it is impossible to deposit a continuous stream since the liquid metal sometimes shrinks to form discrete droplets to minimize its surface energy. Even so, it is still feasible to improve the print quality if the dispensing speed and pressure are well controlled. Zheng et al. [256] have invented a tapping-mode ink delivery system to improve the wetting ability of liquid metals on different substrates. The delivery and printing of liquid metal inks using the tapping-mode technique are shown in Fig. 16. The special printing head is sketched in Fig. 16(a). By applying an external pressure in the inlet, the rolling-bead tightly blocks the gap between the inlet and the outlet, which prevents the liquid metal from flowing out. Once the rolling-bead taps the substrate, the gap is opened, allowing the liquid metal to flow accompanied by rotation of the roller-bead, as shown in Fig. 16(b). With the assistance of the adding pressure, the contact angle of the liquid metal on substrate decreases obviously, thus promoting the wettability. Of the three substrates investigated including PVC film, stainless steel, and office paper, PVC film works most sensitive to the pressure and thus is the best-matching substrate for the tapping-mode system. Figure 16(c) shows the wetting images of the above three materials under different impressed pressures. The contact angle of the liquid metal on the PVC film decreases almost 100° when exerting a pressure of 0.2 N. Figure 16(d) exhibits several printed functional patterns made of liquid metal ink through the tapping-mode printer. This tapping-mode system makes high-precision printed flexible electronics easier and more convenient in people’s daily life. Another desktop printer of flexible electronics on paper also modifies the wettability by reforming the print head as well as utilizing matched coated paper [257].

However, for the elastic soft substrate such as the polydimethylsiloxane (PDMS), even by using the tapping-mode system, the wettability is still poor, resulting in an unsatisfactory print quality. Since PDMS is a superb material to implement biomedical applications due to its excellent elasticity and biocompatibility, a novel dual-trans strategy has been proposed to realize high-precision printing on elastic substrates, as shown in Fig. 17 [258]. Figure 17(a) illustrates main procedures of the dual-trans strategy. At first, the liquid metal ink can be easily printed on the PVC film via using the tapping-mode system. Then covering the PVC film with PDMS solution and cooling down the whole multilayer object to solidify the liquid metal. Since the adhesive force between the metal ink and PVC film is much smaller than that between the metal ink and PDMS film (4770 μN<0.87 N), it is quite easy to peel off the PVC film without destroying the printings. Figure 17(b), (c) shows a transparent band and a flexible temperature measurement circuit printed on PDMS through the dual-trans strategy.

In addition to the above indirect methods which modify the wettability by mechanical design or process management, a straightforward way to enhance the wettability is to decrease the surface tension of RTLM by oxygen absorption. It has been discussed before that the oxides can decrease the surface tension as well as increase the viscosity. Consequently, the wettability and fluidity of RTLM can be easily controlled by carefully adjusting the oxidation reaction [259]. Eutectic GaIn₁₀ alloy with 0.025% oxygen is chosen as the ink of a direct writing pen, and this combination has a perfect wettability with various substrates including epoxy resin board, glass, plastic, silica gel plate, typing paper, cotton paper, cotton cloth, and glass fiber cloth [260]. The liquid metal ink has another potential for bio-electrodes in virtue of its high electrical conductivity, biocompatibility and comfort. Besides, its polarization potential is relatively low (<–1 V). Therefore, it does no harm to animals and people. Researchers in Ref. [261] have adopted the Ga-based liquid metal ink containing an oxygen concentration of 0.34% as the drawing electrocardiogram electrodes for the first time. These unique electrodes have a good adhesion ability on human skin as well as outstanding electrical properties. Owing to the good wettability of such oxidized liquid metal ink, drawing skin circuits on human skin is just as easy as drawing a picture on ordinary paper. Figure 18 shows the liquid metal printed on skin. Figure 18(a) shows a circuit drawn in a palm. Both the bio-electrodes and drawing circuits work well, demonstrating the feasibility and validity of such liquid metal inks as healthcare supplementary means. Figure 18(b) presents the basic principle of fabricating liquid metal electrodes to deliver biological electrical signals, which can be developed to fulfill various specific clinical needs. Although it is extremely simple to fabricate skin electronic circuits by directly drawing with a brush, it is difficult to guarantee that the printing quality can meet the demand, including shapes with sharply defined boundaries, uniform thickness and high precision. A spray-printing strategy combined with a pre-designed stainless mask proposed in Ref. [262] enables printed skin electronics to overcome the above problems. The RTLM ink is atomized through an airbrush powered by an air pump. Figure 18(c) shows that a stream of liquid metal is disrupted by the high-energy gas jets into micro-droplets which are inclined to be oxidized in air. It is found from the SEM images that the surface of liquid metal droplets is likely to have chemically-reactive polar atoms while the skin substrate may contain polar molecules, thus the oxidized micro-droplets rapidly and firmly adhere to the skin substrate due to the Van der Waals force. To manufacture desired electrical circuits, a stainless steel mask pre-designed by chemical etching technique is utilized. A schematic to show the basic principle of fabricating interdigital electrode on skin substrate via a printing mask is represented in Fig. 18(d). During the spray-printing procedure, a stainless steel mask is tightly attached to the substrate to ensure high-accuracy outcome without any fuzzy edges. The printed interdigital array microelectrodes (IDAM) on a piece of pig skin is shown in Fig. 18(e). The accuracy of this spray-printing technique reaches approximately 10 μm, demonstrating the high-precision ability to fulfill various demands. This technique provides a further route for precise and elaborate printed electronics which can be quickly fabricated on almost any desired solid substrate surfaces, whether smooth or rough.

6.3 Oxides of RTLM

Ga-based RTLM is quite easy to be oxidized in air to form a thin layer of Ga₂O₃. According to what have been mentioned before, oxygen pressure can significantly reduce the surface tension of liquid metals. A previous measurement by using the sessile drop method indicates that the surface tension of eutectic Ga-In alloy placed in atmosphere and that of the HCl solution are 624 mN/m and 435 mN/m, respectively [263]. This oxide film severely impairs the unique merits of liquid metal such as fluidity and flexibility. Therefore, it is regarded as an abomination upon most occasions. However, viewed from another perspective, the oxides can be used as a powerful tool to manipulate the surface tension and wettability in a controllable way [58,59].

Researchers have demonstrated the performance of liquid eutectic Ga₇₅In₂₅ alloy for stable structures formation in microchannels depending on the oxides [264]. When injecting it into microchannels by adding a regulated pressure, the liquid metal readily fills the microchannel and retains stable when the pressure is relieved. While in the other microchannel pre-filled with 10% HCl solution used for comparison, the liquid metal stream instantaneously retracts from the microchannel to minimize the surface energy once the pressure returns to atmosphere pressure. It is obvious that the skin of oxides guarantees the structural stability of liquid metal because of the strong enough adhesion force to the microchannel walls [265]. This elastic skin is the base of liquid metal plasticity, allowing it to be molded diversely and steadily even in non-equilibrium shapes [266,267]. Using syringe, such liquid metals can even be injected into the biological body along with packaging material to compose liquid metal wires which are capable of performing electrical physiological functions, as shown in Fig. 19 [268].

By introducing the combined electrical as well as chemical methods, the formation and dissolution of oxides can be utilized as a convenient and effective way to manipulate the shape and wettability of liquid metal reversibly. In Ref. [269], the authors have proposed a synthetically chemical-electrical mechanism to realize reversible large-scale deformation control of liquid metal. When the liquid metal makes contact with an anode which is immersed in the NaOH solution, electrochemical oxidation occurs and a layer of oxide is immediately formed. This oxide film decreases the surface tension and further alters the contact angle at the interface between the liquid metal and the electrolyte at once [270]. The liquid metal sphere disperses to an asymmetric layer whose surface area is almost five times that of its original area as depicted in Fig. 20. After a while, the oxide film is dissolved by the NaOH solution and the surface tension recovers to make the liquid metal form a sphere. Such a reversible procedure relies on the combination of electrical oxidation and chemical removal of the oxides on the liquid metal surface. The oxides behave as intrinsic efficient surfactants, which can vary the surface tension from 600 mN/m to nearly zero broadly and rapidly. Meanwhile it is quite easy for the oxides to form and remove under an extreme low voltage (1–30 V). The above techniques create a new concept that the oxides which are omnipresent for most materials can be used to manipulate surface tension, thereby shedding light on further applications of liquid metals such as MEMS switches [271], flexible electronic skin [272–274], controllable soft robots [275], and etc.

Since oxides can significantly decrease the surface tension of liquid metal, manipulation of the shape and motion of oxidized liquid metal is more flexible and stable at a low voltage. Recently, researchers have discovered that liquid metal can exhibit a lot of peculiar behaviors when it is placed on the graphite surface, as shown in Fig. 21. It becomes a dull, flat puddle by forming a thin oxide layer when it is placed on a graphite surface due to the potential elevation (Fig. 21(a)) [276]. A series of stable shapes could be achieved easily including line, triangle and rectangle (Fig. 21(b)). Placing an anode and a cathode at each side of the liquid metal at different distances, the liquid metal assumes various morphologies (Fig. 21(c)). Besides, with an applied voltage of 10 V in the 0.5×10⁻³ mol/L NaOH solution neither contacting the liquid metal nor the graphite, the liquid metal puddle could move towards the cathode along a flat graphite surface and a 10° graphite lop (Fig. 21(d)). Different from the smooth and quick movement of liquid metal on the glass surface, the liquid metal puddle wriggles slowly in various configurations like a worm. These miraculous phenomena demonstrate the better stretchability and adaptability of the oxidized liquid metal to realize more complicated and flexible applications in potential soft robots and machines.

7 Discussion

With the support of classical thermodynamics and multidisciplinary development, the mechanism of surface tension has become gradually clear. In the existing theoretical models, interfaces are handled as a two-dimensional plane, which is inconsistent with the fact that the thickness of interface reaches up to one or more times the molecular diameter. To further clarify the surface tension accurately, a three-dimensional model is to be built. As aforementioned, EDL plays a significant role in the surface tension performance, while there are obstacles determining the charge distribution and charge quantity. It is imperative to figure out the characterization parameters of EDL on the liquid metal surface in solution.

RTLM possesses commendable virtues particularly its self-actuated behavior which stands a chance of making soft robots. Such soft robots have significant application prospects such as the vascular micro-robot which can remove stasis and the rescue robot which can navigate the ruins to detect survivals. Accordingly, it is valuable to characterize its self-actuated motion. As has been introduced in Section 6, liquid metal can be locomotive in alkaline solution after some treatments. Besides, its motion can be manipulated by electromagnetic field, optical field, chemical filed, and etc. In previous researches, qualitative analysis has been conducted on how parameters in this system affect the movement of liquid metal, like the field intensity, the liquid metal size, the solution composition, and other factors. There is still a long way to go regarding quantitatively manipulating and utilizing the motion of liquid metal because it remains unknown how these external fields work elaborately. In view of the particularity of liquid metal and complexity of the multi-field action, models, formulas and non-dimensional parameters are to be defined to reveal the motion regulations of liquid metal where classical fluid dynamics provides a reference.

Surface tension plays a dominating role in microscale due to the relatively high specific surface area. As the object size increases, the effect of surface tension weakens. It can be seen from the above self-actuated motion of RTLM that the droplet size is limited in centimeter scale. To achieve the terminal goal of self-actuated RTLM soft robot, it is a fundamentally tough problem regarding how to realize self-actuated behaviors in large scale. Furthermore, a specific solution which can remove the oxides on the gallium-based liquid metal surface is indispensable for pre-existing techniques to enforce self-driven motions. However it is impractical to provide such a solution environment for a large-size robot. Consequently, how to disengage the surface tension-induced RTLM motions from solutions also needs further study.

In addition, to extend the usability of the printed electronic circuit using the RTLM ink, efforts should be made in the following aspects. First of all, substrates which are more compatible with the RTLM ink are to be discovered and created. Moreover, it is necessary to enhance the printing precision in order to print microscale electronic circuits which can be used in MEMS systems. Furthermore, the wettability of liquid metal-based ink on various substrates can be further modified through combining the RTLM with other materials to regulate surface tension of the ink.

8 Conclusions

This paper summarizes the basic issues of liquid metals surface tension including theories in three terms to understand the surface tension as well as the main influencing factors induced by the external environment. It elaborates on the experimental methods to measure the surface tension and plenty of examples. It also illustrates the fundamental theory and applications of electrowetting. Besides, it clarifies the impact of surface tension on RTLM in three different aspects. However, it is still a tough problem to quantitate the surface tension of RTLM since it is not in an equilibrium state during the moving process. Additionally, the unbalanced surface tension resulted from the asymmetry charge distribution when the RTLM is immersed in solutions cannot be measured until now. Therefore, more efforts should be made to expound the intrinsic mechanism that how surface tension influences liquid metal behaviors and to deduce formulas to quantify the role of surface tension rather than merely qualitative analysis.

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