Dynamical modeling and experimental research on capillary flow of gallium-based liquid metal for reconfigurable structures Open Access

Gallium (Ga), a unique chemical element discovered by Lecoq de Boisbaudran in 1875, has garnered worldwide attention and is currently experiencing a global revival. As a post-transition metal from group III A of the periodic table of elements, gallium is of great strategic significance in modern industrial civilization.¹ Gallium and its alloys, as emerging multifunctional materials that exist as liquids at room temperature, have attracted immense interest from both academic and industrial communities.^2,3 Combining the remarkable electrical conductivity of metals with the intrinsic fluidity of liquids, gallium-based liquid metals are physically suitable for a diverse spectrum of prospective applications, such as stretchable electronic components,^4,5 self-healing circuitry,^6,7 3D printing and patterning,^8–10 reconfigurable antennas,^11–14 and microfluidic systems.^15–17 In comparison with mercury (Hg), which has a finite vapor pressure and physiological toxicity,¹⁸ gallium-based liquid metals possess near-zero vapor pressure at room temperature as well as low toxicity,^19,20 making them more bio-compatible and safer alternatives utilized in a wide range of biomedical applications, including bio-integrated flexible devices and as carriers for drug delivery.^21–23 

Due to their fluid nature, gallium-based liquid metals can offer versatility in patterning that conventional solid metals cannot match. However, in numerous typical applications that utilize gallium-based liquid metals, a prevalent yet still unresolved physical challenge lies in accurately formulating the relationship between the interface position of the liquid metal column and the corresponding time of motion, which is crucial for precise control and reliable performance in the relevant applications. This is also an indispensable requirement for designing and evaluating gallium-based liquid metal structures and devices. To be specific, for instance, the electrical resistance of stretchable wires is closely related to the dynamic variations in the wire length. The precise regulation of the length of the liquid metal column has a significant impact on enhancing the operational reliability of self-healing circuits. Similarly, when applying positive pressure or vacuum filling in the 3D printing and additive patterning process, gallium-based liquid metals can be injected into capillary tubes. The accurate manipulation in this patterning process, which relies on dynamical analysis of the capillary flow of the gallium-based liquid metal, is rather essential. As for the tuning of the reversibly deformable fluidic antennas, the fundamental mechanism also centers on the capillary flow of the gallium-based liquid metal. When the length of the liquid metal column increases, the corresponding resonance frequency decreases according to $f=143l×1εeff$⁠, in which l denotes the length of the gallium-based liquid metal and ɛ_eff denotes the effective permittivity.¹¹ Consequently, an in-depth exploration into the capillary flow dynamics of gallium-based liquid metals holds paramount importance for the innovative design and fabrication of reconfigurable components and structures, as it paves the way for rapid development in the burgeoning fields of advanced manufacturing and flexible functional materials.

To effectively design reconfigurable structures, it is crucial to accurately describe and manipulate the motion of gallium-based liquid metal. Here, as the most commonly used gallium-based liquid metal, eutectic gallium indium (EGaIn) is selected for the experimental studies. With regard to the flow pattern of EGaIn inside a cylindrical tube, a dynamical model for horizontal capillary flow of EGaIn is proposed. In this model, the cylindrical coordinates are chosen, and the capillary process is assumed to be a one-dimensional laminar flow along the axis of the capillary tube. The EGaIn flow is predominantly resisted by the viscous force due to the increasing liquid column length and the flow resistance exerted by the hydrochloric acid solution within the capillary tube. A photo of the experimental setup for horizontal capillary flow and the free-body diagram for EGaIn within a horizontal capillary tube are illustrated in Fig. 1. Compared with the thin capillary tube, the cross-sectional area of the liquid metal container is large enough that the variations in hydrostatic pressure are negligible. Before each experiment, the EGaIn is subjected to immersion in a hydrochloric acid solution (0.1 mol/l) to eliminate the oxides present within. Moreover, in every capillary flow experiment, the initial hydrostatic height h₀ remains constant by controlling the total volume of EGaIn injected into the container.

In the dynamical model, EGaIn is actuated to flow into the horizontal capillary tube. The forces that govern the motion of EGaIn are the hydrostatic force (F_p), the capillary driving force (F_ca), the viscous drag force (F_v), and the flow resistance (F_e) exerted by the hydrochloric acid within the capillary tube. The air resistance generated by the small amount of air within the capillary tube is negligible.

Using the fourth-order explicit Runge–Kutta method, the numerical solutions of the ordinary differential equation, Eq. (16), under the initial conditions $z0=0$ and $z′0=0$⁠, can be calculated.

The EGaIn (75% gallium and 25% indium by weight) employed in our experimental studies is from Sino Santech Materials Technology Co., Ltd. (Changsha, China). The radius (R) of each copper capillary tube is 0.001 m, and the thickness of the tube wall is 0.0005 m. The lengths of the capillary tubes are 0.10, 0.15, 0.20, 0.25, 0.30, and 0.35 m. The copper capillary tubes are produced by Jiangtong Precision Metal Products Co., Ltd. (Wuxi, China). The liquid metal containers (50 × 50 × 50 mm³ in volume) made of polymethyl methacrylate (PMMA) for measuring the capillary flow are produced by Jingguan Zhuoyi Acrylic Manufacturing Company (Tianjin, China). The hydrochloric acid is from Guangzhou Howei Pharma Tech Co., Ltd. (Guangzhou, China). The HAAKE MARS-40 rotational rheometer is from ThermoFisher Scientific (Karlsruhe, Germany), and the OCA-50 surface tension analysis instrument is from DataPhysics Instruments GmbH (Filderstadt, Germany). The FDR-AX60 high-resolution monitoring camera is from SONY Corporation (Tokyo, Japan).

To improve the accuracy of the dynamical model, a series of experiments are specifically conducted to determine the basic mechanical parameters used in the ordinary differential equation [Eq. (16)]. To measure the dynamic viscosity of EGaIn, a HAAKE MARS-40 rotational rheometer with a rotating bob geometry is employed. It is considered a reliable approach for quantitatively evaluating the viscosities of liquids, particularly for EGaIn, which possesses a remarkably high surface tension and is susceptible to oxidation. The viscosities of EGaIn and the hydrochloric acid are measured at room temperature (25 °C). The photo of the rotational rheometer and the detailed geometrical dimensions of the cylindrical rotor and the cup are shown in the supplementary material.

The rotational rheometer is calibrated using a standard liquid and, subsequently, the time dependency experiments are performed in a high-purity nitrogen atmosphere at a specific temperature of 25 °C and two different angular velocities of 60 and 300 s⁻¹. The measured EGaIn sample volume is 16.1 ml. These experiments aim to verify that the EGaIn is free from any solid contaminants or potential oxidation. Similarly, all viscosity measurements are also conducted in the high-purity nitrogen atmosphere to protect the gallium-based liquid metal from oxidation.

In order to obtain the surface tension of EGaIn, pendant drop experiments, as a standard technique, are carried out. Before measuring the surface tension of EGaIn, calibration of the surface tension analysis instrument (OCA-50) is also conducted with pure water. In the pendant drop experiments, a clean dosing syringe with a needle of 1.65 mm in diameter is used. The dosing syringe is filled with EGaIn and is mounted onto the electronic syringe control unit to dispense a stable EGaIn droplet that is submerged in the hydrochloric acid solution (c_HCl = 1 mol/l). A typical backlit image and the corresponding geometric dimensions of the EGaIn droplet are presented in Fig. 2.

In the process of validating the dynamical model, flow experiments using capillaries of different lengths have been conducted. Given the relatively high wettability of EGaIn on the copper surface in hydrochloric acid,^36,38 copper capillaries are employed in these flow experiments, and all the cylindrical tubes are horizontally placed in the experimental setup. Compared with the capillaries, the cross-sectional area of the liquid metal container is 2500 mm² (50 × 50 mm²), which is 795 times larger than that of the cylindrical capillary tubes (3.14 mm²). The total height of the liquid metal container is 0.05 m, and the initial hydrostatic height is set to be h₀ = 0.025 m. Therefore, the very slight change in the hydrostatic height in the liquid metal container due to capillary filling is negligible.

In the experimental investigations, we first insert the thin sealing plate (thickness: 1 mm) vertically at the bottom of the liquid metal container to seal the capillary inlet and mark a reference line on the sealing plate at the height of the top edge of the liquid metal container. Subsequently, we inject the EGaIn into the container and adjust the angle of the high-resolution monitoring camera to ensure that both the vertical movement of the sealing plate and the state of the capillary outlet can be clearly recorded within the same screen view. Then, we vertically elevate the sealing plate until it departs from the EGaIn pool. As the entire process is recorded, the flow time of the EGaIn can be determined by playing back the video captured using the high-resolution camera in slow motion. Given that the distance between the capillary inlet and the bottom of the liquid metal container is 5 mm, we designate the moment when the reference line on the sealing plate rises vertically by 5 mm as the starting point and the moment when the EGaIn just emerges from the capillary outlet as the end point for measuring the flow time. For each copper capillary tube of a specific length, we utilize a monitoring camera to observe and record the flow time of the EGaIn from the inlet to the outlet of the tube six times. Subsequently, the average flow time for the EGaIn to traverse the entire length of the capillary tube can be calculated.

Figure 3 presents the time dependency curves for EGaIn, which are measured in a high-purity nitrogen atmosphere at a specific temperature of 25 °C and two distinct angular velocities of 60 and 300 s⁻¹. In Fig. 3, both curves, which exhibit nearly zero slope, show that the fluctuations of shear stress are minimal over the range of time, indicating that the EGaIn measured is devoid of solid contaminants or any changes that might be induced by possible oxidation. The corresponding dynamic viscosities of EGaIn and hydrochloric acid are μ_l = 6.928 mPa•s and μ_e = 1.101 mPa•s, respectively.

Based upon the measured surface tension of pure water at 25 °C, which is 72.50 mN/m, it is confirmed that the surface tension analysis instrument has been accurately calibrated. By performing pendant drop experiments and fitting the equilibrium shape of the EGaIn droplet with the Young–Laplace equation, the surface tension of EGaIn in the hydrochloric acid is found to be σ = 507.27 mN/m. Other essential physical parameters concerning capillary flow are listed in Table I. Note that the hydrostatic pressure offered in the liquid metal flow is P = ρ g h₀ = 1538.6 Pa.

According to the experimental measurements, the time taken for the EGaIn to flow from the inlet to the outlet of the horizontal capillary tube is shown in Fig. 4. For the six different lengths of capillary tubes employed, the average flow times of the EGaIn are 0.363, 0.638, 0.942, 1.460, 1.847, and 2.072 s.

The corresponding numerical solutions, combined with the average flow time derived from experimental results, are presented in Fig. 5. These solutions demonstrate that the theoretical calculations of the time required for EGaIn to flow from the inlet to the outlet of the horizontal capillaries are in good agreement with the experimental measurements conducted on capillary tubes of varying lengths. When the density, surface tension, viscosity, and wettability of EGaIn are determined, the dynamical model can describe the flow of EGaIn accordingly. Furthermore, the basic hypotheses of incompressible steady flow and the no-slip boundary condition introduced in the study are validated through the comparison of experimental results with theoretical calculations. Based on the momentum theorem, our mathematical derivation presents the rate of change in the overall momentum of the EGaIn column with varying lengths. In this mathematical treatment, the velocity differences between different points along the radial direction have been incorporated into the rate of change in the momentum of the EGaIn column. It should also be noted that the main objective of this research is to establish the quantitative relationship between the interface position of the EGaIn column and the corresponding time of motion in a very small capillary tube. Given that the capillary tube under investigation has a radius of just 1 mm, the influence of the minor velocity differences between neighboring points on the interface on the overall flow time measurement may exist but is insignificant. Therefore, we consider this dynamical modeling, which is based on the momentum theorem, to be appropriate.

As a quantitative method to describe the gallium-based liquid metal flow in capillary tubes, the dynamical model proposed here can be employed to better design and assess various reconfigurable engineering structures in novel applications, including millimeter-scale flexible robots, reconfigurable circuits and antennas, and medical wearable devices. Moreover, building on this foundation of dynamical modeling, further exploration into the optimized design of reconfigurable liquid metal structures under special environmental conditions—notably under microgravity in space—will present a promising avenue for future development. Consequently, the theoretical framework offers quantitative insights that serve as a valuable guide for the design and fabrication of reconfigurable liquid metal structures.

In conclusion, we have developed a dynamical model based on the momentum theorem to quantitatively evaluate the capillary flow of gallium-based liquid metal. In the dynamical model, the surface tension, the contact angle, the hydrostatic pressure, and the dynamic viscosity are all considered for analyzing the flow behavior of EGaIn. Harnessing the relatively high wettability of EGaIn on the surface of copper with the presence of hydrochloric acid, flow experiments in copper capillaries, along with related measurements of the essential physical parameters, have been conducted to verify the accuracy of the theoretical modeling. The numerical solutions of the dynamical equation obtained by using the fourth-order explicit Runge–Kutta method are basically in good agreement with the experimental measurements performed on capillary tubes of varying lengths. Since the proposed dynamical model is based on a millimeter-scale capillary flow and the no-slip boundary condition, extending it to incorporate the slip boundary condition presents significant potential for the design and fabrication of multiscale micro/nanofluidic structures, representing an intriguing opportunity for gallium-based liquid metal research. Furthermore, this robust theoretical framework is expected to have prospective applications in the analysis of other vital liquid alloys, thereby broadening its impact across the metallurgical industry.

The supplementary material provides a photo of the rotational rheometer (HAAKE MARS-40) and the detailed geometrical dimensions of the cylindrical rotor and the cup.

This work was financially supported by the National Natural Science Foundation of China (Grant No. 12074160) and the Natural Science Foundation of Liaoning Province of China (Grant No. 2024-MS-181).

The authors have no conflicts to disclose.

Chuanyang Jiang: Conceptualization (equal); Data curation (equal); Formal analysis (lead); Investigation (lead); Methodology (equal); Software (lead); Validation (equal); Visualization (lead); Writing – original draft (lead). Xiangpeng Li: Conceptualization (supporting); Investigation (supporting); Methodology (equal); Resources (equal); Validation (equal); Visualization (supporting). Kaixuan Guo: Data curation (equal); Investigation (equal); Software (supporting); Validation (equal); Visualization (equal). Sheng Yang: Data curation (equal); Investigation (equal); Validation (supporting); Visualization (equal). Jiao Yu: Conceptualization (supporting); Data curation (equal); Funding acquisition (equal); Investigation (equal); Software (supporting); Supervision (supporting); Validation (equal); Writing – review & editing (equal). Lu Cao: Conceptualization (equal); Formal analysis (supporting); Funding acquisition (equal); Project administration (lead); Resources (equal); Supervision (equal); Validation (equal); Visualization (supporting); Writing – review & editing (equal). Wenchang Tan: Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Writing – review & editing (lead). Faxin Li: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (supporting); Investigation (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.