# Rotation of Liquid Metal Droplets Solely Driven by the Action of Magnetic Fields
## 2 Materials and Methods  
_Materials and instrumentation_: EGaIn liquid metal and sodium hydroxide (NaOH) were purchased from Santech Materials Co. Ltd, China. An electrolyte solution of NaOH and hydrochloric acid (HCI) were introduced to remove the oxide layer on the surface of LMDs, where NaOH solution was prepared by dissolving solid sodium hydroxide particles in deionized (DI) water, and HCl solution was prepared by diluting concentrated hydrochloric acid with DI water. A small amount (\(\sim\)2 mg) of fine phosphors (Juen technology Co. Ltd, China) was sprinkled on the LMDs to create easily identifiable points for calculating the rotational speed of the LMDs. Blue and yellow dyes were obtained by diluting edible dye (1:5, SUGARMAN) and then dripped into a quartz tube using a syringe pump (LSP02-1B, LONGERPUMP). Self-rotation videos of LMDs were captured using a digital single lens reflex camera (Canon 5D Mark II) equipped with a macro lens (Sigma 105 mm 1:2.8 DG Macro HSM). The sequential snapshots were extracted from these videos.  
_Experimental Setup_: The experimental setup is illustrated in Figure 1a. A pair of permanent magnets were fixed to an aluminum frame, and the aluminum frame was connected to the output shaft of a DC motor (Leadshine 57 HS09) whose speed and direction were controlled by a microcontroller unit (MCU, Arduino Carduino UNO R3). An EGaIn LMD was placed in a quartz tube (diameter: 8 mm, height: 15 mm) filled with NaOH or HCl solution, and the quartz tube was placed at the center point between the magnets.  
_Mechanism_: The mechanism of self-rotation of LMDs is shown in Figure 1b, in which we hypothesize that by moving two permanent magnets around the droplet within an aqueous solution, an EGaIn droplet can be actuated to self-rotate after the introduction of eddy current. On account of the large surface tension of liquid metal, the EGaIn droplet (diameter \(<\) 5 mm) immersed in the aqueous solution can be considered as a sphere. In addition, to simplify the model, the EGaIn droplet is equivalent to several parallel coils located at a different longitude of the droplet. Before the external magnetic field rotates, the magnetic flux through the equivalent coils is almost a constant. When the external magnetic field starts to rotate, relative motion (in other words, phase shift) is formed between the magnetic field and the EGaIn droplet, which further induces the change of the magnetic flux through the equivalent coils \(\varphi\) and eddy current \(I\) within the equivalent coils. The eddy current \(I\) can be explicitly expressed as  
\[I=\frac{\mathrm{d}\Phi}{R_{e}\mathrm{d}t} \tag{1}\]  
where \(\varphi\) is determined by the magnetic flux density \(\mathbf{B}\) as well as the equivalent coils area \(\mathbf{S}\), which can be written as \(\varphi=\mathbf{B}\cdot\mathbf{S}\), and \(R_{e}\) represents the equivalent resistance of the coils. Subsequently, an induced magnetic field is engendered by the eddy current to hinder the change of magnetic flux, according to the [PERSON]'s law [40, 41]. In other words, assuming that the external magnetic field rotates counterclockwise, the induced magnetic field will generate a clockwise torque to hinder the rotation of the magnetic field, no matter whether the magnetic flux through the coils increases or decreases. It is well known that forces are mutual actions of two bodies [38, 42, 43, 44, 45, 46], that is, when the equivalent coils impede the external magnetic field, the external magnetic field also exerts a torque \(\mathbf{M_{e}}\) of the equal magnitude and opposite direction on the EGaIn droplet. According to [PERSON]'s law, the torque \(\mathbf{M_{e}}\) can be expressed as  
\[\mathbf{M_{e}}=2\int_{0}^{\pi R}\sin\frac{l}{R}Idl\times B \tag{2}\]  
where \(R\) is the radius of the equivalent coil, and \(\mathbf{l}\) is the length of the equivalent coil, respectively [42, 43, 44, 45, 46]. The torque \(\mathbf{M_{e}}\) enables the droplet to overcome the friction and commence to self-rotate.