Interaction between the substrate and probe in liquid metal Ga: experimental and theoretical analysis

Interaction between two bodies in a liquid metal is an important topic for development of metallic products with high performance. We conducted atomic force microscopy measurements and achieved the interaction between the substrate and the probe in liquid Ga of an opaque and highly viscous liquid. The interaction cannot be accessed with the normal atomic force microscopy, electron microscopy, and beam reflectometry. We performed a theoretical calculation using statistical mechanics of simple liquids by mixing an experimentally derived quantum effect. From both experiment and theory, we found an unusual behaviour in the interaction between the solvophobic substances, which has never been reported in water and ionic liquids. Shapes of the interaction curves between several solvophobic and solvophilic pairs in liquid Ga are also studied.

Figs. S1a and S1b shows a two-dimensional (2D) frequency shift distribution and an approximate 2D solvation structure (approximate 2D total correlation function), respectively.The 2D solvation structure was calculated from the 2D frequency shift distribution.First, we calculated the 2D force distribution (Fig. 2a) by using the Sader-Jarvis equation: 1 where F is the force between the probe and the substrate, k is the spring constant of the cantilever, a is the amplitude of the cantilever's oscillation, z is distance between the closest surfaces of the substrate and the probe, and t is the variable of the integration with respect to z.
Here, , where is the change in resonant frequency and is its unperturbed resonant frequency.The approximate 2D solvation structure (a function of q) is shown in Fig. S1b.We prepared the function q which is proportional to the total correlation function h (i.e., q ∝ h).The function q was calculated by using an equation q = -∂F/∂z which is an approximate and simple relationship between the structure and the force. 24][5] (b) Approximate 2D solvation structure calculated from (a).7][8][9] The reason why the lowest points in the 2D frequency shift and the 2D solvation structure are not at the same level stems from the roughness of the substrate surface.This result reflects the fact that the substrate surface is covered with spontaneously formed gallium oxide film (GaO x ), which is considered to be nonflat compared with that of a cleaved mica surface.
By the way, eqn (S2) can be simplified depending on the cantilever's amplitude a.When a is sufficiently small, it can be expressed as This greatly simple approximate equation indicates that the frequency shift of the raw AFM data is directly proportional to the solvation structure (total correlation function).We think eqn (S3) is helpful for study of the solvation structure with AFM.We note that eqn (S3) cannot be applied to a situation that the probe and the substrate are directly interacting by their steric repulsive force.
In what follows, we show additional figures.represent the pair potentials between the probe and Ga atom when the affinity parameters are strong and weak.In (b), the diameter of the probe is five times that of the Ga atom (i.e., (2R + σ)/σ = 5).In (c), the diameter of the probe is ten times that of the Ga atom (i.e., (2R + σ)/σ = 10).
(d) The solid and dashed curves denote the pair potentials between the substrate and probe when the probe sizes are ten and five times that of the liquid Ga atom, respectively.k B and T are the Boltzmann constant and absolute temperature, respectively.

Fig. S6
Normalised number densities of the liquid Ga near the probe surface when the probe size is ten times that of the liquid Ga atom.The solid and dashed curves denote the normalised number densities on the solvophilic and solvophobic surfaces, respectively.z PGa is the distance between the centres of the probe and the Ga atom (see Fig. 1a).
For comparison, we show the solvophobic force curves measured and calculated in liquid Ga in Fig. S7.As shown in Fig. S7, shapes of these force curves are qualitatively similar.

Fig. S7
Force curves between the solvophobic probe and substrate.The solid curve is the force curve measured by our AFM, which is the same as that in Fig. 2b.The dashed curve is the force curves calculated by the integral equation theory where the diameter of the probe is five times larger than that that of Ga atom, which is the same as that in Fig. 4a (upper-left).The dotted curve is the force curves calculated by the integral equation theory where the diameter of the probe is ten times larger than that that of Ga atom, which is the same as that in Fig. S5a (upperleft).For visual purposes, the dashed and dotted curves are shifted to the right by a unit length.
Fig. S1 (a) 2D frequency shift distribution.It is 256 frequency shift curves between substrate figure captions.

Fig. S2
Fig. S2 Frequency shift curves measured by our AFM.Black and blue curves represent the

Fig. S4
Fig. S4 (a) Solid and dashed curves are the pair potentials between the substrate and Ga atom