Measuring the Interaction Force Between a Droplet and a Super-hydrophobic Substrate by the Optical Lever Method

The contact between a droplet and a super-hydrophobic surface is very common in daily life and industrial production: water droplets sliding from the surface of lotus leaf^1,2, and a water strider traveling rapidly over the water^3,4,5,6. A super-hydrophobic coating on the exterior surface of a ship can help reduce the corrosion degree of the ship and reduce the resistance of the navigation^7,8,9,10. There is great value for industrial production and bionics research in studying the contact process between a droplet and a super-hydrophobic surface.

To observe the spreading process of droplets on a solid surface, Biance used a high-speed camera to photograph the contact process and found that the duration of the inertial regime is mainly fixed by the drop size¹¹. Eddi photographed the contact process between the droplet and the transparent plate from the bottom and side using a high-speed camera, which comprehensively revealed the variation of the contact radius of the viscous droplet with time¹². Paulsen combined an electrical method with high-speed camera observation, thus reducing the response time to 10 ns^13,14.

Atomic force microscopy (AFM) has also been used to measure the interaction force between the droplet/bubble and solid surfaces. Vakarelski used an AFM cantilever to measure the interaction forces between two small bubbles (approximately 80-140 μm) in aqueous solution during controlled collisions on the scale of micrometers to nanometers¹⁵. Shi used a combination of AFM and reflection interference contrast microscopy (RICM) to simultaneously measure the interaction force and the spatiotemporal evolution of the thin water film between an air bubble and mica surfaces of different hydrophobicity^16,17.

However, since commercial cantilevers used in AFM are too small, the laser spot irradiated on the cantilever would be submerged by droplets or bubbles. The AFM has difficulties in measuring the interaction force between droplets and droplets/substrates in the air.

In this paper, a measurement system based on an optical lever method is designed to measure the interaction force between droplets and super-hydrophobic substrates. The force sensitivity of the optical lever (S_OL) is calibrated by electrostatic force¹⁸, and then the interaction forces between droplets and different super-hydrophobic substrates are measured by the measurement system.

The schematic diagram of the measurement system is shown in Figure 1. The laser and position sensitive detector (PSD) constitute the optical lever system. A millimetric silicon cantilever is used as a sensitive component in the system. The substrate is fixed on the nanopositioning z-stage, which can move in vertical direction. When the substrate approaches the droplet, the interaction force causes the cantilever to bend. Thus, the position of the laser spot on PSD will change, and the output voltage of PSD will change. The output voltage of PSD V_p is proportional to the interaction force F_i, as shown in Eq. (1).

(1)

In order to acquire the interaction force, S_OL must be calibrated first. The electrostatic force is used as the standard force in the calibration of S_OL. As shown in Figure 2, the cantilever and the electrode make up a parallel plate capacitor, which could generate electrostatic force in a vertical direction. The electrostatic force F_es is determined by the voltage of the DC power supply V_s, as shown in Eq. (2)^19,20,21.

(2)

where C is the capacitance of the parallel plate capacitor, z is the displacement of the cantilever free end, and dC/dz is called capacitance gradient. The capacitance could be measured by the capacitance bridge. The mathematical relationship between C and z can be fitted by a quadratic polynomial, as shown in Eq. (3).

(3)

where Q, P and CT are the coefficients of the quadratic term, the primary term and the constant term respectively. Therefore, the electrostatic force F_es can be expressed as Eq. (4).

(4)

Since the overlap area of two plates of the capacitor is very small, the elastic force acted on the cantilever can be expressed as Eq. (5), according to Hooke's law:

(5)

where k is the stiffness of the cantilever.

When the elastic force and electrostatic force applied on the cantilever are equal (i.e., F_i = F_es), the cantilever is in equilibrium. Eq. (6) can be derived from Eqs. (1), (2) and (5):

(6)

In order to reduce the uncertainty of calibration results, a difference method is used to calculate S_OL. The results of two experiments are taken as V_s1, V_p1 and V_s2, V_p2, and are substituted into Eq. (6):

(7)

Transforming the equations and subtracting the lower equation from the upper equation in Eq. (7), the parameters Q and k are eliminated. Then the calibration formula of S_OL is obtained, as shown in Eq. (8):

(8)

Performing a series of experiments, the curve is drawn with P(1/V_p1-1/V_p2) as the ordinate and 2(1/V_s1²-1/V_s2²) as the abscissa. The slope of the curve is S_OL.

After obtaining S_OL, the electrode will be replaced by different super-hydrophobic substrates. The interaction forces between droplets and super-hydrophobic substrates will be measured by the system shown in Figure 1.