Introduction
Theory
Liquid droplet spreading dynamics
At the last stage of spreading, the drop approaches its final equilibrium shape and the radius of the wetted area grows with r ~ 𝑡1 10⁄ .26 This dynamics can be explained by the competition between capillary force from the curvature at the contact point and viscous powers. At the moment of contact, the curvature of the contact area is infinite, resulting in a divergent Laplace pressure jump. This causes a rapid flow in the droplet that replenishes the liquid neck, producing a wet area that grows over time.
Using the relationship between w and r, we can calculate w at the early stage where w< Here, the contact radius in the vertical axis and time in the horizontal axis are normalized by the spherical drop radius and the characteristic inertial time, 𝒕𝒄 = √𝑹𝟑/𝜸, 29, respectively. The Helmholtz model in (a) is the simplest approximation that the surface charges are neutralized by counterions placed at a distance d from the surface. Therefore, the surface charge potential decreases linearly from the surface to the counterions satisfying charge neutrality. The theoretical treatment of the Helmholtz model does not adequately explain all the features because it assumes the rigid layer of opposite charges. In this model, counterions are considered as point charges that form a diffuse double layer in which the change in concentration of the counterions near a charged surface follows the Boltzmann distribution. Here n0 is the bulk concentration, z is the charge of the ion, e is the charge of a proton, k is Boltzmann's constant and T is the temperature. The first layer of solvated ions of finite size is firmly adsorbed on the surface and the subsequent layers consist of point charges as in the Gouy-Chapman model. Therefore, at high surface charge potential, the diffuse layer between the surface and solution charge zones decreases towards zero and d increases. As soon as the top electrode comes into contact with a drop of water sitting on the bottom electrode, electrons flow through the circuit (and the load resistor) to lower the free energy in the system. Due to the voltage difference between the upper and lower electrodes, electrons flow from the bottom up and the opposite ions in the water will redistribute in a direction that favors the elimination of the voltage difference between the upper and lower electrodes. The contact of a water droplet with the upper electrode is equivalent to closing a switch in an electrical circuit. When the top electrode makes contact with the water droplet, electrons flow from the bottom EDLC to the top EDLC through the load resistance and the counter ions in the water are redistributed.18-19 The following equation describes this non-equilibrium behavior. But the contact area of the top interface is changing during bridge formation, so 𝐶𝑇 is time dependent. Since the contact area of the top electrode can be related to the radius of the top contact area, 𝐴𝑇(𝑡) = 𝜋(𝑟(𝑡))2, we can find the relation between the spread radius 𝑟(𝑡) and the measured voltage . 𝑉(𝑡). The increase in charge at the tip after forming a closed circuit can be calculated by measuring the voltage with time using Ohm's law. Here, ℓ is the distance between the surface of the convex lens and the top surface of the [ ITO glass slide and m is. The reflected beam intensity from the top ITO (I1) is fixed. In fact, there are two reflected rays from the ITO: one from the upper surface and the other from the lower surface of the upper ITO glass slide), but both do not depend on the sample height change (h0). But another reflected beam intensity from the plano-convex lens surface (I2) depends on the sample height (h(r=0) = h0). The total intensity of the reflected beam (I) is a function of the phase difference (Ø ) from the difference in the path length between the two reflected beams.39-41. Here, k and 𝛿0 are the wave number of light in air and the initial phase difference between I1 and I2, respectively. Linear motion of the translation stage is confirmed by using a convex lens as shown in Figure 3-4. As a result, the linear motion of the vertical translation stage is consistent with the measured data as shown in Figures 3-4 and 3-6. 4. The dispersion dynamics during the initial period of less than 5ms is due to the mixed effect of the surface energy minimization and electrical energy relaxation. When a convex lens is used as shown in Figure 4-3(a), the interference intensity Iexp. r=0) at the center of the RICM interference pattern agrees with the expected ITheory value, where the deviation (Iexp-ITheory) is negligible (see Figure 4-3(c)). But when a water drop is used, the interference intensity Iexp (r=0) at the center of the RICM interference pattern starts to deviate from the expected ITheory value for the undeformed water drop at the specific height. When there is no time deformation of the spotting surface, the interference intensity should follow equation (2.16). However, when the distance is sufficient, the large attractive force deforms the upper edge of the water drop as in (b). Experimental setup Since the radius of the droplet in this experiment is R=2.78 mm, the expected tc value is approximately 17 ms. To observe the deformation of the water surface, the water droplet is brought to the upper substrate at the same speed as shown in the figure. It means that there is some change in the curvature of the surface of the water drop starting from the specific height. The open blue circles in Figures 4-4(e) and (f) show the interference intensity Iexp of the RICM interference pattern as a function of the radial distance from the center of symmetry. From the interference intensity and Equation (4.2), the morphology of the surface deformation can be obtained using the measured distance between the water droplet and the upper ITO substrate. To understand the spreading dynamics of water droplets, the spreading radius and voltage generation are simultaneously measured. When a plano-convex lens (solid hemisphere) approaches the ITO surface, the observed interference patterns match well with the expected linear motion of the lens surface. Given the balance of forces between inertial and capillary pressure, the radius of expansion should increase with t1/2 according to equation (2.4). The top view image in the upper part (c) and the side view in the lower part (c) are measured simultaneously. When watching the spread only from the sideline, it is difficult to distinguish the exact time of the touchdown. We can measure the spreading dynamics from 10-6 to a few ms, but they did that from a few ms to s. The propagation radius for a time greater than 5 ms mainly comes from relaxation of electrical energy. Calculated charge during the spreading of water droplets using the voltage generation and the equation 𝒒(𝒕) =𝟏. d) Dispersion radius of water droplets as a function of time: the gray curve shows the calculated value from q(t) using equation (2) and the orange curve shows the direct image measurement. But the values of the fit parameters of 𝜀 and d are different from the known values for bulk water and the reported values at the interface. As a calibration, a plano-convex lens is raised at a stage speed of 5μm/s until it contacts the top substrate. The faster swing of the curve means the water point is approaching faster than the phase speed. From the starting point of the deflection shown in Figure 4-3(d), we determine the starting point of the acceleration of the water surface. Interference light intensity as a function of radial distance from the symmetric center in RICM interference pattern. a), (b) Fringe patterns at 4 different heights for convex lens (a) and for water droplets (b). When a convex lens is used, the interference intensity Iexp (open gray circles) agrees with the expected value ITheory (red solid curve). However, when a water droplet is used, the interference intensity Iexp starts to deviate from the expected value ITheory at specific distance. But in (b) at h0 = 100nm, there is a large discrepancy between the measurement and theoretical expectation, and the amount of deviation is reduced by increasing the radius. If we consider the change in distance due to the curvature of the surface, the reduced deviation makes sense. In Figure 4-5(b), the measurement is in good agreement with theory with a curve shift of 28 nm (=∆d). a) Measured strain length (empty black circles) as a function of distance h0. The dashed gray line represents the top position of the drop without deformation, only due to the linear motion of the upward moving translation stage. Using the same method, we can calculate the deformation length from the symmetrical center r=0 to r=200μm and plot the morphology of the surface deformation. Interestingly, deformation amounts are well matched with the exponential fit (solid red curve) than the 2nd order polynomial fit (dashed blue curve). As shown in this figure, the amount of deformation is higher in the middle part of the water surface. In relation to these questions, we must also consider water evaporation and the non-equilibrium deformation morphology of the system. Non-equilibrium dynamics of surface deformation Competition between deformation and evaporation To investigate the solid-liquid interface we build up a home-made experimental system: combination of voltage generation system and RICM. Based on the electric double layer capacitor (EDLC) model, we found a strong correlation between the generated voltage and the distribution radius. To study the surface deformation due to the interaction between the solid and liquid surfaces before the contact, the RICM system with nanometer resolution is used. But when a water droplet surface is used instead, there is a deformation of the droplet surface without any externally applied electric field. Surface morphology indicates water bridge formation depending on the distance between the solid and liquid surfaces. Pak, "Probing surface charge density at solid-liquid interfaces by modulating the electrical double layer," J. Ugaz, “An interference-based nanometer-scale probe of interfacial phenomena between microscopic objects and surfaces” Nature Communications 4, 1919.
Electrical double layer capacitor model
Voltage generation during the water droplet spreading
Reflection Interference Contrast Microscope
Sample preparation
Interference fringe with linear motion
Result and discussion
Relation between water droplet spreading and voltage generation
Comparison between the convex lens calibration and the water droplet deformation
Deformation of the water droplet surface by long range interaction
Physical origin for surface deformation
