Volume Control of Droplets in Vertical Contact-Separation Process Using Radius Difference of Solid–Liquid Interface on Substrate

Figure 2 shows the VCSP of droplets (\(V_{0} = 4\) µL). Here, the top and bottom WP radii, \(R_{\text{L}}\) and \(R_{\text{U}}\), were 1.24 and 1.23 mm, respectively. The upper and lower initial droplets were adjusted to face each other, and the lower and upper substrates were set to the stage and the z-axis control stage, respectively. The upper and lower droplets were opposite each other [Fig. 2(a)]. When the upper substrate was lowered, the droplets contacted each other [Fig. 2(b)]. After the contact, we lifted the upper substrate and observed a double-cone-shaped coalescent droplet immediately before separation [Fig. 2(c)].

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Figure 2. (Color online) VCSP of droplets. Microscopic images of (a) initial droplets before contact, (b) coalesced droplets, (c) double-cone-shaped coalescent droplet immediately before separation, and (d) droplets after separation. \(R_{\text{U}}\) and \(R_{\text{L}}\) are 1.40 and 1.24 mm, respectively. We estimated that \(\Delta V = 0.038\) µL. The interface between water and gas is fitted as a circular dotted line. We highlighted the solid–liquid interface with red dashed line. The radius of curvature of the droplet interface, pattern radius of the hydrophilic region, and droplet height are denoted by r, R, and h, respectively. Scale bar indicates 1 mm.

To obtain the volumes of the lower droplets after the VCSP (\(V_{1}\)) [Fig. 2(d)], we determined the curvature radius of the droplet (r), pattern radius of the hydrophilic region, (R), and droplet height (h) using image analysis. Assuming that the shape of the gas–liquid interface of the droplet is part of a sphere, the volume of the droplet, V, was given as \begin{equation} V = \int_{r-h}^{r} \pi(r^{2}-z^{2})\,\mathrm{d}z = \pi h^{2}\left(r-\frac{h}{3}\right). \end{equation} (1) We calculated the droplet volume prior to and after the VCSP and obtained the volume difference, \(\Delta V = V_{1} - V_{0}\).

Figures 3(a1) and 3(a2) show the photographs of the droplet prior to and after the VCSP, respectively. Here, the radius of the lower WP (\(R_{\text{L}} = 1.24\) mm) was approximately equal to that of the upper WP (\(R_{\text{U}} = 1.23\) mm). The height of the bottom droplet (h) increased from 1.27 to 1.47 mm. We observed that \(\Delta V = 0.95\,\text{$\unicode{x00B5}$L}> 0\) using Eq. (1). This indicates that water moves downward because of the effect of gravity, which correlates with the behavior reported in a study.²⁴⁾ We confirmed that the velocity of the upper substrate does not affect \(\Delta V\) in experimental conditions. Our experimental condition can be regarded as quasi-static. Thus, the dynamic effect such as inertia is negligible.²⁶⁾

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Figure 3. (Color online) Microscopic images of the lower droplet before and after the VCSP. (a) \(R_{\text{U}} < R_{\text{U}}^{*}\). (b) \(R_{\text{U}} = R_{\text{U}}^{*}\). (c) \(R_{\text{U}} > R_{\text{U}}^{*}\). The blue and white dotted lines indicate the gas–liquid interface of the lower droplets before and after the VCSP, respectively.

Next, we increased the radius difference of the solid–liquid interface (\(\delta R = R_{\text{U}} - R_{\text{L}}\)) by designing the radius of the upper WP (\(R_{\text{U}} = 1.40\) mm) to exceed that of the lower WP (\(R_{\text{L}} = 1.24\) mm). Prior to [Fig. 3(b1)] and after the VCSP [Fig. 3(b2)], we did not observe a change in the droplet height, indicating that the droplet volume hardly changed after the VCSP, i.e., \(\Delta V = 0\). Here, we defined the critical radius with \(\Delta V = 0\) as \(R_{\text{U}}^{*}\). \(R_{\text{U}}^{*} = 1.40\) mm for an initial droplet volume of 4 µL.

With a further increase in \(\delta R\) (\(R_{\text{U}} > R_{\text{U}}^{*}\)), the volume of lower droplets after the VCSP [Fig. 3(c2)] exceeded that prior to the VCSP [Fig. 3(c1)]. Thus, we obtained \(\Delta V < 0\), suggesting that the effect of \(\delta R\) overcame the effect of gravity; thus, water was transported against gravity.

The contact radius of droplets on the solid substrate equals the radius of hydrophilic regions because we used WP substrates in the observation. Thus, the contact angle of droplets on WP substrates depends on the relationship between droplet volume and the radius of hydrophilic regions. We set the initial volume of lower droplets so that hemispherical droplets on WP substrates were formed. If the volume of droplets after VCSP increases (decreases), the contact angle \(>\) (\(<\)) 90°, that is, \(\Delta V\) affects the contact angle.

Figure 4 shows the \(R_{\text{U}}\) dependence of the \(\Delta V\). For example, we observed a positive \(\Delta V\sim 1\) µL of a droplet with an initial volume of 4 µL under a symmetric boundary condition (\(R_{\text{L}}\sim R_{\text{U}} = 1.24\) mm). Thus, the \(\Delta V\) monotonically decreased with an increase in \(R_{\text{U}}\), and the volume of the lower droplets was maintained through the VCSP (\(\Delta V = 0\) µL) when \(R_{\text{U}}\sim R_{\text{U}}^{*} = 1.4\) mm. Eventually, a further increase in the \(\delta R\) inverted the sign of \(\Delta V\), suggesting that the effect of \(\delta R\) overcame the effect of gravity.

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Figure 4. (Color online) \(R_{\text{U}}\) dependence of \(\Delta V\). \(R_{\text{L}}\), \(R_{\text{U}}\), and \(\Delta V\) are the lower and upper WP radii and volume difference through the VCSP. The relationship between the initial volume (\(V_{0}\)) and \(R_{\text{L}}\) was given as \(V_{0}=\frac{2\pi}{3}R_{\text{L}}^{3}\).

In addition, when the initial droplet volume (\(\propto R_{\text{L}}^{3}\)) increased, the \(\Delta V\) increased even at a constant \(R_{\text{U}}\). This was because the effect of gravity became dominant for a large \(V_{0}\).²⁴⁾ Regarding \(R_{\text{U}}^{*}\), which realizes \(\Delta V = 0\) µL, \(R_{\text{U}}^{*}\) was observed to monotonically increase as \(R_{\text{L}}\) increased.

To predict \(R_{\text{U}}^{*}\), we constructed a theoretical model based on the force balance of gravity and surface tension. Figure 5 shows the schematic geometry of the proposed theoretical model.

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Figure 5. (Color online) Schematic geometry of the proposed model.

The \(\Delta V\) was determined by the shape of the droplets just before separation. Since we must consider boundary conditions on solid substrates, gravity, and surface tension at the air–liquid interface, the strict determination of the three-dimensional shape of droplets under the effect of gravity is difficult. Thus, for simplicity, we assumed that (i) the shape of coalescent droplets just before separation was a double cone [Fig. 2(c)]. We previously reported that we can predict the volume of droplets after VCSP quantitatively with the approximation (i).^24,25) In addition, we confirmed that droplets wet on hydrophilic regions. Thus, we assumed that (ii) the upper and lower radii of the solid–liquid interface (\(R_{\text{L}}\) and \(R_{\text{U}}\)) were constant during the VCSP. It is necessary for strict analysis to consider the deviation of the droplet shape from the cone shape such as neck structure and curvature.¹⁾

The distance between the WP substrates immediately before the separation was denoted by \(2D\); thus, the height of the double-cone-shaped coalescent droplet was \(2D\) (Fig. 5). Owing to gravity, the contact points of the two cones differed from the center of the two WP substrates, where a deviation of the contact point from the center was denoted by \(\Delta z\). During the observation, we confirmed that \(\Delta z\) was sufficiently small compared with \(R_{\text{L}}\), \(R_{\text{U}}\), and D. The heights of the lower and upper droplets were \(D +\Delta z\) and \(D -\Delta z\), respectively.

Regarding the geometry of coalesced droplets immediately before separation, the force balance between gravity and surface tension was given as \begin{equation} \frac{d}{d(\Delta z)}\{U(\Delta z)+\gamma S(\Delta z)\} = 0, \end{equation} (2) where \(U(\Delta z)\), γ, and \(S(\Delta z)\) are the potential energy, surface tension, and area of the gas–liquid interface, respectively.

\(U(z)\) was given as \begin{equation} U(\Delta z) = \pi\rho g(V_{\text{U}}x_{\text{U}}+V_{\text{L}}x_{\text{L}}), \end{equation} (3) where ρ, g, V, and x are the density of water, gravitational acceleration, volume of each droplet, and center of gravity of each droplet, respectively. We defined the contact point of the double cone as the origin of \(x_{\text{U}}\) and \(x_{\text{L}}\). Subscripts U and L indicate the upper and lower droplets, respectively. The volume and center of gravity of each droplet were obtained by the following equations. \begin{align} V_{\text{U}} &= \frac{1}{3}\pi R_{\text{U}}^{2}(D-\Delta z), \end{align} (4a) \begin{align} V_{\text{L}} &= \frac{1}{3}\pi R_{\text{L}}^{2}(D+\Delta z), \end{align} (4b) \begin{align} x_{\text{U}} &= \frac{3}{4}(D-\Delta z), \end{align} (5a) \begin{align} x_{\text{L}} &= - \frac{3}{4}(D-\Delta z). \end{align} (5b)

The total volume (\(V_{\text{U}}+V_{\text{L}}\)) prior to and after separation was stored in \(2V_{0}\) (\(=4\pi R_{\text{L}}^{3}/3\)).

Substituting Eqs. (4) and (5) into Eq. (3), \(U(\Delta z)\) was given as \begin{equation} U(\Delta z) = \pi\rho g\left(u_{0}+u_{1}\Delta z+\frac{1}{2}u_{2}\Delta z^{2}\right), \end{equation} (6a) where \begin{align} u_{0} &= \frac{R_{\text{U}}^{2}-R_{\text{L}}^{2}}{(R_{\text{U}}^{2}+R_{\text{L}}^{2})^{2}}R_{\text{L}}^{6}, \end{align} (6b) \begin{align} u_{1} &= \frac{R_{\text{L}}^{3}}{2(R_{\text{U}}^{2}+R_{\text{L}}^{2})^{2}}(3R_{\text{U}}^{4}-10R_{\text{U}}^{2}R_{\text{L}}^{2}+3R_{\text{L}}^{4}), \end{align} (6c) \begin{align} u_{2} &= \frac{(R_{\text{U}}^{2}-R_{\text{L}}^{2})^{3}}{2(R_{\text{U}}^{2}+R_{\text{L}}^{2})^{2}}. \end{align} (6d)

Next, the total area of the gas–liquid interface (S) was calculated to obtain the surface energy (\(\gamma S\)). S is the summation of the gas–liquid interface areas of the upper part, \(S_{\text{U}}\), and lower part, \(S_{\text{L}}\), as follows: \begin{equation} S = S_{\text{U}} + S_{\text{L}}, \end{equation} (7a) where \begin{align} S_{\text{U}} &= \pi R_{\text{U}}\{R_{\text{U}}^{2}+(D-\Delta z)^{2}\}^{\frac{1}{2}}, \end{align} (7b) \begin{align} S_{\text{L}} &= \pi R_{\text{L}}\{R_{\text{U}}^{2}+(D+\Delta z)^{2}\}^{\frac{1}{2}}. \end{align} (7c)

We substituted Eqs. (7b) and (7c) into Eq. (7a). Expanding to the second-order term of \(\Delta z\), we obtained \(\gamma S\) by \begin{equation} \gamma S(\Delta z) = \pi\gamma\left\{s_{0}-s_{1}\Delta z+\frac{1}{2}s_{2}\Delta z^{2}\right\}, \end{equation} (8a) where \begin{align} s_{0} &= \frac{R_{\text{U}}\lambda_{1}^{3}+R_{\text{L}}\lambda_{0}^{3}}{R_{\text{U}}^{2}+R_{\text{L}}^{2}}, \end{align} (8b) \begin{align} s_{1} &= \frac{8R_{\text{L}}^{4}}{R_{\text{U}}^{2}+R_{\text{L}}^{2}}\left(\frac{R_{\text{L}}R_{\text{U}}}{\lambda_{1}^{3}}-\frac{R_{\text{U}}^{2}}{\lambda_{0}^{3}}\right), \end{align} (8c) \begin{align} s_{2} &= 4R_{\text{U}}^{3}R_{\text{L}}^{2}(R_{\text{L}}^{2}+R_{\text{U}}^{2})\left(\frac{R_{\text{L}}}{\lambda_{1}^{3}}+\frac{R_{\text{U}}}{\lambda_{0}^{3}}\right), \end{align} (8d) \begin{align} \lambda_{1}^{6} &= 16R_{\text{L}}^{6}+R_{\text{U}}^{2}R_{\text{L}}^{4}+2R_{\text{U}}^{4}R_{\text{L}}^{2}+R_{\text{U}}^{6}, \end{align} (8e) \begin{align} \lambda_{0}^{6} &= 17R_{\text{U}}^{6}+R_{\text{L}}^{4}R_{\text{U}}^{2}+2R_{\text{U}}^{2}R_{\text{L}}^{4}. \end{align} (8f)

Substituting Eqs. (6a) and (8a) into Eq. (2), \(\Delta z\) was given as \begin{equation} \Delta z = \frac{s_{1}-\varepsilon u_{1}}{s_{2}+\varepsilon u_{2}}, \end{equation} (9a) where \begin{equation} \varepsilon = \frac{\rho g}{\gamma}. \end{equation} (9b)

Equation (9b) is a parameter that expresses the competition between gravity and surface tension. \(\varepsilon^{-1/2}\) corresponds to capillary length.¹⁾ To obtain the volume difference, the \(\Delta V = V_{\text{L}}-V_{0}\) of the lower droplet through the VCSP, we substituted Eq. (9a) into Eq. (4b). \begin{equation} \Delta V_{\text{L}} = \frac{2\pi}{3}\frac{R_{\text{L}}^{2}}{R_{\text{U}}^{2}+R_{\text{L}}^{2}}\left(R_{\text{L}}^{3}-R_{\text{L}}R_{\text{U}}^{2}+R_{\text{U}}^{2}\frac{s_{1}-\varepsilon u_{1}}{s_{2}+\varepsilon u_{2}}\right). \end{equation} (10)

Substituting \(\gamma = 77.75\) mN/m and \(\rho = 0.997\) g cm⁻³ into Eq. (9b), we numerically calculated the \(\Delta V = V_{\text{L}} - V_{0}\) under initial conditions of \(V_{0} = 3.5\), 4.0, 4.5, 5.0, and 5.5 µL. Figure 6 shows the \(\Delta V\) as a function of \(R_{\text{U}}\).

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Figure 6. (Color online) Theoretical curve of the \(R_{\text{U}}\) dependence of \(\Delta V\). \(R_{\text{L}}\), \(R_{\text{U}}\), and \(\Delta V\) are the radii of the lower and upper droplets and the volume difference of the lower droplet through the VCSP, respectively.

In the theoretical model, the \(\Delta V\) monotonically decreased with an increase in \(R_{\text{U}}\). In addition, the \(\Delta V\) monotonically increased as the initial volume (\(V_{0}\)) increased. These behaviors qualitatively correlated with the behaviors observed in the experiments.

To perform a quantitative comparison between the experimental and theoretical results, we focused on \(R_{\text{U}}^{*}\), which achieved \(\Delta V = 0\). Figure 7 shows the \(V_{0}\) dependence of \(R_{\text{U}}^{*}\). The theoretical prediction quantitatively correlated with the experimental results, indicating that the simple proposed model based on the force balance equation quantitatively predicted \(\Delta V\). In our calculation, we assumed that \(\Delta V\) is small. Thus, an increase in \(\Delta V\) should cause a deviation between theory and experiment. On the other hand, since \(\Delta V\) is minimized with \(R_{\text{U}} = R_{\text{U}}^{*}\), theoretical prediction quantitatively agrees with experimental results as shown in Fig. 7. The proposed theoretical model can be universally applied to DAST systems comprising liquids with known ρ and γ values. Furthermore, the study findings can be used as criteria to keep the \(\Delta V\) at zero and quantitatively control the volume of droplets.

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Figure 7. (Color online) \(V_{0}\) dependence of \(R_{\text{U}}^{*}\). \(V_{0}\) and \(R_{\text{U}}^{*}\) are the initial volume of the lower droplet and critical radius of the upper gas–liquid interface, respectively.