3.1. Stages of the coalescence process

We tracked the outer surface of the two coalescing immiscible droplets as well as their contact line, and the resulting image sequences are shown in figure 2 (also see supplementary movie 1 available at https://doi.org/10.1017/jfm.2023.685). During droplet coalescence, the governing forces involved are capillary force, inertia and viscous forces but could be dominant in different regions. The time scales which characterize different dynamics, therefore, could vary from around 1 to 2 ms to several minutes (Narhe et al. Reference Narhe, Beysens and Pomeau2008). Based on the time scale and controlling forces (which are analysed in subsequent sections), we divided the coalescence process of two immiscible droplets into three primary stages, namely (I) the growth of the liquid bridge, (II) the oscillation of the coalescing sessile droplet and (III) the formation of a partially engulfed compound sessile droplet (PECSD) and the subsequent retraction. Each stage is dominated by different forces and exhibits different phenomena.

Figure 2.

Typical time sequences of coalescence process of immiscible droplets. The two immiscible liquids are water and $n$-dodecane. The camera was tilted slightly downward (at an angle of $5^\circ$ to the horizontal) to afford better observation. The volume ratio ${{V}^{*}}={{{V}_{{o}}}}/{{{V}_{{w}}}}$ is 0.5117, where ${V}_{o}$ and ${V}_{w}$ are the volumes of the oil droplet and the water droplet, respectively. (A video clip for this process is available as supplementary movie 1.)

(I) The growth of the liquid bridge. The instant of the first contact in the image sequence is defined as $t = 0$, as shown in figure 2(a). After the two droplets contact, a liquid bridge quickly builds up connecting the two droplets, as shown in figure 3(a). The large curvature formed on the liquid–vapour surface in the bridge area induces a large capillary force, which drives the quick growth of the liquid bridge. The growth rate decreases gradually as the bridge curvature reduces and as more liquid is set in motion. When the bridge height is comparable with the height of the oil droplet (see figure 2c), the further growth of the liquid bridge is very slow, which can be seen from the time stamps below the snapshots. For example, it takes only around 1.32 ms for the bridge to grow to half of the oil droplet height, but it takes more than twice as long to reach the height of the initial oil droplet as shown in figures 2(c) and 2(d). When the bridge height increases to the oil droplet height, it indicates the finish of the first stage. Another prominent feature is the three-phase contact line on the surface of the droplets and as shown in figures 1(c) and 1(d), there is a net force along the water surface considering the relative magnitude of the three surface tensions and dynamic contact angles around the three-phase contact line. Consequently, one would expect a precursor film of oil to be driven onto the surface of the water (Cuttle et al. Reference Cuttle, Thompson, Pihler-Puzović and Juel2021; Sanjay et al. Reference Sanjay, Sen, Kant and Lohse2022) due to the surface tension gradient, which is often observed in three-phase flows. However, the precursor film is believed not to show up in this first stage since we can observe a sharp discontinuous interface in the bridge region as can be seen in figures 1(c) and 1(d). In addition, according to Koldeweij et al. (Reference Koldeweij, van Capelleveen, Lohse and Visser2019), the evolution of the leading edge of the oil film is $L(t)\sim \Delta \sigma ^{{1}/{2}}{{t}^{{3}/{4}}}/{{{(\mu \rho )}^{{1}/{4}}}}$, where $\Delta \sigma$ is the surface tension difference and $\mu$ and $\rho$ are the dynamic viscosity and density of the droplet of higher surface tension, respectively. Based on this relation, the time for the leading edge of an oil precursor film to reach the top of the water droplet is calculated to be as long as 3–4 ms, which exceeds the time scale of liquid bridge evolution (less than 2 ms in the present case). This further indicates that the precursor film would not influence the bridge growth.

Figure 3.

(a) The liquid bridge height ${h}_{m}(t)$ is defined as the vertical distance of the bridge neck to the substrate (white dashed line). (b) Illustration of the curvature radius (${r}_{c}$) in the bridge region and is measured by fitting an inscribed circle to the surface shape in the bridge region following the method of Thoroddsen et al. (Reference Thoroddsen, Qian, Etoh and Takehara2007). (c) Snapshots showing the close-up of the liquid bridge of water droplet coalescence with a low-viscosity droplet (bromocyclopentane). The droplet volume ratio ${V}^{*}$ = 0.4315. (d) Snapshots showing the close-up of the liquid bridge of water droplet coalescence with a high-viscosity droplet (silicone oil, 100 cSt). The droplet volume ratio ${V}^{*}$ = 0.635. (e) Snapshots showing the close-up of the liquid bridge of water droplet coalescence with an ethanol droplet. The droplet volume ratio ${V}^{*} = 0.5894$. The overlaid yellow lines on the images are the initial shape of the two droplets. ( f) Evolution of curvature radius (${r}_{c}$) with bridge height. The inset shows the data in a log–log plot, where the solid lines have a slope of 1. Both the bridge height and the curvature radius are rescaled with the initial height of the oil droplet. The scale bars (white solid lines) are 1 mm. (Video clips for these processes are available as supplementary movies 2–4.)

In addition, during the growth of the liquid bridge, surface capillary waves, which are common in the coalescence of miscible droplets on solid surfaces (Eddi et al. Reference Eddi, Winkels and Snoeijer2013; Lee et al. Reference Lee, Kim, Chandra and Yoon2013; Sykes et al. Reference Sykes, Harbottle, Khatir, Thompson and Wilson2020b), are also observed in the present study for immiscible droplets (highlighted by the yellow arrows in figure 3c).

(II) The oscillation of the coalescing sessile droplet. Following the first stage, the coalescing droplet oscillates due to the surface tension and the inertia of the droplet. When the two droplets contact, their pressure difference can be estimated to be positive, $\Delta P={{{\sigma }_{{wa}}}}/{{{R}_{{w}}}}-{{{\sigma }_{{oa}}}}/{{{R}_{{o}}}}>0$, where ${{R}_{{w}}}$ and ${{R}_{{o}}}$ are the initial radii of the water and oil droplets, respectively. Therefore the water droplet first moves towards the oil droplet due to the relatively larger capillary pressure in the water droplet. After the accumulation of the inertia, the droplet overshoots even after the equilibrium point. Surface tension, then, causes the droplet to move backward. This process repeats as the interchange between the surface energy and the kinetic energy with some part of the energy being dissipated both in the bulk and in the contact line. The oscillation process is in a much longer time scale compared with the initial stage of liquid bridge growth.

(III) The formation of PECSD and its subsequent retraction. The third stage ensues as the droplet oscillation gradually relaxes. Driven by capillary forces and resisted by both inertia and viscous forces, the oil is seen surrounding the water droplet and filling under the water droplet. As a consequence, the water droplet can be displaced from the solid by the oil (see figures 2e and 1e), forming a PECSD on the solid surface. A clearer illustration of the displacement process can be seen in figure 1(e), which was obtained through different lighting methods. The PECSD then moves and contracts to decrease the contact area due to the minimization of the surface energy in a much longer time scale (see figure 2f). This slow stage is characterized by the four-phase capillary-driven flow and the retraction of the PECSD.

Since the three stages are controlled by different mechanisms, the dynamics of each stage is analysed in the subsequent sections.

3.2. Growth of liquid bridge

The dynamics of the first stage in the coalescence process is characterized by the rapid growth of the liquid bridge, as shown in figure 3(a). For two sessile droplets on a partial wetting surface, the coalescence of a water droplet with a high-viscosity oil droplet can be different from that with a low-viscosity oil droplet, because the liquid bridge growth could be affected by the extra viscous stress imposed by the substrate and the oil–water–air contact line that is unique to immiscible liquids. Therefore, the bridge growth is expected to exhibit different behaviours for oil droplets with low viscosity and high viscosity. Here, the high-viscosity droplet and low-viscosity droplet were differentiated by the Ohnesorge number ${Oh}={{{\mu }_{o}}}/{\sqrt {{{\rho }_{o}}{{h}_{o}}{{\sigma }_{oa}}}}$, where ${{h}_{o}}$ is the initial height of the oil droplet. Therefore, the low-viscosity oil droplet is identified to have Oh of $O(1)$ while the high-viscosity droplet is identified to have $Oh$ much larger than 1.

We first focus on the coalescence of water droplets with low-viscosity oil droplets. For the coalescence of inviscid spherical droplets of identical liquids, the liquid bridge motion has been proved to be driven by capillary forces and resisted by inertia, and this short-time dynamics can be described as the inertia-controlled regime, where the bridge grows as $t^{1/2}$ (Duchemin et al. Reference Duchemin, Eggers and Josserand2003). While the situation is markedly different in the case of sessile droplets, the evolution of the bridge height ($h_m$ illustrated in figure 3a) is highly dependent on the droplet geometry before contact. The capillary pressure driving the bridge motion is scaled as ${{P}_{{c}}}\propto {\sigma }/{w}$, and the dynamic pressure in the fluids is estimated as ${{P}_{{i}}}\propto \rho {{( {{{h}_{{m}}}}/{t} )}^{2}}$, where, for contact angles $\theta < 90^\circ$, the meniscus scale $w$ was found to be proportional to the bridge height (Eddi et al. Reference Eddi, Winkels and Snoeijer2013): $w\propto {{h}_{{m}}}$. Then, based on the force balance analysis involving the capillary pressure and the dynamic pressure, the bridge growth is predicted to follow ${{h}_{{m}}}\propto {{t}^{{2}/{3}}}$ (Eddi et al. Reference Eddi, Winkels and Snoeijer2013).

In this study of immiscible droplet coalescence, we can see from figure 3(c–e) that the bridge region mainly appears on the lower-surface-tension droplet (i.e. oil), and the bridge height exhibits an almost consistent power-law growth ${{h}_{{m}}}\propto {{t}^{\alpha }}$ as shown in figure 4(a). By referring to the fitting method adopted by Dekker et al. (Reference Dekker, Hack, Tewes, Datt, Bouillant and Snoeijer2022), the exponent was obtained by conducting linear fitting in a log–log plot (to determine the first contact instant accurately, the time was slightly shifted until the fitting can achieve the smallest residual sum of squares). We can see that the exponents are all distributed around 2/3 (see figure 4b). The rapid bridge growth is driven by the large capillary pressure in the bridge region induced by the large curvature. We then trace the radius of curvature during the bridge growth as shown in figure 3( f), and we can see that the curvature follows approximately a linear growth with the bridge height for both the miscible droplet pairs and immiscible droplet pairs. This linear relationship indicates that the immiscible interface does not change the evolving geometrical relationship between the bridge curvature and the bridge height. When incorporating (${{r}_{c}}\propto {{h}_{m}}$) into the balance between capillary pressure ${{P}_{{c}}}\propto {\sigma }/{w}$ and dynamic pressure ${{P}_{{i}}}\propto \rho {{( {{{h}_{{m}}}}/{t} )}^{2}}$, it gives the same 2/3 power law of bridge growth of immiscible droplets. However, when comparing the time that it takes for the liquid bridge height to reach half of the initial height of the oil droplet (see figure 4f), the time for immiscible droplets could be two to three times longer than that for miscible droplets of similar low viscosity. The comparatively longer bridge growth time of immiscible droplets indicates that the growth speed is lower. This lower growth speed can be attributed to the lower kinetic energy converted from the released surface tension energy during the coalescence, as an extra immiscible interface is formed in the immiscible droplets. This immiscible interface would consume part of the released energy that should be converted to kinetic energy. In contrast, this water–oil interface is absent in miscible droplet coalescence.

Figure 4.

(a) Evolution of the liquid bridge height for water droplets with different low-viscosity oil droplets. The inset shows the data in a linear plot. (b) Fitted exponent $\alpha$ under various oil droplets with different viscosities; the dashed line shows $\alpha = 2/3$. (c) Evolution of the liquid bridge height for water droplets with different high-viscosity oil droplets. The inset shows the oil meniscus between the oil droplet and the water droplet. (d) Evolution of the liquid bridge height for water droplets with different miscible droplets made of EWM. (e) Fitted exponent $\alpha$ of the coalescence of miscible droplets with different viscosities. The dashed line shows $\alpha = 2/3$. ( f) Time that it takes for the liquid bridge to reach half of the initial height of the oil droplet.

These results show that for two droplets of approximately inviscid fluids, the immiscibility of the two droplets has no significant effect on the growth scaling of the liquid bridge height, which follows the same power law as that of the inertial coalescence of the same droplets, i.e. ${{h}_{{m}}}\propto {{t}^{{2}/{3}}}$, but it can substantially decrease the growth speed of liquid bridges. This is mainly because the curvature of the bridge is mostly on the lower-surface-tension droplet. Therefore, the large capillary force driving the bridge growth is provided by the oil droplet. Though a water–oil interface is formed after the contact of the two droplets, its size is small and it has a certain distance to the position of the minimum bridge height. Hence, the effect of the water–oil interface on the bridge growth is weak. Therefore, the growth of the liquid bridge height follows the ${{h}_{{m}}}\propto {{t}^{{2}/{3}}}$ power law.

Further, it is noted that, in the present configuration, the two droplets have different contact angles and, therefore, different sizes. To look into the possible size effect, we first, by assuming the length scale that influences the bridge growth to be the initial oil droplet height ${{h}_{o}}$, construct a characteristic time scale ${{t}_{\sigma }}=\sqrt {{{{\rho }_{avg}}{{h}_{o}}^{3}}/{\Delta \sigma }}$, where ${{\rho }_{avg}}={( {{\rho }_{o}}+{{\rho }_{w}} )}/{2}$ is the average density and $\Delta \sigma ={\sigma }_{{wa}}+{\sigma }_{{oa}}-{\sigma }_{{wo}}$ is based on the net interfacial tension as shown in figure 1(d). The bridge growth data in figure 4(a) can then be collapsed together (see figure 5b), which indicates that the bridge growth is predominantly influenced by the size of the oil droplet compared with the water droplet. It also indicates that the properties of the oil droplets affect the time scale of the bridge growth but do not alter the 2/3 scaling. Similarly, in the experiments of coalescence of asymmetric miscible droplets (Pawar et al. Reference Pawar, Bahga, Kale and Kondaraju2019a), where the contact angle of one droplet is smaller than $90^\circ$ and is $90^\circ$ for the other, the bridge growth is found to be dominated by the droplet of contact angle smaller than $90^\circ$ and to follow the same 2/3 power law but with different prefactors. Here, we do not intend to go further into discussing the exact prefactors but aim to test the power-law dynamics of the bridge growth. In our experiments, the contact radii of the two droplets are limited to be approximately the same to minimize the possible relative size effect.

Figure 5.

(a) Linear plot of the data in figure 4(a). (b) Rescaling of the data, where the bridge height was scaled by the initial oil droplet height ${h}^{*}$ = ${h}_{m}/{h}_{o}$ , and the time was rescaled with the constructed time scale ${t}^{*}$ = ${t}/{t}_{\sigma }$.

Moreover, for the coalescence of water droplets with high-viscosity oil droplets (e.g. silicone oils of viscosities ranging from 50 to 500 cSt), the bridge growth is different from that of the low-viscosity droplets in many ways. First, no distinct surface wave is observed on the surface of droplets, while a much sharper curvature in the bridge region is observed, similar to the droplet coalescence scenario of two high-viscosity miscible droplets (Thoroddsen et al. Reference Thoroddsen, Qian, Etoh and Takehara2007). However, seen from the log–log plot in figure 3( f), the curvature radius also evolves nearly linearly within a period, which may account for the 2/3 power-law growth of the bridge. Second, the high viscosity influences the initial growth of the liquid bridge, which leads to a much slower process, as shown in the log–log plot of the data (when $t \ll 1000\ {\mathrm {\mu }}{\rm s}$ in figure 4c). We speculate that this slow motion could be correlated with the extra viscous resistance in the meniscus (see the inset of figure 4c), which forms when the oil droplet contacts the water droplet. The sharp curvature of the meniscus also indicates large stress in that region, which is reminiscent of the case of a water droplet slowly moving on a viscous-oil-impregnated surface (Keiser et al. Reference Keiser, Keiser, Clanet and Quéré2017): during the moving process, the oil meniscus with a sharp curvature was also found surrounding the water droplets and was proposed to be responsible for significant viscous resistance.

3.3. Oscillation of the coalescing droplet

Due to the liquid velocities induced by the bridge growth and the released surface energy resulting from the coalescence, the droplet surface undergoes further deformation, and then the droplet bulk fluid oscillates. Since the two droplets have different surface tensions and different contact angles, the imbalanced Laplace pressure further generates an obvious horizontal movement. Therefore, we trace the horizontal movement of the droplet centroid, $\Delta X$, in the side-view images by image processing, which could represent the bulk movement of the droplet (Somwanshi, Muralidhar & Khandekar Reference Somwanshi, Muralidhar and Khandekar2017; Zhao et al. Reference Zhao, Orejon, Sefiane and Shanahan2021), to investigate the droplet oscillation dynamics. In the present sessile droplet configuration, the values of Bond number ${Bo={{\rho } g{{R}^{2}}}/{\sigma }}$ are rather low, being around 0.034 and 0.078 for 1 mm water droplet and oil droplet in air, respectively. According to the analysis of droplet oscillation in Lamb (Reference Lamb1924), considering the contribution of both surface tension and gravity, the oscillation frequency follows ${f={\sqrt {{g(2l+1)}/{[ 2l(l-1)R ]}+{{\sigma } l(l-1)(l+2)}/{( {\rho }{{R}^{3}} )}}}/{2{\rm \pi} }}$, where $l$ is the oscillation degree mode ($l \geqslant 2$). The effect of gravity on the oscillation frequency is estimated to be less than 1 % in our experiments. This indicates that gravitational forces are less predominant than capillary forces for small droplets, and that capillary forces serve as the dominant restoring force in the present study of the oscillation process.

The oscillation for different immiscible droplet pairs shows different behaviours, as shown in figure 6(a–c). For example, in the case of a silicone oil droplet with a water droplet, the oscillation shows a much larger initial displacement amplitude and a lower oscillation period than that when the oil droplet is $n$-octanol or $n$-decanol. This different oscillation behaviour should not be attributed to the viscous effect in the low-viscosity liquid, because the viscosity of the 10 cSt silicone oil droplet ($9.3\ {\rm mPa}\ {\rm s}$, in figure 6a) is even higher than that of the octanol droplet ($7.4\ {\rm mPa}\ {\rm s}$, shown in figure 6b). Of course, when the oil droplet has a high viscosity, no oscillation behaviour is observed as shown in figure 6(c), which can be explained by the damping effect caused by the high viscosity (Khismatullin & Nadim Reference Khismatullin and Nadim2001). However, for the cases shown in figures 6(a) and 6(b), the fluid viscosities are always low ($3.9\unicode{x2013}12.2\ {\rm mPa}\ {\rm s}$). Considering surface tension is the main cause for the oscillation in droplet coalescence dynamics where it is free of external forces (Chashechkin Reference Chashechkin2019), and the liquids used in figures 6(a) and 6(b) have rather different oil–water interfacial tensions and also different surface tensions, we thus speculate that both the interfacial tension and the surface tension should have a marked influence on the droplet oscillation dynamics, and the influence of interfacial tension on the droplet oscillation dynamics should be further investigated. To avoid the damping effect of high viscosity, low-viscosity oil droplets are then used to analyse the oscillation behaviour.

Figure 6.

Temporal evolution of the normalized horizontal centroid displacement after the coalescence of a water droplet with different oil droplets: (a) silicone oil (10 cSt) and hexadecane; (b) $n$-decanol and $n$-octanal; (c) silicone oil (100 and 500 cSt). Radius $R_c$ is the equivalent radius of the two droplets, ${{R}_{{c}}}=\sqrt [3]{{3( {{V}_{{o}}}+{{V}_{{w}}} )}/({4{\rm \pi} }) }$, and $V_o$ and $V_w$ represent the volumes of water droplets and oil droplets, respectively. Evolution of the outline of the coalescing droplet: (d) water droplet with hexadecane droplet; (e) water droplet with $n$-decanol droplet. The corresponding moments are marked in (a,b).

We first consider the oscillation process from the perspective of energy conservation. The immiscible water–oil interface would contain a non-negligible portion of the released surface energy which is proportional to the interfacial tension ${{\sigma }_{{wo}}}$, resulting in a decreased energy to be converted to the droplets’ kinetic energy given that the viscous dissipation influence is comparably smaller for the low-viscosity droplets used. Therefore, the droplet pair with a larger water–oil interfacial tension would end up in a smaller displacement amplitude, as shown in figure 6(a). Further, by plotting the oscillation periods under different interfacial tensions (see figure 7a), we can see that the oscillation periods show a generally decreasing trend with the interfacial tension. This variation indicates that the interfacial tension, which is unique to the immiscible system, could markedly affect the coalescence process and lead to different oscillation dynamics. Secondly, the surface tension can quantify an oscillating droplet's ability to restore (Deepu, Chowdhuri & Basu Reference Deepu, Chowdhuri and Basu2014), and therefore could also affect the oscillation periods. However, here, both the oil surface tensions and the water–oil interfacial tensions are different, indicating that their frequency differences cannot be attributed to the surface tension or interfacial tension alone. For example, the oscillation period of the water–octanol droplet pair is almost twice as long as that of the water–$n$-hexadecane droplet pair (these two droplet pairs have different surface tensions and interfacial tensions) as shown in figures 6(a) and 6(b). Therefore, both the surface tension and the interfacial tension should be considered in the analysis of the oscillation dynamics of the coalescing droplets.

Figure 7.

(a) Oscillation period $\Delta T$ of the coalescing immiscible droplets versus the water–oil interfacial tension. (b) Oscillation period $\Delta T$ versus the capillary time scale ${t}_{\sigma,e }$ in (3.1).

Though the role of surface tension in the oscillation of simple droplets has been widely studied (Khismatullin & Nadim Reference Khismatullin and Nadim2001; Menchaca-Rocha et al. Reference Menchaca-Rocha, Martínez-Dávalos, Núñez, Popinet and Zaleski2001; Chireux et al. Reference Chireux, Fabre, Risso and Tordjeman2015; Yuan et al. Reference Yuan, He, Fang, Bao and Liu2015; Zhao et al. Reference Zhao, Orejon, Sefiane and Shanahan2021), the oscillation of a coalescing droplet, which is produced after the coalescence of two immiscible droplets, is more complex. Here, we investigate the oscillation dynamics by considering both surface tensions and water–oil interfacial tensions. The mass–spring system is commonly used to model the long-time oscillation behaviour of coalescing droplets, during which surface tension is the restoring force opposing the deformation, and the oscillation of the coalescing droplet can be considered as a capillary–inertia system (Chireux et al. Reference Chireux, Fabre, Risso and Tordjeman2015; Yuan et al. Reference Yuan, He, Fang, Bao and Liu2015). An immiscible liquid system in capillary–inertia controlled processes (such as droplet impact and oscillation) can be regarded as a parallel spring system (Bernard et al. Reference Bernard, Baumgartner, Brenn, Planchette, Weigand and Lamanna2020), and the equivalent spring constant can be represented by the summation of surface tensions and water–oil interfacial tension. With such an analogy, we can first obtain an equivalent spring constant, which is ${{\sigma }_{{e}}}={{\sigma }_{{wa}}}+{{\sigma }_{{oa}}}+{{\sigma }_{{wo}}}$.

Next, in order to see how the oscillation dynamics is related to the interfacial tension, we evaluate the capillary time scale by taking the equivalent spring constant (i.e. the effective capillary force) into consideration:

(3.1) \begin{equation} {{t}_{\sigma ,e}}=\sqrt{\frac{{{m}_{w}}+{{m}_{o}}}{{{\sigma }_{e}}}}=\sqrt{\frac{{\rm \pi} [ \rho_w ( 3{{r}_{w}}^{2}{{h}_{w}}-{{h}_{w}}^{3} )+\rho_o ( 3{{r}_{o}}^{2}{{h}_{o}}-{{h}_{o}}^{3} ) ]}{6{{\sigma }_{{e}}}}}, \end{equation}

where ${{r}_{{w}}}$, ${{h}_{{w}}}$ and ${{r}_{{o}}}$, ${{h}_{{o}}}$ are the initial contact radii and heights of the water and oil droplets, respectively. We then plot the oscillation period versus this proposed capillary time scale ${t}_{\sigma,e}$. As shown in figure 7(b), the oscillation period $\Delta T$ exhibits a good linear relationship with the proposed capillary time scale ${t}_{\sigma,e}$, i.e. $\Delta T\propto {t}_{\sigma,e}$. The linear relationship indicates that ${t}_{\sigma,e }$ properly captures the two immiscible droplets’ oscillation dynamics. It also confirms that apart from the surface tensions of the two liquids, the interfacial tension between the two liquids also acts as a non-negligible restoring force which affects the oscillation.

3.4. Formation and retraction of PECSD

After the contact of the two immiscible droplets, in a much longer time scale, the oil is seen spreading below the water droplet driven by the unbalanced capillary force at the liquid–solid interface. Such a capillary-driven flow will eventually lead to the water droplet being displaced by the oil on the substrate, as illustrated in the inset of figure 8(a), which then forms the PECSD. This PECSD then contracts to further lower the system energy.

Figure 8.

(a) Time taken for the oil droplet to displace the water droplet from the solid substrate ${{t}_{{displace}}}$ under different ${Oh}_s$. The inset illustrates the interface shapes before and after the displacement process, where ${{\sigma }_{s}}$, ${{\sigma }_{ws}}$ and ${{\sigma }_{os}}$ are the interfacial tensions of solid–air, water–solid and oil–solid, respectively, and ${{\sigma }_{oa}}$, ${{\sigma }_{wa}}$ and ${{\sigma }_{wo}}$ are the interfacial tensions of oil–air, water–air and water–oil, respectively. (b) Displacement time ${{t}_{{displace}}}$ scaled with an inertia time scale ${{t}_{\rho }}$ (red circles) and a viscous time scale ${{t}_{\nu }}$ (blue diamonds).

We analyse the droplet displacing process by referring to the capillary-driven flow in the three-phase system, whose dynamics is characterized by the spreading parameter $S={{\sigma }_{ij}}-({{\sigma }_{ik}}+{{\sigma }_{jk}})$ (Adamson & Gast Reference Adamson and Gast1967). It should be noted that for our four-phase flow (i.e. water, oil, air and solid), the interfacial tension between the liquid and the solid also influences the spreading dynamics of the droplet (Bonn et al. Reference Bonn, Eggers, Indekeu, Meunier and Rolley2009), indicating that the wettability of the solid surface should be taken into consideration. We consider this capillary-driven flow from the energetic perspective. When an area $\Delta A$ on the substrate is displaced by the oil (as illustrated in figure 8a), the released surface energy can be written as

(3.2)\begin{equation} \Delta E _\sigma=\Delta A({{\sigma }_{oa}}+{{\sigma }_{ws}})-\Delta A({{\sigma }_{wo}}+{{\sigma }_{os}}), \end{equation}

where $\Delta A$ is the surface area where the water–solid contact area is replaced by the oil–water and oil–solid surfaces. It should be noted that this is a simplified form, as there is a difference between the area along the substrate $\Delta {{A}}$ and the area along the curved surface of the droplet $\Delta {{A}_{2}}$. The two areas can be correlated with a geometrical relation: $\Delta {{A}_{2}}={\Delta {A}/{\cos \alpha }}$, where $\alpha$ is the contact angle of oil below the water droplet. Since $\alpha$ is measured to be small (around ${{15}^{\circ }}$), $\cos {\alpha }$ is close to one. Their respective surface tensions are illustrated in the inset of figure 8(a). The oil–solid and the water–solid interfacial tensions can be obtained from the Young equation, which relates the equilibrium contact angle of a sessile droplet with three surface tensions. For the oil droplet, the oil–solid interfacial tension can be determined as

(3.3)\begin{equation} {{\sigma }_{os}}={{\sigma }_{s}}-{{\sigma }_{oa}}\cos {{\theta }_{1}}. \end{equation}

For the water droplet, the water–solid interfacial tension can be determined as

(3.4)\begin{equation} {{\sigma }_{ws}}={{\sigma }_{s}}-{{\sigma }_{wa}}\cos {{\theta }_{2}}. \end{equation}

Substituting (3.3) and (3.4) into (3.2), $\Delta E_\sigma$ is obtained as

(3.5)\begin{equation} \Delta E _\sigma=\Delta A\left[ {{\sigma }_{oa}}(1+\cos {{\theta }_{1}})-{{\sigma }_{wo}}-{{\sigma }_{wa}}\cos {{\theta }_{2}} \right]. \end{equation}

The capillary-driven flow is favoured when the released surface energy $\Delta E_\sigma >0$. Inspired by the spreading parameter $S$ which characterizes the wetting state of a three-phase system, we here define ${{S}_{\sigma }}={{\sigma }_{oa}}(1+\cos {{\theta }_{1}})-{{\sigma }_{wo}}-{{\sigma }_{wa}}\cos {{\theta }_{2}}$ as the spreading parameter in the present system. In this case, the displacement of water droplets happens under a positive ${{S}_{\sigma }}$, which otherwise would not happen if ${{S}_{\sigma }}$ is negative. We further test this spreading parameter ${{S}_{\sigma }}$ by applying it to the experiments of Rostami & Auernhammer (Reference Rostami and Auernhammer2022), where the merging of two immiscible droplets on solid surfaces was studied. In their experiments, the position of the water droplet was found to show no changes and ${{S}_{\sigma }}$ was calculated to be negative. We, therefore, use ${{S}_{\sigma }}$ to quantify the driving force of this four-phase capillary-driven flow.

We then trace the time taken for the oil droplet to displace the water droplet from the solid surface, ${{t}_{{displace}}}$, which is identified from the moment when the two droplets make contact to the moment when the oil reaches the opposite end of the water droplet. During this process, both inertia and viscous forces could resist the flow. The relative importance of these forces can be measured using a modified Ohnesorge number based on the spreading parameter ${{S}_{\sigma }}$:

(3.6)\begin{equation} {{Oh}_{s}}=\frac{{{\mu }_{o}}}{\sqrt{{{\rho }_{avg}}{{R}_{c}}{{S}_{\sigma }}}}, \end{equation}

where ${R}_{c}$ is the equivalent radius of the two droplets: ${{R}_{{c}}}=\sqrt [3]{{3( {{V}_{{o}}}+{{V}_{{w}}} )}/({4{\rm \pi} }) }$.

The variation of the displacement time against ${Oh}_s$ is shown in figure 8(a). For a small ${Oh}_s$, the displacement time does not show significant variation. In contrast, when ${Oh}_s >1$, the displacement time is much longer and shows significant variations. Further, based on the spreading parameter ${{S}_{\sigma }}$, two relevant time scales can be involved for the flow from dimensional analysis: a viscous time scale ${{t}_{\nu }}={{{\mu }_{o}}{{R}_{c}}}/{{{S}_{\sigma }}}$ and an inertia time scale ${{t}_{\rho }}=\sqrt {{({{m}_{w}}+{{m}_{o}})}/{{{S}_{\sigma }}}}$. Here, in the inertia time scale, we choose the total mass of the oil droplet and the water droplet $({{m}_{w}}+{{m}_{o}})$ to be the characteristic mass, considering that the inertia of both droplets could resist the flow. The measured displacement time is first rescaled with the inertia time scale ${{t}_{\rho }}$, as shown in figure 8(b). The rescaled results show that it is almost a constant for ${Oh}_s<1$, but increases with ${Oh}_s$ when ${Oh}_s >1$. In contrast, by rescaling the data with the viscous time scale ${{t}_{\nu }}$, different behaviour can be observed: the dimensionless displacement time shows a significant decrease with ${Oh}_s$ when ${Oh}_s <1$, and increases when ${Oh}_s >1$. The shaded area in figure 8(b), in which ${Oh}_{s}$ is around 1, indicates a transitional regime where the viscous and inertia contributions are comparable. The rescaled result illustrates that for low-viscosity fluids, the droplets’ inertia is the main resistance in this capillary-driven flow, while with the increase of ${Oh}_s$, both the inertia and viscous forces would have a non-negligible resistance to this four-phase capillary-driven flow.

As the oil displaces the water droplet on the solid surface, a PECSD is formed and is observed to retract at a much slower time scale to further lower the system's energy. Such retraction motion is relatively fast at the beginning of the retraction, and then the retraction speed gradually decreases, as is shown in figure 9(a). Similar retraction processes are common to see in spreading droplets (Andrieu et al. Reference Andrieu, Beysens, Nikolayev and Pomeau2002; Siahcheshm, Goharpey & Foudazi Reference Siahcheshm, Goharpey and Foudazi2018). A measure of the droplet retraction is the relative retraction rate, defined as $\dot {\varepsilon }={\dot {R}(t)}/{R(t)}$, where $R(t)$ is the contact radius of the compound droplet and the overdot indicates the derivative with respect to time. The retraction rate is a critical parameter in characterizing the retraction of sessile droplets (Siahcheshm et al. Reference Siahcheshm, Goharpey and Foudazi2018).

Figure 9.

(a) Evolution of the normalized contact radius for three different liquid pairs. The inset illustrates the maximum contact radius $R_{ max}$ of PECSD during retraction. Radius $R_{{max}0}$ is the maximum contact radius of PECSD at the outset of retraction. (b) Variation of ${{\dot {\varepsilon }}_{m}}{{t}_{i}}{{\theta }_{e}}^{-{10}/{3}}$ as a function of ${Oh}$. The solid line denotes the relationship given in (3.11) with $L$ being of the same order as the droplet size, $\sim$1 mm, and $\lambda$ being of the order of nanometre scale, $\sim$1 nm (de Gennes Reference de Gennes1985; Bonn et al. Reference Bonn, Eggers, Indekeu, Meunier and Rolley2009).

We then investigate the retraction motion from the energy balance. First, by using the parameters of this PECSD defined before along with the maximum retraction velocity ${u}_{m}$, we can have a quick estimation of the Weber number, $We={{{\rho }_{{avg}}}{{R}_{{c}}}{u}_m^{2}}/{\sigma _{{e}}}$, which is of ${O}(10^{-5}$). In this case, due to the low Weber number of this retraction process, the inertia can be safely neglected and the rate of work done by capillary forces can be assumed to be equal to the rate of viscous dissipation (Bonn et al. Reference Bonn, Eggers, Indekeu, Meunier and Rolley2009). Further, with a low capillary number, the viscous dissipation is concentrated in the region close to the contact line (de Gennes Reference de Gennes1985). By considering the flow near the contact line of the retracting droplet, the rate of viscous dissipation in the receding corner per unit length of the contact line is expressed as (de Gennes et al. Reference de Gennes, Brochard-Wyart and Quéré2004)

(3.7)\begin{equation} {{\varPhi }_{d}}=\frac{3{{\mu }_{o}}{{U}^{2}}}{{{\theta }_{r}}}\int_{0}^{\infty }{\frac{{\rm d}\kern0.7pt x}{x}}, \end{equation}

where $U=\dot {R}(t)$ is the retraction velocity. Since the retraction mainly happens in the direction of the major axis, ${R}(t)$ is defined to be the time-dependent contact radius of the compound droplet measured in the direction of the major axis. According to the de Gennes approximation, $\int _{0}^{\infty }{{{\rm d}\kern0.7pt x}/{x} }\approx \int _{\lambda }^{L}{{{\rm d}\kern0.7pt x}/{x}}=\ln ( {L}/{\lambda } )$, where the macroscopic and microscopic cut-off lengths $L$ and $\lambda$ are calculated as the droplet size and the molecular size, respectively (de Gennes Reference de Gennes1985).

The work done by the capillary force per unit length of the contact line can be then expressed as

(3.8)\begin{equation} {{\varPhi }_{f}}={{\sigma }_{oa}}[ \cos {{\theta }_{r}}(t)-\cos {{\theta }_{r,e}} ]U, \end{equation}

where ${{\theta }_{r}}(t)$ is the dynamic receding contact angle and ${{\theta }_{r,e}}$ is the equilibrium receding contact angle and can be simplified as the equilibrium contact angle. Such simplification is valid in this slow process as the capillary number $Ca={{{\mu }_{{o}}}U}/{{{\sigma }_{{oa}}}}$ is estimated to be of ${O}(10^{-4})$, being much smaller than 1 (Edwards et al. Reference Edwards, Ledesma-Aguilar, Newton, Brown and McHale2016). Further, in the limit of ${{\theta }_{r}}<{{\theta }_{e}}\ll 1$, a first-order approximation of the Taylor expansion is used, $\cos {{\theta }_{r}}(t)\approx 1-{{{\theta }_{r}^{2}}}/{2}$, $\cos {{\theta }_{e}}(t)\approx 1-{{{\theta }_{e}^{2}}}/{2}$, and then (3.8) can be rewritten as ${{\varPhi }_{f}}={{{\sigma }_{oa}}[{{\theta }_{e}^{2}}-{{\theta }_{r}^{2}}{{(t)}}]U}/{2}$. The balance between ${{\varPhi }_{f}}$ and ${{\varPhi }_{d}}$ then gives

(3.9)\begin{equation} \frac{\dot{R}(t)}{R(t)}=\frac{{{\sigma }_{oa}}}{6{{\mu }_{o}}\ln{L}/{\lambda }}{{\left( \frac{{\rm \pi} }{3{{V}_{t}}} \right)}^{{1}/{3}}}[ {{\theta }_{e}^{2}}-{{\theta }_{r}^{2}}{(t)} ]{{\theta }_{r}^{{4}/{3}}}{{(t)}}, \end{equation}

where ${{V}_{t}}$ is the total volume of the coalescing droplet, which can be estimated by approximating it as a spherical cap, i.e. ${{V}_{t}}={{\rm \pi} {{\theta }_{r}}(t)R^{3}{{(t)}}}/{4}$. Then, we can obtain the maximum retraction rate by finding ${{\rm d}[ {\dot {R}(t)}/{R(t)} ]}/{{\rm d}t}=0$:

(3.10)\begin{equation} {{\dot{\varepsilon }}_{m}}=\max \left[ \frac{\dot{R}(t)}{R(t)} \right]=\frac{{{\sigma }_{oa}}}{10{{\mu }_{o}}\ln \left( {L}/{\lambda } \right)}{{\left( \frac{{\rm \pi} }{3{{V}_{t}}} \right)}^{{1}/{3}}}{{\left( \frac{2}{5} \right)}^{{2}/{3}}}{{\theta }_{e}}^{{10}/{3}}. \end{equation}

By rearranging (3.10), a relationship for the maximum retraction rate can be obtained:

(3.11)\begin{equation} {{\dot{\varepsilon }}_{m}}{{t}_{i}}{{\theta }_{e}}^{{-10}/{3}}=\frac{1}{10\ln \left( {L}/{\lambda } \right)}{{\left( \frac{1}{25} \right)}^{{1}/{3}}}{{Oh}^{{-}1}}, \end{equation}

where ${{t}_{i}}=\sqrt {{{{\rho }_{avg}}{R}_{c}^{3}}/{{\sigma }_{oa}}}$ is an inertia–capillary time scale and ${Oh}={{{\mu }_{o}}}/{\sqrt {{{\rho }_{avg}}{{R}_{c}}{{\sigma }_{oa}}}}$ is a modified Ohnesorge number for the retraction process.

Comparing this theoretical model in (3.11) with the experimental data, we can see a good agreement as shown in figure 9(b), confirming the above-proposed mechanism that the retraction of the PECSD is dominated by the viscous dissipation occurring in the receding wedge of the contact line.