3.1. Assumptions and correlations

To understand the distinct scaling behaviours in unequal-size droplet coalescence, we next introduce a model for two merging droplets of different but comparable sizes (i.e. $\varDelta \sim O(1)$ , $R_L/R_S \sim O(1)$ and $\theta _L/\theta _S \sim O(1)$ ). A simplified geometry of the liquid-bridge interface is presented in figure 3(a), based on which two physical assumptions can be made.

(i) ‘small bridge’, meaning that the characteristic radii of the bridge, $R_S$ and $R_L$ , are much smaller than the droplet diameters, i.e. $R_S/D_S \sim o(1)$ and $R_L/D_L \sim o(1)$ . This is clearly satisfied during the early-stage coalescence.

(ii) ‘arc-shaped bridge interface’, meaning the interface section between $P_1$ and $P_2$ can be approximated by an arc of the same curvature, where the pressure difference $\varDelta p$ is evenly distributed. As such, we can define the bridge’s principle normal direction $\mathbf{n}_p$ as that pointing from the arc midpoint to the arc centre. This assumption can be considered a reasonable first approximation.

Several geometric correlations can be deduced from the coalescence model in figure 3(a). We begin with two basic ones:

(3.1) \begin{align} & R_S/D_S = \frac {1}{2}\sin {(2\theta _S)} = \theta _S - O(\theta _S^3),\notag\\ & R_L/D_L = \frac {1}{2}\sin {(2\theta _L)} = \theta _L - O(\theta _L^3).\end{align}

Under assumption (i), both $\theta _S$ and $\theta _L$ are of $o(1)$ , so the higher-order terms in (3.1) can be neglected.

A key characteristic of the unequal-size coalescence, which differs from the equal-size situation, is the tilted bridge interface and the misaligned bridge movement from the radial direction, as delineated in figure 3(a). This tilted interface results from the force balance across the interface. Given the arc shape of the bridge interface according to assumption (ii), the total surface stress ( $\Delta p + \mathbf{\tau }$ ) has a symmetric distribution with respect to the arc midpoint, where $\tau = 2\mu \unicode{x1D64E}$ is the viscous stress ( $\unicode{x1D64E}$ is the strain-rate tensor). So the bridge’s principle normal direction $\mathbf{n}_p$ is in line with the integral of the total surface stress over the bridge interface, which is balanced by the surface tension $\sigma$ pulling at its both ends ( $P_1$ and $P_2$ ). Since the two surface tensions are of the same magnitude, $\mathbf{n}_p$ must also be in line with the angular bisector of these two forces. Accordingly, the tilting angle ( $\theta$ ) of the bridge interface satisfies a simple geometric correlation:

(3.2) \begin{equation} \theta = \theta _S - \theta _L. \end{equation}

As $\theta _S$ and $\theta _L$ are of $o(1)$ , $\theta$ is also of $o(1)$ and is thereby treated as the ‘small parameter’ in this model.

Figure 3. (a) Zoomed-in schematics of the liquid-bridge interface between two unequal-size merging droplets. The red arrows represent the forces applied over the bridge interface, and the green arrow indicates the overall movement of the interface. (b) Three main surfaces ( $A_S$ , $A_L$ and $A_B$ ) with significant area changes, respectively corresponding to the three arc segments, $C_S$ , $C_L$ and $C_B$ , in (a).

The width of the bridge ( $S$ ), defined as the axial distance between $P_1$ and $P_2$ , has the correlation

(3.3) \begin{equation} S \approx R_S\theta _S + R_L\theta _L.\end{equation}

Furthermore, the interface geometry in figure 3(a) satisfies $R_L-R_S = S\tan {\theta } \approx S \theta$ , which can be used to derive (3.2) and (3.3) as

(3.4) \begin{align} \theta &\approx \left (D_S^{-1} - D_L^{-1} \right )R, \end{align}

(3.5) \begin{align} S &\approx \left (D_S^{-1} + D_L^{-1} \right )R^2. \end{align}

The detailed derivations of (3.3)–(3.5) are provided in the supplementary material.

With the symmetrically distributed total surface stress and the evenly distributed $\Delta p$ over the arc-shaped bridge interface (assumption (ii)) in figure 3(a), it can be implied that the viscous stress ( $\mathbf{\tau }$ ) also has a symmetric distribution with respect to the arc midpoint. It follows that the liquid-side velocity must also be symmetrically distributed over the bridge interface, so that the overall bridge movement is in line with its principle normal direction $\mathbf{n}_p$ . This has significant physical implications. Consider the very early stage of coalescence when $R_S \approx R_L$ , (3.4) dictates that $\theta$ is a positive value. This explains why the bridge movement is inclined towards the smaller droplet and why $R_S$ is smaller than $R_L$ from the beginning. Thus, the bridge interface movement can be described by an essential kinematic relationship, $\textrm{d} R/\textrm{d} t = V \cos {\theta }$ , where $V$ is the velocity at which the bridge interface expands radially. Applying Taylor expansion, it takes the form

(3.6) \begin{equation} \frac {\textrm{d} R}{\textrm{d} t} \approx V \left (1-\frac {\theta ^2}{2} \right ). \end{equation}

Note that (3.6) recovers the equal-size coalescence kinematics with vanishing $\theta$ .

3.2. Energy balance analysis

From the energy conservation perspective, the movement of the liquid entrained by the expanding bridge is driven by the rapid discharge of the surface energy, which can be formulated as $\Delta E_s + \Delta E_k \approx 0$ , where $\Delta E_s$ and $\Delta E_k$ , respectively, represent the changes in surface and kinetic energies from the initial state of coalescence onset. Note this only applies to the inviscid-dominant regime ( $Oh \sim o(1)$ ) with negligible viscous dissipation, which is valid for the present experiment.

According to figure 3(b), the change in surface energy can be estimated as $\Delta E_s = (-A_S-A_L+A_B) \sigma$ , where $A_S$ , $A_L$ and $A_B$ are the main surfaces with varying areas. It can be further derived as

(3.7) \begin{equation} \Delta E_s \approx -\unicode{x03C0} D_S^2 \theta _S^2 \sigma - \unicode{x03C0} D_L^2 \theta _L^2 \sigma + \unicode{x03C0} S (R_S+R_L) \sigma, \end{equation}

where the third term is one order smaller than the first two terms and can be dropped off (see supplementary material for details). Additionally, $\Delta E_k$ can be estimated as

(3.8) \begin{equation} \Delta E_k \approx \frac {C}{2} \unicode{x03C0} \rho _l R^2S V^2, \end{equation}

where $C$ is a prefactor related to the flow entrainment by the liquid bridge. Equations (3.7) and (3.8) are almost identical to those for the equal-size scenario, except for the terms relating to the size disparity.

With $S$ and $V$ given by (3.5) and (3.6), respectively, we can apply (3.7) and (3.8) to derive the energy balance as

(3.9) \begin{equation} \frac {1}{1-\beta R^2/2} \frac {\textrm{d} R}{\textrm{d} t} \approx \frac {\gamma }{R}, \end{equation}

where $\beta = (D_S^{-1} - D_L^{-1})^2$ and $\gamma = (4 \sigma D_S)^{1/2} [C \rho _l (1+\varDelta ^{-1})]^{-1/2}$ . Given the initial condition $R(t=0) = 0$ , (3.9) has the solution, $R^2 \approx (2 - 2\textrm{e}^{-\beta \gamma t}) \beta ^{-1}$ , with the non-dimensional form

(3.10) \begin{equation} R^{*2} \approx 2 - 2\textrm{e}^{-t^*}, \end{equation}

where $R^* = R \beta ^{1/2}$ and $t^* = t\beta \gamma$ . Interestingly, the bridge evolution governed by (3.10) no longer has a power-law scaling between $R$ and $t$ .

It is worth discussing the features of this solution. Letting $\varDelta \rightarrow 1$ in (3.10), we have $\beta \rightarrow 0$ and $R^2 \approx 2[1 - (1-\beta \gamma t)]/\beta = 2\gamma t$ . This means that the present model is able to recover the inviscid scaling law of $R \sim t^{1/2}$ in the equal-size limit. Likewise, we attain $R^{*2} \approx 2t^*$ when $t^* \rightarrow 0$ , suggesting that in practice the power-law scaling still holds for the early-stage coalescence of unequal-size droplets, owing to the asymmetric effect being negligible. When $t^* \rightarrow \infty$ , it predicts $R^{*2} \approx 2$ , which corresponds to an up-limit radius as if the bridge movement could turn continuously from radial to axial. It should be noted, however, the limiting $R$ in reality is confined by the actual droplet sizes.

3.3. Model validation and discussion

Before verifying the proposed theory, we first check whether the classical scaling law of $R \sim t^{1/2}$ holds for unequal-size droplet coalescence, by plotting the time sequence data of the liquid-bridge radius for various $\varDelta$ and $Oh$ in the parameter space $[(8\sigma /(\rho D_S^3))^{1/2}t, (2R/D_S)^2]$ (Duchemin et al. Reference Duchemin, Eggers and Josseran2003; Aarts et al. Reference Aarts, Lekkerkerker, Guo, Wegdam and Bonn2005) in figure 4. The result indicates that the $R^2 \sim t$ scaling is approximately valid for most cases. However, as $\varDelta$ increases, there is an apparent upward drift of data from the baseline of $\varDelta = 1.0$ , which is associated with an increase of the scaling prefactor from 1.25 to 2.65. Note that, in terms of the scaling relation of $R \sim t^{1/2}$ , where $R$ is scaled by $D_S/2$ and $t$ is scaled by $\tau _i = (\rho D_S^3/(8\sigma))^{1/2}$ , the two corresponding prefactors are 1.12 and 1.63, respectively. The latter, marking the up-limit of the large- $\varDelta$ cases, is significantly higher than those reported in previous equal-size experiments (Aarts et al. Reference Aarts, Lekkerkerker, Guo, Wegdam and Bonn2005). This can be physically understood in that the presence of a larger droplet enhances the expansion speed of the bridge interface. Furthermore, the cases with $\varDelta \gt 3$ display slightly decreased slopes compared with that of $R^2 \sim t$ as time proceeds to the later stage of coalescence when the liquid bridge becomes more asymmetric, indicating a tendency to divert from the classical scaling of $R^2 \sim t$ .

Figure 4. Effect of $\varDelta$ on the scaling of $R^2 \sim t$ . The bridge evolution images for all cases are reported in the supplementary material.

Figure 5. (a) Model validation of (3.10) against experimental data and (b) the performance of (3.10) versus the classical power-law scaling. The top-left inset in (b) shows the deflection of this model from the scaling law in normal coordinates; the bottom-right inset plots the experimental relationship between $R^*$ and $\theta$ .

The comparison of the new theoretical model with the same data in figure 4 is presented in figure 5(a). We can observe the collapse of non-unity- $\varDelta$ data onto a single line given by (3.10), with $C = 6$ given by fitting. In the equal-size limit ( $\varDelta \rightarrow 1$ ), $C = 6$ corresponds to a prefactor of 1.28 in the scaling relation of $R \sim t^{1/2}$ under the aforementioned non-dimensionalisation. This prefactor is in approximate agreement with previous experimental results, e.g. 1.03–1.29 in Wu et al. (Reference Wu, Cubaud and Ho2004) and 1.11–1.24 in Aarts et al. (Reference Aarts, Lekkerkerker, Guo, Wegdam and Bonn2005), which is supportive of our experimental data and model. In this sense, a major significance of the present work lies in its generalisation of Eggers et al. (Reference Eggers, Lister and Stone1999)’s model through the consideration of droplet size disparity. However, it should be noted that this prefactor is slightly smaller than the numerical results, e.g. 1.62 in Duchemin et al. (Reference Duchemin, Eggers and Josseran2003) and 1.5 in Sprittles & Shikhmurzaev (Reference Sprittles and Shikhmurzaev2014). While further study is necessary, the discrepancy between experiment and simulation is possibly related to the ideal initial conditions of the simulations, which does not account for the van der Waals force or potential viscous effect that may influence the initial coalescence behaviours in reality.

Figure 5(a) justifies the universality of our theory in resolving the droplet coalescence of various size ratios, including the late-stage coalescence dynamics at large size ratios. Another important implication is that this theory offers the proper characteristic length and time scales for unequal-size droplet coalescence; these turn out to be $\beta ^{-1/2}$ and $(\beta \gamma )^{-1}$ , respectively, both of which depend on the size ratio $\varDelta$ . So a larger $\varDelta$ corresponds to larger $R^*$ . The good agreement in figure 5(a) also suggests that although the theory was developed for the early-stage coalescence, which generally requires $R/D_S \sim o(1)$ according to assumption (i), the model prediction is potentially useful towards a much later stage—a typical observation in asymptotic analysis (Van Dyke Reference Van Dyke1964). This likely reflects that the geometric and kinematic correlations used in our model offer relatively accurate approximations to later stages. Nevertheless, the extension of this model to very late coalescence stages ( $R/D_S$ approaching 0.5) for large-size-ratio cases should be handled with caution, owing to the non-negligible errors associated with the finite bridge size as well as significant deviation from symmetry.

To assess to what extent the unequal-size droplet coalescence deviates from the power-law scaling of the equal-size case, figure 5(b) compares the data with (3.10) in the $R^{*2}{-}t^*$ diagram. Again, a promising agreement between experiment and theory can be confirmed; an uncertainty analysis is included in the supplementary material. However, the theoretical line here remains almost linear when $R^{*2}\lt 10^{-1}$ , while it exhibits a slight deflection or deviation from the power-law scaling of $R^{*2} = 2t^*$ when $R^{*2}\gt 0.1$ or $R^{*}\gt 0.3$ . According to the present study, this deviation can be interpreted as the result of a strong asymmetry. The bottom-right inset in figure 5(b) demonstrates a general positive correlation between $R^*$ and the liquid-bridge asymmetry measured by $\theta$ . Evidently, the coalescence stage $R^{*}\gt 0.3$ indeed has high degrees of asymmetry, roughly corresponding to $\theta \gt 0.3$ .

In figure 5(b), it also seems that all data do not notably exceed the linear regime. To understand the physical $R$ range where the deviation becomes apparent, we can express $R^{*}$ as $(D_L-D_S)D_L^{-1} R/D_S$ , which is generally much smaller than 0.3 for small- $\varDelta$ cases because $(D_L-D_S)D_L^{-1}$ is rather small. For large $\varDelta$ , $(D_L-D_S)D_L^{-1}$ approaches unity and the scaling line deflects as $R$ reaches the magnitude of approximately $0.3D_S$ , which is rather large. This means that the power-law scaling becomes effectively inaccurate only at large $\varDelta$ and the bridge’s radius is comparable to $0.3D_S$ . On the other hand, $R$ has a geometric up-limit of $0.5D_S$ , which yields a minimum $\varDelta = 2.5$ for one to detect any deviation. This explains why the diversion from $R^2 \sim t$ in figure 4 is observable after a rather late stage of coalescence and when $\varDelta$ is greater than 3 or so. To understand the time range corresponding to the scaling-law deviation, we can use the asymptotic behaviour $R^{*2} \approx 2t^*$ to obtain the criterion, $t^{*}\gt 0.05$ . With the calculation of $t^{*}$ given by (3.9) to (3.10), the criterion can be derived as $t\gt 0.17\tau _i$ for $\varDelta \rightarrow \infty$ . For a finite $\varDelta$ , e.g. $\varDelta = 2$ , we have $t\gt 0.85\tau _i$ . As $\varDelta \rightarrow 1$ , the critical time increases dramatically, which makes it physically impossible to break the scaling law.

We last discuss the viscous effect. From our previous theory (Xia et al. Reference Xia, He and Zhang2019), the viscous regime for equal-size droplet coalescence occurs for $R/(Oh D_S) \lt 1$ . In the present work, as $Oh$ varies from $10^{-3}$ to $10^{-2}$ , the viscous regime corresponds to $R/D_S$ being smaller than $10^{-3}$ – $10^{-2}$ , for which the evolution process is beyond the resolution of this experiment. Note that during such an early stage the liquid bridge is effectively symmetric, which in turn justifies the extension of the equal-size theory to the unequal-size scenario. Given the limited asymmetry, it can be speculated that there are not substantial differences in the viscous regime between the equal-size and unequal-size situations for the present low- $Oh$ conditions. However, if $Oh$ increases to be $O(10^{-1})$ or higher, the viscous term in the energy balance analysis cannot be ignored, which may render the present model inaccurate. In this case, one may adopt a similar modelling approach to Xia et al. (Reference Xia, He and Zhang2019) by integrating the Navier–Stokes equation and then incorporating (3.6) to account for the asymmetric bridge movement; this will be explored in our future work.