3.1. Influence of particle coverage

Particles adsorbed at the droplet interface not only increase resistance to the coalescence process but also create a physical barrier that hinders total coalescence. In this section, we examine the coalescence behaviour of two colliding droplets under varying particle coverages, adjusted by changing the number of interfacial particles.

Figure 3.

Coalescence dynamics of droplet pairs under different Weber numbers ( $\textit{We}$ ) and particle coverage conditions ( $C_1, C_2$ ): (a) clean droplets at $\textit{We}=2$ ; (b) particle-coated droplets ( $C_1=C_2=0.68$ ) at $\textit{We}=2$ ; (c) clean droplets at $\textit{We}=60$ ; (d) particle-coated droplets ( $C_1=C_2=0.68$ ) at $\textit{We}=60$ . The dimensionless time is defined as $t^* = t / \sqrt {\rho _{r} R_d^3 / \sigma }$ .

Figure 3. Long description

Panel A: Sequence of images showing the coalescence of clean droplets at a specific Weber number. The droplets start separate, come together, form a liquid film, and eventually merge into a single larger droplet. Panel B: Sequence of images showing the coalescence of particle-coated droplets at a specific Weber number. The droplets start separate, come together, and remain distinct with particles on their surfaces. Panel C: Sequence of images showing the coalescence of clean droplets at a different Weber number. The droplets start separate, come together, form an elongated shape, and eventually merge into a single larger droplet. Panel D: Sequence of images showing the coalescence of particle-coated droplets at a different Weber number. The droplets start separate, come together, form an elongated shape with particles on their surfaces, and eventually merge into a single larger droplet.

It is noteworthy that simulating such a complex multiphase system necessitates certain compromises to maintain computational tractability. Employing realistic particle sizes would require an intractable number of particles to cover the droplet surface, rendering full-scale simulation infeasible. Taking a representative experimental droplet radius of approximately $100\ {\unicode{x03BC}} \mathrm{m}$ and a particle radius of approximately $0.75\ {\unicode{x03BC}} \mathrm{m}$ , the radius ratio is of order $10^2$ . Retaining this ratio in a particle-resolved LB simulation with $R_{\!p} = 10$ lattice units would require the droplet radius to exceed $10^3$ lattice units and the computational domain to contain more than $10^8$ lattice nodes, even in two dimensions. A corresponding three-dimensional simulation would be far beyond affordable computational cost. The present simulations therefore adopt a reduced droplet-to-particle size ratio $(R_{\!p}/R_d = 1/15)$ and a two-dimensional set-up. We acknowledge that the particle-to-droplet size ratio influences local packing, interface curvature and bridge geometry, and thus that the quantitative findings of this work are specific to the droplet-to-particle size ratio used. Furthermore, since droplet contact and coalescence in our simulations are primarily driven by a collision mechanism, the choice of the Weber number ( $\textit{We}$ ) requires clarification. Based on the aforementioned parameter settings, we select $\textit{We} = 2$ and $\textit{We} = 60$ as representative of the low and high inertia regimes, respectively, to investigate the resulting collision dynamics. Figure 3 illustrates the combined effects of the Weber number and particle coverage on the collision outcome.

At a low Weber number ( $\textit{We}=2$ ), two clean droplets approach each other and initiate coalescence as the intervening liquid film thins and ruptures, as depicted in figure 3(a). During this process, a portion of the ambient fluid remains trapped within the coalescing droplet. In contrast, a markedly different behaviour is observed when each droplet interface is coated with 32 particles, corresponding to the particle coverages of $C_{1}=C_{2}=0.68$ , as shown in figure 3(b). Consistent with previous studies (Aarts & Lekkerkerker Reference Aarts and Lekkerkerker2008; Shen et al. Reference Shen, Chen, Yu and Liu2020), interfacial particles resist droplet deformation and enhance the dissipation of kinetic energy inside the droplet. As a result, during low- $\textit{We}$ collisions of droplets with high particle coverage, the combined effects of spatial obstruction by the particles and the inherently low kinetic energy lead to limited interface deformation. This insufficient deformation prevents the emergence of an adequate particle-free region necessary for effective interfacial contact and coalescence. To avoid non-coalescence outcomes arising from insufficient collision energy, simulations with very low Weber numbers are excluded from our systematic investigation.

At a higher Weber number ( $\textit{We}=60$ ), a larger volume of ambient fluid is retained inside the coalescing clean droplet compared with the $\textit{We}=2$ case, as visible in figure 3(c) at $t^*=6.16$ . This occurs because the shorter drainage time of the liquid film during high-speed collision hinders complete expulsion of the intervening fluid, leading to more entrapment of the ambient fluid. A similar entrapment phenomenon is observed when particles are present at the droplet interfaces (figure 3 d at $t^*=6.16$ ), with the key distinction that the droplet interior encapsulates not only ambient fluid but also a number of particles. Owing to their neutral wettability, most of these particles adsorb at the interface of the entrapped ambient fluid, resulting in a double emulsion structure (Clegg et al. Reference Clegg, Tavacoli and Wilde2016; Whitby & Parthipan Reference Whitby and Parthipan2019). This mechanism is consistent with experimental observations: when particle-stabilised oil droplets are brought into close contact, their interfaces flatten locally (Whitby et al. Reference Whitby, Lotte and Lang2012). As the flattened region expands and the liquid film thins, rupture occurs before some particles can escape, leading to their encapsulation within the coalesced droplet. This inward encapsulation follows from the topological pathway of film rupture and interface closure: as the two interfaces merge and pinch off, particles near the contact region become entrained within the newly formed internal interface. In this sense, the outward expulsion of particles reported in other systems under different mechanical and geometric conditions (Poulichet et al. Reference Poulichet, Huerre and Garbin2017; Gu & Botto Reference Gu and Botto2018) is not contradictory to the present observations, but rather reflects distinct pathways occurring under different configurations.

The physical origin of this entrapment is that the inertial time scale of the collision becomes shorter than the film-drainage time at high $\textit{We}$ , causing the interface to close before the intervening fluid can be fully expelled. The central structures shown in the rightmost column of figures 3(c) and 3(d) are therefore entrapped pockets of the ambient fluid, not gas bubbles, and should not be interpreted as a final steady state (both snapshots are at $t^*=6.16$ ). Similar transient entrapment during rapid coalescence has been experimentally visualised by Fezzaa & Wang (Reference Fezzaa and Wang2008) using ultrafast X-ray phase-contrast imaging and theoretically predicted in inviscid coalescence models (Eggers et al. Reference Eggers, Lister and Stone1999; Duchemin et al. Reference Duchemin, Eggers and Josserand2003). It should be noted, however, that our two-dimensional set-up may overemphasise the quantitative extent of entrapment compared with a three-dimensional system, where out-of-plane drainage pathways facilitate more efficient expulsion of the intervening fluid. Since the focus of this section is the effect of particle coverage on the external droplet morphology, and excessive internal entrapment of particles and fluid would complicate the quantitative analysis of particle coverage, a very high Weber number is not adopted in the subsequent simulations.

Figure 4.

Comparison of droplet morphologies under different particle coverages from (a–d) the experiments of Pawar et al. (Reference Pawar, Caggioni, Ergun, Hartel and Spicer2011) and (e–h) the present simulations. Note that the particle coverages for the left ( $\varPhi _1$ ) and right ( $\varPhi _2$ ) droplets in the experimental panels are consistent with the notations in Pawar et al. (Reference Pawar, Caggioni, Ergun, Hartel and Spicer2011).

Through systematic trial-and-error testing, a moderate Weber number of 13 is adopted for investigating droplet coalescence behaviour across varying particle coverages. This value is chosen as a compromise: it is low enough to avoid excessive ambient-fluid and particle entrapment, yet high enough to provide sufficient impact energy to drive the droplets into contact and generate the interface deformation needed to explore the coverage-dependent regime transition. Under this condition, the droplets approach and deform upon contact, producing a compressed interfacial configuration qualitatively similar to the state induced by micropipette-driven contact in the experiments of Pawar et al. (Reference Pawar, Caggioni, Ergun, Hartel and Spicer2011). More importantly, at this $\textit{We}$ , most droplets with particle coverage below 0.8 undergo complete coalescence with negligible particle entrapment, thereby satisfying the key assumption underlying the derivation of (3.2), i.e. the total number of interfacial particles remains constant during coalescence. To further validate the chosen parameters, we reproduce the droplet coalescence experiments conducted by Pawar et al. (Reference Pawar, Caggioni, Ergun, Hartel and Spicer2011), who employed capillary suction to precisely control interfacial particle coverage. Figure 4 compares the droplet morphologies from their experiments (panels a–d) with our simulation results (panels e–h) for four coverage conditions. As shown in the figure, the simulations reproduce the same coverage-dependent sequence observed experimentally: total coalescence into a circular shape at low coverage (panels d iii and h iii), partially arrested coalescence into a ‘potato-like’ morphology at intermediate coverage (panels c iii and g iii), a stable ‘dumbbell’ shape at higher coverage (panels b iii and f iii) and complete stability with no coalescence at very high coverage (panels a iii and e iii). We emphasise that this comparison serves as a qualitative consistency check rather than a one-to-one quantitative validation, as the experiments and simulations differ in dimensionality, driving protocol and particle-to-droplet size ratio. The time instances shown in the simulated panels are not arbitrary: the three columns correspond to the initial configuration, the moment of closest approach (or shortly after coalescence begins) and the final steady state, respectively. Despite quantitative differences, the present simulations capture the essential particle-crowding mechanism responsible for the coverage-dependent regime transition observed experimentally.

Figure 5.

Interfacial evolution of clean droplets after collision: (a) from initial contact ( $t^{\text{*}} = 0$ ) to an intermediate stage of coalescence ( $t^{\text{*}} = 4.6$ ); (b) local view of the liquid bridge from $t^{\text{*}} = 1.2$ to $t^{\text{*}} = 3.5$ . In (b), $h_{\textit{max}}$ and $h_{\textit{min}}$ denote the maximum and minimum thicknesses of the liquid film, respectively.

Next, we examine the coalescence dynamics across varying particle coverages. Figure 5 illustrates the interfacial evolution following the collision of clean droplets at $\textit{We}=13$ , where the black solid lines ( $\rho ^{N}=0$ contour) represent the liquid–liquid interface. As expected, significant deformation occurs at the droplet contact interface upon collision. From $t^{\text{*}}=1.2$ to $t^{\text{*}} = 3.1$ , the interfaces deform progressively, with the central film thickness ( $h_{\textit{max}}$ ) exceeding that at the rims ( $h_{\textit{min}}$ ) by $t^{\text{*}}=3.1$ . Coalescence initiates as $h_{\textit{min}}$ reaches a critical thickness, resulting in the entrapment of the intervening ambient fluid within the merged droplet. To further quantify the effect of particle coverage on film drainage, we monitor the temporal evolution of the normalised film thickness, denoted as $h_c/R_d$ , where $h_c$ is measured along the line connecting the two droplet centres. As shown in figure 6, the clean droplets exhibit a rapid decrease in $h_c/R_d$ upon approach, reflecting efficient expulsion of the intervening ambient fluid. When particles are present at the droplet interfaces, however, the drainage rate is substantially reduced. For a moderate coverage of $C_1=C_2=0.71$ , the film still thins appreciably, yet $h_c/R_d$ remains consistently larger than that of the clean case at corresponding times. At a higher coverage of $C_1=C_2=0.815$ , the film thinning is strongly suppressed, and $h_c/R_d$ stays at a significantly greater thickness throughout the entire observation period. This coverage-dependent retardation of film drainage provides direct quantitative evidence for the steric hindrance imposed by the interfacial particle layer.

Figure 6.

Temporal evolution of the normalised film thickness $h_c/R_d$ for clean droplets and particle-coated droplets with different coverages.

Figure 7.

Droplet collision and coalescence processes under different particle coverages: (a) $C_{1}=0.44, C_{2}=0.57$ , (b) $C_{1}=0.74, C_{2}=0.72$ and (c) $C_{1}=0.85, C_{2}=0.51$ .

Having established this quantitative film-drainage behaviour, we now examine the corresponding collision outcomes for particle-coated droplets under three distinct coverage conditions in figure 7. Evidently, the initial deformation of the contact interface for particle-coated droplets resembles that of clean droplets. However, at moderate ( $C_{1}=0.74, C_{2}=0.72$ ) and low ( $C_{1}=0.44, C_{2}=0.57$ ) particle coverages, the coalescence mechanism shifts: it no longer initiates solely from the rim rupture of the liquid film but is triggered by an imbalance in the capillary forces acting on the particles. A detailed visualisation of this particle-induced coalescence process is provided in figure 8.

To clarify this mechanism, we define the interface where a particle is originally adsorbed as the ‘original interface’, and the opposing droplet interface that later contacts the particle as the ‘later interface’. At low particle coverage, e.g. $C_1 = 0.44, C_2 = 0.57$ (figures 7 a and 8 a), the original interface is sparsely populated with neutrally wetting particles in equilibrium. During film thinning, when a particle contacts the later interface, the capillary force exerted by the later interface can exceed that from the original interface, disrupting the equilibrium and drawing the particle toward the later interface. This capillary force becomes the primary driver for coalescence, culminating in the merging of the two interfaces. At moderate particle coverage, e.g. $C_1 = 0.74, C_2 = 0.72$ (figures 7 b and 8 b), significant deformation and stretching of the droplet interface after collision create a sufficiently large particle-free region, enabling effective contact between the later interface and particles. The key difference from the case with low particle coverage lies in particle retention: particles are only internalised when the liquid films on both their upper and lower sides rupture nearly simultaneously, as seen in figure 7(a). At very high particle coverage, e.g. $C_1 = 0.85, C_2 = 0.51$ (figure 7 c), the droplet cannot expose a sufficient particle-free region even under significant deformation, preventing interfacial contact. Furthermore, interparticle repulsion prevents the deformed interfaces from achieving the proximity required for coalescence. Over time, the deformed interfaces relax, and the droplets return to their original circular shapes, remaining in a state of total stability.

Another noteworthy phenomenon is arrested coalescence, which occurs under moderate particle coverage. As shown in figure 7(a), at low particle coverage, coalesced droplets deform and eventually stabilise into a circular shape, similar to clean droplets. In contrast, with increased particle loading (figure 7 b), coalescence is arrested before the droplet can relax into a circle. The direct mechanism is particle crowding and jamming at the shrinking interface, which opposes further capillary-driven shape relaxation. After the liquid bridge forms, capillarity acts as the driving force that tends to reduce the interfacial area and restore a circular droplet shape. Since the number of interfacial particles remains nearly constant, this interfacial contraction dynamically increases the particle coverage. When the particle layer becomes sufficiently crowded, it generates mechanical resistance against further contraction and opposes capillary-driven shape relaxation. Arrest occurs once the particles can no longer rearrange to accommodate additional interfacial shrinkage (figure 7 b). This interpretation is consistent with the jamming mechanism discussed in the context of particle-laden interfaces (Stratford et al. Reference Stratford, Adhikari, Pagonabarraga, Desplat and Cates2005).

Figure 8.

Role of individual particles in droplet coalescence under the particle coverages of (a) $C_{1}=0.44, C_{2}=0.57$ and (b) $C_{1}=0.74, C_{2}=0.72$ .

Figure 9.

Evolution of velocity vector fields and $y$ -component velocity contours during droplet coalescence under particle coverages of (a) $C_{1}=0.44, C_{2}=0.57$ and (b) $C_{1}=0.74, C_{2}=0.72$ .

Figure 9 compares the velocity vector fields during total and arrested coalescence of droplets, corresponding to particle coverages of $C_{1}=0.44, C_{2}=0.57$ and $C_{1}=0.74, C_{2}=0.72$ , respectively. Given that the arrest is closely associated with droplet deformation in the $y$ -direction, we also present the $y$ -component of velocity as a contour plot in this figure. After initial coalescence, the liquid bridge expands rapidly in the $y$ -direction, generating high-velocity regions. Under higher particle coverage (figure 9 b), the dense particle arrangement prevents further interfacial deformation by $t^* = 4.0$ , causing the velocity at the ends of the liquid bridge to diminish to nearly zero. The subsequent slow redistribution of particles allows the droplet to stabilise into a ‘dumbbell’ shape by $t^* = 6.2$ . In contrast, under lower particle coverage (figure 9 a), a sufficiently particle-free interface allows particle motion, enabling the droplet to escape the dumbbell shape. It is noted that the elliptical droplet at $t^* = 5.2$ is not the final form, and will contract, oscillate and eventually stabilise into a circular shape under interfacial tension. In summary, high particle coverages impede the conversion of surface energy to kinetic energy during coalescence, maintaining the system in a lower-energy state.

To further elucidate the role of particle coverage in droplet coalescence, it is essential to identify the critical condition for arrested coalescence. In physical systems, interparticle repulsion typically arises from van der Waals, electrostatic and solvation interactions, whereas in our simulations, such repulsion is explicitly modelled to avoid particle overlap and maintain finite separation between densely packed particles at fluid interfaces. Following established numerical practices, the minimum interparticle distance is set to three lattice units to accurately resolve hydrodynamic interactions with the surrounding fluids (Zhang et al. Reference Zhang, Liu and Zhang2020a ). Under these conditions, the maximum attainable particle coverage for a single droplet of initial radius $R_d = 150$ is determined to be $C = 0.87$ , corresponding to 41 adsorbed particles. Theoretically, arrested coalescence occurs if the total interfacial length occupied by particles on both droplets before coalescence exceeds the maximum interfacial length they can occupy on a single, fully coalesced circular droplet. This criterion is expressed as (Pawar et al. Reference Pawar, Caggioni, Ergun, Hartel and Spicer2011)

(3.1) \begin{equation} C\times 2\pi \sqrt {R_1^2+R_2^2}\lt C_1\times 2\pi R_1+C_2\times 2\pi R_2, \end{equation}

where the left side represents the interfacial length occupied by particles after the droplets have fully coalesced into a circular shape, while the right side corresponds to the total interfacial lengths occupied by particles on the two separate droplets before coalescence. Given the equal radii of both droplets ( $R_1=R_2$ ) in our simulations, this expression simplifies to

(3.2) \begin{equation} C_1+C_2\gt 1.23. \end{equation}

To determine the critical condition for arrested coalescence, a total of 110 numerical simulations are performed with the single-droplet particle coverage ranging from 0.40 to 0.85. Figure 10(a) shows the corresponding coalescence phase diagram in the ( $C_1$ , $C_2$ ) parameter space, illustrating the final morphological states after droplet collision (the regime classification is detailed in Appendix E). As expected, the diagram is symmetric about the line $C_1 = C_2$ , reflecting the interchangeable roles of the two droplet coverages. Total coalescence, leading to a circular merged droplet (see examples marked with black circles in figure 10 b), is found to be the dominant outcome, consistent with previous experimental observations (Pawar et al. Reference Pawar, Caggioni, Ergun, Hartel and Spicer2011). However, a distinct wedge-shaped region (marked by red triangles) emerges in the phase diagram under conditions of higher particle coverage. Within this region, droplet coalescence is arrested, resulting in stable non-circular shapes. The final morphology is highly sensitive to particle coverage – even a slight variation can lead to markedly different equilibrium shapes, as shown in the two representative cases highlighted by red triangles in figure 10(b). Moreover, the boundary between total and arrested coalescence observed in the phase diagram agrees well with the theoretical prediction from (3.2), confirming that arrest occurs only when the total particle coverage exceeds a critical threshold. As the particle coverage increases further, droplets become completely stabilised against coalescence when the single-droplet particle coverage exceeds approximately 0.8. This threshold is slightly lower than the theoretical maximum of 0.87, suggesting an enhanced anti-coalescence stability in the simulations. The discrepancy may be attributed to two primary factors: (i) the difficulty in achieving a perfectly uniform particle distribution at the droplet interface, and (ii) limited numerical resolution, which may cause local interparticle distances to fall slightly below the repulsive threshold, effectively increasing the local packing density and thus hindering coalescence.

Figure 10.

(a) Coalescence phase diagram of droplets under different particle coverages, and (b) final droplet morphologies at several representative particle coverages. In (a), the red dashed line, calculated from (3.2), represents the critical threshold between arrested and total coalescence.

Two additional features in figure 10(b) are worth noting in some coalesced cases: asymmetry of the arrested morphology (e.g. the third panel) and persistent inhomogeneity of particle coverage (e.g. the first panel). For the asymmetric case, although the two droplets have the same nominal initial coverage and initially equally spaced particles, the exact positions of individual particles relative to the approaching direction are not identical. During coalescence, such local differences lead to unequal resistance to interface deformation, which is amplified by capillary-driven flow, steric constraints and interfacial crowding, ultimately producing the observed asymmetric shape. The persistent inhomogeneity, on the other hand, arises because the capillary-driven relaxation of the interface proceeds faster than particle rearrangement. As the interface contracts, particles become increasingly crowded. Once the local concentration reaches a critical jamming threshold, further deformation is arrested, ‘freezing’ the non-uniform distribution into a path-dependent configuration. The final morphology is therefore not a uniformly equilibrated particle layer, but a mechanically arrested state determined by the initial particle arrangement and the subsequent coalescence history.

To further quantify the effect of interfacial particles on the effective mechanical response of the droplet interface, we estimate the reduction in effective interfacial tension due to the presence of particles. We select a representative total coalescence case with high particle coverage (the second panel in figure 10 b) that exhibits a nearly circular shape with approximately uniform particle distribution. At steady state, we measure the inner–outer pressure difference and estimate the effective interfacial tension via the Laplace law using the local interface curvature. The results show a reduction of approximately $40\,\%$ relative to the clean interface (from 0.02 to 0.01198 in lattice units), consistent with the particle–particle interaction mechanism reported in previous studies (Gu & Botto Reference Gu and Botto2020; Luo et al. Reference Luo, Qi and Bai2022). This reduction reflects the mechanical resistance of the crowded particle layer against capillary-driven contraction, which is consistent with the jamming interpretation of arrested coalescence discussed above.

3.2. Influence of particle wettability

In § 3.1, the coalescence of particle-coated droplets was systematically investigated under neutral wetting conditions for varying particle coverages. Here, we examine how the variation in the particle wettability influences the collision and coalescence behaviour of droplets, especially under non-neutral wetting conditions. To isolate the influence of the contact angle, simulations are conducted with 28 particles uniformly distributed at the interface of each droplet, corresponding to a particle coverage of $C_1 = C_2 = 0.58$ . Unless otherwise stated, a Weber number of 13 is maintained, and all parameters are held constant except for the contact angle, which is varied from $\theta = 70^\circ$ to $135^\circ$ .

Figure 11.

Droplet collision and coalescence dynamics at $C_1 = C_2 = 0.58$ and $\textit{We} = 13$ for contact angles of (a) $\theta = 70^\circ$ and (b) $\theta = 110^\circ$ , with corresponding close-ups of particle–interface interactions in (c) and (d).

First, two representative contact angles, $\theta =70^\circ$ and $\theta =110^\circ$ , are selected for analysis. Figures 11(a) and 11(b) illustrate the overall process of droplet collision and coalescence under these two wetting conditions. As shown in figure 11(a), for $\theta =70^\circ$ , the two droplets undergo total coalescence after collision, resulting in a single merged droplet that fully encapsulates an individual particle. In contrast, at $\theta =110^\circ$ , the droplets reach a steady state in which they neither coalesce nor separate, but remain connected via a bridge consisting of three particles, as depicted in figure 11(b). To elucidate the origin of these morphological differences, figures 11(c) and 11(d) provide a detailed view of the particle–interface interactions at the two contact angles. A comparison with figure 8(a) reveals that particles preferentially wetted by the droplet phase promote coalescence through a mechanism similar to that of neutrally wetting particles. Specifically, as shown in figure 11(c), their affinity for the droplet phase accelerates migration into the droplet interior, thereby enhancing the coalescence process. Conversely, particles preferentially wetted by the ambient fluid (figure 11 d) inhibit coalescence due to repulsive interactions with the droplet phase. In this case, particles ‘adsorb’ simultaneously onto both interfaces, significantly restricting their mobility and forming a barrier that hinders the drainage of the interstitial liquid film. As a result, the ambient fluid is expelled exclusively through the regions above and below the particles. Notably, this process is accompanied by the re-establishment of bridging as ‘new’ particles (e.g. the particle 1 at $t^{*}=2.3$ in figure 11 d) come into contact with the opposing interface. Consequently, a partially trapped liquid film forms between the bridged particle pairs, further delaying ambient-fluid drainage. The inability of the droplets to fully drain the intervening liquid films prevents direct contact and coalescence, ultimately leading to the formation of a stable, particle-bridged droplet configuration.

As discussed above, particle bridging not only prevents droplet coalescence under significant compression but also sustains a permanent connection between droplets during subsequent relaxation (figure 11 b). In this state, the droplets neither separate nor coalesce because the forces acting on the particle surfaces are in equilibrium. According to Binks & Lumsdon (Reference Binks and Lumsdon2000), the energy barrier for particle desorption from the interfaces is inherently high, thereby inhibiting particle removal and enhancing system stability. Moreover, the sandwich-like structure formed by the particles and the interfaces effectively suppresses outflow of the liquid film while preventing inflow of external fluids, further contributing to the unique droplet morphology.

Figure 12.

Phase diagram of droplet morphology after collision for different Weber numbers and contact angles. A dashed box is included to highlight three representative cases shown below in figure 13.

In § 3.1, we demonstrated that an increase in the Weber number enhances kinetic energy during droplet collision, promoting its conversion to surface energy. This leads to longer-lasting flattened interfaces and increases the probability of particle contact and bridge formation. Here, we further investigate the combined effects of Weber number and contact angle on droplet coalescence, and the corresponding phase diagram is presented in figure 12. It is clear that for $\theta \leqslant 90^\circ$ , the approaching interfaces tend to contact each other and eventually coalesce under capillary forces. Therefore, the following analysis focuses primarily on the cases with $\theta \gt 90^\circ$ .

Figure 13.

Snapshots of liquid film drainage and the formation of particle bridges at $\theta = 106^\circ$ for (a) $\textit{We}=2$ , (b) $\textit{We}=5$ and (c) $\textit{We}=10$ .

As illustrated in figure 12, for $90^\circ \lt \theta \leqslant 100^\circ$ , the collision outcome is governed by the Weber number. At $\textit{We} = 1.4$ , droplets form a stable single-particle-bridge structure, while $2 \leqslant \textit{We} \leqslant 15$ , higher kinetic energy destabilises the bridging structure, leading to complete droplet coalescence. This indicates that within this contact-angle interval, the input of kinetic energy is the key factor that suppresses bridging and promotes coalescence. When the contact angle increases to $\theta = 106^\circ$ , a notable transition occurs: as the Weber number increases, the post-collision droplet structure evolves from single-particle bridging to coalescence, and then to multi-particle bridging. Figure 13 depicts a close-up view of the droplet collision process for three representative Weber numbers within the dashed box in figure 12. At $\textit{We} = 2$ (figure 13 a), the droplets approach each other slowly with curved interfaces and exhibit negligible deformation, ultimately forming a stable bridged configuration. Under such low- $\textit{We}$ conditions, the limited kinetic energy is primarily dissipated by viscous effects, with very little converted into surface energy. At $\textit{We} = 5$ (figure 13 b), only a single particle bridges the two interfaces until $t^*=6.96$ . Although this particle hinders the close approach of the flattened interfaces, the interfaces regain curvature during subsequent relaxation, enabling adjacent particle-free regions to contact and coalesce. At $\textit{We} = 10$ (figure 13 c), the collision induces more pronounced interface flattening, which allows multiple particles to simultaneously contact both interfaces and form multi-particle bridges, thereby enhancing the mechanical stability of the droplet pair. These results indicate that a single-particle bridge provides insufficient stability: the fluid interface near the particle is not fully shielded, and the liquid film in adjacent regions may rupture under flow, allowing coalescence to proceed. Thus, multiple particles are required to effectively prevent coalescence by collectively bridging the interfacial structure.

When $110^\circ \leqslant \theta \leqslant 130^\circ$ and $5 \leqslant \textit{We} \leqslant 15$ , bridging consistently occurs after droplet collision. As the contact angle further increases, the portion of each particle initially submerged in the droplet penetrates further into the ambient fluid. Owing to the lower viscosity of the ambient fluid, viscous resistance on the particles decreases, allowing them to move more freely along the interface during collision. Some particles may even migrate toward the nearest interfacial region between the two droplets prior to coalescence. At $\textit{We} = 2$ (figure 14 a), insufficient kinetic energy prevents significant particle rearrangement, resulting in non-coalescence. However, when $\theta = 135^\circ$ and $\textit{We} \leqslant 10$ , particle motion becomes more pronounced, leading to greater accumulation of particles in the narrow gap between droplets before collision. This increased steric hindrance further inhibits coalescence, as illustrated in figure 14(b). Notably, at $\textit{We} = 15$ (figure 14 c), a unique behaviour is observed: the droplets separate after colliding and forming a particle-bridge structure, suggesting that a larger contact angle does not necessarily imply higher stability within the bridging configuration. The observation is consistent with French et al. (Reference French, Taylor, Fowler and Clegg2015), who showed that an excessively large contact angle reduces the adhesion energy of particles at the interface, thereby weakening the stability of the particle-bridge structure. From our results, relatively stable particle-bridge structures are found to form within the ranges $110^\circ \leqslant \theta \lt 135^\circ$ and $5 \leqslant \textit{We} \leqslant 15$ .

It is important to note that, in the present simulations, no persistent bridge is observed for $\theta \lt 90^\circ$ within the tested two-dimensional collision configuration and parameter range. However, particles with $\theta \lt 90^\circ$ still protrude from the fluid–fluid interface and therefore, in principle, could participate in bridging under different conditions. We cannot exclude the possibility of bridging for $\theta \lt 90^\circ$ in other geometries, with different particle properties or under alternative forcing conditions. Such cases, however, lie outside the scope of the present study and have not been systematically examined here.

Figure 14.

Non-coalescence behaviour after droplet collision at different contact angles and Weber numbers: (a) $\theta = 120^\circ , \textit{We} = 2$ ; (b) $\theta = 135^\circ , \textit{We} = 10$ ; and (c) $\theta = 135^\circ , \textit{We} = 15$ .

3.3. Collision and coalescence of droplet group

In practical applications, emulsion droplets typically exist as droplet groups. Building upon the binary droplet collision and coalescence dynamics discussed in previous sections, we extend our investigation to a droplet group consisting of 12 polydisperse droplets subjected to an initial instantaneous shear flow. This section aims to numerically examine the coalescence and bridging phenomena that arise within such a droplet group under controlled hydrodynamic conditions.

Figure 15.

Schematic diagram of the computational domain ( $L_x \times L_y = 2000 \times 1000$ ) and initial droplet and velocity distributions in the droplet-group simulations. Here, $U_0$ is the initial velocity imposed at the top and bottom boundaries.

The initial geometric configuration is shown in figure 15, consisting of twelve randomly distributed droplets with radii spanning 50–100 lattice units. The average radius, $R_{{ave}}=75$ , is adopted as the characteristic length scale for the droplets. Each droplet interface is uniformly covered with a prescribed number of particles, with particle coverage varying across droplets. To quantitatively characterise the overall particle distribution, we define an average particle coverage $C_{ave}$ as the ratio of the total interfacial length occupied by particles to the total interfacial length of the corresponding clean droplets under identical conditions. Simulations are performed in a computational domain of dimensions $L_x\times L_y=2000\times 1000$ . The boundary conditions are specified as follows: no-slip conditions are imposed at all solid surfaces, including the particle surfaces and the upper and lower domain boundaries, using the half-way bounce-back scheme, while periodic boundary conditions are applied along the left and right boundaries.

Figure 16.

Coalescence of the droplet group at average particle coverages of (a) $C_{ave} = 0$ , (b) $C_{ave} = 0.28$ , (c) $C_{ave} = 0.64$ and (d) $C_{ave} = 0.75$ .

In the absence of external forcing, both the droplet group and the adsorbed particles remain stationary. To trigger relative motion, collision and potential coalescence among the droplets, an instantaneous shear flow is applied at the initial moment, perturbing the quiescent state and initiating hydrodynamic interactions. The initial velocity distribution in the flow field is illustrated in figure 15. Since shear-induced coalescence is frequently encountered in industrial processes (Thivilliers-Arvis et al. Reference Thivilliers-Arvis, Laurichesse, Schmitt and Leal-Calderon2010), the ambient fluid is initialised with a linear velocity profile emulating a fully developed shear flow. This profile varies linearly along the $y$ -direction, establishing a velocity gradient that drives relative droplet motion and thereby promotes subsequent collision and coalescence events. The intensity of the initial shear is quantified by the Reynolds number $\textit{Re}$ , defined as

(3.3) \begin{equation} \textit{Re} =\frac {\rho _{r}\dot {\gamma }R_{{ave}}^2}{\mu _{{r}}}, \end{equation}

where $\dot {\gamma }=2U_0/L_y$ is the shear rate, and $U_0$ denotes the initial velocity imposed at the top and bottom boundaries. In all subsequent simulations, the Reynolds number is maintained at $\textit{Re}=2$ .

In all cases, the initial spatial configuration of droplets remains the same. To facilitate analysis, the disturbance is applied only at the start of the simulation. Figure 16(a) illustrates the coalescence process of a clean-droplet group at $\textit{Re} = 2$ . Under the initial shear, droplets migrate toward each other and make contact. In the absence of particle interference, the rupture of the liquid film between adjacent droplets triggers their coalescence, leading to subsequent oscillation and deformation of the merged droplets. Under the present flow conditions, droplet motion occurs mainly along the $x$ -direction, limiting opportunities for some droplets to come into contact. As a result, even at $t^* = 28.2$ (when the flow is fully stabilised), four individual droplets remain in the domain.

Figure 17.

Time evolution of the normalised interfacial length for a droplet group at average particle coverages $C_{ave} = 0$ , $0.28$ , $0.64$ and $0.75$ . The instants $t_1^*$ and $t_2^*$ denote the onset of coalescence for $C_{{ave}} = 0$ and 0.28, respectively.

We acknowledge that the clean droplets in figure 16(a) coalesce more readily than would be expected for a perfectly sharp-interface system. This behaviour is closely related to the diffuse-interface nature of the present colour-gradient LB model. When two clean droplets approach each other, coalescence can occur once their diffuse interfacial regions overlap. With increasing grid resolution (for a fixed physical droplet size), the effective interface thickness decreases, and the droplets would need to approach more closely before coalescence is triggered; in the sharp-interface limit, coalescence would occur only upon direct contact of the two interfaces. Therefore, the relatively easy coalescence of clean droplets in figure 16(a) is an expected consequence of the diffuse-interface representation and the finite numerical interface thickness. We emphasise that the clean droplet case serves only as a reference within the same numerical framework; the effects of interfacial particles will be examined subsequently by introducing particles at varying coverages, and the main conclusions are drawn from the relative changes with respect to this reference.

The introduction of neutrally wetting particles would significantly alter the coalescence behaviour of the droplet group. Figure 16(b–d) shows the coalescence process at $C_{ave} = 0.28$ , $0.64$ and $0.75$ , representing low, medium and high coverages, respectively. At $C_{ave} = 0.28$ (80 particles), the presence of particles does not substantially hinder coalescence. Five droplets remain at $t^* = 28.2$ , and some droplet surfaces remain only partially covered, indicating limited steric hindrance (figure 16 b). At $C_{ave} = 0.64$ (182 particles), eight independent droplets persist at $t^* = 28.2$ , with some exhibiting arrested coalescence, leading to irregular shapes in the steady state (figure 16 c). At $C_{ave} = 0.75$ (224 particles), droplets are initially highly covered by particles. Under shear, droplets deform and expose more interface, yet the repulsive forces between particles prevent effective droplet contact. Consequently, no coalescence occurs, and all droplets remain intact (figure 16 d).

Figure 18.

Time evolution of the normalised interfacial length at $\textit{Re}=0.5$ for the clean-droplet group ( $C_{ave} = 0$ ) and the droplet group with $C_{ave} = 0.28$ . The instants $t_{3'}^*$ and $t_{1'}^*$ denote the onset of coalescence for $C_{{ave}} = 0$ and 0.28, respectively, while $t_{2'}^*$ represents an intermediate time between $t_{1'}^*$ and $t_{3'}^*$ .

Figure 19.

Coalescence behaviour of droplet groups with $C_{{ave}}=0$ and $C_{{ave}}=0.28$ at (a) $t_{1'}^{*}=21.9$ and (b) $t_{2'}^{*}=23.4$ .

Figure 20.

Snapshots of droplet-group coalescence for contact angles of (a) $\theta = 70^\circ$ , (b) $\theta = 90^\circ$ , (c) $\theta = 110^\circ$ and (d) $\theta = 120^\circ$ , with insets highlighting local droplet morphology and particle distribution.

To quantitatively assess the coalescence process, we analyse the evolution of the total interfacial length, which is normalised by $2\pi R_{{ave}}$ . As shown in figure 17, the total interfacial length of the droplet group decreases with increasing particle coverage $C_{ave}$ , indicating enhanced system stability. This behaviour is attributed to the effective suppression of droplet coalescence at higher values of $C_{ave}$ . As a result, the reduction in total interfacial length becomes less pronounced compared with in systems with lower coverage. In particular, when $C_{ave} = 0.75$ , where coalescence is fully inhibited, the total interfacial length remains nearly constant over time. In addition, we notice in figure 17 that coalescence initiates earlier ( $t_2^* = 5.3$ ) for the $C_{ave} = 0.28$ case than for the clean case ( $t_1^* = 6.5$ ), suggesting that neutrally wetting particles at low coverage can facilitate coalescence. To exclude effects from strong initial shear, we conduct additional simulations at a lower Reynolds number ( $\textit{Re} = 0.5$ ). Figure 18 shows the corresponding evolution of the normalised interfacial length. Even at this lower $\textit{Re}$ , the droplet group with $C_{ave} = 0.28$ still coalesces earlier. Figure 19(a) shows that at $t_{1'}^* = 21.9$ , droplets with sparse particles have already begun to coalesce, while clean droplets are still draining the liquid film. By $t_{2'}^* = 23.4$ (figure 19 b), interfaces in the $C_{ave} = 0.28$ case merge via capillary force, whereas coalescence in the clean group has not yet started. The clean-droplet group begins coalescing only at $t_{3'}^* = 24.8$ . These results demonstrate that at low particle coverage, where steric hindrance is limited, the presence of particles can promote earlier coalescence under initial shear conditions.

We next investigate the effect of particle wettability on droplet-group coalescence at moderate average particle coverage. Following § 3.2, simulations are performed with a fixed number of particles (144) and $\textit{Re} = 2.0$ , varying only the contact angle while keeping the particle distribution unchanged. Figure 20 shows the coalescence process of the droplet group for four different contact angles. Similar to the behaviour observed in binary droplet collisions, the contact angle significantly influences the coalescence process and leads to distinct morphological outcomes. At a small contact angle ( $\theta = 70^{\circ }$ ), as shown in figure 20(a), multiple sparsely covered droplets coalesce into larger ones under shear. During partial coalescence, due to the continuous reduction of the liquid–liquid interfacial length, the physical obstruction by particles results in the stabilisation of droplets in an irregular shape, i.e. arrested coalescence. When the contact angle increases to $\theta = 90^{\circ }$ , a bilayer particle barrier effectively prevents coalescence, as visible in the inset of figure 20(b), maintaining adjacent droplets in a stable, non-coalescing state. At even higher contact angles, particle bridging emerges. For $\theta = 110^{\circ }$ (figure 20 c), fewer droplets coalesce compared with the case with $\theta \leqslant 90^\circ$ , and a stable chain of three droplets connected by particle bridges is formed. When the contact angle is further increased to $\theta = 120^\circ$ (figure 20 d), coalescence is completely suppressed. In this case, more particles are stably adsorbed between droplets, increasing the number of bridged droplets. In summary, for moderately concentrated systems, two stabilisation mechanisms are identified: dense particle layers that inhibit coalescence, and particle bridging (occurring only when particles are preferentially wetted by the ambient fluid) that forms stable droplet chains.

Finally, we perform a quantitative analysis to characterise the stability of particle-bridged droplet clusters under shear. Taking the bridged droplet-group case shown in figure 20(d) as the baseline configuration, where stable bridges are formed at $\textit{Re}=2.0$ , we progressively increase the initial boundary velocity $U_0$ (equivalently increasing $\textit{Re}$ ) to identify the critical condition for bridge failure. Our results show that the particle-bridged droplet group remains stable up to $\textit{Re}=3.0$ , whereas bridge rupture occurs for $\textit{Re}=3.1$ . Thus, for this particular configuration, the bridging strength can be characterised by a critical Reynolds number $\textit{Re}_c\approx 3.0$ for shear-induced breakup. This provides a first quantitative measure of cluster stability, which is directly relevant to rheological characterisations of Pickering emulsion systems.