Reframing droplet coalescence: Identifying the distinctive dynamics of nanofilm evolution

The richness of the dynamics of the thin film formed between two coalescing Brownian droplets is presented. Simulations based on a previously developed model [Phys. Rev. E 81, 051404 (2010)] which solves two coupled Langevin equations in the lubrication limit were performed. The time evolution of the thickness and radius of the cylindrical film is evaluated for the first time in terms of phase space, entropy of permutation, mean-square displacements, and creeping compliance. This new perspective of analysis reveals the specific and distinctive features of the nanofilm formation and breaking: (1) a well-defined attractor-type phase space which slightly become less disperse as they approach instead of single deterministic trajectories observed in hydrodynamic dominated systems, (2) complexity values larger than the ones expected for Fractional Brownian Motion, (3) a super-diffusive behavior of the film thinning but a sub-diffusive behavior of the film growing, and (4) the effective shear forces in the film that strongly depends on its configuration. Remarkably, the major role of thermal fluctuations in to modulate the strong non linear effects of hydrodynamic interactions, forcing the film to explore configurations well delimited by the potential energy.

Figure 1

Truncated spheres geometry model of two colliding emulsion droplets of size $a$ with the cylindrical film formed between them with radius $r$ and thickness $h$. In our model, both parameters vary simultaneously with time.

Figure 2

Configuration space and velocities of the evolving film: (a) $h$ vs. $r$ with energy contour lines (separation between lines 3 $k_{B}T$; the numbers indicate the values of the energy at that configuration), (b) $V_h$ vs. $V_r$, (c) $V_r$ vs. $r$, (d) $V_h$ vs. $r$, (e) $V_r$ vs. $h$, and (f) $V_h$ vs. $h$. In all these cases, the initials conditions used are $h_0=20$ nm and $r_0=30$ nm for 10 numerical coalescence realizations. The time step $\mathrm{\Delta }t\gg m{D}_{ii}^0/k_{B}T$ was selected in accordance with the Brownian dynamic procedure for friction dominated systems (${10}^{-8}$ s). The values of parameters used are: $a=1\mu \mathrm{m}$, interfacial tension $\gamma$=1 mN/m, water viscosity $\eta =8.91\times {10}^{-4}$ kg ${\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$, fundamental electric charge $e=1.60210\times {10}^{-19}$ C, vacuum permitivity ${\epsilon }_0=8.85\times {10}^{-12}{\mathrm{Fm}}^{-1}$, relative dielectric constant of the water at room temperature $\epsilon =80.4$, electrolyte concentration $C_{el}=0.5$ M, surface electric potential ${\mathrm{\Psi }}_0=15$ mV, and emulsion parameter ${\epsilon }_s=0.001$. The velocities were directly calculated from $\mathrm{\Delta }{x}_i/\mathrm{\Delta }t$. Each run corresponded to different initial random seeds, and the averaging was obtained from 1000 runs.

Figure 3

Potential force during the approaching of two deformed droplets with the average distance $\langle h\rangle$. We use the symbols in brackets for the forces to represent their values estimated from the average values of $r$ and $h$. Initial configurations of the film are $h_0=20$ nm and $r_0=$ 10 nm (red), 20 nm (blue), 40 nm (green), 50 nm (brown), and 60 nm (orange). The inset shows the average hydrodynamic friction $F_f$ in the $h$ direction, calculated from $F_f=\zeta ·\mathbf{V}$. Unlike the potential forces, the dispersion exhibited by $F_{fh}$ is due to the random forces expressed through $V$.

Figure 4

Mean-square displacements in (a) $r$ and (b) $h$, for initial values of $h_0=20$ nm and $r_0$ 10 nm (red), 20 nm (blue), 30 nm (green), 40 nm (black), 50 nm (brown), and 60 nm (orange). Inset (a) average value of $r$ for the different initial configurations and 10 realizations, and (b) average value of $h$ (colored transparent curves; the blue envelope curves represent the standard deviation). The curves represented in (c) correspond to the creeping compliance estimated for all the mentioned cases: the inset represents the estimated values of $J$ from $J\left(t\right)=[A_0-A\left(t\right)]/A_0/(\mathrm{\Pi }-{\mathrm{\Pi }}_0)$ ($y$ axis has the same units and magnitudes that the outer curves), while outer curves in (c) correspond to the Maxwell-Kelvin phenomenological model.

Figure 5

Dependence of complexity $C$ with permutation entropy $H$ for $h$ (blue), $r$ (red), $V_h$ (orange), and $V_r$ (green). Black lines represent maximum and minimum complexities, and diamonds represent fractional Brownian motion obtained by Rosso et al. [39] Entropy and complexity were calculated from the distribution permutation of dimension 6 for one realization, as proposed by Bandt and Pompe [38]. From it, the Shannon entropy and complexity were derived. This was applied to the different sets of initial parameters using the averaged values of ten runs.

Figure 6

Evolution of the film radius $r$ for different electrolyte concentrations (a) 0.01 M, (b) 0.1 M, (c) 0.2 M, and (d) 0.5 M. (e) Evolution of the film thickness $h$ leading to coalescence (color curves) and to receding (black curve).