Coarsening of thin films with weak condensation

TL;DR

This work addresses coarsening of weakly volatile thin liquid films via a lubrication PDE and develops a reduced droplet-dynamics model based on nearest-neighbor interactions in the weak condensation limit. By exploiting a quasi-steady droplet ansatz and a parabolic core approximation, the authors derive a low-dimensional ODE system for droplets with positions $\mathbf{X}_k$ and pressures $\mathbf{P}_k$, incorporating both conservative inter-droplet fluxes and a non-conservative condensation flux. The paper reveals three long-time dynamical regimes and associated scaling laws: an early quasi-conservative stage with $N(t)=O(t^{-2/5})$, a mid-stage of drop-wise condensation with $N(t)=O(t^{-1/2})$, and a late-stage filmwise condensation transitioning to a slower, possibly logarithmic coarsening; the total mass $\mathcal{M}(t)$ similarly evolves from $O(t)$ to $O(\sqrt{t})$ and then logarithmically. The findings show that weak condensation fundamentally alters mass exchange and coarsening pathways, producing a transition from collapse-dominated to collision-dominated dynamics with practical implications for heat transfer and desalination contexts where evaporation/condensation plays a key role.

Abstract

A lubrication model can be used to describe the dynamics of a weakly volatile viscous fluid layer on a hydrophobic substrate. Thin layers of the fluid are unstable to perturbations and break up into slowly evolving interacting droplets. A reduced-order dynamical system is derived from the lubrication model based on the nearest-neighbor droplet interactions in the weak condensation limit. Dynamics for periodic arrays of identical drops and pairwise droplet interactions are investigated which provide insights into the coarsening dynamics for large systems. Weak condensation is shown to be a singular perturbation, fundamentally changing the long-time coarsening dynamics for the droplets and the overall mass of the fluid in two additional regimes of long-time dynamics.

Figures (10)

• Figure 1: (Left) The disjoining pressure function $\mathbf{P}=\Pi(h)$, (middle) orbits in the $h,h_x$ phase plane for steady solutions, (right) the homoclinic steady droplet profile for $p=0$ ($\mathbf{P}=\mathcal{P}_*$). All three are drawn with the common vertical axis, $0\le h\le 0.5$.
• Figure 1: (Left) A schematic figure of a system of three quasi-static droplets in a periodic domain $0 \le x \le 150$. (Right) Two quasi-static droplets from a numerical solution of \ref{['Mainpde']} (solid curve) locally satisfying \ref{['eq:quasi-static']} with the pressures $\mathbf{P}=\mathbf{P}_{1,2}$, where the core of droplets can be approximated by parabolas (dot-dashed curves) defined by \ref{['eq:drop_parabolic']}. The shifted pressure $\mathbf{P}=p(x)+\mathcal{P}_*$ (dashed curve) for the precursor layer is given by \ref{['eq:pressure_precursor']} over $x_L\le x\le x_R$ where the fluxes at the edges of droplets are $J^{R,L}$.
• Figure 1: (a) PDE simulation of \ref{['Mainpde']} for a slowly condensing droplet in a periodic domain $0\le x \le L$ with $L=100$ and $\beta=10^{-7}$. (b) The evolution of the droplet pressure in the PDE model (dots) compared against predictions from the dynamical model \ref{['Dpdt_singleDropCondense']} (solid curve) for $t<t_f\approx 1.8\times 10^8$. The last stage of the droplet growth is given by \ref{['eq:condensationLimit']} (dot-dashed curve) as $\mathbf{P} \to P_L$, followed by convergence to film-wise condensation after the domain is filled for $t>t_f$ (dotted curve). (c) The evolution of the mass $\mathcal{M}(\hat{t}\,)$ for a rescaled time variable $\hat{t}=\beta t$ at several values of $\beta$, showing a transition from the scaling $\mathcal{M} = O(\hat{t}\,)$ in the dropwise condensation stage to the scaling $\mathcal{M} = O(\sqrt{\hat{t}\,})$ in the filmwise condensation stage. (d) Parametric plot of droplet half-width $w(t)$ and pressure $\mathbf{P}(t)$ from PDE simulations like (a) at several values of $\beta$. The quasi-steady relation $w(\mathbf{P})=A/\mathbf{P}$\ref{['eq:drop_width']} is shown for comparison (dashed line).
• Figure 1: (Left) PDE simulation of \ref{['Mainpde']} for the dynamics of two equally-sized droplets in a periodic domain. At time $t=0$, two droplets with identical pressure $\mathbf{P}=0.1$ are placed at $\mathbf{X}_1 = 40$ and $\mathbf{X}_2 = 60$. The droplets separate and become more equally spaced as weak condensation occurs. (Right) The evolution of the droplet peak positions $\mathbf{X}_1(t)$ and $\mathbf{X}_2(t)$ (marked by dots) and their contact line positions $\mathbf{X}_{1,2} \pm w(\mathbf{P})$ (solid curves) from the PDE simulation agree well with the predictions (dashed curves) from the dynamical model \ref{['eq:equalPressureSystem']}. The domain size is $L=100$ and $\beta = 10^{-7}$. The shaded area shows the trajectory of two droplets, leading up to filmwise condensation after the droplets collide.
• Figure 1: PDE simulations and pairwise droplet pressure trajectories $(\mathbf{P}_k(t), \mathbf{P}_{k+1}(t))$ with $6$ droplets initially placed with equal spacing in a periodic domain $0\le x \le 450$. In the top panel (a,b), all the droplets grow in time, with pressure trajectories in the pairwise growth region (A). In the bottom panel (c,d), one droplet collapses and the other five droplets grow in time, yielding two pressure trajectories in the growing-shrinking region (B) and the other trajectories in region (A). The initial peak-to-peak spacing $\lambda=75$. The other parameters are identical to those used in Fig. \ref{['fig:twoDropTrajectories']}, $\beta=10^{-7}$, $\hat{\mathcal{D}}_{\min}=5$, $\hat{\mathcal{P}}_{\max}=0.63$.