Nanobubble-induced flow of immersed glassy polymer films

We study the free-surface deformation dynamics of an immersed glassy thin polymer film supported on a substrate, induced by an air nanobubble at the free surface. We combine analytical and numerical treatments of the glassy thin film equation, resulting from the lubrication approximation applied to the surface mobile layer of the glassy film, under the driving of an axisymmetric step function in the pressure term accounting for the nanobubble's Laplace pressure. Using the method of Green's functions, we derive a general solution for the film profile. We show that the lateral extent of the surface perturbation follows an asymptotic viscocapillary power-law behavior in time, and that the film's central height decays logarithmically in time in this regime. This process eventually leads to film rupture and dewetting at finite time, for which we provide an analytical prediction exhibiting explicitly the dependencies in surface mobility, film thickness, and bubble size, among others. Finally, using finite-element numerical integration, we discuss how nonlinear effects induced by the curvature and film profile can affect the evolution.

Figure 1

Schematic of the system (not drawn to scale). A thin glassy polymer film of initial thickness $h_{\text{i}}$, with a thin mobile surface layer of thickness $h_{\text{m}}$ and viscosity $\mu$, is immersed in water. At time $t=0$, an axisymmetric air bubble of radius $r_{\text{b}}$, with origin at $r=0$, is placed atop the glassy film. The excess Laplace pressure inside the bubble drives the mobile layer to flow and deforms the glassy film with vertical deflection $h(r,t)$. ${\gamma }_{\text{LV}}$, ${\gamma }_{\text{SV}}$, and ${\gamma }_{\text{SL}}$ are the surface tensions of the water-air, film-air, and film-water interfaces, respectively. In the mathematical model, we further assume for simplicity that ${\gamma }_{\text{SL}}={\gamma }_{\text{SV}}$ (further noted $\gamma$), with no loss of generality.

Figure 2

(a) Normalized surface profile of a levelling glassy thin film with no bubble, but with a Dirac initial surface perturbation. The dashed line is the normalized Green's function of Eq. (14). The solid colored lines represent the normalized finite-element numerical integration (FENI) of Eq. (5) with $\beta =0$ (no bubble), where the dimensionless times $T$ are given by the color bar. The inset shows the normalized FENI profiles as functions of the radial coordinate $R$, at different times, while the main figure shows the same data plotted as a function of the similarity variable $\xi =R{T}^{-1/4}$. (b) Evolution of the surface profile of a glassy thin film induced by the presence of a bubble. The solid lines are the FENI of Eq. (5) with $\beta =1$, at different times $T$ as indicated, while the dashed lines are the numerical estimates of Eq. (16).

Figure 3

(a) Normalized central height of the bubble-induced perturbation of the film profile as a function of time, computed from Eq. (17). The dashed line indicates the asymptotic expression of Eq. (19), with $C=0.04746$. (b) Half-width of the bubble-induced perturbation as a function of time, defined as the radial coordinate of the maximum of $H(R,T)$ [see Eq. (16) and Fig. 2]. The slope triangle indicates a $1/4$ power-law exponent.

Figure 4

Normalized excess surface energy of the film as a function of time, as computed from Eq. (22) and the solution in Eq. (16), for $\beta =1$. The dashed line is a best fit to Eq. (23) with a numerical prefactor $0.00297$ and an offset 0.22.

Figure 5

Central height of the perturbation of the film profile as a function of time, as obtained from the solutions of the GTFE [Eq. (17)], MGTFE [FENI of Eq. (24) with ${(h_{\text{m}}/r_{\text{b}})}^2$=0.1], and TFE [FENI of Eq. (25)], for three different $\beta$ as indicated.