3.1 Two-dimensional simulations

We investigate the influence of fluctuations on the elastic interface dynamics by solving (2.2) with $N=0$ and $N=1$ for a two-dimensional geometry, providing a three-dimensional description of the film. A direct comparison of the deterministic ( $N=0$ ) and stochastic ( $N=1$ ) simulation is shown in figure 2(a–c), where we first notice that the two blisters appear to evolve in a similar fashion. However, subtracting the two height profiles (figure 2 d) shows that the fluctuations affect the shape of the blister’s leading edge, where a precursor-like film is formed ahead of its bulk as the elastic interface peels away from the pre-wetted substrate, see figure 2(e). The blister dynamics appears to adopt a self-similar profile and the extraction of its height $H(t)$ in figure 2(f) shows that the deterministic simulation $N=0$ follows a power-law behaviour $H(t)\sim t^{-2/11}$ that is disturbed as $N>0$ , see figure 2(f).

Figure 2.

Two-dimensional simulations of (2.2), which represents the three-dimensional dynamics height field $h(\boldsymbol{x},t)=h(x,y,t)$ of the elastic blister. The simulations for $N=\sqrt{(2k_{B}TL^{2})/\unicode[STIX]{x1D707}h_{0}^{2}}$ is the average over 20 simulations. (a–c) The spatio-temporal blister dynamics. (a,d) Deterministic simulation with $N=0$ . (c,f) Inclusion of noise with $N=1$ . The solid contour line shows the height $h(x,t)=3\times \unicode[STIX]{x1D716}=0.03$ , which is three times the height of the pre-wetted film layer $\unicode[STIX]{x1D716}$ . (d) To illustrate the difference between the deterministic and stochastic dynamics, we subtract the height profile for $N=1$ from $N=0$ i.e.  $\unicode[STIX]{x0394}h(\boldsymbol{x},t)=h_{N=0}-\langle h\rangle _{N=1}$ at $t=1.4\times 10^{-4}$ , showing that a precursor-like film is formed ahead of the bulk of the blister. (e) We magnify the spreading front in the area denoted by the dashed lines in (d), to show the local change in curvature and peeling of the elastic interface from the pre-wetted substrate. (f) The spatio-temporal dynamics in (a–c) suggests that the blister adopts to a self-similar profile. We extract $H(t)=\max (h(\boldsymbol{x},\boldsymbol{t}))$ , located at the $\boldsymbol{x}$ -position of the blister’s centre of mass. By plotting $H(t)$ on a logarithmic axis we show that for $N=0$ the height evolves as $H(t)\sim t^{-2/11}$ as in (3.1), which is disturbed by the inclusion of noise $N=1$ .

To understand the blister dynamics in figure 2 we use a scaling analysis (Davidovitch et al. Reference Davidovitch, Moro and Stone2005; Lister et al. Reference Lister, Peng and Neufeld2013), where we start by looking at the limit where fluctuations can be ignored as $N=0$ i.e. purely deterministic dynamics. In this limit, the spreading of the blister is slow and has a nearly uniform pressure $\hat{p}=\hat{p}_{0}=B\hat{\unicode[STIX]{x1D6FB}}^{4}{\hat{h}}(\hat{\boldsymbol{x}},\hat{t})\sim B{\hat{H}}(\hat{t})/\hat{R}^{4}(\hat{t})$ in its bulk (Lister et al. Reference Lister, Peng and Neufeld2013). By integrating $\hat{p}$ we get the blister shape by assuming axial symmetry, as also confirmed in the two-dimensional simulation, and boundary conditions; at the symmetry line $\hat{\unicode[STIX]{x1D735}}{\hat{h}}=\hat{\unicode[STIX]{x1D6FB}}^{3}{\hat{h}}=0$ and at the advancing front ${\hat{h}}=\hat{\unicode[STIX]{x1D735}}{\hat{h}}=O(\hat{\unicode[STIX]{x1D716}})$ (Lister et al. Reference Lister, Peng and Neufeld2013). The blister shape combined with the matching of the local travelling-wave solution ${\hat{c}}$ gives the speed of the peeling tip $\text{d}\hat{R}/\text{d}\hat{t}={\hat{c}}=B\hat{\unicode[STIX]{x1D716}}^{1/2}/12\unicode[STIX]{x1D707}(\hat{\unicode[STIX]{x1D705}}/1.35)^{5/2}$ (Lister et al. Reference Lister, Peng and Neufeld2013) that is dictated by its local curvature $\hat{\unicode[STIX]{x1D705}}$ , the pre-wetted film height $\hat{\unicode[STIX]{x1D716}}$ , membrane stiffness $B$ and viscosity $\unicode[STIX]{x1D707}$ . We scale ${\hat{h}}\sim {\hat{H}}(\hat{t}),\hat{\boldsymbol{x}}\sim \hat{R}(\hat{t})$ together with the solution for ${\hat{c}}$ and impose mass conservation through a constant film volume $\hat{V}=\int _{0}^{L}\int _{0}^{L}({\hat{h}}(\hat{x},\hat{t})-\hat{\unicode[STIX]{x1D716}})\,\text{d}\hat{x}\,\text{d}{\hat{y}}$ (in one dimension given by the constant area $\hat{A}=\int _{0}^{L}({\hat{h}}(\hat{x},\hat{t})-\hat{\unicode[STIX]{x1D716}})\,\text{d}\hat{x}$ ), which gives the self-similar spreading dynamics for a three-dimensional (axisymmetric) blister

(3.1a,b ) $$\begin{eqnarray}\hat{R}(\hat{t})=1.57\left(\frac{B^{2}\hat{\unicode[STIX]{x1D716}}}{\unicode[STIX]{x1D707}^{2}}\hat{V}^{5}\hat{t}\right)^{1/22},\quad {\hat{H}}(\hat{t})=\frac{\hat{V}}{\hat{R}^{2}(\hat{t})}.\end{eqnarray}$$

In non-dimensional form this becomes $R(t)\approx t^{1/11}$ and $H(t)\approx t^{-2/11}$ and in one-dimensional geometries $R(t)\approx t^{2/17}$ and $H(t)\approx t^{-2/17}$ . This is the elastohydrodynamic analogue to Tanner’s law for capillary spreading (two dimensions: $R(t)\sim t^{1/10}$ , one dimension: $R(t)\sim t^{1/7}$ ), but contrasts the classical solution with different exponents in time. Our self-similar solution is verified in the numerical simulation shown in figure 2(f) with $N=0$ . However, for $N>0$ the spreading dynamics is influenced by the fluctuations.

Next, we examine the flow regime where the fluctuation effects dominate over the bending rigidity in (2.1) i.e. a flaccid membrane, and look for its self-similar solution. The stochastic term scales as $\hat{\unicode[STIX]{x1D702}}(\hat{\boldsymbol{x}},\hat{t})\sim 1/(\sqrt{\text{d}\hat{t}}\,\text{d}\hat{x})$ (in one-dimensional geometries as $\hat{\unicode[STIX]{x1D702}}(\hat{x},\hat{t})\sim 1/\sqrt{\text{d}\hat{t}\,\text{d}\hat{x}{\hat{w}}}$ , with ${\hat{w}}$ as the in-plane depth) (Davidovitch et al. Reference Davidovitch, Moro and Stone2005) in (2.1), which combined with the mass conservation equation for the film gives,

(3.2a,b ) $$\begin{eqnarray}\hat{R}_{s}(\hat{t})=\left(\frac{2k_{b}T\hat{V}}{12\unicode[STIX]{x1D707}}\hat{t}\right)^{1/6},\quad {\hat{H}}_{s}(\hat{t})=\frac{\hat{V}}{\hat{R}_{s}^{2}(\hat{t})}.\end{eqnarray}$$

In non-dimensional form this becomes $R_{s}(t)\approx (N^{2}t)^{1/6}$ and $H_{s}\approx R_{s}^{-2}(t)\approx (N^{2}t)^{-1/3}$ and in one-dimensional geometries $R_{s}(t)\approx (N^{2}t/w)^{1/4}$ and $H_{s}(t)\approx (N^{2}t/w)^{-1/4}$ where $w={\hat{w}}/L$ , similar to the capillary films in Davidovitch et al. (Reference Davidovitch, Moro and Stone2005). These scaling predictions (3.1) and (3.2) have been summarized in table 1, for one- and two-dimensional geometries.

Table 1.

Non-dimensional scaling laws for the spreading radius $R(t)$ and height $H(t)$ of the blister, using the scaling: $h={\hat{h}}/h_{0}$ , $x=\hat{x}/L$ , $t=\hat{t}/\unicode[STIX]{x1D70F}=t/(12\unicode[STIX]{x1D707}L^{6})/h_{0}^{3}B$ , where $B$ is the bending stiffness, $\unicode[STIX]{x1D707}$ the viscosity, $L$ the lateral extent of the blister and $h_{0}$ its maximum height at $\hat{t}=0$ . The non-dimensional number $N=\sqrt{(2k_{b}TL^{2})/Bh_{0}^{2}}$ gives the ratio of the force from thermal fluctuations and the elastic bending force, where $k_{b}$ is the Boltzmann constant and $T$ the temperature. In one-dimensional geometries we also have the scaled in-plane width $w={\hat{w}}/L$ , which appear in the pre-factor for the scaling law.

3.2 One-dimensional simulations

In order to characterize the influence of $N=\sqrt{(2k_{b}TL^{2})/Bh_{0}^{2}}$ on the blister dynamics, we systematically vary $N$ in a one-dimensional geometry, see figures 3 and 4. It is clear that fluctuations can significantly speed up the spreading dynamics and we observe a dramatic change in blister shapes. By tracking the spatio-temporal blister shape in figure 4 we identify two distinct regimes, one for the elastohydrodynamic Tanner’s law (3.1) and a stochastic spreading regime (3.2), see table 1.

We extract the spreading radius from the simulations in figure 3 by the averaged second moment to describe the advancing front $r=\langle [(1/A)\int h(x,t)(x-X)^{2}\,\text{d}x]^{1/2}\rangle$ with $X=[\int xh(x,t)\,\text{d}x]/A$ (Davidovitch et al. Reference Davidovitch, Moro and Stone2005; Nesic et al. Reference Nesic, Cuerno, Moro and Kondic2015b ). The numerical simulations show that for $N=0$ the spreading recovers the deterministic self-similar profile in (3.1) and as $N\geqslant 5.3$ the advancing front adopts the stochastic dynamics predicted by (3.2). These regimes are further illustrated by the collapse of blister shapes onto a universal curve by rescaling its numerical profiles with $x/R(t),h/H(t)$ for $N=0$ see figure 5(b) and $x/R_{s}(t),h/H_{s}(t)$ for $N=8$ see figure 5(c).

Figure 3.

The average of 40 numerical simulations of (2.2) in one dimension with $N\in [0,2.5,5.3,7.5,8]$ at time $t=7.8\times 10^{-3}$ , using the scaling for $h(x,t)={\hat{h}}(\hat{x},\hat{t})/h_{0}$ and $x=\hat{x}/L$ . Note that the $x$ -axis is here compressed by a factor $h_{0}/L\ll 1$ . We would like to note that the non-dimensional gradient $\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}x$ for $N=8$ has a maximal absolute value $\max (|\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}x|)\approx O(1)$ for each simulation throughout time and our long wave approximation is still valid as we have $\unicode[STIX]{x2202}{\hat{h}}/\unicode[STIX]{x2202}\hat{x}=(h_{0}/L)\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}x\ll 1$ .

Figure 4.

Comparison of the spatio-temporal blister dynamics for $N=0$ (left side) and $N=8$ (right side). The solid line indicates the height $h(x,t)=3\times \unicode[STIX]{x1D716}=0.03$ , which is three times the pre-wetted film height $\unicode[STIX]{x1D716}$ and the results are obtained from an average of 40 simulations of (2.2) in one dimension.

Figure 5.

(a) The extracted spreading radius $r$ in logarithmic axis for $N\in [0,2.5,5.3,7.5,8]$ , the dashed lines illustrate the spreading laws in (3.1) and (3.2). (b) The height profiles for the times $t=[1:2\times 10^{-2}:10]\times 10^{-3}$ with $N=0$ are rescaled as $x/R(t)$ and $h/H(t)$ in (3.1). (c) The average height profile for the times $t=[1:2\times 10^{-2}:10]\times 10^{-3}$ with $N=8$ are rescaled as $x/R_{s}(t)$ and $h/H_{s}(t)$ in (3.2), where the dashed line is the mean of the scaled profile. The results are obtained from an average of 40 simulations of (2.2) in one dimension.

3.3 Regime transition

Our theoretical model is supported by simulations of (2.2) in both one and two dimensions, but limited to fluctuations with a wavelength $\unicode[STIX]{x1D706}\ll h/\unicode[STIX]{x1D735}h$ , which is asymptotically correct as $\unicode[STIX]{x1D735}h\rightarrow 0$ for $t\rightarrow \infty$ . To determine the threshold in the spreading radius for when the stochastic effects will dominate the flow, we use the scaling prediction in (3.2) and compare the magnitude of the two terms on the right-hand side of (2.2). We start by estimating the contribution from interface bending, which scales as $\unicode[STIX]{x1D735}\boldsymbol{\cdot }(h^{3}\unicode[STIX]{x1D6FB}^{5}h)\sim R^{-14}$ (in one dimension ${\sim}R^{-10}$ ) and the stochastic term scales as $\unicode[STIX]{x1D735}\boldsymbol{\cdot }(Nh^{3/2}\unicode[STIX]{x1D702})\sim N^{2}R^{-8}$ (in one dimension ${\sim}N^{2}w^{-1}R^{-5}$ ). Next, we balance these two terms, giving the condition on the spreading radius $R>1/N^{1/3}$ (in one dimension $R>(w/N^{2})^{1/5}$ ) for stochastic effects to dominate the flow. The numerical simulations in figure 5 illustrates this transition in regimes for $N=[5.3,7.5,8]$ , where there is a clear change in slope for $R\approx 0.2$ in accordance with our analysis. In the simulations, we vary $N\in [0{-}8]$ , which corresponds to a membrane with a bending stiffness $B\in [10{-}100]k_{b}T$ , at 300 k with $k_{b}T=4\times 10^{-21}~\text{Nm}$ , a size $L=5~\unicode[STIX]{x03BC}\text{m}$ and height $h_{0}=10~\text{nm}$ , which gives $N\in 2{-}15$ in accordance with the simulations presented in figures 2–5. In one-dimensional geometries we notice that the in-plane depth $w$ influences the transition from deterministic to stochastic spreading dynamics.

3.4 Scaling estimates of other physical effects

Membrane dynamics at microscopic length scales can also be influenced by additional forces arising from the molecular interaction between the material composing the interface and the solid substrate e.g. a van der Waals disjoining pressure $\hat{p}_{vdW}=A/3{\hat{h}}^{3}$ , where $A$ is the Hamaker constant. In addition, biological adhesion phenomena often involve the attachment of long proteins that can influence the front propagation and by assuming these to be like Hookean springs they give a contribution to the pressure $\hat{p}_{s}=K(1-{\hat{h}}/l_{i})$ (Leong & Chiam Reference Leong and Chiam2010; Carlson & Mahadevan Reference Carlson and Mahadevan2015b ), with the effective spring stiffness $K$ and $l_{i}$ is the natural protein length. To observe the stochastic regime in these other flow situations, their contribution to the pressure must be much smaller than the stochastic contribution in (2.2).

The disjoining pressure would add a contribution to the right-hand side of (2.2) that scales as $\unicode[STIX]{x1D735}\boldsymbol{\cdot }(h^{3}\unicode[STIX]{x1D735}p_{vdW})=\unicode[STIX]{x1D6E4}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(h^{-1}\unicode[STIX]{x1D735}h)\sim \unicode[STIX]{x1D6E4}R^{-2}$ where $\unicode[STIX]{x1D6E4}=AL^{4}/Bh_{0}^{4}$ is a non-dimensional number for the ratio between van der Waals adhesion energy and the bending energy. Performing the same scaling analysis for the contribution from the spring pressure gives $\unicode[STIX]{x1D735}\boldsymbol{\cdot }(h^{3}\unicode[STIX]{x1D735}p_{s})=S\unicode[STIX]{x1D735}\boldsymbol{\cdot }(h^{3}\unicode[STIX]{x1D735}h)\sim SR^{-10}$ (in one dimension ${\sim}SR^{-6}$ ) with $S=KL^{4}/B$ a non-dimensional number for the ratio of the spring pressure and the bending pressure. We individually compare the magnitude of the stochastic fluctuations with the deterministic contributions from these additional physical effect. The contribution from the van der Waals pressure gives $R<(N^{2}/\unicode[STIX]{x1D6E4})^{1/6}$ (in one dimension $R<(N^{2}/\unicode[STIX]{x1D6E4}w)^{1/3}$ ) and the spring pressure gives $R>(S/N^{2})^{1/2}$ (in one dimension $R>(Sw/N^{2})$ ). A typical value for the Hamaker constant is $A\approx 10k_{b}T$ with $\unicode[STIX]{x1D6E4}\in [6.2-0.62]\times 10^{6}$ . If we assume a spring adhesion phenomenon (Carlson & Mahadevan Reference Carlson and Mahadevan2015a ) mediated by proteins with a number density $C\approx 100~(\unicode[STIX]{x03BC}\text{m})^{-2}$ and a stiffness $\unicode[STIX]{x1D6FE}\approx 10^{-7}~\text{N}~\text{m}^{-1}$ , we get that $K=C\unicode[STIX]{x1D6FE}=10^{7}~\text{N}~\text{m}^{-3}$ and $S\in [2{-}15]$ .