Asymptotic regimes in elastohydrodynamic and stochastic leveling on a viscous film
An interface of finite thickness that is elastic and deforms near a solid substrate in a viscous fluid is a situation relevant to various dynamical processes in biology, geophysics and engineering. Here, we study the relaxation dynamics of an elastic plate resting on a thin viscous film that is supported by a solid substrate. By combining scaling analysis, numerical simulations and experiments, we identify asymptotic regimes for the elastohydrodynamic leveling of a surface perturbation of the form of a bump, when the flow is driven by either the elastic bending of the plate or thermal fluctuations. In both cases, two distinct regimes are identified when the bump height is either much larger or much smaller than the thickness of the pre-wetted viscous film. Our analysis reveals a distinct crossover between the similarity exponents with the ratio of the perturbation height to the film height.
The motion of an elastic sheet supported by a thin layer of viscous fluid is a phenomenon that manifests itself in processes spanning wide ranges of time and length scales, from e.g. magmatic intrusion in the Earth’s crust balmforth2005instability (); michaut2011dynamics (), to fracturing and crack formation in glaciers ball2018static (), to pumping in the digestive and arterial systems bilston2003arterial (); takagi2011peristaltic (); borcia2018liquid (), or the construction of 2D crystals for electronic engineering sanchez2018mechanics (). Elastohydrodynamic flows have been studied in model geometries in order to understand their generic features and the inherent coupling between the driving force from the elastic deformations of the material and the viscous friction force resisting motion hosoi2004peeling (); Huang2002 (); Vandeparre2010 (); al2013two (); hewitt2015elastic (); tulchinsky2016transient (); elbaz2016axial (); carlson2016similarity (); arutkin2017elastohydrodynamic (); Kodio2017 (); juel2018instabilities (). For instance, when an elastic sheet deforms onto a wall pre-wetted by a thin viscous film, the dynamics of the advancing front is dictated by the local curvature of the interface lister2013viscous (). This elastohydrodynamic relaxation is reminiscent of the capillary spreading of a viscous drop onto a solid substrate tanner1979spreading (); de1985wetting (); cormier2012beyond (); bergemann2018viscous (). Similar to capillary flows, elastohydrodynamic relaxation processes are not only limited to very thin pre-wetted films. In fact, an elastic sheet with zero spontaneous curvature but with an initial shape of a bump (Fig. 1) with a height much larger than the pre-wetted viscous film will relax towards a flat equilibrium state. Inevitably, the system must then crossover from a situation where the bump height is larger than the pre-wetted film height, to a situation where instead the pre-wetted film becomes thicker than the bump. We are wondering how the elastohydrodynamic leveling changes with the ratio between the bump height and the pre-wetted film thickness. In particular, are there different asymptotic regimes, and how do we transition from one to another? At the nanoscale, thermal fluctuations are expected to contribute and may even dominate the dynamics davidovitch2005spreading (); Demery2011 (); nesic2015fully (); carlson2018fluctuation (); marbach2018transport (), which we would like to quantify further in the leveling dynamics. To answer these questions, we combine numerical solutions of a mathematical model based on the lubrication theory batchelor2000introduction () with scaling analysis and experiments.
Ii Mathematical Model and Numerical Proceedure
We consider the model system depicted in Fig. 1, where we focus on a system where gravity can be neglected, i.e., the bump height is smaller than the elasto-gravity length hosoi2004peeling (). Only situations where the height of the fluid film is small compared its horizontal extent and where the film slopes are small, i.e. , are considered. We describe the viscous flow between the plate and the solid substrate through lubrication theory batchelor2000introduction (). When the deflection of the elastic plate is small compared to its thickness , we can neglect stretching and the pressure reduces to , where is the bending rigidity of the plate, is the Young modulus and is the Poisson ratio landau1959course (). By assuming incompressible flow and imposing no-slip conditions at the two solid substrates, one obtains the governing equation for the evolution of the height profile (see e.g. Ref. hosoi2004peeling ()) and we consider a one-dimensional geometry as we assume no variations along the -direction
where is the fluid’s dynamic viscosity. At small scales, thermal fluctuations can also influence the dynamics, which is described by the last term in Eq. (1), originating from an additional symmetric random stress term in the Navier-Stokes equations grun2006thin (). The amplitude of this term is where is the Boltzmann constant and is the ambient temperature, is the width of the plate along the -direction, and is modelled as a spatiotemporal Gaussian white noise such that and , where indicate average quantities. We non-dimensionalize Eq. (1) by: , , , and . When , this non-dimensionalization procedure gives us a parameter-free partial differential equation for . When , the non-dimensional number appears as a pre-factor in front of the stochastic term, and measures the ratio between thermal and bending energies.
When we solve the dimensionless version of Eq. (1) numerically with the finite element method we split it into three coupled equations for: the bump profile , the linearized curvature , and the bending pressure . These fields are discretized with linear elements and solved by using Newton’s method from the FEniCS library logg2012automated (). For the deterministic case , an adaptive time stepping routine has been used with an upper time step limit of and a discretization in space . For the stochastic case , we have used a constant time step , together with a discretization in space . When , we impose the initial condition: for lister2013viscous () and otherwise. When , we impose the initial condition: with a shorter horizontal length scale in order to reduce the computational time and we checked that using this initial condition for does not affect the results presented here. We further impose the following boundary conditions at the boundary of the numerical domain: , , and . The noise is implemented independently at each nodal point and time step using the ”random” class from the Numpy library numpy (), with zero mean and unit variance.
Iii Experimental Procedure
The samples consist of a fiber of liquid polystyrene (PS) with a glass-transition temperature C deposited on a PS film supported on a silicon (Si) substrate. These samples are capped by a thin sheet of polysulfone (PSU) with C. The sample preparation is carried out as follows: PS fibers (with number-averaged molecular weight kg/mol, and polydispersity index PDI = 1.05, Polymer Source Inc., Canada) are pulled from the melt at 175C by using a glass rod. Thin PS films are spin cast from a toluene solution onto 1010 mm Si substrates, leading to a thickness of 25 to 380 nm that we measure by using ellipsometry (Accurion, EP3). The films are annealed at 110C for at least 12 hours in vacuum to remove any residual solvent and to relax residual stresses. The PS fibers are then transferred onto the PS films and briefly heated above . The heating allows the PS to flow, thereby resulting in a bump. Thin PSU films ( 22 kg/mol, Sigma-Aldrich) are prepared by spin casting from a cyclohexanone solution onto freshly cleaved mica substrates (Ted Pella, USA). The PSU films have a thickness of 160 nm, measured using ellipsometry, and are annealed in vacuum at 200C for at least 12 hours. The PSU films are floated onto water and then transferred onto a supporting apparatus (details are found in Ref. schulman2018 ()), held only by the film edges. These freestanding PSU films can be relaxed to an unstrained state ensuring no in-plane tension and in the last step transferred onto the PS sample.
After the above preparation, the samples are annealed on a hotstage (Linkam, UK) at C, which is above but below . This makes the PS to become a viscous liquid while the capping PSU film remains an elastic solid, composing the effective two-dimensional setup described in Fig. 1. The height profile is imaged during annealing using optical microscopy with a red laser line filter ( = 632.8 nm, Newport, USA), which creates interference fringes in the region of the bump, as shown in Fig. 2(a), due to the reflected light from the Si substrate. Each interference fringe corresponds to a change in height of , where is the average index of refraction of the two polymers that make up the sample (=1.53 and =1.61). This allows the bump profile to be reconstructed by fitting an even power polynomial to the fringe data, as shown in Fig. 2(b) and allows us to track the height dynamics shown in Fig. 2(c). Such profiles can then be used to track the leveling dynamics, and to extract in particular the evolution of the height of the bump with time for various initial geometries.
iv.1 Elastohydrodynamic leveling
We first start by investigating the elastohydrodynamic leveling in the absence of thermal fluctuations (). In Fig. 3 we show the numerical solutions of the dimensionless version of Eq. (1) for and we can see that the aspect ratio controls both the time scale for leveling and the detailed features of height profile. The smaller , the faster the dimensionless leveling process. Also, the dip created near the advancing front of the perturbation is enhanced both in magnitude and lateral extent for smaller . We remark that for each initial aspect ratio there is a transition period of a few numerical time steps preceding the onset of the leveling process. This period is not included in Fig. 4 as it is considered as an influence created by the initial condition, but does not influence the later dynamics. For , the numerical height profiles are further compared with our experiments, which are found to be in good agreement. We recall here that the elastic plate is floating on the liquid film and has edges that are free to move. Therefore, the pressure contribution from bending still largely dominates any contribution from stretching, where Eq. (1) is still valid.
We now turn to a scaling analysis of Eq. (1) for . When , the equation can be linearized and reduces to and we deduce the long-term scaling for the temporal evolution of the horizontal length of the bump: . Since there is area conservation in the -plane, we assume to be constant, that is evaluated to at . By combining these scaling relations we get for
where is the characteristic time scale for the bending-driven leveling dynamics. When , we need to match the curvature of a traveling-wave solution localized near the advancing front with the quasi-static solution lister2013viscous (), i.e., constant pressure in the bump, leading to carlson2018fluctuation ()
By balancing the two asymptotic predictions above, we expect the crossover between them to occur around . In addition, these asymptotic regimes suggest that is essentially a function of only, independent of the value of in particular.
In order to test our scaling predictions, we compute numerical solutions of the dimensionless version of Eq. (1) for , with and extract as a function of . These numerical results are plotted in Fig. 4 and compared to the experimental data. For each sample, the experimental data is matched to the numerical data through one free fitting parameter in front of the time scale . The values of the fluid viscosity and elastic Young modulus are highly sensitive to temperature in the experiments, and we estimate them to be Pa.s mcgraw2012 (); ilton2016 () and GPa springer (), respectively. Since all experiments were carried out at the same temperature and with the same polymer, sample-to-sample variations in result only from uncertainties in the geometrical parameters , , , and . The obtained values are 0.13, 0.7, and 1.3 for the three samples and each of these values are reasonably close to unity. More importantly, the sample-to-sample variations in do not exceed a factor of 10, which is well within the expected relative error arising from the high sensitivity of to the geometrical parameters. The general agreement between the experimental data and the numerical predictions is good, over about 5 orders of magnitude in . The systematic early-time tail in the experimental data might be attributed to the initial compressive thermal stresses in the elastic layer, which arise due to the rapid heating of the samples from room temperature to C, which relax prior to leveling and the time needed for the initial shape to enter the asymptotic regime cormier2012beyond (); Benzaquen2014 ().
The master curve in Fig. 4 confirms that is a function of . Furthermore, the two scaling regimes predicted above are indeed present, with pre-factors close to unity, and the crossover between the two being located near as predicted. Any bump that initially starts in a thin pre-wetted film regime will eventually crossover to a thick-film regime , with the corresponding power laws in time. As a final remark, a similar combination (not included here) of numerical simulations and scaling analysis can been performed for an axisymmetric geometry, leading to for , and for .
iv.2 Stochastic leveling
We investigate here the leveling process when it is dominated by thermal fluctuations (). As shown in Fig. 5, the numerical solutions suggest that the aspect ratio is again essential, as it sets the time sale for leveling where the smaller , the faster the dimensionless leveling process. Moreover, by comparison with the deterministic () case in Fig. 3, the stochastic () profiles exhibit spatiotemporal fluctuations and adopt different average shapes and leveling dynamics.
To go further, we propose a scaling analysis of Eq. (1), inspired by Ref. cormier2012beyond (). We consider specifically the limit, for which the thermal fluctuations are the dominant driving contribution to the dynamics and we assume that we can neglect the bending term so that Eq. (1) reduces to . We consider the average quantities and , where we invoke a scaling davidovitch2005spreading () for the root mean square value of the averaged noise over a space interval and a time interval . By assuming that the average area conservation in the -plan can be expressed as , we get
where is the characteristic time scale for the stochastic leveling dynamics. Interestingly, Eq. (4) describes a complete crossover between two asymptotic regimes in the stochastic leveling dynamics – a situation that is reminiscent of the bending case but with different exponents and time scale: for , we obtain , and thus we recover davidovitch2005spreading (); while for , we get . We expect the crossover between the two asymptotic regimes to occur around , i.e. around .
In order to test the prediction in Eq. (4), we compute the numerical solution of the dimensionless version of Eq. (1) for , with . By averaging over a minimum of 30 realizations, we study specifically as a function of and the results are plotted in Fig. 6. The data from the numerical solutions is in good agreement with Eq. (4) for all , with no adjustable parameter. Our results highlight that Eq. (4) gives an accurate prediction for the stochastic leveling dynamics and show that the missing pre-factor is close to unity. Finally, in order to highlight further the underlying self-similarity associated with each of the two asymptotic regimes, the insets of Fig. 6 show the corresponding full bump height profiles rescaled according to Eq. (4). In each asymptotic regime the height profiles collapse onto a universal shape and confirm the overall self-similarity in the dynamics.
We have described the elastohydrodynamic and stochastic leveling of an elastic plate placed atop a thin viscous film. By combining numerical solutions, scaling analysis, and experiments, we identified various canonical regimes. Our results highlighted the importance of the driving mechanism, either by elastic bending of the plate or thermal fluctuations and the influence of the aspect ratio of the bump height to the film height. For each of these two driving mechanisms, a crossover between two distinct asymptotic regimes is controlled by the aspect ratio. These findings can be helpful to explain relaxation processes in elastohydrodynamic settings, which can be found in biological, engineering, or geological processes.
Acknowledgements.The financial support by the Research Council of Norway (project grant 263056), the Natural Science and Engineering Research Council of Canada and the Joliot Chair of ESPCI Paris is gratefully acknowledged. The authors also thank Tak Shing Chan, Miroslav Kuchta, Corentin Mailliet, René Ledesma-Alonso, Maxence Arutkin, Elie Raphaël, Howard Stone, Pauline Olive, Katelyn Dixon, Paul Fowler, and Mark Ilton, for interesting discussions and preliminary works in different geometries.
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