Symmetrization of Thin Free-Standing Liquid Films via Capillary-Driven Flow
Vincent Bertin, John Niven, Howard A. Stone, Thomas Salez, Elie Raphael, Kari Dalnoki-Veress
SSymmetrization of Thin free-standing Liquid Films via Capillary-Driven Flow
Vincent Bertin,
1, 2, ∗ John Niven, ∗ Howard A. Stone, Thomas Salez,
1, 5
Elie Raphaël, and Kari Dalnoki-Veress
2, 3, † Univ. Bordeaux, CNRS, LOMA, UMR 5798, 33405 Talence, France. UMR CNRS Gulliver 7083, ESPCI Paris, PSL Research University, 75005 Paris, France. Department of Physics & Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada. Department of Mechanical and Aerospace Engineering,Princeton University, Princeton, New Jersey 08544, USA Global Station for Soft Matter, Global Institution for Collaborative Research and Education,Hokkaido University, Sapporo, Hokkaido 060-0808, Japan.
We present experiments to study the relaxation of a nano-scale cylindrical perturbation at one ofthe two interfaces of a thin viscous free-standing polymeric film. Driven by capillarity, the film flowsand evolves towards equilibrium by first symmetrizing the perturbation between the two interfaces,and eventually broadening the perturbation. A full-Stokes hydrodynamic model is presented whichaccounts for both the vertical and lateral flows, and which highlights the symmetry in the system.The symmetrization time is found to depend on the membrane thickness, surface tension, andviscosity.
Surface tension will smooth out small interfacial per-turbations on a thin liquid film, as the curvature ofthe perturbation profile induces a Laplace pressure thatdrives a viscous flow. This capillary-driven levellingcauses the brush strokes on paint to flatten, or the sprayof small droplets to result in a uniform film. Such flowshave been studied in great detail and much of the the-oretical framework is provided by the lubrication ap-proximation, whereby one can assume that the flow inthe plane of the film dominates, and that the velocityvanishes at the solid-liquid interface [1, 2]. In contrast,for a free-standing liquid film there is no shear-stress atboth liquid-air interfaces which modifies the boundaryconditions and results in a different phenomenology [1].These boundary conditions can arise in a variety of situ-ations such as biological membranes [3], soap films [4–9],liquid-crystal films [10–12], fragmentation processes [13],or energy-harvesting technologies [14].The dynamics of liquid sheets has been studied in greatdetail in the past decades [15, 16], and shows similaritieswith the mechanics of elastic plates. The evolution canbe described with two dominant modes, which are thestretching and bending modes associated with momen-tum and torque balances. At macroscopic scales, a vis-cous sheet experiences bending instabilities such as wrin-kling [17–20], and folding [21] when submitted to com-pressive forces. Such viscous buckling phenomena occurin various contexts, like tectonic-plate dynamics [22, 23]and industrial float-glass processes [24–26].In thin free-standing films, surface tension is domi-nant and stabilizes the interfaces against buckling [15].Most theoretical models in this context assume that theinterfaces are mirror-symmetric, and thus focus on thestretching mode, also called the symmetric mode. Thisapproach is employed to study the rupture dynamicsof films in the presence of disjoining forces that desta-bilize long waves in thin film [27–34]. Recently, using ∗ These two authors contributed equally † [email protected] nanometric free-standing polystyrene (PS) films, Ilton etal. observed that a film with initially asymmetric inter-faces symmetrized over short time scales [35]. This sym-metrization was attributed to flow perpendicular to thefilm, but the dynamics was not accessible experimentally.In this Letter we study the viscocapillary relaxationdynamics of a nanoscale cylindrical perturbation initiallypresent on one of the two interfaces of a thin free-standingPS film. Both the symmetric (viscous stretching) andantisymmetric (viscous bending) modes are probed withexperiments (see Fig. 1a-b). Atomic force microscopy(AFM) is used to obtain the profiles of the top and bot-tom interfaces. A full-Stokes flow linear hydrodynamicmodel is developed to characterize the relaxation dynam-ics of the two modes. To provide an intuitive understand-ing of the energy dissipation as the film relaxes, we turnto the schematic plot of the excess surface energy as afunction of time, shown in Fig. 1(c). Initially, the topinterfacial profile, denoted h + , has a high excess energydue to the additional interface that forms the hole, whilethe bottom interfacial profile h − is flat and hence has noexcess surface energy. The excess free energy resultingfrom the perturbation drives a flow that is mediated byviscosity, η . As the film evolves, the total energy dissi-pates as the excess interface decreases. Apart from thatglobal energy dissipation, the symmetrization process re-quires some energy transfer from the top interface to thebottom one – a coupling that is dominated by verticalflow. Once both interfaces are mirror-symmetric, theyrelax in tandem dominated by lateral flow. Remarkably,the temporal evolution of the interfacial profiles, whenappropriately decomposed into their symmetric and anti-symmetric components is found to obey power laws.Thin films of PS are prepared using a method similarto that previously described [35, 36]. PS with molecularweight M w = 183 kg/mol (Polymer Source Inc., poly-dispersity index = 1.06) is dissolved in toluene (FisherScientific, Optima grade) with concentrations of 2 % and7.5 % by weight. Thin films are prepared by spin coat-ing from solution onto freshly cleaved mica sheets (TedPella), and annealed at 130 ◦ C in vacuum (1 × − mbar) a r X i v : . [ c ond - m a t . s o f t ] D ec Figure 1 + antisymmetricsymmetric z r h symmetrization lateral flow vertical flow decompositionprofilea) b) c) Excess surface energy time topbottom symmetricinterfacestime h r h + ( r )
015 102550 100200500 10002000
Figure 2 r /µ m Experimentalinitial condition
HOLE 2
Initial conditionin the theory Elasticbump t exp / min h / n m
13 min [41]).After relaxation of the elastic bump, the flow resultsfrom capillarity and viscosity only. First, there is verti-cal flow to equilibrate the Laplace pressures of the twointerfaces, which results in the symmetrization process.Indeed, two symmetric interfacial profiles at the top andbottom of the film are observed at times larger than ∼ min. Subsequently, the symmetrized interfacialprofiles evolve jointly through lateral uniform flow in or-der to dissipate the excess surface energy [29]. The filmis annealed for ∼ η . Given the axial symmetry of the prob-lem, we introduce cylindrical coordinates ( r, z ) , aswell as the Hankel transforms [42] of the velocityfield (cid:126)u ( r, z, t ) = ( u r , u z ) , and of the interfacial pro-files h ± ( r, t ) : ˜ u r ( k, z, t ) = (cid:82) ∞ d r r u r ( r, z, t ) J ( kr ) , ˜ u z ( k, z, t ) = (cid:82) ∞ d r r u z ( r, z, t ) J ( kr ) , and ˜ h ± ( k, t ) = (cid:82) ∞ d r r h ± ( r, t ) J ( kr ) , where t is time, and the J i arethe Bessel functions of the first kind with indices i =0 , . Injecting these forms into the steady Stokes equa-tions, we find: ∂ z ˜ u r + k∂ z ˜ u z − k ∂ z ˜ u r − k ˜ u z = 0 and ∂ z ˜ u z + k ˜ u r = 0 , which result in the general solution: ˜ u r = − k (cid:18) kA + kzC + D (cid:19) sinh( kz ) − k (cid:18) kB + kzD + C (cid:19) cosh( kz ) , (1a) ˜ u z = (cid:18) A + zC (cid:19) cosh( kz ) + (cid:18) B + zD (cid:19) sinh( kz ) , (1b)where A ( t ) , B ( t ) , C ( t ) and D ( t ) are integration con-stants. The depth of the hole is assumed to be smallin comparison with the total thickness of the film, whichis valid for the experiments, so that we can linearize theproblem by writing the profiles as h ± = ± h / δh ± ,where the perturbations δh ± are small compared to thefilm thickness h at rest. We assume no-shear-stressboundary conditions at both fluid-air interfaces, and ne-glect the nonlinearities from the scalar projections of thenormal and tangential vectors to the interface, whichgives: ( ± kA + C kh kh kB ± D kh kh ± γk η ˜ δh ± , (2a) (cid:18) kA ± C kh D (cid:19) cosh (cid:18) kh (cid:19) + (cid:18) ± kB + D kh ± C (cid:19) sinh (cid:18) kh (cid:19) = 0 , (2b)where γ is the fluid-air interfacial tension. Finally,we invoke the linearized kinematic conditions, ∂ t ˜ h ± =˜ u z ( k, z = ± h / , t ) , and obtain a set of coupled lineardifferential equations. The symmetric-antisymmetric de-composition, through ˜ h sym = ˜ δh + − ˜ δh − and ˜ h anti = .
01 0 . as Figure 3 kh -2 112 11 symmetricantisymmetric ⌘h Figure 3. Dimensionless decay rates of the symmetric andantisymmetric modes (Eqs. (3a) and (3b)) as a function ofthe dimensionless wave number. The slope-triangles indicatepower-law exponents. ˜ δh + + ˜ δh − [see Fig. 1(b)], appears as the natural modaldecomposition for this system. These two modes relaxindependently to equilibrium, with distinct decay rates λ sym and λ anti , since: ∂ t ˜ h sym = − γkη sinh ( kh )sinh( kh ) + kh ˜ h sym = − λ sym ˜ h sym , (3a) ∂ t ˜ h anti = − γkη cosh ( kh )sinh( kh ) − kh ˜ h anti = − λ anti ˜ h anti . (3b)The dimensionless decay rates are plotted in Fig. 3as a function of the dimensionless wave number kh .For each rate, two asymptotic behaviors can be distin-guished. At large kh , both rates exhibit the same limit: lim k →∞ λ ( k ) = γkη . At small kh , the symmetric ratebecomes identical to the one in the symmetric long-wavefree-standing film model: lim k → λ sym = γh k η [29, 35],and thus Eq. (3a) reduces to a heat-like equation in Han-kel space, with a diffusion coefficient γh η . In the samelimit, the antisymmetric rate has a different scaling law: lim k → λ anti = γηh k . Therefore, long waves are quicklydamped for the antisymmetric mode. We note that λ anti has a minimum at k (cid:39) . /h , corresponding to a slow-est mode, which sets the relaxation dynamics.The model relies on the assumption of a Newtonianfluid. As such, it must be compared to experimental pro-files corresponding to annealing times longer than thepolymeric relaxation time. Thus, we take the experi-mental profiles at t exp = 5 min as the initial conditionsfor the model (see Fig. 2). Equations (3a) and (3b) arethen solved, yielding: ˜ h sym/anti ( k, t ) = ˜ h sym/anti ( k,
0) exp (cid:20) − λ sym/anti ( k ) t (cid:21) , (4)where t = t exp − min. The symmetric and antisymmet- HOLE 2 experimenttheory symmetric antisymmetric h / n m r /µ m r /µ m ⌧ = ⌘h = 360 min Figure 4
CHOSEN FIG (a) (b)(d)(c) t exp / min Figure 4. Symmetric (a) and antisymmetric (b) modes ofthe experimental (angular averaged) profiles for various times.The colors correspond to the same times as in Fig 2. Sym-metric (c) and antisymmetric (d) modes of the theoreticalprofiles, according to Eq. (4), for various times, and with theexperimental profiles at t exp = 5 min as the initial conditions( t = 0 ). .
01 0 . . .