Bifurcation study for a surface-acoustic-wave driven meniscus
BBifurcation study for a surface-acoustic-wave driven meniscus
Kevin David Joachim Mitas, ∗ Ofer Manor, and Uwe Thiele
1, 3, † Institut f¨ur Theoretische Physik, Westf¨alische Wilhelms-Universit¨at M¨unster,Wilhelm Klemm Str. 9, 48149 M¨unster, Germany Wolfson Department of Chemical Engineering,Technion - Israel Institute of Technology, Haifa, Israel 32000 ‡ Center of Nonlinear Science (CeNoS),Westf¨alische Wilhelms-Universit¨at M¨unster,Corrensstr. 2, 48149 M¨unster, Germany
Abstract
A thin-film model for a meniscus driven by Rayleigh surface acoustic waves (SAW) is analysed, a prob-lem closely related to the classical Landau-Levich or dragged-film problem where a plate is withdrawn atconstant speed from a bath. We consider a mesoscopic hydrodynamic model for a partially wetting liq-uid, were wettability is incorporated via a Derjaguin (or disjoining) pressure and combine SAW drivingwith the elements known from the dragged-film problem. For a one-dimensional substrate, i.e., neglectingtransversal perturbations, we employ numerical path continuation to investigate in detail how the variousoccurring steady and time-periodic states depend on relevant control parameters like the Weber number andSAW strength. The bifurcation structure related to qualitative transitions caused by the SAW is analysedwith particular attention on the Hopf bifurcations related to the emergence of time-periodic states corre-sponding to the regular shedding of lines from the meniscus. The interplay of several of these bifurcationsis investigated obtaining information relevant to the entire class of dragged-film problems. ∗ Electronic address: [email protected] † Electronic address: [email protected]; URL: ; ORCID ID: 0000-0001-7989-9271 ‡ Electronic address: [email protected] Preprint– contact: [email protected] – June 29, 2020 a r X i v : . [ n li n . PS ] J un . INTRO Over recent years, the interest of industry has grown in applications that control the depositionof a surface coating via the transfer of simple or complex liquids from a bath. Examples are dip-coating processes and Langmuir-Blodgett transfer processes of liquids, solutions and suspensions[23, 27, 40, 56, 71, 72].The transfer of a simple viscous liquid from a bath onto a continuously withdrawn plate is animportant reference case for all investigations of coating phenomena. Where the bath contactsthe plate, a dynamic liquid meniscus is formed resulting from the acting forces of capillarity dueto surface tension, the viscous drag force of the liquid advection and the wettability of the plate.The latter results from the interaction of the liquid, the solid and the ambient gas at the three-phase contact line. Overall, this phenomena is known as the Landau-Levich problem as it wasfirst theoretically analysed by Landau and Levich [33] and Derjaguin [17] in the case of a simpleideally wetting nonvolatile liquid. They show, that a homogeneous macroscopic liquid layer isdeposited on the plate whose thickness h c depends on the capillary number Ca = µv p /γ in theform of a power law, h c ∝ Ca / . Here, v p is the velocity of the plate withdrawal, and µ and γ are the viscosity and surface tension of the liquid, respectively. For partially wetting liquids, atransition from a meniscus state where a macroscopically dry plate emerges from the bath to theLandau-Levich regime occurs at a finite plate velocity [22, 23, 56, 75]. Several different modi oftransition are described and are classified as dynamic unbinding transitions [22]. In the course ofthese transitions beside meniscus states and Landau-Levich states, so-called foot states may occur,where a steady finite-length foot-like protrusion covers part of the moving plate [22, 23, 56, 75].Time-periodic states corresponding to the periodic shedding of liquid ridges, oriented orthogo-nally to the withdrawal direction, from the liquid meniscus are also described [59], similar to linedeposition occurring in Langmuir-Blodgett transfer [28–30]. Here, we refer to this phenomenonas line deposition.The classic Landau-Levich or dragged-plate system is a prominent example of a wide class ofrelated coating systems where a meniscus deforms and film deposition occurs under lateral drivingforces. For instance, plate withdrawal may be replaced by a temperature gradient along the plate[8, 10, 12, 14, 45, 46, 55], by a pressure gradient when propelling an air bubble through a tube(Landau-Levich-Bretherton problem) [11, 31] and by Rayleigh surface acoustic waves (SAWs),which propagate in a resting plate in contact with a bath of liquid. Here, we are particularly2 Preprint– contact: [email protected] – June 29, 2020 nterested in the latter.Briefly, simple vibrations, flexural bulk waves, and Rayleigh surface acoustic waves (SAWs) insolid substrates are capable of displacing, manipulating, and deforming liquid films and drops[5, 9, 44]. In particular, Rayleigh type SAWs, which are of MHz frequency and propagate in asolid substrate, cause the dynamic wetting of the solid surface by liquid films. This was first shownby Rezk et al. [52, 53] for the case of silicon oil. They named the phenomenon acoustowetting that was thoroughly explained in a later study [3]. Similar observations are made in the case ofwater, albeit there capillary stresses compete with and even diminish the SAW induced dynamicwetting effect [4, 41]. Generally, the direction of the SAW-induced dynamic wetting is determinedby the thickness of the liquid film and the wavelength of the SAW. This is consistent with aninterplay between radiation pressure and acoustic flow in oil films. Liquid film thicknesses, whichare small with respect to the wavelength of sound leakage off the SAW, dynamically wet the solidsubstrate along the propagation direction of the SAW. The wavelength of the sound leakage isapproximately the wavelength of the SAW times the ratio between the phase velocities of theSAW and of sound in the liquid and is in the range of tens to hundreds of microns in the case ofMHz-frequency SAWs. In case the thickness of the liquid film is comparable to the wavelengthof the sound leakage, acoustic resonance effects were found to enhance the pressure in the liquidfilm, so that the film may also spread opposite to the direction of the Rayleigh SAW. In addition,when the thickness of the liquid film is greater than a few wavelengths of the sound leakage, adifferent mechanism, Eckart streaming, was found to govern transport in the liquid, yielding againdynamic wetting along the direction of the SAW. The capillary stress at the free liquid surfaceis capable of arresting the dynamic wetting of the liquid film, especially in the case of water.The opposite contributions of capillary and SAW to film dynamics is quantitatively weighed in anacoustic Webber number, which predicts the onset of dynamic wetting and is further discussed inthis paper.Results obtained for these Landau-Levich-type systems are also of relevance for other possiblymore complicated coating geometries [13, 71, 73] and also provide a basis for the understandingof the behaviour of more complex liquids in similar settings [1, 6, 15, 51, 54, 62, 72]. Often,Landau-Levich-type systems are modelled based on mesoscopic hydrodynamic thin-film models(also called long-wave or lubrication models) [15, 47]. It is assumed that length scales orthogonalto the substrate are small compared to the ones parallel to the substrate, e.g., cases of small contactangles and small interface slopes. This allows for the derivation of an evolution equation for the3
Preprint– contact: [email protected] – June 29, 2020 lm thickness profile. Thereby the motion of three-phase contact lines is either modeled through aslip model [56, 75] or a precursor film model [22, 59]. Here, we employ the latter, i.e., wettabilityis modelled by an additional pressure term, the so-called Derjaguin (or disjoining) pressure thatallows for an ultrathin adsorption layer on the macroscopically dry substrate [16, 18, 58, 60].Employing such a precursor film model, Ref. [22] showed that depending on plate inclination angle(and equilibrium contact angle), different transition scenario occur with increasing capillary num-ber (dimensional plate velocity). These are, in particular, four different continuous and discontin-uous unbinding (or dynamic wetting and emptying) transitions that are out-of-equilibrium equiv-alents of equilibrium transitions, which were discussed earlier (see [22] and references therein).Here, we consider how the picture is amended when employing SAW driving to force the meniscusinstead of driving by dragging a plate. SAW driving for this purpose was recently investigatedexperimentally and theoretically [3, 4, 41–43, 52, 53]. In this system, a drop of liquid or liquidmeniscus rests on a smooth solid plate. Then a propagating SAW is excited in the plate. The SAWis a propagating vibration in the substrate in the MHz-frequency range and may induce drops toslide, static menisci to deform and a liquid protrusion or film to spread from a static meniscus.Among other mechanisms, this is caused by a high-frequency periodic flow in a boundary layerclose to the substrate within the liquid that induces a mean convective flow on hydrodynamic lengthand time scales. This flow may overcome the capillary and wettability resistance in the contactline region where the meniscus meets the plate, so that at a critical strength, the acoustically drivenflow results in an advance of the contact line. This is very similar to the transitions, which occurs atcritical capillary numbers in the Landau-Levich problem for partially wetting liquids. It remains anopen question whether SAW driving can be employed to control deposition of structured coatings,which may be a simpler coating technique with no moving parts, compared to driving the coatingfilm by moving the plate [38, 42].In particular, here we combine the precursor-based thin-film model for the Landau-Levich problemfor partially wetting liquids in Refs. [22, 59, 67] with the thin-film model for the SAW-drivenmeniscus in Ref. [43]. With other words, the SAW-model of [43] is expanded by introducinglateral forces due to the withdrawal of an inclined plate and substrate wettability. This allows usto investigate the behavior of the SAW-driven system in direct comparison to results obtained forthe dragged-plate system. The model is numerically investigated employing a combination of pathcontinuation techniques [2, 19, 20] bundled in the software package
PDE PATH [68, 70] and directtime simulation provided by the software package
OOMPH - LIB [25].4
Preprint– contact: [email protected] – June 29, 2020 e consider a one-dimensional substrate, i.e., we neglect transversal perturbations, and investi-gate in detail how the various occurring steady and time-periodic states depend on relevant controlparameters like the Weber number and SAW strength. We uncover the full bifurcation structure re-lated to qualitative transitions caused by the SAW. We pay particular attention to Hopf bifurcations,which bring about the emergence of time-periodic states corresponding to the regular shedding oflines from the meniscus. The interplay of several of these bifurcations is investigated by trackingthe bifurcations in selected parameter planes, thereby discussing the special transition points (socalled codimension-2 bifurcations) where the individual bifurcations emerge and disappear. Ourstudy gives detailed information which is relevant to the entire class of dragged-film problems.The structure of our work is as follows: In section II, we briefly introduce the governing mathemat-ical model, provide a short overview of the employed continuation method, and discuss the usedsolution measures. Next, we reproduce selected bifurcation diagrams for the classical Landau-Levich system [22] in section III and produce, in comparison, bifurcation diagrams for the SAW-driven case [43] in section IV A. The comparison provides us with central reference cases for ourstudy. In continuation, in section IV B the results of [43] for steady states are extended towardspartially wetting liquids. The obtained bifurcation diagrams show a number of Hopf bifurca-tions. Their behavior is analyzed in detail in section IV C together with properties of the emergingbranches of stable and unstable time-periodic states. Finally, we conclude in section V with asummary and outlook.
II. MODEL
In our study we combine the two systems sketched in Fig. 1, namely, a simple nonvolatile par-tially wetting liquid in the classical dip-coating (or Landau-Levich) geometry [22, 59] and a liq-uid meniscus driven by surface acoustic waves (SAW) [43]. Overall the set-up for the two sys-tems is very similar with the main difference beside the driving force being the condition on thebath/meniscus side. The kinetic equation that describes the development of the film thicknessprofile h ( x, t ) when the two systems are combined reads in nondimensional form in the case of aone-dimensional substrate ∂ t h ( x, t ) = − ∂ x (cid:26) Q ( h ) ∂ x (cid:20) s ∂ xx h − f (cid:48) ( h ) (cid:21) − χ ( h ) + h [ (cid:15) s v s ( h ) + U ] (cid:27) , (1) where χ ( h ) = G Q ( h ) ( ∂ x h − α ) , Q ( h ) = h . Preprint– contact: [email protected] – June 29, 2020
IG. 1: Sketches of the meniscus regions of the systems in consideration: (a) a Landau-Levich-type coatingsystem, where liquid is deposited on a plate, which is withdrawn at an angle α from a liquid bath at avelocity of U ; (b) a Landau-Levich-type coating system, where liquid is deposited on a plate from a liquidmeniscus with radius of curvature R onto a horizontal plate, which supports a surface acoustic wave (SAW). Above, α is the substrate inclination angle in long-wave scaling, Q ( h ) is the mobility functionresulting from a no-slip boundary condition at the substrate, We s is the Weber number, and G is adimensionless gravity number. The wetting potential [50, 61] f ( h ) = Ha (cid:18) h p h − h (cid:19) (2)describes wettability for a partially wetting liquid and results in the Derjaguin (or disjoining)pressure Π = − f (cid:48) ( h ) = Ha( h p /h − /h ) [58, 61]. Ha is a nondimensional Hamaker numberthat controls the long-wave equilibrium contact angle.In addition, capillarity and wettability give rise to the Laplace and Derjaguin pressure contributionsin the first bracket of Eq. (1). The next term [ χ ( h ) ] accounts for gravitational contributions thatbalance for the bath where ∂ x h = α and the final bracket gives the driving force for the coatingprocess. It combines the effect of the plate moving at a velocity U and of the SAW at an intensityof (cid:15) s . The corresponding dependence of SAW-driving on the thickness of the liquid films is [43] v s ( h ) = 14 h (cid:18) h sinh(2 h ) − h sin(2 h ) + 2 cos( h ) cosh( h )cos(2 h ) + cosh(2 h ) − (cid:19) , (3)which is illustrated in Fig. 2. The influence of the SAW on the coating process increases with6 Preprint– contact: [email protected] – June 29, 2020 h . . . . . . v s ( h ) FIG. 2: Dependence of the average velocity in the liquid film, v s , on film thickness h (solid line). The thinhorizontal dotted line corresponds to the asymptotic value of v s = 1 / at large h . increasing film thickness and saturates in the limit of large film heights.The employed scaling allows us to use both, the Weber number We s and the SAW strength (cid:15) s , ascontrol parameters. This facilitates comparison to Ref. [43] where We s is the control parameterand also to [22, 59] where the the nondimensional velocity of the dragging plate is used – aparameter that has a similar role as (cid:15) s for the SAW-driving. The scaling is discussed in detail inthe Appendix A and combines aspects of the scalings employed in Refs. [43] and [22, 59].Equation (1) is derived using the well-established lubrication approximation, which is also knownas long-wave or thin-film approximation [15, 47, 60], i.e., it is assumed that all relevant lengthscales orthogonal to the substrate are small compared to the ones parallel to the substrate, e.g.,one considers the case of small contact angle, small plate inclination angle and, in general, smallinterface slopes.The particular combination of dragged-plate driving and SAW-driving has to our knowledge notyet been treated in the literature. The derivation starts from the Navier-Stokes and continuityequations with the usual stress-free conditions at the free surface of the liquid film and with no-slipand no-penetration conditions at the solid-liquid interface. In addition, the solid-liquid interfaceundergoes SAW-induced high-frequency travelling wave oscillations. This implies that fast andslow time-scales have to be separated in a multi-scale approach that then results in the particulareffective SAW-driving term v s that acts on the viscous time scale [3, 4, 43]. A similar approachis employed for liquid films on vertically vibrated substrates where it is used to stabilize various7 Preprint– contact: [email protected] – June 29, 2020 nterface instabilities [34, 65].The resulting Eq. (1) allows us to investigate the different limiting cases studied in the literature:(i) The model in Ref. [43] is recovered without wettability ( f = 0 ) and without gravity ( G = 0 ).(ii) The model in Refs. [22, 59] is recovered without SAW-driving ( (cid:15) s ).We set specific boundary conditions (BC) for h ( x, t ) to investigate the dragged-plate and SAW-driven coating system. We consider the domain Ω = [ − L/ , L/ , where h = h m , and ∂ xx h = 1 at x = L/ (4) ∂ x h = 0 and ∂ xx h = 0 at x = − L/ i.e., on the meniscus side ( x = L/ ) the film profile approaches a meniscus of constant curvatureexactly as in Ref. [43]. Note that the nondimensional radius of curvature is equal to one. For thedragged-plate system this BC is amended to ∂ xx h | x = L/ = 0 as the profile approaches a straightline. The other BC on the bath/meniscus side fixes the film thickness on the boundary of thenumerical domain to h m , this essentially pins the translational degree of freedom and controls theposition of the contact line region within the numerical domain. The BC on the opposite side( x = − L/ ) allow for a flat film of arbitrary h c – the coating height. These boundary conditionsset the necessary conditions to solve Eq. (1). We numerically investigate the model Eq. (1) withthe introduced BC by employing a combination of pseudo-arclength path continuation [2, 19, 20]as bundled in PDE PATH [68, 70] and direct time simulations using
OOMPH - LIB [25].Briefly, the pseudo-arclength path continuation technique is employed to determine steady solu-tions ( ∂ t φ = 0 ) of Eq. (1). This is written as the operator G [ φ, λ ] = 0 , which is discretized usinga finite element scheme. Here, λ stands for a control parameter or a set of them. One follows adiscretized solution φ ( x ) in parameter space varying λ using a prediction-correction scheme: inthe prediction step, one uses the tangent of the solution curve to advance from the known solutionat parameter value λ to a first guess of a solution at a new parameter value λ + ∆ λ , i.e., the controlparameter is used as continuation parameter. Then, in the correction step a Newton procedure isemployed to converge from the guess to a solution of the PDE at λ +∆ λ . This procedure is iteratedto advance step by step along a solution branch. However this simple continuation scheme failswhen the solution branch undergoes a saddle-node bifurcation (or fold). Therefore one employspseudo-arclength continuation, where the arclength ∆ s along the branch is employed as continua-tion parameter while the original control parameter λ is in the correction step adjusted at fixed ∆ s in parallel to the state φ . The arclength ∆ s itself is determined through an approximation, hence8 Preprint– contact: [email protected] – June 29, 2020 he name “pseudo-arclength continuation”. This allows one to follow solution branches throughsaddle-node bifurcations. The package also allows to detect various bifurcations, track them inparameter planes through tow-parameter continuations, switch to bifurcating branches of othersteady states or to branches of time-periodic states [68–70]. For other recent examples where pathcontinuation is applied to thin-film and closely related equations see [20, 21, 30, 35, 36, 59, 64, 66].Here, we used continuation to investigate in detail how the various occurring steady and time-periodic states depend on relevant control parameters like the Weber number and SAW strength.The results are presented in terms of bifurcation diagrams employing as main solution measures(i) the thickness of the coating layer, i.e., the value of h measured on the boundary at x = − L/ ,and (ii) the excess volume V ex = V − V of liquid dynamically extracted from the bath/meniscus.The volume is calculated as the integral V = (cid:82) Ω h ( x )d x and V is the reference volume withoutdriving. In the case of time periodic states the mean value of the measure over one period is shownand the period is used as an additional solution measure. III. DRAGGED FILM WITHOUT SAW
Before we present our main results for the SAW-driven case, we first give an overview of thetransitions occurring for the classical Landau-Levich system [22, 56, 75]. Employing Eq. (1) with We s = 1 , Ha = 1 and (cid:15) s = 0 , we obtain the thin-film equation analysed in [22] with only asmall change in the boundary conditions as discussed in the final part of section II. As explainedin the Appendix A [case (i)], this setting is equivalent to the scaling employed in [22, 59]. Theremaining main control parameters are the plate velocity U and inclination angle α . The geometryis as given in Fig. 1 (a). The bifurcation diagrams in the four main panels of Fig. 3 indicate howthe dependence of the meniscus state on U changes with increasing α . The insets give a selectionof particular steady stable and unstable profiles.In all cases, for U → the steady meniscus approaches the reference state of zero excess volume.The profile shows a smooth continuous transition from the bath to the coating layer of thickness h c .It corresponds to the adsorption layer of equilibrium thickness h c = h p = 1 given by Π( h ) = 0 .The behaviour for increasing U qualitatively depends on the value of α : Fig. 3 (a) shows that atsmall α , the excess volume V ex monotonically increases with U , first slowly than faster until itdiverges at a critical value U c ≈ . of the velocity at U . All states on the branch are linearlystable. The steady meniscus profile deforms as the velocity increases and a foot-like structure of9 Preprint– contact: [email protected] – June 29, 2020
IG. 3: Bifurcation diagrams of steady meniscus states in dependence of plate velocity U for the classicalLandau-Levich system at different inclination angles (a) α = 0 . , (b) α = 1 . , (c) α = 3 . and (d) α = 10 . [Eq. (1)]. The shown solution measure is the excess volume V ex = V − V . The dotted andsolid lines represent unstable and stable states, respectively. Examples of steady state profiles are shownin the insets at loci indicated on the bifurcation curves by correspondingly coloured, filled circles. Notethat the thickness scale is logarithmic. The arrows along the bifurcation curves and in the insets indicatecorresponding directions of change. The remaining parameters are Ha = 1 . , We s = 1 . , G = 0 . , h p = 1 , (cid:15) s = 0 , and L = 800 . Preprint– contact: [email protected] – June 29, 2020 ncreasing length is dragged out from the bath with its length diverging for U → U c (see inset ofFig. 3 (a)). For U > U c , time simulations show that a foot is continuously drawn out of the bath,and the system eventually settles on a Landau-Levich film state (not shown here). The transitionat U c is termed “dynamic continuous emptying transition” [22] in analogy to the equilibriumemptying transition described in [48].The first qualitative change in the bifurcation curve occurs at α = α c ≈ . , see Fig. 3 (b) for anexample of a diagram at α = 1 . > α c . At small U the excess volume monotonically increaseswith U as before, however, then at U ≈ . a saddle-node bifurcation occurs and the bifurcationcurve continues towards smaller U now consisting of unstable states. At a second saddle-nodebifurcation at U ≈ . the states becomes stable again, and the curve folds back towards larger U .Overall the curve undergoes an exponential (or collapsed) snaking [39] about a vertical asymptoteat a critical velocity U = U c ≈ . . We emphasize that at each fold there occurs a change instability. This is analysed in detail in [67]. Looking at the inset of Fig. 3 (b), the steady profilesagain develop a foot whose span now shows undulations. This is related to the snaking bifurcationcurve and the fact that only certain ranges of foot length correspond to stable states. The transitionat U = U c is termed “dynamic discontinuous emptying transition” [22]. Note, that qualitativelyour results agree with the ones of [22], although slightly different BC and domain sizes result indifferent critical values.The second qualitative change in the bifurcation curve occurs at α = α c ≈ . , see Fig. 3 (c) foran example at α = 3 . > α c . At small U the curve behaves as before, then undergoes a singlepair of saddle-node bifurcations; both destabilize the states, i.e., two eigenvalues have positivereal parts after the second one. At further increased U , the steady states become stable again ata Hopf bifurcation, where two complex conjugated eigenvalues cross the imaginary axis. Thisbifurcation was only recently described in Ref. [59] and is discussed in detail for the SAW-drivensystem in section IV C. The branch of stable states continues to arbitrarily large U followingthe classical Landau-Levich power law V ex ∝ U / . Note that in contrast to the cases shown inFig. 3 (a) and (b), no asymptotic value of U is found. Inspecting the steady profiles in the insetof Fig. 3 (c) shows that the transition does not involve an advancing foot. Instead, the coatingthickness h c increases homogeneously. As the film surface after “unbinding” from the substratehomogeneously increases similar to an equilibrium wetting transition, the transition is called a“discontinuous dynamic wetting transition” [22].The final qualitative change in the bifurcation curve can be appreciated when comparing Fig. 3 (c)11 Preprint– contact: [email protected] – June 29, 2020 nd Fig. 3 (d). In Fig. 3 (d) the pair of bifurcations has annihilated and the curve increases mono-tonically. Otherwise the behaviour is as in Fig. 3 (c) - also see the profiles in the inset. In conse-quence, this behaviour is termed “continuous dynamic wetting transition” [22].This brief overview of the classical Landau-Levich system in the case of partially wetting liquidprovides the reference case that we compare to the SAW-driven meniscus, which we consider next.
IV. SAW-DRIVEN MENISCUS
After the brief revision of the transition behaviour in the case of the well-studied Landau-Levichsystem that serves as our reference system, we next consider the SAW-driven system. First, weinvestigate in section IV A the influence of the Weber number that is the main control parameterin Ref. [43]. Then we study the influence of the SAW strength in section IV B.
A. Ideally wetting liquid
We begin our analysis of the influence of the Weber number We s by reproducing the case of acompletely wetting liquid on a resting horizontal plate without the influence of gravity, which isstudied in Ref. [43]. In other words, we consider Eq. (1) with Ha = U = α = G = 0 , fix (cid:15) s = 1 and employ We s as control parameter. As explained in the Appendix A [case (ii)], this settingis equivalent to the scaling employed in [43] [and their Eq. (2.23)]. There, steady profiles wereobtained using a shooting method, albeit here we employ path continuation.The resulting bifurcation diagram is presented in Fig. 4 as a black solid line. The blue dashed curvegives the case where the contribution of the hydrostatic gravitational pressure is added ( G = 10 − and α = 0 ). The steady profiles on these branches correspond to a meniscus, which is smoothlyand monotonically connected to a homogeneous Landau-Levich film or coating layer. For large We s , the film thickness increases monotonically following the power law h c ∝ We / identical tothe power law dependence on capillary number in the classical Landau-Levich case.However, the behaviour of the coating system qualitatively changes for smaller We s . The depen-dence of h c on We s becomes multivalued as each curve features two saddle-node bifurcations.The one occurring at very small h c was not detected in Ref. [43]. No film deposition occurs below We s ≈ . – the threshold value where the crucial saddle-node bifurcation occurs, below whichno film deposition occurs; the meniscus profile ends on the truly dry substrate, a situation not12 Preprint– contact: [email protected] – June 29, 2020
IG. 4: Bifurcation diagram for a SAW-driven meniscus of a completely wetting liquid (
Ha = 0 ) ona resting horizontal plate ( U = α = 0 ), which shows the coating film thickness h c in dependence ofthe Weber number We s at fixed SAW strength ( (cid:15) s = 1 ). The black and blue curve show cases withoutthe contribution of gravity ( G = 0 ) as in Ref. [43] and with the contribution of gravity ( G = 10 − ),respectively, while dotted and solid lines represent unstable and stable states, respectively. Saddle-nodebifurcations (folds) are marked by green circles. The inset zooms onto the saddle-node bifurcation at verysmall h c . The remaining parameters are h m = 8 . and L = 40 . captured by the model in [43] that does not allow for slip at the contact line.When following the bifurcation curve from large to small values of We s , one firsts observes agradual decrease in the coating thickness. Then, passing the saddle-node bifurcation at We s ≈ . , the film deposition state collapses abruptly from the finite thickness h c ≈ . (cf. Fig. 4).Incorporating gravity, the saddle-node bifurcation is shifted towards smaller values of We s , but thecorresponding h c is nearly constant.Following the branch through the saddle-node bifurcation, a sub-branch of unstable solutions con-tinues towards larger We s values until turning back again (and becoming more unstable) at anothersaddle-node bifurcation at rather small h c values (see insets of Fig. 4). The subsequent tiny sub-branch of unstable states ends when h c approaches zero at a critical Weber number. Note that thisis not a bifurcation point. Mathematically, the alternative state of a meniscus ending at a true mi-croscopic contact point is a finite support solution. It has a different topology as the Landau-Levich13 Preprint– contact: [email protected] – June 29, 2020 lm solution and can not be obtained with the numerical methods employed here. However, thissituation can be amended by explicitly incorporating a description of wettability, as in the follow-ing section.
B. Partially wetting liquid
FIG. 5: Bifurcation diagram for a SAW-driven meniscus of partially wetting liquid (
Ha = 2 . , h p = 0 . )on a resting plate ( U = 0 ), which show the coating thickness h c in dependence of the Weber number We s at fixed SAW strength (cid:15) s = 1 . Dotted and solid lines represent unstable and stable states, respectively.Saddle-node and Hopf bifurcations are marked by green and black circles. The inset zooms onto the regionwhere the curve undergoes “snaking”. The remaining parameters are h m = 8 . , G = 10 − , α = 0 . and L = 40 . We consider the full evolution equation (1), while accounting for partially wetting liquids (mainlyusing
Ha = 2 and h p = 0 . ), but keeping the substrate resting and nearly horizontal ( U =0 , α = 0 . ). A truly horizontal substrate gives very similar results but renders details of thecontinuation procedure more cumbersome. This implies that we still use the scaling of case (ii) inthe Appendix A and consider the influence of partial wettability of the results in [43] using We s ascontrol parameter. Inspecting the bifurcation diagram in Fig. 5, we notice that the incorporation ofpartial wettability results in strong changes. For partially wetting liquids described with a wetting14 Preprint– contact: [email protected] – June 29, 2020 nergy, the dry substrate is always covered by a thin adsorption layer of liquid. This implies thatthe Landau-Levich film state and the finite-support meniscus state, discussed in section IV A, arenot topologically different anymore as the former finite-support state becomes a state where themeniscus transitions into the adsorption layer. Therefore, commencing our analysis from large We s and then reducing its magnitude, the branch of solutions in Fig. 5 does not end at finite We s as in Fig. 4, but continues towards We s = 0 . We do not show the branch down to We s = 0 as in this limit the Laplace pressure term diverges what is unphysical. In the following, We s is chosen to be sufficiently large so that to avoid this complication, e.g., in Fig. 5 we start at We s ≈ . . Then, the coating thickness h c slowly increases with increasing We s until a first saddle-node bifurcation occurs at We s ≈ . . There, h c passes the equilibrium precursor film thickness h p = 0 . to increase steadily when following the curve through all bifurcations. Namely, at thefirst saddle-node bifurcation the curve folds back and stability of states switches from stable tounstable as a first real eigenvalue crosses the imaginary axis. Following the branch We s decreases,a second saddle-node bifurcation occurs where an additional real eigenvalue becomes positiveas discussed already at Fig. 3 (c) for the case of the classical Landau-Levich coating system.Subsequently, the branch wiggles through another eight saddle-node bifurcations resulting in statesthat are more and more unstable. This could still be called exponential snaking as the distancebetween subsequent bifurcations exponentially decreases. However, in contrast to [22] in the caseshown here in Fig. 3 (c), the snaking stops after ten saddle-node bifurcations, while in the standardcase, the behaviour continues ad infinitum (domain size permitting). In addition, the increasinginstability is different from the classical Landau-Levich coating system, where stability changes ateach saddle-node bifurcation. However, the finding resembles the behaviour, which is encounteredin Langmuir-Blodgett (LB) transfer [29]. Beyond the final saddle-node bifurcation, h c increasesat nearly constant We s ≈ . , where the curve slightly bends towards larger We s values (seeinset of Fig. 5). On this part of the branch eight Hopf bifurcations occur in close succession. Eachresults in a further coating film destabilization. Following the branch further, we observe anothersequence of 13 Hopf bifurcations that successively stabilize the branch, i.e., beyond the final Hopfbifurcation all states are stable. To summarize, the first ten saddle-node bifurcations and eightHopf bifurcations destabilize the steady states so that 26 eigenvalues cross the imaginary axis;another 13 Hopf bifurcations stabilize the states again. Further increasing We s , the branch followsthe power law h c ∼ We / , which we described before.We have now analysed how the bifurcation curve changes when switching from an ideally wetting15 Preprint– contact: [email protected] – June 29, 2020
IG. 6: A bifurcation diagram showing the excess volume V ex in dependence of the SAW strength (cid:15) s , fordifferent Weber numbers We s as given in the legend. Further, Ha = 1 . . Line styles, symbols and remainingparameters are as in Fig. 5. The tracked loci of all bifurcations in the ( (cid:15) s We s , (cid:15) s )-plane are shown below inFig. 11. liquid (studied in section IV A and Ref. [43]) to a partially wetting liquid using We s as controlparameter. Next, we set Ha = 1 . thereby switching to the scaling of case (iii) in the Appendix Aallowing us to consider the SAW strength (cid:15) s as control parameter. This enables a more directcomparison to the Landau-Levich system where the plate velocity U is used as control parameter(cf. section III). Corresponding bifurcation diagrams for four different fixed values of We s aregiven in Fig. 6 using again the excess volume V ex as solution measure. Overall, all bifurcationcurves are similar to the one shown in Fig. 5, but with decreasing We s the entire curve shiftstowards larger (cid:15) s . In other words, the system needs more SAW power to spread over the substratefor a smaller ratio of convective stress and capillary stress at the surface. Roughly speaking, atlarger We s a lower SAW strength is needed to obtain the same excess volume.We also notice that the number of saddle-node and Hopf bifurcations decreases with decreasing We s . All Hopf bifurcations and all but two saddle-node bifurcations have vanished at We s = 0 . .This process can be well appreciated when tracking the loci of all bifurcations in the parameterplane spanned by We s and (cid:15) s . Before we discuss this further below, first we discuss the steadythickness profiles and the underlying flow field as characterized by the streamlines. In particular,16 Preprint– contact: [email protected] – June 29, 2020 e focus on the profiles belonging to the snaking region of the bifurcation curves using the caseof We s = 4 and Ha = 1 in Fig. 6 as an example. (cid:15) s h c ab cde f FIG. 7: A magnification of the snaking area of the bifurcation for We s = 4 and Ha = 1 in Fig. 6, whichshow the coating thickness h c as function of the SAW strength (cid:15) s . The red filled circles, marked “a” to “f”,indicate loci of steady states, which are presented in Fig. 8. A corresponding magnification of Fig. 6 is given in Fig. 7. The magnification indicates the loci ofthe six states presented in Fig. 8. The plots show the steady thickness profiles and the streamlineswithin the liquid layer. They do not present the complete computational domain but focus on thetransition region from the meniscus to the adsorption layer where the foot-like structure develops.Point “a” in Fig. 7 corresponds to the meniscus solution in Fig. 8 (a), i.e., the profile smoothly andmonotonically connects the meniscus on the right with the adsorption layer on the left, similar tothe black profiles in Figs. 3 (a) to (d). Then, following the snaking curve in Fig. 7 from point “a”to point “d”, a foot-like structure develops and extends. One can see that the SAW induces strongmodulations of the foot thickness, which is related to convection rolls within the foot. At the sametime, the single large-scale convection roll in the meniscus is nearly unchanged. Following thebifurcation curve further, one observes similar modulations. However, in contrast to the Landau-17
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10 20 x h (a) x h (b) x h (c) x h (d) x h (e) x h (f) FIG. 8: Panels (a) to (f) show steady film thickness profiles and the corresponding streamlines within thefluid at parameters which are indicated by the letters “a” to “f” in Fig. 7, respectively. In particular, (a) givesa meniscus solution, (b) to (d) present modulated foot solutions, (e) is a transition state between foot andLandau-Levich film state, and (f) gives a Landau-Levich film state.
Levich case in Figs. 3 (a) and (b), the foot does not possess a clearly defined thickness, but becomesthinner with increasing distance from the meniscus. As a result, the transition from the foot to theadsorption layer is continuous. In parallel, the convection rolls in the foot seem to weaken andpartially fuse. Profiles right at the end of the snaking region and further along the curve are shownin Figs. 8 (e) and (f). The profiles correspond to standard Landau-Levich films in a similar mannerto the red and blue curves in Fig. 3 (c) and (d).18
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IG. 9: Shown are the loci of saddle-node (green lines) and Hopf bifurcations (black and red lines) in theparameter plane spanned by the Weber number We s and the Hamaker number Ha . Further line styles areexplained in the main text. The insets the region where the leftmost Hopf bifurcation in red emerges fromthe saddle-node bifurcation, i.e., where a Bogdanov-Takens bifurcation occurs. The remaining parametersare as in Fig. 5. Up to here, we have investigated individual bifurcation diagrams by employing either the Webernumber We s or the SAW strength (cid:15) s as control parameter. In the course of this study, we havedetected many saddle-node and Hopf bifurcations. The comparison of their occurrence in depen-dence on (cid:15) s for different fixed We s has shown that the number of these bifurcations shows starkchanges. Our next interest is in the appearance and disappearance of bifurcations in our system.Next we track all bifurcations visible in Figs. 6 and 5 in the ( (cid:15) s , We s )- and ( (cid:15) s , Ha )-parameterplanes.First, we show in Fig. 9 the loci of saddle-node (green lines) and Hopf bifurcations (black and redlines) in the parameter plane spanned by the Weber number We s and the Hamaker number Ha , i.e.,we investigate the interplay of capillarity and wettability. The solid green lines denote saddle-nodebifurcations that emerge in pairs in codimension-two hysteresis bifurcations that are cusp-like inthe ( We s , Ha )-plane (see larger-scale inset). In contrast, the two dashed green lines correspond tosaddle-node bifurcations that persist in the entire studied parameter range nearly down to Ha = 0 .Note again, that due to numerical issues our computation does not exactly reach
Ha = 0 . These19
Preprint– contact: [email protected] – June 29, 2020 (cid:15) s . . . (cid:15) s W e s . . . . . FIG. 10: Loci of saddle-node and Hopf bifurcations are shown in the parameter plane spanned by SAWstrength (cid:15) s and its product with the Weber number (cid:15) s We s . The inset magnifies the parameter region wherea Bogdanov-Takens bifurcation occurs. Line styles and remaining parameters are as in Fig. 6. two lines correspond to the saddle-node bifurcations visible in the SAW reference case in Fig. 4.Inspecting the loci of the Hopf bifurcations in Fig. 9, we note that increasing We s , the first twoHopf bifurcations appear pairwise in a codimension-two double Hopf bifurcations at We s ≈ . and Ha ≈ . . The next three pairs appear in the same way. All of them are shown as blacklines. However, the next Hopf bifurcation emerges alone from a line of saddle-node bifurcations(see smaller-scale inset of Fig. 9). There, another type of codimension-two bifurcation occurs,namely, a Bogdanov-Takens bifurcation, where the Hopf bifurcation and a global (homoclinic)bifurcation emerge together at a saddle-node bifurcation. Following the corresponding line ofHopf bifurcations, it separates from the line of saddle-node bifurcations and eventually annihilateswith another Hopf bifurcation that itself emerged from a fifth double Hopf bifurcation. All furtherHopf bifurcations appear in similar ways. Red lines in the figure mark Hopf bifurcations with thistype of emergence.In general, one can say that the number of occurring bifurcations increases with the Weber number(i.e., with decreasing surface tension) and also increases with the Hamaker number Ha (i.e., with20 Preprint– contact: [email protected] – June 29, 2020 ecreasing wettability). The emergence of Hopf bifurcations is further scrutinized in section IV C.
FIG. 11: Bifurcation diagram showing three bifurcation curves from Fig. 6, giving the coating thickness h c over SAW strength (cid:15) s for different We s . Additionally, we show as black and red dashed lines how the lociof the Hopf bifurcations change when We s is changed. The black arrows indicate the direction of increasing We s . The red triangles mark Bogdanov-Takens bifurcations. The remaining parameters are as in Fig. 6. Next, we track the bifurcations in the ( We s , (cid:15) s )-plane. An initial inspection of the curves shows thatfor all tracked bifurcations, We s ∼ /(cid:15) s , and the curves practically coincide in the ( We s , (cid:15) s )-plane.Therefore, in Fig. 10 we show the curves in the ( We s , (cid:15) s We s )-plane where there emergence andrelations are well visible.Furthermore, we illustrate the appearance of the Hopf bifurcations in a different representation byplotting their loci in Fig. 11 as black and red dashed lines together with three bifurcation curvesfrom Fig. 6, that give the coating thickness h c over SAW strength (cid:15) s for different fixed We s . Thearrows in Fig. 11 indicate the direction of increasing We s . The first pair of Hopf bifurcationsemerges at We s = 0 . .The colour of the dashed lines indicates, as introduced above, how the Hopf bifurcations ap-pear when We s is increased. The black dashed lines represent Hopf bifurcations that emergethrough double Hopf bifurcations. The red lines represent Hopf bifurcations that emerge throughBogdanov-Takens bifurcations (marked by red triangles). In particular for the first red line, theabove described sequence of events is well visible: first the Hopf bifurcation emerges in the21 Preprint– contact: [email protected] – June 29, 2020 ogdanov-Takens bifurcation at (cid:15) s ≈ . , moves towards larger We s and smaller (cid:15) s where itannihilates with a second Hopf bifurcation at (cid:15) s ≈ . that itself appeared in a double Hopfbifurcation at (cid:15) s ≈ . . The latter scheme applies to all Hopf bifurcations beyond the first eight.Overall, this intricate emergence of many Hopf bifurcations results in the creation of manybranches of time-periodic states. Each of them connects two bifurcations that are either localor global. In the following, we investigate these branches and their potential reconnections inmore detail. C. Time-periodic states
In our investigation of the bifurcation behaviour of the 1d SAW system we have encountereda large number of Hopf bifurcations when using the Weber number and SAW strength as maincontrol parameters. Tracking the loci of the bifurcations in parameter planes has indicated thatthe Hopf bifurcations are created either in double Hopf bifurcations or Bogdanov-Takens bifurca-tions. Next, we employ
PDE PATH to investigate the branches of time-periodic states (TPS), whichemerge at the Hopf bifurcations. Note that recently such branches were described for the dragged-plate system [59] (also compare section III). Here, we provide a deeper analysis by discussing howthe single-parameter bifurcation diagram (e.g., (cid:15) s ) changes when a second parameter is changed.We are particularly interested in potential reconnections between the different branches of TPS.This is relevant for the present system, but also provides valuable information for the behaviour ofthe entire class of coating systems.Before calculating branches of TPI, we begin by discussing the expected behaviour related to thecodimension-2 bifurcations where the Hopf bifurcations emerge or vanish. Fig. 12 (a) magnifiesthe loci of a number of Hopf bifurcations, highlighting in red a line related with particularly intri-cate behaviour. On the right, panels (b) to (d) sketch qualitatively different bifurcation diagrams(with We s as control parameter) with decreasing (cid:15) s : To the right of the Bogdanov-Takens bifur-cation (red triangle), the branch of steady state is ’bare’; bifurcations, which are related to thered branch in (a) do not yet exist (not shown). Then, decreasing (cid:15) s , when the Bogdanov-Takensbifurcation point marked by a red triangle is passed, a branch of TPS appears which connects aHopf and a homoclinic bifurcation [Fig. 12 (b)]. Decreasing (cid:15) s further, another branch of TPS iscreated in a double Hopf bifurcation at (cid:15) s ≈ . (Fig. 12 (c)). Continuing to decrease (cid:15) s , Hopfbifurcations, where two different branches of TPS end, approach each other until they annihilate at22 Preprint– contact: [email protected] – June 29, 2020
IG. 12: Panel (a) provides a zoom of Fig. 11 focusing on the loci of a number of Hopf bifurcations, whilepanels (b) to (d) show sketches of the bifurcation diagrams of time-periodic states (TPS) related to differentparameter ranges of the red curve in (a). Typical parameter values where cases (b) to (d) occur are indicatedby vertical dotted lines in (a), marked “b”, “c” and “d”. The occurring changes are described in the maintext. In (b) to (d), steady states are indicated as solid black lines, while green [red] lines indicate branchesof TPS that connect two Hopf bifurcations [a Hopf and a homoclinic bifurcation]. (cid:15) s ≈ . in a reverse double Hopf bifurcation effectively, fusing the two branches of TPS into one[Fig. 12 (d)]. Note that the resulting branch may still possess a nontrivial structure, e.g., showingsaddle-node bifurcations. These are not tracked here. Note that the remaining branch inherits aHopf bifurcation and a homoclinic bifurcation as its end points.To numerically study such qualitative changes we determine bifurcation diagrams, which containsteady states and TPS as a function of SAW strength (cid:15) s for an increasing sequence of fixed We-ber numbers, We s . We commence our analysis in Fig. 13 (a) and (b) with two relatively simplediagrams at fixed We s = 0 . and We s = 0 . , respectively. We observe two and six Hopf bi-furcations, respectively, and no global bifurcations. Increasing We s , we observe at We s = 0 . an odd number of nine Hopf bifurcations, see Fig. 14. This implies that a first Bogdanov-Takens23 Preprint– contact: [email protected] – June 29, 2020
IG. 13: Panels (a) and (b) give selected parts of the bifurcation diagrams, which show the (time-averaged)coating film thickness h c as a function of the SAW strength (cid:15) s at the fixed Weber number values We s = 0 . and We s = 0 . , respectively. Blue and green lines represent steady and time-periodic states, respectively.Hopf bifurcations are marked by black circles. The remaining parameters are as in Fig. 11. The insets givethe relevant complete bifurcation curve for the steady states. bifurcation must have occurred and that the additional appearance of a homoclinic bifurcation isexpected.In Fig. 14 we present a bifurcation diagram for We s = 0 . that shows nine Hopf bifurcations onthe branch of steady states. In the snaking region, a very short part of the branch shows linearlystable solutions, namely the part between the Hopf bifurcation and the saddle-node bifurcationin Fig. 14 (b). The related branch of TPS seems unusual as it starts and ends on the same sideof the saddle-node bifurcation, in contrast, to the normal behaviour expected of such a branch,which is created in a Bogdanov-Takens bifurcation [32]. Since the branch has indeed emerged atthis saddle-node bifurcation, it is likely that further codimension-2 events have taken place at thissaddle-node bifurcation, e.g., a fold-Hopf bifurcation. The point could be further investigated inthe future. The overall scenario indicates that the branch of TPS is linearly stable when it emergessupercritically at the Hopf point. This is consistent with the changes in stability observed whenfollowing the branch of steady states with increasing (cid:15) s from small values: One starts with a stablemeniscus state, passes first two destabilizing saddle-node bifurcations (inset of Fig. 14 (a)), thena global bifurcation (not affecting stability) and finally the discussed Hopf bifurcation where bothunstable eigenmodes stabilize. The third saddle-node bifurcation, zoomed onto in Fig. 14 (b) (at24 Preprint– contact: [email protected] – June 29, 2020
IG. 14: Panel (a) gives a bifurcation diagram showing the (time-averaged) coating thickness h c as a func-tion of the SAW strength (cid:15) s at fixed Weber number We s = 0 . . Remaining parameters, line styles, symbolsand inset are as in Fig. 13. The branches of TPS, which connect two Hopf bifurcations are shown in green,while the single branch connecting a Hopf and a homoclinic bifurcation is shown in red. As in (a) it isnearly not visible, panel (b) gives a zoom of the region very close to the saddle-node bifurcation of thesteady states The corresponding dependencies of the period of the TPS on (cid:15) s is presented in Fig. 15. h c ≈ . ) destabilizes the steady state again.Note that the branch of TPS presents a particular numerical challenge when approaching the globalbifurcation since the period T is expected to diverge. With the employed numerical method we arenot able to approach the bifurcation closely. However, Fig. 15 supports our interpretation. Its twopanels show the period T as a function of the SAW strength (cid:15) s for all branches of TPS in Fig. 14.One finds in panel (a) that T for the branch related to the global bifurcation (red line), is muchlarger than T for the branches, which connect two Hopf bifurcations (green branches zoomed ontoin panel (b)). Further, T steeply increases for the red branch indicating the likely divergence whenapproaching the global bifurcation.Next, we further increase the Weber number to We s = 0 . and show the resulting bifurcationdiagram in Fig. 16. There are now six branches of TPS, which connect eleven Hopf bifurcations.The corresponding periods are given in Fig. 17, indicating a strong increase in the period of thebranch that approaches the global bifurcation. Although, overall the picture is similar, there are anumber of important changes. In Fig. 14 at We s = 0 . the global bifurcation had just appeared andconnected to a Hopf bifurcation nearby (all in the vicinity of the saddle-node bifurcation of steady25 Preprint– contact: [email protected] – June 29, 2020
IG. 15: Time period T as function of the SAW strength (cid:15) s for all branches of TPS in Fig. 14. Line stylesare as in Fig. 14. Panel (b) zooms onto the four branches that connect two Hopf bifurcations. states). Now, at We s = 0 . , the global bifurcation connects to the Hopf bifurcation at smallest (cid:15) s , i.e., either many reconnections have taken place or the relevant Hopf bifurcation has passed allother Hopf bifurcations. Inspecting the transition between We s = 0 . and We s = 0 . in detailwe find that there is a combination of moving bifurcations and reconnections of branches of TPSwhen changing the magnitude of We s from 0.8 to 0.9.For clarity we illustrate the complicated sequence of events in the schematic provided in Fig. 18.Panel (a) shows all Hopf bifurcations at We s = 0 . as they occur along the branch of steady statestogether with their connections. That is, the topology of Fig. 18 (a) is identical to Fig. 14: fourdirectly connected pairs of Hopf bifurcations (green) and one Hopf bifurcation connected to aglobal bifurcation (red). This state defines the numbering of the bifurcations in Fig. 18: the pairs“1-1 (cid:63) ” to “4-4 (cid:63) ” and the individual pair “A-g”, where “g” stands for the global bifurcation.The first topological change occurs when the A-g branch approaches the 1-1 (cid:63) branch. First, thebifurcation A passes bifurcation 1, and slightly afterwards the branches reconnect such that after-wards branches 1-A and 1 (cid:63) -g exist, see Fig. 18 (b), i.e., Hopf bifurcation 1 (cid:63) is now connected tothe global bifurcation. This corresponds to the state in Fig. 16.As bifurcation A moves further to the right, it swaps position with bifurcation 2 [Fig. 18 (c)].Then, a bit later another reconnection is triggered and as a result branches 1-A and 2-2 (cid:63) becomebranches 1-2 (cid:63) and 2-A of Fig. 18 (d). Subsequently, more reconnections occur as bifurcation A26 Preprint– contact: [email protected] – June 29, 2020
IG. 16: Panel (a) gives a bifurcation diagram showing the (time-averaged) coating thickness h c in de-pendence of the SAW strength (cid:15) s at fixed Weber number We s = 0 . . Remaining parameters, line styles,symbols and inset as in Fig. 14. Panel (b) gives a zoom of the region very close to the saddle-node bifur-cation of the steady states. The corresponding dependencies of the period of the TPS on (cid:15) s is presented inFig. 17. continues moving to the right [Figs. 18 (d) to (f)]. The picture becomes slightly more involvedwhen between panel (d) and (e) another branch 5-5 (cid:63) appears and gets involved in the reconnectiongame. Roughly speaking, the appearance corresponds to the leftmost fold on the red dashed curvein Fig. 12. The final step, from Fig. 18 (g) to (h) occurs when branches 4-A and 5-5 (cid:63) reconnectinto one branch A-5 (cid:63) , thereby eliminating Hopf bifurcations A and 5. This corresponds to theleftmost fold on the red dashed curve in Fig. 12.The final state Fig. 18 (h) has an appearance very similar to the first one (a), only the connectionto the global bifurcation has moved from the leftmost bifurcation to the rightmost one. Properlycalculating the diagrams, the final state of the described sequence is obtained for We s = 1 . [seeFig. 19]. Note that there is a slight mismatch between sketch and bifurcation diagrams as thesketch focuses on the reconnections, but ignores the additional branch appearing in a double Hopfbifurcation between Figs. 14 and 16.Overall, in this cascade of topological changes of the bifurcation diagram, all branches of TPS arereconnected once, so that finally each surviving Hopf bifurcation n (that was initially connected toHopf bifurcation ( n ) (cid:63) ) is connected to bifurcation ( n +1) (cid:63) . The entire described sequence of events27 Preprint– contact: [email protected] – June 29, 2020
IG. 17: Time period T is shown as a function of the SAW strength (cid:15) s for all branches of TPS in Fig. 16,line styles are identical. Panel (b) provides a zoom. is triggered each time a Bogdanov-Takens bifurcation occurs. Hence, the overall rearrangement ofthe bifurcation diagram involves a large number of codimension-2 events.We exemplify this, by presenting in Fig. 20 a bifurcation diagram at larger We s = 2 . . The corre-sponding dependence of the period T on (cid:15) s is shown in Fig. 20. There exist 11 branches of TPS.Four connect Hopf and homoclinic bifurcations, the other seven pairs of Hopf bifurcations. Of thefour branches involving a global bifurcation, three have already terminated the sequence of recon-nections described above, however, the fourth one still has to undergo the sequence. Inspectingthe individual branches we see that they acquired a more complex structure and now feature manymore saddle-node bifurcations. This tendency is stronger for the outer branches than for the innerone. Also the range of periods increases for the branches connecting two Hopf bifurcations.Finally, we briefly discuss the characteristics of the time-periodic states. Panels (c) to (f) of Fig. 20show four space-time plots of selected TPS at loci indicated in panel (a), all on the same branchof TPS, namely, the “outermost” branch, i.e., the branch that emerges first when decreasing (cid:15) s .In all cases, drops are shed from the foot structure and then move at about constant speed alongthe substrate until they leave the domain. Close to this point they get slightly deformed due toboundary effects that, however, do not influence the main part of their trajectory. Moving alongthe branch of TPS away from the Hopf bifurcation, where it emerges, the number of drops in thedomain becomes smaller as the shedding events that define the period T become less frequent.The foot structure where the drops emerge does not change very much along the branch. This,28 Preprint– contact: [email protected] – June 29, 2020
IG. 18: Illustration of the motion of Hopf bifurcations and the related reconnections of the branches ofTPS. The different steps are schematically represented in parts (a) to (h) with line styles as in Fig. 15. Thevertical axes symbolically represents a solution measure, e.g., period T . Vertical lines indicate connectionto a homoclinic bifurcation. however, is in marked contrast when inspecting the TPS on other branches (not shown). Thedifference between the various branches mainly lies in the the details of the foot structure like itslength and the number of visible undulations.With this we end our investigation of the various branches of time-periodic states that has shownthe richness of the solution structure of the SAW driven system.29 Preprint– contact: [email protected] – June 29, 2020
IG. 19: (a) Bifurcation diagram showing the (time-averaged) coating thickness h c in dependence of theSAW strength (cid:15) s at fixed Weber number We s = 1 . . Remaining parameters, line styles, symbols and insetas in Fig. 14. Panel (b) gives the corresponding dependencies of the period of the TPS on (cid:15) s . V. CONCLUSION
In the present work we have in some detail studied the bifurcation structure corresponding toa liquid meniscus driven by surface acoustic waves (SAW). In particular, we have combined aprecursor-based thin-film model for the classical Landau-Levich (or dragged plate) problem forpartially wetting liquids studied earlier in Refs. [22, 59, 67] with the thin-film model for the SAW-driven meniscus considered in Ref. [43]. This has enabled us to investigate the two systems in acommon framework, first to reproduce selected results for each of them, and then to look at thebehavior of the SAW-driven system in the case of partially wetting liquids in direct comparisonto results obtained for the Landau-Levich system. Our investigation has been based on numericalpath continuation techniques.We have focused one one-dimensional substrates, i.e., we have neglect transversal perturbations,and have investigated in detail how the various occurring steady and time-periodic states dependon the Weber number and the SAW strength. The full bifurcation structure related to qualitativetransitions caused by the SAW has been uncovered with a particular attention on Hopf bifurcationsrelated to the emergence of time-periodic states. The latter correspond to the regular sheddingof lines from the meniscus. The interplay of several of these bifurcations has been investigatedby tracking the bifurcations in selected parameter planes, thereby discussing the codimension-30
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IG. 20: Bifurcation diagram, which shows the (time-averaged) coating film thickness h c as function of theSAW strength (cid:15) s at the fixed Weber number We s = 2 . . The full diagram is shown as inset in panel (b) andpanels (a) and (b) magnify the most interesting regions. Remaining parameters, line styles, and symbolsare as in Fig. 14. Fig. 21 shows the corresponding dependencies of the period of the TPS on (cid:15) s and selectedspacetime plots of TPS. Preprint– contact: [email protected] – June 29, 2020 . t / T (c) h . t / T (d) h −
20 0 2000 . t / T (e) h −
20 0 2000 . t / T (f) h FIG. 21: Time-period T is shown as a function of the SAW strength (cid:15) s for all branches of TPS in Fig. 20,line styles are identical. Panel (b) provides a zoom while panels (c) to (f) show space-time plots of selectedTPS at loci indicated in panel (a) by corresponding labels “c” to “f”, where (c) T = 3 . , (d) T = 4 . , (e) T = 7 . and (f) T = 14 . . between the three types of steady states have been found to be similar when changing the maincontrol parameters Weber number and SAW strength. However, the transitions differ when com-paring them to the dragged-plate system. Here, the foot states are situated on a snaking part ofthe bifurcation curve. Virtually, all of them are unstable, while in the dragged case they consecu-tively switch between stable and unstable. The continuous transition between the snaking part ofthe bifurcation curve and the part on the same curve that corresponds to Landau-Levich films is a32 Preprint– contact: [email protected] – June 29, 2020 eature of the SAW-driven case that has not been observed in the dragged-plate case. There, thesnaking curves always diverge at finite driving accompanied by a diverging foot length. This isnot the case here.Overall, the bifurcation structure has turned out to be rather involved, the general tendency beingthat its complexity increases with increasing Weber number and also with decreasing wettability.More and more saddle-node and Hopf bifurcations appear together with the branches of time-periodic states emerging from the latter. Such branches are also observed for the dragged-platesystem [59], but have here been investigated in more depth. In particular, we have shown howthese branches appear through double Hopf bifurcations and Bogdanov-Takens bifurcations thatalso create global bifurcations. We expect this to be similar in the dragged-plate case and relatedcoating systems. On a qualitative level, our corresponding results are similar to the ones forLangmuir-Blodgett transfer [30] further strengthening the point, put forward, e.g., in [62, 72] thatthese systems share many features and that the investigation of generic models as in [24] is ratherimportant.Our results show that a very finely tuned SAW-driven system could possibly be switched betweenthe deposition of a homogeneous film and the deposition of line or droplet patterns and control bySAW could be used together with other means of control as the usage of prestructured plates [73,74] or variable plate dragging speed [38]. To study the application of SAW control to depositionfrom solutions or suspensions with volatile solvent [42], solute dynamics and solvent evaporationhave to be incorporated into the model by amending corresponding models reviewed in [15, 62,63].However, our results also indicate that the parameter region where all these changes occur is verysmall and might be difficult to access with the present experimental techniques. Therefore, ourresults should not be taken as predictive in a quantitative way but as a catalogue of qualitativetransitions expected in a variety of coating systems. As discussed in the conclusion of Ref. [59],there exists a number of experimental systems that show related qualitative transitions. Theseinclude (i) water drops sliding on an oil film which can be destabilized by an applied voltage andtransform into many small oil droplets underneath the water drop [57] (also see [7]); (ii) recentexperiments on gas bubbles that move through a tube filled with partially wetting liquid where theliquid film between bubble and wall may undergo related instabilities [31], and (iii) the differentdynamic regimes of relatively thick viscous liquid films flowing down a cylindrical fiber [26]. Forthe detailed discussion of these experiments in the context of time-periodic states in the dragged-33
Preprint– contact: [email protected] – June 29, 2020 late case see the conclusion of Ref. [59]. In all these systems, SAW may be employed to stabilizethe Landau-Levich(-Bretherton) films.Finally, we stress again that our investigation has been entirely focused on one-dimensional sys-tems. However, driven contact lines and deposition dynamics often shows transversal instabil-ities [62]. This have been excluded from our study. A first investigation [37] has shown, thatthere exist bifurcations that break the transversal invariance and result in branches of fully two-dimensional states that represent a rich set of structures. However, a systematic study of their prop-erties is a formidable future endeavor that should also aim at an understanding of time-periodictwo-dimensional states occurring in various coating systems.
Acknowledgement
We acknowledge support by the German-Israeli Foundation for Scientific Research and De-velopment (GIF, Grant No. I-1361-401.10/2016), as well as further support by the DeutscheForschungsgemeinschaft (DFG; Grant No. TH781/8-1). We also thank Sebastian Engelnkemperfor the creation of a first tutorial on
PDE PATH usage for dragged-film problems, as well as TobiasFrohoff-H¨ulsmann and Simon Hartmann for support with
PDE PATH and
OOMPH - LIB implemen-tations, respectively.
Appendix A: Nondimensionalization
The appendix discusses our scaling and relates it to the scalings employed in Refs. [43] and [22],respectively. The dimensionless thin-film equation given by Eq. (2.23) of Ref. [43] writes ∂ t h = − ∂ x (cid:20) h s ∂ xxx h + hv s ( h ) (cid:21) . (A1)Re-introducing their scales τ = LχU Re , δ, and L = δ(cid:15) , (A2)for time t , film thickness h , and x -coordinate, respectively, and using the dimensionless numbers We s = χ ReCa (cid:15) and Ca = µUγ , (A3)we obtain the dimensional evolution equation ∂ t h = − ∂ x (cid:20) h µ γ∂ xxx h + U w hv s ( h ) (cid:21) (A4)34 Preprint– contact: [email protected] – June 29, 2020 here now x , t , and h are dimensional and U w = ρχδU /µ is a typical velocity.The complete dimensional thin-film evolution equation that governs the standard Landau-Levichproblem for a partially wetting liquid with additional SAW-driving and a dragged plate is given by ∂ t h = − ∂ x (cid:26) h µ [ ∂ x ( γ∂ xx h + κ Π( h ) − ρgh ) + αρg ] + U w hv s ( h ) + uh (cid:27) (A5)where Π( h ) is a dimensionless Derjaguin pressure and κ the related energy density scale. Theparameter α is the physical inclination angle and u is the velocity of the moving plate.Now we introduce the scales τ = Lν , δ, and L = δ(cid:15) (A6)for time, film thickness, and x -coordinate, respectively. Here ν is a generic velocity scale, that willbe specified later when deriving the different scalings used here and in the literature. We obtainthe nondimensional equation ∂ t h = − ∂ x (cid:26) h ∂ x ( D ∂ xx h + D Π( h )) + D h (cid:16) α(cid:15) − ∂ x h (cid:17) + D hv s ( h ) + D h (cid:27) (A7)with the nondimensional numbers D = (cid:15) γµν , D = (cid:15)κδµν , D = (cid:15)ρgδ µν , D = U w ν , D = uν . (A8) D is an inverse Capillary or Weber number, D may be called a Hamaker number, D is a Gravityor Galileio number, D a velocity ratio corresponding to a SAW strength, and D a velocity ratiocorresponding to a plate dragging strength. Keeping ν general, i.e., not selecting any leadingbalance, keeps all five nondimensional numbers in equation (A7), i.e., with D = We s , D = Ha , D = G , D = (cid:15) s and D = U one has Eq. (1) of the main text. However, specifying ν based ona particular physical effect one recovers the various scalings employed in the literature:(i) Using ν = (cid:15) γ/µ yields D = 1 , D = κL(cid:15)γ , D = ρgL γ , D = U w µ(cid:15) γ , D = uµ(cid:15) γ . (A9)With D = 1 (fixed by choosing L ), D = G , D = 0 and D = U this corresponds to Eq. (1)of [22]. Note that there, Π( h ) is made parameterless by choosing δ . This corresponds to the caseconsidered in the present section III [e.g., Fig. 3] where we keep We s = 1 and effectively recoverthe appropriate scaling. 35 Preprint– contact: [email protected] – June 29, 2020 ii) Using ν = U w gives D = (cid:15) γµU w , D = (cid:15)κδµU w = Ha , D = (cid:15)ρgδ µU w , D = 1 , D = uU w . (A10)With D = D = D = 0 and D = 1 / We s this corresponds to Eq. (2.23) of [43]. Note that there L and δ are chosen by geometry and SAW-intrinsic length, respectively, and are not availableto further reduce the count of dimensionless numbers. This corresponds to the case consideredin the present section IV A and the begin of section IV B [Figs. 4, 5, and 9] as keeping (cid:15) s = 1 we effectively recover the appropriate scaling. In Figs. 5 and 9 the additional effect of partialwettability is added.(iii) Our work mainly shows that increasing the strength of SAW has an effect similar to increasingthe plate velocity in [22]. However, we also want to retain the ability to study the influence of We s ,a parameter crucial in [43]. Using ν = (cid:15)κδ/µ gives D = (cid:15) γκδ = 1We s , D = 1 , D = ρgδκ = G, D = µU w (cid:15)κδ = (cid:15) s , D = µu(cid:15)κδ = U . (A11)The lengths L and δ are chosen as in [43]. This corresponds to the case considered in the mainpart of section IV B [Figs. 6-8 and 10-21] as keeping Ha = 1 effectively implies this scaling.This allows for comparison with the behaviour in both limiting cases, i.e., Refs. [43] and [22],respectively. [1] M. Abo Jabal, A. Egbaria, A. Zigelman, U. Thiele, and O. Manor. Connecting monotonic and os-cillatory motions of the meniscus of a volatile polymer solution to the transport of polymer coils anddeposit morphology.
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