THE EUROPEAN PHYSICAL JOURNAL E. Modelling thin-film dewetting on structured substrates and templates: Bifurcation analysis and numerical simulations

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1 ur. Pys. J., 7 (3) DOI./epje/i3-9- TH UROPAN PHYSICAL JOURNAL Modelling tin-film dewetting on structured substrates and templates: Bifurcation analysis and numerical simulations U. Tiele,a,L.Brusc,M.Besteorn,andM.Bär Max-Planck-Institut für Pysik komplexer Systeme, Nötnitzer Straße 38, D-87 Dresden, Germany Lerstul für Statistisce Pysik und Nictlineare Dynamik, BTU-Cottbus, ric-weinert-strasse, D-36 Cottbus, Germany Received 9 January 3 and Received in final form 6 May 3 / Publised online: July 3 c DP Sciences / SocietàItalianadi Fisica/ Springer-Verlag 3 Abstract. We study te dewetting process of a tin liquid film on a cemically patterned solid substrate (template) by means of a tin-film evolution equation incorporating a space-dependent disjoining pressure. Dewetting of a tin film on a omogeneous substrate leads to fluid patterns wit a typical lengt scale, tat increases monotonously in (coarsening). Conditions are identified for te amplitude and periodicity of te eterogeneity tat allow to transfer te template pattern onto te liquid structure ( pinning ) emerging from te dewetting process. A bifurcation and stability analysis of te possible liquid ridge solutions on a periodically striped substrate reveal parameter ranges were pinning or coarsening ultimately prevail. We obtain an extended parameter range of multistability of te pinning and coarsening morpologies. In tis regime, te selected pattern depends sensitively on te initial conditions and potential finite perturbations (noise) in te system as we illustrate wit numerical integrations in. Finally, we discuss te instability to transversal modes leading to a decay of te ridges into rows of drops and sow tat it may diminis te size of te parameter range were te pinning of te tin film to te template is successful. PACS. 68..e Liquid tin films 8.6.Rf Nanoscale pattern formation 7..Ky Nonlinearity (including bifurcation teory) Tin liquid films on solid substrates often occur in nature and tecnology and ave attracted many researcers []. Tey play important roles, for example, as liquid lining in mammalian lungs [], as tear film in te eye [3], as protective, aestetic, adesive or lubricating coating, as functional layer in eat and mass transfer devices [], or even in te form of lava or mud flow []. Tese liquid layers are caracterised as tin because teir tickness is small compared to typical lengt scales in te direction of te flow. Tis allows a simplification of te Navier-Stokes equations governing te free surface flow to a single nonlinear partial differential equation for te film tickness, a so-called film evolution equation []. Similar equations are also found for quite different pysical problems. For instance, te dewetting of a tin film under te influence of effective molecular interactions [6 ] and te rupture of a tin liquid film on a eated orizontal plate [ 3] can be described by film evolution equations aving a similar form as te Can-Hilliard equation known from te decomposition of a binary mixture []. Te same is true for te respective problems on an inclined plane [, 3, 7] and te driven Can-Hilliard a tiele@mpipks-dresden.mpg.de equation [8]. Altoug te present work focuses on a specific system, namely, tin liquid films on inomogeneous substrates, te main results will also apply to pysically quite different situations. Spontaneous dewetting occurs for liquid films wit ticknesses smaller tan one undred nanometers were substrate-film interactions due to molecular forces become important. Ticker films may also dewet via eterogeneous nucleation. Up to now, most of te studies on dewetting focus on omogeneous substrates. All pases of te process are investigated: te initial rupture [9, ], te growt of individual oles [, ], te evolution of te resulting ole pattern [, 3] and te stability of individual dewetting fronts []. Te final goal is to understand ow to keep tin films stable [] or ow to break tin films in a controlled manner [6]. Tis work addresses te study of te influence of cemically inomogeneous substrates (templates) on te dewetting of a tin liquid film. Conditions for te eterogeneity tat provoke dewetting at pre-determined points (pinning) will be obtained. Several experiments [7 3] involve dewetting processes of tin liquid films on eterogeneous substrates to deposit liquids in a regular manner determined by te template. In most experiments

2 6 Te uropean Pysical Journal pysically or/and cemically striped substrates are used. Reference [33] focuses on te pase ordering and dewetting of a binary mixture on ydropobic and ydropilic stripes, wereas in reference [7] te substrates are cemically striped and pysically grooved. Pure dewetting witout pase ordering is studied in references [3, 3]. Te surface (S) layer of a bacterium membrane can be used to deposit material in a controlled manner [3]. Beside tese intentional eterogeneities, te substrate may also be eterogeneous due to impurities like oxidised patces, dust particles and scratces. Previous teoretical results describe different morpological transitions of liquid layers on substrates wit strong stepwise or nearly stepwise wettability contrasts [3 3]. Te deposited liquid volume, cemical potential or te size of te eterogeneous patces ave been used as control parameters. arlier approaces involve diverging gradients of te cemical potential at te wettability steps [38, ]. Here, we study eterogeneous dewetting on a smootly patterned substrate using te wettability contrast as a control parameter. Our results suggest, owever, tat te actual functional form of te eterogeneity is muc less important tan its lengt scale and its strengt. We employ bifurcation analysis using numerical continuation tecniques [] and illustrate te findings wit simulations in. Continuation is a very effective metod to determine stable and unstable stationary solutions and teir bifurcations by following tem troug parameter space using Newton s metod [, 6]. For tin films continuation was applied recently in studies of dewetting on a omogeneous substrate [, 7, 8], sliding drops on inclined planes [6, 9], transversal front and back instabilities of sliding liquid ridges [] and rivulet instabilities of a tin-film flow over a localized eater []. Te transition from coarsening to pinning tat occurs as te wettability contrast is increased was already subject of a sort earlier study []. Here, we provide a more complete account by extending te bifurcation diagram to large system sizes. Furtermore, te transversal instability of striped liquid patterns is analysed in great detail and numerical simulations in interesting regions of te bifurcation diagram are carried out also wit addition of noise to account for small random perturbations of te system. We use an evolution equation for te film tickness [] tat includes a disjoining pressure to account for te effective molecular interaction between substrate and film. Tis approac can be used over a large range of lengt scales for te eterogeneities to investigate not only te equilibrium configurations but also te dynamics of te dewetting process on te structured surfaces. In particular, te dynamics cannot be predicted from te wellestablised approac based on variations of te interfacial free energy [3 37,,3,3]. However, as a trade-off te tin-film equation is restricted to relatively small contact angles due to te use of te lubrication approximation in its derivation. Depending on te particular problem treated, te disjoining pressure may incorporate long-range van der Waals and/or various types of sort-range interaction terms [ 6]. Tese interactions are responsible for te process of dewetting. Studies of dewetting of a tin liquid film on a substrate are generally based on models involving a disjoining pressure [6,7]. A eterogeneous substrate may be modelled by a spatial variation of te disjoining pressure [38, ]. Recently, Pismen and Pomeau [7] derived a disjoining pressure tat remains finite even for vanising film tickness by combining te long-wave approximation for tin films [] wit a description for te diffuse liquid-gas interface [8]. Tese autors discuss te resulting vertical density profile near a orizontal liquid-gas interface. Te density variation close to te solid substrate due to molecular interactions enters into teir calculation via te boundary condition for te fluid density at te substrate. Te obtained density profile is ten incorporated into a fully consistent teory based on te Stokes equation in te long-wave approximation. Te resulting film tickness equation as te usual form of a tin-film equation wit a disjoining pressure []. Te disjoining pressure is purely ydrodynamic in origin and its form is derived self-consistently rater tan modelled. Pattern formation in tis model was studied bot for a liquid film on a omogeneous orizontal substrate [, 8] and for a film flowing down a sligtly inclined plane [6]. Recent treedimensional studies of sliding drops and liquid ridges on an inclined plane [7, 9, ] revealed transverse instabilities. Te various results available for tis model motivated us to employ it in te present study. A cemically striped substrate is modelled by a periodically modulated disjoining pressure. However, te main results for tis model will not qualitatively differ from results for oter disjoining pressures combining a sort-range destabilising and a long-range stabilising component as, for example, used in [7, 9 6]. Tis as been sown for omogeneous substrates in references [, 7, 8]. Our study is organised as follows. In Section, we introduce te evolution equation for te film tickness, discuss te form of te disjoining pressure used, and nondimensionalise te equations. In Section, we determine stationary solutions and teir linear stability properties. A morpological stability diagram is derived detailing parameter ranges for success or failure of templating. In Section 3, tese results are illustrated by and compared to (two-dimensional) numerical simulations in te parameter range of multistability. Section gives te results for te transverse stability of te liquid ridges in dependence of te eterogeneity. Conclusions are drawn in Section. Film tickness equation Te film tickness evolution equation for a very tin liquid film on a solid substrate is derived from te Stokes equations [] using a long-wave or lubrication approximation. For a two-dimensional geometry, as sketced in Figure, one finds te evolution equation for te space- and

3 U. Tiele et al.: Tin-film dewetting on structural substrates 7 gas respectively. We introduce dimensionless quantities (wit tilde) using new scales: t = 3ηγ κ l t, z x liquid (x) = l, () lγ x = x, κ lγ y = ỹ. κ eterogeneous substrate P et Fig.. Sketc of te two-dimensional geometry. Te differently saded regions of te substrate stand for te different wettability in te eterogeneous case. -dependent film tickness (x, t) on a omogeneous substrate as [] t = {Q() [γ f()]}, () were Q() = 3 /3η is te mobility factor arising for Poiseuille flow. Te term γ xx represents te Laplace or curvature pressure and f() is an additional pressure term. γ and η are te surface tension and viscosity of te liquid. Subscripts t, x and denote te corresponding partial derivatives. Te additional pressure comprises te ydrostatic pressure and te effective molecular interaction between substrate and tin liquid film, i.e. te disjoining pressure: f() =κm(, a)ρg = κ a e/l ( a e/l ) ρg, () were g is te gravitational acceleration and ρ te density of te liquid. Π() = κm(, a) is te disjoining pressure arising from diffuse interface teory [7], κ as te dimension of a spreading coefficient per lengt, a is a small dimensionless positive parameter describing te wetting properties in te regime of partial wetting, l is te lengt scale of te diffuse interface and ρg is te ydrostatic pressure. Te eterogeneous substrate, we are interested in, is modelled by a spatial sinusoidal modulation of te overall strengt of te disjoining pressure in equations (, ) κ = κ ξ(x) =κ ( ɛ cos πx/p et ), (3) amounting to a smoot spatial variation in te wettability of te substrate. Te eterogeneity is caracterised by te amplitude, ɛ, and te imposed periodicity, P et. Te more and less wettable regions of te substrate are tereby cosen to ave identical spatial size wic provided successful templating at strong eterogeneities []. Te maxima and minima of ξ represent te least and most wettable parts, Te ratio κ l/γ is O(a ) [7], i.e. te scale in x-direction is O(l/a). quation () becomes, after dropping te tildes, were t = { 3 [ ξ(x)m(, a) G] }, () G = lρg κ (6) stands for te ratio of gravitation and molecular interactions. Its value is taken to be always positive. It was found tat even a very small G cannot be dropped because te qualitative beaviour of te solution canges setting G = [,8]. However, because te actual value of G</ does not cange te qualitative outcome, we use ere a fixed G =.. Te form of M(, a) allows to transfer te constant a into te mobility factor Q by introducing te sifted film tickness = lna.tis allows to represent te stationary solutions independent of a and direct comparison wit results for omogeneous substrate [, 8]. After dropping te star we find M() =e ( e ), (7) wile te evolution equation () becomes t = {Q(, a) [ f(, x)]} (8) wit Q(, a) =(ln a) 3, f(, x) = ξ(x)m()g and ξ(x) =ɛcos πx/p et. Te problem on te orizontal substrate as a variational structure. In consequence, one can give a Lyapunov function (free-energy potential) [6,,, 6] () = L L L [ ] f(, x) dxdy f (9) tat fulfills d/dt. Te normalising term f denotes te free energy of te flat film of (scaled) tickness on a omogeneous substrate (ɛ = ). We will call sortly energy and use it to compare stationary solutions to determine teir absolute stability. During te evolution of a given initial film tickness profile, tis energy decreases and eventually settles in a minimum wen te system approaces a stationary solution of equation (8). Since te eterogeneity ξ(x) is uniform in te y-direction, we analyse stationary solutions (x) tat are uniform in y

4 8 Te uropean Pysical Journal 6 metastable stable min, max 3 a b unstable metastable.. Fig.. Te ranges for stable, metastable and linearly unstable flat films on te omogeneous substrate in te parameter plane (G, ). Te cross, square and circle indicate parameter values studied ere in detail. G P Fig. 3. Stationary periodic solutions on te omogeneous substrate. Sown are (a) te minimal and maximal film ticknesses and (b) te corresponding energies in dependence on te period for different mean film ticknesses as given in te legend and G =.. Symbols denote solutions tat are analyzed in detail (cf. Figs. and 6, below). Periodic solutions and teir stability. Te omogeneous substrate as well. In contrast to te omogeneous case, tese solutions are ere of direct pysical relevance. For strong eterogeneity tey are te energetically favoured solutions even for te full tree-dimensional problem. At weaker eterogeneity tey may be stable or unstable wit respect to tree-dimensional disturbances as discussed in Section. However, simulations for striped substrates [, 63] indicate tat even in parameter ranges were te system approaces finally a solution depending on bot x and y, te evolution is first attracted by a stripe-like solution tat subsequently decays into an arrangement of drops. One can determine tese stripe-like solutions directly by setting in equation (8) te derivative and all y- derivatives to zero and integrating twice, yielding = xx ξ(x)m() G C. () Te second integration is possible because te constant of te first integration can be set to zero. Tis is due to te fact tat in te stationary states tere is no mean flow in te system (corresponding to te variational structure). Tis is, for example, not te case for a drop on an inclined plate because tere te structures are only stationary in a comoved frame [6]. Here we focus on situations wit conserved liquid volume and use C as a Lagrange multiplier to fulfill tis volume constraint. Alternatively, C corresponds to a cemical potential for situations wit enabled evaporation. Before considering te eterogeneous case, we will sortly review te results for te omogeneous case described by equation (8) wit ɛ = [, 8]. Flat films are (trivial) stationary solutions tat, depending on mean film tickness and te parameter G (see Fig. ), may be unstable wit respect to infinitely small (spinodal dewetting) or finite (nucleation) disturbances of te film surface. Rupture due to te spinodal mecanism occurs for disturbances wit periods larger tan a critical value P c =π/k c wit k c = M G, () wenever M/ < G. Te fastest growing wavelengt is P m = P c. Stationary solutions also exist as a continuous twoparameter family of periodic patterns, i.e. sequences of oles and droplets, parameterised by te mean film tickness =/P x P x (x)dx and te spatial period P of te pattern. Starting from small-amplitude sinusoidal solutions tese brances are calculated using continuation tecniques [] as described in [,6,]. Details on te tecnique can be found in te first section of Appendix A. Tree qualitatively different regimes exist as sown in Figure 3: i) Te mean film tickness corresponds to a linearly unstable flat film. One branc of stationary structured solutions exists wose amplitude (energy) increases (decreases) monotonically wit increasing period. Te energy (q. (9)) is always lower tan te energy of te respective flat film. Tis case is illustrated by te dotdased curves in Figure 3.

5 U. Tiele et al.: Tin-film dewetting on structural substrates 9 ii) Te mean film tickness also corresponds to a linearly unstable flat film. Two brances of stationary solutions exist. Teir respective energies decrease monotonically wit increasing period. Te branc wit lower energy corresponds to te branc found in i) but as now for a small range of periods an energy larger tan te flat film. Its amplitude, owever, still increases monotonically wit increasing period. Te oter branc as always iger energy tan te flat film, its amplitude decreases wit increasing period and it terminates wit small-amplitude solutions at te critical wavelengt of te flat film P c. It represents linearly unstable nucleation solutions tat ave to be overcome to break te film into small droplets wit widt smaller tan P c [, 8]. In Figure 3 te solid curves illustrate tis case. iii) Te mean film tickness corresponds to a metastable flat film. Tere exist two brances of stationary solutions muc as in ii). However, because te flat film is linearly stable now also te branc of nucleation solutions continues towards infinite period. Te nucleation solution at infinite period represents te individual critical ole necessary to break an infinitely extended film at a single site as discussed in [6] for first-order wetting transitions. Te dased curves in Figure 3 correspond to tis case. If dewetting proceeds via spinodal rupture of te film first oles wit a typical distance, P m, are formed as te solutions of te lower branc are approaced. On a slower scale te so formed initial drops coalesce, te pattern coarses and tends to te absolute minimum of te energy at te largest possible P, i.e. te system size. Tis often undesired coarsening may be stopped at an intermediate stage by evaporation of te solvent or by literally freezing te system [6, 66]. Te cemical structuring of te substrate, studied in te next section, opens a tird way. ξ y/p c (a) (b) x/p et (c) (d) (e). Te eterogeneous substrate i (f) We study te influence of te eterogeneity equation (3) by fixing te mean film tickness and te parameter G in te linearly unstable film tickness range as indicated by te cross in Figure. Ten te strengt ɛ and te period P et of te eterogeneity are varied. Te case ɛ< is related by symmetry. Te system size is cosen as multiple of P et, L = np et, and periodic boundary conditions are used. We will subsequently increase n =,,,... wic allows to systematically construct te set of solutions and to understand teir properties for a given system size. Now, suppose a variety of substrates wit fixed period P et but different strengt ɛ of te eterogeneity are available and a dewetting pattern wit spatial period P = P et is desired. One may first coose te mean film tickness suc tat te critical wavelengt of spinodal dewetting P c is of te same order as P et. P et = P c warrants good templating for strong eterogeneity []. In te simplest case, te omogeneous system as only two stationary solutions wit P = P et : te linearly unstable flat film and x/p et Fig.. Scematic overview of te two different longitudinal instability modes. (a) Profiles of stationary solutions to te film tickness equation (8) wit spatial periods P = P et (eavy line) and P =P et (tin line) corresponding to te square and diamond in Figure 3, respectively. Te dotted (dased) arrows indicate te front movements for coarsening via te volume (translation) mode. (b) Te eterogeneity ξ(x) (q.(3)) for acemically structured substrate. Minimaof ξ correspond to te most wettable parts of te substrate. (c) Top view of te pinned solution P = P et. Te liquid (indicated saded areas) is collected on top of te more wettable stripes (indicated by vertical lines). Tis solution can coarse in two ways: (d) by mass transfer from te left to te rigt droplet (mediated by te dotted mode in (f)) or (e) by translation of bot droplets (corresponding to te dased mode in (f)). (f) Te two linear eigenmodes of te pinned solution corresponding to te arrows of identical line styles in (a).

6 6 Te uropean Pysical Journal a periodic large-amplitude solution []. In Figure 3 te square corresponds to tis pinned pattern if P et =. However, all stationary solutions wit larger period ave lower energy. Te representative of tese coarsed solutions wit smallest period, i.e. P =P et, is marked by a diamond in Figure 3. Te spatial profiles of te two competing solutions for a system of size L =P et are sown in Figure (a). On te omogeneous substrate te wanted pattern is always linearly unstable to subsequent coarsening tat transforms adjacent small droplets into a larger drop []. Te coarsening takes one of two possible routes: i) translation of te drops towards eac oter as indicated by te upper line of arrows in Figure (a), or ii) mass transfer from one drop to te oter witout translation as indicated by te lower line of arrows. Te eigenmodes for tese two routes are sown in Figure (f). Tey are calculated via linear stability analysis of te periodic solutions (see App. A). Te different coarsening modes were first discussed in te context of spinodal decomposition of a binary mixture [67]. Solutions wit longer and longer periods result tat ave consecutively lower energies and te final state will always ave a period equal to te system size. However, tis intrinsic tendency towards coarsening enters now in competition wit te eterogeneity (Fig. (b)) tat favors te solution wit P = P et.forl=p et tis competition involves tree solutions (Fig. (c-e)) tat can be stable or unstable depending on te parameters ɛ and P et. min max. 3 (a)...3 ε (b).. System of size L = P et We start our analysis in te smallest system L = P et. Switcing on te eterogeneity implies, as a first consequence, tat te flat film is no longer a solution of equation (). Te flat-film solution is replaced by a periodic solution tat can be given analytically in te limit of weak eterogeneity ɛ. One writes te eterogeneity as ɛ cos(kx) =ɛ/(e ikx c.c.) wit k =π/p et, were c.c. stands for complex conjugate. Te volume conserving ansatz for te film tickness (x) = δ (ei kx c.c.) () is inserted into equation () and gives for δ O(ɛ) and neglecting terms of iger order in ɛ solutions wit k = k and δ = M( ) kc ɛ. (3) k Te above-assumed condition δ is only fulfilled if P et is well separated from P c. M( ) is always positive, ence for k c >k(p c <P et ) te solution is modulated in pase wit te eterogeneity, i.e., te film is ticker on te less wettable parts of te substrate. Tese smallamplitude solutions are linearly unstable like te flat film of te corresponding tickness witout eterogeneity. However, if te critical period is larger tan te period of te eterogeneity (P c >P et ), tis solution as a pase 3 x Fig.. (a) Bifurcation diagram representing maximal (eavy) and minimal (tin) film tickness of periodic stationary solutions. Line styles correspond to te ones of te profiles in (b). Te small-amplitude result equation (3) is denoted by te dot-dased lines. (b) Film tickness profiles of te solutions at ɛ =.. Parameters are G =., =.,P =P et =wic yields δ/ɛ =.733. Te least and most wettable parts of te substrate are at x =andx=, respectively. sift of π wit respect to te eterogeneity, i.e., te film is tinner on te less wettable parts of te substrate. Tis is also te case if te flat film of tickness is linearly stable because tere kc <. Tese small-amplitude solutions are linearly stable. Restricting ourselves again to linearly unstable mean film ticknesses we use te weakly modulated solution equations () and (3) as starting solution for te continuation procedure [] (see App. A). As an example, we coose G =.and =.resulting in P c = 33, and use P et =. Ten te stationary solutions wit period P = P et are computed as ɛ is increased. Te flatfilm and periodic solutions at ɛ = give rise to one and two solution brances, respectively. Figure sows te

7 U. Tiele et al.: Tin-film dewetting on structural substrates (a) (b). ε Re[ β ] (c) (d). ε. Fig. 6. nergy of stationary solutions to equation (8) wit (a) P = P et and (b) additional P =P et versus ɛ. Squareand diamond denote te solutions in te omogeneous case (Fig. 3). (c) igenvalues β wit largest real part of te pinned pattern wit P = P et (lowest branc in (a)). Solid curves correspond to L = P et and broken curves to L =P et. Triangles correspond to period doubling bifurcations in (d). igenvalues for L =P et are denoted by gray lines. (d) Close-up of (b) using identical line styles. Solutions wit P =P et bifurcate from te branc wit P = P et. Stable (unstable) solutions are marked by ( for eac unstable eigenmode). Parameters are =., P et =,G=.anda=.. corresponding bifurcation diagram and profiles of te solutions. Te branc emerging from te flat-film solution is for small ɛ well approximated by te analytical result equation (3). Te solutions on tis branc are indeed in pase wit te eterogeneity wereas te branc of lowest energy tat corresponds to large-amplitude solutions possesses a pase sift, as one would expect from pysical considerations: te drops concentrate on te more wettable patces. Te middle branc is in pase wit te eterogeneity and terminates in a saddle-node bifurcation wit te small-amplitude branc. In Figure 6(a) te relative energies of te solutions sown in Figure (a) are compared. For te cosen values of te parameters te solutions in pase ave a larger energy tan tose out of pase. Wen increasing ɛ te energy of te lower branc decreases furter wereas te energies of te oter brances increase. In a system tat as te size of one period of te eterogeneity, i.e. L = P et, te entire lower branc in Figure 6(a) is linearly stable indicated by a wereas te oter two brances are linearly unstable indicated by or corresponding to te number of unstable eigenvalues. For ɛ = all solutions possess two zero eigenvalues (neutral modes) tat correspond to te symmetries wit respect to translation and cange of mass. As ɛ departs from zero, te translational symmetry is broken and one of te two zero eigenvalues acquires a negative real part, sown for te stable branc as a tin solid line in Figure 6(c)). Te second symmetry corresponds to a sift along te family of solutions wit different tat are all solutions to te same equation (8). Since te eterogeneity does not destroy te possibility of tis sift te second eigenvalue remains equal to zero (tick solid line in Fig. 6(c)). New solutions appear as we increase te system size to L = np et. Beside te already discussed solutions wit period P = P et tere also solutions wit period P = ip et exist, were i and n/i are integers. For instance, in a system of size L =8P et beside te pinned solution wit period P = P et tere appear solutions wit P =P et, P et and 8P et... System of size L =P et Next we consider te doubled system size L =P et.for tis system solutions exist wit two periods: i) P = P et and ii) P = P et as sown in Figure 6 (b). At ɛ = tey are denoted by a square i) and a diamond ii) corresponding to te solutions in Figure 3. For increasing ɛ te solutions wit P = P et are te same as discussed above for te smaller system (L = P et ). However, now te

8 6 Te uropean Pysical Journal (a) 3 ξ (b) (c) 6 8 x Fig. 7. Profiles for asystem of size L =P et. Te dotted and dased lines denote solutions wit P = P et situated on te brances wit corresponding line styles in Figure 6 (b). Te solid lines sow te pinned solution corresponding to te solid line of lowest energy in Figure 6 (b). Solutions drawn wit broken lines in (a) and (b) can be obtained from te pinned solution by mass transfer and translation, respectively. Tereby, te solutions sown as tin lines take te role of a saddle between te respective oter two solutions. Parameters are =., P et =, ɛ =.andg=.. Te panel (c) sows te eterogeneity ξ(x) wit minimacorresponding to te most wettable parts of te substrate. stability analysis sows te appearance of two new eigenvalues (Fig. 6(c)) tat correspond to asymmetric combinations of te neutral modes. Tey correspond to te two different modes of coarsening introduced above. i) Te dased line in Figure 6(c) belongs to a combination of opposite translational modes leading to droplet translation. ii) Te dotted line belongs to a combination of opposite volume modes leading to mass transfer. For te present coice of parameters, Figure 6(c) sows tat te sift mode becomes stable at smaller ɛ values tan te mass transfer mode. So te mass transfer is te dominant coarsening process. As ɛ increases, bot eigenvalues become negative, implying te linear stability of te solution wit P = P et for larger ɛ. At te crossing points, two period doubling bifurcations (triangles) occur were stationary solutions of period P =P et emerge (compare Fig. 6(c,d)). Te two bifurcations are bot subcritical, ence te emerging solutions inerit te respective instability of te unstable solutions at smaller ɛ [68]. Terefore, te four brances of solutions wit P = P et sown in Figure 6 (b) ave te following linear stability. Te upper dotted branc (see also profile in P et /P c coarsening multistability pinning.. ε.6.8 Fig. 8. Morpological pase diagram of templating sows parameter (ɛ, P et/p c) regions wit different beaviour of te tin film on a eterogeneous substrate. Te saded band separates te parameter region of coarsening from te one of pinning. Inside te saded band multistability is present wit te pinned pattern aving te minimal energy inside te dark saded area. Te pinned solution is unstable to longitudinal perturbations (k = ) below te dot-dased curve and unstable to transversal perturbations (k ) below te saded band (solid curve). Parameters are =.andg=.. Fig. 7 (a)) carries two positive eigenvalues and is itself a saddle in function space tat divides evolutions by mass transfer towards te lower dotted branc and te P = P et branc. Te lower dotted branc (profile in Fig. 7 (a)) still as one positive eigenvalue tat leads to a sift of te pattern towards solutions of te lower dased branc. Te upper dased branc (profile in Fig. 7 (b)) as one positive eigenvalue and is a saddle tat divides evolutions by translation of two droplets towards te lower dased branc and te P = P et branc. Te entire lower dased branc (profile in Fig. 7 (b)) is linearly stable in L =P et and represents te coarse solution competing wit te desired pattern. Comparing Figure 6 (b) and (d) we note tat te numbers of unstable eigenvalues for te two bifurcating brances differ near and far from te period doubling bifurcations. Tis is due to two additional pitcfork bifurcations occurring at ɛ =. (diamond symbol, dased branc) and ɛ =. (dotted branc). Te sort dotdased branc stands actually for two brances of identical energy and consists of solutions wit broken mirror symmetry. Tey connect te pitcfork bifurcations and transfer an unstable eigenvalue to te mass transfer branc (dotted). Altoug te additional solutions are all linearly unstable tey may influence te coarsening dynamics in tis ɛ range. Summarising te results on te stability of te solutions in a system of size L =P et at te given parameter values, one can state tat for ɛ<. solutions wit P =P et are te only stable ones and ave lowest energy. For ɛ>. multistability between solutions wit P = P et and P =P et occurs and te initial condition becomes important for te dewetting of a tin film on a eterogeneous substrate. At ɛ =.7 te energies of bot solutions are equal. At larger ɛ te desired pattern as

9 U. Tiele et al.: Tin-film dewetting on structural substrates 63 lowest energy. If suitable initial conditions can be cosen, ten te pinned solution can already be obtained at muc smaller ɛ. However, as illustrated by simulations in in Section 3, coarsening will still occur at muc larger ɛ if disadvantageous initial conditions are cosen. Te pinned solution is te only possible for ɛ>., were te linearly stable P =P et branc ceases to exist. Up to now, we discussed te ɛ ranges for coarsening, multistability and pinning for a single coice of P et.we repeated te above analysis for te full range of P et keeping G =.and =.fixed. Te results are presented in te morpological pase diagram, Figure 8. At low values of ɛp et coarsening prevails wile for large values te pattern is pinned to te eterogeneity. At intermediate values (saded band) te initial condition selects te asymptotic state due to multistability. In particular, it is not possible to pin a pattern to a eterogeneity tat is muc smaller tan te critical wavelengt P c for te surface instability of te corresponding flat film on te omogeneous substrate...3 Large system size In systems of larger size L, new combinations of te neutral modes lead to an increasing number of unstable eigenmodes for te pinned pattern. All new positive eigenvalues lie between tose of L = P et, decrease wit increasing ɛ and eventually become negative as sown in Figure 6 (c) for L =P et. Hence te coarsening instability is most dominant for nearest-neigbour interaction and is already covered by te analysis of te sort system L =P et []. So te pinned pattern in an arbitrarily large system is linearly stable against coarsening for eterogeneities ɛ >.. Tis critical ɛ is determined by te parameters, G and te specific form of Π(). However, since te two symmetries connected wit te coarsening modes are present for arbitrary coices of Π() teresult does not qualitatively depend on tis coice. ac zero crossing of one of tese new eigenvalues corresponds to a bifurcation of a branc of coarsed solutions. Tis explains te massive proliferation of solutions in larger systems. To illustrate tis, Figure 9 sows solutions wit P = P et in te bifurcation diagram for systems of size L = P et. Solutions wit P = P et and P = P et ave to be added according to Figure 6(b) and possess an internal discrete translational symmetry in L =P et. All solid brances in Figure 9 emerge at two period quadrupling bifurcations of te pinned solution P = P et. Bot occur at ɛ. (.338). Following te solutions along tese brances (also troug te saddle node bifurcations), smaller droplets continuously merge into larger ones as is illustrated in te inset for two solutions corresponding to te symbols in te main panel. At ɛ =(.68) te brances connect and te solutions approac te single droplet of system size known from te omogeneous system. At finite ɛ, te coarse solutions possess small satellite drops beside te large drop. Te latter may cover many stripes, but te small satellite drops prevail even at te end of a complete coarsening process x. ε. Fig. 9. Bifurcation diagram for L = P et, solutions wit P =P et and P = P et are excluded. Tin (eavy) solid lines denote unstable (stable) solutions for P =P et tat emerge from te single droplet at ɛ =, =.68. Dased curves are examples of brances tat bifurcate from symmetrical solutions wit P =P et and P = P et atsmallɛ.. Te two lower brances exist up to ɛ =.7, =.9. Te inset sows profiles corresponding to te diamonds wit corresponding line styles for ɛ =.3and te dot-dased line for ɛ =.. Parameters are =., P et =,G=.anda=.. We find -branc segments of linearly stable solutions and denote tem by eavy lines. Four out of tese belong to te brances described above and can be easily computed wen starting from te solution wit a single large drop at ɛ =. Tis procedure is robust and can be generalised to arbitrary system size. Practically, computer resources become demanding as te number of modes in te continuation procedure increases linearly wit te system size. One out of te stable segments belongs to a set of independent brances (denoted by dased curves) tat interconnect bifurcations of brances wit P>P et. Most emerge and terminate at small but always finite ɛ. Note, especially, te two lines terminating in a saddle node at ɛ and.. Tey come close to but do not reac te solution wit P =P et /3 tat exists in te omogeneous system... Canging te film tickness We repeated te above analysis for systems of size L = P et for different coices of te film tickness and concentrate ere on te qualitative canges compared to =. studied above. In Figure te bifurcation diagrams are sown for (a) te lower part of te linearly unstable range at =. coosing P et P c ( ), and (b) in te metastable range at =3., were P et = 3 was cosen. In te metastable range tere exist tree solutions wit P = P et in te omogeneous system. Tis results for ɛ in one additional pair of solutions as compared to te linearly unstable tickness range. Solutions wit P =P et are similar to te case wit =.. Te stable

10 6 Te uropean Pysical Journal (a) (b) ε.. ε. Fig.. nergy of stationary solutions P = P et (solid curves) and P =P et (broken curves) versus ɛ for (a) small film tickness =. (square in Fig. ) and (b) large = 3. in te metastable range (circle in Fig. ). Stable (unstable) solutions are marked by ( for eac unstable eigenmode). Triangles denote period doubling bifurcations were solutions wit P =P et emerge due to te translation mode (dased branc, open triangle) and transfer mode (dotted branc, filled triangle), compare Figure 6. In (b) anoter pair of broken curves (P =P et) is omitted. Tis pair connects two period doubling bifurcations on te P = P et branc maked wit a single and intersects te ɛ = axis at =.7. Parameters are G =.and(a)p et =, (b) P et = 3. branc of coarse solutions also corresponds to te translation of droplets. Hence, for tick films >ln, te stable coarse drop is always centered on a least wettable position and overlaps te adjacent more wettable areas. For small film tickness =.(Fig. (a)) te solutions wit P = P et are similar as for =., since tere are also only two solutions in te omogeneous limit. However, altoug te sequence of te two bifurcations tat stabilise te pinned solution is as for =., te brances wit P = P et excange teir role, i.e. te stable coarse solution is now generated by te transfer mode. Te reason is tat at small <ln te coarse drop P =P et contains an amount of material small enoug to benefit from completely covering a more wettable area and avoiding overlap wit te less wettable areas. Terefore, te additional pitcfork bifurcations tat are needed for larger ticknesses to transfer an eigenvalue between te two coarse brances are no longer present. Moreover, te stabilisation of te pinned solution wit respect to coarsening due to volume transfer occurs at muc larger ɛ.yielding only a narrow range of multistability. In summary, we find tat te morpological pase diagram Figure 8 is qualitatively similar as for film tickness >ln, but only sows a narrow band of multistability. 3 Time evolution We ave studied te stationary solutions and teir bifurcation structure using te strengt of te eterogeneity as control parameter. From tis analysis we predict parameter ranges for pinning, coarsening and multistability (Fig. 8). In order to test tese predictions and to illustrate te occurring processes we study ere te evolution in for some selected parameter values by numerically integrating equation (8) for different initial conditions. Te details of te used numerical procedure are explained in Appendix A. As a first example, Figure sows te long- evolution for a omogeneous system in comparison wit a eterogeneous system were coarsening can still occur, i.e. wit a very small eterogeneity. We sow tese evolutions under te influence of a very small additive noise to take into account te small perturbations of te systems tat can nearly not be excluded on te long scale of te coarsening process. However, we do not intent ere to study systematically te influence of noise and will only note its effects in passing. Details wit respect to te used noise are given in te last section of Appendix A. In Figure te energy equation (9) is plotted in dependence of for different realizations of te noise. Te solid and dased lines indicate te respective evolution witout and wit eterogeneity. Te tin lines indicate te evolution witout noise. We note tat te coarsening starts muc earlier for te omogeneous system tan in te eterogeneous case, corresponding to te result of te linear stability analysis (Fig. 6 (a)). However, te actual coarsening seems to advance faster for te sligtly eterogeneous system.

11 U. Tiele et al.: Tin-film dewetting on structural substrates 6 (a) ε=., η= η (b) 8 x 6 ε=., η= a 6 8 η x 6 3 Fig.. Space- plots sowing te profiles for (a) omogeneous substrate (ɛ =.) and (b) sligtly eterogeneous substrate (ɛ =.). Te remaining parameters are =., P et =, G =.anda=.. Te initial condition is a flat film wit a small dent in te center. A small noise of amplitude η =. is added. Te coordinate x is plotted in units of P et Fig.. nergy for te long- evolution for te omogeneous substrate (solid lines) and a sligtly eterogeneous substrate (dased lines, ɛ =.). Te tin lines represent te respective evolutions witout noise. Te different eavy lines stand for evolutions wit different realizations of noise wit η =.. Te remaining parameters are as in Figure. -.6 b 6 8 Fig. 3. nergy for te long- evolution under te influence of noise of different strengt (as given in te legends) for (a) a eterogeneous substrate wit ɛ =. and (b) a omogeneous substrate. Te remaining parameters are =., P et =, G=.anda=.. Furtermore, for te eterogeneous system te small noise makes no difference wen compared to te evolution witout noise, wereas for te omogeneous system te evolution seems to be sligtly slower witout noise. Tis may indicate tat te coarsening wit eterogeneity is determined by te eterogeneity itself. In Figure it can be clearly seen tat te individual coarsening processes are initiated by mass transfer between te individual drops as was indicated by te linear stability analysis (Fig. 6 (c)). However, it is interesting to note tat te local processes may involve two drops (te two local processes on te rigt side of Figure (a)) or tree drops (te tree local processes on te left side of Fig. (a)). For a larger eterogeneity te pinned solution is linearly stable. Terefore, no coarsening occurs witout te presence of additional influences. However, te system is in te multistable range, i.e. tere exist also coarse solutions tat are linearly stable and for a modest eterogeneity even ave a lower energy tan te pinned solution. Here we use two metods to illustrate te multistability. One way is to add noise to te system as already done above. Figure 3 (a) sows te energy in te course of te long evolution of a system wit a eterogeneity already deep in te linearly stable range for te pinned solution

12 66 Te uropean Pysical Journal (a) (b) (c) (d) ε=., η=. 8 x 6 ε=., η=. 8 x 6 ε=., η= x 6 ε=., η= x Fig.. Space- plots illustrating te long- evolution for different eterogeneity and noise strengts corresponding to selected curves of Figure 3. (a) ɛ =.,η =., (b) ɛ =.,η =., (c) ɛ =.,η =., and (d) ɛ =.,η =.. Te coordinate x is plotted in units of P et. (ɛ =.). One can see tat a noise of a certain strengt can bring te system from te local energy minimum of te pinned solution to te global minimum, or at least a lower minimum of some coarse solution. However, te process depends strongly on te noise. For comparison, we sow in Figure 3 (b) te evolution of te omogeneous system for different noise strengts. Tere te noise sligtly accelerates te coarsening, but te strengt of te noise as little influence. Space- plots are sown for bot cases in Figure. Note tat in te omogeneous case wit very strong noise (Fig. (b)) te local processes ave now a clearly visible contribution from te translational mode. Anoter noteworty effect occurs in te eterogeneous case wit strong noise (Fig. (d)). Te coarsening does not proceed troug local processes but rater troug a global mode, i.e. material is transfered from one alf of te system to te oter. An indication for tis process gives already te bifurcation analysis for te system of size L =P et (Fig. 9), were it was sown tat for ɛ not too large te solution wit P = L as a lower energy tan te solution of period P =P et. By adding noise we studied te consequence of te multistability for te long- evolution. Multistability also affects te sort- evolution as can be seen using different initial conditions. Specifically, we start in a system of size P =P et from a flat film wit a sinusoidal modulation of amplitude d and period P et to give a bias towards te coarse solution tat spans two eterogeneity periods. In te omogeneous system even te very small d =. induces a skipping of te solution wit period P = P et in te evolution: te system goes directly to te P =P et solution as can be seen in Figure (a), were te evolution of te energy is plotted for different amplitudes d. However, in te furter course of te evolution te system will take anoter coarsening step to te P =P et solution. In te eterogeneous system a stronger bias is necessary to favour te coarse solution. So in te multistable sub-range were te coarse solution as te lower energy (ɛ <.for te used parameters) at ɛ =.an amplitude d. is sufficient wereas at ɛ =.(Fig. (b)) already d.7 is needed to go directly to te coarse solution. In te multistable sub-range were te pinned solution as lower energy (ɛ >. for te used parameters) it is still possible to obtain te coarse solution in te sort- evolution. However, a relatively strong bias is needed as sown in Figure (c) for ɛ =.3, were te necessary bias is d.37, i.e. nearly % of te mean film tickness. For ɛ>., te coarse solution does not exist any more (cf. Fig. 6 (b)) and terefore even a very strong bias does not impose te P =P et solution. In a small intermediate range of d, te sort- evolution is attracted towards an unstable tree-drop solution wit te period P =P et, as indicated by te tick dotted lines in Figure (b) and (c). On a sligtly longer scale (t ), te evolving profile is repelled from te unstable solution and approaces a linearly stable solution (corresponding to te second lowest orizontal line in Fig. (c) and to te lowest branc in Fig. 9 at ɛ =.3).

13 U. Tiele et al.: Tin-film dewetting on structural substrates d a d b < >.67 d < > Fig.. nergy for te sort- evolution of asystem of size L =P et for (a) omogeneous substrate and eterogeneous substrates wit (b) ɛ =.and(c)ɛ=.3. Te initial condition is a flat film wit a sinusoidal modulation of period P et and amplitude d (as given in te legends). Te remaining parameters are =., P et =, G =. a nd a =.. In panels (b) and (c) solid lines approac te pinned solution (P = P et) and dased lines te coarse solution wit P =P et, i.e. only two drops remain in te system. Te dotted lines approac unstable solutions wit tree drops. Tese solutions represent saddles in function space tat first attract and subsequently repel te evolution. Tis implies tat evolving solutions can come arbitrarily close to tem during a transient but will finally approac linearly stable solutions. Horizontal dotted (dased) lines indicate te energy values of unstable (stable) periodic solutions as obtained from te bifurcation analysis in Section. corresponding to Figure 6 and Figure 9. We found tat in general te unstable solutions, i.e., te saddle points in function space, may play a crucial role in te evolution. If an evolving profile is attracted by one of tese solutions it may remain a relatively long close to it. Furtermore, te unstable solutions act as nucleation solutions between linearly stable solutions, i.e., tey define te energy barrier between tem. So represent te uppermost orizontal lines in Figure (b) and (c) te respective barriers between te pinned and te coarsed solutions and teir special role can clearly be seen. Screening te initial conditions wit a finer grid reveals more sub-ranges were (different) unstable solutions become important. Transversal instability c So far we investigated two-dimensional drops, i.e. film tickness profiles tat depend on only one spatial dimension. Tis is equivalent to a study of te longitudinal dynamics (in x-direction) of liquid ridges tat are translationally invariant in te transversal direction y. Now we sall address te open question weter tese ridges are stable wit respect to perturbations in te transversal direction. Using linear stability analysis we compute te spectrum of eigenvalues (growt rates) β i of small perturbations δ i (x)exp(β i t)(exp(iky) c.c.) for a large set of transversal wave numbers k. Te focus lies on two aspects: i) ow is te transversal instability influenced by te eterogeneity and ii) wat is te resulting tree-dimensional morpology. First, we consider te omogeneous substrate. Tere an individual liquid ridge is always transversally unstable wit respect to a varicose mode tat emerges from te zero-eigenvalue at k = representing mass conservation. It is reminiscent of te Rayleig instability known from liquid jets [69]. Figure 6 sows te cross-section of suc a ridge in (a), te dispersion relations in (b), and te relevant eigenmodes i (x) in (c). Te second important eigenmode is a zigzag mode emerging from te zero-eigenvalue at k = representing longitudinal translational invariance of te ridge. It is stable on te orizontal substrate, but becomes important for inclined substrates. For details on te omogeneous system see [], were te transition towards an inclined substrate is discussed. Having in mind te sinusoidal form of te eterogeneity one cannot decouple individual ridges because teir widt is of te order of teir distance. Te interaction of te ridges influences also te transversal stability of an array of ridges. Figure 7 sows te part of te eigenvalue spectrum wit largest real part (black solid and dased lines) for te transversal instabilities of a system of two neigbouring ridges. Te corresponding eigenmodes and instability morpologies are depicted in Figure 8 wit corresponding line styles. Te varicose instability (solid lines) can take two forms: te instabilities of te two neigbour ridges may be transversally in-pase or anti-pase (Fig. 8 (a) and (b), respectively). Te in-pase combination is a periodic continuation of te varicose instability of an individual ridge and as te same dispersion relation. Te growt rate goes to zero as te wave number approaces zero and te eigenmode at k = corresponds to te volume neutral mode. However, tis is not te case for te anti-pase mode. For k =, it corresponds to te longitudinal coarsening mode by mass transfer between te two ridges as discussed in Section.. Terefore for k te growt rate does not approac zero but its maximal value (Fig. 7). For te zigzag mode (dased lines, Figure 8 (c) and (d)), one finds tat te in-pase mode is stable as for an individual ridge, but te anti-pase mode as a small band of unstable wave numbers around k =. It approaces for k its largest growt rate corresponding to te one of te coarsening mode by translation. For te sown parameters te longitudinal coarsening by mass transfer is te dominant mode. Combinations of varicose and zigzag modes correspond to linear combinations of te already discussed modes. Studying larger arrays of more tan two ridges allows for new combinations of in-pase