Continuous and discontinuous dynamic unbinding transitions in drawn film flow
When a plate is withdrawn from a liquid bath a coating layer is deposited whose thickness and homogeneity depend on the velocity and the wetting properties of the plate. Using a long-wave mesoscopic hydrodynamic description that incorporates wettability via a Derjaguin (disjoining) pressure we identify four qualitatively different dynamic transitions between microscopic and macroscopic coatings that are out-of-equilibrium equivalents of well known equilibrium unbinding transitions. Namely, these are continuous and discontinuous dynamic emptying transitions and discontinuous and continuous dynamic wetting transitions. We uncover several features that have no equivalent at equilibrium.
pacs:68.15.+e, 47.20.Ky 47.55.Dz 68.08.-p
The equilibrium and non-equilibrium behaviour of mesoscopic and macroscopic drops, meniscii and films of liquid in contact with static or moving solid substrates is not only of fundamental interest but also crucial for a large number of modern technologies. On the one hand, the equilibrium behaviour of films, drops and meniscii is studied by means of statistical physics. A rich substrate-induced phase transition behaviour is described even for simple liquids, e.g., related to wetting and emptying transitions that both represent unbinding transitions. In the former case the thickness of an adsorption layer of liquid diverges continuously or discontinuously at a critical temperature or strength of substrate-liquid interaction, i.e., the liquid-gas interface of the film unbinds from the liquid-solid interface . In the case of the emptying transition a macroscopic meniscus in a tilted slit capillary develops a tongue (or foot) along the lower wall of a length that diverges logarithmically at a critical slit width, i.e., the tip of the foot unbinds from the meniscus .
On the other hand, it is a classical hydrodynamic problem to study how droplets slide down an incline ; ; ; , how moving contact lines (where solid, gas and liquid meet) develop sawtooth shapes at high speeds ; ; , or how the free surface of a bath is deformed when a plate is drawn out, as sketched in Fig. 1(a). Early on it was reported that for sufficiently large plate velocities a homogeneous macroscopic liquid layer is deposited on the drawn plate [Fig. 1(d)]. The resulting coating layer is called a Landau-Levich film. Far away from the bath it has a thickness that depends on the capillary number through the power law  where and are the viscosity and surface tension of the liquid, respectively. This coating technique is widely used and became a paradigm for theoretical (e.g., ; ; ; ; ) and experimental (e.g., ; ; ; ; ) studies.
In contrast, at very low plate velocities no macroscopic film is drawn out but a deformed steady meniscus coexists with the dry plate far away from the bath ; ; ; ;  [Fig. 1(b)]. This meniscus only exists for capillary numbers smaller than a critical one, i.e., . Close to the meniscus develops a foot of a length [Fig. 1(c)] that diverges at either continously  or discontinuously . As the steady free surface meniscus coexists with the dry moving plate, there exists a receding three-phase contact line whose best description is still debated (see, e.g., ; ; ).
Previous works ; ;  employ a slip model that allows the film height to go to zero at the contact line and avoids the contact line singularity through the slip . Although, a slip model allows for a quantitative study of meniscus solutions and Landau-Levich films, it is not able to describe transitions between them as in a slip model they are topologically different [cf. Figs. 1(b) and (c) vs. Fig. 1(d)]. Note that this concerns the actual transition dynamics as well as the description of transitions in dependence of control parameters such as the plate speed.
In contrast, here we employ a long-wave mesoscopic hydrodynamic model that describes wettability via a Derjaguin (disjoining) pressure, i.e., a precursor film model. An investigation of the non-equilibrium transitions between meniscus solutions and Landau-Levich films then allows for an identification of four qualitatively different dynamic unbinding transitions, namely, continuous and discontinuous dynamic emptying transitions and discontinuous and continuous dynamic wetting transitions. Note that far from the transition regions, the predictions of precursor and slip models agree very well and can be quantitatively mapped .
that may be derived as long–wave approximation of the Navier-Stokes and continuity equations with no-slip boundary conditions at the liquid-solid interface and kinematic and stress balance conditions at the liquid-gas interface . Here , and are the scaled plate velocity (Capillary number), gravity (Bond number), and inclination angle of the plate, respectively . The wettability of the partially wetting liquid is described via the Derjaguin (or disjoining) pressure 
It should be noted that the hydrodynamic long-wave model, Eq. (1) with Eq. (2), directly corresponds to a gradient dynamics of an underlying interface Hamiltonian (or free energy) as often used to study the above introduced equilibrium unbinding transitions . This equivalence allows for a natural understanding of the various occuring transitions as non-equilibrium (or dynamic) unbinding transitions (see below).
To calculate steady film and meniscus profiles one sets , then integrates Eq. (1) once and solves the resulting three-dimensional dynamical system in with appropriate boundary conditions: (i) far from the bath for one imposes that the film profile approaches a flat film of unknown height that is determined as part of the solution while (ii) the approach towards the bath for is described by an asymptotic series that can be rigorously derived via a center manifold reduction . Steady profiles and bifurcation diagrams are numerically obtained employing pseudo-arclength continuation . The employed main solution measure is the dynamic excess volume with , where is at . Note that for solutions with a long protruding foot-like structure is approximately proportional to the length of the foot .
An analysis of the changes that steady meniscii undergo with increasing plate speed shows that four qualitatively different cases exist depending on the plate inclination angle . Each case is related to a distinguished nonequilibrium unbinding transition as illustrated in Fig. 2 that shows for all four cases typical bifurcation diagrams in dependence on and steady height profiles for selected values of :
(a) At small , the volume monotonically increases: first slowly, then faster until it diverges at about [Fig. 2(a)]. The corresponding simple meniscus profiles first deform only slightly due to viscous bending before a distinguished foot-like protrusion of a height develops whose length diverges . This corresponds to a continuous dynamic emptying transition, a dynamic analogue of the equilibrium transition discussed in Ref. . One may also say that at the tip of the foot unbinds from the bath. For the foot advances with a constant velocity as shown in Fig. 3.
(b) Above a first critical , the transition changes its character and becomes a discontinuous dynamic emptying transition that has no analogue at equilibrium. As shown in Fig. 2(b), increases first monotonically with until a saddle-node bifurcation is reached at where the curve folds back. Following the curve further, one finds that it folds again at . This back and forth folding infinitely continues at loci that exponentially approach from both sides and that separate linearly stable and unstable parts of the solution branch. This exponential (or collapsed) snaking  results in foot length with where Re and Im determine the exponential approach and the period of the snaking, respectively. Note that for one can always find a critical foot length beyond which the foot advances with a constant velocity . In contrast, for there is always a critical length above which a foot recedes. The two cases are illustrated in Fig. 3 for and , respectively. In both cases, (a) and (b), one finds that the foot height . The limiting velocity coincides with the velocity of a large flat drop (pancake-like drop) sliding down a resting plate of inclination . This allows one to calculate by continuation (see Fig. 5 below). Note that the found relation for the front velocity [Fig. 3(a)] is a direct consequence of the Galilean invariance of the motion of a drop down an incline.
(c) At a second critical , the bifurcation diagram dramatically changes. Above the family of steady meniscii that is connected to does not anymore diverge at a limiting velocity . Instead of a protruding foot of increasing length that unbinds from the meniscus one finds a hysteretic transition [in Fig. 2(c) between and ] towards a coating layer whose thickness homogeneously increases with increasing , i.e., the layer surface unbinds from the substrate in an discontinuous dynamic wetting transition.
(d) With increasing the hysteresis of the discontinuous transition becomes smaller until at a third critical the two saddle-node bifurcations annihilate in a hysteresis bifurcation as further illustrated in Fig. 4. For all one finds a continuous dynamic wetting transition. As in both cases - (c) and (d) - at large the coating layer thickness follows the power law , we identify these unbinding states as Landau-Levich films . The critical velocity where the transition between the microscopic and macroscopic layer occurs, scales as .
Note that the dynamic emptying transitions of cases (a) and (b) and the crossover between them is also observed employing a slip model ; . However, as normally slip models do not take account of the mesoscale wetting behaviour they are unable to describe the discontinuous and continuous dynamic wetting transitions of cases (c) and (d), respectively, as these represent transitions between the topologically different meniscus and film solutions.
To summarize our findings we present in Fig. 5 a phase diagram in the plane spanned by the plate velocity and inclination angle as obtained by tracking the main occuring saddle-node bifurcations (black solid lines) and the limiting velocity (blue dotted line). In region (a), i.e., for , ultimately a simple or extended (foot-like) steady meniscus is found for while for the foot advances at constant speed [cf. Fig. 3(a)]. In region (b), i.e., for , multiple stable foot solutions exist for between the two solid lines. However, for each with there is always a maximal stable foot length towards which a longer foot will retract. For each with there is always a maximal unstable steady foot length beyond which the foot will prolongate continuously.  logarithmically diverges as approaches from below [above].
In region (c), i.e., for , the lines of saddle-node bifurcations limit a region where initial conditions decide whether an ultrathin layer or a macroscopic Landau-Levich coating is obtained. Below [above] the hysteresis range one only finds the ultrathin [Landau-Levich] coating. In region (d), i.e., for , the change between the two coating types is continuous.
Although an extended analysis of the characteristics of the observed qualitative changes with increasing inclination angle is beyond our present scope, we highlight some further important facts. The crossover between regions (a) and (b) at can be understood in terms of a change of the character of the spatial eigenvalues (EV) of a flat film of a height that corresponds to the foot height ; : In region (a) all EV are real while in region (b) only one is real and the other two are a pair of complex conjugate EV. The crossover between regions (c) and (d) at results from a hysteresis bifurcation where two saddle-node bifurcations annihilate. However, the crossover between regions (b) and (c) at that results in the strongest qualitative change, namely, from a dynamic emptying to a dynamic wetting transition cannot be understood by analysing a single family of steady profiles. As illustrated in Fig. 6 the crossover results from a reconnection (reverse necking bifurcation) at that involves two solution families. Both continue to exist on both sides of . This results in intricate behaviour in certain small bands of the (, ) plane and, in particular, around that will be studied in more depth elsewhere. For instance, in the fine grey band around in region (c) [Fig. 5], there exist various stable extended meniscus profiles. They correspond to the left branch in Fig. 6(b). Experimentally, they might only be obtained through a careful control of the set-up at specific initial conditions.
To conclude, we have shown that a long-wave mesoscopic hydrodynamic description of the coating problem for a plate that is drawn from a bath allows one to identify and analyze several qualitative transitions if wettability is modelled via a Derjaguin pressure, i.e., with a precursor film model. As a result we have distinguished four regions where different dynamic unbinding transitions occur, namely continuous and discontinuous dynamic emptying transitions and discontinuous and continuous dynamic wetting transitions. These dynamic transitions are out-of-equilibrium equivalents of well known equilibrium emptying and wetting transitions. Beside features known from the equilibrium versions, our analysis has uncovered several important features that have no equivalents at equilibrium. A future study of the influence of fluctuations might allow one to answer the question which surface profile is selected in the multistable regions.
We acknowledge support by the EU via the FP7 Marie Curie scheme (ITN MULTIFLOW, PITN-GA-2008-214919).
- D. Bonn, J. Eggers, J. Indekeu, J. Meunier, and E. Rolley. Wetting and spreading. Rev. Mod. Phys., 81:739–805, 2009.
- A. O. Parry, C. Rascon, E. A. G. Jamie, and D. G. A. L. Aarts. Capillary emptying and short-range wetting. Phys. Rev. Lett., 108:246101, 2012.
- T. Podgorski, J.-M. Flesselles, and L. Limat. Corners, cusps, and pearls in running drops. Phys. Rev. Lett., 87:036102, 2001.
- U. Thiele, K. Neuffer, M. Bestehorn, Y. Pomeau, and M. G. Velarde. Sliding drops on an inclined plane. Colloid Surf. A - Physicochem. Eng. Asp., 206:87–104, 2002.
- M. Ben Amar, L. J. Cummings, and Y. Pomeau. Transition of a moving contact line from smooth to angular. Phys. Fluids, 15:2949–2960, 2003.
- JH Snoeijer and B Andreotti. Moving contact lines: Scales, regimes, and dynamical transitions. Annu. Rev. Fluid Mech., 45:269–292, 2013.
- T. D. Blake and K. J. Ruschak. A maximum speed of wetting. Nature, 282:489–491, 1979.
- G. Delon, M. Fermigier, J. H. Snoeijer, and B. Andreotti. Relaxation of a dewetting contact line. part 2. experiments. J. Fluid Mech., 604:55–75, 2008.
- L. Landau and B. Levich. Dragging of a liquid by a moving plane. Acta Physicochimica URSS, 17:42, 1942.
- B. Derjaguin. On the thickness of the liquid film adhering to the walls of a vessel after emptying. Acta Physicochim. USSR, 39:13Ã16, 1943.
- S. D. R. Wilson. The drag-out problem in film coating theory. J. Eng. Math., 16:209–221, 1982.
- A. Münch and P. L. Evans. Marangoni-driven liquid films rising out of a meniscus onto a nearly-horizontal substrate. Physica D, 209:164–177, 2005.
- F. S. Goucher and H. Ward. A problem in viscosity: The thickness of liquid films formed on solid surfaces under dynamic conditions. Philosophical Magazine, 44:1002 –1014, 1922.
- P. Groenveld. High capillary number withdrawal from viscous newtonian liquids by flat plates. Chemical Engineering Science, 25(1):33 – 40, 1970.
- JH Snoeijer, G Delon, M Fermigier, and B Andreotti. Avoided critical behavior in dynamically forced wetting. Phys. Rev. Lett., 96:174504, 2006.
- J. H. Snoeijer, J. Ziegler, B. Andreotti, M. Fermigier, and J. Eggers. Thick films of viscous fluid coating a plate withdrawn from a liquid reservoir. Phys. Rev. Lett., 100:244502, 2008.
- M. Maleki, M. Reyssat, F. Restagno, D. Quere, and C. Clanet. Landau-Levich menisci. J. Colloid Interface Sci., 354:359–363, 2011.
- J. Eggers. Hydrodynamic theory of forced dewetting. Phys. Rev. Lett., 93:094502, 2004.
- J. H. Snoeijer, B. Andreotti, G. Delon, and M. Fermigier. Relaxation of a dewetting contact line. part 1. a full-scale hydrodynamic calculation. J. Fluid Mech., 579:63–83, 2007.
- TS Chan, JH Snoeijer, and J Eggers. Theory of the forced wetting transition. Phys. Fluids, 24:072104, 2012.
- J. Ziegler, J. H. Snoeijer, and J. Eggers. Film transitions of receding contact lines. Eur. Phys. J.-Spec. Top., 166:177–180, 2009.
- P.-G. de Gennes. Wetting: Statics and dynamics. Rev. Mod. Phys., 57:827–863, 1985.
- M. G. Velarde (ed.). Discussion and debate: Wetting and spreading science - quo vadis? Eur. Phys. J.-Spec. Top., 197, 2011.
- N. Savva and S. Kalliadasis. Dynamics of moving contact lines: A comparison between slip and precursor film models. Europhys. Lett., 94:64004, 2011.
- The employed scales are , and for the -coordinate, film height and time, respectively. For the dimensional disjoining pressure , with Hamaker constant and short-range interaction strength , the equilibrium precursor height and contact angle are (i.e., the scaled precursor height is ) and (i.e., the scaled equilibrium contact angle is ), respectively. The velocity scale, gravity number and the scaled inclination angle are given by , , and , respectively, where is the density of the liquid, is gravitational accelleration and is the physical plate inclination angle. We use .
- U. Thiele. Structure formation in thin liquid films. In S. Kalliadasis and U. Thiele, editors, Thin Films of Soft Matter, pages 25–93, Wien, 2007. Springer.
- U. Thiele. Thin film evolution equations from (evaporating) dewetting liquid layers to epitaxial growth. J. Phys.: Condens. Matter, 22:084019, 2010.
- A. Oron, S. H. Davis, and S. G. Bankoff. Long-scale evolution of thin liquid films. Rev. Mod. Phys., 69:931–980, 1997.
- V. M. Starov and M. G. Velarde. Surface forces and wetting phenomena. J. Phys.-Condes. Matter, 21:464121, 2009.
- L. M. Pismen. Mesoscopic hydrodynamics of contact line motion. Colloid Surf. A-Physicochem. Eng. Asp., 206:11–30, 2002.
- Here, is the surface area element in long-wave approximation and is an appropriately defined energy containing terms related to wettability and gravity. Note that in the wetting literature is called ”Hamiltonian” as it may be derived from a microscopic Hamiltonian. However, technically it is a Lagrangian as the steady version of Eq. (1) for horizontal resting substrate (without gravity) corresponds to the Euler-Lagrange equation of under the condition of mass conservation.
- The time simulations employ a second order central finite difference scheme in space and a variable-order and variable-step backward differentiation algorithm in time.
- D. Tseluiko, M. Galvagno, and U. Thiele. Heteroclinic snaking near a heteroclinic chain in dragged meniscus problems. Eur. Phys. J. E, 2013.
- H. A. Dijkstra, F. W. Wubs, A. K. Cliffe, E. Doedel, I. F. Dragomirescu, B. Eckhart, A. Y. Gelfgat, A. Hazel, V. Lucarini, A. G. Salinger, E. T. Phipps, J. Sanchez-Umbria, H. Schuttelaars, L. S. Tuckerman, and U. Thiele. Numerical bifurcation methods and their application to fluid dynamics: Analysis beyond simulation. Commun. Comput. Phys., 15:1–45, 2014.
- We define the measure for the foot length , where is the foot height and the coating film height.
- Y. P. Ma, J. Burke, and E. Knobloch. Defect-mediated snaking: A new growth mechanism for localized structures. Physica D, 239:1867–1883, 2010.