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Collaborator: B. Wagner
Cooperation with: A. Münch (HumboldtUniversität zu Berlin, Heisenberg Fellow at WIAS), T.P. Witelski (Duke University, Durham, USA), J. King (University of Nottingham, UK), R. Konrad, K. Jacobs (Universität des Saarlandes, Saarbrücken), M. Rauscher (MaxPlanckInstitut für Metallforschung, Stuttgart), R. Blossey (Interdisciplinary Research Institute, IEMN, Lille, France)
Description:
The occurrence and the nature of slippage of liquids on solid surfaces is vividly discussed in the literature and is of large technological interest since a sliding fluid can flow faster through, e.g., microfluidic devices. In recent years, one major focus of interest has been the process of dewetting of polymer films (typically ranging on the scale of tens to a few hundred nanometers) from hydrophobic substrates. Understanding the dynamics and morphology of this process is particularly important for the design of microelectronic devices. Theoretical progress here depends strongly on the ability to develop, analyze, and solve reduced models, such as lubricationtype models for the film dynamics, thereby addressing the presence of highly separated scales (in space as well as in time) and the high (fourth) order of the involved partial differential equations.
Dewetting typically starts by the formation of holes due to spinodal decomposition or heterogeneous nucleation. As the holes grow, the displaced liquid collects in growing rims surrounding the holes. On longer time scales, the rims themselves are subject to a fingertype instability. In this project, we have derived lubrication models that account for large slip lengths. The slip length can be understood as the length below the solid/liquid interface where the velocity extrapolates to zero. We have developed numerical methods that explore the special structure of the models to yield, in combination with spatial adaptivity and parallelization, efficient codes for the dewetting process. Using linear stability analysis, we showed that, both under noslip and fullslip boundary conditions, perturbations of the dewetting front are amplified, but the effect is greater by orders of magnitude in the fullslip case. Furthermore, the perturbations become much more asymmetrical under fullslip boundary conditions, while they develop symmetrical bulges under noslip conditions, [3]. Additional computations that solve the lubrication model for the full threedimensional flow confirm that these findings carry over to the nonlinear regime [2] and are in good agreement with the experimental findings, [8]. Our recent focus has been the development of multiplescale asymptotic techniques in order to formulate simpler problems that are able to resolve the smallscale structure in the vicinity of the apparent contact line and asymptotically match the inner solution to those solving the largescale outer problem describing the shape and dynamics of the rim. As a consequence, a further model reduction of the lubrication model to a corresponding sharpinterface model could be achieved. Here, the linear stability analysis is greatly simplified and could be investigated purely analytically, [6], [5], confirming our earlier numerical results on the linear stability of the evolving rims.
In this project, we derived lubrication models for dewetting thin films from the NavierStokes equations for incompressible flow, which are characterized by the orders of magnitude of the slip length. As one traverses the ranges of the slip length, the flow field will change from a parabolic flow field to what is essentially plug flow, which implies a change from the balance of the pressure gradient with the horizontal velocity to the balance of the pressure gradient with the vertical velocity and thereby also a change of the velocity scale. For small slip lengths (at most of the order of the height of the film), we obtained the velocity scale = / together with the wellknown dimensionreduced lubrication equation for the profile h(x, t)
where , , , denote surface tension, viscosity, capillary number, and interfacial potential, respectively. If, however, the slip length tenses much larger than the height of the film, we obtain a new lubrication model, with the velocity scale = / and the systemRe^{*}f + ff = + h h  h  ,h =  hf,  (2) 
As a first application we found that the strong slip regime could capture the rim profiles of dewetting nanoscale polymer films observed in experiments by R. Konrad and K. Jacobs.
In this project, we concentrated on the rim morphology affected by slippage. Our collaborators R. Konrad and K. Jacobs have performed experiments to compare the dewetting behavior of liquid polymer films on silicon/silicon oxide wafers that have been coated with either octadecyltrichlorosilane (OTS) or dodecyltrichlorosilane (DTS). The experiments show that the dewetting rates for DTS are significantly larger than for OTS. They also compared the profile of the rim that forms as the film dewets and found that it develops a spatially decaying oscillatory structure on the side facing the undisturbed film if an OTScoated wafer is used, but is monotonically decaying for DTS. For this situation, only the solid/liquid friction coefficient can be different, suggesting that slippage plays a role in this transition. For the first time, we showed that this transition is in fact captured by a lubrication model that can be derived from the NavierStokes equations with a Navierslip boundary condition at the liquid/solid interface, and accounts for large slip lengths; it is the strong slip model (2) introduced above.
For this model, an approximate description of the portion of the profile of the dewetting rim that connects to the undistributed uniform film h = 1 for x was developed to find the evolution of the film near the flat state. The dominant contributions could be derived to obey the ODE  Re = 4 +  for the perturbation about the flat state. Here, = x  s(t) and s(t) denotes the position of the contact line. The normal mode solutions () = e^{} of this ODE, which decay for , are spatially oscillating due to a pair of complex conjugate , if the discriminant corresponding to the dispersion relation for ,
is positive. Hence, the rim passes over into a damped capillary wave. But if it is negative, the complex conjugate is replaced by two real modes, which allows the solution () to decay monotonically for . In our case, the Re number is extremely small and can be neglected, so that the transition from complex conjugate to real decaying modes occurs when > () = (3/4)/(4)^{1/3}. This threshold is shown in Figure 1. We also found the transition from complex conjugate to real decaying modes can also be found from the full Stokes equation. In Figure 1, we also show the corresponding threshold that was numerically obtained from the normal mode solution of the linearized Stokes problem.
One observes that as b is increased, the dewetting rate increases, and for slip lengths larger than the threshold, the dashed line in Figure 1 leaves the region where capillary waves are expected. Inspection of the profile in a semilog plot of the profiles in Figure 2 showed a second maximum for the smallest b, indicating that we have the oscillatory structure of a wave.
These findings compared very well with the experimental results where on OTScovered substrates, the rim of the dewetting PS film exhibits an oscillatory shape, whereas on DTScovered surfaces, at the same temperature, no oscillation is observed, [7].
In the range where the polymer chain length begins to be comparable with the film thickness, the dynamics of the thin films becomes nonNewtonian, leading to novel dynamical signatures. In order to describe such effects, various different models have been postulated in an ad hoc or, at best, a phenomenological manner, leading to controversies about the interpretation of experimental results.
Given the success of the lubrication approximation for the dynamics of thin films of Newtonian character, we were prompted to look at this issue for the case of nonNewtonian liquids and addressed the question of the derivation of thinfilm equations based on the lubrication approximation for viscoelastic fluids, [10]. We introduced a general model class of viscoelastic fluids which was the basis for our derivation. Subsequently, we demonstrated how to derive thinfilm equations for the nonNewtonian liquid, stressing the underlying physical assumptions. For the situations we have in mind, we assumed the liquid to be incompressible with mass density . With this, the equation of mass conservation reduces to ^{ . } = 0, with the velocity field = (u_{x}, u_{y}, u_{z}). The equation of momentum conservation is =  p_{R} + ^{ . }, with the reduced pressure p_{R} = p + V. The hydrostatic pressure is p, the pressure induced by external fields such as gravity or van der Waalstype dispersion forces V, and the deviatoric (traceless) part of the stress tensor is (which is symmetric). With = + ^{ . }, we denote the materials (or total) derivative and with = (,,) the gradient operator.
Starting with the corotational Jeffreys model, we found that in order to obtain a closedform lubrication model, only the linear versions
with i = x, y, respectively, are in general allowed. For this we found for the lubrication approximation the following systemWe presently discuss how the models we obtained relate to the phenomenological models discussed in the literature so far.
References:



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