Derivation, Analysis and Numerics of Reduced ODE Models Describing Coarsening Dynamics of Liquid Droplets

In this dissertation we consider the topic of derivation, analysis and numerics of reduced ODE models corresponding to a family of one-dimensional lubrication equations derived in Münch et al. [1]. This family describes the dewetting process of nanoscopic thin liquid films on hydrophobized polymer substrates due to the presence of the long-range attractive van der Waals and short-range Born repulsive intermolecular forces and takes account of all possible ranges of slip-lengths at the polymer substrate interface. The final stages of the dewetting process are characterized by a slow-time coarsening dynamics of the remaining droplets that are separated and interact with each other through a nanoscopic ultra thin liquid layer of thickness ε. Reduced ODE models derived from underlying lubrication equations allow for an efficient analytical and numerical investigation of the coarsening process. One of our main interests in this study is to investigate the influence of slip-length on the coarsening dynamics using the derived reduced ODE models. In the first part of this study using asymptotic methods we derive reduced ODE models for lubrication equations that describe the evolution without slippage (no-slip equation) small slip-lengths (weak-slip equation), intermediate slip-lengths (intermediate slip equation) and large slip-lengths (strong-slip equation). By that we generalize the results of Glasner and Witelski [2], where a reduced ODE model for the no-slip lubrication equation was derived. The resulting reduced ODE model describes the evolution in time for a set of pressures and positions for an array of droplets. We find that the difference between the reduced models for the no-, intermediateand weak-slip equations, and the one for the strong-slip equation lies in their dependence on the slip-length. In the strong-slip case we find a unique critical slip-length, which decides the direction of migration of droplets. If the slip-length is smaller than this critical value the droplet migrates opposite to the direction of the applied effective flux. If the slip-length is bigger than the critical value the droplet migrates in the direction of the flux. This result is new and establishes an interesting property especially in the light of a recent work of Glasner et al. [3], where it was established that migration of droplets is opposite to the applied effective flux in the noand intermediate-slip cases. Next, we numerically solve the system of reduced ODE models and find a good agreement of their results with those given by numerical solutions of the corresponding lubrication equations. The second part of this study is devoted to a new method for derivation and justification of reduced ODE models based on a center-manifold reduction approach recently applied by Mielke and Zelik [4] to a certain class of semilinear parabolic equations. We first give an alternative derivation of the reduced ODE model for the no-slip case using a formal reduction onto an ’approximate invariant’ manifold parameterized by a set of pressures and positions of droplets in an array. Then we find a good agreement of the new reduced ODE model with the previously asymptotically derived one. One of the main problems for the rigorous justification of this formal approach is the description of the asymptotics for the spectrum of the no-slip lubrication equation linearized at the stationary solution, which corresponds physically to a single droplet, with respect to the small parameter ε tending to zero. We find that the corresponding eigenvalue problem (EVP) turns out to be a singularly perturbed one. For its spectrum we show rigorously the existence of an ε-dependent spectral gap, which may happen to be an important property for the rigorous justification of our formal reduction approach in future. Besides, using a modified implicit function theorem first suggested by Recke and Omel’chenko [5] we show the existence of eigenvalues with prescribed asymptotics, in particular of an exponentially small one, for the above linearized singularly perturbed EVP. Here our results offer a new technique for solving of a certain type of singularly perturbed EVPs.