Thin film evolution equations from (evaporating) dewetting liquid layers to epitaxial growth

In the present contribution we review basic mathematical results for three physical systems involving self-organising solid or liquid films at solid surfaces. The films may undergo a structuring process by dewetting, evaporation/condensation or epitaxial growth, respectively. We highlight similarities and differences of the three systems based on the observation that in certain limits all of them may be described using models of similar form, i.e., time evolution equations for the film thickness profile. Those equations represent gradient dynamics characterized by mobility functions and an underlying energy functional. Two basic steps of mathematical analysis are used to compare the different system. First, we discuss the linear stability of homogeneous steady states, i.e., flat films; and second the systematics of non-trivial steady states, i.e., drop/hole states for dewetting films and quantum dot states in epitaxial growth, respectively. Our aim is to illustrate that the underlying solution structure might be very complex as in the case of epitaxial growth but can be better understood when comparing to the much simpler results for the dewetting liquid film. We furthermore show that the numerical continuation techniques employed can shed some light on this structure in a more convenient way than time-stepping methods. Finally we discuss that the usage of the employed general formulation does not only relate seemingly not related physical systems mathematically, but does as well allow to discuss model extensions in a more unified way

Liquid transport generated by a flashing field-induced wettability ratchet

We develop and analyze a model for ratchet-driven macroscopic transport of a continuous phase. The transport relies on a field-induced dewetting-spreading cycle of a liquid film with a free surface based on a switchable, spatially asymmetric, periodic interaction of the liquid-gas interface and the substrate. The concept is exemplified using an evolution equation for a dielectric liquid film under an inhomogeneous voltage. We analyse the influence of the various phases of the ratchet cycle on the transport properties. Conditions for maximal transport and the efficiency of transport under load are discussed.Comment: 10 pages, 5 figure

Depinning of three-dimensional drops from wettability defects

Substrate defects crucially influence the onset of sliding drop motion under lateral driving. A finite force is necessary to overcome the pinning influence even of microscale heterogeneities. The depinning dynamics of three-dimensional drops is studied for hydrophilic and hydrophobic wettability defects using a long-wave evolution equation for the film thickness profile. It is found that the nature of the depinning transition explains the experimentally observed stick-slip motion.Comment: 6 pages, 9 figures, submitted to ep

Modelling the evaporation of thin films of colloidal suspensions using Dynamical Density Functional Theory

Recent experiments have shown that various structures may be formed during the evaporative dewetting of thin films of colloidal suspensions. Nano-particle deposits of strongly branched `flower-like', labyrinthine and network structures are observed. They are caused by the different transport processes and the rich phase behaviour of the system. We develop a model for the system, based on a dynamical density functional theory, which reproduces these structures. The model is employed to determine the influences of the solvent evaporation and of the diffusion of the colloidal particles and of the liquid over the surface. Finally, we investigate the conditions needed for `liquid-particle' phase separation to occur and discuss its effect on the self-organised nano-structures

Experimental observation of nanoscale radiative heat flow due to surface plasmons in graphene and doped silicon

Owing to its two dimensional electronic structure, graphene exhibits many unique properties. One of them is a wave vector and temperature dependent plasmon in the infrared range. Theory predicts that due to these plasmons, graphene can be used as a universal material to enhance nanoscale radiative heat exchange for any dielectric substrate. Here we report on radiative heat transfer experiments between SiC and a SiO2 sphere which have non matching phonon polariton frequencies, and thus only weakly exchange heat in near field. We observed that the heat flux contribution of graphene epitaxially grown on SiC dominates at short distances. The influence of plasmons on radiative heat transfer is further supported with measurements for doped silicon. These results highlight graphenes strong potential in photonic nearfield and energy conversion devices.Comment: 4 pages, 3 figure

Pointwise convergence of vector-valued Fourier series

We prove a vector-valued version of Carleson's theorem: Let Y=[X,H]_t be a complex interpolation space between a UMD space X and a Hilbert space H. For p\in(1,\infty) and f\in L^p(T;Y), the partial sums of the Fourier series of f converge to f pointwise almost everywhere. Apparently, all known examples of UMD spaces are of this intermediate form Y=[X,H]_t. In particular, we answer affirmatively a question of Rubio de Francia on the pointwise convergence of Fourier series of Schatten class valued functions.Comment: 26 page

Dewetting of thin films on heterogeneous substrates: Pinning vs. coarsening

We study a model for a thin liquid film dewetting from a periodic heterogeneous substrate (template). The amplitude and periodicity of a striped template heterogeneity necessary to obtain a stable periodic stripe pattern, i.e. pinning, are computed. This requires a stabilization of the longitudinal and transversal modes driving the typical coarsening dynamics during dewetting of a thin film on a homogeneous substrate. If the heterogeneity has a larger spatial period than the critical dewetting mode, weak heterogeneities are sufficient for pinning. A large region of coexistence between coarsening dynamics and pinning is found.Comment: 4 pages, 4 figure

Solidification fronts in supercooled liquids: how rapid fronts can lead to disordered glassy solids

We determine the speed of a crystallisation (or more generally, a solidification) front as it advances into the uniform liquid phase after the system has been quenched into the crystalline region of the phase diagram. We calculate the front speed by assuming a dynamical density functional theory model for the system and applying a marginal stability criterion. Our results also apply to phase field crystal (PFC) models of solidification. As the solidification front advances into the unstable liquid phase, the density profile behind the advancing front develops density modulations and the wavelength of these modulations is a dynamically chosen quantity. For shallow quenches, the selected wavelength is precisely that of the crystalline phase and so well-ordered crystalline states are formed. However, when the system is deeply quenched, we find that this wavelength can be quite different from that of the crystal, so that the solidification front naturally generates disorder in the system. Significant rearrangement and ageing must subsequently occur for the system to form the regular well-ordered crystal that corresponds to the free energy minimum. Additional disorder is introduced whenever a front develops from random initial conditions. We illustrate these findings with results obtained from the PFC.Comment: 14 pages, 7 figure

Dynamical density functional theory for the dewetting of evaporating thin films of nanoparticle suspensions exhibiting pattern formation

Recent experiments have shown that the striking structure formation in dewetting films of evaporating colloidal nanoparticle suspensions occurs in an ultrathin `postcursor' layer that is left behind by a mesoscopic dewetting front. Various phase change and transport processes occur in the postcursor layer, that may lead to nanoparticle deposits in the form of labyrinthine, network or strongly branched `finger' structures. We develop a versatile dynamical density functional theory to model this system which captures all these structures and may be employed to investigate the influence of evaporation/condensation, nanoparticle transport and solute transport in a differentiated way. We highlight, in particular, the influence of the subtle interplay of decomposition in the layer and contact line motion on the observed particle-induced transverse instability of the dewetting front.Comment: 5 pages, 5 figure