Complex interfacial and wetting dynamics

Consider interface evolution in bounded and unbounded settings, namely in the spreading of droplets and stratified gas-liquid flows. A typical prototype consists of the surface-tension-dominated motion of a two-dimensional droplet on a substrate. The case of chemically heterogeneous substrates was examined here. Assuming small slopes, a single evolution equation for the droplet free surface was derived from the Navier-Stokes equations, with the singularity at the contact line being alleviated using the Navier slip condition. The chemical nature of the substrate is incorporated into the system by local variations in the microscopic contact angle. By using the method of matched asymptotic expansions, the flow in the vicinity of the contact lines is matched to that in the bulk of the droplet to obtain a set of coupled ordinary differential equations for the location of the two contact points. The solutions obtained by asymptotic matching are in excellent agreement with the solutions to the full governing evolution equation. The dynamics of the droplet is examined in detail via a phase-plane analysis. A number of interesting features that are not present in homogeneous substrates are observed: multiple droplet equilibria, pinning of contact points on localised heterogeneities, unidirectional motion of droplet and the possibility of stick-slip behaviour of contact points. Unbounded gas-liquid flows are also often encountered in natural phenomena and applications. The prototypical system considered here consists of a liquid film flowing down an inclined planar substrate in the presence of a co-flowing turbulent gas. The gas and liquid problems are solved independently by making certain reasonable assumptions. The influence of gas flow on the liquid problem is analysed by developing a weighted integral-boundary-layer (WIBL) model, which is valid up to moderate Reynolds numbers. We seek solitary-wave solutions of this model using a pseudo-arclength continuation approach. As a general trend, it is found that the wave speed increases with increasing gas shear and the liquid flow rate. Further insight into the problem is provided by time-dependent computations of the WIBL model. Finally, the absolute-convective instability of a falling film that is in contact with a counter-current turbulent gas is analysed. The Orr--Sommerfeld (O-S) problem is formulated from the full governing equations and boundary conditions. The O-S problem along with low-dimensional models, namely, a long-wave and WIBL models are used to explore the linear stability of the gas-liquid system. It is found that for a fixed liquid Reynolds number, at low and high gas flow rates, the system is convectively unstable, and for a range of intermediate gas flow rates we have absolute instability. We supplemented our analysis by doing time-dependent computations of the linearised WIBL model subject to a localised initial condition which showed good agreement. The upper limit of the absolute instability regime predicted by our linear analysis is close to the flooding point obtained from the fully non-linear computations of the WIBL model.Open Acces

Basin bifurcations, oscillatory instability and rate-induced thresholds for AMOC in a global oceanic box model

The Atlantic Meridional Overturning Circulation (AMOC) transports substantial amounts of heat into the North Atlantic sector, and hence is of very high importance in regional climate projections. The AMOC has been observed to show multi-stability across a range of models of different complexity. The simplest models find a bifurcation associated with the AMOC `on' state losing stability that is a saddle node. Here we study a physically derived global oceanic model of Wood {\em et al} with five boxes, that is calibrated to runs of the FAMOUS coupled atmosphere-ocean general circulation model. We find the loss of stability of the `on' state is due to a subcritical Hopf for parameters from both pre-industrial and doubled CO${}_2$ atmospheres. This loss of stability via subcritical Hopf bifurcation has important consequences for the behaviour of the basin of attraction close to bifurcation. We consider various time-dependent profiles of freshwater forcing to the system, and find that rate-induced thresholds for tipping can appear, even for perturbations that do not cross the bifurcation. Understanding how such state transitions occur is important in determining allowable safe climate change mitigation pathways to avoid collapse of the AMOC.Comment: 18 figure

A simplified model of the Martian atmosphere - Part 2: a POD-Galerkin analysis

In Part I of this study Whitehouse et al. (2005) performed a diagnostic analysis of a simplied model of the Martian atmosphere, in which topography was absent and in which heating was modelled as Newtonian relaxation towards a zonally symmetric equilibrium temperature field. There we derived a reduced-order approximation to the vertical and the horizonal structure of the baroclinically unstable Martian atmosphere, retaining only the barotropic mode and the leading order baroclinic modes. Our objectives in Part II of the study are to incorporate these approximations into a Proper Orthogonal Decomposition-Galerkin expansion of the spherical quasi-geostrophic model in order to derive hierarchies of nonlinear ordinary differential equations for the time-varying coefficients of the spatial structures. Two different vertical truncations are considered, as well as three different norms and 3 different Galerkin truncations. We investigate each in turn, using tools from bifurcation theory, to determine which of the systems most closely resembles the data for which the original diagnostics were performed

Dynamics of Patterns

This workshop focused on the dynamics of nonlinear waves and spatio-temporal patterns, which arise in functional and partial differential equations. Among the outstanding problems in this area are the dynamical selection of patterns, gaining a theoretical understanding of transient dynamics, the nonlinear stability of patterns in unbounded domains, and the development of efficient numerical techniques to capture specific dynamical effects

Design of a bistable switch to control cellular uptake

International audienceBistable switches are widely used in synthetic biology to trigger cellular functions in response to environmental signals. All bistable switches developed so far, however, control the expression of target genes without access to other layers of the cellular machinery. Here, we propose a bistable switch to control the rate at which cells take up a metabolite from the environment. An uptake switch provides a new interface to command metabolic activity from the extracellular space and has great potential as a building block in more complex circuits that coordinate pathway activity across cell cultures, allocate metabolic tasks among different strains or require cell-to-cell communication with metabolic signals. Inspired by uptake systems found in nature, we propose to couple metabolite import and utilization with a genetic circuit under feedback regulation. Using mathematical models and analysis, we determined the circuit architectures that produce bistability and obtained their design space for bistability in terms of experimentally tuneable parameters. We found an activation–repression architecture to be the most robust switch because it displays bistability for the largest range of design parameters and requires little fine-tuning of the promoters' response curves. Our analytic results are based on on–off approximations of promoter activity and are in excellent qualitative agreement with simulations of more realistic models. With further analysis and simulation, we established conditions to maximize the parameter design space and to produce bimodal phenotypes via hysteresis and cell-to-cell variability. Our results highlight how mathematical analysis can drive the discovery of new circuits for synthetic biology, as the proposed circuit has all the hallmarks of a toggle switch and stands as a promising design to control metabolic phenotypes across cell cultures

Modeling the reactive magnetron sputtering process

Reactive magnetron sputtering is a versatile plasma technique to deposit thin layers of compound material on all kind of objects. The purpose of these thin films is to add or enhance interesting properties to the object. The term ``thin'' should here be interpreted as ranging from a few nanometers up till several microns. The technique is well appreciated in industry and found its application in numerous technological products ranging from big building glass windows, over car parts and drill chucks to the touch panel of smart phones. The basics of the technique are conceptually simple. A mostly metallic target is bombarded by the ions from a low-pressure noble gas plasma. These ions are accelerated over the applied negative voltage difference between the chamber and the target. The particles that are sputtered due to this bombardment, condense on surfaces within the vacuum chamber and ideally on the object to be coated. To modify this metallic layer into a compound layer, one or more reactive gases are added to the process. This reactive gas chemically reacts with the deposited material to form the desired compound. The purpose of the magnetron setup is to enhance this method by the creation of a magnetic field in the close neighborhood of the target surface. This magnetic field locally intensifies the plasma density and the ion production to optimize the deposition flux as function of the electrical power provided to the process. In the first chapter the technique is qualitatively explained and the most important aspects are introduced. Over the decades that the technique has been used, several magnetron setups came to existence. These designs can differ in the target geometry, the shape of the magnetic field and the electrical operation characteristics. The more general known design decisions are touched on. Characteristic for reactive sputtering as deposition technique is the occurrence of hysteresis phenomena in the process curves. These hystereses cause that the operation conditions are not uniquely determined by their instantaneous operation parameters, but will be history dependent. These hystereses can be investigated by so called direct controlled or feedback controlled hysteresis experiments. They differ in the way how an operation point is established. For direct control, the operation parameters are manually set irrespective on how the system will behave. For feedback control these parameters are automatically adapted by system monitoring in order to realize a certain system state. The technological relevance of these hysteresis phenomena is the impact on the process efficiency through the deposition rate and on the operation stability. Modeling of these hystereses is then interesting to understand their origin, their dependencies and their impact. Two modeling approaches are here recognized to do this. The first ``atomistic'' approach individually models in great detail and with high quantitative power the many physical and chemical subprocesses which are involved in the technique. Combining them would result in a strong predictive tool to cut away unnecessary trial-and-error experiments. The goal of this approach is very ambitious but is hampered by the big difference in temporal and spatial scales which have to be combined across the subprocesses. The second ``holistic'' approach starts to model the technique as a whole but in a way that strongly (over)simplifies the reality. Its primary goal is the basic qualitative understanding and modeling of the whole process and only in second order, the exact quantification of it. It is the second approach, also coined the top-bottom approach, which will be followed in this work. This contrasts with the former ``atomistic'' or bottom-up approach. The second chapter sets off with a historical overview of simple reactive sputtering models which prelude the original Berg model. The Berg model can be viewed as the ancestor of the Reactive Sputtering Deposition (RSD) models which follow. The development of the Berg model fits the ``holistic'' approach as it models the essence of the hysteresis during reactive sputtering deposition with a minimum in model complexity and parameters. The dependency of the hysteresis in the reactive gas pressure (system observable), as function of the introduced reactive flow (operation parameter), is studied with respect to the material and the operation parameters within the Berg model. In the thesis, new solution strategies for the original Berg model are proposed which enables the simulation of the hysteresis as function of the pumping speed, for example. Starting from the original Berg model, two kinds of extensions can be formulated. The first kind embodies extensions which add physical mechanisms to the model or differently describe included mechanisms. The second kind of extensions add spatial or temporal resolution to the model without touching the essence of the model. For the original Berg model, the inclusion of a deposition profile (spatial), an ion current profile (spatial) and the time dynamics (temporal) are examples of second kind extensions. In this work, the modeled time dynamics are shown to qualitatively correspond with the experimental behavior. In chapter three, a first kind extension of the original Berg model results in the RSD2007 model. The new mechanism that the original RSD2007 model introduces is reactive implantation and subsurface reaction. The derivation of this original RSD2007 model has here been investigated more thoroughly and lead to an analytical solution form of the subsurface implantation and reaction equations. For a uniform implantation profile an analytical closed form is achieved. In this work it is shown that these subsurface mechanisms can resolve at least three limitations of the original Berg model with experiments. The backside is the introduction of more model parameters which partially can be retrieved from experiments or other models. In this thesis, the remaining unknown parameters are estimated by fitting two RSD implementations to experimental hystereses of an aluminum and yttrium target sputtered in an argon/oxygen atmosphere. The first implemented RSD model which assumes an uniform current profile, is not able in reproducing the correct experimental oxide sputter yields. The second RSD implementation with a spatial resolved current profile however succeeds. The correlation between these sputter yields and the subsurface reaction rate is investigated in this work and the sputter yield ratio between the two metal systems showed to be constant and independent of the chosen model. Chapter four proposes a new formulated RSD2013 model elaborated within this thesis. The complete model is given in all its details together with the implementation of the model in a public available and user-friendly software package, equally called RSD2013. The major extensions over the previous version RSD2009 is the remodeling of redeposition, the addition of a second subsurface layer, a saturation limit for implanted reactive species and a complete equivalent steady state model for all model options including a multi-cell description and/or redeposition. Furthermore the option to include or exclude certain aspects of the model is been realized. The latter is illustrated with a series of simulations where the detail of the modeled system is gradually incremented. But the major physical enhancement to the RSD model is the remodeling of redeposition, the deposition of sputtered material back on the sputtering target surface. The dependency of redeposition on the metal-gas combination, the target-sample distance, the angular sputtering distribution and the gas density is examined with a Monte Carlo transport code SiMTra. The relevance of redeposition during reactive sputtering deposition is put forward. In a case study, the influence of the redeposition fraction on the reactive sputtering system is investigated by RSD2013 modeling. More specific, the influence on the target condition, the effect on the hysteresis and the shape modification of the racetrack and the sputter profile is examined. In the final chapter some future suggestions for next generation RSD models are put forward. Some of these suggestions form contemporary research topics of the research group DRAFT while others are disparately waiting to be tackled

Exploration-Exploitation in Multi-Agent Learning: Catastrophe Theory Meets Game Theory

Exploration-exploitation is a powerful and practical tool in multi-agent learning (MAL), however, its effects are far from understood. To make progress in this direction, we study a smooth analogue of Q-learning. We start by showing that our learning model has strong theoretical justification as an optimal model for studying exploration-exploitation. Specifically, we prove that smooth Q-learning has bounded regret in arbitrary games for a cost model that explicitly captures the balance between game and exploration costs and that it always converges to the set of quantal-response equilibria (QRE), the standard solution concept for games under bounded rationality, in weighted potential games with heterogeneous learning agents. In our main task, we then turn to measure the effect of exploration in collective system performance. We characterize the geometry of the QRE surface in low-dimensional MAL systems and link our findings with catastrophe (bifurcation) theory. In particular, as the exploration hyperparameter evolves over-time, the system undergoes phase transitions where the number and stability of equilibria can change radically given an infinitesimal change to the exploration parameter. Based on this, we provide a formal theoretical treatment of how tuning the exploration parameter can provably lead to equilibrium selection with both positive as well as negative (and potentially unbounded) effects to system performance.Comment: Appears in the 35th AAAI Conference on Artificial Intelligenc