Macroscopic limit of the Becker-D\"oring equation via gradient flows

This work considers gradient structures for the Becker-D\"oring equation and its macroscopic limits. The result of Niethammer [17] is extended to prove the convergence not only for solutions of the Becker-D\"oring equation towards the Lifshitz-Slyozov-Wagner equation of coarsening, but also the convergence of the associated gradient structures. We establish the gradient structure of the nonlocal coarsening equation rigorously and show continuous dependence on the initial data within this framework. Further, on the considered time scale the small cluster distribution of the Becker--D\"oring equation follows a quasistationary distribution dictated by the monomer concentration

Point island models for nucleation and growth of supported nanoclusters during surface deposition

Point island models (PIMs) are presented for the formation of supported nanoclusters (or islands) during deposition on flat crystalline substrates at lower submonolayer coverages. These models treat islands as occupying a single adsorption site, although carrying a label to track their size (i.e., they suppress island structure). However, they are particularly effective in describing the island size and spatial distributions. In fact, these PIMs provide fundamental insight into the key features for homogeneous nucleation and growth processes on surfaces. PIMs are also versatile being readily adapted to treat both diffusion-limited and attachment-limited growth and also a variety of other nucleation processes with modified mechanisms. Their behavior is readily and precisely assessed by kinetic Monte Carlo simulation

The Becker-Döring equations with monomer input, competition and inhibition

We investigate the Becker-Döring model of nucleation with three generalisations; an input of monomer, an input of inhibitor and finally, we allow the monomers to form two morphologies of cluster. We assume size-independent aggregation and fragmentation rates. Initially we consider the problem of constant monomer input and determine the steady-state solution approached in the large-time limit, and the manner in which it is approached. Secondly, in addition to a constant input of monomer we allow a constant input of inhibitor, which prevents clusters growing any larger and this removes them from the kinetics of the process; the inhibitor is consumed in the action of poisoning a cluster. We determine a critical ratio of poison to monomer input below which the cluster concentrations tend to a non-zero steady-state solution and the poison concentration tends to a finite value. Above the critical input ratio, the concentrations of all cluster sizes tend to zero and the poison concentration grows without limit. In both cases the solution in the large-time limit is determined. Finally we consider a model where monomers form two morphologies, but the inhibitor only acts on one morphology. Four cases are identified, depending on the relative poison to monomer input rates and the relative thermodynamic stability. In each case we determine the final cluster distribution and poison concentration. We find that poisoning the less stable cluster type can have a significant impact on the structure of the more stable cluster distribution; a counter-intuitive result. All results are shown to agree with numerical simulation

Fokker-Planck approach of Ostwald ripening: simulation of a modified Lifschitz-Slyozov-Wagner system with a diffusive correction

International audienceWe propose a well-balanced scheme for the modified Lifshitz-Slyozov equation, that incorporates a size-diffusion term. The method uses the Fokker-Planck structure of the equation. In turn, large time simulations can be performed with a reduced computational cost, since the time step constraints are relaxed. The simulations bring out the critical mass threshold and the relaxation to equilibrium, which can be expected from the formal analogies with the Becker-Döring system

Application of Scaling and Kinetic Equations to Helium Cluster Size Distributions: Homogeneous Nucleation of a Nearly Ideal Gas

A previously published model of homogeneous nucleation [Villarica et al., J. Chem. Phys. 98, 4610 (1993)] based on the Smoluchowski [Phys. Z. 17, 557 (1916)] equations is used to simulate the experimentally measured size distributions of 4He clusters produced in free jet expansions. The model includes only binary collisions and does not consider evaporative effects, so that binary reactive collisions are rate limiting for formation of all cluster sizes despite the need for stabilization of nascent clusters. The model represents these data very well, accounting in some cases for nearly four orders of magnitude in variation in abundance over cluster sizes ranging up to nearly 100 atoms. The success of the model may be due to particularities of 4He clusters, i.e., their very low coalescence exothermicity, and to the low temperature of 6.7 K at which the data were collected

Modeling the interplay of mechanics and self-assembly in the actin cytoskeleton

Many cellular processes such as cell migration or division require a trade-off between structural integrity and dynamic reorganization of the load-bearing elements. The actin cytoskeleton has evolved to provide this function for animal cells, but a physical understanding of the interplay between its mechanics and self-assembly is missing. Here I model theoretically two paradigmatic situations of this kind. First, I consider the self-assembly of non-muscle myosin II minifilaments, with a special focus on the stochastic effects that arise due to the small system size of around 30 load bearing elements that turn-over simultaneously to producing contractile force. The self-assembly model follows a consensus architecture, thereby relating the geometrical neighborhood relations of the myosin II monomers with associated binding energies. I find that the turn-over of monomers depends on the mechanochemistry of the cross-bridge cycle by simulating the associated master equation explicitly and by a mean-field approach that maps the complex assembly structure to a simple monomer-addition scheme. Using a rheological framework, I characterize the distinct mechanical properties of non-muscle myosin II minifilaments that arise due to differences in the cross-bridge cycle of the different myosin II isoforms, that can co-assemble in one hetero-filament. Quantitative analysis of the frequency dependent response by a complex modulus, reveals a cross over from viscous to elastic behavior as the ratio of slow to fast isoforms working together is increased. Second I consider the dynamical stability of a peripheral stress fiber, that depends on the interplay of contraction by myosin II minifilaments, self-assembly of new actin filaments at both ends of the fiber and cortical tension. In collaboration with an experimental group, we could show how the myosin II isoform content is differentially reflected by the phenotype of peripheral stress fibers and show their position in a stability phase diagram of the stress fiber. These results demonstrate quantitatively how mechanics and self-assembly interact on different scales in the actin cytoskeleton

Electric and Magnetic Coupling Phenomena at Oxide Interfaces

Perovskit-Oxide weisen eine große Bandbreite an physikalischen Eigenschaften bei gleichzeitig hoher struktureller Qualität in kleinsten Dimensionen auf. Die dramatischen Veränderungen ihrer Eigenschaften bei nur geringer Variation der stöchiometrischen Zusammensetzung sind sowohl für ein tieferes physikalisches Verständnis als auch für mögliche Anwendungsperspektiven interessant. In der vorliegenden Arbeit wurde der Einfluss von Ladungsübertragung an Grenz- flächen, Anisotropiemodifikation durch Verspannung und Oberflächeneffekte sowie magnetische und strukturelle Kopplung untersucht. Aufgrund ihrer kontrastierenden Eigenschaften im Hinblick auf Ferromagnetismus und Ladungstransport wurden dotiertes Lanthanmanganat und Strontiumruthenat (SRO) für die Untersuchungen ausgewählt. Durch ihre hervorragenden Wachstumseigenschaften mit fehlerlosen Grenzflächen auf atomarer Ebene erlauben sie als Modellsystem die Untersuchung elektronischer, magnetischer und struktureller Kopplung in Perovskit-Oxiden – mit folgenden Ergebnissen: Durch Ladungsübertragung an Grenzflächen wird Ferromagnetismus in Schichten von weniger als vier Einheitszellen in Manganaten stabilisiert. Die mikroskopische Struktur der Systeme kann aus der Analyse der durch die Anisotropie bedingten Symmetrie der winkelabhängigen Magnetotransport- messungen erschlossen werden. Bei abnehmender Schichtdicke verringert sich die intrinsische orthorhombische Symmetrie in SRO zugunsten einer tetragonalen aufgrund der Symmetriebrechung an der Grenzfläche. Die Untersuchungen des anormalen Hall Effekts unterstreichen seine Tensor-Natur und zeigen eine Abhängigkeit des Vorzeichens sowohl von der magnetischen Anisotropie als auch der mikroskopischen Schichtqualität. Die Beobachtung einer Anisotropie oberhalb der Übergangstemperatur von SRO in Manganatschichten einer Dicke von zwei bis sechs Einheitszellen weist auf eine strukturelle Kopplung über die Sauerstoffoktaederrotationen hin. Die komplexe Wechselwirkung zwischen antiferromagnetischer Kopplung und schichtdickenabhängiger Anisotropie und dem magnetischen Moment werden in einem 2-Schichten-Modell beschrieben. Übergitter mit Einzelschichten von weniger als drei Einheitszellen lassen sich nicht mehr mit individuellen Einzelschichten beschreiben sondern stellen einen künstlichen Ferrimagneten dar.Perovskite oxides show a range of physical properties in combination with high structural quality in small dimensions. The dramatic change of their properties upon small variation in stoichiometry or external influences as pressure/strain are interesting for both a deeper understanding of fundamental condensed matter physics as well as electronic applications. In the present thesis the influence of charge transfer at interfaces, modification of the magnetic anisotropy by strain and surface effects, as well as magnetic and structural coupling was studied. In virtue of their contrasting ferromagnetic and transport properties, charge doped lanthanum manganite and strontium ruthenate (SRO) were chosen for this study. Their superior growth properties allowing atomically flat defect free interfaces make them a model system to study electronic magnetic and structural coupling phenomena in perovskite oxides − with the following findings: Charge transfer at interfaces stabilizes ferromagnetism in single layers of manganites down to one unit cell thickness similar to finite size scaling in ordinary transition metal ferromagnets. The microscopic structure of crystalline layers can be obtained from an analysis of the symmetries present in angle dependent magnetotransport measurements, which are determined by the anisotropy. Upon thickness reduction, the intrinsic orthorhombic symmetry in SRO is reduced in favour of a tetragonal one owing to the symmetry breaking at the interface. Studies on the anomalous Hall effect underline its tensorial nature and show a sign dependence on both magnetic anisotropy and microstructural quality. The observation of an in-plane anisotropy in manganite layers in the thickness range of two to six unit cells indicates a structural coupling via the oxygen octahedra. The complex interplay of antiferromagnetic coupling and layer thickness dependent anisotropy and magnetic moment are described in a bilayer model. Superlattices with individual layers of less than three unit cells cannot be described by the individual layer properties but represent an artificial ferrimagnet