Pattern formation with a conservation law

Pattern formation in systems with a conserved quantity is considered by studying the appropriate amplitude equations. The conservation law leads to a large-scale neutral mode that must be included in the asymptotic analysis for pattern formation near onset. Near a stationary bifurcation, the usual Ginzburg--Landau equation for the amplitude of the pattern is then coupled to an equation for the large-scale mode. These amplitude equations show that for certain parameters all roll-type solutions are unstable. This new instability differs from the Eckhaus instability in that it is amplitude-driven and is supercritical. Beyond the stability boundary, there exist stable stationary solutions in the form of strongly modulated patterns. The envelope of these modulations is calculated in terms of Jacobi elliptic functions and, away from the onset of modulation, is closely approximated by a sech profile. Numerical simulations indicate that as the modulation becomes more pronounced, the envelope broadens. A number of applications are considered, including convection with fixed-flux boundaries and convection in a magnetic field, resulting in new instabilities for these systems.

Controlling Pattern Formation in Nanoparticle Assemblies via Directed Solvent Dewetting

We have achieved highly localised control of pattern formation in two dimensional nanoparticle assemblies by direct modification of solvent dewetting dynamics. A striking dependence of nanoparticle organisation on the size of atomic force microscope-generated surface heterogeneities is observed and reproduced in numerical simulations. Nanoscale features induce rupture of the solvent-nanoparticle film, causing the local flow of solvent to carry nanoparticles into confinement. Microscale heterogeneities instead slow the evaporation of the solvent, producing a remarkably abrupt interface between different nanoparticle patterns.

Hexagonal patterns in finite domains

In many mathematical models for pattern formation, a regular hexagonal pattern is stable in an infinite region. However, laboratory and numerical experiments are carried out in finite domains, and this imposes certain constraints on the possible patterns. In finite rectangular domains, it is shown that a regular hexagonal pattern cannot occur if the aspect ratio is rational. In practice, it is found experimentally that in a rectangular region, patterns of irregular hexagons are often observed. This work analyses the geometry and dynamics of irregular hexagonal patterns. These patterns occur in two different symmetry types, either with a reflection symmetry, involving two wavenumbers, or without symmetry, involving three different wavenumbers. The relevant amplitude equations are studied to investigate the detailed bifurcation structure in each case. It is shown that hexagonal patterns can bifurcate subcritically either from the trivial solution or from a pattern of rolls. Numerical simulations of a model partial differential equation are also presented to illustrate the behaviour.

Coerced Mechanical Coarsening of Nanoparticle Assemblies

Coarsening is a ubiquitous phenomenon [1-3] that underpins countless processes in nature, including epitaxial growth [1,3,4], the phase separation of alloys, polymers and binary fluids [2], the growth of bubbles in foams5, and pattern formation in biomembranes6. Here we show, in the first real-time experimental study of the evolution of an adsorbed colloidal nanoparticle array, that tapping-mode atomic force microscopy (TM-AFM) can drive the coarsening of Au nanoparticle assemblies on silicon surfaces. Although the growth exponent has a strong dependence on the initial sample morphology, our observations are largely consistent with modified Ostwald ripening processes [7-9]. To date, ripening processes have been exclusively considered to be thermally activated, but we show that nanoparticle assemblies can be mechanically coerced towards equilibrium, representing a new approach to directed coarsening. This strategy enables precise control over the evolution of micro- and nanostructures.