351 research outputs found
Non-local dispersal and bistability
The scalar initial value problem [ u_t = ho Du + f(u), ] is a model for dispersal. Here represents the density at point of a compact spatial region and time , and is a function of with values in some function space . is a bounded linear operator and is a bistable nonlinearity for the associated ODE . Problems of this type arise in mathematical ecology and materials science where the simple diffusion model with is not sufficiently general. The study of the dynamics of the equation presents a difficult problem which crucially differs from the diffusion case in that the semiflow generated is not compactifying. We study the asymptotic behaviour of solutions and ask under what conditions each positive semi-orbit converges to an equilibrium (as in the case ). We develop a technique for proving that indeed convergence does hold for small and show by constructing a counter-example that this result does not hold in general for all
Nonlinear effects for island coarsening and stabilization during strained film heteroepitaxy
Nonlinear evolution of three-dimensional strained islands or quantum dots in
heteroepitaxial thin films is studied via a continuum elasticity model and the
development of a nonlinear dynamic equation governing the film morphological
profile. All three regimes of island array evolution are identified and
examined, including a film instability regime at early stage, a nonlinear
coarsening regime at intermediate times, and the crossover to a saturated
asymptotic state, with detailed behavior depending on film-substrate misfit
strains but not qualitatively on finite system sizes. The phenomenon of island
stabilization and saturation, which corresponds to the formation of steady but
non-ordered arrays of strained quantum dots, occurs at later time for smaller
misfit strain. It is found to be controlled by the strength of film-substrate
wetting interaction which would constrain the valley-to-peak mass transport and
hence the growth of island height, and also determined by the effect of elastic
interaction between surface islands and the high-order strain energy of
individual islands at late evolution stage. The results are compared to
previous experimental and theoretical studies on quantum dots coarsening and
saturation.Comment: 19 pages, 12 figures; submitted to Phys. Rev.
Some remarks on stability for a phase-field model with memory
The phase field system with memory can be viewed as a phenomenological extension of the classical phase equations in which memory effects have been taken into account in both fields. Such memory effects could be important for example during phase transition in polymer melts in the proximity of the glass transition temperature where configurational degrees of freedom in the polymer melt constitute slowly relaxing "internal modes" which are di±cult to model explicitly. They should be relevant in particular to glass-liquid-glass transitions where re-entrance effects have been recently reported [27]. We note that in numerical studies based on sharp interface equations obtained from (PFM), grains have been seen to rotate as they shrink [35, 36]. While further modelling and numerical efforts are now being undertaken, the present manuscript is devoted to strengthening the analytical underpinnings of the model
Unimodality of steady state distributions of growing cell populations
We consider an equation for the evolution of growing and dividing cells, and show, using a result of Kato and McLeod, that the probability density function for the stationary size distribution is necessarily unimodal
Finite-time Singularities in Surface-Diffusion Instabilities are Cured by Plasticity
A free material surface which supports surface diffusion becomes unstable
when put under external non-hydrostatic stress. Since the chemical potential on
a stressed surface is larger inside an indentation, small shape fluctuations
develop because material preferentially diffuses out of indentations. When the
bulk of the material is purely elastic one expects this instability to run into
a finite-time cusp singularity. It is shown here that this singularity is cured
by plastic effects in the material, turning the singular solution to a regular
crack.Comment: 4 pages, 3 figure
Interplay of internal stresses, electric stresses and surface diffusion in polymer films
We investigate two destabilization mechanisms for elastic polymer films and
put them into a general framework: first, instabilities due to in-plane stress
and second due to an externally applied electric field normal to the film's
free surface. As shown recently, polymer films are often stressed due to
out-of-equilibrium fabrication processes as e.g. spin coating. Via an
Asaro-Tiller-Grinfeld mechanism as known from solids, the system can decrease
its energy by undulating its surface by surface diffusion of polymers and
thereby relaxing stresses. On the other hand, application of an electric field
is widely used experimentally to structure thin films: when the electric
Maxwell surface stress overcomes surface tension and elastic restoring forces,
the system undulates with a wavelength determined by the film thickness. We
develop a theory taking into account both mechanisms simultaneously and discuss
their interplay and the effects of the boundary conditions both at the
substrate and the free surface.Comment: 14 pages, 7 figures, 1 tabl
Influence of uniaxial stress on the lamellar spacing of eutectics
Directional solidification of lamellar eutectic structures submitted to
uniaxial stress is investigated. In the spirit of an approximation first used
by Jackson and Hunt, we calculate the stress tensor for a two-dimensional
crystal with triangular surface, using a Fourier expansion of the Airy
function. crystal with triangular surface in contact with its melt, given that
a uniaxial external stress is applied. The effect of the resulting change in
chemical potential is introduced into the standard model for directional
solidification of a lamellar eutectic. This calculation is motivated by an
observation, made recently [I. Cantat, K. Kassner, C. Misbah, and H.
M\"uller-Krumbhaar, Phys. Rev. E, in press] that the thermal gradient produces
similar effects as a strong gravitational field in the case of dilute-alloy
solidification. Therefore, the coupling between the Grinfeld and the
Mullins-Sekerka instabilities becomes strong, as the critical wavelength of the
former instability gets reduced to a value close to that of the latter.
Analogously, in the case of eutectics, the characteristic length scale of the
Grinfeld instability should be reduced to a size not extremely far from typical
lamellar spacings. In a Jackson-Hunt like approach we average the undercooling,
including the stress term, over a pair of lamellae. Following Jackson and Hunt,
we assume the selected wavelength to be determined by the minimum undercooling
criterion and compute its shift due to the external stress. we realize the
shifting of the wavelength by the application of external stress. In addition,
we find that in general the volume fraction of the two solid phases is changed
by uniaxial stress. Implications for experiments on eutectics are discussed.Comment: 8 pages RevTex, 6 included ps-figures, accepted for Phys. Rev.
Non-local dispersal
We consider a model of spatial spread that has applications in both material science and biology. The classical models are based upon partial differential equations, in particular reaction-diffusion equations. Here the dispersal term is given in terms of an integral operator and we restrict ourselves to the scalar case
Modelling silicosis : structure of equilibria
We analyse the structure of equilibria of a coagulation–fragmentation–death model ofsilicosis. We present exact multiplicity results in the particular case of piecewise-constantcoefficients, results on existence and non-existence of equilibria in the general case, as wellas precise asymptotics for the infinite series that arise in the case of power law coefficient
Distributional fixed point equations for island nucleation in one dimension: a retrospective approach for capture zone scaling
The distributions of inter-island gaps and captures zones for islands
nucleated on a one-dimensional substrate during submonolayer deposition are
considered using a novel retrospective view. This provides an alternative
perspective on why scaling occurs in this continuously evolving system.
Distributional fixed point equations for the gaps are derived both with and
without a mean field approximation for nearest neighbour gap size correlation.
Solutions to the equations show that correct consideration of fragmentation
bias justifies the mean field approach which can be extended to provide
closed-from equations for the capture zones. Our results compare favourably to
Monte Carlo data for both point and extended islands using a range of critical
island size . We also find satisfactory agreement with theoretical
models based on more traditional fragmentation theory approaches.Comment: 9 pages, 7 figures and 1 tabl
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