360 research outputs found
Geometry of phase separation
We study the domain geometry during spinodal decomposition of a 50:50 binary
mixture in two dimensions. Extending arguments developed to treat non-conserved
coarsening, we obtain approximate analytic results for the distribution of
domain areas and perimeters during the dynamics. The main approximation is to
regard the interfaces separating domains as moving independently. While this is
true in the non-conserved case, it is not in the conserved one. Our results can
therefore be considered as a first-order approximation for the distributions.
In contrast to the celebrated Lifshitz-Slyozov-Wagner distribution of
structures of the minority phase in the limit of very small concentration, the
distribution of domain areas in the 50:50 case does not have a cut-off. Large
structures (areas or perimeters) retain the morphology of a percolative or
critical initial condition, for quenches from high temperatures or the critical
point respectively. The corresponding distributions are described by a tail, where and are exactly known. With increasing time,
small structures tend to have a spherical shape with a smooth surface before
evaporating by diffusion. In this regime the number density of domains with
area scales as , as in the Lifshitz-Slyozov-Wagner theory. The
threshold between the small and large regimes is determined by the
characteristic area, . Finally, we study the
relation between perimeters and areas and the distribution of boundary lengths,
finding results that are consistent with the ones summarized above. We test our
predictions with Monte Carlo simulations of the 2d Ising Model.Comment: 10 pages, 8 figure
Long-time asymptotics for polymerization models
This study is devoted to the long-term behavior of nucleation, growth and
fragmentation equations, modeling the spontaneous formation and kinetics of
large polymers in a spatially homogeneous and closed environment. Such models
are, for instance, commonly used in the biophysical community in order to model
in vitro experiments of fibrillation. We investigate the interplay between four
processes: nucleation, polymeriza-tion, depolymerization and fragmentation. We
first revisit the well-known Lifshitz-Slyozov model, which takes into account
only polymerization and depolymerization, and we show that, when nucleation is
included, the system goes to a trivial equilibrium: all polymers fragmentize,
going back to very small polymers. Taking into account only polymerization and
fragmentation, modeled by the classical growth-fragmentation equation, also
leads the system to the same trivial equilibrium, whether or not nucleation is
considered. However, also taking into account a depolymer-ization reaction term
may surprisingly stabilize the system, since a steady size-distribution of
polymers may then emerge, as soon as polymeriza-tion dominates depolymerization
for large sizes whereas depolymerization dominates polymerization for smaller
ones-a case which fits the classical assumptions for the Lifshitz-Slyozov
equations, but complemented with fragmentation so that " Ostwald ripening "
does not happen.Comment: https://link.springer.com/article/10.1007/s00220-018-3218-
Boundary value for a nonlinear transport equation emerging from a stochastic coagulation-fragmentation type model
We investigate the connection between two classical models of phase
transition phenomena, the (discrete size) stochastic Becker-D\"oring, a
continous time Markov chain model, and the (continuous size) deterministic
Lifshitz-Slyozov model, a nonlinear transport partial differential equation.
For general coefficients and initial data, we introduce a scaling parameter and
prove that the empirical measure associated to the stochastic Becker-D\"oring
system converges in law to the weak solution of the Lifshitz-Slyozov equation
when the parameter goes to 0. Contrary to previous studies, we use a weak
topology that includes the boundary of the state space (\ie\ the size )
allowing us to rigorously derive a boundary value for the Lifshitz-Slyozov
model in the case of incoming characteristics. The condition reads where is the volume distribution
function, solution of the Lifshitz-Slyozov equation, and the
aggregation and fragmentation rates, the concentration of free particles
and a nucleation constant emerging from the microscopic model. It is
the main novelty of this work and it answers to a question that has been
conjectured or suggested by both mathematicians and physicists. We emphasize
that this boundary value depends on a particular scaling (as opposed to a
modeling choice) and is the result of a separation of time scale and an
averaging of fast (fluctuating) variables.Comment: 42 pages, 3 figures, video on supplementary materials at
http://yvinec.perso.math.cnrs.fr/video.htm
A nonlinear theory of non-stationary low Mach number channel flows of freely cooling nearly elastic granular gases
We use hydrodynamics to investigate non-stationary channel flows of freely
cooling dilute granular gases. We focus on the regime where the sound travel
time through the channel is much shorter than the characteristic cooling time
of the gas. As a result, the gas pressure rapidly becomes almost homogeneous,
while the typical Mach number of the flow drops well below unity. Eliminating
the acoustic modes, we reduce the hydrodynamic equations to a single nonlinear
and nonlocal equation of a reaction-diffusion type in Lagrangian coordinates.
This equation describes a broad class of channel flows and, in particular, can
follow the development of the clustering instability from a weakly perturbed
homogeneous cooling state to strongly nonlinear states. If the heat diffusion
is neglected, the reduced equation is exactly soluble, and the solution
develops a finite-time density blowup. The heat diffusion, however, becomes
important near the attempted singularity. It arrests the density blowup and
brings about novel inhomogeneous cooling states (ICSs) of the gas, where the
pressure continues to decay with time, while the density profile becomes
time-independent. Both the density profile of an ICS, and the characteristic
relaxation time towards it are determined by a single dimensionless parameter
that describes the relative role of the inelastic energy loss and heat
diffusion. At large values of this parameter, the intermediate cooling dynamics
proceeds as a competition between low-density regions of the gas. This
competition resembles Ostwald ripening: only one hole survives at the end.Comment: 20 pages, 15 figures, final versio
Growth in systems of vesicles and membranes
We present a theoretical study for the intermediate stages of the growth of
membranes and vesicles in supersaturated solutions of amphiphilic molecules.
The problem presents important differences with the growth of droplets in the
classical theory of Lifshitz-Slyozov-Wagner, because the aggregates are
extensive only in two dimensions, but still grow in a three dimensional bath.
The balance between curvature and edge energy favours the nucleation of small
planar membranes, but as they grow beyond a critical size they close themselves
to form vesicles. We obtain a system of coupled equations describing the growth
of planar membranes and vesicles, which is solved numerically for different
initial conditions. Finally, the range of parameters relevant in experimental
situations is discussed.Comment: 13 pages and 5 postscript figures. To appear in Phys. Rev
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