151 research outputs found

### Systematically extending classical nucleation theory

The foundation for any discussion of first-order phse transitions is
Classical Nucleation Theory(CNT). CNT, developed in the first half of the
twentieth century, is based on a number of heuristically plausible assumtptions
and the majority of theoretical work on nucleation is devoted to refining or
extending these ideas. Ideally, one would like to derive CNT from a more
fundamental description of nucleation so that its extension, development and
refinement could be developed systematically. In this paper, such a development
is described based on a previously established (Lutsko, JCP 136:034509, 2012 )
connection between Classical Nucleation Theory and fluctuating hydrodynamics.
Here, this connection is described without the need for artificial assumtions
such as spherical symmetry. The results are illustrated by application to CNT
with moving clusters (a long-standing problem in the literature) and the
constructrion of CNT for ellipsoidal clusters

### Nonlinear diffusion from Einstein's master equation

We generalize Einstein's master equation for random walk processes by
considering that the probability for a particle at position $r$ to make a jump
of length $j$ lattice sites, $P_j(r)$ is a functional of the particle
distribution function $f(r,t)$. By multiscale expansion, we obtain a
generalized advection-diffusion equation. We show that the power law $P_j(r)
\propto f(r)^{\alpha - 1}$ (with $\alpha > 1$) follows from the requirement
that the generalized equation admits of scaling solutions ($f(r;t) =
t^{-\gamma}\phi (r/t^{\gamma})$). The solutions have a $q$-exponential form
and are found to be in agreement with the results of Monte-Carlo simulations,
so providing a microscopic basis validating the nonlinear diffusion equation.
Although its hydrodynamic limit is equivalent to the phenomenological porous
media equation, there are extra terms which, in general, cannot be neglected as
evidenced by the Monte-Carlo computations.}Comment: 7 pages incl. 3 fig

### A microscopic approach to nonlinear Reaction-Diffusion: the case of morphogen gradient formation

We develop a microscopic theory for reaction-difusion (R-D) processes based
on a generalization of Einstein's master equation with a reactive term and we
show how the mean field formulation leads to a generalized R-D equation with
non-classical solutions. For the $n$-th order annihilation reaction
$A+A+A+...+A\rightarrow 0$, we obtain a nonlinear reaction-diffusion equation
for which we discuss scaling and non-scaling formulations. We find steady
states with either solutions exhibiting long range power law behavior (for
$n>\alpha$) showing the relative dominance of sub-diffusion over reaction
effects in constrained systems, or conversely solutions (for $n<\alpha<n+1$)
with finite support of the concentration distribution describing situations
where diffusion is slow and extinction is fast. Theoretical results are
compared with experimental data for morphogen gradient formation.Comment: Article, 10 pages, 5 figure

### Nucleation of colloids and macromolecules: does the nucleation pathway matter?

A recent description of diffusion-limited nucleation based on fluctuating
hydrodynamics that extends classical nucleation theory predicts a very
non-classical two-step scenario whereby nucleation is most likely to occur in
spatially-extended, low-amplitude density fluctuations. In this paper, it is
shown how the formalism can be used to determine the maximum probability of
observing \emph{any} proposed nucleation pathway, thus allowing one to address
the question as to their relative likelihood, including of the newly proposed
pathway compared to classical scenarios. Calculations are presented for the
nucleation of high-concentration bubbles in a low-concentration solution of
globular proteins and it is found that the relative probabilities (new theory
compared to classical result) for reaching a critical nucleus containing $N_c$
molecules scales as $e^{-N_c/3}$ thus indicating that for all but the smallest
nuclei, the classical scenario is extremely unlikely.Comment: 7 pages, 5 figure

### The effect of the range of interaction on the phase diagram of a globular protein

Thermodynamic perturbation theory is applied to the model of globular
proteins studied by ten Wolde and Frenkel (Science 277, pg. 1976) using
computer simulation. It is found that the reported phase diagrams are
accurately reproduced. The calculations show how the phase diagram can be tuned
as a function of the lengthscale of the potential.Comment: 20 pages, 5 figure

### Phase behavior of a confined nano-droplet in the grand-canonical ensemble: the reverse liquid-vapor transition

The equilibrium density distribution and thermodynamic properties of a
Lennard-Jones fluid confined to nano-sized spherical cavities at constant
chemical potential was determined using Monte Carlo simulations. The results
describe both a single cavity with semipermeable walls as well as a collection
of closed cavities formed at constant chemical potential. The results are
compared to calculations using classical Density Functional Theory (DFT). It is
found that the DFT calculations give a quantitatively accurate description of
the pressure and structure of the fluid. Both theory and simulation show the
presence of a ``reverse'' liquid-vapor transition whereby the equilibrium state
is a liquid at large volumes but becomes a vapor at small volumes.Comment: 13 pages, 8 figures, to appear in J. Phys. : Cond. Mat

### Stability of Uniform Shear Flow

The stability of idealized shear flow at long wavelengths is studied in
detail. A hydrodynamic analysis at the level of the Navier-Stokes equation for
small shear rates is given to identify the origin and universality of an
instability at any finite shear rate for sufficiently long wavelength
perturbations. The analysis is extended to larger shear rates using a low
density model kinetic equation. Direct Monte Carlo Simulation of this equation
is computed with a hydrodynamic description including non Newtonian rheological
effects. The hydrodynamic description of the instability is in good agreement
with the direct Monte Carlo simulation for $t < 50t_0$, where $t_0$ is the mean
free time. Longer time simulations up to $2000t_0$ are used to identify the
asymptotic state as a spatially non-uniform quasi-stationary state. Finally,
preliminary results from molecular dynamics simulation showing the instability
are presented and discussed.Comment: 25 pages, 9 figures (Fig.8 is available on request) RevTeX, submitted
to Phys. Rev.

### The low-density/high-density liquid phase transition for model globular proteins

The effect of molecule size (excluded volume) and the range of interaction on
the surface tension, phase diagram and nucleation properties of a model
globular protein is investigated using a combinations of Monte Carlo
simulations and finite temperature classical Density Functional Theory
calculations. We use a parametrized potential that can vary smoothly from the
standard Lennard-Jones interaction characteristic of simple fluids, to the ten
Wolde-Frenkel model for the effective interaction of globular proteins in
solution. We find that the large excluded volume characteristic of large
macromolecules such as proteins is the dominant effect in determining the
liquid-vapor surface tension and nucleation properties. The variation of the
range of the potential only appears important in the case of small excluded
volumes such as for simple fluids. The DFT calculations are then used to study
homogeneous nucleation of the high-density phase from the low-density phase
including the nucleation barriers, nucleation pathways and the rate. It is
found that the nucleation barriers are typically only a few $k_{B}T$ and that
the nucleation rates substantially higher than would be predicted by Classical
Nucleation Theory.Comment: To appear in Langmui

### Diffusion in a Granular Fluid - Simulation

The linear response description for impurity diffusion in a granular fluid
undergoing homogeneous cooling is developed in the preceeding paper. The
formally exact Einstein and Green-Kubo expressions for the self-diffusion
coefficient are evaluated there from an approximation to the velocity
autocorrelation function. These results are compared here to those from
molecular dynamics simulations over a wide range of density and inelasticity,
for the particular case of self-diffusion. It is found that the approximate
theory is in good agreement with simulation data up to moderate densities and
degrees of inelasticity. At higher density, the effects of inelasticity are
stronger, leading to a significant enhancement of the diffusion coefficient
over its value for elastic collisions. Possible explanations associated with an
unstable long wavelength shear mode are explored, including the effects of
strong fluctuations and mode coupling

### Density functional theory of inhomogeneous liquids. I. The liquid-vapor interface in Lennard-Jones fluids

A simple model is proposed for the direct correlation function (DCF) for
simple fluids consisting of a hard-core contribution, a simple parametrized
core correction, and a mean-field tail. The model requires as input only the
free energy of the homogeneous fluid, obtained, e.g., from thermodynamic
perturbation theory. Comparison to the DCF obtained from simulation of a
Lennard-Jones fluid shows this to be a surprisingly good approximation for a
wide range of densities. The model is used to construct a density functional
theory for inhomogeneous fluids which is applied to the problem of calculating
the surface tension of the liquid-vapor interface. The numerical values found
are in good agreement with simulation

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