Searcharxiv⌕ Search

arXiv subjects

E. Trizac

Publications and source records attributed to E. Trizac.

At least 37 records · Page 2Linked to original sources
arXiv2015

Poisson-Boltzmann thermodynamics of counter-ions confined by curved hard walls

We consider a set of identical mobile point-like charges (counter-ions) confined to a domain with curved hard walls carrying a uniform fixed surface charge density, the system as a whole being electroneutral. Three domain geometries are considered: a pair of parallel plates, the cylinder and the sphere. The particle system in thermal equilibrium is assumed to be described by the nonlinear Poisson-Boltzmann theory. While the effectively 1D plates and the 2D cylinder have already been solved, the 3D sphere problem is not integrable. It is shown that the contact density of particles at the charged surface is determined by a first-order Abel differential equation of the second kind which is a counterpart of Enig's equation in the critical theory of gravitation and combustion/explosion. This equation enables us to construct the exact series solutions of the contact density in the regions of small and large surface charge densities. The formalism provides, within the mean-field Poisson-Boltzmann framework, the complete thermodynamics of counter-ions inside a charged sphere (salt-free system).

cond-mat.soft↗
arXiv2015

Towards an $H$-theorem for granular gases

The $H$-theorem, originally derived at the level of Boltzmann non-linear kinetic equation for a dilute gas undergoing elastic collisions, strongly constrains the velocity distribution of the gas to evolve irreversibly towards equilibrium. As such, the theorem could not be generalized to account for dissipative systems: the conservative nature of collisions is an essential ingredient in the standard derivation. For a dissipative gas of grains, we construct here a simple functional $\mathcal H$ related to the original $H$, that can be qualified as a Lyapunov functional. It is positive, and results backed by three independent simulation approaches (a deterministic spectral method, the stochastic Direct Simulation Monte Carlo technique, and Molecular Dynamics) indicate that it is also non-increasing. Both driven and unforced cases are investigated.

cond-mat.stat-mech↗
arXiv2015

Effective charge of cylindrical and spherical colloids immersed in an electrolyte: the quasi-planar limit

We consider the non-linear Poisson-Boltzmann theory for a single cylindrical or spherical macro-ion in symmetric 1:1, together with asymmetric 1:2 and 2:1 electrolytes. We focus on the regime where $κa $, the ratio of the macro-ion radius $a$ over the inverse Debye length in the bulk electrolyte, is large. Analyzing the structure of the analytical expansion emerging from a multiple scale analysis, we uncover a hidden structure for the electrostatic potential. This structure, which appears after a heuristic resummation, suggests a new and convenient expansion scheme that we present and work out in detail. We show that novel exact results can thereby be obtained, in particular pertaining to effective charge properties, in complete agreement with the direct numerical solution to the problem.

cond-mat.soft↗
arXiv2014

Memory effect in uniformly heated granular gases

We evidence a Kovacs-like memory effect in a uniformly driven granular gas. A system of inelastic hard particles, in the low density limit, can reach a non-equilibrium steady state when properly forced. By following a certain protocol for the drive time dependence, we prepare the gas in a state where the granular temperature coincides with its long time value. The temperature subsequently does not remain constant, but exhibits a non-monotonic evolution with either a maximum or a minimum, depending on the dissipation, and on the protocol. We present a theoretical analysis of this memory effect, at Boltzmann-Fokker-Planck equation level, and show that when dissipation exceeds a threshold, the response can be coined anomalous. We find an excellent agreement between the analytical predictions and direct Monte Carlo simulations.

cond-mat.stat-mech↗
arXiv2014

Counter-ions between or at asymmetrically charged walls: 2D free-fermion point

This work contributes to the problem of determining effective interaction between asymmetrically (likely or oppositely) charged objects whose total charge is neutralized by mobile pointlike counter-ions of the same charge, the whole system being in thermal equilibrium. The problem is formulated in two spatial dimensions with logarithmic Coulomb interactions. The charged objects correspond to two parallel lines at distance d, with fixed line charge densities. Two versions of the model are considered: the standard "unconstrained" one with particles moving freely between the lines and the "constrained" one with particles confined to the lines. We solve exactly both systems at the free-fermion coupling and compare the results for the pressure (i.e. the force between the lines per unit length of one of the lines) with the mean-field Poisson-Boltzmann solution. For the unconstrained model, the large-d asymptotic behaviour of the free-fermion pressure differs from that predicted by the mean-field theory. For the constrained model, the asymptotic pressure coincides with the attractive van der Waals-Casimir fluctuational force. For both models, there are fundamental differences between the cases of likely-charged and oppositely-charged lines, the latter case corresponding at large distances d to a capacitor.

cond-mat.stat-mech↗
arXiv2014

Kovacs-like memory effect in driven granular gases

While memory effects have been reported for dense enough disordered systems such as glasses, we show here by a combination of analytical and simulation techniques that they are also intrinsic to the dynamics of dilute granular gases. By means of a certain driving protocol, we prepare the gas in a state where the granular temperature $T$ coincides with its long time limit. However, $T$ does not subsequently remain constant, but exhibits a non-monotonic evolution before reaching its non-equilibrium steady value. The corresponding so-called Kovacs hump displays a normal behavior for weak dissipation (as observed in molecular systems), but is reversed under strong dissipation, where it thus becomes anomalous.

cond-mat.stat-mech↗
arXiv2014

Exact non-equilibrium solutions of the Boltzmann equation under a time-dependent external force

We construct a novel class of exact solutions to the Boltzmann equation, in both its classical and quantum formulation, for arbitrary collision laws. When the system is subjected to a specific external forcing, the precise form of which is worked out, non equilibrium damping-less solutions are admissible. They do not contradict the $H$-theorem, but are constructed from its requirements. Interestingly, these solutions hold for time-dependent confinement. We exploit them, in a reverse-engineering perspective, to work out a protocol that shortcuts any adiabatic transformation between two equilibrium states in an arbitrarily short time-span, for an interacting system. Particle simulations of the direct Monte Carlo type fully corroborate the analytical predictions.

cond-mat.stat-mech↗
arXiv2013

Generalized Onsager theory for strongly anisometric patchy colloids

The implications of soft `patchy' interactions on the orientational disorder-order transition of strongly elongated colloidal rods and flat disks is studied within a simple Onsager-van der Waals density functional theory. The theory provides a generic framework for studying the liquid crystal phase behaviour of highly anisometric cylindrical colloids, which carry a distinct geometrical pattern of repulsive or attractive soft interactions localised on the particle surface. In this paper, we apply our theory to the case of charged rods and disks for which the local electrostatic interactions can be described by a screened-Coulomb potential. We consider infinitely thin rod-like cylinders with a uniform line charge and infinitely thin discotic cylinders with several distinctly different surface charge patterns. Irrespective of the backbone shape, the isotropic-nematic phase diagrams of charged colloids feature a generic destabilization of nematic order, a dramatic narrowing of the biphasic density region and a reentrant phenomenon upon reducing the electrostatic screening. At higher particle density the electrostatic repulsion leads to a complete suppression of nematic order in favour of spatially inhomogeneous liquid crystals.

cond-mat.soft↗
arXiv2013

On phase behavior and dynamical signatures of charged colloidal platelets

We investigate the competition between anisotropic excluded-volume and repulsive electrostatic interactions in suspensions of thin charged colloidal discs, by means of Monte-Carlo simulations and dynamical characterization of the structures found. We show that the original intrinsic anisotropy of the electrostatic potential between charged platelets, obtained within the non-linear Poisson-Boltzmann formalism, not only rationalizes the generic features of the complex phase diagram of charged colloidal platelets such as Gibbsite and Beidellite clays, but also predicts the existence of novel structures. In addition, we find evidences of a strong slowing down of the dynamics upon increasing density.

cond-mat.soft↗
arXiv2013

Ground state structure of a bilayer Wigner crystal with repulsive dielectric images

We study the ground-state structures of identical classical point charges with Coulomb interactions, confined between two symmetric parallel charged walls. For the well understood homogeneous dielectric case with no electrostatic images, the charges evenly condense on the opposite walls, thereby forming a bilayer Wigner crystal; five structures compete upon changing the inter-wall separation. Here, we consider a dielectric jump between the walls and a solvent in which charges are immersed, implying repulsive images. Using recently developed series representations of lattice sums for Coulomb law, we derive the complete phase diagram. In contrast to the homogeneous dielectric case, the particles remain in a hexagonal Wigner monolayer up to a certain distance between the walls. Beyond this distance, a bifurcation occurs to a sequence of Wigner bilayers, each layer having a nonzero spacing from the nearest wall. Another new phenomenon is that the ground-state energy as a function of the wall separation exhibits a global minimum.

cond-mat.stat-mech↗
arXiv2012

Linear hydrodynamics for driven granular gases

We study the dynamics of a granular gas heated by the stochastic thermostat. From a Boltzmann description, we derive the hydrodynamic equations for small perturbations around the stationary state that is reached in the long time limit. Transport coefficients are identified as Green-Kubo formulas obtaining explicit expressions as a function of the inelasticity and the spacial dimension.

cond-mat.stat-mech↗
arXiv2012

Like-charge colloidal attraction: a simple argument

By a length scale analysis, we study the equilibrium interactions between two like-charge planes confining neutralising counter-ions. At large Coulombic couplings, approaching the two charged bodies leads to an unbinding of counter-ions, a situation that is amenable to an exact treatment. This phenomenon is the key to attractive effective interactions. A particular effort is made for pedagogy, keeping equations and formalism to a minimum.

cond-mat.soft↗
arXiv2012

Critical phenomena and phase sequence in classical bilayer Wigner crystal at zero temperature

We study the ground-state properties of a system of identical classical Coulombic point particles, evenly distributed between two equivalently charged parallel plates at distance $d$; the system as a whole is electroneutral. It was previously shown that upon increasing d from 0 to infinity, five different structures of the bilayer Wigner crystal become energetically favored, starting from a hexagonal lattice (phase I, d=0) and ending at a staggered hexagonal lattice (phase V, d -> infinity). In this paper, we derive new series representations of the ground-state energy for all five bilayer structures. The derivation is based on a sequence of transformations for lattice sums of Coulomb two-particle potentials plus the neutralizing background, having their origin in the general theory of Jacobi theta functions. The new series provide convenient starting points for both analytical and numerical progress. Its convergence properties are indeed excellent: Truncation at the fourth term determines in general the energy correctly up to 17 decimal digits. The accurate series representations are used to improve the specification of transition points between the phases and to solve a controversy in previous studies. In particular, it is shown both analytically and numerically that the hexagonal phase I is stable only at d=0, and not in a finite interval of small distances between the plates as was anticipated before. The expansions of the structure energies around second-order transition points can be done analytically, which enables us to show that the critical behavior is of the Ginzburg-Landau type, with a mean-field critical index beta=1/2 for the growth of the order parameters.

cond-mat.str-el↗
arXiv2012

Universal reference state in a driven homogeneous granular gas

We study the dynamics of a homogeneous granular gas heated by a stochastic thermostat, in the low density limit. It is found that, before reaching the stationary regime, the system quickly "forgets" the initial condition and then evolves through a universal state that does not only depend on the dimensionless velocity, but also on the instantaneous temperature, suitably renormalized by its steady state value. We find excellent agreement between the theoretical predictions at Boltzmann equation level for the one-particle distribution function, and Direct Monte Carlo simulations. We conclude that at variance with the homogeneous cooling phenomenology, the velocity statistics should not be envisioned as a single-parameter, but as a two-parameter scaling form, keeping track of the distance to stationarity.

cond-mat.stat-mech↗
arXiv2012

Ground state of classical bilayer Wigner crystals

We study the ground state structure of electronic-like bilayers, where different phases compete upon changing the inter-layer separation or particle density. New series representations with exceptional convergence properties are derived for the exact Coulombic energies under scrutiny. The complete phase transition scenario --including critical phenomena-- can subsequently be worked out in detail, thereby unifying a rather scattered or contradictory body of literature, hitherto plagued by the inaccuracies inherent to long range interaction potentials.

cond-mat.str-el↗
arXiv2012

A two-dimensional one component plasma and a test charge : polarization effects and effective potential

We study the effective interactions between a test charge Q and a one-component plasma, i.e. a complex made up of mobile point particles with charge q, and a uniform oppositely charged background. The background has the form of a flat disk, in which the mobile charges can move. The test particle is approached perpendicularly to the disk, along its axis of symmetry. All particles interact by a logarithmic potential. The long and short distance features of the effective potential --the free energy of the system for a given distance between Q and the disk-- are worked out analytically in detail. They crucially depend on the sign of Q/q, and on the global charge borne by the discotic complex, that can vanish. While most results are obtained at the intermediate coupling Gamma = beta q^2 = 2 (beta being the inverse temperature), we have also investigated situations with stronger couplings: Gamma=4 and 6. We have found that at large distances, the sign of the effective force reflects subtle details of the charge distribution on the disk, whereas at short distances, polarization effects invariably lead to effective attractions.

cond-mat.stat-mech↗
arXiv2011

Wigner-Crystal Formulation of Strong-Coupling Theory for Counter-ions Near Planar Charged Interfaces

We present a new analytical approach to the strong electrostatic coupling regime (SC), that can be achieved equivalently at low temperatures, high charges, low dielectric permittivity etc. Two geometries are analyzed in detail: one charged wall first, and then, two parallel walls at small distances, that can be likely or oppositely charged. In all cases, one type of mobile counter-ions only is present, and ensures electroneutrality (salt free case). The method is based on a systematic expansion around the ground state formed by the two-dimensional Wigner crystal(s) of counter-ions at the plate(s). The leading SC order stems from a single-particle theory, and coincides with the virial SC approach that has been much studied in the last 10 years. The first correction has the functional form of the virial SC prediction, but the prefactor is different. The present theory is free of divergences and the obtained results, both for symmetrically and asymmetrically charged plates, are in excellent agreement with available data of Monte-Carlo simulations under strong and intermediate Coulombic couplings. All results obtained represent relevant improvements over the virial SC estimates. The present SC theory starting from the Wigner crystal and therefore coined Wigner SC, sheds light on anomalous phenomena like the counter-ion mediated like-charge attraction, and the opposite-charge repulsion.

cond-mat.stat-mech↗