arXiv subjects

Emmanuel Trizac

Publications and source records attributed to Emmanuel Trizac.

89 records · Page 5Linked to original sources

arXiv2003

On the first Sonine correction for granular gases

We consider the velocity distribution for a granular gas of inelastic hard spheres described by the Boltzmann equation. We investigate both the free of forcing case and a system heated by a stochastic force. We propose a new method to compute the first correction to Gaussian behavior in a Sonine polynomial expansion quantified by the fourth cumulant $a_2$. Our expressions are compared to previous results and to those obtained through the numerical solution of the Boltzmann equation. It is numerically shown that our method yields very accurate results for small velocities of the rescaled distribution. We finally discuss the ambiguities inherent to a linear approximation method in $a_2$.

cond-mat.stat-mech↗

arXiv2003

Random inelasticity and velocity fluctuations in a driven granular gas

We analyze the deviations from Maxwell-Boltzmann statistics found in recent experiments studying velocity distributions in two-dimensional granular gases driven into a non-equilibrium stationary state by a strong vertical vibration. We show that in its simplest version, the ``stochastic thermostat'' model of heated inelastic hard spheres, contrary to what has been hitherto stated, is incompatible with the experimental data, although predicting a reminiscent high velocity stretched exponential behavior with an exponent 3/2. The experimental observations lead to refine a recently proposed random restitution coefficient model. Very good agreement is then found with experimental velocity distributions within this framework, which appears self-consistent and further provides relevant probes to investigate the universality of the velocity statistics.

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arXiv2003

Kinetics of Fragmenting Freely Evolving Granular Gases

We investigate the effect of fragmentation on the homogeneous free cooling of inelastic hard spheres, using Boltzmann kinetic theory and Direct Monte Carlo simulations. We analyze in detail a model where dissipative collisions may subsequently lead to a break-up of the grains. With a given probability, two off-springs are then created from one of the two colliding partners, with conservation of mass, momentum and kinetic energy. We observe a scaling regime characterized by a single collisional average, that quantifies the deviations from Gaussian behaviour for the joint size and velocity distribution function. We also discuss the possibility of a catastrophe whereby the number of particles diverges in a finite time. This phenomenon appears correlated to a ``shattering'' transition marked by a delta singularity at vanishingly small grains for the rescaled size distribution.

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arXiv2003

Some exact results for Boltzmann's annihilation dynamics

The problem of ballistic annihilation for a spatially homogeneous system is revisited within Boltzmann's kinetic theory in two and three dimensions. Exact analytical results are derived for the time evolution of the particle density for some isotropic discrete bimodal velocity modulus distributions. According to the allowed values of the velocity modulus, different behaviors are obtained: power law decay with non-universal exponents depending continuously upon the ratio of the two velocities, or exponential decay. When one of the two velocities is equal to zero, the model describes the problem of ballistic annihilation in presence of static traps. The analytical predictions are shown to be in excellent agreement with the results of two-dimensional molecular dynamics simulations.

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arXiv2002

On the bulk modulus of the cell model of charged macromolecules suspensions

We study theoretically the bulk modulus (inverse of the compressibility) of a suspension of charged objects (macro-ions), making use of a cell model to account for the finite density of macro-ions. The diffuse layer of charged micro-species around a macro-ion is described by a generic local density functional theory. Within this general framework, we obtain the condition for a positive bulk modulus, which is fulfilled by several proposals made in the literature and rules out the possibility of a critical point. We show that a sufficient condition for a positive compressibility also ensures that the same theory produces repulsive effective pair potentials.

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arXiv2002

Analytical estimate of effective charges at saturation in Poisson-Boltzmann cell models

We propose a simple approximation scheme to compute the effective charge of highly charged colloids (spherical or cylindrical with infinite length). Within non-linear Poisson-Boltzmann theory, we start from an expression of the effective charge in the infinite dilution limit which is asymptotically valid for large salt concentrations; this result is then extended to finite colloidal concentration, approximating the salt partitioning effect which relates the salt content in the suspension to that of a dializing reservoir. This leads to an analytical expression of the effective charge as a function of colloid volume fraction and salt concentration. These results compare favorably with the effective charges {\em at saturation} (i.e. in the limit of large bare charge) computed numerically following the standard prescription proposed by Alexander {\it et al.} within the cell model.

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arXiv2002

A simple approach for charge renormalization of highly charged macro-ions

We revisit the popular notion of effective or renormalized charge, which is a concept of central importance in the field of highly charged colloidal or polyelectrolyte solutions. Working at the level of a linear Debye-Hückel like theory only, we propose a simple, efficient and versatile method to predict the saturated amount of charge renormalization, which is however a non-linear effect arising at strong electrostatic coupling. The results are successfully tested against the numerical solutions of Poisson-Boltzmann theory for polyions of various shapes (planar, cylindrical and spherical), both in the infinite dilution limit or in confined geometry, with or without added electrolyte. Our approach, accurate for monovalent micro-ions in solvents like water, is finally confronted against experimental results, namely the crystallization of charged colloidal suspensions and the osmotic coefficient of B-DNA solutions.

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arXiv2002

Effective charge saturation in colloidal suspensions

Because micro-ions accumulate around highly charged colloidal particles in electrolyte solutions, the relevant parameter to compute their interactions is not the bare charge, but an effective (or renormalized) quantity, whose value is sensitive to the geometry of the colloid, the temperature or the presence of added-salt. This non-linear screening effect is a central feature in the field of colloidal suspensions or polyelectrolyte solutions. We propose a simple method to predict effective charges of highly charged macro-ions, that is reliable for mono-valent electrolytes (and counter-ions) in the colloidal limit (large size compared to both screening length and Bjerrum length). Taking reference to the non linear Poisson-Boltzmann theory, the method is successfully tested against the geometry of the macro-ions, the possible confinement in a Wigner-Seitz cell, and the presence of added salt. Moreover, our results are corroborated by various experimental measures reported in the literature. This approach provides a useful route to incorporate the non-linear effects of charge renormalization within a linear theory for systems where electrostatic interactions play an important role.

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arXiv2002

Dynamics of ballistic annihilation

The problem of ballistically controlled annihilation is revisited for general initial velocity distributions and arbitrary dimension. An analytical derivation of the hierarchy equations obeyed by the reduced distributions is given, and a scaling analysis of the corresponding spatially homogeneous system is performed. This approach points to the relevance of the non-linear Boltzmann equation for dimensions larger than one and provides expressions for the exponents describing the decay of the particle density n(t) ~ t^{-ξ} and the root mean-square velocity ${\bar v} ~ t^{-γ}$ in term of a parameter related to the dissipation of kinetic energy. The Boltzmann equation is then solved perturbatively within a systematic expansion in Sonine polynomials. Analytical expressions for the exponents $ξ$ and $γ$ are obtained in arbitrary dimension as a function of the parameter $μ$ characterizing the small velocity behavior of the initial velocity distribution. Moreover, the leading non-Gaussian corrections to the scaled velocity distribution are computed. These expressions for the scaling exponents are in good agreement with the values reported in the literature for continuous velocity distributions in $d=1$. For the two dimensional case, we implement Monte-Carlo and molecular dynamics simulations that turn out to be in excellent agreement with the analytical predictions.

cond-mat.stat-mech↗

arXiv2002

Molecular dynamics simulations of vibrated granular gases

We present molecular dynamics simulations of mono- or bidisperse inelastic granular gases driven by vibrating walls, in two dimensions (without gravity). Because of the energy injection at the boundaries, a situation often met experimentally, density and temperature fields display heterogeneous profiles in the direction perpendicular to the walls. A general equation of state for an arbitrary mixture of fluidized inelastic hard spheres is derived and successfully tested against numerical data. Single-particle velocity distribution functions with non-Gaussian features are also obtained, and the influence of various parameters (inelasticity coefficients, density...) analyzed. The validity of a recently proposed Random Restitution Coefficient model is assessed through the study of projected collisions onto the direction perpendicular to that of energy injection. For the binary mixture, the non-equipartition of translational kinetic energy is studied and compared both to experimental data and to the case of homogeneous energy injection (``stochastic thermostat''). The rescaled velocity distribution functions are found to be very similar for both species.

cond-mat.stat-mech↗

arXiv2002

Kinetics and scaling in ballistic annihilation

We study the simplest irreversible ballistically-controlled reaction, whereby particles having an initial continuous velocity distribution annihilate upon colliding. In the framework of the Boltzmann equation, expressions for the exponents characterizing the density and typical velocity decay are explicitly worked out in arbitrary dimension. These predictions are in excellent agreement with the complementary results of extensive Monte Carlo and Molecular Dynamics simulations. We finally discuss the definition of universality classes indexed by a continuous parameter for this far from equilibrium dynamics with no conservation laws.

cond-mat.stat-mech↗

arXiv2001

Electrostatically swollen lamellar stacks and adiabatic pair potential of highly charged plate-like colloids in an electrolyte

We show that describing the screened electrostatic interactions in a periodic stack of rigid parallel identical charged plate-like colloids within a local density functional theory approach, generically leads to a swelling behaviour. Within the same framework, we find that the effective potential between a pair of such plates immersed in an electrolyte is repulsive at all distances. This general result is in contradiction with a theory put forward by Sogami, Shinohara and Smalley, that we criticize and thoroughly reconsider to show how the two approaches can be reconciled.

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arXiv2001

Heated granular fluids: the random restitution coefficient approach

We introduce the model of inelastic hard spheres with random restitution coefficient $α$, in order to account for the fact that, in a vertically shaken granular system interacting elastically with the vibrating boundary, the energy injected vertically is transferred to the horizontal degrees of freedom through collisions only, which leads to heating through collisions, i.e. to inelastic horizontal collisions with an effective restitution coefficient that can be larger than 1. This allows the system to reach a non-equilibrium steady state, where we focus in particular on the single particle velocity distribution $f(v)$ in the horizontal plane, and on its deviation from a Maxwellian. Molecular Dynamics simulations and Direct Simulation Monte Carlo (DSMC) show that, depending on the distribution of $α$, different shapes of $f(v)$ can be obtained, with very different high energy tails. Moreover, the fourth cumulant of the velocity distribution quantifying the deviations from Gaussian statistics is obtained analytically from the Boltzmann equation and successfully tested against the simulations.

cond-mat.soft↗

arXiv2000

Universal free energy correction for the two-dimensional one-component plasma

The universal finite-size correction to the free energy of a two-dimensional Coulomb system is checked in the special case of a one-component plasma on a sphere. The correction is related to the known second moment of the short-range part of the direct correlation function for a planar system.

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arXiv1998

Optimal packing of polydisperse hard-sphere fluids

We consider the effect of intermolecular interactions on the optimal size-distribution of $N$ hard spheres that occupy a fixed total volume. When we minimize the free-energy of this system, within the Percus-Yevick approximation, we find that no solution exists beyond a quite low threshold ($η\thickapprox 0.260$). Monte Carlo simulations reveal that beyond this density, the size-distribution becomes bi-modal. Such distributions cannot be reproduced within the Percus-Yevick approximation. We present a theoretical argument that supports the occurrence of a non-monotonic size-distribution and emphasizing the importance of finite size effects.

cond-mat.soft↗

arXiv1998

A coalescence model for freely decaying two-dimensional turbulence

We propose a ballistic coalescence model (punctuated-Hamiltonian approach) mimicking the fusion of vortices in freely decaying two-dimensional turbulence. A temporal scaling behaviour is reached where the vortex density evolves like $t^{-ξ}$. A mean-field analytical argument yielding the approximation $ξ=4/5$ is shown to slightly overestimate the decay exponent $ξ$ whereas Molecular Dynamics simulations give $ξ=0.71\pm 0.01$, in agreement with recent laboratory experiments and simulations of Navier-Stokes equation.

cond-mat.stat-mech↗

arXiv1997

A Wigner-Seitz model of charged lamellar colloidal dispersions

A concentrated suspension of lamellar colloidal particles (e. g. clay) is modelled by considering a single, uniformly charged, finite platelet confined with co- and counterions to a Wigner-Seitz (WS) cell. The system is treated within Poisson-Boltzmann theory, with appropriate boundary conditions on the surface of the WS cell, supposed to account for the confinement effect of neighbouring platelets. Expressions are obtained for the free energy, osmotic and disjoining pressures and the capacitance in terms of the local electrostatic potential and the co- and counterion density profiles. Explicit solutions of the linearized Poisson-Boltzmann (LPB) equation are obtained for circular and square platelets placed at the centre of a cylindrical or parallelepipedic cell. The resulting free energy is found to go through a minimum as a function of the aspect ratio of the cell, for any given volume (determined by the macroscopic concentration of platelets), platelet surface charge and salt concentration. The optimum aspect ratio is found to be nearly independent of the two latter physical parameters. The osmotic and disjoining pressures are found to coincide at the free energy minimum, while the total quadrupole moment of the electric double-layer formed by the platelet and the surrounding co- and counterions vanishes simultaneously. The osmotic equation-of-state is calculated for a variety of physical conditions. The limit of vanishing platelet concentration is considered in some detail, and the force acting between two coaxial platelets is calculated in that limit as a function of their separation.

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