Postdoctoral researcher

## Prof. Dr. Nikolai Brilliantov

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Department of Mathematics, University of Leicester, UK

Postdoctoral researcher

Department of Mathematics, University of Leicester, UK

Physical Review E **87**, 039904(E)

(2013)

European Journal of Environmental and Civil Engineering **12**, 827-870

(2008)

We discuss several models for granular particles commonly used in Molecular

Dynamics simulations of granular materials, including spheres with linear dashpot force, viscoelastic

spheres and adhesive viscoelastic spheres. Starting from the vectorial interaction

forces we derive the coefficients of normal and tangential restitution as functions of the

vectorial impact velocity and of the material constants. We review the methods of

measurements of the coefficients of restitution and characterize the coefficient of normal

restitution as a fluctuating quantity. Moreover, the scaling behavior and the influence of

different force laws on the dynamical system behavior are discussed. The powerful method of

event-driven Molecular Dynamics is described and the algorithmic simulation technique is

explained in detail. Finally we discuss the limitations of event-driven MD.

Int. J. Mod. Phy. C, **18**, 701-711

(2007)

We present a universal description of the velocity distribution function of granular gases, f(v), valid for both, small and intermediate velocities where v is close to the thermal velocity and also for large v where the distribution function reveals an exponentially decaying tail. By means of large-scale Monte Carlo simulations and by kinetic theory we show that the deviation from the Maxwell distribution in the high-energy tail leads to small but detectable variation of the cooling coefficient and to extraordinary large relaxation time.

Physical Review Letters, **98**, 128001

(2007)

In a granular gas of rough particles the axis of rotation is shown to be correlated with the translational velocity of the particles. The average relative orientation of angular and linear velocities depends on the parameters which characterize the dissipative nature of the collision. We derive a simple theory for these correlations and validate it with numerical simulations for a wide range of coefficients of normal and tangential restitution. The limit of smooth spheres is shown to be singular: even an arbitrarily small roughness of the particles gives rise to orientational correlations.

Physical Review E, **76**, 051302

(2007)

We investigate the collision of adhesive viscoelastic spheres in quasistatic approximation where the adhesive interaction is described by the Johnson, Kendall, and Roberts (JKR) theory. The collision dynamics, based on the dynamic contact force, describes both restitutive collisions quantified by the coefficient of restitution ε as well as aggregative collisions, characterized by the critical aggregative impact velocity g_cr. Both quantities, ε and g_cr, depend sensitively on the impact velocity and particle size. Our results agree well with laboratory experiments.

Modelling and numerics of kinetic dissipative systems (Pareschi, L. and Russo, G. and Toscani, G.), Nova Science, Hauppauge NY

(2006)

The most striking phenomenon in the dynamics of granular gases is the formation of clusters and other structures. We investigate a gas of dissipatively colliding particles with a velocity dependent coefficient of restitution where cluster formation occurs as a transient phenomenon. Although for small impact velocity the particles collide elastically, surprisingly the temperature converges to zero.

Physical Review E, **74**, 041302

(2006)

The velocity distribution function of granular gases in the homogeneous cooling state as well as some heated granular gases decays for large velocities as f ∼ exp(−const. v). That is, its high-energy tail is overpopulated as compared with the Maxwell distribution. At the present time, there is no theory to describe the influence of the tail on the kinetic characteristics of granular gases. We develop an approach to quantify the overpopulated tail and analyze its impact on granular gas properties, in particular on the cooling coefficient. We observe and explain anomalously slow relaxation of the velocity distribution function to its steady state.

Europhysics Letters, **74**, 424-430

(2006)

The velocity distribution of a granular gas is analyzed in terms of the Sonine polynomials expansion. We derive an analytical expression for the third Sonine coefficient a_3. In contrast to frequently used assumptions this coefficient is of the same order of magnitude as the second Sonine coefficient a_2. For small inelasticity the theoretical result is in good agreement with numerical simulations. The next-order Sonine coefficients a_4, a_5 and a_6 are determined numerically. While these coefficients are negligible for small dissipation, their magnitude grows rapidly with increasing inelasticity for 0< ε < 0.6. We conclude that this behavior of the Sonine coefficients manifests the break down of the Sonine polynomial expansion caused by the increasing impact of the overpopulated high-energy tail of the distribution function.

Powders & Grains 2005: Proceedings of the 5th International Conference on Micromechanics of Granular Media (Garcia-Rojo, R., Herrmann, H. J., McNamara, S.), 1247-1253, Taylor & Francis

(2005)

We develop an analytical theory of adhesive interaction of viscoelastic spheres in quasistatic approximation. Deformations and deformation rates are assumed to be small, which allows for the application of the Hertz contact theory, modified to account for viscoelastic forces. The adhesion interactions are described by the Johnson, Kendall, and Roberts theory. Using the quasistatic approximation we derive the total force between the bodies which is not sufficiently described by the superposition of elastic, viscous and adhesive contributions, but instead an additional cross-term appears, which depends on the elastic, viscous and adhesive parameters of the material. Using the derived theory we estimate the contribution of adhesive forces to the normal coefficient of restitution and derive a criterion for the validity of the viscoelastic collision model.

Powders & Grains 2005: Proceedings of the 5th International Conference on Micromechanics of Granular Media (Garcia-Rojo, R., Herrmann, H. J., McNamara, S.), **2**, 505-509, Taylor & Francis

(2005)

The rolling motion of a rigid cylinder on an inclined flat viscous surface is investigated and the nonlinear resistance force against rolling, F_R(v), is derived. For small velocities F_R(v) increases with velocity due to increasing deformation rate of the surface material. For larger velocity it decreases with velocity due to decreasing contact area between the rolling cylinder and the deformed surface. The cylinder is, moreover, subjected to a viscous drag force and stochastic fluctuations due to a surrounding medium (air). For this system, in a wide range of parameters we observe bistability of the rolling motion. Depending on the material parameters, increasing the noise level may lead to increasing or decreasing average velocity.

Chaos **15**, 026108

(2005)

We study the diffusion of tracers (self-diffusion) in a homogeneously cooling gas of dissipative particles, using the Green-Kubo relation and the Chapman-Enskog approach. The dissipative particle collisions are described by the coefficient of restitution which for realistic material properties depends on the impact velocity. First, we consider self-diffusion using a constant coefficient of restitution, =const, as frequently used to simplify the analysis. Second, self-diffusion is studied for a simplified stepwised dependence of on the impact velocity. Finally, diffusion is considered for gases of realistic viscoelastic particles. We find that for =const both methods lead to the same result for the self-diffusion coefficient. For the case of impact-velocity dependent coefficients of restitution, the Green-Kubo method is, however, either restrictive or too complicated for practical application, therefore we compute the diffusion coefficient using the Chapman-Enskog method. We conclude that in application to granular gases, the Chapman-Enskog approach is preferable for deriving kinetic coefficients.

Europhysics Letters, **69**, 371-377

(2005)

We investigate the motion of a hard cylinder rolling down a soft inclined plane. The cylinder is subjected to a viscous drag force and stochastic fluctuations due to the surrounding medium. In a wide range of parameters we observe bistability of the rolling velocity. In dependence on the parameters, increasing noise level may lead to increasing or decreasing average velocity of the cylinder. The approximative analytical theory agrees with numerical results.

J.Phys.: Condens. Matter, **17**, S2705–S2713

(2005)

The Physics of Granular Media (Hinrichsen, H. and Wolf, D. E.), 189-209, Wiley, Amsterdam

(2004)

The collision of convex bodies is considered for small impact velocity, when plastic deformation and fragmentation may be disregarded. In this regime the contact is governed by forces according to viscoelastic deformation and by adhesion. The viscoelastic interaction is described by a modified Hertz law, while for the adhesive interactions, the model by Johnson, Kendall and Roberts (JKR) is adopted. We solve the general contact problem of convex viscoelastic bodies in quasi-sstatic approximation, which implies that the impact velocity is much smaller than the speed of sound in the material and that the viscosity relaxation time is much smaller than the duration of a collision. We estimate the threshold impact velocity which discriminates restitutive and sticking collisions. If the impact velocity is not large as compared with the threshold velocity, adhesive interaction becomes important, thus limiting the validity of the pure viscoelastic collision model.

Physical Review Letters, **93**, 134301

(2004)

A force-free granular gas is considered with an impact-velocity-dependent coefficient of restitution as it follows from the model of viscoelastic particles. We analyze structure formation in this system by means of three independent methods: molecular dynamics, numerical solution of the hydrodynamic equations, and linear stability analysis of these equations. All these approaches indicate that structure formation occurs in force-free granular gases only as a transient process.

Oxford University Press, Oxford

(2004)

Granular Gas Dynamics (Lecture Notes in Physics) (Pöschel, T. and Brilliantov, N. V), **624**, 131-162, Springer, New York

(2003)

The gaskinetic theory, including the theory of Granular Gases is based on the Boltzmann equation with the collision integral. Many properties of the gas, from the characteristics of the velocity distribution function to transport coefficients may be expressed in terms of functions of the collision integral which we call kinetic integrals. Although evaluation of these functions is conceptually straightforward, technically it is rather cumbersome. We report here a method of analytic evaluation of kinetic integrals based on the symbolic programming. The method is illustrated for various properties of the Granular Gas, ranging from the moments of the velocity distribution function to the transport coefficients. Most of these quantities and may not be found in practice manually.

Physical Review E, **67**, 061304

(2003)

The hydrodynamics of granular gases of viscoelastic particles, whose collision is described by an impact-velocity dependent coefficient of restitution, is developed using a modified Chapman-Enskog approach. We derive the hydrodynamic equations and the according transport coefficients with the assumption that the shape of the velocity distribution function follows adiabatically the decaying temperature. We show numerically that this approximation is justified up to intermediate dissipation. The transport coefficients and the coefficient of cooling are expressed in terms of the elastic and dissipative parameters of the particle material and by the gas parameters. The dependence of these coefficients on temperature differs qualitatively from that obtained with the simplifying assumption of a constant coefficient of restitution which was used in previous studies. The approach formulated for gases of viscoelastic particles may be applied also for other impact-velocity dependencies of the restitution coefficient.

Biophys. J., **85**, 3460-3474

(2003)

We study the kinetics of prion fibril growth, described by the nucleated polymerization model analytically and by means of numerical experiments. The elementary processes of prion fibril formation lead us to a set of differential equations for the number of fibrils, their total mass and the number of prion monomers. In difference to previous studies we analyze this set by explicitely taking into account the time dependence of the prion monomer concentration. The theoretical results agree with experimental data whereas the generally accepted hypothesis of constant monomer concentration leads to a fibril growth behavior which is not in agreement with experiments. The obtained size distribution of the prion fibril aggregates is shifted significantly towards shorter lengths as compared to earlier results, which leads to a enhanced infectivity of the prion material. Finally we study the effect of filtering of the inoculated material on the incubation time of the disease.

Int. J. Mod. Phys. C, **13**, 1263-1272

(2002)

Numerical simulations of a dissipative hard sphere gas reveal a dependence of the cooling rate on correlation of the particle velocities due to inelastic collisions. We propose a coefficient which characterizes the velocity correlations in the two-particle velocity distribution function and express the temperature decay rate in terms of this coefficient. The analytical results are compared with numerics.

Philosophical Transactions of the Royal Society of London A, **360**, 415-428

(2002)

Our study examines the long-time behaviour of a force-free Granular Gas of viscoelastic particles, for which the coefficient of restitution depends on the impact velocity, as it follows from the solution of the impact problem for viscoelastic spheres. Starting from the Boltzmann equation, we derived the hydrodynamic equations and obtained microscopic expressions for the transport coefficients in terms of the elastic and dissipative parameters of the particle material. We performed the stability analysis of the linearised set of equations and found that any inhomogeneities and vortices vanish after long time and the system approaches the flow-free stage of homogeneous density. This behaviour is in contrast to that of a gas consisting of particles which interact via a (non-realistic) constant coefficient of restitution, for which inhomogeneities (clusters) and vortex patterns have been proven to arise and to continuously develop.

Physica A, **325**, 274-283

(2002)

A gas of particles which collide inelastically if their impact velocity exceeds a certain value is investigated. In difference to common granular gases, cluster formation occurs only as a transient phenomenon. We calculate the decay of temperature due to inelastic collisions. In spite of the drastically reduced dissipation at low temperature the temperature surprisingly converges to zero.

Granular Gases (Lecture Notes in Physics) (Pöschel, T. and Luding, S), **564**, 203-212, Springer, Berlin, Heidelberg, New York

(2001)

Given a chain of viscoelastic spheres with fixed masses of the first and last particles. We raise the question: How to chose the masses of the other particles of the chain to assure maximal energy transfer? The results are compared with a chain of particles for which a constant coefficient of restitution is assumed. Our simple example shows that the assumption of viscoelastic particle properties has not only important consequences for very large systems (see [1]) but leads also to qualitative changes in smallsystems as compared with particles interacting via a constant restitution coefficient.

Physical Review Letters, **63**, 021505

(2001)

The transmission of kinetic energy through chains of inelastically colliding spheres is investigated for the case of constant coefficient of restitution ε = const and impact-velocity dependent coefficient ε(v) for viscoelastic particles. We derive a theory for the optimal distribution of particle masses which maximize the energy transfer along the chain and check it numerically. We found that for ε = const the mass distribution is a monotonous function which does not depend on the value of ε. In contrast, for ε(v) the mass distribution reveals a pronounced maximum, depending on the particle properties and on the chain length. The system investigated demonstrates that even for small and simple systems the velocity dependence of the coefficient of restitution may lead to new effects with respect to the same systems under the simplifying approximation ε = const.

Granular Gases (Lecture Notes in Physics) (Pöschel, T. and Luding, S), **564**, 100-124, Springer, Berlin, Heidelberg, New York

(2001)

We consider collisional models for granular particles and analyze the conditions under which the restitution coefficient might be a constant. We show that these conditions are not consistent with known collision laws. From the generalization of the Hertz contact law for viscoelastic particles we obtain the coefficient of normal restitution ε as a function of the normal component of the impact velocity v_imp. Using ε(v_imp) we describe the time evolution of temperature and of the velocity distribution function of a granular gas in the homogeneous cooling regime, where the particles collide according to the viscoelastic law. We show that for the studied systems the simple scaling hypothesis for the velocity distribution function is violated, i.e. that its evolution is not determined only by the time dependence of the thermal velocity. We observe, that the deviation from the Maxwellian distribution, which we quantify by the value of the second coefficient of the Sonine polynomial expansion of the velocity distribution function, does not depend on time monotonously. At first stage of the evolution it increases on the mean-collision time-scale up to a maximum value and then decays to zero at the second stage, on the time scale corresponding to the evolution of the granular gas temperature. For granular gas in the homogeneous cooling regime we also evaluate the time-dependent self-diffusion coefficient of granular particles. We analyze the time dependence of the mean-square displacement and discuss its impact on clustering. Finally, we discuss the problem of the relevant internal time for the systems of interest.

Coherent Structures in Complex Systems (Lecture Notes in Physics) (D. Reguera, L.L. Bonilla, M. Rubi), **567**, 408-419, Springer, Berlin, Heidelberg, New York

(2001)

We investigate the evolution of the velocity distribution function of a granular gas composed of viscoelastic particles in the homogeneous cooling state, i.e. before clustering occurs. The deviation of the velocity distribution function from the Maxwellian distribution is quantified by a Sonine polynomials expansion. The first non-vanishing Sonine coefficient a_2(t), reveals a complex time dependence which allows to assign the granular gas the property of an age. We discuss the possibility to measure the age of a granular gas.

Physical Review E, **61**, 2809 – 2812

(2000)

We analyze the velocity distribution function of force-free granular gases in the regime of homogeneous cooling when deviations from the Maxwellian distribution may be accounted only by the leading term in the Sonine polynomial expansion, quantified by the second coefficient a_2. We go beyond the linear approximation for a₂ and find three different values (three roots) for this coefficient which correspond to a scaling solution of the Boltzmann equation. The stability analysis performed showed, however, that among these three roots only one corresponds to a stable scaling solution. This is very close to a_2, obtained in previous studies in a linear with respect to a_2 approximation.

Physical Review E, **61**, 1716-1721

(2000)

The coefficient of self-diffusion for a homogeneously cooling granular gas changes significantly if the impact-velocity dependence of the restitution coefficient epsilon is taken into account. For the case of a constant epsilon the particles spread logarithmically slowly with time, whereas a velocity-dependent coefficient yields a power law time dependence. The impact of the difference in these time dependences on the properties of a freely cooling granular gas is discussed.

Physical Review E, **61**, 5573-5587

(2000)

The velocity distribution in a homogeneously cooling granular gas has been studied in the viscoelastic regime, when the restitution coefficient of colliding particles depends on the impact velocity. We show that for viscoelastic particles a simple scaling hypothesis is violated, i.e., that the time dependence of the velocity distribution does not scale with the mean square velocity as in the case of particles interacting via a constant restitution coefficient. The deviation from the Maxwellian distribution does not depend on time monotonically. For the case of small dissipation we detected two regimes of evolution of the velocity distribution function: Starting from the initial Maxwellian distribution, the deviation first increases with time on a collision time scale saturating at some maximal value; then it decays to zero on a much larger time scale which corresponds to the temperature relaxation. For larger values of the dissipation parameter there appears an additional intermediate relaxation regime. Analytical calculations for small dissipation agree well with the results of a numerical analysis.

Stochastic Processes in Physics, Chemistry, and Biology (Lecture Notes in Physics) (Freund, J. A. and Pöschel, T.), **557**, 107-117, Springer, Berlin, Heidelberg, New York

(2000)

In most of the literature on granular gases it is assumed that the restitution coefficient ε, which quantifies the loss of kinetic energy upon a collision is independent on the impact velocity. Experiments as well as theoretical investigations show, however, that for real materials the restitution coefficient depends significantly on the impact velocity. We consider the diffusion process in a homogeneous granular gas, i.e. in a system of dissipatively colliding particles. We show that the mean square displacement of the particles changes drastically if we take the impact velocity dependence of ε into account. Under the oversimplifying assumption of a constant coefficient one finds that the particles spread in space logarithmically slow with time, whereas realistic particles spread due to a power law.

European Physical Journal B, **10**, 169-174

(1999)

The resistance against rolling of a rigid cylinder on a flat viscous surface is investigated. We found that the rolling-friction coefficient reveals strongly non-linear dependence on the cylinder’s velocity. For low velocity the rolling-friction coefficient rises with velocity due to increasing deformation rate of the surface. For larger velocity, however, it decreases with velocity according to decreasing contact area and deformation of the surface.

European Physical Journal B, **12**, 299-301

(1999)

We show that two basic mechanical processes, the collisionof particles and rolling motion of a sphere on a plane, are intimately related. According to our recent findings, the restitution coefficient for colliding spherical particles ε, which characterizes the energy loss upon collision, is directly related to the rolling friction coefficient µ_roll for a viscous sphere on a hard plane. We quantify both coefficients in terms of material constants which allows to determine either of them provided the other is known. This relation between the coefficients may give rise to a novel experimental technique to determine alternatively the coefficient of restitution or the coefficient of rolling friction.

Physical Review E, **60**, 4465-4472

(1999)

We perform a dimension analysis for colliding viscoelastic spheres to show that the coefficient of normal restitution epsilon depends on the impact velocity g as ε= 1-gamma_1 g^(1/5) + gamma_2 g^(2/5) …, in accordance with recent findings. We develop a simple theory to find explicit expressions for coefficients gamma1 and gamma2. Using these and few next expansion coefficients for ε (g) we construct a Padé approximation for this function which may be used for a wide range of impact velocities where the concept of the viscoelastic collision is valid. The obtained expression reproduces quite accurately the existing experimental dependence ε(g) for ice particles.

Europhysics Letters, **42**, 511-516

(1998)

A first-principle continuum-mechanics expression for the rolling friction coefficient is obtained for the rolling motion of a viscoelastic sphere on a hard plane. It relates the friction coefficient to the viscous and elastic constants of the sphere material. The relation obtained refers to the case when the deformation of the sphere is small, the velocity of the sphere V is much less than the speed of sound in the material and when the characteristic time is much larger than the dissipative relaxation times of the viscoelastic material. To our knowledge this is the first first-principle expression of the rolling friction coefficient which does not contain empirical parameters.

Physical Review E, **53**, 5382-5392

(1996)

We propose a model for collisions between particles of a granular material and calculate the restitution coefficients for the normal and tangential motion as functions of the impact velocity from considerations of dissipative viscoelastic collisions. Existing models of impact with dissipation as well as the classical Hertz impact theory are included in the present model as special cases. We find that the type of collision (smooth, reflecting or sticky) is determined by the impact velocity and by the surface properties of the colliding grains. We observe a rather nontrivial dependence of the tangential restitution coefficient on the impact velocity. *©1996 The American Physical Society.*

Physica A, **231**, 417-424

(1996)

Collisions between granular particles are irreversibleprocesses causing dissipation of mechanical energy by fragmentation or heating of the colliders. The knowledge of these phenomena is essential for the understanding of the behaviour of complex systems of granular particles. We have developed a model for inelastic collisions of granular particles and calculated the velocity restitution coefficients, which describe all possible collisions in the system. The knowledge of these coefficients allows for event-driven many-particle simulations which cannot be performed in the frame of molecular dynamics. The benefit of this approach is to treat very large particle numbers necessary for the understanding of intrinsic large-scale phenomena in granular systems.

Bull. Am. Astron. Soc., **26**, 1143-1144

(1994)