PHYSICAL REVIEW E 85, 041306 (2012) Negative coefficient of normal restitution Patric M¨ uller, Dominik Krengel, and Thorsten P¨ oschel Institute for Multiscale Simulation, Universit ¨ at Erlangen-N¨ urnberg, N¨ agelsbachstraße 49b, 91052 Erlangen, Germany (Received 15 February 2012; published 27 April 2012) This paper shows that negative coefficients of normal restitution occur inevitably when the interaction force between colliding particles is finite. We derive an explicit criterion showing that for any set of material properties there is always a collision geometry leading to negative restitution coefficients. While from a phenomenological point of view, negative coefficients of normal restitution appear rather artificial, this phenomenon is generic and implies an important overlooked limitation of the widely used hard sphere model. The criterion is explicitly applied to two paradigmatic situations: for the linear dashpot model and for viscoelastic particles. In addition, we show that for frictional particles the phenomenon is less pronounced than for smooth spheres. DOI: 10.1103/PhysRevE.85.041306 PACS number(s): 45.70.−n, 45.50.Tn I. INTRODUCTION Both, kinetic theory of granular matter, based on the Boltzmann equation [1–3], as well as highly efficient event- driven molecular dynamics (eMD) simulation of granular matter [4–6] are based on the hard sphere model of particle collisions. Hard sphere collisions are characterized by δ- shaped interaction forces. Therefore, in a collision the particles instantaneously exchange momentum, while their positions stay invariant. Due to the instantaneous character of the collisions, the dynamics of any finite hard sphere system is represented by a sequence of binary collisions (events), leading to the main concept of eMD. As only momentum is transferred during a collision, each event is characterized by only two scalar values: The coefficient of normal restitution ε n relating the post- and precollisional normal component of the particles’ relative velocity and the corresponding coefficient of tangential restitution for the tangential component. For colliding hard spheres located at  r 1 and  r 2 , traveling at velocities ˙  r 1 and ˙  r 2 , the coefficient of normal restitution ε HS n is, thus, defined by ( ˙  r ′ 1 − ˙  r ′ 2 ) · ˆ e 0 r =−ε HS n ( ˙  r 0 1 − ˙  r 0 2 ) · ˆ e 0 r , (1) where for each quantity X, the symbol X 0 denotes the value of X at the beginning of the impact at time t 0 and X ′ is the value at time t 0 + τ , when the collision terminates. Equation (1) addresses hard spheres implying instantaneous collisions, τ → 0. Note that the unit vector ˆ e r ≡ ( r 2 − r 1 )/ | r 2 − r 1 | remains unchanged during a collision, ˆ e ′ r = ˆ e 0 r , due to the instantaneous character of hard sphere collisions. Therefore, ˆ e 0 r appears on both sides of Eq. (1). For soft sphere collisions characterized by finite interaction forces and contact durations τ , this may be invalid: Oblique impacts may lead to variations of the unit vector ˆ e ′ r = ˆ e 0 r . Consequently, for soft spheres the coefficient of normal restitution ε n is defined by ( ˙  r ′ 1 − ˙  r ′ 2 ) · ˆ e ′ r =−ε n ( ˙  r 0 1 − ˙  r 0 2 ) · ˆ e 0 r , (2) which reduces to Eq. (1) in the limit τ → 0. The variation of the unit vector ˆ e r during a collision may be expressed by the angle ϕ ′ ≡ cos −1 ( ˆ e 0 r · ˆ e ′ r ) . (3) For both, kinetic theory based on the Boltzmann equation as well as eMD, it is essential to assume that ϕ ′ is negligible. Then, Eq. (1) allows for the computation of the post-collisional particle velocities from the impact velocities, that is, the collision rule needed in eMD simulations and also the Jacobian ∂ (  v 1 ,  v 2 ) /∂ (  v ′ 1 ,  v ′ 2 ) needed for the integration of the Boltzmann equation. It was shown, however, that the assumption ϕ ′ ≈ 0 is not always justified [7]. In textbooks, the coefficient of normal restitution is fre- quently assumed to be a material constant [8], 0  ε n  1. While from experimental (e.g., [9–13]) and theoretical (e.g., [14–18]) studies it is known that the coefficient of normal restitution depends on the impact velocity or is a fluctuating quantity (e.g., [19]) it was still assumed to not fall below zero. However, for the case of oblique impacts of nanoclusters it was shown recently [20] that the rotation of the unit vector ϕ ′ may lead to negative values of the coefficient of normal restitution defined by Eq. (1). This surprising result was attributed to the softness of nanoclusters leading to relatively long contact durations which, in turn, may lead to a significant reorientation of the particles’ normal vector during a collision. While the investigation in [20] refers to nanoclusters colliding at very large impact velocity (1850 m/s), here we show that negative coefficients of normal restitution are a much more general phenomenon. We show that negative values occur inevitably for any type of collision whose dynamics is governed by finite interaction forces leading to finite duration of contact: For any set of material properties there is always a collision geometry leading to negative values of ε HS n . We wish to mention that there are alternative definitions of the coefficient of restitution (e.g., [21–25]), which avoid the problem addressed here. However, only the definition Eq. (1) or (2), respectively, referred to as Newton’s model [26], allows for the computation of the post-collisional vectorial velocities which are needed for both event-driven simulations and kinetic theory. In Sec. II we introduce our notation. In Sec. III we derive a general condition for negative coefficients, while in Secs. IV and V we specify our findings to particles which interact via a linear dashpot model and to smooth viscoelastic spheres, respectively. Section VI discusses the role of frictional forces between the colliding particles. Finally, in Sec. VII we summarize our results and present some outlook. II. COLLISION OF SMOOTH SPHERES The collision of a pair of smooth spheres of masses m 1 and m 2 , located at  r 1 and  r 2 , is governed by Newton’s equation 041306-1 1539-3755/2012/85(4)/041306(8) ©2012 American Physical Society