Coefficient of restitution for one-dimensional harmonic solids
Anthony G. Basile
Department of Math and Natural Sciences, D’Youville College, Buffalo, New York 14201-1084
Randall S. Dumont
Department of Chemistry, McMaster University, Hamilton, Ontario, Canada L8S 4M1
Received 18 June 1999
Using a numerical algorithm based on the time evolution of normal modes, we calculate the coefficient of
restitution for various one-dimensional harmonic solids colliding with a hard wall. We find that, for a
homogeneous chain, =1 in the thermodynamic limit. However, for a chain in which weaker springs are
introduced in the colliding front half, remains significantly less than one even in the thermodynamic limit,
and the ‘‘lost’’ energy goes mostly into low frequency normal modes. An understanding of these results is
given in terms of how the energy is redistributed among the normal modes as the chain collides with the wall.
We then contrast these results with those for collisions of one-dimensional harmonic solids with a soft wall.
Using perturbation theory, we find that =1 for all harmonic chains in the extremely soft wall limit, but that
inelasticity grows with increasing chain size in contrast to hard wall collisions.
PACS numbers: 45.10.-b, 45.50.Tn
I. INTRODUCTION
Collisions have long been studied in physics and the de-
velopment of their dynamics has led to the formulation of
numerous conservation laws. Historically, the conservation
of energy remained elusive because it holds only for systems
in which conservative forces act; yet, most macroscopic sys-
tems have some dissipative forces at work and so tend to
‘‘loose’’ energy with time. Nonetheless, at the microscopic
level, the forces holding ordinary matter together are electro-
magnetic in nature and therefore conservative. Dissipation is
then understood as energy which was originally in a few
macroscopic degrees of freedom, but which somehow gets
diffused among many microscopic degrees of freedom as the
interaction progresses. However, while it is easy to under-
stand how conservative forces conserve energy, it is not so
easy to see how they can dissipate it.
Recent studies of the scattering of thermal clusters from
solid surfaces raise such questions 1–4. These studies focus
on collisions with significant deposition of translational ki-
netic energy into internal modes, and concern themselves
with collision-activated energy-threshold processes, such as
chemical reactions of species imbedded in inert clusters, or
cluster evaporation and shattering. In connection to these
studies, in this paper we investigate the more elementary
question of the dynamical mechanism for internal energy
deposition, without the complications associated with a ini-
tial internal energy, b activation of threshold processes, and
c interactions between rotational and vibrational degrees of
freedom. To this end, we consider the coefficient of restitu-
tion of one dimensional harmonic solids initially at 0 K.
The literature’s treatment of the coefficient of restitution
is largely concerned with collisions of macroscopic bodies.
In particular, there is a focus on the role of rotational dynam-
ics, to the extent that some studies consider only rigid body
collisions 5. In general, there is a reliance on simulation
based on a phenomenological description of the interaction
between small elements within the colliding bodies which
include intrinsically dissipative forces 6. In contrast to
these studies, our approach here is to focus solely on conser-
vative translational to vibrational energy transfer, as a model
for low energy microscopic cluster collisions.
The coefficient of restitution is introduced to measure the
translational kinetic energy ‘‘lost’’ during a collision:
=
K
a
K
b
, 1
where K
b
and K
a
are the translational kinetic energies before
and after the collision, respectively. Succinctly put then, our
question is this: If one maintains that at a sufficiently micro-
scopic level the forces holding the solid together are conser-
vative, then no energy is lost during the collision and the
difference, K
b
-K
a
, must go into the internal degrees of
freedom of the elastic solid. What internal degrees of free-
dom are excited and how are they excited? Below, we ad-
dress this question by considering the dynamics of one di-
mensional solids of identical point masses connected by
harmonic springs which are made to collide with a hard or
soft wall. While this model represents a strong idealization
of real physical systems—in particular, it is one dimensional
and only takes into account harmonic interactions—it does
yield exact analytical solutions and so makes explicit at least
one mechanism for the dissipation of energy by conservative
forces.
Since our aim is to gain insight into how varies with the
internal structure of the solid, we first investigate an ideal
case in which =1 and then see how deviation from it can
lead to 1. We show that the homogeneous chain, in
which all the spring constants are equal 7, has =1 in the
thermodynamic limit when it collides with a hard wall, while
the introduction of a ‘‘cushion’’ in the form of weaker
springs placed in the colliding front half of the chain gener-
ally leads to 1. Augmenting our numerical study with a
heuristic analysis, we argue why this should be the case and
arrive at some intuitive understanding of a mechanism by
which energy can be dissipated by conservative forces. We
then contrast hard wall collisions with soft wall collisions
PHYSICAL REVIEW E FEBRUARY 2000 VOLUME 61, NUMBER 2
PRE 61 1063-651X/2000/612/20159/$15.00 2015 ©2000 The American Physical Society