Parameter-free dissipation in simulated sliding friction

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Information about Parameter-free dissipation in simulated sliding friction

Published on January 5, 2017

Author: AndreaBenassi3

Source: slideshare.net

1. Parameter‐free
dissipation

 in
simulated
sliding
friction

 A.
Benassi
1,2




G.E.
Santoro
1,2,3




A.
Vanossi
1,2



and



E.
Tosatti
1,2,3
 Abstract
 Non‐equilibrium 
 molecular 
 dynamics 
 simulations,
 of 
 crucial 
 importance 
 in 
 sliding 
 friction, 
 are
 hampered
by
arbitrariness
and
uncertainties
in
the
 way
Joule
heat
is
removed.
We
implement
in
a
 realistic
frictional
simulation
a
parameter‐free,
non‐ markovian,
stochastic
dynamics,
which,
as
expected
 from
theory,
absorbs
Joule
heat
precisely
as
a
semi‐ infinite
harmonic
substrate
would.
Simulating
stick‐ slip
friction
of
a
slider
over
a
2D
Lennard‐Jones
 solid, 
 we 
 compare 
 our 
 virtually 
 exact 
 frictional
 results 
 with 
 approximate 
 ones 
 from 
 commonly
 adopted
empirical
dissipation
schemes.
While
the
 latter
are
generally
in
serious
error,
we
show
that
 the
exact
results
can
be
closely
reproduced
by
a
 viscous
Langevin
dissipation
at
the
boundary
layer,
 once 
 the 
 back‐reflected 
 frictional 
 energy 
 is
 variationally
optimized.

 1.  SISSA
Scuola
Internazionale
Superiose
Studi
Avanzati,
Trieste
(Italy)
 2.  DEMOCRITOS
National
Simulation
Center,
Trieste
(Italy)
 3.  International
Center
for
Theoretical
physics
(ICTP),
Trieste
(Italy)
 Simulating
a
2D
semi‐infinite
Lenard‐Jones
substrate
 i)  Following
Adelman
we
write
the
Hamilton’s
 
equations
for
all
the
atoms,
 distinguishing
between
3
different
regions
(fig.
2a)
 ii)  Under
the
hypothesis
of
an
harmonic
heat
bath
with
dynamical
tensor φ we
 avoid
to
simulate
explicitly
the
heat
bath
(fig.
2b)
accounting
for
its
presence
 through
effective
equations
of
motion
for
the
atoms
in
the
dissipation
layer
 iii)  These
equations
allow
us
to
dissipate
the
energy
injected
in
the
substrate
 has
if
the
substrate
was
really
a
semi‐infinite
object
 iv)  The
effective
equations
are
non‐markovian
Langevin
equations
with
many

 memory
kernels

K
and
stochastic
forces
R.
 fig.
2
(b)
 … z x fig.
2
(a)
 1 2 3 Dissipation
 layer
 Explicitly

 simulated
 atoms

 Infinite

 heat
bath
 (not
simulated)
 i, j = 1, 2, 3, ... µ, ν = x, z m¨qi µ = + j,ν qj ν(t) Ki,j µ,ν(0) − φi,j µ,ν − m j,ν t 0 ˙qi µ(s)Ki,j µ,ν(t − s)ds + Ri µ(t) Direct
interaction
 Indirect
interaction
+
 self
interaction
 Heat
bath
contribution
 ∈ Comparison
with
other
dissipation
schemes
 We
compared
the
results
for
the
semi‐infinite
substrate
with
other
two
dissipation
schemes
based
on
markovian
 Langevin
equations:
 The
memory
kernels
and
the
stochastic
noise
 The
kernels
are
not
chosen
a
priori,
they
come
from
the
microscopic
 theory
too:
 λi
and
ωi 2
being
the
eigenvectors
and
eigenvalues
of
the
dynamic

matrix
 φ
of
the
heat
bath.
All
this
kernels
are
oscillating
and
decaying
functions,
 an
example
is
given
in
fig.
3.
 Kk,m µ,ν = i (λi · φk µ,µ)(λi · φm ν,ν) ω2 i cos(ωit) R(t)i µ = 0 R(t)i µR(t )j ν = mKBTKi,j µ,ν(t − t ) fig.
3
 Accordingly
to
the
fluctuation
dissipation
theory,
memory
kernels
are
 also 
 needed 
 to 
 correlate 
 the 
 stochastic 
 noise 
 that 
 arise 
 at 
 finite
 temperature:
 k v0 fig.
4
 Incommensurate
dry
friction
 Putting
a
slider
on
the
free
surface
of
our
semi‐infinite
substrate
(fig.
4)
enable
us
to
 study 
 friction 
 phenomena 
 without 
 any 
 a 
 priori 
 assumption 
 on 
 the 
 shape 
 of 
 the
 dissipative
force.
 The
energy
of
the
slider
is
dissipated
exciting
the
phononic
modes
of
the
substrate,
 once
that
the

phonons
reach
the
dissipation
layer
they
are
absorbed
as
if
they
where
 continuing
to
propagate
in
the
non
simulated
part
of
the
substrate.
 • 
The
slider
is
driven
through
a
spring
connected
to
the
slider
center
of
mass.

 • 
The
slider
is
slightly
incommensurate
with
respect
to
the
substrate,
an
anti‐kink
 appears
moving
backward
with
jumps
of
5‐7
atoms
at
once

 • 
Periodic
boundary
conditions
are
applied
along
the
sliding
direction
 A
tipical
stick‐slip
profile
is
shown
in
fig.5
(a)
 where 
 the 
 friction 
 force 
 is 
 plotted 
 against
 time.
 9
atoms
slider
over
a
10
x
20
Lennard‐Jones
 substrate
 KBT
=
0.035


few
Kelvin
degrees,
v0=0.01,
 k=5.0,
vertical
load=10.0


(LJ
units)
 fig.
5
 Bibliography
and
Acknowledgments
Conclusions
 i)
 
 
Through
a
non‐markovian
Langevin
dissipation
scheme
we
can
simulate
the
dissipation
of
semi‐infinite
harmonic
 substrates
in
a
rather
small
simulation
cell
 ii)


Friction
related
phenomena
can
be
exactly
simulated
within
this
framework,
with
no
need
for
empirical
parameters
 iii)

A
comparison
with
viscous
damping
dissipation
schemes
shows
a
strong
dependence
of
the
friction
force,
and
related
 quantities,
on
the
empirical
parameters
 iv)
Using
 
the
exact
results
as
a
reference,
we
demonstrated
that
even
a
viscous
damping
dissipation
scheme
can
be
 tailored
in
such
a
way
to
reproduce
the
correct
friction
force,
once
that
the
damping
parameter
is
chosen
according
to
a
 simple
and
self
standing
procedure.
 This
activity
has
been
funded
 
by
ESF
Eurocore
FANAS‐AFRI
 [1]

S.
Adelman
and
J.
Doll,


J.Chem.Phys.


64


2375
(1976)
 






R.
J.
Rubin,
J.
Math.
Phys.



1



309
(1960)
 [2]

X.
Li
and
W.
E,



Phys.
Rev.
B



76



104107
(2007)
 [3]

L.
kantorovich,


Phys.Rev.B


78


094304
(2008)





 






L.
kantorovich
and
N.
Rompotis,


Phys.Rev.B


78


094305
(2008)
 Thanks
to:
 Alexander
Filippov
‐‐
Donetsk
Institute
for
Physics
and
Engineering
of
NASU
(Ukraine)
 Rosario
Capozza
‐‐
Universita’
degli
studi
di
Modena
e
Reggio
Emilia
(Italy)
 Giovanni
Bussi
‐‐
SISSA
Scuola
Internazionale
Superiose
Studi
Avanzati
(Italy)
 for
interesting
and
helpful
discussions.

 k v0 k v0 (b)
Viscous
damping
applied
to
the
slider
atoms


 while
the
substrate
atoms
are
frozen
 (equivalent
to
a
Frenkel‐Kontorova
model)
 (c)
Viscous
damping
applied
to
the

 substrate
atoms
only
 −γ(vi − vCM ) i i −γvi The
friction
force
now
depends
on
the
choice
of
the
damping
parameter
γ.
Fig.6
shows
this
dependence
for
the
 average
friction
force
and
for
its
variance:
dashed
line
for
case
(b)
and
dotted
line
for
case
(c),
the
blue
stripes
indicate
 the
exact
values
obtained
with
the
non‐markovian
aproach.

 fig.
6
 numbers
refer
to
fig.5
where
some
selected
 stik‐slip
profiles
are
shown
in
comparison
with
 the
exact
result
(a)


 When
we
place
a
too
high
viscous
damping
on
the
moving
slider
or
 too
close
to
the
slider‐substrate
interface,
we
prevent
the
slider
 from
exchanging
the
right
quantity
of
energy
with
the
substrate.
 This
results
in
a
too
large
friction
force.
 If
we
place
a
too
small
viscous
damping
on
the
substrate
atoms,
we
 are
not
removing
the
energy
efficiently.
The
substrate
heats
up
and
 the
friction
force
usually
results
to
be
smaller
than
the
exact
value.
 The
viscous
damping
must
be
switched
on
far
from
the
sliding
 interface:


 top:
layer
resolved
kinetic
energy
for
a
 slip
event
on
a

semi‐infinite
substrate
 (up)
and
on
a

finite
substrate
(down)
 Bottom:
phonons
excited
by
a
slip
event
 in
a
2D
substrate
 Averaging 
 over 
 many 
 long 
 simulations, 
 the
 average
friction
force
is
1.17
(LJ
units),
we
can
 now
compare
this
exact
result
(a)
with
the
 ones 
 obtained 
 employing 
 other 
 dissipation
 schemes.
 (d)
Viscous
damping
applied
to
the

 last
substrate
atoms
only
 k v0 i In
the
latter
case
(d),
it
exists
a
range
of
γ
values
(between
2
and
20)
in
which
the
friction
force
and
its
variance
are
 independent
of
γ
and
are
really
close
to
the
exact
results.

 More
interestingly
the
exact
result
is
reproduced
by
those
γ
values
which
minimize
the
substrate
average
internal
 energy
W
(see
fig
6
(c)):
 Now
even
without
the
exact
result
as
a
reference,
the
optimal
γ value
of
can
be
variationally
obtained.

 W = E(T, γ, v0) − E(T, γ, 0) −γvi fig.
1



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