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2. mathematics
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Journal of Molecular Liquids, 34 ( 1987) 269-283
ElsevierSciencePublishersB.V.,Amsterdam-PrintedinThe
THE DYNAMICS
OF LIQUID WATER
: ELECTRIC
269
Netherlands
FIELD EFFECTS
M. W. EVANS*
University
College
of Swansea,
* Current Address:
(Received
Singleton
IBM, Dept. 48B/428,
22 September
Park, Swansea,
Neighborhood
SA2 8PF
Rd., Kingston,
NY 12401,U.S.A.
1986)
ABSTRACT
The technique
investigate
of field-effect
the decoupling
related
non-linear
adapted
for this purpose
transient
effect;
effects
acceleration
computer
the fall transient
in liquid water
at electric
Langevin
function.
This is interpreted
acceleration;
and
Using a pair potential
and persistent;
field strengths
to mean
and that the system
has been used to
ST2, it is found that fall
is pronounced
decoupling
are very strong,
at 300 K.
from the well known
in water
moderate
water
simulation
sufficient
and accompanies
to saturate
that non-linear
is statistically
a
the
effects
in
non-Markovian
and non-Gaussian.
INTRODUCTION
The technique
of field-effect
computer
identify
and characterise
matter.
These are the field decoupling
Evans
Cl1
article
and confirmed
[21
using
the modifications
fundamental
try to explain
framework,
recently
of motion,
Langevin
the decoupling
effect means
effect analytically
0 1987ElsevierSciencePub1ishersB.V.
link between
and the
equation
falls outside
at all.
that
I51 as Reduced
equation,
If the diffusion
the effect might not be predicted
0167-7322/87/$03.50
and shows in detail
a rigorous
such as the Liouville
equations.
C31 and
theory in order to begin
summarised
seems to be able to provide
acceleration
A review
methods.
The field decoupling
class of equations,
equations
much simpler-to-solve
diffusion
state of
by
and co-workers
and numerical
to conventional
of the new effects.
(R.M.T.),
theoretically
by Oxtoby
on the work in this area [41
necessary
only one particular
Model Theory,
algorithms
Cl] has been used to
of the condensed
effect and fall transient
numerically
independent
is now available
to take account
simulation
properties
The first of these was predicted
or deceleration.
Grigolini
two general
used to
the R.M.T.
For example,
the
the
270
diffusion
elegant
equation
solution
reproduce
used by Praestgaard
to the problem
the decoupling
with the inherently
acceleration
removal
or deceleration
from a molecular
difficult
to describe.
acceleration
potential
nature
potential
in nature.
at all it is necessary
ensembles
involved
of the fall transient
orientational
following
between
C41
in comparison
Furthermore
the acceleration
behaviour
or using strong fields in a numerical
theory of molecular
at all, C71
Furthermore,
strong applied
Debye's
to inter - molecular
diffusion
other alteration
possible
electric
equations
dissipation
simulation.
a fall transient
C81,either
a decoupling
For
theorem holds rigorously.
effects
and adapted
effect for arbitrary
effect
[2,51
equations
of
but if no
diffusion,
it becomes
but there is still no
and equilibrium
dependence
no
coefficient
function,
of rotational
The fall transient
but linear diffusion
acceleration
D.C. or A.C.
If the friction
by a memory
the field decoupling
in this case retain an identical
for non-Markovian
field
The effect is not a
is that the theory contains
potentials.
is replaced
acceleration.
equilibrium
for inertial
fields,
is made to the equation
to describe
fall transient
a.c.f.
- will not provide
The reason
171
or deceleration
theory of Debye - the great "classical"
theory will not provide
field strength.
reference
Debye's
diffusion
ensemble.
even when the theory is corrected
for arbitrarily
external
of the molecular
diffusion
with
tells us a lot about the normal,
free, equilibrium
the rotational
equations
or rotational
with the appropriate
function
and interacting
the fall transient
this must be done for
banal consequence
example,
on the recognition
Langevin
in the translational
In other words
auto-correlation
the instantaneous
diffusing
to supplement
The
field is much more
In order to describe
in that equation.
at equilibrium.
and
but does not
of the liquid state of matter.
of a strong applied
terms that are -non-linear
coordinates
by these authors,
It relies for its description
is not linear
is a plausible
It is known now that this has to do
of fall transients
ensemble
of the fact that the effective
molecules
considered
effect at all.
non-Markovian
161
and van Kampen
upon time.
orientational
In other words
the first fluctuation
-
The same is true for translational
diffusion.
The
level of sophistication
primary
of fall transient
acceleration
side of these diffusion
effective
is reached
equations
inter-molecular
which
irrespective
translational
coordinate,
or not with a memory
141
for an analytical
by a representation
is non-linear
of whether
function.
description
when the terms on the left hand
are supplemented
potential
been replaced
v = - v. cos8(t)
needed
of the
in the rotational
the friction
coefficient
One of the simplest
or
has
of these is:
(1)
271
which
could for example
between
two molecular
This automatically
represent
dipoles
the analytical
0
$'(t-r)8(r)dr + Vosin8(t)
where
$ is the memory
Mori,
C91,and W(t) is a Wiener
rotational
function
diffusion
an elementary
acceleration
analysis.
coordinate
of the
However,
the original
An equation
the introduction
equation
of a model
inside a diffusing
the diffusion
truncation
of the angular velocity
the full equivalence
linear itinerant
oscillator
when the original
so called
"non-linear
representation
subjected
and co-workers
to describe
numerically
simulation
behaviour
using
algorithms.
model"
pair potential
have shown C41
function.
Recent work
Cl47
and also that the
cage and encaged molecule
In this eventuality
becomes
a relatively
diffusion
the
simple
of molecules
that this model
can be used
and decelerations
molecular
dynamics
link between
pair - potential.
of this type are necessary,
predictions
fraction
of type (1).
three fully independent
work of Grigolini
an
and more transparent
accelerations
and the effective
Cl11
bound harmonically
corresponded
This has forged an analytical
and analytical
the theoretical
between
of the rotational
observations
oscillator.
of the Mori continued
correlation
oscillator
level of the
in two dimensions
of type (1).
fall transient
of fall transients
that real experimental
numerical
itinerant
in two dimensions
Grigolini
of a molecule
of the two approaches,
binding
potential
to an inter-molecular
successfully
observed
harmonic
as solving
can be met,
itinerant
takes on a more plausible
by a non-linear
Cl01
by Coffey and Calderwood
that this model
has demonstrated
is replaced
almost as difficult
cage, treated as a rigid entity,
to a particular
the need
analysis
to go to the next higher
motion
into the
implies
and moment
This difficulty
first introduced
It was shown by Evans Cl21
Cl31
becomes
"ab initio".
it is possible
and aimed at describing
expansion
function
procedures,
C51 and to deal with the non-linear
framework
This is a version
mathematically
and fall transient
field is introduced
of the memory
(2), so that the problem
but alternatively,
annulus.
electric
truncation
by
the random torque of the
of type (2) is capable of providing
of both the decoupling
fractions,
Liouville
type first introduced
representing
effects when the external
to solve eqn.
meaning
t.
to two dimensions.
= W(t)
of the simple
process
equation.
description
to use continued
R.M.T.
them at the instant
consideration
0 is also the rotational
t
I
energy
equation:
B(t) +
Cl01
of the potential
on the angle 8 between
restricts
In this case the coordinate
Langevin
the dependence
in real molecular
and co-workers
liquids.
[4,5,131
computer
the
It follows
to test the
Furthermore,
and of Coffey
and
212
co-workers
[11,141
apparently
disparate
with
as V. of eqn (1).
acceleration
paper,
this model has shown how it may be used to inter-relate
phenomena
Among
effects.
in terms of a few well defined
these are the decoupling
This implies
Section
methods
and of the parameters
subjected
to saturate
done in practice
radiation
electric
field strengths
measurements
remains
Lennard
function.
that characterises
In section
the principal
The original
molecular
moment
noting
effect.
tetrahedral
of inertia
that the analytical
to show up deviations
in particular
from
function
computer
acceleration
and
that
In this respect we
.
and Rahman
disposition
Cl81
to
of partial
charges
1191
in terms of the fall transients
to coincide with the I axis of
frame.
velocity
The decoupling
effect
is
in axes 1,2 and 3 of the moment
it is shown that the complete
is statistically
description
equation
non-Markovian
of the computer
that is non-linear,
These three characteristics
in section
the
a.c.f. 's such as those of the angular velocity,
using the model of the non-linear
Finally
of Stillinger
in this section
have to rely on a diffusion
also non-Gaussian.
even with
Cl61 accurate
auto correlation
cl71 by other methods.
J+,,which happens
a.c.f. 's investigated
This implies
with electro-
However,
to liquids such as water
are presented
and rotational
Finally
frame
This could not be
of .such investigations,
work
ST2 has been retained.
using equilibrium
frame.
equilibrium
absence
and
Jones terms, and this gets rid of their artificial
11 the results
angular momentum,
inertia
orientational
the simulation
of the water dipole moment,
illustrated
might be expected
exhaustively
in terms of
algorithms
in Kerr effect experiments,
the simple ST2 potential
involve atom-atom
function.
the only way of looking at fall transient
to extend
have been investigated
have adapted
simulation
laser technology.
of the equivalent
In the continued
it is important
switching
available
in this
of 108 water molecules
field, but might be plausible
on the fall transient
at equilibrium.
simulation
the Langevin
using contemporary
the time dependence
dynamics
field in the z axis of the laboratory
Cl,151
with electric
magnetic
the computer
the molecular
to a static electric
strong enough
to be described
self consistently
it contains.
1 of this paper describes
used to investigate
such
and fall transient
that the results
for water at 300 K, could also be described
the same model,
parameters
itinerant
of
set of
in nature.
simulation
non Markovian,
will
and
can now be inter-related
[Sl
oscillator.
three some suggestions
the need for experimental
for further work are given,
data on the fall transient
273
SECTION
1:
COMPUTER
A complete
SIMHLATION
description
those of Stillinger
available
of the algorithm
employed
and comparison
of results with
Cl81 and of Clementi et al., C201
and Rahman
in an accompanying
the methods
ALGORITHM
paper.
in treating
Here we restrict
Cl91
is
the description
the sample with very intense
external
to
electric
fields.
The key to such methods
Cl1
is that the kinetic
is always kept constant,
both during
place in the picoseconds
after the field is applied,
transient
process
is achieved
easy to control
energy)
during
equilibrium
no different
in a standard
the fall transient
of the external
strong enough
release
external
thermodynamic
transient
Table
dynamics
does not affect
resealing
C211
and the equivalent
ensembles
kinetic
terms
part of the Hamiltonian,
is reduced
considerably
by the
in the ensemble.
to equilibrium
infinitesimally
This theorem actually
equilibrium
With
after its
dissipation-theorem,
removed
implies
from
that the fall
a.c.f. must be identical
in their
1
Mean Energies,
Kinetic
Field Off and Field On.
Potential
(kJ / mole)
(kJ / mole)
This
It is
at field free
In energetic
at all by the first fluctuation
for statistical
equilibrium.
field.
routines.
here, and the procedure
the kinetic
the regression
takes
the fall
and translational
simulation.
torque -8xg on each molecule
fields $
which
of the electric
of interest
in strong contrast,
cannot be described
which was derived
process
process,
and also during
from the control of temperature
molecular
process
energy,
removal
temperature
(i.e. the rotational
the fall transient
but the potential
release
with standard
the temperature
is, in actuality,
the rise transient
after the instantaneous
numerically
energy of the ensemble
uE(kJ / mole)
Translation
Rotational
3.78
3.88
-35.77
3.85
3.76
-35.27
0.50
3.85
3.81
-25.12
10.65
3.85
3.74
-17.34
18.43
3.90
3.73
-15.58
20.19
0.00
214
time dependence.
investigations
perturbation
Note that this is assumed
of molecular
dynamics
of the ensemble
The application,
with external
therefore,
implicitly
in all the experimental
based on spectroscopy
122,231
i.e. the
radiation.
of a z axis electric
field &,
produces
the
torque
-txE
on each H 0 molecule.
If the unit vector E, is defined in
%Z
2
the dipole moment axis t the equilibrium average <e12(t)>
becomes non-zero
a result of this torque.
over the 108 molecules
Here the average
at each instant
dependence
of this average
saturates,
Cl1
can be removed
t after field application.
instantaneously
to generate
was continuously
table 1, together with the rotational
the external
the fall transient.
monitored
The
This eventually
level, after which
a plateau
as
is just a simple average
on time is the rise transient.
i.e. reaches
energy in the simulation
<e12(t)>
and is summarised
and translational
kinetic
field
The potential
in
energies.
RESULTS AND DISCUSSION
With the use of computer
Langevin
function
saturation
for water
simulation
This function
(fig. (1)).
level of each rise transient
this figure is the theoretical
Langevin
it is possible
plotted
function
against
to saturate
was constructed
uE/kT.
the
from the
The curve in
and the points are obtained
L
Fig. 1.
The Langevin
a)
of rise transients
b)
Rise transients
(1) - (5)
from the computer
<eZ>
Increasing
corresponding
function
simulation
Saturation
see fig. (lb).
plotted vs. time in picoseconds.
field strength;
respectively
L (nE/kT).
(6) - (7) fall transients
to (3) and (2).
levels
‘IO
from the simulation
stronger
Fig.
dipole
(fig. (lb)).
the field the shorter
The rise transients
(2) plots three fall transients
a.c.f.
the a.c.f. and are therefore
large and is also persistent.
characteristic
available
I
I
Fig. 2.
water
Field-off
Table
water
1
potential
(potential
the strongest
constant
Although
energy
C
Qtr
and translational
= - 35.77 kJ/mole)
field
the atom-atom
applied
for
the field free
in the presence
The sample
of
is clearly
of liquid water.
of the H20 molecule
<l$(t)~T(0)>/(<W2(0)> t <v2(o)> 11
This
field and the fall - transient
a property
functions
field.
between
is small.
ways of seeing this is through
time cross-correlation
in
of fig. (3).
structure
of the strongeat
energy
functions
p.d.f. 's are little affected
properties
kinetic
and equilibrium
(-15.58 kJ/mole)
of these is:
=
function
for all E, but the mean
pair distribution
in equilibrium
of fig. (2) is therefore
field, the dynamical
simplest
auto-correlation
in this simulation
by the atom-atom
applied
One of the clearest
field induced
dipole
is more than halved by the strongest
still a liquid in the presence
acceleration
with
,
equilibrium
These show that the difference
case
can be detected
Cl61
0.24
1 shows that the rotational
energy
is very
that this is
field strength.
is approximately
point is emphasised
to it for
effect
. . . . . . . Fall transients <es> , (see text).
<e(t).y(o)>/<p'>.
(l)-(4): Decreasing
with respect
and that these accelerations
0.12
equilibrium
to zero than
acceleration
to assume
field Kerr effect apparatus.
0
- the
at field free equilibrium.
considerably
The fall transient
It is reasonable
of real water,
electric
the appropriate
decay much more rapidly
accelerated
shown in the figure.
against
of course,
computed,
<#t).~(o)>/+
It can be seen that the fall transients
all pE/kT
are field dependent
the rise time.
even by the strongest
are changed
significantly.
the appearance
C24-271
(c.c.f.'s).
of electric
The
276
Atom atom pair distribution
Fig. 3.
Field-off
. . . .
where
Field-on
equilibrium
equilibrium
the subscript
T
arguments.
(potential
denotes
have shown analytically
symmetry
(potential
that $&
However
vector
functions.
energy = -35.77 kJ/mole)
energy
= -15.58 kJ/mole).
transposition,
vanishes
RycLaert
for all t when B = 2
fig. (4) shows clearly
et al. [281
using general
that the (x,y) and (y,x)
elements
of C
appear directly
in the laboratory frame (x,y,z) for E > 2
str
liquid water.
This result confirms others in the literature C241 obtained
using two independent
speaking
therefore,
condensed
molecular
numerical
computer
cross correlation
matter whenever
simulation
functions
the sample
algorithms.
is subjected
Fig. 4.
(x,y) and (y,z) elements of the cross-correlation
<wy(t)vxW
<oxWvy(o)>
(X,Y) z
I
; (y,x):
,
I
.,~,o)>f<";,o)>~
<p(o)>~<"~(o,>~
at field on equilibrium,
Strictly
such as these appear
energy as for fig. (3).
in
to a symmetry
function
C
Qtr
in
(x,y) and (y,x) elements
Fig. 5.
<(e(t)
(X,Y)
2
at field-on
breaking
liquids
;
in dielectric
takes no account
in frame
(x,y,z).
C25-271,
that may exist in
molecular
Another
moments
example
of
is shown
of the second order c.c.f.
<lJJ2(o)><v2(o+
Coriolis
In fig.
(x,y,z).
of fig.
acceleration
(5) the two elements
and its own angular
velocity
again seem to be mirror
to pick up from the noise
images,
EFFECTS
in this section.
frame a.c.f.'s,
The decoupling
and some of these are
effect
in
than the mirror-image
(4).
These show up in laboratory
illustrated
and Kerr effect
x $t)&T(O)>
the molecular
THE DECOUPLING
c.c.f.'s
(1,2,3) of the principal
or for $ > 2, directly
=
diffusion
Recent work has shown furthermore
(5) in the shape of two elements
but are much more difficult
elements
etc.
theory of rotational
in frame
<t(t)
frame
;
field, as, for example,
The simple
of these effects.
c,(t)
between
1
is only one out of very many possible
inertia,
in fig,
x ~w)x~yb)’
<w*(o)xv2(o)>
external
that C
%tr
molecular
$2
equilibrium.
spectroscopy.
whatsoever
of the cross-correlation
can be identified
by
278
looking at the envelope
of the oscillations
induced by the external
It is present when the time dependence
of this envelope
than that of the equivalent
a.c.f.
C61 fail to produce
analytical
models
inherently
Markovian
independent
virtually
in statistical
analytical
no attempt
investigation
and numerical
frequencies,
in a liquid and monitoring
fast detector
well-defined
Fig. 6.
confirmations,
however,
the former are
The effect has received many
[4,51
but there has been
Its experimental
by the use of induced birefringence
using a giga watt
laser to induce birefringence
with a submillimeter
systems.
theoretically
is out of balance
this effect because
nature.
The present
and numerically,
laser or an interferometer
situation,
where
and un-investigated
of the decoupling
functions.
1.
Angular
velocity
a.c.f. at field off equilibrium.
2.
Angular
velocity
a.c.f. at field on equilibrium.
. . . . . . Envelope of the field oscillations
Rotational
kinetic
energy a.c.f. at field off equilibrium.
4.
Rotational
kinetic
energy a.c.f. at field on equilibrium.
of the field oscillations
the extension
field free time dependence
a.c.f.
<x(0).8(t)>
<W2>
<t(t)
in 4).
of the oscillation
envelope
of the a.c.f. is visible
with respect
.$gt)@(o) .~(O)>/d>
to the
in the angular velocity
(fig. (6)) and in the second order rotational
energy a.c.f.
auto-
in 2).
3.
For water
is
experimentally,
effect with the angular velocity
correlation
Envelope
with
the phenomenon
and unsatisfactory.
Illustration
---_-
field.
to be slower
It is known now that some
to look for it experimentally.
could be achieved,
at far infra-red
suitably
field-free
begins
kinetic
279
1.0
1.0
_$.5
d
0.5
d
0
0
OO
-0.5
0
0.2 tcpS1
0.1
1.0
0.05
0
ICI
,D
0.5
‘t.
$0
-0.5
+
0
0.05
0.10
tcps1
Fig. 7.
a)
Field decoupling
effect
in
<,$1(t)
. &T(O)>/ <&f >
;
b)
Field decoupling
effect
in
c,&(t)
. g2(0)>/
<eg >
;
cl
Field decoupling
effect
in
c&(t)
. &(o)>/<
62 >
.
Fig. 8.
Field decoupling
effect
3
in the angular moment
a.c.f.
0.10
t(ps1
280
It is confirmed
in the three rotational
<&W . &(o)>/<q>
;
4,(t)
. &(o)>/<e,2
> ;
<g,(t)
. $3(o)>/42>
;
(fig. (7)); in the angular momentum
angular
all aspects
of water,
E simultaneously
of the molecular
needing
investigation
to express
[4,14l
acceleration,
the centre of mass linear velocity,
velocity,
motion
experimental
a.c.f.'s
(fig. (8)) and in the first and
a.c.f.
second order a.c.f. 's of the Coriolis
considers
velocity
C25-271 which
x, and the resultant
(fig. (9)).
Therefore
and is clearly
Our future
investigation.
and to provide
results
interpretation
occurs
in
property
analytical
itinerant
in terms of the parameters
a self consistent
molecular
decoupling
a fundamental
of these results will use the non-linear
these numerical
automatically
oscillator
of that model
of the results of this
paper.
Fig. 9.
Field decoupling
Coriolis
acceleration
NON-MARKOVIAN
in the first and second order a.c.f.'s
AND NON-GAUSSIAN
PROPERTIES
OF LIQUID WATER
This section deals finally with these properties,
and defined
in the literature.
[4,12,14,22,231
properties
of liquid water have been known
Stillinger
and Rahman.
velocity
a.c.f.
118,19,201
of the
(see text).
Cl81
in frames
which
dynamical
since the first simulations
It is well known,
for example,
(x,y,z) and (1,2,3) has a negative
and is not a simple exponential.
are characterised
The non-Markovian
by
that the linear
overshoot
The simple Langevin
equation
would
281
produce
Doob's
an exponential
theorem.
the whole
decay,
[22,231
range of correlation
The non-Gaussian
Berne and Harpe
This property
[291
nature
can be detected
and comparing
a.c.f.
C41
confirms
that the dynamics
in the well-defined
in describing
the evolution
thermodynamic
equilibrium
Clearly
capable
frame
rigorously
Gaussian
for this work and the
in the liquid state at 300 K are
sense that Gaussian
of describing
builds
statistics
do not succeed
self-consistently.
becomes
this assumption
data reported
the complete
At
Gaussian
into the
here must be interpreted
range of new phenomena
capable
(1,2,3).
in the absence
implies
(1,2,3)
[4,141
the existence
For example,
of which A
of an electric
This implies
(1,2,3) and these are illustrated
elements)
must vanish
out of the three vectors
therefore
theoretically
the external
in the product
on the grounds
electric
field.
that the diagonal
and all elements
are equal.
simulation
of molecular
in
survives
Cl91
of this tensor must vanish
for all the molecules
shows that the computer
which
elsewhere.
This is indeed what is found in the simulation.
the tripleproduct
functions
is reducible
is the only property
that all other elements
field
of cross correlation
the triple product
1
of anisotropy.
(the off diagonal
self
itinerant
of this.
liquid sample in the absence
exist in frame
The group theory
exist
from the first order
Future work aims to show that the non-linear
after averaging
theory
analytically
assuming
simulation
(1,2,3) as nine elements,
(x,y,r).
such
(i.e. t + m) their behaviour
is qualitatively
in the moving
in frame
second order a.c.f.'s
energy:
of water
theory can be used to predict
elements
by
liquid.
of these two a.c.f.'s
computer
For an isotropic
frame
in nature.
for carbon monoxide
out this exercise
the range of numerical
consistently.
group
non-Markovian
C301
using a model
oscillator
simulation
<~(t).F(O)>I&
not Gaussian,
algorithm.
through
show that
liquids was first detected
that calculated
We have carried
once more, because
process
for liquid water
is similarly
by investigating
kinetic
the result with
linear velocity
statistics.
of the Markov
. I>/&,
. x(t)x(o)
<x(t)
functions
of molecular
using computer
as that of the translational
result
characteristic
These and other results
in frame
At the time origin
in the ensemble
because
The application
of group
provides
symmetry
the elements
in the absence
of
two
which
282
ACKNOWLEDGEMENTS
The University
Fellowship
of Wales
is thanked
for the award of the Pilcher
and IBM for the award of a Visiting
Senior
Professorship.
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