Brazilian Journal of Physics, vol. 39, no. 2A, August, 2009
439
q-exponential distribution in time correlation function of water hydrogen bonds
M. G. Campo, G. L. Ferri, G. B. Roston
Departamento de Fı́sica, Universidad Nacional de la Pampa
Uruguay 151,(6300) Santa Rosa, La Pampa, Argentina
(Received on 23 December, 2008)
In a series of molecular dynamics simulations we analyzed structural and dynamics properties of water at different temperatures (213 K to 360 K), using the Simple Point Charge-Extended (SPC/E) water. We detected a
q-exponential behavior in the history-dependent bond correlation function of hydrogen bonds. We found that q
increases with T −1 below approximately 300 K and is correlated to the increase of the tetrahedral structure of
water and the subdiffusive motion of the molecules.
Keywords: Water hydrogen bonds, q-statistics, Cage effect
1.
INTRODUCTION
Water physical properties have been investigated in far
greater details, because water is one of the most ubiquitous
substances on Earth and most life processes are related with
their properties [1–3]. In particular, structural and dynamic
properties of the hydrogen bonds have been investigated using different experimental and theoretical techniques, nevertheless several issues remain unsolved. X-Ray diffraction,
neutron diffraction and proton magnetic shielding tensor measurements can be related to the oxygen-oxygen radial distribution, hydrogen bonds geometry of water and other structural properties [4–6].
A number of experimental and simulation techniques were
used to study the elemental dynamics process of rupturing
and forming of hydrogen bonds in water, characterized by an
average bond lifetime τHB and caused by diffusion and librational motions of water molecules on a very fast time scale
[7–13]. In molecular dynamics (MD), a “history-dependent”
bond correlation function P(t) is used to obtain τHB [14, 15].
P(t) represents the probability that an hydrogen bond formed
at a time t = 0 remained continuously unbroken and will be
broken at time t.
τHB =
Z∞
tP(t)dt
(1)
0
Both MD simulations and experimental data for depolarizated light scattering show an Arrhenious behavior of τHB ,
where τHB ∝ exp(−E ∗ /RT ) and E ∗ is an activation energy
[13–15]. E ∗ can be interpreted as the energy required to break
a hydrogen bond (≈ 10 KJ/mol). However, the functional behavior of P(t) has not been clearly established to date. In
previous works it was found that P(t) not to verify either an
exponential or a power law behavior in the liquid and supercooled region of water, regardless of the hydrogen bond definition used [14, 15].
In other works the dynamics of liquid and supercooled water was studied by analyzing the behavior of the mean-square
displacement time series (M(t) )
M(t) = h[r(t) − r(0)]2 i
(2)
where r(t) is the position of de oxygen atoms of water at
time t, r(0) is their initial position and the brackets hi denote ensemble average. At room temperature the motion of
water molecules at short times is ballistic, where M(t) ∝ t α
and α ≈ 2 for pure ballistic motion. At later times the water
molecules move diffusively (α ≈ 1 according to Einstein relation) and the diffusion coefficient D can be calculated. However, a plateau emerges in M(t) (0 < α < 1) at intermediate
times in the supercooled region of water (T . 273 K) where
the movement of the water molecules is called subdiffusive
[16, 17]. This phenomenon is attributed to a caging behavior
of water, in which a water molecule is temporarily trapped
by its neighbors and then moves in short burst due to nearby
cooperative motion.
The diffusion is associated with a non-Gaussian statistics,
when the Einstein relation is not satisfied [18–20]. In particular, the caging behavior of water is studied by Mazza et.
al [17] calculating the non-Gaussian parameter α2 (∆t) of the
displacement of water molecules.
Constantino Tsallis and collaborators introduced the qexponential probability distribution [21]. This can be defined through their ”complementary” distribution functions,
also called ”survival ” functions:
(1 − q) x 1/(1−q)
Pq,κ (X > x) = 1 −
κ
(3)
Tsallis et al. proposed these distributions to handle
statistical-mechanical systems with long-range interactions,
necessitating a non-extensive generalization of the ordinary
Gibbs-Shannon entropy. Following Jaynes’s procedure [22]
of maximizing an entropy subject to constraints on expectation values, they got the q-exponential ditributions, in wich
κ enforces the constrains, and q measures the departure from
extensivity, Boltzmann-Gibbs statistics being recovered as q
→ 1. Sometimes, a q-exponential behavior of dynamic variables has been observed in cases of particles with a subdifussive behavior [18, 23, 24]. However, this last behavior has
not been considered in water dynamic variables of molecular
dynamics simulations.
In this work, using a MD method, we studied the structural
and dynamics behavior of the extended simple point charge
(SPC/E) model of water over a wide range of temperatures
(213-360 K). As a result, we will show that the bond correlation function, P(t), has a non-exponential behavior and fits
to a q-exponential function. Besides, we analyzed possible
correlations between this q-exponential behavior of P(t), the
hydrogen bonds distributions and the displacement of water
molecules.
were 360, 343, 323, 313, 303, 293, 283, 273, 263, 253, 233
and 213 K.
We assigned the velocities of the molecules
440
M. G. according
Campo ettoal.
Boltzmann’s distribution in the system at 360 K. The equili273 1.11 0.02 0.994 -4.125 -1.243 353.3
bration method was similar for all systems: initially, we made
283 1.10 0.02 0.994 -4.309 -1.211 325.2
a 200 steps
of ∼
energy
continuing
with energy
a equi- as
libration
run of
1 ns minimization,
using the constant
potential
TABLE I: Fitting293
parameters
of 0.993
the bond-4.992
correlation
function
1.11 0.02
-1.047
286.4correlibrationcriterion.
run of ∼ 1Then,
ns using
constant the
potential
energyfor
as an
stability
we the
calculated
trajectory
sponding to the different simulated systems.
stability criterion.
Then,
we calculated
the trajectory
for an
303 1.10 0.02 0.993 -5.174 -1.025 266.3
additional
3
ns.
The
preceding
higher
temperature
liquid
conT / K q ∆q
r2 a . 103 fs−1
b
τHB / fs
additionalwas
3 ns.used
The as
preceding
higherpoint
temperature
liquid
configuration
the starting
for each
successive
313 1.09 0.02 0.992 -5.412 -0.985
245.3
213 1.28 0.02 0.997 -4.798 -0.780 576.4
figuration was
as of
thethe
starting
pointwere
for each
successive
simulation.
The used
values
densities
according
to the
323 1.09 0.02 0.991 -5.555 -0.975 238.8
233 1.23
0.01 0.997 -4.840 -0.842 456.7
simulation.
The
values
of
the
densities
were
according
the 2
bibliography [16, 29, 30]. The simulation time stepto was
343
1.08
0.02
0.993
-6.300
-0.855
201.8
253 1.16 0.02 0.996 -4.457 -1.068 386.3
bibliography [16, 29, 30]. The simulation time step was 2
fs and the trajectories were collected every 10 fs for all sim3600.01
1.060.996
0.01 0.995
-0.884
188.7
263 1.13
-4.280-6.342
-1.149
362.8
fs and the trajectories were collected every 10 fs for all simulations.
The sampling time is shorter than the typical time
273 1.11 0.02 0.994 -4.125 -1.243 353.3
ulations. The sampling time is shorter than the typical time
during
which
canbe
bedestroyed
destroyedbyby
libration
283 1.10 0.02 0.994 -4.309 -1.211 325.2
during whichaahydrogen
hydrogen bond
bond can
libration
movements.
to a q-exponential
function.
Besides,
we analyzed
293
1.11 0.02 0.993
-4.992
-1.047
286.4 possible
movements.
correlations
between
q-exponential
behavior
303
1.10 0.02
0.993this -5.174
-1.025
266.3of P(t), the
hydrogen
distributions
and the
displacement
313
1.09 bonds
0.02 0.992
-5.412
-0.985
245.3 of water
q
molecules.
323
1.09 0.02 0.991 -5.555 -0.975 238.8
1.3
343 1.08 0.02 0.993 -6.300 -0.855 201.8
360 1.06 0.01 0.995 -6.342 -0.884 188.7
233 1.23 0.01 0.997
-4.840
-0.842 456.7
253 1.16 0.02 0.996
263 1.13 0.01 0.996
-4.457
-4.280
-1.068 386.3
-1.149 362.8
2. THEORY AND METHOD
2. THEORY
AND METHOD
We performed
MD simulations
of the SPC/E model of water using the GROMACS package [25, 26]. This water model
assumed a MD
rigidsimulations
geometry of of
thethe
molecules,
with anof
O–H
We performed
SPC/E model
wa-dis◦ . The hydrogen
tance
of
0.1
nm
and
a
H–O–H
angle
of
109.47
ter using the GROMACS package [25, 26]. This water model
charge
is qH =of0.4238e
and the oxygen
charge
assumedatom
a rigid
geometry
the molecules,
with anatom
O–H
dis- is
−2q
.
The
parameters
of
the
Lennard-Jones
interactions
◦
tance of 0.1Hnm and a H–O–H angle of 109.47 . The hydro-between the oxygen atoms were those of the AMBER-99φ force
gen atom charge is qH = 0.4238e and the oxygen atom charge
field [27].
logq(P(t))
is −2qH . The parameters of the Lennard-Jones interactions
between the oxygen atoms were those of the AMBER-99φ
force field [27].
0
-2
323 K
1.2
1.1
1.0
2.5
3.0
3.5
4.0
-3
4.5
5.0
-1
1/T /10 K
283 K
FIG. 2: Non-extensivity parameters q values as a function of 1/T .
The2:possible
significanceparameters
of the change
of q behavior
with T isofdisFIG.
Non-extensivity
q values
as a function
1/T .
cussed
in thesignificance
text.
The
possible
of the change of q behavior with T is discussed in the text.
-6
We used a geometric definition of hydrogen bond [10] with
aWe
maximum
distance between
their of
oxygen
atomsbond
of 3.5[10]
Å and
used a geometric
definition
hydrogen
with
-8
145◦ as thedistance
minimum
angle formed
by the atoms
donor
a maximum
between
their oxygen
atomsOof
3.5–H–
Å and
O◦acceptor
. In
order to obtain
the hydrogen
bonds
distribution
145
as the
minimum
angle formed
by the
atoms
Odonor –H–
-10
function,
each to
simulation,
wehydrogen
calculatedbonds
a histogram
of
Oacceptor . Ininorder
obtain the
distribution
the
quantity
of
hydrogen
bonds
for
each
analyzed
molecule
function, in each simulation, we calculated a histogram of
-12
then normalized
it withbonds
the number
of trajectories
the
theand
quantity
of hydrogen
for each
analyzed and
molecule
0.0
0.5
1.0
1.5
2.0
2.5
number of water molecules in the system. Then, we obtained
and then normalized it with the number of trajectories and the
t / ps
f (n) with n = 1, 2, ..., 5, being f (n) the probability of occurnumber
of water molecules in the system. Then, we obtained
rence of n hydrogen bonds.
f (n)The
withbond
n =correlation
0, 1, ..., 5,function
being fP(t)
(n) the
of simoccurwasprobability
obtained from
of
n
hydrogen
bonds.
FIG. 1: Fitting of P(t) of the systems at to 283 (¤) and 323 K (O). rence
ulations by building a histogram of the hydrogen bonds lifeThe blue lines correspond to the calculated P(t).
The bond
correlation
function
P(t)wewas
obtained
fromfolsimtimes
for each
configuration.
Then,
analyzed
if P(t)
-4
FIG. 1: Fitting of P(t) of the systems to 283 () and 323 K (O). The
lines correspond to the calculated P(t).
We carried out MD simulations with a constant number of
water molecules (1158), in which we used the Berendsen’s
thermostat to apply a thermal and a hydrostatic bath to the
system, obtaining isobaric-isothermal ensembles at 1 atm of
pressure [28]. The values of the systems’ temperatures were
360, 343, 323, 313, 303, 293, 283, 273, 263, 253, 233 and
213 K.
We assigned the velocities of the molecules according to
Boltzmann’s distribution in the system at 360 K. The equilibration method was similar for all systems: initially, we made
a 200 steps of energy minimization, continuing with an equi-
ulations by building a histogram of the hydrogen bonds’ lifetimes for each configuration. Then, we analyzed if P(t) follows a Tsallis distribution of the form
expq (t) = [1 + (1 − q)t]1/(1−q)
(4)
being t the hydrogen bonds lifetime and q the nonextensivity
parameter. In the limit when q → 1, expq (t) → exp(t).
The non-Gaussian behavior of the displacement of water
molecules was studied calculating the time t ∗ , the time at
which the non-Gaussian parameter α2 (t) reaches a maximum.
The non-Gaussian parameter is
3 r4 (t)
α2 (t) =
−1
5 hr2 (t)i
(5)
(4)
being t the
hydrogen
bonds lifetime
the August,
nonextensivity
Brazilian
Journal
of Physics,
vol. 39,and
no.q2A,
2009
ps
parameter. In the limit when q → 1, expq (t) → exp(t).
270 K
Occurrence percentag
expq (t) = [1 − (1 − q)t]
4 HB
40
5 HB
441
30
20
320 K
10
0
0
210
10
240
270
300
330
360
T / K
-1
10
0.8
1.0
T
/T
MD
/K
1.2
-1
Occurrence percentage of HB
0
(b)
4 HB
0
2 HB
1 HB
0
300 K
0
0
0
3.0
3.5
1/ T / 10
-3
4.0
-1
4.5
K
FIG. 3: Average bond lifetime τHB from Montrose et al. depolar-
FIG.
3: light
Average
bondexperiments
lifetime τHB
from
et al.
depolarizated
scattering
[13]
(°)Montrose
and from our
simulation
izated
light
scattering
experiments
[13]
()
and
from
our
simulation
(¤). To facilitate comparison of the results, we scaled the temper(). To facilitate comparisonSPC/E
of the results, we scaled the temperaature of our simulation by TSPC/E
= 245 K, and the temperature of
MD
HMD
ture
of
our
simulation
by
T
= 245
and the temperature of
2O
the experimental data by TMD = 277
K (TK,
MD : Temperature at which
H2 O
the
by TMD
= 277 K (TMD : Temperature at which
theexperimental
water densitydata
displays
a maximum).
the water density displays a maximum).
The non-Gaussian behavior of the displacement of water
molecules
4 was
studied
calculating the time t ∗ , the time at
where
r (t)
and r2 (t) parameter
are the fourth
second
moments
which the
non-Gaussian
α2 (t) and
reaches
a maximum.
ofThe
thenon-Gaussian
displacementparameter
distribution,
is respectively. α2 (t) is known
q
1.3
(c)
1.2
273 K
1.1
to be zero for a Gaussian distribution [17, 31].
­
®
3 r4 (t)
α2 (t) =
−1
(5)
3. RESULTS AND
DISCUSSION
5 hr2 (t)i
­ 4 ®
­ 2 ®
where
r (t)theand
r (t) q-exponential
are the fourth behavior
and second
To study
possible
of moments
the bond
of
the
displacement
distribution,
respectively.
α
(t)
known
2
correlation function we followed this procedure: Weisassigned
to
be
zero
for
a
Gaussian
distribution
[17,
31].
successive values to q ( 1 < q . 1.3, with a step of 0.01) and
made a linear fitting 1 of logq P(t), choosing the q value with
the highest correlation
coefficient r2 . Below the 130 fs the be3. RESULTS AND DISCUSSION
havior of P(t) is strongly influenced by the libration of atoms.
Above 2.5 ps the statistics is poor, because of the unlikely
To study the possible q-exponential behavior of the bond
fact that two water molecules remain bonded by a hydrogen
correlation function we followed this procedure: We assigned
bond
at this time. Hence, we fitted all the P(t) functions in
successive values to q ( 1 < q . 1.3, with a step of 0.01) and
the
interval
from
1301 fs
fs, finding that P(t) can be
made a linear
fitting
of to
log2500
q P(t), choosing the q value with
acceptably fitted with a q-exponential and may be written as
follows
1
logq (x) =
1−x1−q
1
P(t) = P01−q
expq (at + b) = P0 [1 + (1 − q)(at + b)] 1−q
(6)
where P0 , a and b are constants. The values of q and others
parameters of the fitting for different temperatures are indicated in Table I. Fig. 1 shows, as an example, the P(t) fitting
of the systems at 283 and 323 K.
Fig. 2 shows that q ∝ T −1 for temperatures below ∼300
K, whereas this parameter is ∼1 for temperatures above this
value. This fact indicates that a clear non-exponential behavior of P(t) occurs in the region below approximately 300 K.
1
logq (x) =
1−x1−q
1−q
1.0
20
30
40
50
Occurrence percentage of HB
FIG. 4: (a) Hydrogen bonds distribution functions for n=1 to n=5
versus T . (b) Hydrogen bonds distribution functions for n=1,2 and 4
versus T −1 . (c) q versus probability of occurrence of four hydrogen
bonds.
the highest correlation coefficient r2 . Below the 130 fs the behavior of P(t) is strongly influenced by the libration of atoms.
Above the 2.5 ps the statistical is poor, because of the unlikely
FIG.that
4: two
(a) Hydrogen
bonds distribution
functions
n=0 to n=5
fact
water molecules
remain bonded
by for
a hydrogen
versusatTthis
. (b)time.
Hydrogen
bonds
distribution
functions
for n=1,2 and
bond
Hence,
we
fitted
all
the
P(t)
functions
in 4
versus
T −1 . from
(c) q versus
occurrence
fourcan
hydrogen
the
interval
130 fsprobability
to 2500 fs,of finding
thatofP(t)
be
bonds.
acceptably
fitted with a q-exponential and may be written as
follows
The q values higher than one indicate the increase of the probability that two molecules remain bonded much longer.
Although the average bond lifetimes τHB obtained using
Eq. 1 and 6 do not coincide with those of experimental data
from polarizated light scattering experiments (see Fig. 3),
both curves can be fitted by an Arrhenius behavior (τHB ∝
exp(−E ∗ /RT )). This fact was observed in previous works
[14] [15].
The hydrogen bonds distribution functions for different
temperatures are shown in Fig. 4(a). In this graphic we
can distinguish three regions : The first one between 213
and ∼270 K in which predominates the four hydrogen bonds
1.4
q
(b)
M. G. Campo et al.
1.3
M(t) / nm
2
from polarizated light scattering experiments (see Fig. 3),
both curves can be fitted by an Arrhenius behavior (τHB ∝
442
exp(−E ∗ /RT )). This fact was observed in previous works
[14] [15].
1.2
0
10
1.1
-1
10
1
1
10
t
-2
*
/ ps
f(4 HB)
10
(c)
1.2
-3
10
1
t / ps
10
100
0.8
FIG. 5: Mean square displacement of water in the systems at 213 K
(solid line), 273 K (dashed line) and 360 K (dotted line).
FIG. 5: Mean square displacement of water in the systems at 213 K
(solid line), 273 K (dashed line) and 360 K (dotted line).
0.4
The hydrogen bonds distribution functions for different
temperatures are shown in Fig. 4(a). In this graphic we can
distinguish three regions : A first one between 213 and ∼270
20
30
40
50
K in which (predominates
four hydrogen
bonds
probability
f (4)) that isthehigher
than f (3)
andprobability
f (2), a sect* / ps
( f (4))
that is
higher than
f (3)and
and∼320
f (2), aK,
second
region beond
region
between
∼270
in which
f (3) >
tween
andfinally,
∼320 K,
in which
f (3)
> f (4)
> f (2)
f (4)
> f∼270
(2) and
a third
region
above
∼320
K inand
which
finally,
third
in which f (3)regime
> f (2)almost
>
FIG. 6: (a) time when α2 (t) reaches a maximum t ∗ vs. 1/T . (b)
f (3)
> fa(2)
> region
f (4). above
In the∼320
lowerKtemperature
1
f
(4).
In
the
lower
temperature
regime
almost
50
%
of
wa50 % of water molecules have four hydrogen bonds, this per-Non-extensivity parameter q vs. t ∗ . (c) g(t ∗ ) vs f (4), g(t ∗ ) = (t ∗ )− 4
ter
molecules
have
four
hydrogen
bonds,
this
percentage
decentage decreases when the temperature increases, reaching.
creases when the temperature increases, reaching 20 % when
20 % when T = 360 K. In Fig. 4(b) we can observe the ocT = 360 K. In Fig. 4(b) we can observe the occurrence of a
currence
of relation
a reciprocal
relation
(2),and
f (1)
reciprocal
between
f (2), between
f (1) and ff(4)
the and
tem- f (4) To study the correlation between the structural transition
and
the
temperature.
These
results
coincide
only
qualitatively
and the change of q we analyzed the behavior of q versus f (4)
perature. These results coincide only qualitatively with the
with
ones by
obtained
Sutmann
and According
Vallauri [32].
onesthe
obtained
Sutmannbyand
Vallauri [32].
to the Ac-(see Fig. 4 (c)). We can observe that for temperatures below
cording
to theand
water
model and
thedefinition
hydrogenused,
bond
definition300 K there are a linear correlation between both variables.
water model
the hydrogen
bond
amongst
used,
other
things,
results
from different
simula-In other words, the produced structural change below approxotheramongst
things, the
results
from the
different
simulations
may quantions
may differ.
quantitatively differ.
imately 320 K causes the change in the statistical behavior
titatively
To study the correlation between the structural transition
and the change of q we analyzed the behavior of q versus
f (4) (see Fig. 4 (c)). We can observe that for temperatures
below 300 K there is a linear correlation between both variables. In other words, the produced structural change below
approximately 320 K causes the change in the statistical behavior of P(t) and, as a consequence, a significative increase
of the probability of two water molecules be bonded much
longer.
Fig. 5 shows the mean square displacements for the systems at 213, 273 and 360 K. M(t) shows some evidence
of the caging behavior for water at temperatures below 300
K, where we can observe the appearance of an intermediate
plateau at times between 0.1 and 100 ps that indicates a subdiffusive motion of the water. At longer times, there is a recovery of diffusive motion. We can observe from Fig. 6 (a) that
t ∗ increases with the decrease of T , according to the results
obtained by Mazza et al. [17]. Clearly, t ∗ ∼ T −1 at temperatures below 300 K. This fact is attributable to the increase of
the subdiffusive motion of the water.
Fig. 6 (b) shows that q and t ∗ are correlated when the temperature drops below 300 K, which is an expected result because both parameters have a similar behavior with this variable. These changes in q and t ∗ are produced by a modification in the water structure when f (4) outnumbers f (2) at
FIG. 6: (a) time when α2 (t) reaches a maximum t ∗ vs. 1/T . (b)
1
Non-extensivity parameter q vs. t ∗ . (c) f (4)− 4 vs t ∗ .
approximately 320 K, and a structure in which prevails three
and four hydrogen bonds begins to be the most frequent one
in the water.
Moreover, in Fig. 6 (c) we can see the correlation between
t ∗ and f (4) for values corresponding to the systems below
1
300 K, in which it is observed that t ∗ ∼ f (4)− 4 .
Considering these results, the four hydrogen bonds probability associated to a tetrahedral structure in water is a critical
variable in the phenomenon of ”cage effect” that occurs at
Brazilian Journal of Physics, vol. 39, no. 2A, August, 2009
low temperatures and the dynamics of the hydrogen bonds
that changes from an exponential to a non-exponential distribution.
4.
CONCLUSIONS
We have shown that the temporal correlation function of
hydrogen bonds P(t) has a q-exponential behavior. The
nonextensivity parameter q takes values above 1 below approximately 300 K. This increase of q indicates the increase
of the probability that two molecules remain bonded during
a long time t. The transition of P(t) from an exponential to
[1] D. Eisenberg and W. Kauzmann, The Structure and Properties
of Water (Clarendon Press, Oxford, 1969).
[2] F. H. Stillinger, Science 209, 451 (1980).
[3] O. Mishima and H. E. Stanley, Nature (London) 396, 329
(1998).
[4] A. H. Narten, M. D. Danford and H. A. Levy, Faraday Discuss.,
43, 97 (1967).
[5] A.K. Soper, F. Bruni and M. A. Ricci, J. Chem. Phys. 106, 247
(1997).
[6] K. Modig, B. G. Pfrommer and B. Halle, Phys. Rev. Lett.,
90(7), 075502 (2003).
[7] C.A. Angell and V. Rodgers, J. Chem. Phys. 80, 6245 (1984).
[8] J.D. Cruzan, L.B. Braly, K. Liu, M.G. Brown, J.G. Loeser and
R.J. Saykally, Science 271, 59 (1996).
[9] S. Woutersen, U. Emmerichs and H. Bakker, Science 278, 658
(1997).
[10] A. Luzar and D. Chandler, Phys. Rev. Lett. 76, 928 (1996).
[11] A. Luzar, J. Chem. Phys. 113, 10663 (2000).
[12] F. Mallamace, M. Broccio, C. Corsaro, A. Faraone, U. Wandrlingh, L. Liu, C. Mou, and S.H. Chen, J. Chem. Phys. 124, 124
(2006).
[13] C.J. Montrose, J.A. Búcaro, J. Marshall-Coakley and T.A.
Litovitz, J. Chem. Phys. 60, 5025 (1974).
[14] F.W. Starr, J.K. Nielsen, and H.E. Stanley, Phys. Rev. Lett. 82,
2294 (1999).
[15] F.W. Starr, J.K. Nielsen, and H.E. Stanley, Phys. Rev. E. 62,
579 (2000).
443
a q-exponential behavior occurs in parallel with a structural
modification in water, where the probability of occurrence of
four hydrogen bonds outnumbers the one of two hydrogen
bonds. The increase of q is also correlated with the increase
of the Non-Gaussian behavior of water displacement. This
fact is associated which the increase of t ∗ , the time at which
the Non-Gaussian parameter α2 (∆t) reaches a maximum.
Acknowledgment
M.G. Campo, G.L. Ferri and G.B.Roston are grateful for
the financial support by PICTO UNLPAM 2005 30807 and
the Facultad de Ciencias Exactas de la UNLPam.
[16] S. Chatterjee, P.G. Debenedetti, F.H. Stillinger and R.M.
Lynden-bell, J. Chem. Phys. 128, 124511 (2008).
[17] M.G. Mazza, N. Giovambattista, H.E. Stanley and F.W. Starr,
Phys. Rev. E 76, 031203 (2007).
[18] B. Liu and J. Goree, Phys. Rev. Lett. 100, 055003 (2008).
[19] T. H. Solomon, E. R. Weeks, and H. L. Swinney, Phys. Rev.
Lett. 71, 3975 (1993).
[20] S. Ratynskaia et al., Phys. Rev. Lett. 96, 105010 (2006).
[21] C. Tsallis, J. of Statistical Phys. 52, 479 (1988).
[22] E. T. Jaynes, Essays on Probability, Statistics, and Statistical
Physics (Reidel, London, 1983).
[23] C. Tsallis et al., Phys. Rev. Lett. 75, 3589 (1995).
[24] C. Tsallis, Brazilian J. of Phys., 29(1), (1999).
[25] H.J.C.Berendsen, J.R. Grigera, T.P. Straatsma, J. Phys.Chem.
91, 6269 (1987).
[26] H.J.C. Berendsen, D. van der Spoel and R.V. Drunen, Comp.
Phys. Comm. 91, 43 (1995).
[27] E.J. Sorin and V.S. Pande, Biophys. J. 88 (4), 2472 (2005).
[28] H.J.C. Berendsen, J. Postma, W. van Gusteren, A. Di Nola and
J. Haak, J. Chem. Phys. 81, 3684 (1984).
[29] L.A. Báez and P. Clancy, J. Chem. Phys. 101(11), 9837 (1994).
[30] B. Taras and A.D.J. Haymet, Molecular Simulation 30, 131
(2003).
[31] A. Raman, Phys. Rev. 136, 405 (1964).
[32] G. Sutmann and R. Vallauri, J. of Mol. Liq. 98-99, 213 (2002).