q-exponential distribution

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UNIVERSIDAD NACIONAL DE LA PAMPA Facultad de
Ciencias Exactas y Naturales
q-exponential distribution in time correlation
function of water hydrogen bonds Campo, Mario
G., Ferri, Gustavo L., Roston, Graciela
B. Departamento de Física. Facultad de Ciencias
Exactas y Naturales de la UNLPam. Uruguay 151.
Santa Rosa (L.P.) Argentina.
V Workshop de Mecánica Estadística y Teoría de la
Información Mar del Plata Abril 2009
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Water structure Whats hydrogen bond?
Hydrogen bond (HB)
In water the HB energy 23.3 kJ mol-1 compared
with 492.2148 kJ mol-1 energy in covalent bond.

• HB in water is 90 electrostatic
• and 10 covalent.
• HB restricts the water neighboring.
• The HB direction is that of the shorter
• O-H (Odonor Oaceptor )

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• Two criteria to define HB
• Energetic
• O-O distance?3.5 Å
• O-O interaction energy gt EHB
• Geometric
• O-O distance?3.5 Å
• O-HO angle gt ?HB

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Water structure HB distribution

• Water is connected by a random tetrahedral
  network of HB.
• HB distribution.

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Whats the importance of the hydrogen bonds?

• Anomalous properties of water are influenced by
  the behavior of hydrogen bonding.

residence time
10 fs
20 fs
30 fs
40 fs
Measurements of lifetimes are made depolarized
light scattering techniques
P(t) can be obtained from simulations by building
a histogram of the HB residence times.
C.J. Montrose, et al., J. Chem. Phys. 60, 5025
(1974).
Starr F.W., Nielsen J.K., and Stanley H.E., Phys.
Rev. Lett., 82, 2294-2297, (1999).
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Behavior of P(t) do not have neither power-law
nor exponential behavior.
Starr F.W., Nielsen J.K., and Stanley H.E., Fast
and slow dynamics of hydrogen bonds in liquid
water, Phys. Rev. Lett., 82, 2294-2297, (1999).
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Molecular dynamics simulation
GROMACS package. System with 1185 SPC/E water
molecules. 12 independent systems at different
temperatures(213 to 360 K) and 1 atm. Cut-of
radius for the interaction potentials 1.3
nm. Berendsens bath of temperature and
pressure. 2.5 ns for equilibration. 5 ns
aditional simulation ? results. ?t simulation
2 fs. ?t data collection 10 fs.
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P(t) do not have neither power-law nor
exponential behavior.
T273 K
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We found that P(t) can be fitting with a
q-exponential function
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T/K q
?q 360 1.04 0.01 343 1.06 0.01 323 1.05 0.01 313
1.07 0.01 303 1.09 0.01 293 1.09 0.01 283 1.1 0.02
273 1.13 0.01 263 1.15 0.01 253 1.18 0.02 233 1.2
2 0.02 213 1.27 0.02
q(T) behavior

• q increase with the decrease of T.

• Above 300 K, P(t) decays exponentially with T
  (q1)

• qT-1 (Tlt300 K)

300K
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Changes in the hydrogen bond structure with
temperature
270 K
300 K
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reciprocal relation between HBs and T (similar to
q(T) at Tgt300 K).
When T decrease, at 300 K 4 HB percentages
exceeds that 2 HB
structural transition of 4 HB -tetrahedral
structure to 3 HB -2 HB structure
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300 K
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below 300 K there are a linear correlation
between the tretrahedral structure of water and
q.
300 K
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Cage effect
mean square displacement (MSD)

• Cage effect occurs
• in SPC/E model
• simulations
• (Chaterjee et al.,
• J. Chem. Phys. 128, 124511
• (2008).
• Cage effect increase with
• the decrease of T

Subdiffusive behavior ? cage effect
qGaussian distribution of the displacement of
particles correlated with anomalous diffusion.
Liu and Goree, Phys. Rev. Lett. 100, 055003
(2008)
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MSD in our MD simulations
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The non-Gaussian behavior of the displacement of
water molecules was studied calculating the time
t, the time at which the non-Gaussian parameter
a2(t) reaches a maximum. The non-Gaussian
parameter is Where r 4(t) and r 2(t) are the
fourth and second moments of the displacement
distribution, respectively. a2(t) is known to be
zero for a Gaussian distribution M.G. Mazza et
al. Phys. Rev. E 76, 031203 (2007).
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• t is correlated with f (4) for values
  corresponding to the systems below 300K. It is
  observed that f(4) (t)-1/4.
• The increase of q is also correlated with the
  increase of the non-Gaussian behavior of water
  displacement.

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CONCLUSION

• The temporal correlation function of hydrogen
  bonds P(t), has a q-exponential
• behavior.
• q have values above 1, below a characteristic
  temperature.
• The increase of q is associated with the increase
  of the probability of two molecules
• remain bonded during a longer time t.
• The temperature (300 K), at which the transition
  of q 1 to q gt 1 occurs ,
• coincides with that at which the tetrahedral
  structure of water and the cage effect in
• the MSD begins to prevail.

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Thank you !
q-exponential distribution in time correlation
function of water hydrogen bonds Campo, Mario G.
, Ferri, Gustavo L., Roston Graciela
B. Departamento de Física. Facultad de Ciencias
Exactas y Naturales de la UNLPam. Uruguay 151.
Santa Rosa (L.P.) Argentina.