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. UNIVERSIDAD NACIONAL DE LA PAMPA Facultad de Ciencias Exactas y Naturales V Workshop de Mecánica Estadística y Teoría de la Información – Mar del Plata – Abril 2009

Water structure: What’s hydrogen 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 (O donor – O aceptor ) A B H Hydrogen bond (HB) In water the HB energy ~23.3 kJ mol-1 compared with kJ mol -1 energy in covalent bond.

Two criteria to define HB: Energetic: O-O distance  3.5 Å O-O interaction energy > E HB Geometric O-O distance  3.5 Å O-H…O angle >  HB

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

Arrhenius behavior of  HB depolarized light scattering experiments Molecular dynamics What’s the importance of the hydrogen bonds? Starr F.W., Nielsen J.K., and Stanley H.E., Phys. Rev. Lett., 82, , (1999). Anomalous properties of water are influenced by the behavior of hydrogen bonding. 10 fs20 fs30 fs40 fs residence time =  HB is the mean of the distribution of HB lifetimes time t P(t): History-dependent HB correlation function: probability that an initially bonded pair remains bonded at all times up to time t. energetic geometric P(t) can be obtained from simulations by building a histogram of the HB residence times. Measurements of lifetimes are made depolarized light scattering techniques C.J. Montrose, et al., J. Chem. Phys. 60, 5025 (1974).

Behavior of P(t) do not have neither power-law nor exponential behavior. t/ps Starr F.W., Nielsen J.K., and Stanley H.E., “Fast and slow dynamics of hydrogen bonds in liquid water”, Phys. Rev. Lett., 82, , (1999). T K

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. Berendsen’s bath of temperature and pressure. 2.5 ns for equilibration. 5 ns aditional simulation  results.  t simulation = 2 fs.  t data collection = 10 fs. Molecular dynamics simulation e e e

P(t) do not have neither power-law nor exponential behavior. T=273 K Dynamics due libration Geometrical definition of hydrogen bond: minimum O-O separation of 3.5 Å minimum O H· · ·O of 145° > 145° < 3.5 Å

We found that P(t) can be fitting with a q-exponential function ln 1 (x)  ln(x)

q(T) behavior ~300K q increase with the decrease of T. q~T -1 (T<300 K) T/K q  q Above 300 K, P(t) decays exponentially with T (q~1)

~270 K ~300 K Changes in the hydrogen bond structure with temperature 4 HB above the 3 and 2 HB 4 HB between the 3 and 2 HB 4 HB below of 2 HB and 3 HBs

reciprocal relation between HBs and T (similar to q(T) at T>300 K). ~300 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

~300 K

below 300 K there are a linear correlation between the tretrahedral structure of water and q.

Cage effect q–Gaussian distribution of the displacement of particles correlated with anomalous diffusion. [Liu and Goree, Phys. Rev. Lett. 100, (2008)] mean square displacement (MSD) Subdiffusive behavior  cage effect Cage effect occurs in SPC/E model simulations [(Chaterjee et al., J. Chem. Phys. 128, (2008)]. Cage effect increase with the decrease of T

MSD in our MD simulations Cage effect Slope < 1

The non-Gaussian behavior of the displacement of water molecules was studied calculating the time t*, the time at which the non-Gaussian parameter α 2 (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. α 2 (t) is known to be zero for a Gaussian distribution [M.G. Mazza et al. Phys. Rev. E 76, (2007)]. ® 2 ( t ) = 3 h r 4 ( t )i 5 h r 2 ( t )i ¡ 1

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.

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 > 1 occurs, coincides with that at which the tetrahedral structure of water and the cage effect in the MSD begins to prevail. CONCLUSION

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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.