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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

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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.

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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

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

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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).

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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

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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

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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

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

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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)

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~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

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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

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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.

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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

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MSD in our MD simulations Cage effect Slope < 1

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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 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

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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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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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Angell C.A., Water: A Comprehensive Treatise, Plenum Press, New York, (1981). Angell C.A. and Rodgers V., “Near infrared spectra y the disrupted network model of normal y supercooled water”, J. Chem. Phys., 80, , (1984). Berendsen H.J.C., Grigera J.R., Straatsma T.P., “The missing term in effective pair potentials”, J. Phys.Chem., 91, , (1987). Berendsen H., Postma J., van Gusteren W., Di Nola A. and Haak J., “Molecular dynamics with coupling to an external bath”, J. Chem. Phys., 81, , (1984). Berendsen H.J.C., van der Spoel D. and Drunen R.V., “GROMACS: a message passing parallel molecular dynamics implementation”, Comp. Phys. Comm., 91, 43-56, (1995). Cruzan J.D., Braly L.B., Liu K., Brown M.G., Loeser J.G., and Saykally R.J., “Quantifying Hydrogen Bond Cooperativity in Water: VRT Spectroscopy of the Water Tetramer”, Science, 271, 59-62, (1996). Debenedetti P.G., Metastable Liquids, Princeton University Press, Princeton, (1996). Eisenberg D. and Kauzmann W., The Structure y Properties of Water, Oxford University Press, New York, (1969). Mallamace F., Broccio M., Corsaro C., Faraone A., Wandrlingh U., Liu L., Mou C., and Chen S.H., “The fragile-to-strong dynamics crossover transition in confined water: nuclear magnetic resonance results”, J. Chem. Phys., 124, , (2006). Mishima O. and Stanley H.E., “The Relationship between Liquid, Supercooled and Glassy Water”, Nature, 396, , (1998). Montrose C.J., Búcaro J.A., Marshall-Coakley J. and Litovitz T.A., “Depolarized Rayleigh scattering y hydrogen bonding in liquid water”, J. Chem. Phys., 60, , (1974). Luzar A. and Chandler D., “Hydrogen bond kinetics in liquid water”, Nature, 379, 55-57, (1996a). Luzar A. and Chandler D., “Effect of Environment on Hydrogen Bond Dynamicsin Liquid Water”, Phys. Rev. Lett., 76, , (1996b). Sciortino F. and Fornili S.L., “Hydrogen bond cooperativity in simulated water: Time dependence analysis of pair interactions”, J. Chem. Phys., 90, , (1989). Stillinger F.H., “Theory y molecular models for water”, Adv. Chem. Phys., 31, 1-102, (1975). 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). Starr F.W., Nielsen J.K. and Stanley H.E., “Hydrogen-bond dynamics for the extended simple point-charge model of water”, Phys. Rev. E., 62, , (2000). Sutmann G., and Vallauri, R., “Dynamics of the hydrogen bond network in liquid water”, Journal of Molecular Liquids, 98–99, 213–224, (2002). Tsallis C., “Possible generalization of Boltzmann-Gibbs statistic”, Journal of Statistical Physics, 52, , (1988). Walpole R. and Myers R., Probabilidad y Estadística, 4ª Ed. McGraw Hill, México, (1992). Woutersen S., Emmerichs U. and Bakker H., “Femtosecond Mid-Infrared Pump-Probe Spectroscopy of Liquid Water: Evidence for a Two-Component Structure”, Science, 278, 658, (1997). References

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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.

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