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 PAMPAFacultad 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 (Odonor – Oaceptor )

AB

HHydrogen bond (HB)

In water the HB energy ~23.3 kJ mol-1 compared with 492.2148 kJ mol-1 energy

in covalent bond.

• Two criteria to define HB: Energetic:

O-O distance3.5 Å

O-O interaction energy > EHB

Geometric

O-O distance3.5 Å

O-H…O angle > HB

Water structure: HB distribution

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

• HB distribution.

HB0 HB1 HB2 HB3 HB4 HB50.00

8.44

16.88

25.32

33.76

42.19

Per

cent

age

of o

ccur

ence

of H

B p

er m

olec

ule

HB

Arrhenius behavior of HB

depolarized lightscattering 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, 2294-2297, (1999).

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

10 fs20 fs30 fs40 fs

residence time =

0

)( dttPtHB

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 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, 2294-2297, (1999).

T 350-200 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 simulation0.4238 e

-0.8476 e

0.4238 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

0 300 600 900 1200 1500 1800 2100

1E-5

1E-4

1E-3

P(t

)

t / fs

10 100 1000

1E-5

1E-4

1E-3

P(t

)

t / fs

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

0 300 600 900 1200 1500 1800 2100 2400 2700-35

-30

-25

-20

-15

-10

-5

0

5

t / fs

log q

(P(t))

t / fs

q=1,136

(a)

0 300 600 900 1200 1500 1800 2100 2400 2700-35

-30

-25

-20

-15

-10

-5

0

5

t / fs

log q

(P(t))

t / fs

q=1,136

(a) q/q xqx 1111exp

1

1ln

1

q

xx

q

q

ln1(x) ln(x)

qq tqt 1/1'11'exp

xbatt '

batAtP q exp

q(T) behavior

~300K

•q increase with the decrease of T.

• q~T-1 (T<300 K)

T/K q q360 1.04 0.01 343 1.06 0.01323 1.05 0.01313 1.07 0.01303 1.09 0.01293 1.09 0.01283 1.1 0.02273 1.13 0.01263 1.15 0.01253 1.18 0.02233 1.22 0.02213 1.27 0.02

•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

1.0 1.1 1.2 1.3

16.9

33.8

50.7

4 H

B %

q

~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, 055003 (2008)]

Dtrtrt

6)0()(lim 2

mean square displacement (MSD)

tMSD

Subdiffusive behavior cage effect

•Cage effect occurs in SPC/E modelsimulations[(Chaterjee et al., J. Chem. Phys. 128, 124511(2008)].

•Cage effect increase with the decrease of T

MSD in our MD simulations

Cage effectSlope < 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 α2(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. α2(t) is known to be zero for a Gaussian distribution [M.G. Mazza et al. Phys. Rev. E 76, 031203 (2007)].

®2(t) = 3hr 4(t)i5hr 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

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, 6245-6252, (1984).Berendsen H.J.C., Grigera J.R., Straatsma T.P., “The missing term in effective pair potentials”, J. Phys.Chem., 91, 6269-6271, (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, 3684-3690, (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, 124-127, (2006).Mishima O. and Stanley H.E., “The Relationship between Liquid, Supercooled and Glassy Water”, Nature, 396, 329-335, (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, 5025-5029, (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, 928-931, (1996b).Sciortino F. and Fornili S.L., “Hydrogen bond cooperativity in simulated water: Time dependence analysis of pair interactions”, J. Chem. Phys., 90, 2786-2792, (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, 2294-2297, (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, 579-587, (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, 479-487, (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

Thank you !

q-exponential distribution in time correlation function of water hydrogen bondsCampo, 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.