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Structural and dynamical heterogeneity of stable and metastable water
Physica A 314 (2002) 477 – 484
www.elsevier.com/locate/physa
Structural and dynamical heterogeneity of stable and metastable water G.G. Malenkov Institute of Physical Chemistry, Russian Academy of Sciences, Leninskii Prospect 31, Moscow, 119991, Russia
Abstract Computer simulation of water of various densities over wide range of temperatures (including ◦ far below 0 C) revealed structural heterogeneity of the simulated systems. This heterogeneity is manifested by non-uniform space distribution of molecules with close values of quantities, characterizing their local environment. Tetrahedricity index, Vorono01 polyhedron volume and total potential energy were chosen as such quantities. Molecules with high or low values of these quantities group together forming ramifying clusters, piercing the volume of the system. Correlation between values of di2erent characteristics of local environment is feeble, if any. Dynamics (amplitude of vibration, di2usion coe4cient) of molecules with high and low values of characteristics of local environment is di2erent. c 2002 Published by Elsevier Science B.V. Keywords: Structure of water; Molecular dynamics; Supercooled water
1. Introduction Since the pioneering paper by R0ontgen [1], many models explaining unusual properties of water have been proposed. R0ontgen’s model was the
E-mail address: [email protected] (G.G. Malenkov). c 2002 Published by Elsevier Science B.V. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 1 0 8 5 - 3
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G.G. Malenkov / Physica A 314 (2002) 477 – 484
most typical example of which was NHemethy and Scheraga’s model [12], was buried by Stanley’s works [13–15] in which the ideas of percolation theory were introduced to the science about water structure. The picture created by NHemethy and Scheraga was reversed: if non-bonded molecules ever existed, they could not form the “sea” surrounding the clusters. There could be at most rare patches of such molecules embedded in the continuous network. Such a picture was consistent with the results of computer simulations and
G.G. Malenkov / Physica A 314 (2002) 477 – 484
479
on the values of both parameters was found for the two modi
2. Model and simulation method The same algorithm, computer program and potential functions as described in Refs. [16,17,20 –22] were used for molecular dynamics simulation of ice III model in NVE (microcanonical) ensemble. The initial model contained 768 rigid water molecules O The in tetragonal periodic unit cell with parameters a = b = 26:704 and c = 27:82 A. program allows to choose the total energy values E. The systems were adjusted to the chosen E using a conventional procedure of velocity rescaling. It was possible to change the temperature of the system by changing the E value. After the equilibration E remained constant within the limits less than 0.002%. The temperature was raised and when the ice III model was melted, the unit cell was converted to a cubic one with O This unit cell corresponds to the density 1:17 g=cm3 . Several the edge equal to 26:98 A. runs at di2erent temperatures were done for this density. The unit cell was expanded to obtain densities 0.999 and 0:94 g=cm3 . Several instantaneous con
5 6
(li − lj )2 =(15l2 ); where li and lj are edges of the tetrahedron
i=1 j=i+1
with the oxygen atom of the molecule whose is being calculated with the vertices in which oxygen atoms of the four nearest water molecules are placed. l is the average length of edges of this tetrahedron. The less is, the more regular is the tetrahedron. The VVP values were calculated using the program kindly provided by Dr. V.P.Voloshin. Etot is the interaction energy of the particular molecule with all the others.
3. Results and discussion Very slight correlation was found between and VVP values [21]. As is seen from Fig. 1, there is no correlation between VVP and Etot values at all. We analyzed coordination of water molecules in the obtained con
480
G.G. Malenkov / Physica A 314 (2002) 477 – 484
Fig. 1. Absence of correlation between Vorono01 polyhedron, volume (vVP) and potential energy (Etot ) of water molecules. Solid line—linear regression. Its slope is practically zero.
bonds per molecule and distribution of coordination types in the low-temperature con
G.G. Malenkov / Physica A 314 (2002) 477 – 484
481
Table 1 In the symbol of the coordination type dm an [20] m means number of hydrogen bonds in which a water molecule participates as a donor, n—as acceptor of a proton. Molecules with m ¿ 2 are involved in bifurcated bonds
NHB d 1 a1 d 1 a2 d 1 a3 d 2 a1 d 2 a2 d 2 a3 d 2 a4 d 3 a1 d 3 a2 d 3 a3 d 4 a2
T = 109 K; d = 0:94
T = 99 K; d = 0:999
T = 105 K; d = 1:17
4.06 0.7 5 0.3 7.3 75 9.5 0.3 0.4 4.2 0.8 0
4.15 0 1.6 0 6.5 71 11 0.5 0.5 6.5 1.7 0
4.70 0 0.3 0 3.1 42 22.4 2.5 1.1 16.5 6.9 2.9
Fractions of molecules of each coordination type aregiven in percent.
formed by the molecules with close values of parameters which characterize their local environment. It is quite natural that dynamical properties of molecules with high and low values of the parameters describing their environment are di2erent. Many examples of such dependence were given in our publications [16 –21]. Here such a dependence for liquid water at temperature 300 K is given for illustration (Fig. 3). I shall not discuss here temperature dependence of di2usion coe4cient at di2erent densities. I only want to remind that rare changes of hydrogen bonded partners occur at temperatures between 100 and 230 K for all densities and these events became more frequent when temperature increased. Di2usion determined as the slope of R2 (t) in the time interval between 0.5 and 2 ps is very di4cult to be observed for LDA at temperatures below 230 K. But in the case of HDA, slow but visible di2usion at temperatures higher than 140 K can be detected. Well-pronounced di2usion starts at T ¿ 230 K both for LDA and HDA. These problems are discussed in our other publication [23].
4. Conclusions We describe in this paper some manifestations of heterogeneity found in computer simulated aqueous systems. It should be noted that regions in which molecules with low tetrahedricity index predominate do not coincide either with regions with small or large values of Vorono01 polyhedron volume, or with regions where molecules with high- or low-potential energy are found. But all these regions are of irregular and ramifying form. Dynamical properties of molecules in these regions are di2erent as well.
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G.G. Malenkov / Physica A 314 (2002) 477 – 484
Fig. 2. Space distribution of water molecules with small (red balls) and large (blue balls) values of VVP (left), Etot (centre) and (right). (a) d = 0:94, T = 109 K; (b) d = 1:0, T = 100 K; (c) d = 1:17, T = 104 K; (d) d = 1:0, T = 297 K. 20% of molecules with smallest and largest values were chosen in each case.
Acknowledgements The author is grateful to V.P.Voloshin for providing the program for calculating VVP values and to Drs. Yu.I. Naberukhin and E.A. Zheligovskaya for the other help and valuable discussions. The work was supported by the Russian Foundation for Basic Research, project no. 00-03-32283.
G.G. Malenkov / Physica A 314 (2002) 477 – 484
483
Fig. 3. Dependence of mean square displacement on time for water molecules with di2erent local environment. (a) for large and small tetrahedricity indices; (b) for large and small Vorono01 polyhedron volumes. The slope of r 2 (t) at t ¿ 400 fs is proportional to di2usion coe4cient.
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G.G. Malenkov / Physica A 314 (2002) 477 – 484
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
W.C. R0ontgen, Ann. Phys. Chem. 45 (1892) 91. J.D. Bernal, R.H. Fowler, J. Chem. Phys. 1 (1933) 515. J.A. Pople, Proc. Roy. Soc. London A 202 (1950) 323. M.G. Sceats, M. Stavola, S.A. Rice, J. Chem. Phys. 70 (1979) 3927. M.G. Sceats, S.A. Rice, J. Chem. Phys. 71 (1979) 973. M.G. Sceats, S.A. Rice, J. Chem. Phys. 72 (1980) 3236. M.G. Sceats, S.A. Rice, J. Chem. Phys. 72 (1980) 6183. O.Ya. Samoilov, Zh. Fiz. Khimii 20 (1946) 1411. L. Pauling, The Hydrogen Bonding, Vol. 1, Pergamon Press, London, 1959 G.G. Malenkov, Dokl. Akad. 137 (1961) 220. H.S. Frank, A.S. Quist, J. Chem. Phys. 34 (1961) 604. G. NHemethy, H.A. Scheraga, J. Chem. Phys. 36 (1962) 3382. H.E. Stanley, J. Phys. A 12 (1979) L329. H.E. Stanley, H.J. Teixeira, J. Chem. Phys. 73 (1980) 3403. A. Geiger, H.E. Stanley, Phys. Rev. Lett. 49 (1982) 1895. G.G. Malenkov, E.A. Zheligovskaya, A.A. Averkiev, I. Natkaniec, L.S. Smirnov, L. Bobrowicz-Sarga, S.I. Bragin, High Pressure Res. 17 (2000) 273. G.G. Malenkov, E.A. Zheligovskaya, A.A. Averkiev, J. Struct. Chem. 42 (2001) 10. Yu.I. Naberukhin, V.A. Luchnikov, G.G. Malenkov, E.A. Zheligovskaya, J. Struct. Chem. 38 (1997) 593. I.I. Vaisman, M.L. Berkovitz, J. Am. Chem. Soc. 114 (1992) 7889. G.G. Malenkov, D.L. Tytik, E.A. Zheligovskaya, Mol. Liquids 82 (1999) 27. V.P. Voloshin, E.A. Zheligovskaya, G.G. Malenkov, Yu.I. Naberukhin, J. Struct. Chem. 42 (2001) 948. V.P. Voloshin, E.A. Zheligovskaya, G.G. Malenkov, Yu.I. Naberukhin, J. Struct. Chem. 43 (2002), in press. G.G. Malenkov, New kinds of phase transitions: transformations in disordered substances, V.V. Brazhkin et al. (Eds.), Kluwer Academic, Dordrecht, 2002, pp. 423–435.
www.elsevier.com/locate/physa
Structural and dynamical heterogeneity of stable and metastable water G.G. Malenkov Institute of Physical Chemistry, Russian Academy of Sciences, Leninskii Prospect 31, Moscow, 119991, Russia
Abstract Computer simulation of water of various densities over wide range of temperatures (including ◦ far below 0 C) revealed structural heterogeneity of the simulated systems. This heterogeneity is manifested by non-uniform space distribution of molecules with close values of quantities, characterizing their local environment. Tetrahedricity index, Vorono01 polyhedron volume and total potential energy were chosen as such quantities. Molecules with high or low values of these quantities group together forming ramifying clusters, piercing the volume of the system. Correlation between values of di2erent characteristics of local environment is feeble, if any. Dynamics (amplitude of vibration, di2usion coe4cient) of molecules with high and low values of characteristics of local environment is di2erent. c 2002 Published by Elsevier Science B.V. Keywords: Structure of water; Molecular dynamics; Supercooled water
1. Introduction Since the pioneering paper by R0ontgen [1], many models explaining unusual properties of water have been proposed. R0ontgen’s model was the
E-mail address: [email protected] (G.G. Malenkov). c 2002 Published by Elsevier Science B.V. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 1 0 8 5 - 3
478
G.G. Malenkov / Physica A 314 (2002) 477 – 484
most typical example of which was NHemethy and Scheraga’s model [12], was buried by Stanley’s works [13–15] in which the ideas of percolation theory were introduced to the science about water structure. The picture created by NHemethy and Scheraga was reversed: if non-bonded molecules ever existed, they could not form the “sea” surrounding the clusters. There could be at most rare patches of such molecules embedded in the continuous network. Such a picture was consistent with the results of computer simulations and
G.G. Malenkov / Physica A 314 (2002) 477 – 484
479
on the values of both parameters was found for the two modi
2. Model and simulation method The same algorithm, computer program and potential functions as described in Refs. [16,17,20 –22] were used for molecular dynamics simulation of ice III model in NVE (microcanonical) ensemble. The initial model contained 768 rigid water molecules O The in tetragonal periodic unit cell with parameters a = b = 26:704 and c = 27:82 A. program allows to choose the total energy values E. The systems were adjusted to the chosen E using a conventional procedure of velocity rescaling. It was possible to change the temperature of the system by changing the E value. After the equilibration E remained constant within the limits less than 0.002%. The temperature was raised and when the ice III model was melted, the unit cell was converted to a cubic one with O This unit cell corresponds to the density 1:17 g=cm3 . Several the edge equal to 26:98 A. runs at di2erent temperatures were done for this density. The unit cell was expanded to obtain densities 0.999 and 0:94 g=cm3 . Several instantaneous con
5 6
(li − lj )2 =(15l2 ); where li and lj are edges of the tetrahedron
i=1 j=i+1
with the oxygen atom of the molecule whose is being calculated with the vertices in which oxygen atoms of the four nearest water molecules are placed. l is the average length of edges of this tetrahedron. The less is, the more regular is the tetrahedron. The VVP values were calculated using the program kindly provided by Dr. V.P.Voloshin. Etot is the interaction energy of the particular molecule with all the others.
3. Results and discussion Very slight correlation was found between and VVP values [21]. As is seen from Fig. 1, there is no correlation between VVP and Etot values at all. We analyzed coordination of water molecules in the obtained con
480
G.G. Malenkov / Physica A 314 (2002) 477 – 484
Fig. 1. Absence of correlation between Vorono01 polyhedron, volume (vVP) and potential energy (Etot ) of water molecules. Solid line—linear regression. Its slope is practically zero.
bonds per molecule and distribution of coordination types in the low-temperature con
G.G. Malenkov / Physica A 314 (2002) 477 – 484
481
Table 1 In the symbol of the coordination type dm an [20] m means number of hydrogen bonds in which a water molecule participates as a donor, n—as acceptor of a proton. Molecules with m ¿ 2 are involved in bifurcated bonds
NHB d 1 a1 d 1 a2 d 1 a3 d 2 a1 d 2 a2 d 2 a3 d 2 a4 d 3 a1 d 3 a2 d 3 a3 d 4 a2
T = 109 K; d = 0:94
T = 99 K; d = 0:999
T = 105 K; d = 1:17
4.06 0.7 5 0.3 7.3 75 9.5 0.3 0.4 4.2 0.8 0
4.15 0 1.6 0 6.5 71 11 0.5 0.5 6.5 1.7 0
4.70 0 0.3 0 3.1 42 22.4 2.5 1.1 16.5 6.9 2.9
Fractions of molecules of each coordination type aregiven in percent.
formed by the molecules with close values of parameters which characterize their local environment. It is quite natural that dynamical properties of molecules with high and low values of the parameters describing their environment are di2erent. Many examples of such dependence were given in our publications [16 –21]. Here such a dependence for liquid water at temperature 300 K is given for illustration (Fig. 3). I shall not discuss here temperature dependence of di2usion coe4cient at di2erent densities. I only want to remind that rare changes of hydrogen bonded partners occur at temperatures between 100 and 230 K for all densities and these events became more frequent when temperature increased. Di2usion determined as the slope of R2 (t) in the time interval between 0.5 and 2 ps is very di4cult to be observed for LDA at temperatures below 230 K. But in the case of HDA, slow but visible di2usion at temperatures higher than 140 K can be detected. Well-pronounced di2usion starts at T ¿ 230 K both for LDA and HDA. These problems are discussed in our other publication [23].
4. Conclusions We describe in this paper some manifestations of heterogeneity found in computer simulated aqueous systems. It should be noted that regions in which molecules with low tetrahedricity index predominate do not coincide either with regions with small or large values of Vorono01 polyhedron volume, or with regions where molecules with high- or low-potential energy are found. But all these regions are of irregular and ramifying form. Dynamical properties of molecules in these regions are di2erent as well.
482
G.G. Malenkov / Physica A 314 (2002) 477 – 484
Fig. 2. Space distribution of water molecules with small (red balls) and large (blue balls) values of VVP (left), Etot (centre) and (right). (a) d = 0:94, T = 109 K; (b) d = 1:0, T = 100 K; (c) d = 1:17, T = 104 K; (d) d = 1:0, T = 297 K. 20% of molecules with smallest and largest values were chosen in each case.
Acknowledgements The author is grateful to V.P.Voloshin for providing the program for calculating VVP values and to Drs. Yu.I. Naberukhin and E.A. Zheligovskaya for the other help and valuable discussions. The work was supported by the Russian Foundation for Basic Research, project no. 00-03-32283.
G.G. Malenkov / Physica A 314 (2002) 477 – 484
483
Fig. 3. Dependence of mean square displacement on time for water molecules with di2erent local environment. (a) for large and small tetrahedricity indices; (b) for large and small Vorono01 polyhedron volumes. The slope of r 2 (t) at t ¿ 400 fs is proportional to di2usion coe4cient.
484
G.G. Malenkov / Physica A 314 (2002) 477 – 484
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
W.C. R0ontgen, Ann. Phys. Chem. 45 (1892) 91. J.D. Bernal, R.H. Fowler, J. Chem. Phys. 1 (1933) 515. J.A. Pople, Proc. Roy. Soc. London A 202 (1950) 323. M.G. Sceats, M. Stavola, S.A. Rice, J. Chem. Phys. 70 (1979) 3927. M.G. Sceats, S.A. Rice, J. Chem. Phys. 71 (1979) 973. M.G. Sceats, S.A. Rice, J. Chem. Phys. 72 (1980) 3236. M.G. Sceats, S.A. Rice, J. Chem. Phys. 72 (1980) 6183. O.Ya. Samoilov, Zh. Fiz. Khimii 20 (1946) 1411. L. Pauling, The Hydrogen Bonding, Vol. 1, Pergamon Press, London, 1959 G.G. Malenkov, Dokl. Akad. 137 (1961) 220. H.S. Frank, A.S. Quist, J. Chem. Phys. 34 (1961) 604. G. NHemethy, H.A. Scheraga, J. Chem. Phys. 36 (1962) 3382. H.E. Stanley, J. Phys. A 12 (1979) L329. H.E. Stanley, H.J. Teixeira, J. Chem. Phys. 73 (1980) 3403. A. Geiger, H.E. Stanley, Phys. Rev. Lett. 49 (1982) 1895. G.G. Malenkov, E.A. Zheligovskaya, A.A. Averkiev, I. Natkaniec, L.S. Smirnov, L. Bobrowicz-Sarga, S.I. Bragin, High Pressure Res. 17 (2000) 273. G.G. Malenkov, E.A. Zheligovskaya, A.A. Averkiev, J. Struct. Chem. 42 (2001) 10. Yu.I. Naberukhin, V.A. Luchnikov, G.G. Malenkov, E.A. Zheligovskaya, J. Struct. Chem. 38 (1997) 593. I.I. Vaisman, M.L. Berkovitz, J. Am. Chem. Soc. 114 (1992) 7889. G.G. Malenkov, D.L. Tytik, E.A. Zheligovskaya, Mol. Liquids 82 (1999) 27. V.P. Voloshin, E.A. Zheligovskaya, G.G. Malenkov, Yu.I. Naberukhin, J. Struct. Chem. 42 (2001) 948. V.P. Voloshin, E.A. Zheligovskaya, G.G. Malenkov, Yu.I. Naberukhin, J. Struct. Chem. 43 (2002), in press. G.G. Malenkov, New kinds of phase transitions: transformations in disordered substances, V.V. Brazhkin et al. (Eds.), Kluwer Academic, Dordrecht, 2002, pp. 423–435.