J. Phys. Chem. B 2008, 112, 1729-1735
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A Diffusive Anomaly of Water in Aqueous Sodium Chloride Solutions at Low Temperatures Jun Soo Kim and Arun Yethiraj* Department of Chemistry and Theoretical Chemistry Institute, UniVersity of Wisconsin, Madison, Wisconsin 53706 ReceiVed: August 21, 2007; In Final Form: NoVember 19, 2007
Molecular dynamics simulations are presented for the self-diffusion coefficient of water in aqueous sodium chloride solutions. At temperatures above the freezing point of pure water, the self-diffusion coefficient is a monotonically decreasing function of salt concentration. Below the freezing point of pure water, however, the self-diffusion coefficient is a non-monotonic function of salt concentration, showing a maximum at approximately one molal salt. This suggests that sodium chloride, which is considered a structure-making salt at room temperature, becomes a structure-breaking salt at low temperatures. A qualitative understanding of this effect can be obtained by considering the effect of ions on the residence time of water molecules near other water molecules. A consideration of the freezing point depression of aqueous sodium chloride solutions suggests that the self-diffusion coefficient of water in supercooled sodium chloride solutions is always higher than that in pure (supercooled) water at the same temperature.
1. Introduction The structure and dynamics of water in aqueous solutions play an important role in many important applications including the crystallization of proteins and the conformational behavior of peptides and nucleic acids. The effect of ions on the structure of water has been extensively studied, and the concept of structure-breaking and structure-making, where the ions are believed to either disrupt the hydrogen-bonding network or form the ionic hydration structure in water, has been widely used to provide a qualitative understanding of the effect of ions on the structure of water.1-5 The Hofmeister series, for example, which rates ions on their effectiveness in precipitating proteins, has been interpreted as a qualitative ranking of the tendency of ions to make (or break) the structure of water. One measure of the structure-altering nature of salts is the self-diffusion coefficient (DW) of the water molecules.5-9 In structure-creating salts, DW decreases monotonically as the salt concentration is increased, which is interpreted as the tendency for the salt ions to increase the structure of water. In structurebreaking salts, however, DW is a non-monotonic function of salt concentration, increasing for low salt concentrations, going through a maximum before decreasing as the salt concentration is increased further.6-8 In this work, we are interested in the effect of ions on the dynamics of water near and below the freezing point, i.e., under supercooled conditions. The dynamics of water under these conditions has implications in the nucleation and growth of ice crystals; in classical theories, the crystal growth rate is proportional to the self-diffusion coefficient of the water molecules.10 Crystal growth is important in a variety of applications including cryogenics, and increasing the freezer life of ice cream. Water displays several diffusive anomalies in this regime,11-13 and salt effects are of interest. We investigate, using molecular dynamics simulations, the self-diffusion of water as a function of temperature and salt concentration. We find that sodium chloride (NaCl), which is believed to be structure-
creating at all temperatures,4,7,8,14 becomes structure-breaking below the freezing point of pure water. For aqueous ionic solutions at room temperature, the effect of ions on the mobility of water molecules can be understood in a qualitative fashion by considering the hydrogen-bonded structure of water and the effect of ion hydration.1,4,7,9 There are strong intermolecular correlations due to the hydrogenbonding network. The presence of ions could weaken these correlations, and thus result in an increase in the mobility of water. On the other hand, the hydration of the ions should decrease the mobility of the water molecules in the hydration shell, specifically the first hydration shell.15 Whether an ion is structure-breaking or structure-creating depends on a subtle balance between these two effects. It has been argued that, for structure-breaking ions such as Rb+, Cs+, Br-, and I-, the effect of weakening the hydrogen-bond structure is dominant at low concentrations, resulting in a reduced viscosity and increased mobility of water, while the effect of ion hydration becomes more prominent at higher salt concentrations. On the other hand, for the structure-making ions such as Li+ and Ca2+, the formation of hydration structure is argued to be dominant at all compositions, leading to the increased viscosity and the slower diffusion of water in a monotonic fashion. There have been some experiments on the temperature dependence of salt effects on water mobility, but most of these have considered aqueous solutions at room temperature and higher temperatures. The viscosity is roughly inversely proportional to the self-diffusion coefficient of water and can be used to infer the structure-breaking and structure-creating effect of salts. In aqueous CsCl solutions, the viscosity is a nonmonotonic function of salt concentration at 0 °C but increases monotonically with salt concentration at 100 °C,16 which can be interpreted as structure-breaking ions becoming structurecreating as the temperature is increased. Similar behavior is also seen for many other aqueous solutions, including KCl, KBr, KI, and RbBr.4,8,14,17 Recently, Kanno et al.18 concluded that structure-making ions become structure-breakers in glassy
10.1021/jp076710+ CCC: $40.75 © 2008 American Chemical Society Published on Web 01/24/2008
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TABLE 1: Parameters of the Lennard-Jones Interactions atom
σ (Å)
(kJ/mol)
Na Cl Na-Cl O Na-O Cl-O
3.33 4.42 3.84 3.12 3.22 3.71
0.012 0.493 0.076 0.670 0.088 0.575
aqueous electrolyte solutions based on the hydrogen-bond strength inferred from the uncoupled Raman OD stretching spectra. A transition from structure-creating to structure-breaking behavior with changing temperatures has never been reported for NaCl (to our knowledge), which is believed to be structurecreating at all temperatures.4,8,14,17 An examination of the viscosity isotherms in the review by Angell,19 however, suggests that LiCl is structure-breaking at T ) -30 °C and T ) -35 °C in contrast to the structure-creating behavior at higher temperatures. Since NaCl is usually known to be less structure-creating than LiCl, we may also expect that such a transition might also occur for NaCl, although we are not aware of any such studies. In this paper, we present molecular dynamics simulations results for the self-diffusion of water in NaCl solutions near the freezing point of water. There have been a large number of simulation studies of aqueous solutions, but most of them studied systems with only one ion or a pair of ions with an emphasis on properties of hydrated ions and water molecules within the hydration shells. Although some papers investigate the selfdiffusion of water as a function of salt concentration,20-27 none of them study temperatures near the freezing point (for the water model employed). Our simulations demonstrate that NaCl is structure-creating at temperatures above the freezing point of pure water but becomes structure-breaking at lower temperatures. We attempt to provide a qualitative understanding of this effect by considering the residence time of co-ordinated water molecules and the effect of ions on the residence time and mobility of water. Finally, based on the freezing point depression of aqueous NaCl solutions,28 we suggest that, under the supercooled conditions, the self-diffusion coefficient in the NaCl solution is always greater than that of pure water at the same temperature. 2. Simulation Details We use the TIP5P model for water29-31 which has been shown to provide a reasonable description of the properties of water near its freezing point. For example, this model gives a freezing point of 274 K32 which is much closer to the experimental value (273.15 K) than other popular water models such as the TIP4P model,33 which has a freezing point of 232 K,32 or the SPC/E model,34 which has a freezing point of 215 K.32 We also perform some simulations of the SPC/E model for comparison. For Na+ and Cl- ions, the interaction potentials are obtained from A˙ qvist35 and the OPLS force field,36 respectively. The parameters of the Lennard-Jones collision diameter (σ) and well-depth () are given in Table 1. All of the molecular dynamics (MD) simulations are carried out with the GROMACS v3.3.1 program.37,38 The equations of motion are integrated numerically using a leapfrog algorithm with a time step of 1 fs. The Berendsen coupling method39 is used to keep the temperature at the desired value and the pressure at 1 bar, with coupling constants of 0.1 and 0.5 ps, respectively, and the SETTLE algorithm40 is used to keep the water molecules rigid. A cutoff distantce of 1 nm is used for
TABLE 2: Summary of State Points Studied number of number Na+ and of H2O salt concn simulation time simulation time (SPC/E)c (ns) Cl- ions molecules (molality)a (TIP5P)b (ns) 0 6 12 18 24 30 36 42
512 500 488 476 464 452 440 428
0 0.666 1.365 2.099 2.870 3.683 4.540 5.446
0.4-1.6 0.4-1.6 0.4-1.4 0.5-1.5 0.6-1.9 0.6-1.9 0.7-2.5 0.9 ns - 2.9 ns
1.3-7.0 1.5-7.9
-
a
The concentration is given in molality, i.e., moles/1000 g of solvent. At temperatures of 260, 265, 270, 275, 280, and 285 K. Longer simulations are at lower temperatures. c At temperatures of 210, 215, 220, 225, 230, and 235 K.
b
Lennard-Jones interactions, and the particle-mesh Ewald method41,42 is used for the long-range electrostatic interactions. Initial configurations of pure water with N ) 512 molecules are created in cubic simulation cells with a linear dimension of 25 Å. These configurations are equilibrated for 2 ns at the desired temperatures. During the equilibration at each temperature, configurations are saved every 400 ps to generate five different initial configurations for longer trajectories over which properties are averaged. For aqueous NaCl solutions, randomly chosen water molecules are replaced by ions, with the number of ions depending on the concentration. Initial configurations for each temperature are obtained from 2 ns equilibration runs in the same way as for pure water. We investigate six temperatures: 260, 265, 270, 275, 280, and 285 K. The state points investigated are summarized in Table 2. For each state point, trajectories are obtained using the five different initial configurations, each of which has a duration such that the water molecules have moved (on average) roughly two-thirds of the simulation box length. The simulation duration used is given in Table 2, with longer simulations required at lower temperatures. The mean square displacement (MSD) is calculated from each run, and the Einstein relation is used to determine the diffusion coefficient.43 Properties are averaged over the five independent trajectories, and statistical uncertainties are one standard deviation. For the SPC/E model, we perform simulations for two concentrations, 0 and 1.365 m, and for temperatures (T) of 210, 215, 220, 225, 230, and 235 K, which are in the neighborhood of the freezing point for this model. The relaxation times at such low temperatures are much longer, however, and these simulations are more computationally intensive. Properties were averaged over five trajectories each of which was of a duration such that the water molecules moved roughly half the simulation box length, on average. We calculate the residence time (τR) of water molecules near other water molecules using the protocol of Impey et al.,44 which has been used previously to calculate the residence time of water molecules in the ionic hydration shell.23,44,45 For the calculation of τR, we introduce the function Pij(t,t0;t*) which has a value of 1, if the water molecules i and j are within a cutoff distance of 3.3 Å at time t0 and t + t0 and in between they do not separate to a greater distantce than the cutoff distance for longer than the excursion time (t*) of 2 ps, and has a value of 0 otherwise (the cutoff distance of 3.3 Å is the position of the first minimum in the oxygen-oxygen pair correlation function of pure water). The average number of surviving water molecules in the vicinity of a water molecule at time t is defined as
Self-Diffusion Coefficient of Water
n(t) )
1
tcorr-t N
1
tcorr - t N
J. Phys. Chem. B, Vol. 112, No. 6, 2008 1731 N
∑ ∑ ∑ Pij(t,t0;t*)
(1)
t0)1 i)1 j)1 j*i
where N is the number of water molecules and tcorr is the total correlation time, set to 250 ps in this work. Ideally, tcorr should be as large as possible so that n(t) has decayed completely during this time. We find that, with tcorr ) 250 ps, n(t) decays rapidly to less than 0.02 n(0) for all concentrations at the temperatures considered. The residence time is obtained by numerically integrating the ratio n(t)/n(0) from 0 to tcorr, i.e.,
τR )
∫0t
n(t) dt n(0)
corr
(2)
Simulation results for the diffusion coefficient of pure water (using the TIP5P model) are compared to experimental data,46 together with the original TIP5P model calculation by Mahoney and Jorgensen,30 in Figure 1. Under supercooled conditions, the model tends to predict values of the self-diffusion coefficient that are smaller than experiment, and this discrepancy increases as the temperature is decreased. Tan et al.47 have compared the self-diffusion coefficient of water predicted by several nonpolarizable models to experimental data. None of the models accurately reproduces the diffusion coefficient over a wide range of temperatures, but of the models they tested the TIP5P model was in the best agreement with experiment (along with their own model). Note that the diffusion coefficients obtained in our simulations are slightly lower than that in the previous work.30 This is because we include long-range electrostatic interactions which were not included in the original work.30 We are interested in the behavior of supercooled water. Although ice is the most stable state below the freezing point of water (273.15 K), supercooled water is metastable for temperatures above the homogeneous nucleation temperature (Th). For pure water, Th ) 231 K,11 and for aqueous solutions, Th is even lower.48 Although Th is not known for the TIP5P model, we expect that the temperature range studied is higher than Th because the melting point of the model is very close to the experimental value. We do not observe any ice crystallites in our simulation but do not expect to in any case because this process occurs on much larger time scales. In a simulation of TIP4P water at 230 K, it took 200-300 ns before homogeneous nucleation was observed.49 3. Results and Discussion The main result of the paper is the variation of the water self-diffusion coefficient (DW) with salt concentration at different temperatures. Figure 2 depicts the relative diffusion coefficient DW/D0, where D0 is the self-diffusion coefficient of pure water at the same temperature, as a function of NaCl concentration for various temperatures. For temperatures greater than or equal to 275 K (which is slightly above the freezing point of water, 274 K,32 in the TIP5P model), the relative diffusion coefficient decreases monotonically as the salt concentration is increased. This is in qualitative agreement with experimental observations of diffusion and viscosity at temperatures over 273.15 K for NaCl solutions.4,6,8,14,17 The behavior is qualitatively different, however, for temperatures 270 K and lower. In this case, the initial slope of the plot of DW/D0 with concentration is positive, the relative diffusion coefficient goes through a maximum, and then decreases for higher salt concentrations. The value of this maximum in DW/D0 is higher and occurs at higher salt concentrations as the temperature is decreased. This increase
Figure 1. Comparison of the self-diffusion coefficient of pure TIP5P water to experimental results. Differences between the two TIP5P model simulations arise from the inclusion of long-range interactions in our simulations.
in DW relative to pure water in the dilute solutions may be interpreted as NaCl behaving as a structure-breaker at low temperatures (below the freezing point of pure water), although it is a structure-maker at higher temperatures. The anomalous behavior in DW cannot be inferred from the structural properties such as pair correlation functions, the hydration number of ions, and the instantaneous number of hydrogen bonds. Figure 3a depicts the oxygen-oxygen pair correlation function (gOO(r)) for 260 and 280 K. In both cases, gOO(r) shows the signature of water structure disruption; i.e., the heights of the peaks become lower and the shoulder develops next to the first peak, as the salt concentration is increased, as has been observed in previous studies.22,26,50 The instantaneous number of hydrogen bonds decreases with ion concentration at both temperatures, as seen in Figure 3b. The important point is that the effect of salt on the structural properties is similar at both temperatures but the effect of salt on DW is qualitatively different. The hydration number of ions calculated by the integration of the ion-oxygen pair correlation functions up to the first minimum also shows a similar change with concentration at 260 and 280 K, with no distinctive behaviors observed (not shown). The translational and tetrahedral order parameters, q and t,12 also display the same trend with salt concentration at 260 and 280 K (not shown). In order to understand the origin of the anomalous diffusion at low temperatures, we investigate the residence time (τR) of water molecules near other water molecules. One would expect that the greater the residence time, the lower the self-diffusion coefficient. Figure 4 depicts τR as a function of salt concentration for temperatures of 260 and 280 K. At 280 K, the calculated residence time does not show any evidence of structure disruption and increases slightly with increasing salt concentration, implying that the water structure gets slightly more stable with the addition of salt. At 260 K, the calculated residence time decreases sharply as the salt concentration is increased from zero, and then starts to rise again for larger salt concentrations. The sharp decrease can be attributed to the disruption of water structure by ions in dilute solutions. Note, however, that τR does not increase to values greater than that of pure water for high salt concentrations, and therefore does not capture all of the features seen in the water self-diffusion coefficient.
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Figure 2. Self-diffusion coefficient of water in aqueous NaCl solutions relative to that of pure water at the same temperature, as a function of salt concentration (in molality) for various temperatures (as marked). Solid lines are fifth-order polynomial fits to the simulation results.
The residence time is an average quantity, and we find it useful to calculate the fraction of “long resident” water molecules. We use a subjective criterion to define a long resident water molecule; i.e., we keep track of the water molecules that are within the cutoff distance of another water molecule (with 2 ps excursions allowed as before) for times greater than a “reference time”. These molecules are termed long resident. We use a reference time of 40 ps at 260 K; this value is approximately equal to τR at this temperature. For 280 K, we use a reference time of 18 ps for reasons explained shortly. Once a water molecule has been classified as “long resident”, it is further classified as a “hydration shell” water molecule if it is within a cutoff distance of rNa-O and rCl-O from Na+ and Clions, respectively, and classified as “bulk” water otherwise. These cutoff distances, rNa-O and rCl-O, correspond to the position of the first minimum in the ion-oxygen pair correlation functions and increase with concentration (3.15-3.18 for Na-O at both 260 and 280 K, and 3.91-4.33 and 3.98-4.41 for Cl-O at 260 and 280 K, respectively). This classification of long resident water molecules is useful for understanding the origin of the diffusive behavior in Figure 2. Parts a and b of Figure 5 depict the fraction of water molecules that are long resident molecules as a function of salt concentration at 260 and 280 K, respectively. As the salt concentration is increased, the fraction of bulk long resident water molecules decreases and the fraction of hydration shell long resident water molecules increases. The decrease in the fraction of long resident water molecules in the bulk can be attributed to the breaking of hydrogen bonds between water molecules which is caused by the reorganization of water molecules in the vicinity of ions due to the strong electrostatic interaction. The strong electrostatic interaction also causes the water molecules within the hydration shell to be held together strongly by ions, resulting in the increase of the fraction of hydration shell long resident water molecules.
For 280 K, we choose a reference time so that the fraction of long resident water molecules in the hydration shell is roughly the same as for 260 K. This allows us to determine the effect of ions on the persistence of the bulk water structure under conditions where the hydration effects are similar. In any case, the arbitrariness in choosing the reference time precludes a quantitative comparison of the results at different temperatures. The main difference in the plots in Figure 5a and b is that in dilute solutions the effect of the ions on the bulk long resident water molecules is much greater at 260 K; i.e., this quantity drops much faster with increasing salt concentration than at 280 K. This drop is sharper than the increase in long resident water molecules in the hydration shell and causes the overall fraction to become a non-monotonic function of concentration similar to what is seen for τR. We speculate that this is the origin of the behavior in the dilute solutions seen in Figures 2 and 4; i.e., the structure-breaking effect of ions on the water structure is greater at lower temperatures because there is more persistent hydrogen-bonded structure in the water (with longer hydrogenbond lifetime at lower temperatures51). The above analysis does not explain why the diffusion coefficient decreases at higher salt concentrations. Figure 6 depicts the fraction of long resident water molecules as a function of the reference time for T ) 280 K and 4.54 m NaCl. As the reference time is increased from 10 to 50 ps, the fraction of long resident water molecules decreases, as expected, in this case from 0.93 to 0.18. The decrease is much sharper for bulk water molecules than for hydrated water molecules, which implies that at high salt concentrations the majority of slow water molecules are in the hydration shell of one or more ions. At high concentrations, the self-diffusion coefficient of Na+ and Cl- ions is much lower than that of water molecules (for 4.54 m, the self-diffusion coefficient of Na+ and Cl- ions is approximately 25 and 39% that of water molecules, respectively)
Self-Diffusion Coefficient of Water
Figure 3. Structural properties: (a) the pair correlation function between oxygen atoms for T ) 260 and 280 K (note that the data for 280 K have been displaced vertically for clarity of presentation); (b) the instantaneous number of hydrogen bonds per molecule.
Figure 4. Residence time of water around another water molecule as a function of NaCl concentration (in molality) for 260 and 280 K.
and therefore the mobility of water molecules around them is smaller than that of bulk water molecules, resulting in the decrease in DW with increasing salt concentration.
J. Phys. Chem. B, Vol. 112, No. 6, 2008 1733
Figure 5. Fraction of long resident water molecules determined with a reference time of (a) 40 ps for 260 K and (b) 18 ps for 280 K as a function of NaCl concentration. Error bars are of the same size with symbols and therefore omitted.
The range of salt concentrations and temperatures where the diffusive anomaly is observed is summarized in Figure 7. The curves in Figure 2 are fit to extract the position of the maximum in the curve (circular symbols) and the point at which DW/D0 ) 1 (cross symbols). The solid lines are fitted to circular and cross symbols and meant to guide the eye. To construct these lines, an additional data point is placed at (0 m, 272.5 K) motivated by the fact that we do not have data at infinite dilution and the diffusive behavior is clearly different at 270 K compared to 275 K. There are three regions in the figure: in the colored regions, DW > D0 with the slope of DW/D0 with salt concentration greater than 1 in the dark gray region and smaller than 1 in the light gray region. The line with open circles is therefore the locus of the maximum in the plot of DW/D0 versus salt concentration (Figure 2). Outside of these two regions, DW < D0. There is a large range of temperature and salt concentrations where the diffusive anomaly is predicted, and could therefore be measured experimentally. The dashed line is a fit to the experimental data for the freezing point of aqueous NaCl solution.28 Interestingly, the freezing point curve is very close to the outer boundary. We therefore speculate that the selfdiffusion coefficient of water in supercooled aqueous NaCl solutions will always be greater than that of pure water at the same temperature.
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Kim and Yethiraj 4. Conclusions
Figure 6. Fraction of long resident water molecules for reference times varying from 10 to 50 ps. The molecules are identified as bulk, in the hydration shell of only one ions, and in the overlapped hydration shells of two or more ions based on their final position.
Figure 7. Map outlining the region of diffusive anomalies. The relative diffusion coefficient is greater than 1 in the colored regions, less than one elsewhere, increases with concentration in the innermost dark gray region, and decreases with increasing concentration elsewhere. The circular and cross symbols represent the maximum points and the intercepts with DW/D0 ) 1 in Figure 2, respectively. The solid lines are boundaries of the colored regions fitted to circles and crosses and meant to guide the eye, and the dashed line is experimental data28 for the freezing point of water as a function of salt concentration.
It is of interest to see if the behavior observed here is also seen in other models of liquid water. To test this, we investigated another popular water model, namely, the SPC/E model.34 In this model, the freezing point of water is 215 K.32 For the SPC/E model, we observe a transition from structure-making to structure-breaking between temperatures of 215 and 220 K which suggests that this transition occurs near the freezing point of the water for both models. The fact that the transition occurs near the freezing point implies that the disruption of strong hydrogen-bond structure plays an important role in determining the diffusive behavior. As has been suggested for the disordering effect on water structure,52 the anomalous diffusion induced by salt is therefore similar to the effect of applied pressure.12,13
We present molecular dynamics simulations for the selfdiffusion coefficient of water in aqueous NaCl solutions near the freezing point. At temperatures below the freezing point of water, the self-diffusion coefficient displays a non-monotonic dependence on salt concentration; i.e., it increases with increasing salt concentration for low salt concentrations, goes through a maximum, and then decreases for high salt concentrations. One can interpret this in terms of the disruptive effect of salt on the structure of water. In this sense, the anomalous diffusion induced by salt is similar with that observed when the pressure is applied. The diffusive behavior of water greatly influenced by the presence of NaCl salts is dependent on the temperature with qualitatively different behaviors occurring near the freezing point of water. Above the freezing point, NaCl is a structure-creating salt, but below the freezing point, it is a structure-breaking salt. This prediction could be tested experimentally. There is no signature of this transition in the structural properties such as pair correlation functions. An analysis of the residence time of water molecules (around other water molecules) suggests that, under supercooled conditions and low salt concentrations, the ions have a more drastic effect on the persistent water structure than at higher temperatures. The range of salt concentrations and temperatures where the diffusive anomaly is observed correlates with the freezing point of aqueous NaCl solutions. From this correlation, we propose that the self-diffusion coefficient of water in aqueous NaCl solutions under supercooled conditions is always greater than that of pure water at the same temperature. This speculation can be tested via experiment. A computational determination of the freezing point of aqueous NaCl solutions for the water model employed is also of interest. There are many interesting questions related to the effect of ions at low temperatures. An understanding of salt effects could be important to a molecular level understanding of the freezing point depression, the formation of brine pockets in polar ice, and cryogenic effects on biological cells. Our work suggests that salt effects can be quite different in supercooled states compared to room temperature and, hopefully, will inspire more research in the area. Acknowledgment. This material is based upon work supported by the United States Department of Agriculture National Research Initiative Program (Grant No. 2006-35503-16998). We acknowledge the National Science Foundation (Grant No. CHE0717569) for partial support of this research and Professor S. Damodaran for useful discussions. Note Added after ASAP Publication. This article was released ASAP on January 24, 2008. In the Simulation Details section, paragraph 2, the last sentence has been revised. The corrected version posted on January 29, 2008. References and Notes (1) Franks, F. Water : a matrix of life, 2nd ed.; The Royal Society of Chemistry: Cambridge, U.K., 2000. (2) Burgess, J. Metal Ions in Solution; John Wiley & Sons: Sussex, U.K., 1978. (3) Koryta, J.; Dvorˇa´k, J.; Kavan, L. Principles of Electrochemistry, 2nd ed.; John Wiley & Sons: West Sussex, U.K., 1993. (4) Jenkins, H. D. B.; Marcus, Y. Chem. ReV. 1995, 95, 2695. (5) Ohtaki, H.; Radnal, T. Chem. ReV. 1993, 93, 1157. (6) Mu¨ller, K. J.; Hertz, H. G. J. Phys. Chem. 1996, 100, 1256. (7) McCall, D. W.; Douglass, D. C. J. Phys. Chem. 1965, 69, 2001. (8) Endom, L.; Hertz, H. G.; Thu¨l, B.; Zeidler, M. D. Ber. BunsenGes. Phys. Chem. 1967, 71, 1008.
Self-Diffusion Coefficient of Water (9) Akai, J. A.; Jonas, J. J. Solution Chem. 1976, 5, 563. (10) Frenkel, J. Phys. Z. Sowjet Union 1932, 1, 498. (11) Debenedetti, P. G.; Stanley, H. E. Phys. Today 2003, 56, 40. (12) Errington, J. R.; Debenedetti, P. G. Nature 2001, 409, 318. (13) Water Anomalies, http://www.lsbu.ac.uk/water/anmlies.html. (14) Out, D. J. P.; Los, J. M. J. Solution Chem. 1980, 9, 19. (15) Omta, A. W.; Kropman, M. F.; Woutersen, S.; Bakker, H. J. Science 2003, 301, 347. (16) Bernal, J. D.; Fowler, R. H. J. Chem. Phys. 1933, 1, 515. (17) Afzal, M.; Saleem, M.; Tarlq Mahmood, M. J. Chem. Eng. Data 1989, 34, 339. (18) Kanno, H.; Yonehama, K.; Somraj, A.; Yoshimura, Y. Chem. Phys. Lett. 2006, 427, 82. (19) Angell, C. A. Chem. ReV. 2002, 102, 2627. (20) Lyubartsev, A. P.; Laaksonen, A. J. Phys. Chem. 1996, 100, 16410. (21) Aouizerat-Elarby, A.; Dez, H.; Prevel, B.; Jal, J. F.; Bert, J.; DupuyPhilon, J. J. Mol. Liq. 2000, 84, 289. (22) Chowdhuri, S.; Chandra, A. J. Chem. Phys. 2001, 115, 3732. (23) Laudernet, Y.; Cartailler, T.; Turq, P.; Ferrario, M. J. Phys. Chem. B 2003, 107, 2354. (24) Egorov, A. V.; Komolkin, A. V.; Chizhik, V. I.; Yushmanov, P. V.; Lyubartsev, A. P.; Laaksonen, A. J. Phys. Chem. B 2003, 107, 3234. (25) Mudi, A.; Chakravarty, C. J. Phys. Chem. B 2006, 110, 8422. (26) Harsa´nyi, I.; Pusztai, L.; Soetens, J. C.; Bopp, P. A. J. Mol. Liq. 2006, 129, 80. (27) Du, H.; Rasaiah, J. C.; Miller, J. D. J. Phys. Chem. B 2007, 111, 209. (28) Potter, R. W., II; Clynne, M. A.; Brown, D. L. Econ. Geol. 1978, 73, 284. (29) Mahoney, M. W.; Jorgensen, W. L. J. Chem. Phys. 2000, 112, 8910. (30) Mahoney, M. W.; Jorgensen, W. L. J. Chem. Phys. 2001, 114, 363. (31) Lı´sal, M.; Kolafa, J.; Nezbeda, I. J. Chem. Phys. 2002, 117, 8892. (32) Vega, C.; Sanz, E.; Abascal, J. L. F. J. Chem. Phys. 2005, 122, 114507.
J. Phys. Chem. B, Vol. 112, No. 6, 2008 1735 (33) Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. D.; Impey, R. W.; Klein, M. L. J. Chem. Phys. 1983, 79, 926. (34) Berendsen, H. J. C.; Grigera, J. R.; Straatsma, T. P. J. Phys. Chem. 1987, 91, 6269. (35) A˙ qvist, J. J. Phys. Chem. 1990, 94, 8021. (36) Chandrasekhar, J.; Spellmeyer, D. C.; Jorgensen, W. L. J. Am. Chem. Soc. 1984, 106, 903. (37) Lindahl, E.; Hess, B.; van der Spoel, D. J. Mol. Model. 2001, 7, 306. (38) van der Spoel, D.; Lindahl, E.; Hess, B.; Groenhof, G.; Mark, A. E.; Berendsen, H. J. C. J. Comput. Chem. 2005, 26, 1701. (39) Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; DiNola, A.; Haak, J. R. J. Chem. Phys. 1984, 81, 3684. (40) Miyamoto, S.; Kollman, P. A. J. Comput. Chem. 1992, 13, 952. (41) Darden, T.; York, D.; Pedersen, L. J. Chem. Phys. 1993, 98, 10089. (42) Essmann, U.; Perera, L.; Berkowitz, M. L.; Darden, T.; Lee, H.; Pedersen, L. G. J. Chem. Phys. 1995, 103, 8577. (43) McQuarrie, D. A. Statistical Mechanics; Harper & Row: New York, 1976. (44) Impey, R. W.; Madden, P. A.; McDonald, I. R. J. Phys. Chem. 1983, 87, 5071. (45) Koneshan, S.; Rasaiah, J. C.; Lynden-Bell, R. M.; Lee, S. H. J. Phys. Chem. B 1998, 102, 4193. (46) Gillen, K. T.; Douglass, D. C.; Hoch, M. J. R. J. Chem. Phys. 1972, 57, 5117. (47) Tan, M. L.; Brooks, B. R.; Ichiye, T. Chem. Phys. Lett. 2006, 421, 166. (48) Kanno, H.; Miyata, K.; Tomizawa, K.; Tanaka, H. J. Phys. Chem. A 2004, 108, 6079. (49) Matsumoto, M.; Saito, S.; Ohmine, I. Nature 2002, 416, 409. (50) Mountain, R. D.; Thirumalai, D. J. Phys. Chem. B 2004, 108, 19711. (51) Sciortino, F.; Poole, P. H.; Stanley, H. E.; Havlin, S. Phys. ReV. Lett. 1990, 64, 1686. (52) Leberman, R.; Soper, A. K. Nature 1995, 378, 364.
