J. Phys. Chem. B 1999, 103, 1991-1994
1991
Thermal Offset Viscosities of Liquid H2O, D2O, and T2O C. H. Cho,† J. Urquidi,† S. Singh,‡ and G. Wilse Robinson*,†,§ SubPicosecond and Quantum Radiation Laboratory, Department of Chemistry and Biochemistry, and Department of Physics, Texas Tech UniVersity, Lubbock, Texas 79409-1061, and HNC Software Inc., 5930 Cornerstone Court West, San Diego, CA 92121-3728 ReceiVed: October 30, 1998; In Final Form: January 19, 1999
As suggested in a previous study under the title “Simple Relationship Between the Properties of Isotopic Water”, viscosity results verify the fact that the structural properties of liquid H2O and D2O are nearly identical once a zero-point-energy-induced thermal offset effect is taken into account. This means that the viscosities of these two isotopic forms must be compared at different temperatures, rather than at the same temperature. Only in this way can the expected (MD2O/MH2O)1/2 viscosity ratio be retrieved. Application of this most simple idea, with no additional parameter adjustment, to H2O viscosity data, or equivalently to any of the existing empirical viscosity equations for H2O, leads to D2O viscosities having better than 1% accuracy over a wide temperature range. This isotopic correlation concept has also been used here to predict viscosities of liquid T2O, no viscosity data apparently being available for this substance.
I. Introduction paper,1
A 1991 Walrafen and Chu discussing the shear viscosity of liquid water, put emphasis on the equilibrium between bonded (B) and unbonded (U) species. Later in their paper, concerning the strange isotope effects (H2O vs D2O) on the viscosity, it is pointed out that for high temperatures the shear viscosity isotope ratios are roughly the square-root of the mass ratio (MD2O/MH2O)1/2, but for supercooled water, these ratios are even much larger than 1.381, the square root of the ratios of the moments of inertia. Many other papers have dealt with this anomalous isotope effect without providing much enlightenment. For example, in a 1977 DeFries and Jonas paper2 commenting on the dynamics of H2O and D2O, it was said that the only way to understand the isotope effects within a Stokes/ Einstein framework was to make the backward sounding assumption that water changed from “slipping” boundary conditions at low temperatures to “sticking” boundary conditions at high temperatures. Continuing with this dilemma, a 1985 paper by Kestin et al.,3 this group over the years contributing a great deal to the measurement and data refinement of liquid water viscosities, said that, concerning the viscosities of the two isotopic forms, representative equations which result from the principle of corresponding states give standard deviations that are “well outside the accuracy of measurements, even in the limited temperature and pressure range covered”. A few years ago, we introduced4 a thermal shift or offset concept for correlating the properties of liquid H2O, D2O, and T2O. This concept, which was an outgrowth of our type-Ih/ type-II outer structure mixture model for water,5 depends on achieving a structural equivalence of these isotopic forms for comparison of the densities and the dynamics. Surprisingly, this had never been considered when attempting to explain isotope effects in water. It is true that attempts to bring the thermodynamics of H2O and D2O into line for comparison of the viscosities have been made.6 However, the results of those †
Department of Chemistry and Biochemistry, Texas Tech University. HNC Software Inc. § Department of Physics, Texas Tech University. ‡
attempts, which used differences between freezing point, boiling point, and critical point temperatures, were never very convincing. The clear reason for the failure of this type of thermodynamic equivalence is the incorrect introduction of thermodynamic properties of the solid and gaseous forms of water into the description of the liquid. The neglect of the correct structural thermal offset effect has held back the formulation of accurate relationships between properties of the isotopic forms of water. This purely liquid state effect is directly responsible for the increased temperatures of maximum density for D2O (11.185 °C) and T2O (∼13.4 °C) compared with H2O (3.984 °C). When this effect is taken into account for the densities, the correct mass ratio MD2O/MH2O arises.4 When it is applied to the viscosities, a vibrational7 (MD2O/ MH2O)1/2, not a rotational, mass effect has been seen to emerge.8-10 As will be shown in the present paper, the thermal offset concept can also be used to estimate unknown T-dependent viscosities of D2O and T2O, or when such data are known, this concept can act as a validity check using the higher precision data sometimes available from H2O measurements. More interestingly, perhaps, as will be shown here, when this concept is used to convert any of the existing empirical viscosity relationships for H2O, remarkably good D2O viscosities arise without additional parameter adjustment. II. Experimental Viscosity Data While there exist many experimental papers concerned with the viscosity of liquid H2O over normal to high temperature ranges for atmospheric and elevated pressures,3,6,11-22 far fewer data are available at supercooled temperatures9,13,16,21 and for D2O.3,6,9,12,23,24 The most extensive data sets for both H2O and D2O at supercooled temperatures are those from Osipov et al.,9 but because of the use of a very small capillary diameter (1 µm),10 these viscosities are certainly too large for very low temperatures (_35 e t < _20 °C). See the ref 8 citation in the present paper. The Osipov et al. data also seem too small for some temperature points above _20 °C. Through a large number of detailed examinations of H2O
10.1021/jp9842953 CCC: $18.00 © 1999 American Chemical Society Published on Web 03/02/1999
1992 J. Phys. Chem. B, Vol. 103, No. 11, 1999
Cho et al.
viscosity data from about _10 to +150 °C, it was concluded that the following five-parameter fitting equation was best able to correlate the viscosities of water to a four significant figure level,
η(T) ) A[∆T + a∆T2 + b∆T3 + c∆T4]-γ
TABLE 1: Smoothed Experimental Viscosity Data in cP Units from eq 1 Using Values from Refs 12 and 16 for H2O and Ref 12 for D2O t °C _
30 _25 _ 20 _ 15 _10 _ 5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125
(1)
where A, a, b, c, and γ are adjustable parameters and ∆T ) T - T0, T0 being a fixed reference temperature. This equation possesses a more realistic curvature than other equations used over the years for this purpose. For example, the equations used by Kestin et al.21 have too weak a curvature, giving low viscosity values when extended into the higher or especially the lower temperature ranges. Looking at the root-mean-square deviations of eq 1 with respect to the experimental points reported by the original authors, it was concluded that for H2O the Eicher/ Zwolinski data16 from about _5 to +30 °C were the most reliable, while for +40 to +125 °C, the old Hardy/Cottington data12 were the best. Data from refs 14, 15, and 17 were also good. For supercooled H2O, the extensive Hallet data,13 reported just to three significant figures, were found to match the smoothed data from eq 1 pretty well, even though the rootmean-square deviations were fairly large. These large deviations merely illustrate the difficulty of achieving reproducibility in this temperature range. For D2O, from +5 to +125 °C, again the Hardy/Cottington12 data, adjusted to 100% D2O, had the lowest root-mean-square deviations from the smoothed values of eq 1. The more recent Millero et al. data,23 performed on 99.88% D2O and extrapolated to 100% D2O, while being systematically a few parts lower in the third decimal place, were otherwise of comparable quality as those from ref 12. Our “standard” viscosities were therefore taken to be the smoothed eq 1 data of ref 16 from _5 to +30 °C and of ref 12 from +40 to +125 °C for H2O, and those of ref 12 from +5 to +125 °C for D2O. These smoothed data for 5 °C intervals from the extrapolated low temperature of _30° up to +125 °C are given in Table 1, and the parameters for these fits are shown in Table 2. For comparison, the IAPS recommended H2O viscosities,22 while agreeing exactly to the fourth decimal place with the values from 0 to 20 °C in Table 1, are about 0.0004 cP higher from 30 to 50 °C then become about 0.0004 cP lower between 80 and 100 °C.
Tables 1 and 2 contain smoothed experimental data and fitting parameters using eq 1 for H2O and D2O viscosities. Thermal offset viscosities for D2O obtained from the H2O data in Table 1 and from representative empirical equations in the literature will now be discussed. As already emphasized, these D2O thermal offset data are obtained from the H2O viscosity equations without employing any new parameters, only making the necessary modifications to T0 and to the molecular mass factor, as discussed in earlier work.4 As an example, application of the thermal offset effect to eq 1 requires that a, b, c, and γ be taken the same as they are for H2O, but T0 and A must be scaled. From Table 1 of ref 25, T0 for D2O is taken to be 231.832 K, giving a thermal offset compared with H2O of 6.498 °C; see Table 2. The square-root-mass factor (MD2O/MH2O)1/2 ) 1.054371 must then be applied to the resulting temperature-scaled viscosities. Thus, to obtain the converted eq 1 for the thermal offset Viscosities of D2O means that AD2O ) 1.054371 × AH2O ) 845.87228 cPKγ, instead of the empirically determined value of 885.60402 cPKγ in Table 2. Applying this concept to experimental H2O viscosity data points is even simpler, though the temperatures of the D2O data
D2O 19.57 10.89 7.076 5.025 3.782 2.967 2.400 1.988 1.679 1.440 1.251 1.100 0.9759 0.8733 0.7872 0.7143 0.6519 0.5981 0.5513 0.5104 0.4744 0.4425 0.4141 0.3887 0.3658 0.3452 0.3266 0.3096 0.2942 0.2800 0.2670 0.2550
TABLE 2: Fitting Parameters and Root-Mean-Square Deviations Ω cPKγ
A, a, K-1 b, K-2 c, K-3 γ T0a, K Ω a
III. Thermal Offset Viscosities
H2O 8.989 6.033 4.372 3.336 2.642 2.152 1.792 1.519 1.307 1.138 1.002 0.8903 0.7973 0.7192 0.6528 0.5959 0.5467 0.5039 0.4665 0.4334 0.4041 0.3781 0.3548 0.3338 0.3149 0.2977 0.2822 0.2680 0.2550 0.2430 0.2321 0.2219
H2O
D2O
802.25336 3.4741 × 10-3 -1.7413 × 10-5 2.7719 × 10-8 1.53026 225.334 1.056 × 10-4
885.60402 2.7990 × 10-3 -1.6342 × 10-5 2.9067 × 10-8 1.55255 231.832 7.636 × 10-5
Parameter fixed.
may not be the most convenient. For example, the viscosity of D2O at 26.498 °C would be the viscosity of H2O at 20 °C multiplied by 1.054371 or 1.0565 cP, probably with about three decimal place precision. As seen in the ensuing tables, the uncertainty is more evident at higher temperatures (t > 30 °C), where the outer structure mixture model concept, on which the thermal offset effect is based, must slowly fail. This was pointed out in earlier work,4 and is caused by different thermal expansivities of the Ih and II bonding components, a change in the structure of the type-II component to more varied bonding forms, and eventually to a breakdown of the entire two-state mixture model concept with increasing temperature. Table 3 contains these thermal offset values and compares them with the smoothed experimental viscosity values for D2O from Table 1, truncated to three significant figures. Note that for temperatures below 50 °C, the percentage errors in the thermal offset values are less than 1%, which is comparable to the experimental uncertainty. We note particularly that at very low temperatures the huge ηD2O/ηH2O viscosity ratios, which have been a concern in the past,1 become modified to a normal square-root-mass ratio when the thermal offset effect is applied. It should be pointed out that any standard graph comparing such closely coincident
Thermal Offset Viscosities of H2O, D2O, and T2O
J. Phys. Chem. B, Vol. 103, No. 11, 1999 1993
TABLE 3: Theoretical Offset Viscosities of D2O in cP with % Errors Compared with Experimental Values from Table 1a t °C
expt
theor
%
-30 -20 -10 0 10 20 30 40 50 60 70 80 90 100
19.57 7.08 3.78 2.40 1.68 1.25 0.976 0.787 0.652 0.551 0.474 0.414 0.366 0.327
19.52 7.10 3.80 2.41 1.68 1.25 0.972 0.782 0.645 0.544 0.467 0.407 0.358 0.319
-0.2 +0.3 +0.4 +0.3 +0.1 -0.2 -0.4 -0.7 -1.0 -1.3 -1.6 -1.8 -2.0 -2.3
a % errors here and in Table 4 were obtained from an additional significant figure and do not accurately reflect the differences between the truncated values given in the tables.
TABLE 4: Theoretical Offset Viscosities of D2O in cP from Converted Kestin et al.21 Equations t °C
expt
theor
%
0 10 20 30 40 50 60 70 80 90
2.40 1.68 1.25 0.976 0.787 0.652 0.551 0.474 0.414 0.366
2.40 1.68 1.25 0.972 0.782 0.646 0.545 0.468 0.407 0.359
0.0 +0.1 -0.2 -0.4 -0.7 -1.0 -1.2 -1.5 -1.7 -2.0
experimental and calculated values would show exact agreement, so it is important when testing these ideas to present the tabular values. Since this procedure can be used for eq 1 and for experimental data, it must also work for any of the empirical H2O viscosity equations in the literature. All that needs to be done to convert the equation to D2O viscosities is to substitute (t _ 6.498) for t in the empirical H2O equation, then multiply the result by 1.054371 to obtain the viscosity of D2O at temperature t. For example, converting the Kestin et al.21 H2O viscosity equations (their eq 15 for t e 40 °C and their eq 16 for t > 40 °C), Table 4 shows the comparison of D2O data from these equations with the Table 1 truncated experimental D2O viscosities. There appear to be no experimental data for the viscosities of T2O. These can be calculated in the same way as those for D2O and are listed in Table 5. For T2O, the square-root-mass factor is 1.105862, and in this paper we will estimate the thermal offset to be 8.766 °C. IV. Conclusions In conclusion, the “strange” isotope effects in liquid water have fallen out of our two-state outer structure mixture model quite naturally and quite simply. Because the two-state contributions change with temperature, pressure, and isotopic mass, an explicit incorporation of these effects must be included in any theoretical or computational study of liquid water. This has never been done, so confusion about this problem continues to persist. In fact, the bringing together of the properties of H2O and D2O should be considered further corroboration of the outer neighbor mixture model for water and thus to represent an important breakthrough in this field.
TABLE 5: Estimated Thermal Offset Viscosities of T2O in cP, with Offset ) 8.766 °C t °C _
30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45
T2O
t °C
T2O
29.14 14.53 8.93 6.13 4.50 3.47 2.77 2.27 1.90 1.62 1.40 1.22 1.08 0.957 0.859 0.776
50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125
0.706 0.645 0.592 0.547 0.506 0.471 0.440 0.412 0.386 0.364 0.343 0.325 0.308 0.293 0.279 0.266
Understanding isotope effects on the viscosity is particularly important since the interpretation of so many experiments in chemistry and biology depend on knowledge gained from isotopic substitution, D2O for H2O. In fact, indicating their usefulness, the ideas used here to clarify the viscosity relationships have already started to be used by other laboratories to help understand experimental results for liquid H2O and D2O concerning dielectric relaxation26 and solubilities27 and virial coefficients28 of macromolecules in these solvents. Transport and conformational changes of macromolecules depend centrally on the structure and viscosity of water and how these characteristics might be affected in the vicinity of the macromolecule or other “surface”. Reacting ions and neutral solutes also depend on these characteristics. It is therefore felt that the understanding presented in this paper will be extremely important to future work in solution chemistry, electrochemistry, and biology, including of course protein folding/unfolding reactions.29,30 We note from Tables 3 and 4 that the thermal offset viscosities become consistently smaller than the experimental viscosities as the temperature increases. This is caused by the breakdown of the mixture model idea, the additional softening of the structure and normal lowering effects on the viscosity with increasing temperature, complementing the mixture model transformations. Exact agreement with the D2O experimental viscosities in Table 1 can be obtained by empirically adjusting T0 or the square-root-mass factor to give these parameters a temperature dependence. A parallel concept could be used for T2O viscosities, which are also expected to be too low at higher temperatures. As far as the T2O viscosities reported here, the thermal offset of 8.766 °C was an estimate based on the ∼9.4 °C difference in the temperature of maximum density with that of H2O. However, the viscosities of T2O in Table 5 should still be reliable to better than two significant figures. More importantly perhaps, only one experimental viscosity of T2O for a single temperature needs to be known to adjust all the values in Table 5 so that the nearly three decimal place precision of Table 3 is obtained for the T2O viscosities. If indeed there ever arises an interest in high-precision viscosities of pure T2O, they can be found in this way. More likely, isotopic mixtures of H2O and T2O will be of more interest, so some of our future efforts will be directed toward obtaining good theoretical viscosities of isotopic mixtures. Mixtures of H2O and D2O are certainly important as outlined earlier in the 1985 Kestin et al. paper.3 However, these three-component mixtures, H2O, D2O, and HDO, are somewhat more difficult to describe by the methods outlined here, though with a good set of densities of various
1994 J. Phys. Chem. B, Vol. 103, No. 11, 1999 H2O/D2O mixtures over a range of temperatures and pressures, this might be possible to do. Acknowledgment. Though its importance is recognized in environmental, biological, and many technically related problems, it is still sad to say (see page vii of ref 8) that our water research has not been able to attract adequate funding. For this reason, we have been particularly grateful to the Robert A. Welch Foundation and the Petroleum Research Fund for helping to keep this research alive so that the inevitable complete understanding of this water problem can be finalized in our laboratory. References and Notes (1) Walrafen, G. E.; Chu, Y. C. J. Phys. Chem. 1991, 95, 8909. (2) DeFries, T.; Jonas, J. J. Chem. Phys. 1977, 66, 896. (3) Kestin, J.; Imaishi, N.; Nott, S. H.; Nieuwoudt, J. C.; Sengers, J. V. Physica A 1985, 134, 38. (4) Vedamuthu, M.; Singh, S.; Robinson, G. W. J. Phys. Chem. 1996, 100, 3825. (5) Cho, C. H.; Singh, S.; Robinson, G. W. J. Chem. Phys. 1997, 107, 7979 and papers cited. (6) Agayev, N. A. Proceedings of the 9th International Conference on the Properties of Steam, September 10-14, 1979, Technische Universita¨t Mu¨nchen, F.R.G., p 362. (7) The vibrational force constant description forms the basis for standard gas-phase calculations of the viscosity from intermolecular potential functions. See, for example, the problem on page 561 of the following: Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; John Wiley & Sons: New York, 1954 and discussions preceding this problem. (8) Robinson, G. W.; Zhu, S.-B.; Singh, S.; Evans, M. W. Water in Biology, Chemistry and Physics: Experimental OVerViews and Computational Methodologies; World Scientific: Singapore, 1996, pp 113-114. Even though the Osipov et al. data9 used in Table 4.6 of this book are flawed because of the use of far too small10 a capillary diameter (1 µm), roughly the correct H2O-D2O trends seem to occur. This may indicate that not
Cho et al. only the properties of bulk phase liquid water but also of surface perturbed water experience the same type of thermal offset effect. (9) Osipov, Yu. A.; Zheleznyi, B. V.; Bondarenko, N. F. Russ. J. Phys. Chem. 1977, 51, 748. (10) Hare, D. E.; Sorensen, C. M. J. Chem. Phys. 1987, 87, 4840. (11) White, G. F.; Twining, R. H. Am. Chem. J. 1913, 50, 380. (12) Hardy, R. C.; Cottington, R. L. J. Res. Natl. Bur. Stand. 1949, 42, 573. (13) Hallett, J. Proc. Phys. Soc. 1963, 82, 1046. (14) Korosi, A.; Fabuss, B. M. Anal. Chem. 1968, 40, 157. (15) Korson, L.; Drost-Hansen, W.; Millero, F. J. J. Phys. Chem. 1969, 73, 34. (16) Eicher, L. D.; Zwolinski, B. J. J. Phys. Chem. 1971, 75, 2016. (17) Kingham, D. J.; Adams, W. A.; McGuire, M. J. J. Chem. Eng. Data 1974, 19, 1. (18) Kestin, J.; Khalifa, H. E.; Ro, S.-T.; Wakeham, W. A. J. Chem. Eng. Data 1977, 22, 207. (19) Nagashima, A. J. Phys. Chem. Ref. Data 1977, 6, 1133. (20) Kestin, J.; Khalifa, H. E.; Sookiazian, H.; Wakeham, W. A. Ber. Bunsen-Ges. Phys. Chem. 1978, 82, 180. (21) Kestin, J.; Sokolov, M.; Wakeham, W. A. J. Phys. Chem. Ref. Data 1978, 7, 941. (22) Sengers, J. V.; Watson, J. T. R. J. Phys. Chem. Ref. Data 1986, 15, 1291. (23) Millero, F. J.; Dexter, R.; Hoff, E. J. Chem. Eng. Data 1971, 16, 85. (24) Matsunaga, N.; Nagashima, A. J. Phys. Chem. Ref. Data 1983, 12, 933. (25) Vedamuthu, M.; Singh, S.; Robinson, G. W. J. Phys. Chem. 1994, 98, 8591. (26) Barthel, J.; Buchner, R.; Hoelzl, C. Isotope Effects in Dielectric Relaxation: A Study of Normal and Supercooled H2O and D2O. Institut fu¨r Physikalische Chemie, Universita¨t Regensburg, Germany. Private communication. (27) Gripon, C.; Legrand, L.; Rosenman, I.; Vidal, O.; Robert, M. C.; Boue´, F. J. Cryst. Growth 1997, 177, 238. (28) Gripon, C.; Legrand, L.; Rosenman, I.; Vidal, O.; Robert, M. C.; Boue´, F. J. Cryst. Growth 1997, 178, 575. (29) Wiggins, P. Physica A 1997, 238, 113. (30) Robinson, G. W.; Cho, C. H. The Role of Hydration Waters in Protein Unfolding. In preparation.
