J. Phys. Chem. B 2000, 104, 7179-7182
7179
Refractive Index Mysteries of Water G. Wilse Robinson,*,†,‡ Chul Hee Cho,† and Gregory I. Gellene§ SubPicosecond and Quantum Radiation Laboratory, Department of Chemistry and Biochemistry, Texas Tech UniVersity, Mail Stop 41061, Lubbock, Texas 79409-1061, Department of Physics, Texas Tech UniVersity, Mail Stop 41051, Lubbock, Texas 79409-1051, and Department of Chemistry and Biochemistry, Texas Tech UniVersity, Mail Stop 41061, Lubbock, Texas 79409-1061 ReceiVed: March 8, 2000; In Final Form: May 11, 2000
Even though the refractive index is dominated by single molecule effects, such as the polarizability, there are some aspects of this property for water that are different from the refractive index behavior of most other liquids. Here, we discuss, then explain, these differences in terms of a two-state bonding model for this liquid. A simple two-parameter formulation based on this model is found capable of describing the refractive index of water, including its thermal maximum near 0 °C, from slightly supercooled temperatures to about +45 °C within a four- to five-decimal place accuracy. The agreement with experimental data achieved here using this simplest of possible models provides further strong support for the two-state description of this liquid.
TABLE 1: Data for Refractive Index Calculationsa
I. Introduction From studies concerning the properties of water as a function of temperature and pressure,1-4 it is becoming evident that this liquid on the average is composed of dynamically interconverting microdomains of two very different structural types. Unlike other ideas, both mixture components in this model5 have hydrogen-bonded inner tetrahedral structure with the four neighbors to the central molecule all having roughly 2.8 Å nearest neighbor O‚‚‚O distances. However,3 the O‚‚‚O‚‚‚O angles can be either the regular tetrahedral 109.5° as in ice-Ih, or strongly bent to angles around 80° as they are in the moderately dense ice polymorphs. Because of this bending, the outer next-nearest-neighbor O‚‚‚O distances cluster around roughly two values in the liquid, 4.5 Å (2 × 2.8 sin 1/2 109.5°) as in ordinary ice, and the shorter outer-neighbor O‚‚‚O distance of 3.5 Å (2 × 2.8 sin 1/2 80°) as in the moderately dense ice forms, ice-II in particular. Transformations between these two bonding forms in the pure liquid are evidenced experimentally by an increase in the outer second-neighbor O‚‚‚O amplitude near 3.5 Å with a concurrent decrease of the ordinary 4.5 Å outer O‚‚‚O regular tetrahedral bonding on increasing the temperature6 or the pressure.7 In fact, the good structural isosbestic characteristics of the radial distribution function with changing temperature and pressure4 have now fully confirmed a precise two-state description of this liquid at the second O‚‚‚O neighbor level. Changes with temperature in the volume fractions fI and, for a purely twostate description, fII ) 1 - fI, of Ih-type and II-type bonding in pure liquid water are directly related to the density.2 From that study, numerical values of fI and the densities FI and FII of Ihtype and II-type structure in the liquid as a function of temperature were obtained. These values are summarized in Table 1. These changes in water structure with temperature of * E-mail:
[email protected]. Fax: 806-742-3590. † SubPicosecond and Quantum Radiation Laboratory, Department of Chemistry and Biochemistry. ‡ Department of Physics. § Department of Chemistry and Biochemistry.
t, °C
fI
FI
FII
Lcalc
Lexp
nexp
105∆n
-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +15 +20 +25 +30 +35 +40 +45
0.49628 0.49180 0.48739 0.48306 0.47880 0.47460 0.47046 0.46639 0.46237 0.45841 0.45451 0.45065 0.44685 0.44310 0.43939 0.43573 0.41806 0.40135 0.38549 0.37039 0.35598 0.34219 0.32898
0.90196 0.90156 0.90115 0.90075 0.90035 0.89994 0.89954 0.89914 0.89874 0.89833 0.89793 0.89753 0.89713 0.89673 0.89633 0.89593 0.89394 0.89196 0.88998 0.88802 0.88606 0.88411 0.88217
1.11808 1.11671 1.11534 1.11398 1.11262 1.11127 1.10991 1.10856 1.10722 1.10588 1.10454 1.10320 1.10187 1.10054 1.09921 1.09789 1.09132 1.08483 1.07842 1.07208 1.06582 1.05963 1.05351
0.20652 0.20651 0.20650 0.20649 0.20648 0.20647 0.20646 0.20645 0.20644 0.20643 0.20642 0.20641 0.20640 0.20639 0.20638 0.20637 0.20632 0.20628 0.20624 0.20620 0.20617 0.20613 0.20610
0.20656 0.20654 0.20653 0.20651 0.20649 0.20648 0.20647 0.20645 0.20644 0.20643 0.20641 0.20640 0.20639 0.20638 0.20637 0.20636 0.20631 0.20626 0.20623 0.20620 0.20617 0.20615 0.20613
1.33427 1.33430 1.33432 1.33433 1.33434 1.33434 1.33434 1.33433 1.33431 1.33429 1.33427 1.33424 1.33420 1.33416 1.33412 1.33407 1.33376 1.33335 1.33286 1.33229 1.33165 1.33095 1.33019
+7 +6 +5 +4 +3 +2 +2 +1 0 0 0 -1 -1 -2 -2 -2 -3 -3 -3 -1 0 +3 +6
a The values of f , F and F are obtained from the fit ii parameters I I II in ref 2 and are used here and in our other work on water without change. The values of the two specific refraction parameters derived from this analysis are LI ) 0.20779 and LII ) 0.20527 cm3 g-1. These are the only new parameters required for this refractive index analysis. In the last column, the deviation, ∆n ) nexp - ncalc. The experimental data nexp for t e 20 °C are from ref 18, while the higher temperature values are from ref 27.
course have to be numerically invariant with respect to all the properties of pure watersdensity,2 viscosity,8 X-ray diffraction structure,4 or the refractive index to be described in the present paper. These same structural changes in the solvent must also be included in a full thermodynamic description of any hydration problem, including protein hydration.9 Despite the obvious and broad successes of this realistic starting point for the study of liquid water, our explicit structural approach has been strongly criticized, not only for being
10.1021/jp000913f CCC: $19.00 © 2000 American Chemical Society Published on Web 06/29/2000
7180 J. Phys. Chem. B, Vol. 104, No. 30, 2000 oversimplified and “too structural”, but because the terminology ice-II-type seems to ignore the wide array of other structures that could be involved. However, from the beginning,1 all those structures have been considered in our papers, including the amorphous solid forms. The advantage of using ice-type bonding as a guide for understanding the bonding in the liquid is that these structures are more precisely known than the structures of the amorphs. Furthermore, ice-II has the lowest energy structure of any of the moderately dense crystalline polymorphs and would be expected to be the most likely structure to grow in with increasing temperature, replacing the slightly more stable ice-Ih-type structure. In this regard, as described below, it is simply incorrect to obtain information about the relative stabilities of the possible ice-type packing configurations in the liquid from the phase diagram, or equivalently from the Gibb’s free energies, of the pure crystalline forms. Since crystalline ice-II has ordered hydrogens and the neighboring phases do not,10 the Pauling entropy11,12 of proton disorder R ln 3/2 ) 3.3712 J K-1 mol-1 stabilizing ice-Ih is not present in crystalline ice-II. Since there is little chance of having ordered hydrogens in the ice-II-type structure in the liquid, the P∆V term ≈ 830 J mol-1 in the crystalline ∆G free energy difference compensating for the lack of this entropy in ice-II would be expected to be much smaller, or absent, in the liquid. In fact, at atmospheric pressure near 0 °C, the ∆G difference between the ice-Ih-type structure and the ice-II-type structure in the liquid is very likely within the thermal energy of kT. Thus, it is not surprising that at this temperature there are approximately equal probabilities for the occurrence of these two bonding forms, as verified from compositional determinations in our density analysis,2 and also very much in line with other earlier5 and more recent13 proposals. The transformation from ice-Ih to ice-II can be described structurally as follows:10,14 If the hexagonal columns in ice-Ih are detached, moved relatively up and down parallel to the c-axis, then rotated about 30° (∼109.5° - 80°) around their c-axes, they can be relinked together in a more compact way. As Kamb did earlier,5 we have continued to call this detaching and relinking a bending of the hydrogen bonds, or equivalently in the words of Eisenberg and Kauzmann (ref 10, page 79) a distortion of those bonds. This bending transformation has been clearly defined in our work3,15 by two distinct fairly narrow angular-dependent potential minima separated by a thermally accessible barrier. We have never proposed a single wide angle potential, for which criticisms13 of bending would be valid. Since the other moderately dense forms of ice, -!II, -V and -VI, have similar local oxygen-atom and next-nearest-neighbor structures as ice-II,3 but, ignoring the Pauling entropy term, are not as energetically accessible, is the reason for our continuing “insistence” on ice-II bonding as the type of dense structure in the liquid. From these considerations then, it is seen that the most important feature in the liquid is the probability of having highly bent intermolecular O‚‚‚O‚‚‚O bonds from a transformation between the ice-Ih-type and ice-II-type structural forms. II. Experimental Refractive Index The refractive index n of water as well as many other properties of this liquid continue to attract the attention of various international organizations.16 One of the goals of those activities has been to create the simplest empirical equations that best fit what are considered to be the most reliable experimental data. Using an improved equation of state, eight adjustable parameters and two fixed wavelength parameters, a recent reformulation of the refractive index equation for water
Robinson et al. as a function of temperature, pressure and wavelength of light has been published.17 One of the main objectives of this latest work was to achieve better agreement with the conceived best experimental data18 at supercooled temperatures down to -12 °C. An earlier empirical equation19 was found capable of matching the experimental refractive indices for visible light from about 10-60 °C at ambient pressure within the experimental uncertainty of nearly one part in 105. However, this earlier formulation at -10 °C, for example, gave a value of n lower than the experimental value by 16 × 10-5. Moreover, below 0 °C, nexp - ncalc was found to rise uniformly, approaching 25 × 10-5 at -12 °C. The new ten-parameter equation17 reduced this error to about 6 × 10-5 at -12 °C, but again a uniformly increasing difference was evident below 0 °C. Formulations20 that focus mainly on higher temperatures would not notice this type of discrepancy. The differences between the experimental and calculated refractive index data could be caused by some sort of systematic error in either or both of those determinations. If it develops that it is in the formulation, very likely such a systematic error results from the inability of any of the empirical equations to match the curvature in the supercooled region of this property of water. In other work,8 we have found this to be the case for widely accepted empirical viscosity equations for water.21 We believe that this type of deficiency is directly related to the “twostate” bonding characteristics of this liquid,3 where the compositional variations and physical properties of the two distinct forms are strongly temperature and pressure dependent. To reduce the number of empirical parameters, future internationally adopted equations for the various properties of water will themselves more than likely have to use some sort of a twostate formulation, especially when considering temperatures below 0 °C. III. The Two-State Formulation In this paper, only the temperature dependence of the refractive index will be considered and only for a single wavelength, the average 589.262 nm of the Na-D doublet. It is important to note that the Na-D doublet is well removed from any complicating absorption resonances in water. As far as the pressure dependence of the refractive index is concerned, this will be considered in future work using pressure-dependent compositional variations derived from the isothermal compressibility22 and other pressure/volume data for water.23 Isotope effects on the refractive index, D2O vs H2O, can be directly obtained from the approximately 6.5 °C “thermal offset effect” described in earlier work.24 To provide the most transparent presentation, the two-state procedure used here will be the simplest one possible, on which future elaborations can be made. The two bonding components in water, as usual, are considered separately. Each of their indices of refraction is assumed to follow the Lorentz-Lorenz equation,10 2 1 n(λ) - 1 ) L(λ) F n(λ)2 + 2
(1)
where F is the density and L(λ) is sometimes referred to as the specific refraction. Once the specific refraction L is known, eq 1 can be used to obtain n, the λ dependence of n and L in eq 1 henceforth being dropped because of our consideration of only a single wavelength. The explicit outer-neighbor Ih-type/II-type two-state bonding model for water would specify3 that the specific refraction L of
Refractive Index Mysteries of Water
J. Phys. Chem. B, Vol. 104, No. 30, 2000 7181
the liquid can be expressed as a weighted combination of LI and LII of the two contributing forms,
L ) fILI + (1 - fI)LII
(2)
See ref 25 for a discussion of this approximation. As mentioned in our earlier papers,2 this two-state description is expected to break down at higher temperatures because of the intrusion by other structures, including broken hydrogen bonds. For this reason, only temperatures to the vicinity of +45 °C will be considered here in any detail. IV. Results The compositional factors in eq 2 as well as the temperaturedependent densities FI and FII in eq 1 for this analysis are expected to be identical to those derived from the earlier twostate density study2 and can therefore be used here without additional adjustment. Notice then from eqs 1 and 2 that there are only two new parameters that need consideration, LI and LII. Within the spirit of simplification in the present paper, and in keeping with experimental data for other substances (see, for example, the data in ref 26 when converted to specific refraction through eq 1), these specific refraction values will be assumed constant for the temperature range considered. This simple procedure would apply to other wavelengths as well, as long as they are not too close to the water resonances near 229 nm in the ultraviolet and 5.43 µm in the infrared. Since both LI and LII should be slightly temperature dependent,26 a precise fit of the experimental refraction data would not be expected from this ultra-simple procedure. Even so, a 1 × 10-4 to 1 × 10-5 precision was attained for the refractive index using this procedure. See Table 1. In addition, other interesting results emerged. For instance, the specific refraction LI of the Ih-type component coming from this analysis was found to have a value of 0.20779 cm3 g-1. This is fairly close to the specific refraction of crystalline ice-Ih,10 about 0.2101 cm3 g-1. Unfortunately, the specific refraction of ice-II seems not to have been measured. However, by comparing the specific refractions of ice-Ih and the liquid, where Lliq ) 0.20648 cm3 g-1 at 0 °C (see Table 1), a slightly lower value of LII compared with LI would be expected. This is found to be the case with LII from this simple two-state analysis being 0.20527 cm3 g-1. Perhaps the most interesting characteristic of the temperature dependence of the refractive index of water is its thermal maximum, which occurs, not at 4 °C as does the density maximum, but rather near 0 °C. This weak maximum has a similar shape and occurs at nearly the same temperature in both the experimental data and in the two-state model. See Figure 1. Why is this maximum shifted about 4 °C below the density maximum? Using a single state approach, solving eq 1 for n with a single L (∼0.2065), and using the known temperaturedependent densities of water, gives the refractive index maximum very close to 4 °C, so something else must be taken into account. As we have been finding for the other “strange” properties of water3,4 and interfacial water,9 this refractive index maximum is directly related to the steep variation of fI with temperature. At low temperatures, the fILI term of eq 2 dominates the (1 - fI)LII term, but just below 0 °C, because of the growth of (1 - fI) with increasing temperature, the situation reverses. This behavior, when combined with the additional fI dependence of the density, is directly responsible for the refractive index maximum shifting from near 4 °C to near 0 °C.
Figure 1. Comparison of the experimental refractive indices (symbols) with the calculated values (solid line) as a function of temperature.
V. Conclusions Because of the great precision of refractive index measurements, the small variations caused by liquid state contributions can be accurately separated from the dominant single molecule contributions. These refractive index measurements therefore allow yet a different way of looking at the properties of liquid water. Nevertheless, our very simple two-state description emerges quantitatively from these experimental data. This indicates once again3 the ability to unify all the O atom related properties of water under the same theoretical “roof”. Remarkably, using the simplest rendition of this theory, the quantitative features of the subtle thermal maximum near 0 °C in the refractive index of H2O automatically arise. Though this simplest of ideas was found here and in our earlier work to give a good quantitative description of this supposedly mysterious material, room is certainly left for further improvement. For instance, in the present analysis, one should take account of the temperature dependence of LI and LII. This will contribute more curvature to the calculated refractive index expression, giving better agreement with the experimental data at both high and very low temperatures. However, even without such elaborations, this is the first time that the properties of any liquid have been so thoroughly characterized by structural features derived from intermolecular bonding properties. It is perhaps ironic that this type of extremely simple explanation would emerge for any liquid, particularly one universally believed to be one of the most complicated liquids in existence. Acknowledgment. Acknowledged for financial support of this work are the R. A. Welch Foundation (D-0005 and D-1094) and the Petroleum Research Fund (ACS-PRF# 32253-AC6). Private financial support from Dr. Joel Kwok, Dr. and Mrs. Jamine Lee and the Chemistry Support Fund at Texas Tech is also acknowledged. References and Notes (1) Bassez, M.-P.; Lee, J.; Robinson, G. W. J. Phys. Chem. 1987, 91, 5818. (2) Vedamuthu, M.; Singh, S.; Robinson, G. W. J. Phys. Chem. 1994, 98, 2222. (3) Cho, C. H.; Singh, S.; Robinson, G. W. J. Chem. Phys. 1997, 107, 7979. (4) Urquidi, J.; Cho, C. H.; Singh, S.; Robinson, G. W. J. Mol. Struct. 1999, 485-486, 363. For a more complete discussion of the structural
7182 J. Phys. Chem. B, Vol. 104, No. 30, 2000 isosbestic points, see Robinson, G. W.; Cho, C. H.; Urquidi, J. J. Chem. Phys. 1999, 111, 698. (5) Kamb, B. In Structural Chemistry and Molecular Biology; Rich, A., Davidson, N., Eds.; W. H. Freeman and Co.: San Francisco, 1968; pp 507-542. (6) Bosio, L.; Chen, S.-H.; Teixeira, J. Phys. ReV. A 1983, 27, 1468. (7) Okhulkov, A. V.; Demianets, Yu. N.; Gorbaty, Yu. E. J. Chem. Phys. 1994, 100, 1578. (8) Cho, C. H.; Urquidi, J.; Robinson, G. W. J. Chem. Phys. 1999, 111, 10171. (9) Robinson, G. W.; Cho, C. H. Biophys. J. 1999, 77, 3311. (10) Eisenberg, D.; Kauzmann, W. The Structure and Properties of Water; Oxford University Press: London, 1969. (11) Pauling, L. J. Am. Chem. Soc. 1935, 57, 2680. (12) Nagle, J. F. J. Math. Phys. 1966, 7, 1484. (13) Soper, A. K.; Ricci, M. A. Phys. ReV. Lett. 2000, 84, 2881. (14) Kamb, B. Acta Crystallogr. 1964, 17, 1437. (15) Cho, C. H.; Singh, S.; Robinson, G. W. Chem. Phys. 1998, 232, 329. (16) Thermodynamic formulation release; International Association for the Properties of Water and Steam: Fredericia, Denmark, 1996.
Robinson et al. (17) Harvey, A. H.; Gallagher, J. S.; Levelt Sengers, J. M. H. J. Phys. Chem. Ref. Data 1998, 27, 761. (18) Saubade, Ch. J. Physique 1981, 42, 359. (19) Schiebener, P.; Straub, J.; Levelt Sengers, J. M. H.; Gallagher, J. S. J. Phys. Chem. Ref. Data 1990, 19, 677. (20) Djurisˇic´, A. B.; Staniæ, B. V. Appl. Opt. 1999, 38, 11. (21) Kestin, J.; Sokolov, M.; Wakeham, W. A. J. Phys. Chem. Ref. Data 1978, 7, 941. (22) Vedamuthu, M.; Singh, S.; Robinson, G. W. J. Phys. Chem. 1995, 99, 9263. (23) Cho, C. H.; Urquidi, J.; Robinson, G. W. Pressure Effect on the Density of Water. J. Chem. Phys. (manuscript in preparation). (24) Cho, C. H.; Urquidi, J.; Singh, S.; Robinson, G. W. J. Phys. Chem. B 1999, 103, 1991. (25) Heller, W. J. Phys. Chem. 1965, 69, 1123. (26) Aralaguppi, M. I.; Jadar, C. V.; Aminabhavi, T. M. J. Chem. Eng. Data 1999, 44, 435. (27) Tilton, L. W.; Taylor, J. K. J. Res. Nat. Bur. Stand 1938, 20, 419.
