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J . Phys. Chem. 1994, 98, 8591-8593
Accurate Mixture-Model Densities for DzO Mary Vedamuthu, Surjit Singh, and G. Wilse Robinson’ Subpicosecond and Quantum Radiation Laboratory, Departments of Physics and Chemistry, Box 41 061, Texas Tech University, Lubbock, Texas 79409- I061 Received: March 9, 1994; In Final Form: May 24, 1994’
Recently, we presented arguments strongly supporting a type of mixture model-actually, a mixed bonding model-for liquid water (H2O) through an analysis of its extremely precise temperature-dependent density data. This model is now applied to the densities of liquid D20 in the temperature range -20 to +lo0 OC. It is found that the available experimental data can be fit to the reported five-decimal-point precision. As in the case of H 2 0 , the output parameters indicate the existence of two general types of bonding characteristics in the liquid, a bulky form similar to that in ice-Ih and a dense form such as that found in ice-11. Independent support for the model is obtained from the temperature-differential X-ray scattering data of Bosio et al. on D20.
TABLE 1:
Introduction
The structure of liquid water has been the subject of much experimental and theoretical research for more than a century. In the late 1800’s,R8ntgen considered liquid water to be a mixture of two kinds of “molecules”, one icelike, whose mass fractions are dependent on pressure and temperature.’ In 1933, Bernal and Fowler suggested the simultaneous existence of two crystalline forms in the liquid statee2Mixture models based on the supposed existence in water of various crystalline and/or pseudocrystalline domains have continued to claim attention up to recent time~.3-~ A mixture model proposed earlierS for liquid H20 to explain its density anomaly provided a quantitative assessment of experimental densities for -34 It 5 +70 O C within the reported four- to seven-decimal-point precision. According to this model, there exists in liquid water two intermixed bonding structures. One type of bonding is “open” or “bulky” and has a density similar to that of ordinary ice-Ih. The other has a morecompact structure with a density close to that of the dense polymorphs of ice (such as ice-11, -111, -V, and -VI). As the temperature or is raised, the bulky structure breaks down, creating more of the dense structure. Strong evidence is now accumulating3g7 that this increased density is accomplished by an increase in nonnearest-neighbor coordination by bending of the hydrogen bonds and not by a deformation or breaking of nearest-neighbor hydrogen bonding.
tit ic H20
fit id D20
fit i? D20
4.54866E-2 3.6522E-4 8.69196E-2 225.334 1.08761 0.9195 0.84632 1.1816 4.78005E-4 1.29473E-3
3.91129E-2 3.02581 E-4 7.34949B-2 23 1.832 0.9802P.b 1.0202 0.76854 1.3012 4.00995E-4 1.300458-3
3.97129E-2’ 3.02581E-4’ 7.34949E-2” 231.832‘ 0.98007 1.0203 0.15357 1.3270 1.04376E-3 4.47931E-4 6.39610E-6 3.1458 1 E-6
00
oa
oa 0’
Parameter fixed. * See text. Exact algebraic fit ofeight H2Odensities between -30 and + 4 OC. Exact algebraic fit of seven D20 densities between -20 and +11.18 “C. Least-squares fit of 13 DzO densities between -20 and +70 OC. /The units are as follows: TO(K); Vr(To), VrdTo) (cm3g-’); PI(To),PII(TO)(g ~ m - ~A),;C,QI, (YII (K-’); B, 81,811 (K-2).
reason5 for this behavior is that, a t low temperatures, the proportion of the dense form rapidly increases with increasing T. Density Data
The aim here is to obtain, through eq 1, precise quantitative fits of the temperature-dependent density data of D20 and to compare the results with those obtained earlier for H20. The density a t atmospheric pressure of liquid H20 from -30 to 150 O C is well represented by Kell’s rational function with seven parameters8 A similar function with one less parameter and one less figure of accuracy was employed by KeH9 to represent the densityof D20fromO to+lOl OCat atmospheric pressure. Using the coefficients for the rational function given in Table 3 of ref 9, we calculated the densities for temperatures below 0 O C . These data (with five-decimal-point precision) from -20 to + l o 0 O C are presented in Table 2. Some fairly recent experimental work on supercooled D20 has been performed by Hare and SorensenIo on samples confined in glass capillaries of 25 pm diameter. These density values, in the temperature range from -20 to +40 O C , are presented in Table 2. However, these data should not be taken at face value because of systematic errors arising from surface energy effects.1’ It has been argued that the effect of surface energy on the free energy of samples in small capillaries can be considerable. Because of this, the measured apparent density increases with decreasing sample size below about 50 pm. Comparison of Kell’s data with Hare and Sorensen’s data from 0 to +40 “C shows that the agreement is fairly good, with
+
Fitting Procedure
To test this model, we considered in an earlier paper5 the experimental densities of H20 in the temperature range -34 to +70 O C . These data could be fit to the mixture model with very high precision. Furthermore, the output parameters from these fits were found to have a close relationship with the open and dense polymorphs of ice (see Tables 1 and 2 of ref 3). In order to fit Kell’s8 smoothed experimental density data, the following equation was used:
where V(P,T) is the specific volume of water and Vl(P,T) and VII(P,T)are the specific volumes of the open and dense components, respectively. The factorf(P,T) is the mass fraction of the liquid that is composed of the open ice-I-type bonding. A volume minimum (density maximum) occurs in eq 1 in spite of the fact that both V1(P,T) and VII(P,T)increase with T . The @
parad
Abstract published in Advance ACS Abstracts, July 15, 1994.
0022-365419412098-8591$04.50/0
0 1994 American Chemical Society
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Vedamuthu et al.
The Journal of Physical Chemistry, Vol. 98, No. 34, 1994
TABLE 2 Liquid DzO Densities (gm ~ m - at ~ )1 atm from Kell’s Equation Compared with Experimental H-S Densities, Fit i and Fit ii t “C H-Sa Kellb fit i fit ii -20 -19 -18 -17 -15 -10 -5 0 10 11.18 20 30 40 50 60 70 80 90 100
1.09240 1.093 62 1.094 80 1.09575 1.097 52 1.10094 1.103 33 1.104 79 1.10606 1.105 28 1.103 19 1.100 01
1.091 81 1.092 99 1.094 09 1.095 11 1.096 96 1.100 58 1.103 07 1.104 69 1.105 99 1.106 00 1.105 34 1.103 24 1.099 96 1.095 70 1.090 60 1.084 75 1.078 24 1.071 12 1.063 46
1.091 81 1.092 99 1.094 09 1.095 11 1.096 96 1.100 58 1.103 07 1.104 69 1.105 99 1.106 00 1.105 34 1.103 25 1.100 02 1.095 85 1.090 90 1.085 29 1.079 13 1.07248 1.065 43
1.091 81 1.092 99 1.094 09 1.095 11 1.096 96 1.100 58 1.103 07 1.104 69 1.105 99 1.106 00 1.105 34 1.103 24 1.099 96 1.095 70 1.090 60 1.084 75 1.078 24 1.071 11 1.063 43
Reference 10 with linear interpolation. Reference 9. deviations only of the order of 1-2 X 10-4 g cm-3. However, at temperatures lower than about -5 OC, larger deviations are evident, with the Hare/Sorensen density values being greater than Kell’s. This is what would be expected from surface energy effects. If a surface energy correction were applied, these two sets of density data in the supercooled regime would no doubt be in much better agreement. For example, comparing Hare and Sorensen’s density data for H2Oin capillariesof 25 and 300pmdiameters,1°J2scaling the temperature to the same HzO/D20 ( T - TO)values (see below), and applying the H2O error ratio to the D20 data gives almost perfect agreement with the Kell extrapolated D20 density data. In any case, for the present purposes, we will use Kell’s smoothed data9to represent the experimental D20 densities for the analysis in this paper. Density Fits
In order to fit the experimental density data of DzO using eq 1, analytical forms for Vl(P,T),VII(P,T), andf(P,T) in terms of a set of fitting parameters are required. The forms for VI and V I ,used are5
Through the use of the procedure in ref 5 , some preliminary least-squares fitting was performed, with the B’s equal to zero, using all temperature points in Table 2 from -20 to +11.18 “C. Through these fits it was found that the parameter TOwas about 232 K, which is close to the singular temperature of 236 K for D20 given by Angell.14 However, such fits, as would be expected because of the small number of significant figures and the large number of adjustable parameters for so few data points, gave unstable values of the output parameters VI and VII. Hence, it was decided to fix the value of VI(TO),which should correspond roughly to the D 2 0 ice-Ih density at 232 K. In the earlier work,5 this correspondence was not exact. The density of the open ice-Ih-type bonding structure in the liquid, because of missing long-range order, could beslightly lower than in the crystal at the same temperature. To incorporate this correction, the ratio, 1.0222/0.9213, of the actual densities15 of D20 ice-Ih at 232 K [To(D20)] and H2O ice-Ih at 225 K [To(HzO)] was multiplied by the density, 0.9195 g ~ m - of~ the , ice-Ih component (at 225 K) used earlier5 for the liquid H2O density fits. With this procedure, the fixed value of VI(TO)to be used in the liquid D20 fitting procedure is 0.98020 cm3 g-I. This corresponds to p = 1.0202g cm-3, compared with the actual density,l5 p = 1.0222 g ~ m - of ~ ,D2O ice-Ih at 232 K. Fit i. Similar to the procedure in the earlier paper,5 an exact algebraic fit was performed by utilizing seven roughly equally spaced points between -20 and +11.18 OC. This creates the seven required equations to solve algebraically for the seven output parameters ( P I fixed and ps = 0). The density values obtained from this fit exactly match the five-decimal-point density precision given by the Kell equation, not only for the data points used in the fit but also for others out to nearly +30 OC. The resulting parameters and density values from fit i are given in Tables 1 and 2. Fit ii. In addition to the above algebraic fit, a least-squares fit to include higher temperature density data was also performed. In this fit, the parameters A-C and TOare fixed at their output values from the previous fit, and the ps are included. This fit utilizes 13 of Kell’s density values from -20 to +70 OC, with an equal number of density points above and below the density maximum. The analysis is carried out by varying the six remaining adjustable parameters VI, V I I (,Y I , ( Y I I , @ I , and 011. The output parameters so obtained are given in Table 1, and the resulting density values are found in Table 2. Discussion
In these equations, TOis the temperature at which f ( P , T ) = 1, i.e. where all the bonding in the liquid is represented by the lowdensity, open hydrogen-bonded network structure of ice-Ih. However, in the liquid state, this structure is expected to be highly disordered, similar to that in amorphous solid water.l3 As stressed in the earlier work,5 the proposed model, because of missing longrange order, is by no means to be conceived as a “mixture of ices”. The parameters ( Y I and ( Y I I in eqs 2 and 3 are the linear, and PI and P11 are the quadratic, thermal expansion coefficients. A rather arbitrary analytical function, [ l - f ( P , T ) ] = tanh
[
+
A( T - To) B( T - To)2 1 +C(T-To)
is chosen for f ( P , T ) , bearing in mind that its value lies in the range &le5 Substituting eqs 2-4 into eq 1 gives a function with 10 adjustable parameters for each pressure P .
In the present paper, the main attention concerns the isotope effect on the density anomaly. The density maximum in D 2 0 occursatatemperature(11.18 OC)over7 OChigherthaninH20 (3.94 “C). We believe that the major factor contributing to this isotope effect is that the parameter f ( P , T ) is sensitive to intermolecular frequencies and zero-point energies in the temperature-dependent potentials,6causing the open structure in H 2 0 to break up more easily at a given temperature than that in D20. ThusflD2O) >f(H20) at all temperatures, pushing the density maximum to a higher temperature in D20. The isotope effects on the librations, which are closely related to activation energies, are also responsible for isotope effects on the viscosity, heat capacity, and the relaxation times.6 The densities obtained through fit i agree very well with the experimental density data, even beyond the fitting range. With theadditionofthePparametersinfitii,itisseen that thecalculated densities are excellent out to nearly 100 OC. Considering the number of fitting parameters used in the model, the attainment of such agreement is in itself not very surprising. After all, Kell’s fitting function for D2O contains only six parameters. At this point then, the question that naturally arises does not concern whether or not the density data can be fit to high precision by the proposed model, but rather, it concerns whether the fitting parameters make sense.
Accurate Mixture-Model Densities for D2O A first concern has to do with D20/H2O density ratios. Ignoring zero-point effects and the often unreported variations in isotopic composition in the experiments, the ratio of D20 to H20 densities would be close to the molecular weight ratio of D2016/H20"5 = 1.1117. We have seen above, however, that, mainly because of the higher value of To(D20), the required ratio from the actual ice-Ih crystal densitie@ is closer to 1.1095. In fact, we have fixed the ratio PI(HZO)/PI(D~O) for fit i at exactly this value. What about the ratio? From Table 1, it is seen to be 1,1012, somewhat lower than the P I ratio. Note also that the ratios p l l / p ~for H20 (fit i) and D2O (fit i) are 1.2850 and 1.2754, respectively, close, as they should be, since only the mass, not the lowest order bonding structure, is changed by the isotopic substitution. Thus, it is seen that, by fixing the value of pl(D20) in fit i, reasonable values are obtained for pll(D20). As we found to be the case from the H20 analysis, when higher temperature data are included in the fits, as in fit ii, there is a tendency for TO)to tend toward the density of denser solid forms (ice-V and -VI), and when ps are included, the a's change to accomodate them. The same effects are observed here for D20. As a further check on the reasonableness of the fitting parameters, it is interesting to calculate the fI values from the A-C and TO parameters using eq 4 for the two isotopic modifications a t their temperatures of maximum density. They differ by only about +0.01, which is within their uncertainty. This reveals that, when the temperatures are scaled according to the temperatures of maximum density (or for that matter, their TOvalues), the bonding fractions in liquid D2O are not very different from those in liquid H20. This is in line with the observationl5 that the difference between H20 and D2O ice-Ih forms is hardly significant, both with respect to the hydrogenbond lengths and the tetrahedral angle. As mentioned in the earlier paper,5 further support for this model is obtained from the differential X-ray diffraction data of Bosio et al.' on liquid D20. These data show that, as the temperature of the liquid is increased, a peak around 4.5 A, which corresponds to the first non-nearest-neighbor 0-0 distance in ice-Ih, decreases considerably, while a peak around 3.3 8,strongly emerges. This emerging structural form is in fact very close to that which would be expected for the first non-nearest-neighbor O.-O distance in ice-I1 and the other high-density forms of ice.
The Journal of Physical Chemistry, Vol. 98, No. 34, 1994 8593
In conclusion, as stated in the previous a n a l y ~ i sthe , ~ excellent density fits that can be obtained from the ice-Ih/dense-ice mixedbonding model for liquid water certainly provide a necessary condition for the reality of the model. To complete the substantiation of the model, more types of data, of course, need to be considered. This will be attempted in the future, though already in earlier work6 it was found that, besides the density, many other experimental data, such as the viscosity, various relaxation times, and intermolecular vibrational frequencies, for both liquid H2O and D20, were quantitatively consistent with the picture presented here. If nothing else, such a description will tie together and make more understandable most properties of these important liquids within a semiempirical framework where the parameters have some physical content derived from the ice structures. Acknowledgment. Financial support for this work has been shared by the National Science Foundation (Grant CHE91 12002) and the Robert A. Welch Foundation (Grants D-0005 and D-1094). References and Notes (1) Rbntgen, W. C. Ann. Phys. 1892, 45, 91. (2) Bernal, J. D.; Fowler, R. H. J . Chem. Phys. 1933, 1, 515. (3) Kamb, B.InStructural Chemistry andMolecular Biology; Rich, A., Davidson, N., Eds.; W. H. Freeman and Co.: San Francisco, 1968; pp 507542. (4) Davis, D. M.; Jarzynski, J. In Water and AqueousSolutions; Horne, R. A., Ed.; Wiley-Interscience: New York, 1972; p 377. (5) Vedamuthu, M.; Singh, S.; Robinson, G. W. J . Phys. Chem. 1994, 98, 2222. (6) Bassez, M.-P.; Lee, J.; Robinson, G. W. J. Phys. Chem. 1987, 91, 5818. (7) Bosio, L.; Chen, S.-H.; Teixeira, J. Phys. Rev. 1983, A27, 1468. (8) Kell, G. S. J. Chem. Eng. Data 1975,20,97; in particular, see Table 3. (9) Kell, G. S. J . Chem. Eng. Data 1967, 12, 66. (10) Hare, D. E.; Sorensen, C. M. J . Chem. Phys. 1986, 84, 5085. (11) Leyendekkers, J. V.; Hunter, R. J. J . Chem. Phys. 1985,82, 1440 1447. (12) Hare, D. E.; Sorensen, C. M. J. Chem. Phys. 1987,87,4840. (13) Stanley, H. E.; Teixeira, J. J . Chem. Phys. 1980, 73, 3404. (14) Angell, C. A. In Water: A Comprehensive Treatise; Franks, F.,Ed.; Plenum: New York, 1982; 7, pp 1-82. (15) Lonsdale, K. Proc. R . SOC.London, A 1958, 247, 424.
