Zero point energy and the origin of the density maximum in water

Physics Letters A 372 (2008) 1551–1554 www.elsevier.com/locate/pla

Zero point energy and the origin of the density maximum in water F.A. Deeney ∗ , J.P. O’Leary Physics Department, National University of Ireland, Cork, Ireland Received 24 April 2007; received in revised form 10 October 2007; accepted 11 October 2007 Available online 18 October 2007 Communicated by V.M. Agranovich

Abstract The density maximum in water must arise due to the opposing action of two independent physical processes. Here we calculate the effects of zero point energy on water near room temperature, to show that the phenomenon, acting in competition with classical expansion/contraction, can explain the existence of the density anomaly. © 2007 Elsevier B.V. All rights reserved. PACS: 03.65.-w; 61.20.-p

Over the past two decades there has been an upsurge of interest in the structure and properties of water. This has been due partly to the continued fundamental role of water in the sciences, particularly in biology and chemistry, but also to the increasingly powerful computational platforms that have become available. In addition there has been more sophisticated use of experimental methods such as x-ray [1–4] and neutron scattering [5,6], together with various advanced spectroscopic techniques [7–9]. Because of these investigations, there is now a good knowledge of the different types of molecular clusters that exist in water, together with information on the hydrogen bond structure and how it varies under the action of different external factors such as temperature, pressure and added solutes [10, 11]. These results have been incorporated into different computational models to account for the thermodynamical properties of water, including its behaviour in the supercooled state. The density maximum in water is probably its best-known anomaly and attempts to account for this phenomenon go back to Roentgen [12] in 1892 and possibly further. His idea was that water at any temperature is made up of two separate types of clusters, one ice-like, open and low-density (LD), the other more compact and of higher density (HD). He proposed that as the temperature of water changes, the relative amount of

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E-mail address: [email protected] (F.A. Deeney). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.10.031

each type also changes; as the temperature falls, the fraction of the LD type present increases until eventually this open structure dominates and the overall water density decreases. The crossover point below which this change occurs is then the maximum density temperature. This model is usually referred to as the two-state or mixture model and it has been developed sporadically over the last century. There has been a particular renewal of interest in this approach during the last decade [13,14] when modern computational techniques were used, in conjunction with a more sophisticated use of the mixture concept, with some success. A different computational model [15, 16] has been to devise, using some of the findings from recent experiments, a suitable inter-molecular potential, and thereby an empirical equation of state, to account for the density maximum and other anomalous properties. The directionality of the attractive hydrogen bonds, in particular, is included in these models. By appropriate parameterisation, this method has also yielded reasonable agreement between calculation and experiment. (See Refs. [17] and [18], for example, for recent reviews of computer simulations of water structure.) We present here a different approach to the problem of the density maximum in water that concentrates on the nature of the underlying physical mechanisms involved during the cooling/heating processes rather than on details of the intermolecular potential. From classical thermodynamics one knows that as the temperature falls, the thermal fluctuations of a system are reduced. The system then contracts and the density increases. In

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the case of water, the fact that the density below 4 ◦ C actually gets progressively smaller can only be due to the dominance under these conditions of an independent physical mechanism that causes an increase in thermal fluctuations with falling temperature. This will act in opposition to the classical contraction and there will be a critical temperature at which the two effects cancel giving rise to a density maximum. Consider the possible origin of this second mechanism. At the present time there is just one known physical phenomenon that gives rise to a decrease in density as the temperature of a substance falls and that is the quantum zero point energy (ZPE). This effect can be described in several different ways. (See the recent review by Mehra and Rechenberg [19].) For example, using the Heisenberg uncertainty principle for a single particle of mass m in one dimension, one has xpx ∼ h¯ and thus K =

h¯ 2 px2 , ∼ 2m 2m(x)2

where K is a measure of the fluctuations in the value of the kinetic energy K. For a liquid this means that as the temperature is lowered (or the pressure increased) the spatial fluctuations x decrease with the result that K must increase [20]. This is in opposition to the classical contraction and one of the known consequences is that liquid helium does not freeze at one atmosphere of pressure however much its temperature is lowered [21]. The increase in the ZPE fluctuations K as the temperature falls, increases the entropy thereby disrupting the ordering process associated with the freezing process and preventing the formation of helium in the solid state at one atmosphere. In a recent communication [22] we showed that the existence of the density maximum in liquid He4 could be explained in terms of this ZPE effect acting in opposition to the classical expansion/contraction. Specifically, as the temperature falls, more and more helium atoms drop into the ground state, causing the fluctuations in kinetic energy K to increase and hence the system to expand. The density maximum then occurs at the crossover point for the two processes. We now investigate whether this same disruptive ZPE effect, acting on water at the maximum density temperature of 4 ◦ C, is large enough to counterbalance the contraction of the liquid at the same temperature thereby providing an explanation for the density anomaly just as in the case of liquid He4 . To this purpose, we calculate the rate of density change with temperature αZ = (dρ/dT )Z in water at ambient temperatures due to ZPE and compare it with the classical αC = (dρ/dT )C evaluated at the same temperature. If the model is correct, these two quantities should cancel at 4 ◦ C to give a maximum in the ρ–T curve, i.e., (dρ/dT )total = 0. To determine αZ , we look to recent detailed experimental and computational results that have been obtained by several groups of workers in determining the effects of the ZPE on the structure of water at room temperature [1,2]. The importance of ZPE fluctuations at these temperatures has been recognised for some time [23] but it is only in the last decade or so that experimental techniques have become sufficiently sophisticated to measure the full extent of the effect without ambiguity. In addition, the computational work required was impossible until modern terascale computing platforms became available. Tomberli et al. [1] have

used high intensity, high-energy synchrotron radiation (energies ∼ 100–120 keV) to measure the structural differences between H2 O and D2 O to high precision. These differences are interpreted in terms of the isotopic mass difference in the ZPE effect for the two liquids. Most importantly for our purposes, they have also found that this difference, due to isotopic H2 O/D2 O substitution, is equivalent to a change in thermal disorder, i.e., in entropy, caused by a temperature change of 5.5 ◦ C. [See also Ref. [24].] More recently, Badyal et al. [2] have extended this work, again using high intensity, high-energy scattering techniques, and have confirmed these findings. We now use these results to determine the quantitative effect of ZPE fluctuations on the density of water near room temperature. We note also the work of Chen et al. [25] as reported in a recent paper, where they used high-power computational methods to carry out a detailed simulation of the effects of ZPE on water structure using first principle molecular dynamics. In this way they found that the effect of introducing quantum effects is to enhance the water dipole moment by ∼ 4% with respect to its classical value. This causes a strengthening in the intermolecular bonding. The effect of substituting D2 O for H2 O is then to weaken the bonding, since the increase in the dipole moment of D2 O due to ZPE will be less than in the case of H2 O as the ZPE effect for D2 O is smaller. Thus the density of D2 O must decrease by a small amount. One can calculate this ZPE-related density drop as follows: consider the densities of H2 O and D2 O at 25 ◦ C, i.e., at a point away from either of their maximum density points. The two densities are known accurately [26] at this temperature: ρH2 O = 997.05 kg m−3 and ρD2 O = 1104.36 kg m−3 . If there were no ZPE isotopic effects acting, we would expect this difference in density to be due only to the different values measured for the two molecular masses mD2 O and mH2 O . The ratio of these two masses is listed as 1.1117 so that the expected density of D2 O, based on classical mass difference alone, should be 997.05 kg m−3 multiplied by this factor, i.e., ρD2 O = 1108.42 kg m−3 . There is a shortfall, therefore, of 1108.42 − 1104.36 = 4.06 kg m−3 , to be accounted for. It has been shown by Neuefeind et al. [24] and by others, that any differences in the molecular structure of H2 O and D2 O are largely quantum mechanical in origin so that this density shortfall can be identified with differences in quantum isotopic ZPE effects alone. We thus have the following: • Isotopic substitution, as already mentioned, is assumed to give the same ZPE disruptive effect and hence density change as would be produced by a temperature difference T = 5.5 ◦ C. (See, however, the discussion below.) • Using the results of Chen et al., we estimate this density change to be ρ = 4.06 kg m−3 . Combining these two results gives that the density change per degree, αZ = (dρ/dT )Z , due to ZPE effects at ambient temperatures, is αZ ≈ 4.06/5.5 = 0.74 kg m−3 K−1 . We wish now to compare this with its classical counterpart αC . One can estimate this latter quantity by extrapolating from tabulated values of the density of water at high temperatures [27]. It is important

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to take values at temperatures that are as high as possible where classical expansion/contraction should dominate the effects of the ZPE. If we examine the range 90–100 ◦ C, for example, we estimate αC ≈ 0.70 kg m−3 K−1 . To obtain this we take tabulated values [27] of water density in this temperature range and divide by the temperature difference to get an approximate value for the slope αC , e.g., averaging over the density range 965.32–958.37 kg m−3 between 90 ◦ C and 100 ◦ C gives αC ≈ 0.70 kg m−3 K−1 . Comparing the values of the two quantities we then have • αC ≈ 0.70 kg m−3 K−1 and αZ ≈ 0.74 kg m−3 K−1 at around 25 ◦ C. We thus get values of the two parameters that are very similar. There are potentially significant error bars, however, on both of these quantities, as we shall discuss next, so that the very close agreement is fortuitous. We have carried out the above calculations for a temperature of 25 ◦ C. There were several reasons for doing so. As we have already mentioned already, we wished to avoid the temperatures at which either H2 O or D2 O had a density maximum. In addition the best measurements for the densities of the two liquids have been carried out fairly recently at 25 ◦ C [26]. Many of the scattering experiments referred to have also been conducted at or near this temperature. We have shown, therefore, that the two opposing physical processes, one quantum and one classical, give rise to values of α = (dρ/dT ), with opposite signs, that approximately cancel at 25 ◦ C thereby providing a mechanism whereby a density maximum should be observed in that general region. The next step should be to carry out a similar estimation of αZ and αC at 4 ◦ C to see whether one gets a similar cancellation at that point. There is a major problem in doing this, however, based on recent work of Hart et al. [28]. They report the results of further precision photon diffraction experiments involving H2 O and D2 O, together with comparisons of their new experimental results with recent quantum molecular dynamics simulations. One of the main conclusions that they draw is that the assumption that structural differences observed in the scattering experiments for H2 O and D2 O are equivalent to the differences produced by a simple temperature offset in the state functions, is only reliable at temperatures above ∼ 310 K. Below this temperature the assumption becomes increasingly invalid. This temperature equivalence is key to our estimation of αZ , and since our calculations were based on measurements at 25 ◦ C, there will be some error in our calculated value. It is apparent, however, that any attempt to make similar estimates of αZ at 4 ◦ C might give seriously inaccurate results. There are, therefore, significant possible sources of error in our estimates of both αZ and αC . In the former case, the potential error is in the assumption of exact equivalence between the disruptive/entropy effects of isotope exchange and that of a temperature change of 5.5 ◦ C. In the latter the problem is in determining an accurate value of αC at 4 ◦ C by extrapolating from high-temperature data. In spite of these and other possible limitations, we may still conclude from our calculations that somewhere in the region of room temperature, the rate of density change with temperature due to ZPE fluctuations, should be

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close in value and opposite in sign to the classical value. Hence a maximum should occur in the density. Thus our proposed model that two independent opposing physical mechanisms, i.e., ZPE fluctuations and classical expansion/contraction, govern the thermodynamical behaviour of water and give rise to a maximum in the density, is reasonably well supported by calculations. The model, although developed specifically for water and liquid He4 , contains, in fact, no reference to the nature of the molecular bonding. This is different to the approach used in previous computational modelling methods for water [17,18] in which hydrogen bonding in that liquid is taken to be a key factor in constructing an intermolecular potential that will yield a maximum in the water density. By contrast our model should, in principle, be valid for all liquids. If that were the case, density maxima should then be commonly observed. That this does not happen (molten SiO2 is one of the few other examples [29, 30]), may, however, just be due to the maximum density points occurring below the freezing temperatures Tfp for most liquids, making them undetectable. After all, even for water, Tmd occurs just 4 ◦ C above its freezing point. It is well known experimentally that the relationships between the freezing points and the maximum density temperatures for various water/solute mixtures are such that the Tmd drops more rapidly than the freezing point with increasing solute concentration. Fig. 1 illustrates this in the case of water/ethanol mixtures. One sees that at a certain ethanol concentration, the Tmd vs. c curve intersects the Tfp vs. c line. At higher concentrations, Tmd drops below Tfp and becomes undetectable (except, perhaps, under conditions of extreme supercooling). Extrapolating this data then shows that if there is a density maximum in pure ethanol, it must occur at a temperature that is much lower than its freezing point. In other words, as the bonding network is weakened by the introduction of ethanol, the Tmd falls. One might thus attribute the limited number of liquids in which a density maximum is observed, simply to the fact that the maximum density temperatures for most liquids occur too far below their freezing points to be detectable. Water and molten SiO2 are then exceptions in having Tmd > Tfp probably due to the specific nature of their bonding networks. The other exception, liquid He4 , with its simple bond structure, may have its visible maximum because of the particular effects of the ZPE at these very low temperatures. In conclusion we have shown that the occurrence of a density maximum in water can be accounted for, just as in the case of liquid He4 , in terms of the opposing action of two independent physical processes, one of these being the classical expansion/contraction effect and the other quantum ZPE fluctuations. Our calculations show that around 25 ◦ C, estimates of the effects of the ZPE on the density roughly counterbalance those of the classical thermodynamical effects thereby indicating a density maximum somewhere in that temperature region. At lower temperatures, the ZPE will dominate, so that further cooling will cause increasing disruption of the molecular bonds and thereby a progressive decrease in the density. The theory also suggests that maxima might be detected in the densities of most other liquids as well but for the fact that they freeze be-

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Fig. 1. Comparison of the concentration dependence of the maximum density temperature Tmd with that of the freezing temperature Tfp for water–ethanol mixtures. Maximum density data (Q) are from Ref. [31]; freezing point data (2) are from our own measurements.

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