Increasing correlation length in bulk supercooled H₂O, D₂O, and NaCl solution determined from small angle x-ray scattering

Abstract

Using small angle x-ray scattering, we find that the correlation length of bulk liquid water shows a steep increase as temperature decreases at subzero temperatures (supercooling) and that it can, similar to the thermodynamic response functions, be fitted to a power law. This indicates that the anomalous properties of water are attributable to fluctuations between low- and high-density regions with rapidly growing average size upon supercooling. The substitution of H₂O with D₂O, as well as the addition of NaCl salt, leads to substantial changes of the power law behavior of the correlation length. Our results are consistent with the proposed existence of a liquid-liquid critical point in the deeply supercooled region but do not exclude a singularity-free model.

INTRODUCTION

The thermodynamic properties of liquid water, such as isothermal compressibility and heat capacity, show many anomalies that are strongly enhanced in the supercooled regime.¹ Their temperature dependence can be decomposed into that of a normal liquid background and an anomalous contribution where the latter shows an apparent power law divergence upon approaching a singular, but experimentally inaccessible, temperature of about 228 K at an ambient pressure.^{2, 3} There are many theories proposed to account for this apparent divergence,^{4, 5} out of which four are most commonly discussed: the stability limit (SL) conjecture,⁶ the liquid-liquid critical-point (LLCP) scenario,⁷ the singularity-free (SF) scenario,⁸ and the critical-point free scenario.⁹ It has been found that these four scenarios can be unified in the framework of a microscopic cell model with distinctions based on the relative strength between the cooperative and the directional components of hydrogen-bonding (H-bonding) interactions.⁵ Experimental data on H-bond energies are, in the microscopic cell model, only consistent with the LLCP scenario.⁵ Recent findings of a density minimum at low temperature in confined water¹⁰ are proposed to be an indication for the existence of a LLCP. For bulk supercooled water, however, there is a need for further experimental information related to the microscopic structure. To shed light on the microscopic origin of the apparent divergences in response functions, the most direct way is to determine the correlation length ξ in the Ornstein–Zernike (OZ) theory,¹¹ which is related to the asymptotic decay of the pair-correlation function.¹² However, previous small angle x-ray scattering (SAXS) or small angle neutron scattering measurements have not shown any widespread agreement due to limitations in the experimental data.^{13, 14, 15} The most recent SAXS study reported an almost constant ξ, varying from 3.6 Å at 273 K to 3.8 Å at 239 K.¹⁵ In a recent simulation study, a power law could, on the other hand, be fitted to the temperature dependence of a structural correlation length, increasing from 1.6 Å at 300 K to 3.6 Å at 210 K.¹⁶

In this study, we exploit improvements in synchrotron radiation sources to perform a precise and quantitative SAXS study of supercooled H₂O, D₂O, and 1m NaCl solution with significantly improved statistics and signal to empty cell background ratio compared to previous studies, with the range of momentum transfer Q extended to also include the wide-angle scattering (WAXS) domain. The new high quality data indeed show an apparent power law divergence of ξ with decreasing temperature in supercooled water, which is furthermore observed to be isotope dependent. NaCl solution exhibits a substantially reduced correlation length but with a steeper power law divergence. Since adding monovalent salt and applying pressure give analogous effects on H-bonding in water,¹⁷ our combined results provide an indication that a potential LLCP may exist in bulk water at a pressure higher than 1 atm. However, our data do not exclude that the location of the LLCP could be shifted to extremely low temperatures, close to absolute zero, which would allow for consistency also with the SF scenario.^{5, 8}

METHODS

The SAXS measurements were performed at beamline 4-2 at the Stanford Synchrotron Radiation Lightsource (SSRL) using a beam energy of 11 keV and an optical fiber coupled detector (Rayonix225HE). This detector has a larger active area (225×225 mm²) than that used in our earlier study of ambient water¹⁸ so that a single sample-detector distance of 0.7 m is large enough to cover an extended Q range of 0.04–1.0 Å⁻¹. A quartz capillary with an inner diameter of 1.5 mm was integrated into a samplecooling holder kept in a nitrogen gas atmosphere to eliminate water condensation. The total volume of supercooled water for measurement is around 5 μl. The scattering of the empty capillary at each temperature was measured separately and subtracted. Water scattering is approximately 40%–50% of the total signal at Q=0.04–0.3 Å⁻¹. A quantum mechanically calculated molecular scattering factor of an isolated H₂O molecule¹⁹ was employed to separate the scattering structure factor S(Q) from the total scattering intensity where the scattering momentum transfer Q is defined as Q=4π sin θ∕λ, with λ being the wavelength and θ being one-half of the scattering angle. The same molecular scattering factor was used for D₂O due to the fact that H₂O and D₂O are indistinguishable in terms of intramolecular electron density.²⁰ The total structure factor for NaCl solution was calculated by using the weighted sum of scattering factors of H₂O, Na⁺, and Cl⁻ ions.

RESULTS AND DISCUSSION

Figure 1 depicts the scattering structure factor, S(Q), for (a) H₂O at 252–284 K and (b) D₂O at 258–284 K; the lowest temperature is set by beam-induced crystallization. All scattering curves exhibit an enhancement approaching Q=0 after experiencing a minimum around 0.4–0.5 Å⁻¹. The enhancement at low Q increases rapidly as temperature decreases and the increasing trend is more dramatic than that observed recently for ambient water.¹⁸ Isosbestic points, i.e., where all S(Q) at different temperatures have the same value, are apparent at Q=0.39 Å⁻¹ for H₂O and at Q=0.37 Å⁻¹ for D₂O but only below 283 K. In spectroscopy, the presence of isosbestic points is generally regarded as a sign of a transition from one component to another, characterized by distinct spectral features, although the uniqueness of this interpretation has been questioned.²¹ Such isosbestic points have also been observed in scattering measurements on, e.g., the transition from high-density amorphous to low-density amorphous ice confined in high-pressure cryocooled protein crystals²² and on deeply supercooled water under high pressures.²³ The enhancement of the small angle signal as temperature decreases gives direct experimental evidence of density fluctuations of growing magnitude; also in Ref. 24, the strongest enhancement at low Q was interpreted as indicating the largest density heterogeneity.

Figure 1.

Experimental structure factor, S(Q), derived from small angle scattering of (a) H₂O, (b) D₂O, and (c) 1m NaCl solution. Temperatures, from top to bottom at the low Q side, (a) for H₂O are 252, 254, 258, 263, 268, 273, 278, and 284 K, (b) for D₂O are 258, 263, 268, 273, 278, and 284 K, and (c) for 1m NaCl solution are 253, 258, 263, 268, 273, and 278 K. The temperature range is labeled in each case. Data were collected covering an extended Q range between 0.04 and 1.0 Å⁻¹, which represents a significant improvement compared to earlier results with Q range of 0.05–0.28 Å⁻¹ in Ref. 15 and 0.15–0.85 Å⁻¹ in Ref. 13, respectively.

To understand the microscopic origin of the observed density fluctuations, the experimental S(Q) was analyzed by decomposing it into two contributions using two different schemes. In method I [Fig. 2a] the scattering background due to thermal density fluctuations, present in any liquid and predominant at high temperatures, is first described by a term, S^ref, monotonically increasing with Q in the relevant range. In the framework of the Percus–Yevick, the approximation for a hard-sphere fluid²⁵S^ref is given by

$$1∕S^{\text{ref}}\propto 1–12\eta \frac{\left[\eta \left(3-{\eta }^2\right)-2\right]}{{\left(1-\eta \right)}^4}\frac{j_1\left(Q\sigma \right)}{Q\sigma },$$ (1)

where j₁ is the first-order spherical Bessel function, σ is the hard-sphere diameter, and η is the volume fraction of water that can be related to the number density n by η=πnσ³∕6. The excess scattering at small angles, S^OZ (increasing with decreasing temperature) is described in the OZ theory¹¹ by a Lorentzian,

$$S^{\text{OZ}}\propto 1∕\left({\xi }^{-2}+Q^2\right),$$ (2)

where ξ is the OZ correlation length related in real space to the asymptotic decay of the pair-correlation function of the system.¹² The total S(Q) is thus fitted by a linear combination of S^ref and S^OZ, with relative fractions depending on water temperature; the lower the temperature, the larger the obtained S^OZ fraction. Figure 3a shows the derived ξ for H₂O and D₂O, which both appear to increase following a power law as water is cooled below 283 K; the increase is even more prominent in deuterated water. We estimate that the relative uncertainty in the derived values of ξ is almost negligible (<±0.02 Å). The increase of ξ for H₂O is about 30% from 283 to 252 K compared to 7% from 340 to 283 K.¹⁸ We also observe that ξ for D₂O varies about 36% from 283 to 258 K compared to 10% from 330 to 283 K. We compare the variation of ξ with the corresponding 30% increase in the isothermal compressibility of H₂O between 283 and 252 K,² demonstrating a very similar magnitude in the anomalous enhancements of the two properties.

Figure 2.

Comparison between (a) method I and (b) method II (see text) used to decompose the total scattering structure factor into the normal S^ref and the anomalous S^OZ contributions, as described in the text. The arrows indicate the trend with temperature decrease.

Figure 3.

Dependence of correlation length ξ on the temperature T. (a) Comparison between H₂O (dots), D₂O (squares), and 1m NaCl solution in H₂O (diamonds) with the best-fit power laws (solid lines) for each case. The deviations from the power law at highest temperature of 324 K are clear for both H₂O and D₂O. Inset: dependence of the logarithm of ξ on the logarithm of the T difference from 228 K for H₂O and 1m NaCl solution. The steeper slope of the NaCl data indicates a larger critical exponent ν compared to that for H₂O. (b) Comparison of H₂O ξ derived from the method based on the Percus–Yevick approximation in Eq. 1 (dots) and from the method used in Ref. 18 with an assumption of the cutoff Q_max≈0.9 Å⁻¹ (diamonds) and 0.8 Å⁻¹ (squares). A very similar increase of ξ as T decreases in the supercooled region is obtained in all three cases in spite of the small differences in slope, whereas the absolute magnitude of ξ depends on the method.

The decomposition of S(Q) is not unique, and in Ref. 18, another method (method II), based on decomposing the isothermal compressibility into two contributions,²⁶ was used to separate the scattering background [see Fig. 2b]. We consider method I to be closer to reality based on the fact that the Percus–Yevick approximation describes the temperature dependence of S^ref(Q), in good agreement with that of S(Q) of simple liquids, e.g., CCl₄ (Ref. ¹⁸) and pure ethanol. The best-fit hard-sphere diameter σ for H₂O in Eq. 1 was found to continuously increase from 2.52 to 2.61 Å with corresponding number density n decreasing from 0.0334 to 0.0325 molecule∕ų as temperature decreases from room temperature to 254 K. The close agreement of the fit σ and n with established values and their temperature dependence further adds confidence in the validity of method I. The derived ξ from these two methods, testing also two choices of cutoff, Q_max≈0.9 and 0.8 Å⁻¹ (in method II, no anomalous contribution is assumed beyond Q_max) are compared in Fig. 3b for H₂O scattering. The derived increase of ξ with supercooling is independent of the method used and ξ can, below 283 K, be fitted within measurement uncertainty to a power law as

$$\xi ={\xi }_0{\epsilon }^{-\nu },$$ (3)

where ε=T∕T_s−1 is the reduced temperature with T_s being a singular temperature for the system, ξ₀ is a constant, and ν is the critical exponent (ν>0). This form of power law behavior has been used to phenomenologically describe the thermodynamic anomalies of supercooled water.²⁶ It should, however, be noted that the reduced temperatures extend over less than a decade in both the current study and in previous measurements on thermodynamic anomalies^{2, 3} as limited by spontaneous homogeneous nucleation in liquid water. In addition, isobaric cooling is measured here, different from the isochoric cooling generally used to define the critical exponents; at an ambient pressure, this furthermore presumably implies an approach toward the Widom line T_W,²⁷ where ξ shows a maximum but does not diverge. We therefore use the notion of an apparent power law behavior and make the assignment T_S=T_W since the present measurements are far from a critical point in temperature and presumably also in pressure, and the exponent ν is thus not expected to coincide with the true critical exponent, should it exist.

We note that the limited temperature range available is inadequate to simultaneously and reliably determine both T_W and ν so that ν is fit separately by fixing T_W to the best estimate from earlier thermodynamic response function measurements with larger temperature range [T_s=228 K for H₂O and 233 K for D₂O (Ref. ²⁶)]. The fit power laws are summarized in Table 1. The best-fit ν (0.32 for H₂O and 0.44 for D₂O) are rather small, but visibly enhanced small angle scattering can still be observed even at ambient conditions.¹⁸ We observe that ξ for D₂O exceeds that for H₂O at supercooled temperatures, but above 280 K, they become comparable. The difference in ξ between H₂O and D₂O at subzero temperatures cannot be described exclusively by a thermal offset as also observed in earlier WAXS measurements,²⁸ quantum simulations,²⁹ and from x-ray spectroscopy.³⁰ The density fluctuations in D₂O, indicated by the temperature dependence of ξ, furthermore grow more quickly in size as the temperature decreases compared to H₂O.

Table 1.

Summary of the best-fit parameters ξ₀ and ν in Eq. 3. T_W was fixed by using the best-fit results from an earlier thermodynamics measurement (Ref. ²⁶).

	ξ0 (Å)	TW (K)	ν
H2O	1.287	228	0.32
D2O	1.011	233	0.44
1m NaCl solution	0.552	228	0.52

Our results differ from the earlier study by Xie et al.,¹⁵ where a nearly constant ξ with temperature was found. This inconsistency is not caused by a difference in S^ref subtraction, but is due to the data itself; a direct comparison of the total S(Q) at the same temperatures shows a factor of 2 larger enhancements in Ref. 15 with a much smaller variation with temperature (see supplementary material³¹). We speculate that there may be an additional Q-dependent background in the data of Xie et al. caused by their relatively small water-to-capillary scattering ratio.¹⁵ Subtracting a Q-independent S^ref from our data, as done in Ref. 15, instead gives an even steeper temperature dependent increase in ξ. Moreover, it is interesting to note that the derived ν=0.32 for H₂O is close to one-third of the exponent γ=0.89–0.91, which has been derived from a power law fit to the isothermal compressibility after subtraction of the normal contribution and using similar T_W values as in the present study.^{26, 32} The general relationship of ν=1∕2γ as expected at a critical point¹² is thus not exactly fulfilled in the current temperature range and pressures far from the predicted LLCP.⁷

The effect of monovalent salts on the water structure is studied by measuring 1m NaCl solution (1.75 mol % in H₂O), with results shown in Fig. 1c. An isosbestic point is also observed at Q=0.52 Å⁻¹, compared to Q=0.39 Å⁻¹ for H₂O. The small angle enhancements of NaCl solution display a more well-defined plateau at low Q compared to pure water at the same temperature, in a similar manner as when applying pressure to pure water in a recent pressure-dependent SAXS measurement.³³ Due to the small mole fraction of ions at 1m concentration, the temperature dependence in S(Q) of the NaCl solution is assumed to be mainly attributable to the structural change in “free water,” i.e., water molecules not directly bonded to ions. Using the method described by Eqs. 1, 2, ξ for NaCl solution was calculated and shown in Fig. 3a. We find that the NaCl solution also exhibits an apparent power law behavior in ξ upon cooling, but with a rather smaller magnitude compared to pure H₂O; this implies that the characteristic length scale of density fluctuations is reduced by the presence of ions.

There are also concentration fluctuations in the salt solution but these are expected to be of more importance at higher temperatures and should have only a minor effect on the power law dependence. For example, the concentration fluctuation in 1m NaCl solution at 273 K was estimated to be only 10% of the total scattering intensity at Q=0 Å⁻¹ following Bhatia and Thornton’s theory for binary-component systems.³⁴ Upon close inspection, the NaCl solution exhibits a steeper increase of ξ compared to pure H₂O at low temperatures. Using the same fixed T_W=228 K as for water, we obtained ν=0.52 for the NaCl solution, which is larger than ν=0.32 for pure H₂O; a lower T_W for the NaCl solution would lead to even larger ν. The difference in ν can be readily visualized from comparison of the logarithmic plot of ξ as function of the logarithm of the temperature relative to T_W=228 K in Fig. 3a (inset), where the slope represents ν.

Our observation of increased density fluctuations in water and monovalent salt solutions upon supercooling is consistent with the thermodynamic anomalies of water as being related to rapidly increasing fluctuations in density upon cooling in the subzero temperature region.^{2, 26} The apparent power law divergence of ξ is indicative of a rapidly increasing mean size of low-density liquid (LDL) regions upon supercooling, which have been spectroscopically assigned by Huang et al.¹⁸ as characterized by an open tetrahedral H-bonded network; strict, well-defined boundaries between low- and high-density regions are, however, not expected.³⁵ The larger and more rapidly increasing ξ for D₂O compared to H₂O indicates that quantum effects influence the enhanced density fluctuations in supercooled water more than the temperature offset predicts. Furthermore, the smaller ξ with NaCl compared to pure water can be understood in terms of smaller regions of LDL analogous to pressure effects.¹⁷ The reduced tendency to form LDL regions induced by ions also explains the fact that H-bond-breaking impurities (such as salts, alcohols, etc.) push the anomalous region in supercooled water systematically to lower temperatures.²⁶ This is supported by a recent molecular dynamics simulation that finds a LLCP for a NaCl solution at lower pressure compared to pure water.³⁶

CONCLUSION

To conclude, we have shown the apparent power law divergence of ξ of bulk liquid water and monovalent salt solutions as temperature decreases in the supercooled regime. It indicates that water anomalies are attributable to the increased characteristic length scale of density fluctuations. It is consistent with the interesting possibility that earlier reported structural heterogeneities in ambient water¹⁸ are remnants of the strongly anomalous behavior in the supercooled regime. We emphasize that the apparent power law behavior of ξ, as observed in the currently accessible temperature range, does not discriminate between a finite maximum of ξ connected to the Widom line²⁷ in the framework of the LLCP and SF scenarios, and a divergence of ξ at the limit of liquid stability in the SL scenario. We note that, however, for the SF scenario, approximated as the earlier percolation model,³⁷ Xie et al. estimated the intensity enhancement to be too small to account for the experimental data.¹⁵ Furthermore, if the analogy between adding monovalent salts to water and increasing the pressure holds, then the larger ν observed for NaCl solutions compared to that for pure water would favor the LLCP and SF scenarios since ν is expected to increase as a LLCP at a pressure higher than 1 atm is approached.