Understanding the Structure Factor and Isothermal Compressibility

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Understanding the Structure Factor and Isothermal Compressibility of Ambient Water in Terms of Local Structural Environments S. D. Overduin and G. N. Patey*

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Department of Chemistry, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1 ABSTRACT: We determine contributions to the structure factor of ambient water (modeled using the TIP4P/2005 potential) from molecules with different local structural environments, distinguished based on a tetrahedral order parameter. The small wavenumber behavior of the partial structure factors indicates the presence of effective attractive interactions between molecules with similar local environments and effective repulsive interactions between molecules with different local environments. These effective interactions lead to significant concentration fluctuations under ambient conditions. The concentration fluctuations correlate with density fluctuations, and provide an explanation for the recently reported (Huang et al. Proc. Natl. Acad. Sci. U.S.A. 2009, 36, 15214) increase in the structure factor at small wavenumber, as well as for the wellknown anomalous temperature dependence of the isothermal compressibility. molecules. We find that contributions to the structure factor from like correlations increase at low wavenumber, indicating effective attractive interactions between molecules with similar local structural environments. In contrast, the unlike contribution shows a corresponding decrease, consistent with effective repulsive interactions between molecules with different structural environments. The presence of effective attractive and repulsive interactions leads to concentration fluctuations, very similar to those observed in binary liquid mixtures of “antagonistic” species. We show that these concentration fluctuations couple with density fluctuations, and can account for both the increase in the structure factor at low wavenumber and the anomalous behavior of the isothermal compressibility.

I. INTRODUCTION The often anomalous behavior of water is believed to be due to competition between the formation of energetically favored tetrahedral structures and entropically favored higher density arrangements.1−3 Several theories have been proposed to explain how competition between these structures leads to water’s anomalous properties.1,4−6 One idea, that has received considerable support,7−12 suggests that the different structural possibilities open to water can lead to two metastable, coexisting liquid states in the supercooled region.4 From this perspective, water’s anomalous properties can be interpreted as echoes of underlying critical behavior.1,4 However, a complete singularity-free description of the properties of water is also possible.5 Independent of the nature of water in the supercooled region, a better microscopic understanding of water’s anomalous properties under ambient conditions is needed. Recently, Huang et al.13 reported an enhancement in the structure factor of water at small wavenumber, under ambient conditions. They found that the enhancement grows with decreasing temperature, and suggest that it signals the presence of anomalous density fluctuations.11,13−15 However, apart from simulation results suggesting tetrahedral clusters of some minimum size have lower density,16 there is little direct evidence for such density fluctuations in ambient water.17−20 In this paper, we investigate the behavior of the structure factor obtained from simulations of the TIP4P/2005 water model at 1 bar, over a range of temperatures in the ambient liquid regime. Our main objective is to establish explicitly how molecules with more- and less-tetrahedral local environments contribute to the structure factor and isothermal compressibility. To accomplish this, we distinguish two types of water molecules based on the local order of their immediate surroundings, and decompose the total structure factor into contributions from correlations between like and unlike © 2012 American Chemical Society

II. SIMULATION AND ANALYSIS Molecular dynamics simulations of the TIP4P/2005 water model 21 were performed using the GROMACS 4.5.3 simulation package.22 The TIP4P/2005 water model, described elsewhere,21 nearly quantitatively reproduces many of water’s properties, including its anomalies.21,23 In particular, both quantities discussed in this work, the enhancement of the structure factor at low wavenumber, and the anomaly in the isothermal compressibility are well captured by this model. All simulations were performed with N = 32 000 particles at constant temperature using the Nosé−Hoover thermostat.24,25 Short-range interactions were truncated at 1.4 nm, and the particle mesh Ewald method26 was used to account for longrange electrostatic terms. In order to determine the equilibrium density at 1 bar, simulations were first performed in the NPT ensemble for at least 2 ns, employing the Berendsen barostat.27 Received: July 31, 2012 Revised: September 7, 2012 Published: September 10, 2012 12014

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S(k). We emphasize that the values of S(0) calculated using eq 5 are in quantitative agreement with those obtained through other methods. For example, at P = 1 bar, we obtain S(0) = 0.061, 0.067, and 0.073 for 278, 320, and 340 K, respectively. These are in excellent agreement with the corresponding values (0.062, 0.067, and 0.073, respectively) obtained by Sedlmeier et al.19 from differentiation of the density with respect to pressure. The structure factor can also be determined directly from the simulation trajectory, using the expression

Simulations at the equilibrium density were then equilibrated in the NVT ensemble for an additional 2 ns, followed by production runs of at least 6 ns. All results reported here are averages over ∼6000 configurations collected during the production runs. Following Errington and Debenedetti,28 we quantify the local order experienced by each molecule using the tetrahedral order parameter q=1−

3 8

3

4

2 ⎛ 1⎞ ⎜cos(ϕ ) + ⎟ ij ⎝ 3⎠ j=i+1

∑ ∑ i=1

(1)

S(k) =

where ϕij is the angle between lines connecting the oxygen atom of a given water molecule to the oxygen atoms of its ith and jth neighbors. Note that only the four nearest neighbors are included. For our analysis, we classify molecules as either high-q (H) or low-q (L), which depends of course on how we choose to divide the sample. We are interested in how high-q and low-q molecules contribute to the structure factor of water. How we divide the molecules into these categories is clearly somewhat arbitrary, the key consideration being to make a division that reveals the influence of the different local structures. After considering a number of possibilities, we found it sufficient to distinguish between more- and less-ordered local structures, simply using the median value of the tetrahedral order parameter q (eq 1): for a given configuration, molecules with a q value greater than the median q are designated H, and the remainder, or molecules with a q value less than the median q value, are designated L. Of course, local order is transient, and in contrast to a true mixture, here the label of any given molecule can and will change with time. The structure factor can be measured by scattering experiments, and for an isotropic fluid is given by S(k) = 1 + 4πρ

∫0

∞

drr

sin(kr ) h(r ) k

S(0) 4 πρR3u(kR ) N 3

(6)

S(k) = x HSHH(k) + x LSLL(k) + 2 x Hx L SHL(k)

(7)

where Sαβ(k) = δαβ + 4πρ xαxβ

∫0

∞

drr

sin(kr ) hαβ (r ) k

(8)

and xα is the mole fraction of species α. Since we distinguish H and L molecules based on the median q value, xH = xL = 0.5. The partial structure factors can be corrected for finite-size effects, using a procedure analogous to that outlined above for the total structure factor. Sαβ(k) can also be determined directly by summing only the relevant terms in eq 6. In analogy with binary mixtures, additional insight can be gained by considering density−density, SNN(k), concentration− concentration, SCC(k), and density−concentration, SNC(k), structure factors as defined by Bhatia and Thornton.31 In the present case

(2)

SCC(k) = x Hx L[x LSHH(k) + x HSLL(k) − 2 x Hx L SHL(k)] (9)

⎡ ⎤ x − xH S NC(k) = x Hx L⎢SHH(k) − SLL(k) + L SHL(k)⎥ ⎢⎣ ⎥⎦ x Hx L

(3)

(10)

where S N(k , R ) = 1 + 4πρ

∫0

R

drr

sin(kr ) h(r ) k

and SNN(k) = S(k). Physically, SNN(k) and SCC(k) are related to density and concentration fluctuations, respectively, and SNC(k) is a measure of the coupling between density and concentration fluctuations.31 For an ideal solution, SCC(k) = xHxL, and SNC(k) = 0, which are also the limiting values approached as k → ∞. For any binary mixture, SCC(0) will diverge at a demixing transition. This is also true of SNC(0) and SNN(0), provided that the partial molar volumes of the two components are not identical. However, for demixing, SCC(0) is the primary indicator, and usually gives an earlier indication of an underlying demixing transition than does SNN(0).31−35 Bhatia and Thornton31 also define a dilation function Δ(k) = −SNC(k)/SCC(k) and a dilation factor Δ(0) = −SNC(0)/ SCC(0) = ρ(vH − vL), where ρ is the total density, and for a true binary mixture vH and vL would denote the partial molar volumes of the two components. Both Δ(k) and Δ(0) appear in the analysis given below.

(4)

is obtained from an N particle simulation and u(x) = (3/ x3)(sin x − x cos x). If R is sufficiently large, S(k, R) ≃ S(k), and for k = 0, eq 3 gives S(0) ≃

+

N ⎤2 1 ⎡⎢ ∑ cos(k·ri)⎥ ⎥⎦ N ⎢⎣ i = 1

As discussed by Sedlmeier et al.,19 the direct method gives better results at small k than those obtained from the uncorrected Fourier transform of h(r). We find good agreement between the direct method and eq 3. The total structure factor can be written as a sum of partial structure factors30

where h(r) = g(r) − 1, g(r) is the radial distribution function, and ρ is the number density. In order to obtain accurate values at low wavenumbers, the structure factor must be corrected for finite size effects. Salacuse et al.29 have shown that, for sufficiently large N and R, the correction is given by S(k , R ) ≃ S N(k , R ) +

N ⎤2 1 ⎡⎢ ∑ sin(k·ri)⎥ ⎥⎦ N ⎢⎣ i = 1

S N(0, R ) 1−

1 4 πρR3 N3

(5)

Equations 3 and 5 are valid only if R is larger than any correlation length present in the system, but this condition is easily met by the 32 000 particle simulations employed in the present analysis. Additionally, to reduce the influence of any statistical noise, S(0) is determined by averaging the right-hand side of eq 5 over a range of R, between 2.5 and 3.5 nm. Similarly, eq 3 is averaged over the same range of R to obtain 12015

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III. RESULTS AND DISCUSSION As discussed above, we use the median q value of each configuration to distinguish high- and low-q molecules. The probability distribution of the tetrahedral order parameter is shown for different temperatures in Figure 1. Average median q

is possible that averaging masks longer ranged correlations from a smaller subset of molecules. To test this possibility, we calculated partial correlation functions for smaller subsets, defined by limiting q values. As the structural distinction becomes more extreme, we find enhanced differences between the high- and low-q subsets, but longer ranged correlations never appear. These observations are consistent with the absence of density heterogeneities reported by English and Tse17 for the SPC/E water model, and with the view20,38 that ambient water is a homogeneous liquid on length scales ≳1 nm. Similar differences in partial distribution functions obtained for the TIP4P/2005 model were recently reported by Wikfeldt et al.39 These authors employed a different order parameter, and divided the water sample based on an inherent structure analysis, rather than on the order parameter distribution at the actual temperature considered. Despite these differences, their results and ours are in qualitative agreement. We next consider the partial structure factors, shown in Figure 3, and on an expanded scale at small k in Figure 4.

Figure 1. Normalized probability distribution function of the order parameter q (eq 1), for different temperatures. The red and black vertical lines indicate the average median values of q for 260 and 360 K, respectively.

values for T = 260 and 360 K are indicated by vertical lines; median q values for all other temperatures lie between these two values. As noted by others,14,19,28,36 the distribution is bimodal at ambient temperatures; the high-q peak becomes more prominent at low temperature, while the low-q peak gradually disappears as the temperature is decreased. As recently emphasized by Perera,37 pair correlations in water are relatively short ranged. Following Perera, we plot the oxygen−oxygen functions r2hαβ(r) in Figure 2. Structural features can be discerned in all partial pair correlation functions up to ∼1.8 nm but are weak beyond ∼1 nm. Since our method of labeling molecules is based simply on the median q value, it

Figure 3. Partial [SHH(k), SLL(k), SHL(k)] and total [S(k)] oxygen− oxygen structure factors calculated using the corrected Fourier transform method (eq 3), for different temperatures as labeled.

Interestingly, both HH and LL contributions to the structure factor, in other words contributions from like-q molecules, display a pronounced minimum at ∼11 nm−1 (Figure 3). We note that Wikfeldt et al.39 observed a similar minimum in the partial structure factors of molecules with similar inherent local structure, as measured by their order parameter. In contrast, no such minimum is observed in the HL contribution. Thus, the weak minimum observed in the total structure factor at ∼4−5 nm−1 (Figure 4) can be attributed to a balance between positive contributions from correlations between like species (structures) and the negative contribution from correlations between unlike species. Note the excellent agreement between results from the direct method and eq 3 shown in Figure 4.

Figure 2. Partial oxygen−oxygen pair correlation functions, r2hLL(r) (red), r2hHH(r) (blue), and r2hHL(r) (gold), together with the total r2h(r) (black). Results for temperatures of 270 K (top), 300 K (middle), and 360 K (bottom) are shown. 12016

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Figure 5. The ideal (light blue squares), excess HH (black triangles), LL (dark blue circles), and HL (pink squares) contributions to the total isothermal compressibility (red crosses). Note that the total is also shown (inset) on an expanded scale to highlight the anomaly. Lines are drawn to guide the eye.

Figure 4. Partial and total oxygen−oxygen structure factors at small k, for temperatures 270 K (red), 278 K (black), 300 K (blue), 320 K (magenta), 340 K (light blue), and 360 K (gold). The symbols represent values obtained by the direct method (eq 6), and the solid lines are from eq 3.

the entire temperature range, the HH part remains rather flat, decreasing slightly at higher temperatures, and increasing slightly at the lower end. The behavior of the like contributions reflects the effective attractive interactions noted above. The temperature dependence of these terms, combined with the increasing ideal contribution, results in the anomalous increase in κT at lower temperatures. A different, perhaps more penetrating, perspective is provided by SCC(k) and SNC(k), which together with SNN(k) = S(k) are plotted in Figure 6. As noted above, in a mixture where interactions between like species are more favorable than interactions between unlike species, one expects SCC(0) to give an earlier indication of any underlying demixing behavior than SNN(0).32,33,35 Our results clearly show a greater increase in SCC(k) than in S(k) as k → 0. Furthermore, the increase in SCC(k) at low wavenumber is apparent for all temperatures, indicating that concentration fluctuations are significant, even at the highest temperature (360 K) considered. Correlations between density fluctuations and concentration fluctuations are also evident, with SNC(0) growing in magnitude with decreasing temperature. Note also that Δ(0) = −SNC(0)/SCC(0) is positive for all temperatures considered. If we risk possibly overstretching the binary mixture analogy, and view Δ(0) = ρ(vH − vL) as measuring the difference in “partial molar volumes” of high-q and low-q species, then positive values of Δ(0) are consistent with the intuitive view that local structures with high tetrahedral order should be less dense than those that are less tetrahedrally ordered. Again following Bhatia and Thornton,31 it is instructive to formally divide the structure factor into two terms

Clearly, the small k behavior of the partial structure factors is sensitive to rather subtle variations in the pair correlation functions. For example, the significant increases in SHH(k) and SLL(k) at low wavenumber come about despite relatively shortrange density correlations. Moreover, although h(r) and hHL(r) appear quite similar in Figure 2, S(k) and SHL(k) are very different at low wavenumber. In particular, S(k) increases as k → 0, while SHL(k) decreases monotonically. It is interesting to compare our partial structure factors with results obtained for true binary mixtures.40,41 If in a binary mixture unlike interactions are less favorable than like interactions, the partial structure factors can closely resemble those shown in Figure 3, even well away from a demixing critical point. Thus, our results can be interpreted in terms of effective attractive interactions between like-q molecules, or more precisely between similar local structures, and effective repulsive interactions between unlike-q molecules or unlike local structures. A subtle balance of these attractive and repulsive interactions gives rise to the low wavenumber minimum in the total structure factor. The isothermal compressibility is related to the structure factor, κT = S(0)/(ρkBT). In “normal” liquids, κT decreases monotonically with decreasing temperature, whereas in water at 1 bar κT passes through a minimum at ∼320 K, and then increases for a range of lower temperatures. Because the ideal part, κT° = 1/(ρkBT), is included in the SHH(0) and SLL(0) contributions to κT, the influence of the HH, LL, and HL correlations can be seen more clearly if we consider their contributions to the excess compressibility, κex ° . These T = κT − κT excess contributions together with κT° and κT are plotted in Figure 5. We see that the temperature dependence of the like and unlike contributions is strikingly different. The HL contribution decreases monotonically with decreasing temperature, whereas, in contrast, the LL contribution increases over

S(k) = θ(k) + [Δ(k)]2 SCC(k)

(11)

where Δ(k) = −SNC(k)/SCC(k) is the dilation function noted above and θ(k) = xHxL[SHH(k)SLL(k) − SHL2(k)]/SCC(k). The 12017

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Figure 6. Concentration−concentration [SCC(k)], density−density [S(k)], and density−concentration [SNC(k)] structure factors for different temperatures as denoted in the legend. Note the distinct peak in SCC(k) at k = 0.

Figure 7. Top panel: The total oxygen−oxygen structure factor S(k) and the function θ(k) obtained at 300 and 360 K. Bottom panel: The small k behavior of S(k) (solid thin lines) and θ(k) (dashed thick lines) for three temperatures, as labeled.

second term in eq 11 represents an explicit contribution from concentration fluctuations and the coupling between fluctuations in density and concentration. If fluctuations in density and concentration are not coupled (SNC(k) = 0), then Δ(k) = 0 and S(k) = θ(k). Thus, θ(k) gives an indication of what S(k) might look like in the absence of coupled fluctuations in concentration and density. As shown in the top panel of Figure 7, S(k) and θ(k) are very similar. However, small differences are present, as highlighted in the bottom panel of Figure 7. In particular, there is no increase in θ(k) as k → 0 for T ≲ 320 K, while there is in S(k). In other words, under ambient conditions, we can attribute the observed increase in the structure factor at small wavenumber (i.e., the anomalous scattering, discussed by Huang et al.13) to the second term in eq 11. Note that the effect is very strongly dependent on the magnitude of [Δ(k)]2. For example, at 360 K, [Δ(0)]2 ≃ 0.0007, which is nearly 2 orders of magnitude smaller than its value at 270 K (∼0.04). Consequently, S(k) and θ(k) are virtually indistinguishable at 360 K (Figure 7). Huang and co-workers11,13,14 have attempted to separate the structure factor into normal and anomalous contributions, and then used Ornstein−Zernike (OZ) theory to obtain a correlation length for the anomalous component. If we take θ(k) to be the normal contribution to the structure factor, we can identify SA(k) = S(k) − θ(k) as the anomalous part. Then, assuming the OZ form SA(k) = C(ξ−2 + k2)−1, where C is a temperature dependent constant, we can fit our results to estimate a correlation length ξ. The fits are not sufficiently accurate to establish an unambiguous temperature dependence over the temperature range we consider, but we obtain correlation lengths on the order of 0.3 nm, which are similar to estimates that others have reported.11,13,14,38 Of course, our simulations are not in the near vicinity of a known critical point, so the meaning of correlation lengths estimated in accordance with OZ theory is not entirely clear. Nevertheless, the relatively

small values we obtain are consistent with the short-range nature of the pair correlation functions, noted above. Considering the k = 0 limit of eq 11, we can also divide the isothermal compressibility into two contributions κT = θ(0)/(ρkBT ) + [Δ(0)]2 SCC(0)/(ρkBT )

(12)

where the second term again depends only on concentration fluctuations and the dilation factor Δ(0). Both κT and θ(0)/ (ρkBT) are shown in Figure 8. We note that θ(0)/(ρkBT)

Figure 8. Comparison of κT and θ(0)/(ρkBT) as functions of temperature. 12018

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Research Council of Canada is gratefully acknowledged. This research has been enabled by the use of WestGrid and Compute/Calcul Canada computing resources, which are funded in part by the Canada Foundation for Innovation, Alberta Innovation and Science, BC Advanced Education, and the participating research institutions. WestGrid and Compute/ Calcul Canada equipment is provided by IBM, HewlettPackard, and SGI.

decreases monotonically with temperature, over the temperature range considered. Thus, in the present analysis, the anomalous increase in κT below ∼320 K comes entirely from correlated fluctuations in concentration and density.

IV. SUMMARY AND CONCLUSIONS The physical picture that emerges from our analysis is that the different structural arrangements present in water over relatively short length scales lead to effective attractive interactions between molecules with similar local environments and effective repulsive interactions between molecules with different local environments. These effective interactions result in significant concentration fluctuations over the entire temperature range (260−360 K) considered. Under ambient conditions, concentration and density fluctuations are sufficiently coupled to give rise to both an increasing structure factor at small wavenumber and the anomaly in the temperature dependence of the isothermal compressibility. One might speculate that evidence for attraction between like, and repulsion between unlike, local structures, together with accompanying concentration fluctuations, suggests that a demixing-like transition is possible. In fact, there is considerable, albeit indirect, evidence that the TIP4P/2005 model has a liquid−liquid transition, with the critical point estimated10 to be at 1350 bar and 193 K, which we note is quite far removed from the ambient conditions considered here. Also, the recent work of Limmer and Chandler42 involving other water-like models casts some doubt on any conclusion that a liquid− liquid critical point exists in supercooled water based solely on indirect simulation evidence. The results of Limmer and Chandler42 demonstrate that, without explicit consideration of free energy basins, it is very difficult to be sure if two metastable liquid states actually exist in the supercooled region. However, we do note that very convincing evidence of liquid−liquid coexistence in the supercooled region has been obtained for the ST2 water model.43−45 This being said, our partial structure factors do resemble what one would expect for a binary mixture that is likely to have a demixing transition somewhere in its phase diagram. However, while this is suggestive, one must keep in mind that water is not a true mixture where the components have fixed identities, and the composition and temperature are independent variables. Thus, it is entirely possible that, while water resembles a system one might expect to have a “demixing” transition, no transition actually occurs because, as the temperature is lowered toward a potential critical value, the q distribution also changes such that a critical “composition” is never achieved. Such a scenario would provide a simple explanation as to why water exhibits signals of an underlying liquid−liquid critical point, yet the existence of such a critical point has proved difficult to establish.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS G.N.P. would like to thank Dr. A. Perera and Dr. Y. Koga for many interesting discussions of water and its properties. The financial support of the Natural Science and Engineering 12019

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dx.doi.org/10.1021/jp3075749 | J. Phys. Chem. B 2012, 116, 12014−12020