Proton Momentum Distribution and Diffusion Coefficient in Water: Two

Letter pubs.acs.org/JPCL

Proton Momentum Distribution and Diffusion Coefficient in Water: Two Sides of the Same Coin? F. Bruni,† A. Giuliani,† J. Mayers,‡ and M. A. Ricci*,† †

Dipartimento di Fisica “E. Amaldi”, Università degli Studi “Roma Tre”, Via della Vasca Navale 84, 00146 Roma, Italy STFC, Rutherford Appleton Laboratory, ISIS Facility, Didcot, Oxfordshire OX11 0QX, United Kingdom

‡

S Supporting Information *

ABSTRACT: Water, the prototype of a liquid to ordinary people, is the most anomalous liquid to physicists, showing regions of the temperature−density (T,ρ) plane where its microscopic structure, diffusion coefficient, and density have anomalous behaviors. Structural anomalies occur over a broad bell-shaped T,ρ region. This region contains, as a matryoshka, two smaller regions, one delimiting dynamical and the other delimiting thermodynamic anomalies. Water anomalous behavior in each of these regions manifests itself as a decrease of order or an increase of the diffusion coefficient upon increasing pressure and as a decrease of density upon cooling. Here, we show that the radial momentum distribution of water protons and their mean kinetic energy have a peculiar, theoretically unpredicted anomaly in the region of dynamical anomalies. This anomaly can be rationalized as due to two distinct “families” of water protons, experiencing quite distinct local environments, leading to an enhancement of the momentum fluctuations along with an increase of kinetic energy. SECTION: Liquids; Chemical and Dynamical Processes in Solution

T

proton radial momentum distribution, n(p), and to the proton mean kinetic energy, ⟨Ek⟩, that is proportional to its variance, 3σ2

he origin of water anomalies is ascribed to the pronounced directionality of the H-bonds, which dominate the water−water interaction and build up an open, disordered, and dynamic network of molecules.1−3 In a simplistic view, the maximum of the diffusion coefficient D at ∼200 MPa at temperatures ≤ 268 K can be rationalized in terms of enhanced molecular mobility due to breaking of Hbonds with increasing pressure, until the competing hindrances due to increased density eventually get the better of it. At odds with this interpretation, the average number of H-bonds per molecule, as measured by neutron diffraction,4 does not sensibly change upon compression at 268 K, although the overall microscopic structure of the liquid undergoes clear modifications. In particular, the modifications with pressure of the second neighboring shell have been recently interpreted in terms of two identical molecular species, separated by a common H-bond, forming two interpenetrating molecular networks.5 In this picture, the second neighbor of each molecule belongs to a different H-bond network. These networks are characterized by a fast H-bond breaking and making dynamics, which slips to the observation in a typical time-averaged diffraction experiment. The diffusion of water molecules in the liquid does instead proceed through local fluctuations of energy and volume, with both quantities dependent on the H-bond dynamics.6 Contrarily to neutron diffraction, the deep inelastic neutron scattering technique, DINS, is particularly suited for studies of the local environment (r ≤ 1 Å) of protons and their short-time (t ≤ 10−15 s) dynamics as it is based on measurements of the dynamic structure factor, S(q,ω), at high-energy, ℏω, and highmomentum, ℏq, transfers.7 This technique gives access to the © 2012 American Chemical Society

⟨E k ⟩ =

3ℏ2 2 σ 2M

(1)

In order to look for possible correlations between the anomaly of the diffusion coefficient and the short-time fluctuations of the local environment of the water protons, we have performed a DINS experiment on water at 268 K, between 26 and 400 MPa, that is, in the region where the diffusion anomaly (along with anomalies of other dynamical quantities) is most visible. The radial momentum distribution functions, n(p), obtained at the six pressure states investigated are shown in Figure 1, and the pressure dependence of the corresponding mean kinetic energy is compared with that of the diffusion coefficient in Figure 2. At all pressure states, the radial momentum distribution is non-Gaussian, as already found in both aqueous8 and nonaqueous systems,9 due to the anisotropicity and anharmonicity of the proton motion. In addition and more interestingly, we notice that the n(p) develops a clear shoulder, centered at ∼15 Å−1, in the same pressure range where the diffusion coefficient has a maximum. Consequently, at these pressures, the mean kinetic energy shows a maximum too. This maximum is another anomaly of water as, according to both Received: July 24, 2012 Accepted: August 29, 2012 Published: August 29, 2012 2594

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coefficient of water suggests that the two phenomena may have a common quantum origin. In order to get a better understanding of the observed phenomenon, we have fitted the momentum distribution functions with three Gaussian lines. We remark that this procedure, although arbitrary, is indeed a reasonable way to single out components of a distribution function. The fit of two n(p) functions, representative of the two distinct groups (shown in the left and right panels of Figure 1, respectively) are reported in Figure 3. Within the two groups, the distribution

Figure 1. The radial momentum distribution functions of water protons at 268 K. These functions can be separated into two groups, plotted in the left and right panels, respectively. The functions of the first group, corresponding to P = 26 (black line and circles), 53 (red line and squares), and 400 MPa (green line and up-triangles), are moderately non-Gaussian and superimpose on each other, despite the large pressure difference. The functions of the second group (right panel) correspond to pressures where the diffusion coefficient shows a maximum, namely, 100 (blue line and down-triangles), 150 (cyano line and diamonds), and 200 MPa (magenta line and crosses). The latter functions also superimpose on each other but are clearly different from those plotted in the left panel and show an evident shoulder at ∼15 Å−1. It is consequently expected that the two groups of distributions have different variance (kinetic energy), as shown in Figure 2.

Figure 3. Comparison between the fits of the momentum distributions at P = 26 (top panel) and at 100 MPa (bottom panel). These distribution functions are representative of the two distinct groups identified in Figure 1, respectively. Both are fitted by using three Gaussian lines, two centered at ∼6.5−6.8 Å−1 (dot−dashed lines, red and cyano, respectively, at the two pressure states) and at ∼4 Å−1 (dotted lines, green and magenta at the two pressure states) and a third (dashed line), which moves from ∼10 (blue, top panel) to ∼12 Å−1 (orange, bottom panel). The standard deviation of the first two Gaussian lines is almost independent of the pressure, while the standard deviation of the third is ∼5.5 Å−1 for data reported in the top panel and ∼6.5 Å−1 for data reported in the bottom panel.

functions, and consequently the three Gaussian required to reproduce their line shapes, are almost the same. More interestingly, we notice that two out of the three Gaussians are centered at the same p value and have the same standard deviation, within the experimental uncertainty, at all six pressure states investigated. The third Gaussian moves at higher p values and becomes broader in the pressure range between 100 and 200 MPa. This observation suggests that all n(p) functions can be fitted by using two line shapes, representing the momentum distribution functions of two distinct “families” of protons. The first line shape is nonGaussian, centered at lower average p value, and is given by the sum of the two common Gaussian terms; the second is Gaussian, centered at higher average p value, moves with pressure, and broadens in the pressure region of the diffusion anomaly. This decomposition of the radial momentum distribution functions, shown in Figure 4, along with ⟨Ek⟩ values shown in Figure 2, indicates that at pressures below 53 MPa and at 400 MPa, the water proton population, characterized by a lower mean kinetic energy, can be split in two families, with 60% of the population having an average momentum of ∼6 Å−1 and the remaining 40% with an higher average momentum (∼10 Å−1). Conversely, at pressures in the range of 100 ≤ P ≤ 200 MPa, the water proton population,

Figure 2. Comparison of the pressure dependence of the diffusion coefficient of water and of the mean kinetic energy of water protons. Data for D have been taken from ref 10 and are reported as triangles. The proton mean kinetic energy (solid circles), calculated from the n(p) functions according to eq 1, reaches a maximum (∼180 meV) at the same pressure where the diffusion coefficient is maximum. The solid lines are a polynomial fit, with weights on data inversely proportional to their standard deviation. Error bars cannot be drawn, being smaller than the size of the graphic symbols.

classical and semiclassical models, the kinetic energy should depend on temperature, not on pressure. The observed anomaly of the kinetic energy is a manifestation of quantum behavior of the water protons; the observation that it falls within the same T,ρ region as the anomaly of the diffusion 2595

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On the other hand, extremes in the response functions of a homogeneous fluid are observed crossing the Widom line. This is defined as the locus of the maximum correlation length,13 extending the coexistence line between two phases into the homogeneous region. Consequently, the observed maxima of both the diffusion coefficient and mean kinetic energy may be interpreted as signatures of the existence of a second critical point of water,1 the coexistence line ending at much lower temperatures between an high-density and a low-density phase of water. Finally, it is relevant to notice that anomalous behavior of the water proton mean kinetic energy has also been observed at ordinary pressure as a function of temperature, both for H2O and D2O, around the temperature of maximum density14−16 and in the supercooled phase.17 The proton mean kinetic energy is in excess compared to the theoretically predicted value whenever other dynamical, structural, or thermodynamic properties are anomalous, thus confirming that the H-bond network, where protons are localized/delocalized, its structure, and dynamics are key issues that determine water properties.

Figure 4. Comparison of the two components of the momentum distribution functions. The top panel reports the decomposition of the proton momentum distribution at P = 26 (black), 53 (red), and 400 MPa (green); the solid line and symbols or the dashed lines are for the low p and high p components, respectively. The bottom panel reports the decomposition for the “anomalous” states at P = 100 (blue), 150 (cyano), and 200 MPa (magenta), and the solid line and symbols or the dashed lines are for the low p and high p components, respectively. The area below the low p component (yellow shaded) is the 60and 70% of the total area for the top and bottom distributions, respectively.

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MATERIALS AND METHODS The theoretical background of DINS can be found in detail in refs 7 and 11, and references therein. Here, we briefly emphasize its main features and application to water protons. The theoretical basis of DINS is the impulse approximation (IA); this approximation treats the scattering event as singleatom scattering with conservation of the momentum and of the kinetic energy of the system (neutron plus target hydrogen atom).18 In the IA, it is assumed that neutron scattering is incoherent and occurring within time scales much shorter than the typical relaxation times of the collective excitations of the system. Under these conditions, the struck atom recoils freely from the collision, with interparticle interaction in the final state being negligible (i.e., the wave function of the particle in its final state assumed to be a plane wave). In a molecular system, as, for instance, water, the contribution to the scattering cross section of protons can be easily distinguished from that of oxygens due to the large mass difference. The recoil energy, ℏωr, is linked to the hydrogen mass, M, and to the wave vector transfer q, by the relation ℏωr = (ℏq)2/2M. The IA is strictly valid in the limit of q → ∞, where the neutron scattering function, SIA(q⃗,ω), is linked to the proton momentum distribution n(p) by the relation

characterized by a higher mean kinetic energy, can be split in two families, but now, 70% of the population has an average momentum of ∼6 Å−1, and the remaining 30% becomes more mobile, with an average momentum of ∼12 Å−1. As a consequence, for pressures in the range of 100−200 MPa, the proton momentum fluctuations increase, resulting in a maximum in the proton mean kinetic energy. Large fluctuations of the momentum, associated with a secondary structure of the distribution function, are usually ascribed to proton coherence along the H-bond and correlated to the local energy landscape experienced by these protons.11 Moreover, the measured excess of energy with respect to bulk water at ambient pressure provides evidence of a variation of the ground state of the proton stretch mode (its excited states are at much larger energies compared to the ground state). This in turn changes because the configuration of the hydrogen bond changes, thus affecting the interparticle potential that the proton experiences. In this picture, protons with higher mean kinetic energy see a lower energy barrier between adjacent positions along the bond; consequently, their breaking/making H-bond dynamics is enhanced. On the other hand, this mechanism is a prerequisite for diffusion of water molecules;6 thus, we suggest that the enhanced mobility of protons along the H-bonds may be the microscopic origin of the maximum of the diffusion coefficient with increasing pressure. In this view, the anomalies of the proton momentum radial distribution and mean kinetic energy and that of the diffusion coefficient of water may be considered as two sides of the same coin, as suggested in the title of this Letter. Incidentally, because we have introduced the concept of “two families” of protons, it is worth clarifying that direct relation of these results with the recent postulation of two species of Hbonds in water12 (weak and strong, respectively) is not straightforward, the latter based on evidence from classical computer simulations, where the quantum nature of the proton dynamics is missing.

SIA(q ⃗ , ω) = =

⎛ ℏq2 p ⃗ ·q ⃗ ⎞ ⎟ dp ⃗ n(p ⃗ )δ ⎜ω − − 2M M ⎠ ⎝

∫

M J (q ̂ , y ) q IA

(2)

where y = (M/q)(ω − (ℏq /2M)). In an isotropic system, there is no dependence on q̂, and the response function becomes 2

∞

JIA (y) = 2π

∫|y|

pn(p) dp

(3)

To extract the proton mean kinetic energy, ⟨Ek⟩, and the proton momentum distribution, n(p), a general expansion of the response function in Hermite polynomials Hn(x) is used. This can be written as JIA (y) = 2596

2 2 e−y /2σ ⎡⎢ 1+ 2π σ ⎢⎣

∞

∑ n=2

⎛ y ⎞⎤ ⎟⎥ H 2n ⎜ ⎝ 2 σ ⎠⎦⎥ 22nn! cn

(4)

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where σ is the standard deviation and the cn are the Hermite coefficients. ⟨Ek⟩ and σ are directly related through the equation σ2 = (2M/3ℏ2)⟨Ek⟩. The coefficients cn and σ, appearing in the series expansion, can be determined by a leastsquares fitting procedure.7 Small deviations from IA at finite q values can be accounted for in terms of additive corrections to the asymptotic form as J(y,q) = JIA(y) + ΔJ(y,q), where ΔJ(y,q) ≈ H3(y/21/2σ)/q. The proton momentum distribution n(p) can be expressed in terms of the generalized Laguerre polynomials, L1/2 n , and of the Hermite coefficients, cn, by n(p) =

2 2 ⎛ p2 ⎞ e−p /2σ · c ( −1)n Ln1/2⎜ 2 ⎟ 3 ∑ n ( 2π σ ) n ⎝ 2σ ⎠

at Very Low Temperature. Phys. Chem. Chem. Phys. 2000, 2, 1551− 1558. (2) Errington, J. R.; Debenedetti, P. G. Relationship between Structural Order and the Anomalies of Liquid Water. Nature 2001, 409, 318−321. (3) Stanley, H. E.; Teixeira, J. Interpretation of the Unusual Behavior of H2O and D2O at Low Temperatures: Tests of a Percolation Model. J. Chem. Phys. 1980, 73, 3404−3422. (4) Soper, A. K.; Ricci, M. A. Structures of High-Density and LowDensity Water. Phys. Rev. Lett. 2000, 84, 2881−2884. (5) Soper, A. K. Water: Two Liquids Divided by a Common Hydrogen Bond. J. Phys. Chem. B 2011, 115, 14014−14022. (6) Ricci, F. P.; Ricci, M. A.; Rocca, D. Self Diffusion in Liquid Water. J. Chem. Phys. 1977, 66, 5509−5512 and references therein.. (7) Andreani, C.; Colognesi, D.; Mayers, J.; Reiter, G. F.; Senesi, R. Measurement of Momentum Distribution of Light Atoms and Molecules in Condensed Matter Systems Using Inelastic Neutron Scattering. Adv. Phys. 2005, 54, 377−469. (8) Flammini, D.; Pietropaolo, A.; Senesi, R.; Andreani, C; McBride, F.; Hodgson, A.; Adams, M. A.; Lin, L.; Car, R. Spherical Momentum Distribution of the Protons in Hexagonal Ice from Modeling of Inelastic Neutron Scattering Data. J. Chem. Phys. 2012, 136, 024504/− 024504/8. (9) Andreani, C.; Degiorgi, E.; Senesi, R.; Cilloco, F.; Mayers, J.; Nardone, M.; Pace, E. Single Particle Dynamics in Fluid and Solid Hydrogen Sulphide: An Inelastic Neutron Scattering Study. J. Chem. Phys. 2001, 114, 387−398. (10) Harris, K. R.; Newitt, P. J. Self-Diffusion of Water at Low Temperatures and High Pressures. J. Chem. Eng. Data 1997, 42, 346− 348. (11) Reiter, G. F.; Mayers, J.; Platzman, P. Direct Observation of Tunneling in KDP Using Neutron Compton Scattering. Phys. Rev. Lett. 2002, 89, 135505/1−135505/4. (12) Nilsson, A.; Pettersson, L. G. M. Perspectives on the Structure of Liquid Water. Chem. Phys. 2011, 389, 1−34. (13) Franzese, G.; Stanley., H. E. The Widom Line of Supercooled Water. J. Phys.: Condens. Matter 2007, 19, 205126. (14) Flammini, D.; Ricci, M. A.; Bruni, F. A New Water Anomaly: the Temperature Dependence of the Proton Mean Kinetic Energy. J. Chem. Phys. 2009, 130, 236101/1−236101/2. (15) Giuliani, A.; Bruni, F.; Ricci, M. A.; Adams, M. A. Isotope Quantum Effects on the Water Proton Mean Kinetic Energy. Phys. Rev. Lett. 2011, 106, 255502/1−255502/4. (16) Giuliani, A.; Ricci, M. A.; Bruni, F.; Mayers, J. Quantum Effects and the Local Environment of Water Hydrogen. A Deep Inelastic Neutron Scattering Study. Phys. Rev. B 2012, submitted. (17) Pietropaolo, A.; Senesi, R.; Andreani, C.; Botti, A.; Ricci, M. A.; Bruni, F. Excess Proton Mean Kinetic Energy in Supercooled Water. Phys. Rev. Lett. 2008, 100, 127802/1−127802/4. (18) West, G. B. Electron Scattering from Atoms, Nuclei and Nucleons. Phys. Rep. 1975, 18, 263−323. (19) ISIS. www.isis.stfc.ac.uk (2012). (20) Pietropaolo, A.; Tardocchi, M.; Schooneveld, E. M.; Senesi, R. Characterization of the γ Background in Epithermal Neutron Scattering Measurements at Pulsed Neutron Sources. Nucl. Instrum. Methods Phys. Res., Sect. A 2006, 568, 826−838. (21) Pietropaolo, A.; Andreani, C.; Filabozzi, A.; Pace, E.; Senesi, R. Resolution Function in Deep Inelastic Neutron Scattering Using the Foil Cycling Technique. Nucl. Instrum. Methods Phys. Res., Sect. A 2007, 570, 498−510. (22) Mayers, J.; Reiter, G. The VESUVIO Electron Volt Neutron Spectrometer. Meas. Sci. Technol. 2012, 23, 045902/1−045902/18.

(5)

DINS experiments have been carried out in the range of 2.5 ≤ ℏωr ≤ 30 eV, by using the VESUVIO spectrometer.19 The actual configuration of VESUVIO has 64 scintillator detectors for neutrons; these detectors are located between 0.5 and 0.75 m from the sample position, covering the angular range of 33 ≤ 2θ ≤ 73°. At each scattering angle, the energy of the scattered neutrons, En, is selected by using resonance detectors and foilcycling techniques with Au analyzers (En = 4897 meV), providing a resolution of ∼2 Å−1 fwhm in y-space, along with a complete removal of the sample-independent background.20,21 For each detector, the time-of-flight data sets were corrected for multiple scattering, heavy atom (all atoms in the samples but hydrogen, and atoms of the sample container) recoil signals, and residual gamma background, using standard routines available on VESUVIO.22 The sample was contained in the high-pressure Ti−Zr cell available at ISIS, which can withstand pressures up to 700 MPa at ambient temperature. The sample container was connected to a pressure rig, which allowed hydrostatic pressure control. Temperature was controlled by a helium cryostat. At each pressure, data were collected for 36 h, with ISIS running at 180 μA/h.

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ASSOCIATED CONTENT

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AUTHOR INFORMATION

* Supporting Information S

Experimental data of J(y) and their fit according to eq 4 at P = 200 MPa and T = 268 K are shown as an example. Two tables report the fit parameters of the J(y) functions (according to eq 4) and the parameters of the three Gaussian lines required to reproduce the n(p) functions (see Figure 3). This material is available free of charge via the Internet at http://pubs.acs.org. Corresponding Author

*Tel.: + 39-06-57337226. Fax: + 39-06-57337102. E-mail: riccim@fis.uniroma3.it. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge discussions with Prof. H. E. Stanley. This work has been performed within the Agreement No. 01/ 9001 between CCLRC and CNR, concerning collaboration in scientific research at the spallation neutron source ISIS and with partial financial support of CNR. Financial support from PRIN2008 (WALTER) is acknowledged.

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REFERENCES

(1) Stanley, H. E.; Buldyrev, S. V.; Canpolat, M.; Mishima, O.; SadrLahijany, M. R.; Scala, A.; Starr, F. W. The Puzzling Behavior of Water 2597

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