VOLUME 79, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 1 SEPTEMBER 1997 Mixing of Longitudinal and Transverse Dynamics in Liquid Water M. Sampoli,1 G. Ruocco,2 and F. Sette3 1Universitá di Firenze and Istituto Nazionale di Fisica della Materia, I-50139, Firenze, Italy 2Universitá di L'Aquila and Istituto Nazionale di Fisica della Materia, I-67100, L'Aquila, Italy 3European Synchrotron Radiation Facility, B.P. 220 F-38043 Grenoble Cedex, France (Received 13 March 1997) Molecular dynamics simulations of liquid water in the 1.3 35 nm21 momentum transfer (Q) region show two excitations with a Q dependent symmetry character. The symmetry analysis suggests that the observed anomalies in the high frequency collective dynamics originate from relaxation processes responsible for (i) the appearance of a propagating transverse dynamics at high frequency, (ii) the transition from normal to fast sound, and (iii) a mixed symmetry of the two modes at large Q. [S0031-9007(97)03759-9] PACS numbers: 61.20.Ja, 61.20.Lc, 61.25.Em The investigation of large wave vector excitations in liq- like. In the opposite limit, at Q larger than 4 nm21, the uid water has been a challenging task since the pioneer- dynamics is solidlike; there are two modes with energies ing computational [1] and experimental [2] studies on its close to the longitudinal and transverse phonon branches dynamic structure factor, S Q, v . These works revealed in ice. Here, however, contrary to ice, both modes have the existence of acoustic-like excitations propagating with a large mixing of longitudinal and transverse symmetry. a speed, y 3300 m s, corresponding to a value more A propagating transverse dynamics starts to appear in than twice the hydrodynamic sound (yo 1500 m s). the same intermediate Q region where the longitudinal Subsequent works studied this issue by molecular dynam- branch acquires a transverse component, and its sound ics (MD) [3,4], inelastic neutron-(INS) [5], and inelastic velocity changes from yo to y . The transition between x-ray scattering (IXS) [6,7]. The high frequency picture the two regimes is found in the Q-v region, corresponding emerging from S Q, v of H2O can be summarized as fol- to the length scale and lifetime of local order in liquid lows: (i) The acousticlike mode propagates with y in the water. Therefore, these results link the anomalies in 4 to 14 nm21 Q range. (ii) For Q larger than 4 nm21, the high frequency collective dynamics of liquid water there is a second, weakly dispersing mode with an energy to relaxation processes originating from locally ordered of 5 meV. (iii) Both modes involve the motion of the molecular assemblies. molecular center of mass. (iv) At Q 4 nm21, the en- The MD simulation was carried out considering N ergy of the two modes becomes comparable, and in the 1 to 4000 "D2O" SPC E [9] molecules enclosed in a cubic 4 nm21 Q region only one mode is observed; the sound ve- box with periodic boundary conditions [10]. The molar locity of this mode changes, decreasing Q, from y toward volume was 18 cm3 and the temperature 250 K, i.e., the yo. This picture indicates the existence of two branches, temperature of maximum density for the SPC E potential one strongly dispersing and the other weakly dispersing model [11]. The electrostatic long-range interactions are with Q. The first one is identified as the sound branch taken into account with the tapered reaction field method, with a bend up in the region below Q 4 nm21. The and the rotational equations of motion were integrated second one, on the basis of MD simulations [1,4] and of using an improved algorithm [12]. After thermalization, INS and IXS results on ice crystals [7,8], can be related the molecular trajectories have been followed by about to a localized motion reminiscent of the transverse dynam- 100 ps and stored every 10 fs, i.e., every five integration ics in the crystal, and to the bending motion between three time steps. From the stored configurations, we have hydrogen-bonded water molecules. The most important evaluated the instantaneous Q component of the density point, however, is not yet settled: is the physical mecha- fluctuations of the center of mass rQ t : nism responsible for the bending of the sound branch and p X for the observation of a second mode at Q larger than rQ t 1 N exp iQ ? Rj t , 4 nm21 a feature common to a large class of liquids or j is it specific to water? where Rj t is the instantaneous position of the molecular In this Letter we report a numerical MD investigation center of mass [13]. The dynamic structure factor is cal- on the symmetry character of the modes observed in liquid culated from the power spectrum of rQ t , i.e., S Q, v water. The MD results in the Q range of the IXS and INS jF T rQ t j2. To reduce the noise in S Q, v , the Welsh experiments show the existence of two different dynamic method [15] with a Hanning window has been employed. regimes. In the small Q limit, Q , 2 nm21, the dynamics The time window was Dt 20 ps, giving rise to an is liquidlike; there are pure longitudinal modes propagating energy resolution of 0.04 meV. The S Q, v has been with y yo, and the transverse dynamics is relaxational- averaged over independent directions of Q 2pL h, k, l 1678 0031-9007 97 79(9) 1678(4)$10.00 © 1997 The American Physical Society VOLUME 79, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 1 SEPTEMBER 1997 (where L 4.93 nm is the box length) at several Q although less pronounced, is found in the T modes branch. values in the 1.3 to 35 nm21 range. The longitudinal Figures 2(b) and 2(c) show the small momentum transfer current spectra, CL Q, v , are obtained as CL Q, v region in an expanded Q scale. In Fig. 2(c) the change v2S Q, v Q2. in slope of the L branch marks the transition between yo The comparison between the calculated S Q, v and and y . the inelastic x-ray scattering data measured at 278 K [6] The MD results agree well with available INS and IXS is very good for any of the Q values used in the IXS data on liquid water, and they can be directly compared experiment. As a consequence of the capability of the with measurements and lattice dynamics calculations of present potential model to represent well the dynamics solid water in its ice Ih form [7,8]. In ice crystals, as in of real water at normal density, we expect that also the liquid, there are two phonon modes with energies below the transverse dynamics, a quantity not experimentally 30 meV [18]. These are the longitudinal and transverse accessible, can be reliably determined from the MD data. phonon branches, linearly dispersing at small Q with ve- We therefore calculated the transverse current spectra locities y 4000 m s and y 2000 m s, respectively. CT Q, v jF T jTQ t j2 with Moreover, the transverse branch becomes almost flat at Q p X larger than 7 nm21. Therefore, the MD results not only jTQ t 1 N Q 3 Q 3 vj t exp iQ ? Rj t . confirm that at Q larger than 4 nm21 the longitudinal dy- j namics in liquid water become similar to that of the crys- In Fig. 1 we report examples of longitudinal [Figs. 1(a) talline solid, but also show that the transverse dynamics and 1(b)] and transverse [Figs. 1(c) and 1(d)] current spec- acquires a solidlike behavior in about the same Q region. tra at selected Q values. Both CL Q, v and CT Q, v , There is, however, an important difference between liquid show the existence of two excitations. The high frequency and solid in the symmetry character of the two branches. excitation disperses with Q, and its sound velocity changes In the solid, a dominant longitudinal or transverse charac- from 2000 m s to 3300 m s. This excitation ap- ter is preserved throughout the considered Q region. In the pears at each Q value in the longitudinal current spectra, liquid, the MD results show that, as a consequence of lack while it is found in the transverse current spectra only at of translational invariance, the pure symmetry character of Q . 4 nm21. We assign this feature to a quasilongitudi- the two modes is rapidly lost, and at large Q both modes nal sound branch, and we call it L mode for its longitudinal contribute in similar amounts to CL Q, v and CT Q, v . character in the Q ! 0 limit. The behavior of the low fre- The MD results obtained at low Q, i.e., below 2 nm21, quency excitation is in some sense opposite; it is always can be compared with both experiment and common wis- present in the transverse current spectra, while it appears in dom for the collective dynamics of simple liquids in the the longitudinal current spectra only at Q . 4 nm21. This Q ! 0 limit; i.e., only one mode survives in each cur- low frequency feature is from now on referred to as the T rent spectrum. In CL Q, v , the L mode propagates with mode. At small Q, the T mode disperses with a sound ve- a velocity of sound approaching the hydrodynamic value locity of 1500 m s and becomes almost nondispersing measured in the macroscopic time- and length-scale lim- at Q . 7 nm21. its. In CT Q, v , the T mode loses its propagating charac- The calculated spectra have been fitted with simple mod- ter observed in the 2 4 nm21 region, and one could infer els to summarize the excitations in terms of their energy that the transverse dynamics becomes progressively relax- [Vh Q , h L, T , i.e., the position of the current spec- ational from the pileup of spectral intensity toward zero tra maxima], their energy width [Gh Q ], and their inte- frequency. The transition of the L mode to a pure lon- grated intensity [Ih Q ]. Different line shapes have been gitudinal symmetry propagating with the yo, and the dis- used, ranging from a simple damped harmonic oscillator appearance of the T mode, bring the collective dynamics (DHO) [16] to more refined viscoelastic models [17]. It is of liquid water back to the behavior expected for a simple beyond the aim of the present work to discuss the physi- liquid in the macroscopic limit. cal meaning and the reliability of the different approaches. The transition from the low frequency and low Q All these models, however, give fitting parameters which hydrodynamic behavior to the high frequency solidlike are consistent among each other within their uncertain- regime has already been observed in many glass-forming ties, and we report here the parameters obtained using the liquids [19]. This transition takes place at a Q value where DHO line shape for the T mode and the viscoelastic the- the pulsation of the sound wave v equals the inverse of ory with an exponential memory function for the L mode. the structural relaxation time t. Manifestations of this The corresponding best fits to the MD data are shown in transition, among other phenomena, are (i) the increase of Figs. 1(a)­1(d). the longitudinal sound velocity from yo toward y , and (ii) The values of the excitation energies, VL Q and the appearance of a propagating transverse dynamics. This VT Q , are reported in Fig. 2. Figure 2(a) shows the behavior is very similar to the one found here for liquid complete investigated Q region. The dispersion relation water, as shown by the Q dependence of the parameters of the L modes follows the universal behavior of liquid reported in Fig. 2, which can be scaled to the typical a- systems; i.e., a minimum appear at about the Q value relaxation process [19]. At variance with other known where the S Q attains its maximum. A similar minimum, glass-forming systems, where the condition vt 1 is 1679 VOLUME 79, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 1 SEPTEMBER 1997 FIG. 1. The longitudinal (a),(b) and transverse (c),(d) current spectra are reported with the error bars at the indicated Q values. The full lines are the best fits to the spectra as indicated in the text. Dashed (dotted) line is the individual contribution to the fit coming from the L mode (T mode). fulfilled at frequencies below 1010 Hz, in water the tran- besides from dynamic relaxational processes, may also sition is found at much higher frequencies, 2.5 3 come from the local structural order existing in liquid wa- 1011 Hz, i.e., t 0.6 ps. Finally, the wavelength mark- ter. A temperature study would discriminate between the ing the transition, lt 2pyot 3 nm, is comparable to dynamic and structural effects. the structural correlation length derived from S Q mea- In conclusion, the analysis of the longitudinal and trans- surements by Bosio et al. [20]. 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