VOLUME 80, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 9 FEBRUARY 1998 Low-Energy Excitations in Water: A Simple-Model Analysis Tsuneyoshi Nakayama Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan (Received 13 August 1997) This paper demonstrates that characteristics of low-energy excitations in water observed in Raman scattering and inelastic neutron and x-ray scattering experiments follow naturally from a simple- model analysis, taking into account essential features of water structure in the center-of-mass system. It is shown that the modes associated with the lower band near 60 cm21 are strongly localized with localization lengths of the order of their wavelength and the higher band near 180 cm21 attributes to mesoscopically localized modes within the network connected by hydrogen bonds. [S0031-9007(97)05166-1] PACS numbers: 62.30.+d, 63.20.Pw, 63.90.+t The realization of two broad bands in Raman spectra When discussing the dynamic properties of water, there near 60 and 180 cm21 in water is as old as the discov- is an important time scale tc, which is of the order ery of the Raman scattering itself [1,2]. A qualitative un- of 10211 sec 3.3 cm21 , distinguishing harmonic vi- derstanding has long been available for these low-energy brations and relaxation processes [15]. On time scales bands. However, many key aspects of a detailed quanti- shorter than tc, thermal motions of water molecules can tative description, at the molecular level, are still miss- be regarded as being of vibrations on a slowly moving ing despite focused experimental efforts in this decade network, while at time scales longer than tc slower dis- [3]. Sette et al. [4] have very recently suggested the ex- placements of the equilibrium positions become relevant. istence of collective excitations with a velocity of sound The energy regime v ¿ 1 tc should be primarily as- 3200 6 100 m s, more than twice larger than the ordi- sociated with harmonic vibrations. Namely, liquid wa- nary speed of sound 1400 m s, by measuring the dy- ter at shorter time scales than 10211 sec resides in an namic structure factor S q, v up to 200 cm21 in the approximately harmonic potential energy minimum (in q 3 14 nm21 range by means of high resolution in- configuration space) and the molecular motion can be de- elastic x-ray scattering. The equivalent dispersive excita- scribed as being vibrational. tions at low energies for D2O had been observed also by Walrafen et al. [16] observed the noticeable varia- the coherent inelastic neutron scattering up to 120 cm21 tion of intensities of 60 and 180 cm21 bands with tem- and q 6 nm21 [5]. However, different interpretations perature and adding electrolytes in water, in which the for inelastic neutron scattering data are given [6] claim- Raman spectra of the 180 cm21 band is insensitive to ing that excitations observed in Refs. [4,5] are rather op- effects produced by hydration of cations, but it is very ticlike (dispersionless) excitations. Thus, the nature of sensitive to the effects resulting from the hydration of low-energy collective excitations in water still remains certain anions. This result not only suggests that the controversial [7]. Since the energy range spanned by connectivity of tetrahedral network via hydrogen bonds Raman spectra and inelastic x-ray and neutron-scattering is much influenced, but also most importantly indicates data overlaps, the origin of low-energy excitations should that the modes associated with the 180 cm21 band are re- be elucidated on the same basis. The present paper sets lated to the connectivity of the network. More recently, forth a new interpretation of low-energy excitations in wa- Tominaga and Miyoshi-Takeuchi [17] have performed ter by a simple-model analysis. The model is abstract, but high resolution Raman experiments and confirmed that when discussing the physical origin of low-energy spectra the complete disappearance of the 180 cm21 band with in water, its significance becomes clear. the concentration of dioxane in water of 0.7 0.8 molar The commonly accepted overall picture of water struc- fraction. The molar fraction 0.7 0.8 indicates an almost ture is a tetrahedral random network totally connected complete disconnectivity of a hydrogen bonded tetrahe- by hydrogen bonds, isomorphous with that in quartz. dral network. In addition, they have found [17] that This structural model originally proposed by Bernal and the lower band near 60 cm21 is not much affected by Fowler [8] has stimulated a lot of subsequent works on the increasing the concentration of dioxane. The equivalent microscopic structure of water. Recent experiments [9­ evidences had been obtained from the temperature de- 13] and computer simulations [14] have suggested that the pendence of the intensities of the 60 cm21 band [16] and random tetrahedral network is not perfect but must con- from data with aqueous LiCl solutions [18]. These results tain defects which are characterized geometrically by the give a new insight to the description of the modes asso- presence of an extra fifth molecule in the first coordina- ciated with 60 and 180 cm21, namely, the modes attribut- tion shell, or topologically by the presence of bifurcated ing to the 60 cm21 band are "strongly" localized at least hydrogen bonds (up to 40%). within several tetrahedral units and not associated with 1244 0031-9007 98 80(6) 1244(4)$15.00 © 1998 The American Physical Society VOLUME 80, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 9 FEBRUARY 1998 the connectivity of tetrahedral network, while the modes between v2min to v2max. The distribution of v2i arises attributing to the 180 cm21 band are related to the connec- from the local distortion and strain due to the disordered tivity via hydrogen bonds and not strongly localized ones. structure of water. We should emphasize that molecules Here strongly localized modes imply the modes with the connected by the spring constant ki in Fig. 1(a) do localization length lc v of the order of its wavelength not directly correspond to the true molecular arrange- l v . It will be clarified later that our view mentioned ment in water. It simply means some kinds of low- above is consistent with recent inelastic neutron and x-ray energy vibrational states. It should be associated with scattering experiments [3­6]. extra vibrational states in the local potential minima with We propose, at first, a simple structural model, tak- different curvatures ki Mv2i due to the different en- ing account of essential features of water in terms of the vironment of each defect. It is not worthwhile to give a center-of-mass system. This is because low-energy ex- strict picture for these entities from the two reasons: It citations treated here, below the Debye cutoff frequency depends on such factors as the particular configuration of vD (corresponding to that of ice Ih), are surely those molecules surrounding the extra vibrational state on local associated with the center-of-mass dynamics. There is strains and potential energies (this is the very reason no long-range order in liquid water and the equilibrium for taking the distributed v2i). positions are quite irregularly distributed in space. Fig- The Hamiltonian for our model is expressed by ure 1(a) is the schematic illustration of our quasi-1D " X model, which takes into account characteristics of water P2 p2 K H i 1 i 1 Qi 2 Qi21 2 structure such as disordered network-forming structure at i 2M 2M 2# any instant. This model consists of two main chains with ki a constant mass M of molecules and these are connected 1 q , (1) 2 i 2 Qi 2 to their nearest neighbors by linear springs with constant strength K. The central hypothesis of our model is that where molecules have mass M, and Qi and qi are in water there should be a certain number of extra vi- generalized coordinates representing displacements or brational states (e.g., corresponding to defect molecules changes of angle variable. The corresponding momenta suggested in Refs. [9­13] or librational motions of tetra- are described by Pi and pi, respectively. Capital letters hedra itself), which we attach to each main chain with denote quantities for main chains and lower letters are the mass M by linear springs with the strength ki at site for additional vibrational states. Note that vibrations i. The mass M and the force constant ki are related to lose their pure longitudinal or transverse character in the characteristic frequency v2i ki M of these addi- disordered systems like water, and direct coupling to tional states, where the parameters v2i are random quanti- density fluctuations with neutrons or x-rays probes the ties and assumed to be uniformly distributed in the range one-component projection of the vibrations. Thus, the scalar approximation in Eq. (1) is valid. The dynamical structure factor S q, v is expressed in terms of the Fourier transform of the correla- tion function of density fluctuations as S q, v 1 2p s dt e2ivt dr2q 0 drq t , where drq t is the q component of the density fluctuation. The angular brackets · · · mean the thermal average. Decomposing Qi t [or qi t ] defined in Eq. (1) into normal modes, one P obtains drq t 2ieivlt i qei l e2iqri 1 O Qi t 2 , where ri is the equilibrium position of the molecule at site i. Here ei l is the eigenvector belonging to the eigenfrequency vl. Substitution of this expression into the definition of S q, v yields Ç Ç n 1 1 X X 2 S q, v d v 2 v qe , vN l i l e2iqri l i (2) where n is the Bose-Einstein distribution function and N FIG. 1. Schematic illustration of a simplified (quasi-1D) water the total number of atoms, respectively. We employ the structure in the center-of-mass system. The network structure numerical method given in Ref. [19] to calculate S q, v . is composed of unit cells with six or eight molecules. (a) The Calculated results for S q, v with a system size N number of extra vibrational states is 40% of the total number of main molecules, and they are randomly attached to two main 12 000 are given in Fig. 2 in units of M 1, and K 1, chains. (b) The broken network model in which one bond in where periodic boundary conditions are taken. The lower each unit cell with six or eight molecules is missing. and upper limits for the distribution of the frequency v2i 1245 VOLUME 80, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 9 FEBRUARY 1998 are taken to be v2min 14 (or equivalently kmin 14) and in the frequency region vmin , v , vmax. It should be v2max 1 kmax 1 . The force constant K 1 should noted that defect molecules and molecules belonging to correspond to the (largest) stretching force constant Kr main chains vibrate in antiphase (opticlike modes) [5]. 1 eV A22 (in Keating's notation) between O-O within the The "strong" localization in the range vmin , v , vmax network and the force constant ki should be smaller than originates from the resonance between excitations along the strength of K. It is not easy to estimate meaningfully main chains and extra vibrational states [20]. Note that the value of ki. However, for instance, the relevant force the distributed v2i is the key element for the strong constant aside from Kr for network-forming structures is localization. Figure 3(b) is the eigenmode with vl the bending one Ku 0.5 eV rad22 . Thus, the above 1.699 10 belonging to the higher band in Fig. 2. The choice of the parameters ki seems to be reasonable. mode pattern possesses quite different characteristics from An important conclusion drawn from Fig. 2 is that those given in Fig. 3(a), indicating that only molecules there appear fairly two bands in the calculated spectra in main chains vibrate remarkably, and extra vibrational as observed in Raman scattering experiments. The lower freedoms do not follow the vibrations of main chains. peaks, being broadened by increasing the wave number q, In addition, these modes are dispersive [3,4] and are are almost independent of q (nondispersive). The higher mesoscopically localized. Excitations below vmin will be band depends strongly on q, indicating the modes con- dominated by a viscoelastic effect of water and this is tributed are "dispersive." By taking the spring constant beyond the scope of the present Letter. K 1 in Eq. (1) as the actual value of the stretching We have calculated S q, v for a "broken" network force constant Kr 1 eV A22 and M for the mass of structure illustrated in Fig. 1(b). This is a model structure water molecule, the peak energy of the higher band at 2 for the situation in which the network via hydrogen in Fig. 2 becomes 200 cm21 and that of the lower band bonds are disconnected due to the hydration of anions at 0.5 in Fig. 2 does 50 cm21. Both of these values [17,18] or increasing temperature [16]. Figure 4 clearly coincide with observed peak energies [16­18] in spite of demonstrates that, with decreasing the connectivity of the a simple-model analysis. In addition, the velocity of the network, the intensities of higher band (corresponding to fast sound is estimated from the dispersion relation of the the 180 cm21 band) noticeably decrease and, in contrast, higher band depicted in Fig. 2 to be y 3 3 103 m sec. the lower band (60 cm21 band) is insensitive to the This value agrees fairly with observed values [4,5]. connectivity. These results surely recover the tendency Figure 3(a) shows the eigenmode with vl 0.498 01, observed in Raman scattering experiments for 60 and which provides the evidence of strongly localized modes 180 cm21 bands [16­18]. Dispersive excitations observed in Refs. [4,5] had been interpreted as modes propagating within the hydrogen- bonded patches (O-O stretching) predicted by molecu- lar dynamics calculations [21­25]. 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