VOLUME 81, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 7 SEPTEMBER 1998 Viscous versus Elastic Response of Hydrogen-Bonded Liquids: Collective Dynamics in HF D. Bertolini Istituto di Fisica Atomica e Molecolare del Consiglio Nazionale delle Ricerche, I-56100 Pisa, Italy G. Sutmann Dipartimento di Fisica dell'Universitą di Trento, I-38050 Povo (Trento), Italy A. Tani Dipartimento di Chimica e Chimica Industriale, Universitą di Pisa, I-50126 Pisa, Italy R. Vallauri* Istituto Nazionale per la Fisica della Materia and Dipartimento di Fisica dell'Universitą di Trento, Via Sommarive 14, I-38050 Povo (Trento), Italy (Received 23 April 1998) The role of hydrogen bonding in the determination of the dynamical properties of liquids is investigated by a computer simulation of HF, and compared with water. Sound dispersion is found to be much smaller and interpretable in terms of the viscoelastic approximation. Some dynamical solidlike features are found even in this liquid and interpreted in terms of localized motions of molecules along topological chains present in the liquid. The present investigation answers some relevant questions concerning the dynamical behavior of associated liquids. [S0031-9007(98)07004-5] PACS numbers: 61.25.Em, 62.60.+v, 63.20.Pw A full understanding of the dynamical properties spectra at v 10 ps21, reinforces the overall picture of hydrogen-bonded liquids has certainly not yet been that the response of water is similar to that of a solid achieved, since the appearance of features not present in at length scales of the order of 1 nm. A collective simple monatomic liquids has still to be given a robust excitation at the same energy has been measured in ice physical basis. Water is certainly the liquid which has close to the melting point both by neutron [2] and x-ray been more extensively investigated both experimentally [3] scattering experiments. Regarding the nature of these [1­3] and by computer simulation [4­7], and the presence dynamical properties several important questions remain of modes at wave vectors k . 1 nm21, propagating with a to be answered: Are these features peculiar of water? velocity 2.5 times larger than the ultrasonic sound velocity, What is the role of hydrogen bonding? has been the subject of a large debate. It is now currently In the present Letter we try to answer these fundamental believed that this extraordinary large anomalous disper- questions in order to pave the way to a more general sion can be explained by the fact that, on a length scale understanding of the dynamical properties of hydrogen- of a few molecular clusters, the behavior of water is more bonded liquids. To this end we have analyzed in detail similar to that of a solid rather than of a liquid. Therefore by computer simulation the relevant dynamical correlation the dynamical features at short wavelengths are dominated functions of a model system of liquid HF. Here the by the elastic response, which appears to be much more hydrogen bond energy ( 225 kJ mol21) between two rigid than in simple liquids. This effect has been traced molecules is even stronger than in water ( 220 kJ mol21) back to the structure set up by the presence of hydrogen so that its role should appear even more evident [12]. bonds [8]. Nearest neighbor oxygen atoms can approach Recently we have implemented both a nonpolarizable at an average distance smaller than that characteristic of [13] and a polarizable [14] model and analyzed the the atomic radius determined by the repulsive part of the structural properties [12,14]. The results were compared potential (e.g., s in the Lennard-Jones potential). This with ab initio calculations [15], and we found that, even if implies that the second moment of the longitudinal current the polarizable model gives better results for the partial [9] is much higher than one would expect in a simple radial distribution functions, the nonpolarizable model liquid. Correspondingly C turns out to be very high is sufficiently realistic when compared with neutron ( 4500 5000 m s), i.e., the same magnitude of the ve- diffraction data [16]. In view of these results we have locity of propagation of the longitudinal acoustic modes in chosen to adopt the nonpolarizable model in the present ice. This argument is supported by the observation that at study of the dynamical quantities, since it appears less k . 2 nm21 transverse waves can propagate in the liquid, demanding from the point of view of computational time. as demonstrated by computer simulations [7,10,11]. In our computer simulation of HF we have studied a The presence of an optical-like mode, which appears system of 512 molecules interacting through the three-site at k 10 nm21 both in the longitudinal and transverse model developed in Ref. [13]. The trajectory length was 2080 0031-9007 98 81(10) 2080(4)$15.00 © 1998 The American Physical Society VOLUME 81, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 7 SEPTEMBER 1998 450 ps where a time step of integration of 2 fs was applied. Periodic boundary conditions with a tapered reaction field method were used to take long range interactions into ac- count. For the computation of the spectra of the correla- tion functions we used time intervals ranging between 1 a) and 6 ps (depending on the convergence) leading to mini- mum frequencies vmin 0.167 ps21 or vmin 1 ps21. We have evaluated the correlation functions of the longitudinal [JL k, t ] and transverse [JT k, t ] currents 1 X J L k, t p k k ? v N i exp ikri t ; i 1 X (1) J T k, t p k 3 v N i exp ikri t , i where k indicates the unit vector along k, ri t and vi t are the position and velocity of the center of mass of the ith molecule, respectively, and N is the number of molecules in the sample. The range of wave vectors b) which was explored starts from the minimum accessible value 2.5 nm21. Two thermodynamic states have been investigated: state point A at r 1.178 g cm3, T 203 K and B at r 1.015 g cm3, T 273 K. The spectra of the longitudinal and transverse current correlation functions are presented in Fig. 1 for state point A. The presence of collective modes is signaled by the ap- pearance of peaks in the spectra. By reporting the fre- quency of these peaks as a function of k (as shown in Fig. 2) one can start to make an assignment of the nature of FIG. 1. Surface plots of the longitudinal (a) and transverse (b) these collective modes. One notices that the low frequency current spectra of HF at state point A. Units are nm21 (for k) longitudinal mode changes almost linearly at small wave and ps21 (for v). The amplitude is in arbitrary units. vectors, thus indicating its acoustic origin. The corre- sponding sound velocity is reported in Fig. 3 and compared at relatively high wave vector. The fact that transverse with what one would expect for the corresponding isother- p modes cannot be supported (at least at small wave vectors) mal counterpart C0 kBT MS k [17], where M is the is again an indication that the collective excitations are mass of the molecule and S k is the center of mass struc- dominated by the viscous response. ture factor. Since the experimental adiabatic sound veloc- The second peak present in both longitudinal and trans- ity is found to be Cs 483.3 m s at state point B, one verse spectra appears to be almost at the same frequency can conclude that a slight sound dispersion is present, but and not much affected by the temperature change (v nothing similar to what is found in water. We have evalu- 40 ps21 at T 273 K and v 45 ps21 at T 203 K). ated C k through the second moment of the longitudinal Moreover, it results to be almost constant at increasing current correlation function and compared it with C k in wave vectors. In order to understand its origin, we com- Fig. 3. One can surely conclude that the propagation of pare these findings with light and neutron scattering results this collective longitudinal mode is dominated by the vis- for solid HF. cous rather than elastic response of the system, contrary As is well known HF crystallizes in the orthorhombic to what happens in water (it is instructive to see a simi- structure with two molecules per primitive cell (in the fer- lar comparison for water reported in Fig. 5 of Ref. [8]). roelectric configuration) [18]. The molecules form infin- An analogous situation is found even at state point A, as itely long zigzag chains. Raman and infrared spectra have shown in Fig. 3. An experimental value at this state point been measured [19,20]; in particular, two bands have been is not known but since at T 240 K, Cs 623.8 m s we observed centered at v 35 and 67 ps21, both of them may safely state that again at increasing wave vector (i.e., infrared active. They have been attributed to translational out of the hydrodynamic regime) a slight increase of this motions of the molecules, where the low frequency mode velocity is present which remains in any case much smaller involves oscillations perpendicular to the chain, whereas than C k . the higher frequency mode involves oscillations in the di- As far as the transverse current spectra are concerned, rection of the chain. Both these modes are then charac- only at low temperature a low frequency peak appears terized by a stretching of the hydrogen bond. Because the 2081 VOLUME 81, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 7 SEPTEMBER 1998 5000 50 40 10 4000 T=203 K T=203 K 8 30 6 3000 4 ] 20 2 -1 2000 0 5 10 15 20 25 10 1000 (k) [ps 050 0 40 c(k) [m/s] 4000 T=273 K 8 30 T=273 K 3000 6 4 20 2 2000 00 5 10 15 20 25 10 1000 0 0 5 10 15 20 25 0 0 5 10 15 20 25 k [nm-1] k [nm-1] FIG. 2. Frequency values of the peak positions of the longi- tudinal (dots) and transverse (squares) current spectra. Upper FIG. 3. Sound dispersion of HF at two state points A (upper figure: state point A. Lower figure: state point B. The in- figure) and B (lower figure). Dashed line: C0 k isothermal sets show the details of the longitudinal and transverse acoustic sound velocity. Solid line: C k evaluated through the second modes. The arrows indicate the high frequency peak position moment of the longitudinal current correlation function. The of the VACF spectrum. arrow indicates the value of the experimental sound velocity. two modes are infrared active they can safely be labeled The present study shows that even in the case of liquid as optical. HF, when the system makes a transition from the solid We believe that the high frequency mode apparent in to the liquid phase, some topological features persist, in both the longitudinal and transverse spectra can be ascribed particular, the arrangement of molecules in the form of to the same type of oscillating motion of the molecules. By linear chains, which determine the dynamical behavior. examining the structure of liquid HF we have found [12] The chains are found to move rather free, so that the that topological chains still persist in the liquid phase. In viscosity is much lower than in other molecular liquids fact, an overwhelming percentage ( 82%) of molecules (e.g., in water). This observation is confirmed by the fact has two nearest neighbors, and by a combined geometrical that the isothermal compressibility of HF is very high as and energetic definition of the hydrogen bond we found shown by the k ! 0 behavior of the structure factor S k that HF molecules are arranged in paths that resemble long reported in Fig. 4 for state points A and B. chains, even if not strictly linear. The oscillatory motion An estimate of the shear viscosity can be made by of the molecules along these chains is therefore responsible the time dependence of the transverse current correlation for the appearance of the well defined second peak in the function at small wave vectors which shows a monotonic longitudinal and transverse current spectra. decay, typical of the hydrodynamic regime. One can This conclusion is confirmed by the analysis of the assume that ! center of mass velocity autocorrelation (VACF) function k h k k2 and its corresponding spectrum for the two examined state C BT T k, t exp 2 t (2) points. A well defined peak is present at a frequency M nM which is marked as an arrow in Fig. 2. Since the VACF [h k being a generalized shear viscosity coefficient and n spectrum is a measure of the density of states, it is the number density] and derive a value for h of 0.16 3 useful to compare the present result with that obtained 1023 Pa s at T 203 K, about 10 times smaller than the by inelastic neutron diffraction measurements from solid corresponding value for water close to the melting point and liquid HF [21]. The spectrum of solid HF presents [22]. This result can explain the different behavior of a shoulder at about 26.82 meV (i.e., v 40.6 ps21) in HF with respect to water. In fact, by adopting a simple close agreement with the present finding. In the liquid viscoelastic model for the analysis of the dynamics of the phase a large plateau is present at the same energy. transverse current it turns out that the Maxwell relaxation 2082 VOLUME 81, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 7 SEPTEMBER 1998 1,4 lar clusters hydrogen-bonded liquids exhibit common fea- tures which can be understood in terms of the dynamical 1,2 behavior of the corresponding solid. We thank Dr. U. Balucani, Professor R. Righini, and 1,0 Professor M. Suhm for helpful discussions. 0,8 S(k) *Author to whom correspondence should be addressed. 0,6 T=203 K [1] J. Teixeira, M. C. Bellissent-Funel, S. H. Chen, and T=273 K B. Dorner, Phys. Rev. Lett. 54, 2681 (1985). 0,4 [2] F. J. Bermejo, M. Alvarez, S. M. Bennington, and R. Vallauri, Phys. Rev. E 51, 2250 (1995). 0,2 [3] F. Sette, G. Ruocco, M. Krish, U. Bergmann, C. Mas- 0 10 20 30 40 50 ciovecchio, V. Mazzacurati, G. Signorelli, and R. Verbeni, k [nm-1] Phys. Rev. Lett. 75, 850 (1995). [4] A. Rahman and F. H. Stillinger, Phys. Rev. A 10, 368 FIG. 4. Center of mass structure factor of HF at the two state (1974). points. [5] M. Wojcik and E. Clementi, J. Chem. Phys. 85, 6085 (1986). [6] M. A. Ricci, D. Rocca, G. Ruocco, and R. Vallauri, Phys. time tM k h k G k [G k being the wave vector Rev. A 40, 7226 (1989). dependent rigidity modulus proportional to the normalized [7] F. Sciortino and S. Sastry, J. Chem. Phys. 100, 3881 second moment of the transverse current, v2T k ] is so (1994). short that v2T k t2M is always ų1. Consequently in this [8] U. Balucani, G. Ruocco, A. Torcini, and R. Vallauri, Phys. liquid the dynamics of collective properties is dominated Rev. E 47, 1677 (1993). by the viscous response. Evidently the difference between [9] U. Balucani and M. Zoppi, Dynamics of the Liquid State water and HF has to be found in the fact that, whereas G (Clarendon Press, Oxford, 1994). is only a few percent smaller in HF than in water, the [10] U. Balucani, J. P. Brodholt, and R. Vallauri, J. Phys. shear viscosity h is 10 times smaller. Condens. Matter 8, 9269 (1996). This argument can also be applied to the analysis of [11] M. Sampoli, G. Ruocco, and F. Sette, Phys. Rev. Lett. 79, 1678 (1997). the longitudinal currents. Here the relaxation time tL k [12] P. Jedlovszky and R. Vallauri, Mol. Phys. 93, 15 (1998). starts from a value [23] [13] P. Jedlovszky and R. Vallauri, Mol. Phys. 92, 331 (1997). 1 4 [14] P. Jedlovszky and R. Vallauri, J. Chem. Phys. 107, 10 166 t 3 h 1 hB L 0 , (3) (1997). nM C2 2 C20 [15] U. Röthlisberger and M. Parrinello, J. Chem. Phys. 106, where h 4658 (1997). B is the bulk viscosity. Again comparing with water, whereas C [16] M. Deraman, J. C. Dore, J. G. Powles, J. H. Holloway, and is of the same order of magnitude, the shear viscosity of HF (and most likely even the bulk P. Chieux, Mol. Phys. 55, 1351 (1985). [17] The ratio g c viscosity) is much smaller so that in the case of HF this p cy of the specific heat is experimentally found to be 1.1. This is in agreement with our simula- relaxation time is shorter and the liquid cannot respond in tion value which remains close to one at increasing wave a rigid fashion in contrast to water [8]. vectors. We then may safely compare the isothermal and In conclusion, the so-called fast sound phenomenon is adiabatic sound velocities. not a common feature of hydrogen-bonded liquids, but it [18] M. W. Johnson, E. Sandor, and E. Arzi, Acta Crystallogr. is strictly dependent on the actual connectivity properties Sec. B 31, 1998 (1975). of the system. This phenomenon appears evident in water, [19] J. S. Kittelberger and D. F. Hornig, J. Chem. Phys. 46, a three dimensional percolating system, whereas in HF, 3099 (1967). where the presence of topological chains reduces the per- [20] A. Anderson, B. H. Torrie, and W. S. Tse, Chem. Phys. colating character to one dimension, the velocity of sound Lett. 70, 300 (1980). away from the hydrodynamics region does not show such [21] H. Boutin, G. J. Safford, and V. Brajovic, J. Chem. Phys. 39, 3135 (1963). an exceedingly large dispersion. On the other hand, the [22] D. Bertolini and A. Tani, Phys. Rev. E 51, 1091 (1995); appearance in HF of the optical-like high frequency mode 52, 1699 (1995). in both the longitudinal and transverse spectra confirms [23] J. P. Boon and S. Yip, Molecular Hydrodynamics (Dover the fact that on a length scale characteristic of molecu- Publications, Inc., New York, 1991). 2083