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Further evidence for the existence of two kinds of H-bonds in ice Ih
5 September 1994 PHYSICS LETTERS A
ELSEVIER
Physics Letters A 192 (1994) 295-300
Further evidence for the existence of two kinds of H-bonds in ice Ih J . C . L i a, S . M . B e n n i n g t o n
b, D . K . ROSS a
Department of Pure and Applied Physics, University of Salford, Salford, M6 4WT, UK b ISIS Facilities, Rutherford-Appleton Laboratory, Didcot, Oxon, OXI I, OQX, UK
Received 15 April 1994;accepted for publication 28 June 1994 Communicated by A. Lagendijk
Abstract
The hydrogen defects in the host D20 lattice of ice have been studied using a high resolution inelastic neutron spectrometer. Three main peaks at ~ 105, ~ 185 and ~ 408 meV, associated with three localised modes (#b #2 and #3) oftbe H-defect centre, have been observed in the inelastic neutron spectra. The peak associated with the/t, mode at ~ 105 meV has an asymmetrical shape from which two Gaussian components centred at 103 _ 0.5 and 107 + 0.5 meV for different concentrations of HDO can be resolved, confirming the existence of two different strengths for the H-bonds. A lattice dynamic calculation for the H defects in a perfect ice Ih lattice, based on the two H-hood force constants model of Li and Ross [Nature 365 (1993) 327 ], shows the three modes in very similar positions to those observed in the measured data and predicts the splitting of the g~ peak into modes at 103 and 107 meV. Hydrogen bonding is one o f the most important and intriguing interactions, which dominate our daily lives, and scientists across different disciplines for long have endeavoured to understand the complex nature o f water and o f other H-bonded systems. There are not as yet, however, completely acceptable explanations o f some properties of water, often referred to as its "anomalous" properties. The large bond energy and the asymmetrical geometry o f the H-bond, combined with the fact that the electrons in the sp 3 orbitals of the oxygen atoms can rehybridise to respond to the relative configurations of adjacent molecules, give rise to a large number o f abnormal properties o f water/ice which cannot be explained by the ordinary rules o f physics and chemistry. As a consequence, a large number o f models have been proposed [2,3 ] in attempts to interpret some o f these properties o f water, such as the high heat capacity [4], the extended possible supercooling o f the liquid state [ 5 ], and the large density and entropy fluctuations [ 6 ].
Meanwhile, a large number of H-bond potentials have also been proposed. Some o f these are based on ab initio quantum mechanical calculations and some are very arbitrary. Some are good at reproducing the structure o f water and some are good at reproducing its thermodynamic properties [ 7,8 ]. The recent development o f the high flux pulsed neutron source for ISIS and o f the high resolution inelastic neutron scattering instruments at the Rutherford-Appleton Laboratory, have enabled us to examine the dynamic properties of different exotic high pressure (up to 40 kbar) forms o f ice by measuring their inelastic incoherent spectra [ 9-11 ]. Also, because neutron scattering gives a measure o f all the vibrational modes with equal sensitivity, the data can easily be converted to the amplitude-weighted phonon density o f states ( P D O S ) , particularly for measurements carried out at very low temperatures. Lattice dynamic calculations o f this function will therefore provide a more direct test o f existing potentials. In
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J.c. Li et aL / Physics Letters A 192 (I 994) 295-300
order to reproduce the two measured molecular optic bands centred at 28 and 37 meV for ice I [9], two empirical force constants have been proposed [ 10 ]. The two types of H-bonds are related to two different types of local configurations. The resulting dispersion curves and PDOS calculated for a large superlattice constructed so as to represent the proton disordering in the local environments give good agreement with the experimental data [10]. The results also agree well when this model was applied to other forms of ice, such as ice VIII [ 11 ] which due to the proton ordering in this structure, there is only one type of local proton arrangement, corresponding to the weak H-bonds. The lack of any dependence of the spectrum of ice Ih on crystal orientations rules out existing dynamic models for this phase which assume the existence of different H-bond force constants in different crystal directions. Other explanations of the two molecular optic bands at 27 and 36 meV are based on TO (transverse optic) and LO (longitudinal optic) splitting [12] (as is found in ionic crystals such as sodium chloride, where the stiffer force constant corresponds to the LO vibrations). However, because the LO modes have higher frequencies and should have less intensity than the TO modes, the ratio of the TO to the LO modes should be 2: 1. However, the data show the opposite ratio. Also this splitting exists only at q = 0 and not near the zone boundary, the region that dominates the IINS. In attempts to reproduce these IINS spectra, using lattice dynamic (LD) calculations, we have tested some of existing potentials, but the results of these calculations, as expected, are far from satisfactory because most molecular dynamics calculations using these potentials do not yield the two optic bands in the translational region [ 13 ], since these potentials always consist of a repulsive part (often Lennard-Jones or exponential types) plus electrostatic terms due to charges on oxygens and protons [7]. However, the contribution from the electrostatic terms could not produce the large difference in the H-bond force constant (the maximum effect would be 15% among different dipole-dipole arrangements). After looking at the problem from several points of view, we are forced to the conclusion that the existence of two molecular optic bands in the spectrum leads inevitably to the conclusion that there are two strengths of H-bonds randomly distrib-
uted in every direction [ 1 ]. However, the true cause of the large difference in the force constants between the strong and weak is still unresolved. The large difference of the H-bond strength force constants used for the lattice dynamic calculations cannot be explained by any classical considerations. Therefore, the true cause must lay on the quantum mechanical level. Considering the H-bond are combination (or hybridisation) of 2s and 2p orbital from O atom, the nature of the H-bond is very sensitive to the surroundings. We assume that the different electronic configurations alter the interaction strengths through cooperative effects. The two well-separated molecular optic bands in the spectrum imply that there are two strengths of Hbonds. A simple calculation using the relationship 0)-~ k v / ~ (where 0) is the vibrational frequency and k is the force constant) gives a ratio of the two Hbond force constants as k I :k2= (0)1:0)2) 2= 272:372= 1: 1.9. Based on the above knowledge, a number of lattice dynamic calculations were made using two H-bond force constants in a large superlattice unit cell where the protons are distributed in a random way, but subject to the Bernal-Fowler ice rules [ 14]. This empirical model treats the H 2 0 units as point masses. There are therefore only three force constants, the O - O - O bending force constant and the O - O bond stretching force constant, k, where k will have two values listed in Table 1. These are statistically distributed through the lattice as shown in Fig. 1. We have used the hypothesis that the two force constants may be related to the relative orientations of the two adjacent molecules involved. These are described in terms of the orientation of the dipoles associated with the molecules as shown in Fig. I. In ice Ih, there are arrangements A, B, C and D while, in ice Ic, only C and D are found. If the interaction were purely electrostatic, B and C would be strong interactions while A and D would be weak. Moreover, in a random lattice, these two types would appear in the required ratio of 2/3: 1/3 in ice. The above hypothesis for the existence of "two kinds of H-bond" in ice is based on the model used to interpret the inelastic incoherent neutron scattering (IINS) spectra [ 10]. The model, therefore, requires further tests to demonstrate its uniqueness. An obvious experiment is to measure the defect modes, due to isolated H atoms in a ice D20 lattice (and vice
J.C Li et al. / Physics Letters A 192 (1994) 295-300
297
Table 1 Force constants for ice Ih and their main effects Force constant
Value
Main effect of force constant
k KI /(2 g G~ G2 Ga
35.9 2.1 1.1 3.2 0.78 0.45 0.16
affects the internal modes affects the optic modes in the translational region same as for the K1 affects the internal modes affects the acoustic modes and the width of the librational modes level of low energy cutoff of the lihrational band affects the acoustic modes
O-H stretching strong O...H stretching weak O...H stretching H - O - H bending H...O-H bending O---H-O bending O"-O..-O bending
Units: stretching force constants, K and k, are expressed in eV/A 2, bending force constants, G and g, in eV/Rad 2.
(~ •
g/g
H~
%°/
~,.
J
tl/
~/t
H"
~'H~O
t~
/
H',-W, "H
\o/
H"
Fig. 2. Schematic illustration of the bending and stretching modes, 0~, 02 and o3, and the three defect modes, gl,/z2 and #3, for ice Ih. The eigenvectors which define the orientations of the vibrations are obtained from the lattice dynamic calculations.
(A) Fig. 1. The upper diagram shows a section of the ice I network; the strong H-bonds are indicated by the solid bonds and the weak bonds are shown in double lines. The lower diagram shows four possible orientations of molecule pairs in ice Ih. In ice Ic only the (C) and (D) types of molecule pairs are found.
versa). Because H atoms are bonded in the same way as D in D 2 0 ice, forming H D O molecules, but have a different mass, the translational properties of the lattice are broken by the H atoms. Hence the H atoms will vibrate with three strong localised modes, gl, #2 and #3 as shown in Fig. 2. These modes are almost decoupled from the band frequencies of the surrounding D 2 0 molecules and involve three degrees of freedom. Therefore, these three modes could pro-
vide direct information on the force constants associated with H-bond stretching, H - O - H (or H - O - D ) bending and H - O covalent stretching without the requirement need of a particular lattice dynamic model, or the uncertainties associated with dispersion effects. Thus the interpretation of such data is largely model independent. A series of experiments were, therefore, carried out on the (high energy transfer) spectrometers on the ISIS pulsed neutron source at the Rutherford-Appleton Laboratory. Because a direct geometry was used on this instrument, it was possible to make the measurements at relatively low Q values with a resolution about 1% (AE/E). Since H has a much larger incoherent cross section than D (its incoherent cross section is some twenty times greater than for D ) and a large amplitude of vibration, there is a considerable advantage in making measurements on H in D20 ice.
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J.C Li et al. / Physics Letters A 192 (1994) 295-300
After allowing for the ratio of the masses, some forty times greater sensitivity is obtained for modes involving only hydrogen motion, a separation with which no other techniques can compete. Hence a very small amount of H impurity can be easily detected in D20 ice. This is dramatically evident in our H contaminated D20 samples [ 10]. Clearly, H contamination is very important for inelastic incoherent neutron scattering for non-hydrogenous samples. The defect modes do not, on average, have any crystal orientational dependence. Therefore, the measurements can be made using polycrystalline ice with different H20 concentrations in order to investigate the concentration dependence of the defect modes. In order to have better resolution at the energy transfers in which we are interested, two incident energies, Ei = 170 and 500 meV were used. Fig. 3 shows the spectra measured with E i = 170 meV. When the data for all of the low angle detectors (angle from 4° to 7° from the sample position) are summed, the spectra show some structures in the #l
1
0
~ 60 100 1S0 200 ENERGY TRANSFER (meV)
flO ENER JyOOTRANSFI ~20(mev)140
Fig. 3. A series of measurements of the inelastic incoherent neutron scattering for polycrystalline D20 ice Ih with different concentrations of H20, using the HET spectrometer. The incident energy is 170 meV: The spectra show an asymmetrical peak associated with the/tt mode, which is due to the two possibilities for the H-bond strengths. These spectra are fitted with the two Gaussian functions, demonstrating clear evidence for the two components of#l, which are centered at 102.6 and 106.6 meV for the 2% H20 sample; 102.7 and 107.3 meV for the 10% H20 sample and 103.5 and 107.6 meV for the 30% H20 sample.
mode at ~ 105 meV. Because this mode is inactive in infrared and Raman spectroscopy, this phenomenon has not been observed previously [ 15 ]. The spectrum for < 0.5% H20 clearly shows two peaks at 103+0.5 and 107_+0.5 meV, although only four points are available for each peak and the data are only slightly greater than statistics. However, when the spectra for higher H20 concentrations (e.g. 2%, 10%, 20% and 30% H20) are examined, the peak at ~ 105 meV shows asymmetry and clearly consists of two components. By fitting two Gaussian functions, we were able to determine the separation of the two frequencies as shown in Fig. 3. The lower energy component, centred at ~ 103 meV, has the higher intensity and is about twice as intense as the higher energy component centred at ~ 107 meV. The separations of the two components are about 4 meV for all the concentrations used. When the spectrum from the higher Q (angle from 10° to 30 °) band detectors is examined, the data show that the relative intensities of these two components are not changed significantly. This implies that they are all the fundamental modes, because the intensities of combination modes vary with Q4e x p [ - 2 W ( Q ) ] and the intensities of the normal modes vary with Q 2 e x p [ - 2 W ( Q ) ] , where W(Q) is the Debye-Waller factor. By increasing the incident energy, the details of two high energy modes #2 and #3 can be examined. The spectrum (see Fig. 4) for the sample with < 0.5% H20 shows no indication of#2. Because the bending modes are always very broad, it is not surprising that it should be submerged in the background for this concentration. For the higher concentration spectra, the energy of this peak associated with the #2 mode at 185 meV is considerably lower than that for the ordinary H - O - H bending modes, u2 (200 meV). The difference between these two types of mode is that for the #2 mode, the H atom in the HDO molecule vibrates towards the D but both O and D are stationary. For the u2 mode, both H atoms in the H20 molecule are vibrating in phase towards each other (see Fig. 2). However, it is still unclear why, in common with H - O - H bending modes in general, this particular mode (#2) is so ill-defined and concentration independent, even for measurements at 20 K where proton movement (or exchange) will be almost eliminated. In fact, there are indications from neutron [ 16 ] and infrared and Raman [ 15,17 ] spectra that
J.C. Li et aL / Physics Letters A 192 (1994) 295-300
i v lV3
it3
•
i 0
0
.1
100
i
400 200 300 ENERGY TRANSFER (meV)
500
Fig. 4. IINS spectra for D 2 0 ice with different concentrations of H20 for incident energy of 500 meV. The spectra show peaks associated with the #2 mode at 185 meV (related to the bending force constant, g) and the #3 mode at 400 meV (related to the stretching force constant, k).
this peak actually gets narrower when the temperature increases. In going to the higher energy transfer, the peak due to the stretch mode,/z3, is very narrow ( ~ 10 meV) in comparison with the O - D stretch modes (containing ~ and P3) in normal ice ( ~ 5 0 meV). The measured width for the defect mode is indeed approximately the same as the resolution at this energy transfer, indicating that the resolution corrected peak would be very narrow and there is no dispersion over large q values across the first Brillouin zone in the reciprocal space. Using the two H-bond model [10], the defect modes can be calculated by introducing H atoms in the D20 lattice. Because we are interested only in the three defect modes, the large super-lattice which was used in our lattice dynamic model can be avoided by using an unit cell with four oxygens (space group for O only is P 6 / m m c ) and 12 hydrogens with mixtures of the strong H-bond (SHB) and weak H-bond (WHB). The force constants used are shown in Table 1. By putting the H atom on different site H-bonds, the phonon dispersion curves are calculated across the first Brillouin zone (BZ) and the integrated phonon
299
density of states are calculated (the details will be published shortly). Fig. 5 shows a few such examples of calculated PDOS. As can be seen, the phonon band at ~ 300 meV represents the stretching modes u~ and z,3, while the peak at ~ 150 mcV is due to bending mode of v2 for D20 molecules. When one H atom is introduced in the D20 lattice, three more frequencies are added in the spectrum at ~ 105, ~ 185 and ~ 400 meV, associated with the three defect modes,/~,/~2 and/~3 (see Fig. 5b). The two frequencies at higher energy transfers,/z2 and/z3, are of very flat dispersion curves and were very little influenced by the value of the H-bond force constant (i.e. K). This is because the intramolecular force constants (bending, g, and stretching, k) are an order of magnitude higher than the intermolecular force constants (such as H-bond force constants K~ and K2). Hence the intramolecular modes are decoupled from the lattice modes. The lowest frequency at ~ 105 meV has a slight dispersion and is very sensitive to the strength of the Hbond interaction. If an H atom is introduced on the 70
''''1''''1''~'1''''1
'''
60 100%H20 30%
~30
20
01.0 0
100 200 300 400 ENERGY TRANSFER (meV)
500
Fig. 5. The calculated phonon density of states, G(co) for different sites of the H atoms. The inserted cell shows the possible positions (Pb P2 and P3) of the H defects. The H atoms arc indicated by the open circles and the D are solid circles. The dashed lines indicate the weak H-bonds (K2). (a) For a D 2 0 lattice without H defect; (b) H atom is introduced on the strong H-bond (P~ position); (c) two H atoms are introduced, one on strong Hbond (P~) and another on weak H-bond (P2), but on two different water molecules; (d) three H atoms are introduced (PI, P2 and P3), two o f them on the same water molecule.
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J.C. Li et al. / Physics Letters A 192 (1994) 295-300
SHB (see Fig. 5c), the vibrational frequency of the #~ is about 4 meV higher than the frequency of the mode for a H atom on the WHB. This is very close to what was observed in the n e u t r o n spectrum. If two H atoms are introduced in the cell, one on a SHB and another on a WHB, but on different molecules, the frequencies of the/L 2 a n d the/13 for both H atoms are identical, but the/tl frequencies are the same as if they are introduced separately. If the two H atoms are on the same molecule (i.e. form an H 2 0 molecule), one is a SHB a n d the other is on a WHB, the resulting/t~ frequencies are similar, but the #2 value is the same as for the ordinary b e n d i n g mode u2 at 200 meV. This is about 15 meV higher than the/z2 mode for the H D O molecule (see Fig. 5d), a n d again this agrees with the experimental observation. In this Letter, we have reported the first n e u t r o n spectroscopic measurements of H defect modes in ice Ih. The sharp peaks for these point defect modes are isotopically decoupled from the modes of neighbouring vibrators, and hence the defect modes provide direct i n f o r m a t i o n about interactions associated with the defects. The two c o m p o n e n t s in t h e / t l can be considered as evidence for the "two strengths of Hbonds" which have not been seen because of the inactivity of the modes in IR a n d R a m a n spectra. The implications of this discovery have been discussed in an earlier paper [ 1 ] and are of considerable relevance to water science. The authors would like to thank the Science and Engineering Research Council ( U K ) for financial
support and the Rutherford-Appleton Laboratory for the use of neutron facilities.
References [ 1] J.C. Li and D.K. Ross, Nature 365 (1993) 327. [2 ] R.A. Hone, Water and aqueoussolutions (Wiley,New York, 1972). [3] H.H.G. Jellinek, Water structure at the water-polymer interface (Plenum, New York, 1972). [4] J.A. Pople, Proc. R. Soc. A 205 (1951) 163. [5] H.S. Frank and W.-Y. Wen, Disc. Faraday Soc. 24 (1957) 133; H.S. Frank, Proc. R. Soc. A 247 (1958) 481. [6] H.E. Stanleyand J. Teixeia,J. Chem. Phys. 73 (1980) 3404. [7] M.D. Morse and S.A. Rice, J. Chem. Phys. 76 (1982) 560. [8] J.L. Finney, J.E. Quinn and J.O. Baum, in: Water science review, Vol. 1, ed. F. Franks (Cambridge Univ. Press, Cambridge, 1985) p. 93. [9]J.-C. Li, J.D. Londono, D.K. Ross, J.L. Finney, J. Tomkinson and W.F. Sherman, J. Chem. Phys. 94 ( 1991 ) 6770. [ 10] J.-C. Li and D.K. Ross, in: Physics and chemistry of ice, eds. N. Maeno and T. Hondoh (Hokkaida Univ. Press, Sapporo, 1992) p. 27. [ 11 ] A.I. Kolesnikov, J.-C. Li, D.K. Ross, V.V. Sinitsin, O.I. Barkalov, E.L. Bokhenkov and E.G. Ponyatovskii, Phys. Lett. A 168 (1992) 308. [ 12] D.D. Klug and E.J. Whalley,J. Glaciology85 ( 1978) 55. [ 13 ] M. Marchi, J.S. Tse and U Klein, J. Chem. Phys. 60 ( 1986) 2414. [ 14] J.D. Bernal and R.H. Fowler,J. Chem. Phys. 1 ( 1933) 515. [15] J.P. Devlin, Intern. Rev. Phys. Chem. 9 (1990) 29. [16] S.H. Chen, K. Toukan, C.-K. Loong, D.L. Price and J. Teixeira, Phys. Rev. Lett. 53 (1984) 1360. [ 17] Y. Marechal, J. Chem. Phys. 95 ( 1991 ) 5565.
ELSEVIER
Physics Letters A 192 (1994) 295-300
Further evidence for the existence of two kinds of H-bonds in ice Ih J . C . L i a, S . M . B e n n i n g t o n
b, D . K . ROSS a
Department of Pure and Applied Physics, University of Salford, Salford, M6 4WT, UK b ISIS Facilities, Rutherford-Appleton Laboratory, Didcot, Oxon, OXI I, OQX, UK
Received 15 April 1994;accepted for publication 28 June 1994 Communicated by A. Lagendijk
Abstract
The hydrogen defects in the host D20 lattice of ice have been studied using a high resolution inelastic neutron spectrometer. Three main peaks at ~ 105, ~ 185 and ~ 408 meV, associated with three localised modes (#b #2 and #3) oftbe H-defect centre, have been observed in the inelastic neutron spectra. The peak associated with the/t, mode at ~ 105 meV has an asymmetrical shape from which two Gaussian components centred at 103 _ 0.5 and 107 + 0.5 meV for different concentrations of HDO can be resolved, confirming the existence of two different strengths for the H-bonds. A lattice dynamic calculation for the H defects in a perfect ice Ih lattice, based on the two H-hood force constants model of Li and Ross [Nature 365 (1993) 327 ], shows the three modes in very similar positions to those observed in the measured data and predicts the splitting of the g~ peak into modes at 103 and 107 meV. Hydrogen bonding is one o f the most important and intriguing interactions, which dominate our daily lives, and scientists across different disciplines for long have endeavoured to understand the complex nature o f water and o f other H-bonded systems. There are not as yet, however, completely acceptable explanations o f some properties of water, often referred to as its "anomalous" properties. The large bond energy and the asymmetrical geometry o f the H-bond, combined with the fact that the electrons in the sp 3 orbitals of the oxygen atoms can rehybridise to respond to the relative configurations of adjacent molecules, give rise to a large number o f abnormal properties o f water/ice which cannot be explained by the ordinary rules o f physics and chemistry. As a consequence, a large number o f models have been proposed [2,3 ] in attempts to interpret some o f these properties o f water, such as the high heat capacity [4], the extended possible supercooling o f the liquid state [ 5 ], and the large density and entropy fluctuations [ 6 ].
Meanwhile, a large number of H-bond potentials have also been proposed. Some o f these are based on ab initio quantum mechanical calculations and some are very arbitrary. Some are good at reproducing the structure o f water and some are good at reproducing its thermodynamic properties [ 7,8 ]. The recent development o f the high flux pulsed neutron source for ISIS and o f the high resolution inelastic neutron scattering instruments at the Rutherford-Appleton Laboratory, have enabled us to examine the dynamic properties of different exotic high pressure (up to 40 kbar) forms o f ice by measuring their inelastic incoherent spectra [ 9-11 ]. Also, because neutron scattering gives a measure o f all the vibrational modes with equal sensitivity, the data can easily be converted to the amplitude-weighted phonon density o f states ( P D O S ) , particularly for measurements carried out at very low temperatures. Lattice dynamic calculations o f this function will therefore provide a more direct test o f existing potentials. In
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J.c. Li et aL / Physics Letters A 192 (I 994) 295-300
order to reproduce the two measured molecular optic bands centred at 28 and 37 meV for ice I [9], two empirical force constants have been proposed [ 10 ]. The two types of H-bonds are related to two different types of local configurations. The resulting dispersion curves and PDOS calculated for a large superlattice constructed so as to represent the proton disordering in the local environments give good agreement with the experimental data [10]. The results also agree well when this model was applied to other forms of ice, such as ice VIII [ 11 ] which due to the proton ordering in this structure, there is only one type of local proton arrangement, corresponding to the weak H-bonds. The lack of any dependence of the spectrum of ice Ih on crystal orientations rules out existing dynamic models for this phase which assume the existence of different H-bond force constants in different crystal directions. Other explanations of the two molecular optic bands at 27 and 36 meV are based on TO (transverse optic) and LO (longitudinal optic) splitting [12] (as is found in ionic crystals such as sodium chloride, where the stiffer force constant corresponds to the LO vibrations). However, because the LO modes have higher frequencies and should have less intensity than the TO modes, the ratio of the TO to the LO modes should be 2: 1. However, the data show the opposite ratio. Also this splitting exists only at q = 0 and not near the zone boundary, the region that dominates the IINS. In attempts to reproduce these IINS spectra, using lattice dynamic (LD) calculations, we have tested some of existing potentials, but the results of these calculations, as expected, are far from satisfactory because most molecular dynamics calculations using these potentials do not yield the two optic bands in the translational region [ 13 ], since these potentials always consist of a repulsive part (often Lennard-Jones or exponential types) plus electrostatic terms due to charges on oxygens and protons [7]. However, the contribution from the electrostatic terms could not produce the large difference in the H-bond force constant (the maximum effect would be 15% among different dipole-dipole arrangements). After looking at the problem from several points of view, we are forced to the conclusion that the existence of two molecular optic bands in the spectrum leads inevitably to the conclusion that there are two strengths of H-bonds randomly distrib-
uted in every direction [ 1 ]. However, the true cause of the large difference in the force constants between the strong and weak is still unresolved. The large difference of the H-bond strength force constants used for the lattice dynamic calculations cannot be explained by any classical considerations. Therefore, the true cause must lay on the quantum mechanical level. Considering the H-bond are combination (or hybridisation) of 2s and 2p orbital from O atom, the nature of the H-bond is very sensitive to the surroundings. We assume that the different electronic configurations alter the interaction strengths through cooperative effects. The two well-separated molecular optic bands in the spectrum imply that there are two strengths of Hbonds. A simple calculation using the relationship 0)-~ k v / ~ (where 0) is the vibrational frequency and k is the force constant) gives a ratio of the two Hbond force constants as k I :k2= (0)1:0)2) 2= 272:372= 1: 1.9. Based on the above knowledge, a number of lattice dynamic calculations were made using two H-bond force constants in a large superlattice unit cell where the protons are distributed in a random way, but subject to the Bernal-Fowler ice rules [ 14]. This empirical model treats the H 2 0 units as point masses. There are therefore only three force constants, the O - O - O bending force constant and the O - O bond stretching force constant, k, where k will have two values listed in Table 1. These are statistically distributed through the lattice as shown in Fig. 1. We have used the hypothesis that the two force constants may be related to the relative orientations of the two adjacent molecules involved. These are described in terms of the orientation of the dipoles associated with the molecules as shown in Fig. I. In ice Ih, there are arrangements A, B, C and D while, in ice Ic, only C and D are found. If the interaction were purely electrostatic, B and C would be strong interactions while A and D would be weak. Moreover, in a random lattice, these two types would appear in the required ratio of 2/3: 1/3 in ice. The above hypothesis for the existence of "two kinds of H-bond" in ice is based on the model used to interpret the inelastic incoherent neutron scattering (IINS) spectra [ 10]. The model, therefore, requires further tests to demonstrate its uniqueness. An obvious experiment is to measure the defect modes, due to isolated H atoms in a ice D20 lattice (and vice
J.C Li et al. / Physics Letters A 192 (1994) 295-300
297
Table 1 Force constants for ice Ih and their main effects Force constant
Value
Main effect of force constant
k KI /(2 g G~ G2 Ga
35.9 2.1 1.1 3.2 0.78 0.45 0.16
affects the internal modes affects the optic modes in the translational region same as for the K1 affects the internal modes affects the acoustic modes and the width of the librational modes level of low energy cutoff of the lihrational band affects the acoustic modes
O-H stretching strong O...H stretching weak O...H stretching H - O - H bending H...O-H bending O---H-O bending O"-O..-O bending
Units: stretching force constants, K and k, are expressed in eV/A 2, bending force constants, G and g, in eV/Rad 2.
(~ •
g/g
H~
%°/
~,.
J
tl/
~/t
H"
~'H~O
t~
/
H',-W, "H
\o/
H"
Fig. 2. Schematic illustration of the bending and stretching modes, 0~, 02 and o3, and the three defect modes, gl,/z2 and #3, for ice Ih. The eigenvectors which define the orientations of the vibrations are obtained from the lattice dynamic calculations.
(A) Fig. 1. The upper diagram shows a section of the ice I network; the strong H-bonds are indicated by the solid bonds and the weak bonds are shown in double lines. The lower diagram shows four possible orientations of molecule pairs in ice Ih. In ice Ic only the (C) and (D) types of molecule pairs are found.
versa). Because H atoms are bonded in the same way as D in D 2 0 ice, forming H D O molecules, but have a different mass, the translational properties of the lattice are broken by the H atoms. Hence the H atoms will vibrate with three strong localised modes, gl, #2 and #3 as shown in Fig. 2. These modes are almost decoupled from the band frequencies of the surrounding D 2 0 molecules and involve three degrees of freedom. Therefore, these three modes could pro-
vide direct information on the force constants associated with H-bond stretching, H - O - H (or H - O - D ) bending and H - O covalent stretching without the requirement need of a particular lattice dynamic model, or the uncertainties associated with dispersion effects. Thus the interpretation of such data is largely model independent. A series of experiments were, therefore, carried out on the (high energy transfer) spectrometers on the ISIS pulsed neutron source at the Rutherford-Appleton Laboratory. Because a direct geometry was used on this instrument, it was possible to make the measurements at relatively low Q values with a resolution about 1% (AE/E). Since H has a much larger incoherent cross section than D (its incoherent cross section is some twenty times greater than for D ) and a large amplitude of vibration, there is a considerable advantage in making measurements on H in D20 ice.
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J.C Li et al. / Physics Letters A 192 (1994) 295-300
After allowing for the ratio of the masses, some forty times greater sensitivity is obtained for modes involving only hydrogen motion, a separation with which no other techniques can compete. Hence a very small amount of H impurity can be easily detected in D20 ice. This is dramatically evident in our H contaminated D20 samples [ 10]. Clearly, H contamination is very important for inelastic incoherent neutron scattering for non-hydrogenous samples. The defect modes do not, on average, have any crystal orientational dependence. Therefore, the measurements can be made using polycrystalline ice with different H20 concentrations in order to investigate the concentration dependence of the defect modes. In order to have better resolution at the energy transfers in which we are interested, two incident energies, Ei = 170 and 500 meV were used. Fig. 3 shows the spectra measured with E i = 170 meV. When the data for all of the low angle detectors (angle from 4° to 7° from the sample position) are summed, the spectra show some structures in the #l
1
0
~ 60 100 1S0 200 ENERGY TRANSFER (meV)
flO ENER JyOOTRANSFI ~20(mev)140
Fig. 3. A series of measurements of the inelastic incoherent neutron scattering for polycrystalline D20 ice Ih with different concentrations of H20, using the HET spectrometer. The incident energy is 170 meV: The spectra show an asymmetrical peak associated with the/tt mode, which is due to the two possibilities for the H-bond strengths. These spectra are fitted with the two Gaussian functions, demonstrating clear evidence for the two components of#l, which are centered at 102.6 and 106.6 meV for the 2% H20 sample; 102.7 and 107.3 meV for the 10% H20 sample and 103.5 and 107.6 meV for the 30% H20 sample.
mode at ~ 105 meV. Because this mode is inactive in infrared and Raman spectroscopy, this phenomenon has not been observed previously [ 15 ]. The spectrum for < 0.5% H20 clearly shows two peaks at 103+0.5 and 107_+0.5 meV, although only four points are available for each peak and the data are only slightly greater than statistics. However, when the spectra for higher H20 concentrations (e.g. 2%, 10%, 20% and 30% H20) are examined, the peak at ~ 105 meV shows asymmetry and clearly consists of two components. By fitting two Gaussian functions, we were able to determine the separation of the two frequencies as shown in Fig. 3. The lower energy component, centred at ~ 103 meV, has the higher intensity and is about twice as intense as the higher energy component centred at ~ 107 meV. The separations of the two components are about 4 meV for all the concentrations used. When the spectrum from the higher Q (angle from 10° to 30 °) band detectors is examined, the data show that the relative intensities of these two components are not changed significantly. This implies that they are all the fundamental modes, because the intensities of combination modes vary with Q4e x p [ - 2 W ( Q ) ] and the intensities of the normal modes vary with Q 2 e x p [ - 2 W ( Q ) ] , where W(Q) is the Debye-Waller factor. By increasing the incident energy, the details of two high energy modes #2 and #3 can be examined. The spectrum (see Fig. 4) for the sample with < 0.5% H20 shows no indication of#2. Because the bending modes are always very broad, it is not surprising that it should be submerged in the background for this concentration. For the higher concentration spectra, the energy of this peak associated with the #2 mode at 185 meV is considerably lower than that for the ordinary H - O - H bending modes, u2 (200 meV). The difference between these two types of mode is that for the #2 mode, the H atom in the HDO molecule vibrates towards the D but both O and D are stationary. For the u2 mode, both H atoms in the H20 molecule are vibrating in phase towards each other (see Fig. 2). However, it is still unclear why, in common with H - O - H bending modes in general, this particular mode (#2) is so ill-defined and concentration independent, even for measurements at 20 K where proton movement (or exchange) will be almost eliminated. In fact, there are indications from neutron [ 16 ] and infrared and Raman [ 15,17 ] spectra that
J.C. Li et aL / Physics Letters A 192 (1994) 295-300
i v lV3
it3
•
i 0
0
.1
100
i
400 200 300 ENERGY TRANSFER (meV)
500
Fig. 4. IINS spectra for D 2 0 ice with different concentrations of H20 for incident energy of 500 meV. The spectra show peaks associated with the #2 mode at 185 meV (related to the bending force constant, g) and the #3 mode at 400 meV (related to the stretching force constant, k).
this peak actually gets narrower when the temperature increases. In going to the higher energy transfer, the peak due to the stretch mode,/z3, is very narrow ( ~ 10 meV) in comparison with the O - D stretch modes (containing ~ and P3) in normal ice ( ~ 5 0 meV). The measured width for the defect mode is indeed approximately the same as the resolution at this energy transfer, indicating that the resolution corrected peak would be very narrow and there is no dispersion over large q values across the first Brillouin zone in the reciprocal space. Using the two H-bond model [10], the defect modes can be calculated by introducing H atoms in the D20 lattice. Because we are interested only in the three defect modes, the large super-lattice which was used in our lattice dynamic model can be avoided by using an unit cell with four oxygens (space group for O only is P 6 / m m c ) and 12 hydrogens with mixtures of the strong H-bond (SHB) and weak H-bond (WHB). The force constants used are shown in Table 1. By putting the H atom on different site H-bonds, the phonon dispersion curves are calculated across the first Brillouin zone (BZ) and the integrated phonon
299
density of states are calculated (the details will be published shortly). Fig. 5 shows a few such examples of calculated PDOS. As can be seen, the phonon band at ~ 300 meV represents the stretching modes u~ and z,3, while the peak at ~ 150 mcV is due to bending mode of v2 for D20 molecules. When one H atom is introduced in the D20 lattice, three more frequencies are added in the spectrum at ~ 105, ~ 185 and ~ 400 meV, associated with the three defect modes,/~,/~2 and/~3 (see Fig. 5b). The two frequencies at higher energy transfers,/z2 and/z3, are of very flat dispersion curves and were very little influenced by the value of the H-bond force constant (i.e. K). This is because the intramolecular force constants (bending, g, and stretching, k) are an order of magnitude higher than the intermolecular force constants (such as H-bond force constants K~ and K2). Hence the intramolecular modes are decoupled from the lattice modes. The lowest frequency at ~ 105 meV has a slight dispersion and is very sensitive to the strength of the Hbond interaction. If an H atom is introduced on the 70
''''1''''1''~'1''''1
'''
60 100%H20 30%
~30
20
01.0 0
100 200 300 400 ENERGY TRANSFER (meV)
500
Fig. 5. The calculated phonon density of states, G(co) for different sites of the H atoms. The inserted cell shows the possible positions (Pb P2 and P3) of the H defects. The H atoms arc indicated by the open circles and the D are solid circles. The dashed lines indicate the weak H-bonds (K2). (a) For a D 2 0 lattice without H defect; (b) H atom is introduced on the strong H-bond (P~ position); (c) two H atoms are introduced, one on strong Hbond (P~) and another on weak H-bond (P2), but on two different water molecules; (d) three H atoms are introduced (PI, P2 and P3), two o f them on the same water molecule.
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J.C. Li et al. / Physics Letters A 192 (1994) 295-300
SHB (see Fig. 5c), the vibrational frequency of the #~ is about 4 meV higher than the frequency of the mode for a H atom on the WHB. This is very close to what was observed in the n e u t r o n spectrum. If two H atoms are introduced in the cell, one on a SHB and another on a WHB, but on different molecules, the frequencies of the/L 2 a n d the/13 for both H atoms are identical, but the/tl frequencies are the same as if they are introduced separately. If the two H atoms are on the same molecule (i.e. form an H 2 0 molecule), one is a SHB a n d the other is on a WHB, the resulting/t~ frequencies are similar, but the #2 value is the same as for the ordinary b e n d i n g mode u2 at 200 meV. This is about 15 meV higher than the/z2 mode for the H D O molecule (see Fig. 5d), a n d again this agrees with the experimental observation. In this Letter, we have reported the first n e u t r o n spectroscopic measurements of H defect modes in ice Ih. The sharp peaks for these point defect modes are isotopically decoupled from the modes of neighbouring vibrators, and hence the defect modes provide direct i n f o r m a t i o n about interactions associated with the defects. The two c o m p o n e n t s in t h e / t l can be considered as evidence for the "two strengths of Hbonds" which have not been seen because of the inactivity of the modes in IR a n d R a m a n spectra. The implications of this discovery have been discussed in an earlier paper [ 1 ] and are of considerable relevance to water science. The authors would like to thank the Science and Engineering Research Council ( U K ) for financial
support and the Rutherford-Appleton Laboratory for the use of neutron facilities.
References [ 1] J.C. Li and D.K. Ross, Nature 365 (1993) 327. [2 ] R.A. Hone, Water and aqueoussolutions (Wiley,New York, 1972). [3] H.H.G. Jellinek, Water structure at the water-polymer interface (Plenum, New York, 1972). [4] J.A. Pople, Proc. R. Soc. A 205 (1951) 163. [5] H.S. Frank and W.-Y. Wen, Disc. Faraday Soc. 24 (1957) 133; H.S. Frank, Proc. R. Soc. A 247 (1958) 481. [6] H.E. Stanleyand J. Teixeia,J. Chem. Phys. 73 (1980) 3404. [7] M.D. Morse and S.A. Rice, J. Chem. Phys. 76 (1982) 560. [8] J.L. Finney, J.E. Quinn and J.O. Baum, in: Water science review, Vol. 1, ed. F. Franks (Cambridge Univ. Press, Cambridge, 1985) p. 93. [9]J.-C. Li, J.D. Londono, D.K. Ross, J.L. Finney, J. Tomkinson and W.F. Sherman, J. Chem. Phys. 94 ( 1991 ) 6770. [ 10] J.-C. Li and D.K. Ross, in: Physics and chemistry of ice, eds. N. Maeno and T. Hondoh (Hokkaida Univ. Press, Sapporo, 1992) p. 27. [ 11 ] A.I. Kolesnikov, J.-C. Li, D.K. Ross, V.V. Sinitsin, O.I. Barkalov, E.L. Bokhenkov and E.G. Ponyatovskii, Phys. Lett. A 168 (1992) 308. [ 12] D.D. Klug and E.J. Whalley,J. Glaciology85 ( 1978) 55. [ 13 ] M. Marchi, J.S. Tse and U Klein, J. Chem. Phys. 60 ( 1986) 2414. [ 14] J.D. Bernal and R.H. Fowler,J. Chem. Phys. 1 ( 1933) 515. [15] J.P. Devlin, Intern. Rev. Phys. Chem. 9 (1990) 29. [16] S.H. Chen, K. Toukan, C.-K. Loong, D.L. Price and J. Teixeira, Phys. Rev. Lett. 53 (1984) 1360. [ 17] Y. Marechal, J. Chem. Phys. 95 ( 1991 ) 5565.