Prediction of FIR absorption for liquid water with hot H2Odipoles as the cause of the translational band

摘要

Mendeleev Communications Electronic Version, Issue 2. 1997 (pp. 47ndash;86) Prediction of FIR absorption for liquid water with hot HO2 dipoles as the cause of the translational band Vladimir I. Gaiduk* and Vladimir V. Gaiduk Institute of Radio Engineering and Electronics, Russian Academy of Sciences, 141120 Fryazino, Moscow Region, Russian Federation. E-mail: vig169@ire216.msk.su The analytic theory of wideband (0ndash;1000 cmndash;1 region) dielectric spectra is elaborated for liquid water; it is assumed that H2O molecules librate/rotate in a rectangular potential box with flat edges and the librational/translational absorption bands at ca. 700 and 200 cmndash;1 are due to slow and fast reorienting particles, respectively. The study1ndash;3 of wideband (0ndash;THz) dielectric spectra of liquid water yields valuable information on water structure and the timescales of molecular events.The imaginary peak e''D of the complex permittivity e(w) = e' ndash; ie'' is found in the microwave region while in the FIR spectral region are observed the librational (near 700 cmndash;1) and the so-called translational (near 200cmndash;1) absorption bands a(w). Here w is the angular frequency of radiation; the frequency n = w/(2pc) is measured in cmndash;1, c is the velocity of light.The existence of two absorption peaks is evidence of some bimodalality. Its nature has not yet been established since the molecular theory of this band has not been elaborated. This problem is the main purpose of this communication. The dielectric spectra of water have been previously described1ndash;3 using analytical theory elaborated for the combination of the confined rotator (CR) and extended diffusion (ED) models.In each model the energy H of the particles is varied in the interval 0,7 and different types of molecular rotation during lifetime t are considered since two independent molecular fractions (L and R) are introduced, the proportion r- of R-particles being the fitting parameter (the proportion rL of the other fraction is equal to 1 ndash; r).The CR model describes the libration of water molecules in a H-bond network. The L-molecules are found in an infinitely deep rectangular potential well. The dielectric response of another (R) fraction is described by the ED model by the consideration of free rotation of dipoles1ndash;3 (in ref. 2 ndash; by consideration of their rotation in a conservative potential well with the cosine squared profile).In this communication we attempt to prove the validity of the following suggestion:4 one can describe the two-humped absorption/low frequency Debye spectrum of liquid water with use of an adequate (lsquo;hatrsquo;) intermolecular potential profile, i.e. without employing two independently introduced molecular ensembles.Besides, we suggest a simple calculation that is able to predict, in its main features, the orientational spectrum of liquid water in the frequency region 0,1000 cmndash;1. In particular, the influence of temperature T on this spectrum was studied. We also show that it is possible to calculate the critical temperature Tcr of liquid water, if one regards Tcr to be the point where the employed model of molecular rotation ceases to exist.The so called hybrid model (HM)5,6 was applied to liquid water that was previously used for the description of 0ndash;THz spectra of simple nonassociated liquids, such as CH3F. In the HM the potential profile has the form of two rectangular potential boxes with the depth U0 and angular width 2b. One box is turned relative to the other by the angle p (Figure 1).The two subensembles (L and R) appear inherently in this model. L-molecules, with energy H U0, perform librations while hot R-particles, with energy H U0, perform free rotation, their proportion r = r (U0,b) being a function of the wellrsquo;s geometry. The low frequency (Debye) spectrum of a liquid appears to be due to equalization of the induced population of dipoles via their transition, under the influence of strong collisions, from one well to another.The mean rotational frequency of L- and R-particles determine the positions nL and nR of the two absorption peaks observed in water. The formulae, based on the linear dielectric response theory,6 for the absorption a(w) and complex permittivity e* = e' + ie'' (* is the complex conjugation symbol).a and e* are related as follows where is the refraction index. The dipole moment m of a molecule in a liquid (water) is related to the moment m0 of an isolated molecule: km is the correction coefficient (km= ca. 1.12) and nyen; is the optical refraction index (nyen; 2 = ca. 1.7). The dielectric response, related to the U potential, is anisotropic and is calculated for the local (during lifetime t) configuration. This response is characterized by the spectral functions (SF) Kverbar;verbar;(z) and K^(z). These correspond to orientations of the radiation amplitude Em along and across the symmetry axis of the potential U.The argument z of SFs is the dimensionless complex frequency z, I is the moment of inertia of a molecule that for simplicity we regard to be a linear one, kB is the Boltzmann constant.In accordance with usual approach we define I as follows: 1/I = 1/2(1/Ix + 1 / Iy), where Ix and Iy are the moments of inertia of an asymmetrical top molecule about the principal axis perpendicular to the dipole moment vector. The influence of the molecule form is very important for the description of the discrete rotational spectrum in the water vapour.In the condensed (liquid) state, in which the potential well is deep, the main spectral peculiarities of water are related to the influence of the intermolecular potential on molecular rotation, viz. to the U0 /(kBT) A B ndash;b 0 b p A B C C D D E E 0 90 180 270 U0 /(kBT) (a) (b) Figure 1 The geometric scheme pertaining to the double well rectangular potential employed in the hybrid model.Rectangular (a) and polar (b) coordinate systems. For the spatial configuration Figure (b) transforms into two cones joined by the spherical surface. * c � ( )Im *( ) Re *( ) --------------------------------- 2p n ------------- , = = '' (1) w e n e e a n Re *() ordm; e 0k n2yen; 2 +( ) 3 ---------------------- = (2) m m m z x iy; y + , � I 2kBT( ) � 1 2 � , = = = (3) lsquo;h t lsquo;hMendeleev Communications Electronic Version, Issue 2. 1997 (pp. 47ndash;86) transformation of almost free rotors into librators. Correspondingly, in water the absorption peak considerably shifts to higher frequencies compared to the position of maximum absorption in vapour. The main object of our theoretical study is the complex orientational susceptibility c*.An isotropic polar medium and isothermal collision model (see ref. 6) is given by The averaged (over all orientations) SF L(z) is given by This SF is applied to an isotropic polar medium. The general approach for the calculation of this L(z) SF is described in ref. 6. For small-amplitude (b p/4) librations of dipoles in a cone the and SFs were determined in ref. 6; the SF of free rotors was determined in ref. 6 (here the designations are slightly changed compared to ref. 6). Analogous expressions are derived for the SF of the hybrid model (the theory of the HM was elaborated in collaboration with B. M. Tseitlin): where NA is the Avogadro constant, N and r are the number and mass density, M is the molecular mass. The Kirkwood correlation factor g is calculated with the experimental values of es, nyen; and G: The complex permittivity e* is determined by c* and nyen; 2 as the solution of the quadratic equation The proportion r of R-particles is given by The experiential low frequency e*(n) spectrum is taken from ref. 7. In particular, the static permittivity es and the relaxation time tD are estimated from the empiric formulae The parameters b, t, u, km and employed molecular constants are given in Table 1.Having only a few free parameters, we tried to fit the following features of the observed7ndash;9 dielectric spectra. (a) The position nL of the libration peak. The latter is mainly determined by the libratiob (b is near 20deg;). The value amax of the absorption maximum is corrected by the value of km. (b) The two-humped absorption curve. This curve appears in theory when the contributions of the free rotors and of the librators are commensurable in the vicinity of the frequency nR.The correct result is obtained, if the potential U is rather deep (u = ca. 5.5). (c) The coincidence of the calculated and theoretical positions nD of the loss maximum eD''. For chosen b and u the calculated nD value increases when the lifetime t rises.The calculated and observed spectra qualitatively agree (Figure 2). We interpret the fitted parameters of the model as follows. (A) The libration-band frequency nL is close to the mean frequency of librations,6 since the relation approximately holds. (B) One may estimate the lifetime t from the formula6 that follows from the Debye rotational diffusion theory.10 The lifetime t substantially decreases when T rises.(C) In the interval 1, 50 deg;C the well-depth U0 is nearly constant (U0 = ca. 3kcal molndash;1) and is commensurable with the energy of H-bond (about 5 kcal molndash;1). The constancy of U0 corresponds to the known picture of a rigid water structure. (D) The rise of T from 1 to 50 deg;C is accompanied by the weakening of water structure since the proportion r of hot particles doubles and the number mL of librations during the lifetime t decreases 3 times (see Table 1); mL = yc /(2y), where Consequently, the rotational mobility of some H2O molecules, viz.of R-particles, in this T-interval significantly increases although the water structure is relatively hard. The drawbacks of the HM probably result from the roughness of the chosen intermolecular potential profile, i.e.(1) Too slow transition of the absorption curve a(n) to the transparency near 1000 cmndash;1 cf. solid/dashed curves in Figure 2(a). (2) The calculated absorption minimum between 200 and 300 cmndash;1 is deeper than that observed experimentally, cf. solid and dashed curves for T = 27 deg;C, Figure 2(a). * x( ) gGzL z ( ) gx iyL z ( ) 1 gxz +( ) + 1 ndash; = (4) c L z( ) 2 3 � ( )K^ 1 3 � ( )Kverbar;verbar; K.+ + = o (5) K^ Kverbar;verbar; K o Kverbar;verbar; z( ) 24 p5 ------C u,( ) 4 sin()2 t3e t2 ndash; t2 z p � ( )2 ndash; ------------------------------ t, d 0 u ograve; = (6) b b b b K^ z( ) 192 p5 ---------C u,( ) 4 cos()2 t3e t2 ndash; t2 2 z p � ( )2 ndash; --------------------------------- t, d 0 u ograve; = (7) b b b b K z( ) 6e u ndash; C u,( ) t3e t2 ndash; t2 z2 ndash; -------------- t, d 0 yen; ograve; = o (8) b u U0 kBT ---------, C u,( ) 1 e u ndash; 1 ndash;() cos + 1 ndash; , = = b b G 2N 3kBT ------------,N NA M ---------- , = = (9) m r g s n2yen; ndash;( ) 2 s n2 yen; +() 12� pG s .= (10) e e e * n2yen; ndash;( ) 2 * n2yen; +() 12 ndash; p * * 0. = (11) e e e c r e u ndash; C u,( ). = (12) b s T( ) 77!66 103!3 , D T( )ndash;2p D T( ) 1 ndash; , = = (13a) e t g q D 20!27 146!5 , + 1 300 T.� ndash; = = (13b) q q g L 3 2 � ( ) 4 c 1 ndash; raquo; (14a) n hb T( ) 1 2 � ( ) 2 T ( ) D T( ) , = (14b) t b t a Constants: s = 2.9 Aring;, I = 1.483times;10ndash;40 g cm2, m0 = 1.84 D. Table 1 Molecular constantsa of liquid water, the fitted and other parameters of the hybrid model. Parameters of the HM Other quantities T/deg;C b/degree u U0/kcalmolndash;1 t/ps from (14b) km r y r/gcmndash;3 mL g l /Aring; Fitted from (14a) 1 17.98 20.84 5.6 3.05 0.851 1.161 0.071 0.052 1.000 16.0 2.24 0.205 27 19.93 21.14 5.7 3.04 0.475 1.118 0.093 0.089 0.999 8.2 2.35 0.209 50 22.09 22.10 4.6 3.0 0.366 1.097 0.122 0.112 0.988 5.7 2.38 0.218 yc C u,( ) p 2 -------erf u( ) ue u ndash; ndash;egrave; oslash; aelig; ouml; p 2 ------- e u ndash; + sin .= (15) b bMendeleev Communications Electronic Version, Issue 2. 1997 (pp. 47ndash;86) (3) For the range n Icirc; 10, 100cmndash;1 the calculated submillimeter absorption is less than the observed one.This property of the HM probably confirms the idea7 that the second Debye relaxation region exists in water. However, an alternative physical mechanism may be responsible for this lsquo;excessrsquo; absorption. (4) The theory of an isotopic effect (the coincidence of the translational peak nR in ordinary and heavy water) is worthy of special study.The HM cannot explain this effect. Our preliminary consideration shows that one needs to employ for this purpose a hat potential with curved brims (not with flat Figure 2 FIR absorption (a) and low frequency (Debye) spectra of the permittivity components e', e'' (b),(c). Liquid water for T = 1, 27 and 50 deg;C; for the last two temperatures the curves are shifted downwards.Solid linesndash;calculation for the hybrid model, dashed lines ndash; observed data.7ndash;9 a /cmndash;1 4000 2000 0 ndash;2000 ndash;4000 0 200 400 600 800 1000 (a) e'' 60 40 20 0 ndash;20 0.1 1 10 100 (b) e' 30 20 10 0 ndash;10 ndash;20 1 10 100 (c) n/cmndash;1 a /cmndash;1 300 200 100 0 10 20 30 40 50 3000 2000 1000 200 400 600 800 1000 3500 3000 2500 2000 1500 1000 15 20 25 30 35 n/cmndash;1 T/deg;C (a) (b) (c) 0 a/cmndash;1 Figure 3 The predicted frequency (a), (b) and temperature (c) dependencies of the absorption coefficient in water.(a), (b): dashed lines for T = 25deg;C, solid lines for T = 20, 15 and 10deg;C; (c): curves 1, 2, 3, 4, 5 and 6 for n =150, 200, 250, 650, 700 and 750 cmndash;1. 1, 2 3 4 5 6 a /cmndash;1Mendeleev Communications Electronic Version, Issue 2. 1997 (pp. 47ndash;86) edges as it was assumed in the present study). We hope to return later to this question. The following calculation scheme is suggested for the prediction of dielectric spectra in water (0 T 50deg;C). (1) Formulae (1)ndash;(11) are employed. (2) U0 and km are kept constant (3.05 kcal molndash;1 and 1.12, respectively). (3) The libration amplitude b is found by the interpolation of the values presented in Table 1.(4) The relaxation time tD and the lifetime t are calculated from equations (13) and (14b), respectively. First example. For T Icirc; 15, 30deg;C and n Icirc; 1,100 cmndash;1 the absorption a(n) is calculated. In this region the Debye spectrum transforms into quasi-resonance. It is seen from Figure 3(a) that the absorption rises monotonically when T increases.For n Icirc; 5, 30cmndash;1 the Debye-like plateau gives way to an intensive lsquo;over- Debyersquo; absorption. Second example. a(n) was calculated for the same temperature interval and n Icirc;100, 1000 cmndash;1, where the translational (near aR) band transfers to the librational (near aL) one. It can be seen from Figure 3(b),(c) that with the rise of T the aR-peak increases and the aL-peak decreases.Figuratively speaking, the increase of the lsquo;vapourrsquo; fraction in water accompanies the melting of its lsquo;icersquo; fraction. Let us consider finally the lsquo;rotationalrsquo; cell model of a liquid. We can express5,6 the libration angle b by a few steady-state parameters, viz. (comparing with refs. 5,6 the formula is slightly simplified). Here mH is the proton mass and s is the effective diameter of a molecule.We choose the s value (s = ca. 2.9 Aring;) from the demand that for T = 50 deg;C the estimated and above fitted b values should coincide (ca. 22.1deg;). This cell model is applicable for T 50 deg;C when the density r and the amplitude b substantially alter with the rise of T (Figure 4). One may wait from physical reasoning that the maximum libration angle bmax is about p/2.For water this b-value is reached if temperature Tmax = 628 K, just near the critical temperature Tcr = 647 K, the difference being only 3. One may also determine Tcr by this way in nonassociated liquids.12 We have found that for T 50 deg;C the rotational cell model gives qualitative estimations (see Figure 4) since equation (16) yields b amplitudes which vary with T more slowly than those fitted for the HM.Consequently, for low temperatures (T 50 deg;C) molecular rotation is determined sooner by the ice-like tetrahedral water structure; moleculhis case does not correspond to a simple gas-like picture of molecular rotation that determined the liquid cell model. Thus, we have discovered the physical mechanism responsible for the existence of R-particles.The latter was always1ndash;3 employed for the calculation of the orientational spectrum in water. R-molecules have an elevated rotational mobility and inherently appear in the hybrid model as overbarrier lsquo;hotrsquo; particles. These molecules are the cause of the lsquo;translationalrsquo; absorption peak near nR = ca. 200 cmndash;1. In frames of the HM we have elaborated a simple method for the prediction of wideband (n 1000 cmndash;1) dielectric spectra of a(n), e'(n), e''(n) and n(n), i.e.of absorption, components of permittivity and of refraction index. We have also shown that the critical temperature Tcr corresponds to that limiting state of a liquid, in which the libration amplitude b reaches its maximum possible value bmax raquo; p/2. It seems that our idea that hot H2O dipoles are the origin of the translational band does not contradict the other (see e.g., ref. 13) reasoning according to which this band is related to intermolecular vibrations of H-bonded H2O and D2O molecules. These vibrations accompany the rotational motion of H2O dipoles. Unfortunately, the existing H-bond theory is not capable of describing the wideband dielectric spectra of water even if on the level of this study.The elaboration of the method capable to predict the water spectra/its critical temperature was accomplished in frames of the grant no. 95-03-0-8214a, while the potential well effect on spectra was studied in frames of the grant no. 96-05-65379. Both grants were supported by the Russian Foundation for Basic Research. References 1 V.I. Gaiduk, T. A. Novskova and V. V. Brekhovskikh, J. Chem. Soc., Faraday Trans., 1993, 89, 1975. 2 V. I. Gaiduk and V. V. Gaiduk, Physica A, 1995, 222, 46. 3 V. I. Gaiduk, V. D. Gusrsquo;kova and T. A. Novskova, Izvestiya Vyssh. Uchebn. Zaved., Radiofiz., 1988, 31, 799 (in Russian). 4 V. I. Gaiduk, Mendeleev Commun., 1994, 15. 5 V. I. Gaiduk, T. A. Novskova and V. V. Brekhovskikh, J. Chem. Soc., Faraday Trans., 1991, 87, 559. 6 V. I. Gaiduk and B. M. Tseitlin, Adv. Chem. Phys., 1994, 87, 125. 7 H. J. Liebe, J. A. Hufford and T. Manabe, J. Infrared and Millimeter Waves, 1991, 12, 659. 8 H. D. Downing and D. Williams, J. Geophys. Res., 1975, 80, 1656. 9 L. W. Pinkley, P. P. Sethna and D. Williams, J. Opt. Soc. Am., 1976, 67, 494. 10 P. Debye, Polyarnye moleculy (Polar molecules), GONTI, Moscowndash; Leningrad, 1931 (in Russian). 11 I. T. Goronoskii, Yu. P. Nazarenko and E. F. Necryach, Kratkii spravochnik khimika (Short handbook of chemist), ed. O. D. Kurilenko, Naukova Dumka, Kiev, 1965 (in Russian). 12 V. V. Brekhovskikh and V. I. Gaiduk, Radiotekhnika i Elektronika (in press) (in Russian). 13 B. Curnutte and J. Bandeker, J. Mol. Spectrosc., 1972, 41, 500. T( ) p 8 --- mHM I ------------- M NA T( ) ------------------- 1 3 � ndash;egrave; oslash; aelig; ouml; , = (16) r s b Figure 4 The temperature dependence of the libration amplitude b obtained for the rotational cell model of water. Points ndash; the b-values fitted for the hybrid model at T =1, 27 and 50 deg;C (from left to right). b/deg; 200 150 100 50 0 0.2 0.4 0.6 0.8 1 Reduced temperature TR Received: Moscow, 27th September 1996 Cambridge, 17th January 1997; Com. 6/0666