Hat-curved models

Application to Strongly Absorbing Nonassociated Liquid V. Hat-Curved Model and Its Application for Polar Fluids... 

Hat-Curved Model as Symbiosis of Rectangular-Well and Parabolic-Well Models... 

Form factor of the hat-curved model Normalized concentration of molecules Kirkwood correlation factor Steady-state energy (Hamiltonian) of a dipole Dimensionless energy of a dipole Moment of inertia of a molecule Longitudinal and transverse components of the spectral function Complex propagation constant Elasticity constant (in Section IX)... 

S V 0 Normalized steepness in the hat-curved model Effective potential Current deflection of a dipole moment from the symmetry axis... 

Section VI. It is possible to unblock the first drawback (i), if to assume a nonrigidity of a dipole—that is, to propose a polarization model of water. This generalization roughly takes into account specific interactions in water, which govern hydrogen-bond vibrations. The latter determine the absorption R-band in the vicinity of 200 cm-1. A simple modification of the hat-curved model is described, in which a dipole moment of a water molecule is represented as a sum of the constant (p) and of a small quasi-harmonic time-varying part p(/j. 

The latter is determined by the oscillation frequency, decaying coefficient, and vibration lifetime. This nonrigid dipole moment stipulates a Lorentz-like addition to the correlation function. As a result, the form of the calculated R-band substantially changes, if to compare it with this band described in terms of the pure hat-curved model. Application to ordinary and heavy water of the so-corrected hat-curved model is shown to improve description (given in terms of a simple analytical theory) of the far-infra red spectrum comprising superposition of the R- and librational bands. 

However, the so-corrected hat-curved model still does not give a perfect agreement with the experiment, since it does not allow us to eliminate the second drawback (ii), namely, disagreement with experiment of the calculated complex permittivity in the submillimeter wavelength region. 

A general approach (VIG, GT) to a linear-response analytical theory, which is used in our work, is viewed briefly in Section V.B. In Section V.C we consider the main features of the hat-curved model and present the formulae for its dipolar autocorrelator—that is, for the spectral function (SF) L(z). (Until Section V.E we avoid details of the derivation of this spectral function L). Being combined with the formulas, given in Section V.B, this correlator enables us to calculate the wideband spectra in liquids of interest. In Section V.D our theory is applied to polar fluids and the results obtained will be summarized and discussed. 

Section VI will present a first step for description (again in terms of the hat-curved model) of the collective (cooperative) effects in water due to the H bonds (i.e., following Walrafen [16]), resulting from the specific interactions. The dielectric spectra of ordinary and heavy water will be calculated in this section. For this purpose we shall apply (with some changes) recent investigation [6, 8] based on the concept of a nonrigid dipole. Other applications of the hat-curved model to water will be described in Sections VII, IX, and X. 

The coefficient D, being proportional to a normalizing coefficient C of the Maxwell-Boltzmann energy distribution W = C exp(- / ), is determined by the parameters of the hat-curved model as... 

Parameters of the Hat-Curved Model Pertinent to Liquid Water... 

Figure 24. Absorption coefficient (a, c) and wideband diecltric loss (b, d) calculated for liquid H20 water at 22.2°C (a, b) and 27°C (c, d) for the hat-curved model (solid lines). The experimental a(v) dependencies [17, 42, 56] are shown by dashed lines. The horizontal lines in Figs, (a) and (c) denote the maximum absorption recorded in the librational band.

As a second example, we consider liquid fluoromethane CH3F, which is a typical strongly absorbing nonassociated liquid. For our study we choose the temperature T 133 K near the triple point, which is equal to 131 K. The relevant experimental data [43] were summarized in Table IV. As we see in Table VIII, which presents the fitted parameters of the model, the angle p is rather small. At this temperature the density p of the liquid, the maximum dielectric loss and the Debye relaxation time rD are substantially larger than they would be, for example, near the critical temperature (at 293 K). At such small (5 the theory given here for the hat-curved model holds. For calculation of the complex permittivity s (v) and absorption a(v), we use the same formulas as for water. 

The calculated spectra are illustrated by Fig. 25. In Fig. 25a we see a quasiresonance FIR absorption band, which, unlike water, exhibits only one maximum. Figure 25b demonstrates the calculated and experimental Debye-relaxation loss band situated at microwaves. Our theory satisfactorily agrees with the recorded a(v) and e"(v) frequency dependencies. Although the fitted form factor/is very close to 1 (/ 0.96), the hat-curved model gives better agreement with the experiment than does a model based on the rectangular potential well, where / = 1 (see Section IV.G.3). 

Figure 25. Frequency dependence of the absorption coefficient (a) and dielectric loss (b). Liquid fluoromethane CH F at 133 K calculated for that hat-curved model (solid lines). Dashed curve in Fig. (a) refers to the experimental [43] data, vertical line in Fig. (b) marks the experimental position of the maximum dielectric loss. The parameters of the hat-curved model are presented in Table VIII.

Figure 26. Comparison of the FIR absorption spectra of liquid water calculated for the hat-curved model (solid lines) and the rectangular-well model (dashed lines) with the same u and p values at 22.2°C (a) and 27°C (b). The parameters w. p. r./ of the hat-curved model are given in Table VII.

The hat-curved model also gives a satisfactory description of the wideband dielectric/FIR spectra of a nonassociated polar fluid (CH3F) (Fig. 25). It is worthwhile mentioning that only a poor description of the low-frequency (Debye) spectrum could be accomplished, if the rectangular potential were used for such a calculation [32] see also Section IV.G.3. Unlike Fig. 25b, the estimated peak-loss frequency does not coincide38 in this case with the experimental frequency vD. 

Thus, evolution of semiphenomenological molecular models mentioned in Section V.A (items 1-6) have led to the hat-curved model as a model with a rounded potential well. This model combines useful properties of the rectangular potential well and those peculiar to the field models based on application of the parabolic, cosine, or cosine-squared potentials. Namely, the hat-curved model retains the main advantage of the rectangular-well model—its possibility to describe both the librational and the Debye-relaxation bands. 

The integrals (197b) will be calculated in Section V.E.7 in the planar libration-regular precession (PL-RP) approximation as functions of the parameters u, (5, / of the hat-curved model. For further consideration it is convenient to introduce the coefficient... 

This section presents the continuation of Section V. In the latter a new model [10] termed the hat-curved model was described, where a rigid dipole reorients in a hat-like intermolecular potential well having a rounded bottom. This well differs considerably from the rectangular one, which is extensively applied to polar fluids. Now the theory of the hat-curved model will be generalized, taking into account the non-rigidity of a dipole that is, a simplified polarization model of water is described here. 

Spectral properties of water in the R-band range (near 200 cm-1), which are directly determined by H-bonding of water molecules, were termed [16] as stipulated by specific interactions. All other ones we shall term unspecific interactions. The latter were considered in Section V devoted to a pure hat-curved model capable of description of the Debye, librational, and partly R-bands (the ongoing background was considered in items 1-6 in Section V.A.1). 

Here we try to study specific interactions in water in terms of slightly modified hat-curved model with a simplified account of collective (cooperative) effects in water in relation to SWR spectra. Below, in items A-D, we shall shortly describe how the problem of these effects was gradually recognized in our publications [6-9, 11]. At first, we shall draw attention on a small isotope shift of the R-band—that is, on practical coincidence of the peak absorption frequencies vR 200 cm 1 for both ordinary (H20) and heavy (D20) water. 

A. Previous models of water (see 1-6 in Section V.A.l) and also the hat-curved model itself cannot describe properly the R-band arising in water and therefore cannot explain a small isotope shift of the center frequency vR. Indeed, in these models the R-band arises due to free rotors. Since the moment of inertia I of D20 molecule is about twice that of H20, the estimated center of the R-band for D20 would be placed at y/2 lower frequency than for H20. This result would contradict the recorded experimental data, since vR(D20) vR(H20) 200 cm-1. The first attempt to overcome this difficulty was made in GT, p. 549, where the cosine-squared (CS) potential model was formally (i.e., irrespective of a physical origin of such potential) applied for description of dielectric response of rotators moving above the CS well (in this work the librators were assumed to move in the rectangular well). The nonuniform CS potential yields a rather narrow absorption band this property agrees with the experimental data [17, 42, 54]. The absorption-peak position Vcs depends on the field parameter p of the model given by... 

We regard the concept of elasticity to be the key aspect of the problem of specific interaction. A preliminary study [12, 12a] cast light on this problem in terms of dynamics of water molecules and of the relevant vibration frequencies. We consider this question in Section IX. On the other hand, the work [12b] will be improved in Section VII in terms of the hat-curved model. 

The main purpose of this section is consideration of the FIR spectra due to the second dipole-moment component, p(f). However, for comparison with the experimental spectra [17, 42, 51] we should also calculate the effect of a total dipole moment ptot. In Refs. 6 and 8 the modified hybrid model44 was used, where reorientation of the dipoles in the rectangular potential well was considered. In this section the effect of the p(f) electric moment will be found for the hat-curved, potential, which is more adequate than the rectangular potential pertinent to the hybrid model. In Section VI.B we present the formula for the spectral function of the hat-curved model modified by taking into account the p(f) term (derivation of the relevant formula is given in Section VI.E). The results of the calculations and discussion are presented, respectively, in Sections VI.C and VI.D. 

Figure 31. Normalized absorption versus normalized frequency. Solid curve Calculation for the parameters of the hat-curved model typical for liquid D20 (these parameters are presented in Table XIII, lower line.) Dashed curve The lifetime Tvib becomes two times longer, other parameters remain the same. Dashed-and-dotted curve Calculation, when s becomes two times larger, other parameters remain the same.

The fitted parameters of the hat-curved model are given in Table XIIIA the parameterization procedure was described in Section V.D. The solid lines in... 

Figure 32. Absorption coefficient (a, b, c) and dielectric loss (d, e, f). Water H20 at 27°C (a, d), water H20 at 22.2°C (b, e), and water D20 at 22.2°C (c, f) Solid lines Calculation for the hat-curved model experimental [42, 51] values of absorption (squares) and loss (dashed lines), calculation from empirical formula [17] (dashed-and-dotted lines). Contribution to absorption due to nonrigidity of dipoles is shown by dots.

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