Debye molecular model

The discussion above is a description of problem that requires answers to the following (1) the determination of the distribution of ions around a reference ion, and (2) the determination of the thickness (radius) of the ionic atmosphere. Obviously this is a complex problem. To solve this problem Debye and Huckel used a rather general approach they suggested an oversimplified model in order to obtain an approximate solutions. The Debye-Huckel model has two basic assumptions. The first is continuous dielectric assumption. In this assumption water (or the solvent) is a continuous dielectric and is not considered to be composed of molecular species. The second, is a continuous charge distribution in the ionic atmosphere. Put differently, charges of the ions in the ionic surrounding atmosphere are smoothened out (continuously distributed). 

Considerable progress has been made in going beyond the simple Debye continuum model. Non-Debye relaxation solvents have been considered. Solvents with nonuniform dielectric properties, and translational diffusion have been analyzed. This is discussed in Section II. Furthermore, models which mimic microscopic solute/solvent structure (such as the linearized mean spherical approximation), but still allow for analytical evaluation have been extensively explored [38, 41-43], Finally, detailed molecular dynamics calculations have been made on the solvation of water [57, 58, 71]. 

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. 

Here multiplier 2 approximately accounts for doubling of integrated absorption due to spatial motion of a dipole, which is more realistic than motion in a plane to which LCs(Z) corresponds. For representation (235), only one (Debye) relaxation region with the relaxation time rD is characteristic. At this stage of molecular modeling it was not clear (a) why the CS potential, which affects motion of a dipole in a separate potential well, is the right model of specific interactions and (b) what is physical picture corresponding to a solid-body-like dipole moment pcs. 

The Bjerrum treatment suffers from the oversimplifications of the Debye-Hlickel model, particularly regarding the correct method of calculating the energy of an ion pair at small distances when molecular structure is surely important. Nevertheless it is certainly a step in the right... 

As detailed in Chapter 2, van der Waals interactions consist mainly of three types of long-range interactions, namely Keesom (dipole-dipole angle-averaged orientation, Section 2.4.3), Debye (dipole-induced dipolar, angle-averaged, Section 2.5.7), and London dispersion interactions (Section 2.6.1). However, only orientation-independent London dispersion interactions are important for particle-particle or particle-surface attractions, because Keesom and Debye interactions cancel unless the particle itself has a permanent dipole moment, which can occur only very rarely. Thus, it is important to analyze the London dispersion interactions between macrobodies. Estimation of the value of dispersion attractions has been attempted by two different approaches one based on an extended molecular model by Hamaker (see Sections 7.3.1-7.3.5) and one based on a model of condensed media by Lifshitz (see Section 7.3.7). 

These are models which attempt to take into consideration the molecular structure of the solvent molecules. Previous treatments based on the Debye-Hiickel model have approximated the solvent to a continuous medium characterised simply by the macroscopic relative permittivity. Qualitatively, the Bom-Oppenheimer models could be taken as replacing the Gurney terms and the cavity terms of the previous model (Sections 10.16.3 to 10.16.6). 

We assume that the above-indicated drawback of the present model can be avoided (or at least reduced) if a new paradigm [mentioned below in Section X.B.4(ii)] of the molecular model will be constructed. In our opinion, this drawback of the present model is stipulated by the following. In view of Eq. (11) the libration lifetime T0r is determined by the experimental Debye relaxation time td, so variation of Tor cannot be used for other corrections of the calculated spectra. In the proposed new paradigm it is desirable to use Tor for the latter purpose, while a correct describing of the low-frequency Debye spectrum is assumed to be reached by variation of additional parameter(s). 

Calculation of the dielectric permittivity of an isotropic polar material involves the problem of the permanent dipole contribution to polarizability and the problem of calculation of the local field acting at the molecular level in terms of the macroscopic field applied. Debye s model for static permittivity considers the local field equal to the external field. This assumption is valid only for gases at low density or dilute solutions of polar molecules in nonpolar solvents. Several workers... 

In contrast with condensed phases, intermolecular interactions in gases are negligibly small. The dipole moment found in the gas phase at low pressure is usually accepted as the correct value for a particular isolated molecule. The molecular dipole moment calculated for pure liquid using Debye s model gives values which are usually very different from those obtained from gas measurements. Intermolecular interactions in liquids produce deviations from Debye s assumptions. 

Debye has developed a theory for evaluating gas exposures which, for given atomic coordinates, represents the intensity of the scattered beam as a function of the angle of diffraction. It is obviously not possible in general to deduce a definite molecular model from one experimental diagram without further assumptions (harmonic analysis). Frequently the evaluation of the diffraction pattern must be based on a tentatively assumed plausible model, the interferences given by it calculated and th( model compared with the diagram actually obtained. 

Equation 12.26 is known as Debye equation. Figure 12.3 gives the frequency dependence of the real and imaginary (loss) part of the Debye function, s shows a steplike decay with increasing frequency where s" presents a symmetric peak with a maximum cOp = 27ifp = 1/Xp and a half width of 1.14 decades. The Debye equation can be justified by different molecular models like in the framework of a simple double potential model or the rotational diffusion approach. 

Molecular models for chloromethane, dichloromethane, and trichloromethane are given to show the direction of the dipole of molecule more clearly. Calculated dipoles for these three molecules are 2.87, 2.50, and 1.72 Debye, respectively, and it is clear that the directional nature of the individual bond dipoles plays a role in the overall magnitude of the dipole moment for the molecrde. With three chlorine atoms directed to different regions of space, chloroform is the least polar of the three molecules, despite the presence of three polarized bonds. 

An important result of Debye s model is that the relationship between the macroscopic, X, and molecular relaxation times, t, is given by Eq. (9). For example, Eq. (9) predicts that materials with Eg 100 and e = 2 the ratio of times is 25.5, with the molecular time being the faster one. 

NEWTON - Even if one adopts the concept of Tj based on a simple Debye dispersion model, one must recognize the ambiguity in evaluating associated with the choice of a value for (infrared or optical ). Furthermore, Wolynes has recently argued for the likely occurrence of an additional relaxation time of magnitude larger than T,, arizing from the molecularity of the solvent in the immediate vicinity of the ion. 

In the above discussion of the frequency dependent permittivity, the analysis has been based on either the single particle rotational diffusion model of Debye, or empirical extensions of this model. A more general approach can be developed in terms of time correlation functions [6], which in turn have to be interpreted in terms of a suitable molecular model. While using the correlation function approach does not simplify the analysis, it is useful, since experimental correlation functions can be compared with those deduced from approximate theories, and perhaps more usefully with the results of molecular dynamics simulations. Since the use of correlation functions will be mentioned in the context of liquid crystals, they will be briefly introduced here. The dipole-dipole time correlation function C(t) is related to the frequency dependent permittivity through a Laplace transform such that ... 

Tbe form of the response function may be derived from molecular models or from other considerations. The simplest Ametion. introduced by Debye, is r) 1 exp(—(frX where the parameter t is referred to as tbe relaxation time of the assembly of dipoles. 

The Debye model is more appropriate for the acoustic branches of tire elastic modes of a hanuonic solid. For molecular solids one has in addition optical branches in the elastic wave dispersion, and the Einstein model is more appropriate to describe the contribution to U and Cj from the optical branch. The above discussion for phonons is suitable for non-metallic solids. In metals, one has, in addition, the contribution from the electronic motion to Uand Cy. This is discussed later, in section (A2.2.5.6T... 

The SPC/E model approximates many-body effects m liquid water and corresponds to a molecular dipole moment of 2.35 Debye (D) compared to the actual dipole moment of 1.85 D for an isolated water molecule. The model reproduces the diflfiision coefficient and themiodynamics properties at ambient temperatures to within a few per cent, and the critical parameters (see below) are predicted to within 15%. The same model potential has been extended to include the interactions between ions and water by fitting the parameters to the hydration energies of small ion-water clusters. The parameters for the ion-water and water-water interactions in the SPC/E model are given in table A2.3.2. 

Our approach in this chapter is to alternate between experimental results and theoretical models to acquire familiarity with both the phenomena and the theories proposed to explain them. We shall consider a model for viscous flow due to Eyring which is based on the migration of vacancies or holes in the liquid. A theory developed by Debye will give a first view of the molecular weight dependence of viscosity an equation derived by Bueche will extend that view. Finally, a model for the snakelike wiggling of a polymer chain through an array of other molecules, due to deGennes, Doi, and Edwards, will be taken up. 

Equation (2.61) predicts a 3.5-power dependence of viscosity on molecular weight, amazingly close to the observed 3.4-power dependence. In this respect the model is a success. Unfortunately, there are other mechanical properties of highly entangled molecules in which the agreement between the Bueche theory and experiment are less satisfactory. Since we have not established the basis for these other criteria, we shall not go into specific details. It is informative to recognize that Eq. (2.61) contains many of the same factors as Eq. (2.56), the Debye expression for viscosity, which we symbolize t . If we factor the Bueche expression so as to separate the Debye terms, we obtain... 

The concentration of salt in physiological systems is on the order of 150 mM, which corresponds to approximately 350 water molecules for each cation-anion pair. Eor this reason, investigations of salt effects in biological systems using detailed atomic models and molecular dynamic simulations become rapidly prohibitive, and mean-field treatments based on continuum electrostatics are advantageous. Such approximations, which were pioneered by Debye and Huckel [11], are valid at moderately low ionic concentration when core-core interactions between the mobile ions can be neglected. Briefly, the spatial density throughout the solvent is assumed to depend only on the local electrostatic poten-... 

As the density of a gas increases, free rotation of the molecules is gradually transformed into rotational diffusion of the molecular orientation. After unfreezing , rotational motion in molecular crystals also transforms into rotational diffusion. Although a phenomenological description of rotational diffusion with the Debye theory [1] is universal, the gas-like and solid-like mechanisms are different in essence. In a dense gas the change of molecular orientation results from a sequence of short free rotations interrupted by collisions [2], In contrast, reorientation in solids results from jumps between various directions defined by a crystal structure, and in these orientational sites libration occurs during intervals between jumps. We consider these mechanisms to be competing models of molecular rotation in liquids. The only way to discriminate between them is to compare the theory with experiment, which is mainly spectroscopic. 

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

The results of the Debye theory reproduced in the lowest order of perturbation theory are universal. Only higher order corrections are peculiar to the specific models of molecular motion. We have shown in conclusion how to discriminate the models by comparing deviations from Debye theory with available experimental data. 

Finally, it must be recalled that the transport properties of any material are strongly dependent on the molecular or ionic interactions, and that the dynamics of each entity are narrowly correlated with the neighboring particles. This is the main reason why the theoretical treatment of these processes often shows similarities with models used for thermodynamic properties. The most classical example is the treatment of dilute electrolyte solutions by the Debye-Hiickel equation for thermodynamics and by the Debye-Onsager equation for conductivity. 

In the Ising-type model, the change of molecular volume AV due to the LS<->HS transformation leads to a change of phonon frequencies of the lattice. The effect may be treated within the Debye approximation which requires that the interaction parameters and J2 are replaced by J and J 2 where ... 

NFS spectra of the molecular glass former ferrocene/dibutylphthalate (FC/DBP) recorded at 170 and 202 K are shown in Fig. 9.12a [31]. It is clear that the pattern of the dynamical beats changes drastically within this relatively narrow temperature range. The analysis of these and other NFS spectra between 100 and 200 K provides/factors, the temperature dependence of which is shown in Fig. 9.12b [31]. Up to about 150 K,/(T) follows the high-temperature approximation of the Debye model (straight line within the log scale in Fig. 9.12b), yielding a Debye tempera-ture 6x) = 41 K. For higher temperatures, a square-root term / v/(r, - T)/T,... 

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