Rectangular well model

Rectangular Well Model. A compromise between the oversimple hard sphere model, which is mathematically tractable, and the Lennard-Jones potential, which is very difficult to treat mathematically, is the rectangular well model. The force between two molecules is everywhere zero except at two distances of separation, era, at which the force is infinitely large and attractive, and err, at which the force is infinitely large and re- 

This model is characterized by three parameters the well depth C/o, the range of the attractive forces o-a, and the hard sphere radius Cr. Probably because of this it is able to represent many equilibrium and transport properties of real molecules with semiquantitative accuracy. 

In our development of the kinetic theory we shall begin with the simplest of these models, that of the point particle. 

For an ideal gas we shall take the point particle with no forces between molecules. A real gas will approach this behavior when its density is such that the average distance between molecules is large compared to their diameter a. This means that the volume of the vessel is very large compared to the total volume occupied by the molecules themselves, or V Ntct /Qj where N = number of molecules in the volume V. 

If now we assume that the molecules move completely at random and independently of each other and that the system is at equilibrium and isolated (no exchange of energy between the gas and its environment) then the total energy of the gas is simply the kinetic energy attributable to the random motion of the molecules. This total energy and the volume then fix completely the thermodynamic properties of the gas. If now we could know the probability distribution function for the molecular velocities (at equilibrium), that would determine uniciuely the properties of the system. 

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Our rather voluminous chapter could conditionally be divided into two parts (a possibility exists to read them independently one from another). In the first part (Sections II-IV) written mostly by B.M.T., a brief review of the ACF method is given and two basic rectangular-well models are described. The other part (Sections I,V-X), written mostly by V.I.G., concerns substantial complication of these models and their application for description, sometimes quantitative, of wideband dielectric and far-infrared (FIR) spectra of strongly polar fluids. A schematic diagram (Fig. 1) illustrates the main topics of both above-mentioned parts, which are here marked A and B. 

Figures 15a and 15b show the wideband absorption and loss spectra, respectively, calculated for the PL-RP version. Figures 15c and 15d demonstrate similar spectra pertinent to the hybrid model. It is seen that the spectra calculated for both versions of the rectangular-well model agree in their main features with the experimental dielectric/FIR spectra recorded in the region 0-1000 cm-1.

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.

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. 

Figure 6.3 The one-dimensional Kronig-Penney potential. The rectangular wells model of the interionic distances, whereas the barriers of a Vq height model the one-dimensional ion lattice.

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

In the second period, which was ended by review GT after the average perturbation theorem was proved, it became possible to get the Kubo-like expression for the spectral function L(z) (GT, p. 150). This expression is applicable to any axially symmetric potential well. Several collision models were also considered, and the susceptibility was expressed through the same spectral function L(z) (GT, p. 188). The law of motion of the particles should now be determined only by the steady state. So, calculations became much simpler than in the period (1). The best achievements of the period (2) concern the cone-confined rotator model (GT, p. 231), in which the dipoles were assumed to librate in space in an infinitely deep rectangular well, and applications of the theory to nonassociated liquids (GT, p. 329). 

It will be calculated in Section IV.F for an example of a finite-depth rectangular well (viz., for the hat-flat model), where also a more general definition of this quantity will be given. 

The spectral function L(z) involved in Eq. (142) is determined by the profile of the model potential well (in this section it is the rectangular well). It follows from Eq. (148) that if we fix the dimensional quantities, such as frequency v and temperature, then the spectral function L(z) depends also on the lifetime x and the moment of inertia of a molecule I. We consider a gas-like reorientation of a polar molecule determined by a dipole moment p of a molecule in a liquid. Calculation of the moment of inertia I deserves special discussion. 

We employ the following equations Eq. (142) for the complex susceptibility X, Eq. (141) for the complex permittivity , and Eq. (136) for the absorption coefficient a. In (142) we substitute the spectral functions (132) for the PL-RP approximation and (133) for the hybrid model, respectively. In Table IIIB and IIIC the following fitted parameters and estimated quantities are listed the proportion r of rotators, Eqs. (112) and (127) the mean number m of reflections of a dipole from the walls of the rectangular well during its lifetime x, Eqs. (118)... 

However, serious drawbacks of model 3 are that (i) the proportion r of the rotators should be fitted that is, it is not determined from physical considerations and (ii) the depth of the well, in which a polar particle moves, is considered to be infinite. Both drawbacks were removed in VIG (p. 305, 326, 465) and in Ref. 3, where it was assumed that (a) The potential is zero on the bottom of the well (/(()) = 0 at [ fi < 0 < P], where an angle 0 is a deflection of a dipole from the symmetry axis of a cone, (b) Outside the well the depth of the rectangular well is assumed to be constant (and finite) U(Q) = Uq at [— ti/2 < 0 < ti/2]. Actually, two such wells with oppositely directed symmetry axes were supposed to arise in the circle, so that the resulting dipole moment of a local-order region is equal to zero (as well as the total electric moment in any sample of an isotropic medium). 

Inside a rectangular well a dipole rotates freely until it suffers instantaneous collision with a wall of the well and then is reflected, while in the field models a continuously acting static force tends to decrease the deflection of a dipole from the symmetry axis of the potential. Therefore, if a dipole has a sufficiently low energy, it would start backward motion at such a point inside the well, where its kinetic energy vanishes. Irrespective of the nature of forces governing the motion of a dipole in a liquid, we may formally regard the parabolic, cosine, or cosine squared potential wells as the simplest potential profiles useful for our studies. The linear dielectric response was found for this model, for example, in VIG (p. 359) and GT (p. 249). 

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... 

A brief description of the hybrid model based on application of the rectangular well potential was given in Section V.E. 

To extend the usefulness of the model to permit a description of chemical reactions, we must introduce another parameter, the effective duration of a collision. The rectangular well or central force models do this automatically by permitting molecular interaction over a range of distances. However, they are both more complex than the hard sphere model. We can rescue the hard sphere model by specifying a parameter era, the effective diameter for chemical interaction, while keeping hard sphere core diameter. When the centers of two identical molecules are a distance effective reaction volume is 7r([Pg.155]

In order to exemplify the above results we calculate the transmission coefficient vs energy in multibarrier resonant tunneling structures which may be modeled by a one-dimensional system formed by alternating N-1-1 rectangular barriers with N rectangular wells. We assume that the electrons possess the same effective mass through the system and that the tunneling process is coherent i.e., elastic [61]. 

The figures show the curves for D = 1.2 and o = 1,2. As we can see. D controls the well depth and or its width, (b) The Morse oscillator is a kind of compromise between the harmonic oscillator (bl) and a rectangular well (b2). Both potentials correspond to exact solutions of the Schriidinger equatiotL Model b2 gives the discrete spectrum, as well as the eontinuum and the resonance states. The latter ones are only very rarely considered for Morse oscillators, but they play an important role in scattering phenomena (primarily in reactive collisions). 

The solution flow is nomially maintained under laminar conditions and the velocity profile across the chaimel is therefore parabolic with a maximum velocity occurring at the chaimel centre. Thanks to the well defined hydrodynamic flow regime and to the accurately detemiinable dimensions of the cell, the system lends itself well to theoretical modelling. The convective-diffiision equation for mass transport within the rectangular duct may be described by... 

While the smooth substrate considered in the preceding section is sufficiently reahstic for many applications, the crystallographic structure of the substrate needs to be taken into account for more realistic models. The essential complications due to lack of transverse symmetry can be dehneated by the following two-dimensional structured-wall model an ideal gas confined in a periodic square-well potential field (see Fig. 3). The two-dimensional lamella remains rectangular with variable dimensions Sy. and Sy and is therefore not subject to shear stresses. The boundaries of the lamella coinciding with the x and y axes are anchored. From Eqs. (2) and (10) one has... 

We consider the problem of liquid and gas flow in micro-channels under the conditions of small Knudsen and Mach numbers that correspond to the continuum model. Data from the literature on pressure drop in micro-channels of circular, rectangular, triangular and trapezoidal cross-sections are analyzed, whereas the hydraulic diameter ranges from 1.01 to 4,010 pm. The Reynolds number at the transition from laminar to turbulent flow is considered. Attention is paid to a comparison between predictions of the conventional theory and experimental data, obtained during the last decade, as well as to a discussion of possible sources of unexpected effects which were revealed by a number of previous investigations. 

Numerical results of the heat transfer inside four 1 cm heat sinks with 150 and 200 channels were presented by Toh et al. (2002). Their calculation predicted the local thermal resistanee very well. The micro-heat sink modeled in the numerical investigation by Li et al. (2004) consisted of a 10 mm long silicon substrate. The rectangular micro-channels had a width of 57 pm, and a depth of 180 pm. The heat... 

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