Harmonic oscillator cell model

The harmonic oscillator cell model (Prigogine and Garikian [1950], Rowlinson [1952], Prigogine and Mathot [1952], Prigogine, Trappenbbrs and Mathot [1953]) is obtained by retaining only the first term in this expansion. As can be ihferred from Fig. 7.3.1 this modd is only valid at low temperatures and may be of interest in the study of solid solutions. For an example of application to solid solution see Sarolea [1953]. 

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Potential Cell Model 127 4 Harmonic Oscillator Cell Model 130, 5. ICirkwood s... 

Bormulae (8.2.7) and (8.2.8) are much simplified if the cell partition functions depend only on the density or more generally if the ratio aI aa (and WbI bb) is temperature independent. This simplifying feature is realized in both the smoothed potential and the harmonic oscillator cell models (cf. Ch. VII, 3, 4). 

Another defect of the cell model that we have already mentioned is the neglect of correlations. This appears very clearly in the limit of high densities and low temperatures. In this case we may use a harmonic oscillator cell potential. All molecules perform independent harmonic oscillations in their cells and the result is a specific heat curve of the Einstein type with an exponential decrease for T- 0 instead of a specific heat curve of the Debye type with a characteristic 7 law. Furthermore quantum statistical effects (exchange effects) cannot be introduced in a one-partide model. For this reason the cell model cannot be applied to problems in which such exchange effects play a dominant role (e.g. the X point of liquid He ). However, multiple cell occupations do take account of these correlations and exchange effects. IDgher multiple cell occupations prodde a continuous transition from a one-particle model to the correct 2V-partide model. 

An Excel spreadsheet comparing potential energy curves calculated for HCl for Morse and harmonic oscillator models with ab initio quantum mechanical results obtained with the program Gaussian. The example illustrates the use of cell formulas and some of the text Format options, such as bold and italic fonts of various sizes, subscripts and superscripts, and Greek and other special characters. 

Dipole oscillations in an assembly of molecules in the membrane of cells can be modeled as phase-locked solid state oscillators by a basic circuit as in Figure 1. Loose coupling between such circuits imposes an eigenvalue problem from which significant mode softening can be shown to result and this has been suggested to be an important requirement in the energetics associated with the reproduction and mutation of cells. As each individual unit oscillator can operate at subharmonics as well as harmonics, the above model is consistent with the idea that in vivo a number of discrete frequencies exist in the cell. 

Two forms of the cell model (CM) are then developed harmonic oscillator approximation and square-well approximation. Both forms assnme hexagonal closed packing (HCP) lattice structure for the cell geometry. The model developed by Paul and Di Benedetto [13] assumes that the chain segments interact with a cylindrical symmetric square-well potential. The FOV model discnssed in the earlier section uses a hard-sphere type repulsive potential along with a simple cubic (SC) lattice structure. The square-well cell model by Prigogine was modified by Dee and Walsh [14]. They introduced a numerical factor to decouple the potential from the choice of lattice strncture. A universal constant for several polymers was added and the modified cell model (MCM) was a three-parameter model. The Prigogine cell EOS model can be written as follows. 

Cell Model forr-mers 331. 5. Harmonic Oscillator Model 332. 6. Smoothed... 

Starting with the cell partition function (16.3.17) and the lattice energy (16.4.2) we may easily obtain the explicit expressions for the thermodynamic properties. The main difference is that the cell partition function depends only on the voliune but not on the temperature. The calculations are the same as for the harmonic oscillator model and will not be repeated. Instead of (16.5.4) we obtain the equation of state... 

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