Partition function cell model

As pointed out before when discussing structured models, physiological functions like the single cell growth rate r(m, S), cellular division rate T(m, S), and cell mass partition function p(m, m, S) are difficult to measure and this adds to the complexity of using the model. The mathematical complexity is also a limitation for optimization and control purposes that demand online measurements and calculations. 

Just as in our abbreviated descriptions of the lattice and cell models, we shall not be concerned with details of the approximations required to evaluate the partition function for the cluster model, nor with ways in which the model might be improved. It is sufficient to remark that with the use of two adjustable parameters (related to the frequency of librational motion of a cluster and to the shifts of the free cluster vibrational frequencies induced by the environment) Scheraga and co-workers can fit the thermodynamic functions of the liquid rather well (see Figs. 21-24). Note that the free energy is fit best, and the heat capacity worst (recall the similar difficulty in the WR results). Of more interest to us, the cluster model predicts there are very few monomeric molecules at any temperature in the normal liquid range, that the mole fraction of hydrogen bonds decreases only slowly with temperature, from 0.47 at 273 K to 0.43 at 373 K, and that the low... 

The cell model is a commonly used way of reducing the complicated many-body problem of a polyelectrolyte solution to an effective one-particle theory [24-30]. The idea depicted in Fig. 1 is to partition the solution into subvolumes, each containing only a single macroion together with its counterions. Since each sub-volume is electrically neutral, the electric field will on average vanish on the cell surface. By virtue of this construction different sub-volumes are electrostatically decoupled to a first approximation. Hence, the partition function is factorized and the problem is reduced to a singleparticle problem, namely the treatment of one sub-volume, called cell . Its shape should reflect the symmetry of the polyelectrolyte. Reviews of the basic concepts can be found in [24-26]. 

The Ross melting rule [155] is expressed instead in terms of a threshold value of an excess free energy. It is based on a postulate of invariance of a scaled form of the partition function. A useful formulation of the statement requires the further assumption of a cell model for the solid. In terms of the free energy, the Ross melting rule has... 

Most cell model calculations to-date have been performed on electrostatically stabilized dispersions. The canonical partition function for such a system is given by... 

The Simha and Somcynsky (S-S) [1969] cell-hole theory is based on the lattice-hole model. The molecular segments of an -mer occupy ay-fraction of the lattice sites, while the remaining randomly distributed sites, /i = 7 — y, are left as empty holes. The fraction /i is a measure of the free-volume content. The goal was to provide improved description of fluids, ranging from low-molecular-weight spherical molecules (such as argon) to macromolecular chains. The S-S configurational partition function is... 

Equations of state derived from statisticai thermodynamics arise from proper con-figurationai partition functions formuiated in the spirit of moiecuiar modeis. A comprehensive review of equations of state, inciuding the historicai aspects, is provided in Chapter 6. Therefore, we touch briefly in oniy a few points. Lennard-Jones and Devonshire [1937] developed the cell model of simple liquids, Prigogine et al. [1957] generalized it to polymer fluids, and Simha and Somcynsky [1969] modified Pri-gogine s cell model, allowing for more disorder in the system by lattice imperfections or holes. Their equations of state have been compared successfully with PVT data on polymers [Rodgers, 1993]. 

To obtain the thermodynamic functions of the hard-sphere erystal we use the cell model of Lennard-Jones and Devonshire [17]. The idea of the eeU model is that a given particle moves in a free volume v set by its neighbours whieh are located on their lattice positions (see Fig. 3.2). Then the partition funetion Q takes the form... 

O is evaluated on the basis of the cell model. To make it simpler a onedimensional sy.stem is considered, consisting of N impenetrable spherical particles of diameter partition function for such a system is ... 

Polymer molecules are modeled as having two distinct sets of modes contributing to the partition function in the cell models. The two modes are internal and external modes. The internal modes are used to represent the internal motions of the molecules, and the intermolecular interactions are accounted for by the external modes. Prigogine and Kondepudi [12] proposed the conceptual separation of the two modes. The PVT properties of the polymer systems will be affected by the external modes. A polymer molecule is divided into r repeat units. Each repeat unit has 3 degrees of... 

Nitta et. al. ( 7) extended the group interaction model to thermodynamic properties of pure polar and non-polar liquids and their solutions, including energy of vaporization, pvT relations, excess properties and activity coefficients. The model is based on the cell theory with a cell partition function derived from the Carnahan-Starling equation of state for hard spheres. The lattice energy is made up of group interaction contributions. 

We know that a second way of calculating the macroscopic values is to use the canonical partition function. This is the method that we shall use from hereon in. To do so, we must construct a structure of the liquid, in order to be able evaluate the terms of interaction in the canonical partition function. Various techniques are used. We shall describe four such techniques Guggenheim s and Mie s models, extrapolated respectively from the gas and solid models, the Lennard-John and Devonshire cellular model and the cell/vacancy model. 

There are several model theories that treat a liquid like a disordered solid. In the cell modefi each atom of a monatomic fluid such as liquid argon is assumed to be confined in a cell whose walls are made up of its nearest neighbors. In the simplest version, this cell is approximated as a spherical cavity inside which the potential energy of the moving atom is constant and outside of which the potential energy is infinite. Because each atom moves independently, the classical canonical partition function can be written as a product of molecular partition functions. The classical canonical partition function is... 

The choice of the upper limit (a — a) is rather arbitrary. In so far as we are interested in excess properties of solutions, we could as well take (a — constant term to the free energy and disappears in the excess functions (cf., however, Ch. XVIII for the quantum case). As in the Lennard-Jones and Devonshire model, we shall use (7.1.23) for the lattice energy" of our system. The cell partition function W is clearly of the same form as for hard spheres (cf. 7.1.11) and depends only on the density. It may be written in the form... 

Let us now briefly consider the thermodjmamic properties which correspond to this model. The cell partition function (7.1.1) takes the simple form... 

The constant factor t v y) may be absorbed in the kinetic part of the partition functions. From this relation for the cell partition function all relevant thermodynamic properties may easily be deduced. The two models we have considered in 3 and 4 are dearly ovnsimplifled. Their intor t is that they retain the most important features of the Lennard-Jones and Devonshire cell model of condensed states with an appreciable gain in simplidty. 

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). 

In the smoothed potential cell model the cell partition function depends only on the density and is the same as for pure liquids (7.3,2)... 

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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