The Cell Models

In the absence of convective effect, the profiles of > between any two adjacent bubbles exhibits an extremum value midway between the bubbles. Therefore, there exists around each bubble a surface on which d jdr = 3(C )/3r = 0, and hence the fluxes are zero. Using the cell model [Eqs. (158) or (172)] one obtains the following boundary conditions For t > 0... 

In a cellular automata model of a solution, there are three different types of cells with their states encoded. The first is the empty space or voids among the molecules. These are designated to have a state of zero hence, they perform no further action. The second type of cell is the water molecule. We have described the rules governing its action in the previous chapter. The third type of cell in the solution is the cell modeling a solute molecule. It must be identified with a state value separate from that of water. 

Fig.2. ApAT of the potentiometric titration of PGA ( 0.011 monomol dm ) with strong base as a function of a ( ) theoretically calculated curve on the basis of the cell model, (+) LiOH, ( a ) (C//3 NOH, and ( O ) C H NOH.

Thus we prefer to make the cell model our reference for density fluctuations. Density fluctuations are thereby truncated but in a well-defined way open to correction. 

We recently conducted experiments to extract mRNA from a cell model (MBA-MB-486 cell line of human breast cancer) processed in both frozen and FFPE blocks in a comparable fashion (unpublished data). The cell model system was prepared in three ways for comparison (1) Positive Control Fresh Cell Pellet Two flasks of cells were collected in a pellet, and stored at -70°C until use (2) Frozen Cell Block Two flasks of cells was embedded in OCT... 

Compound (Miles Laboratories, Elkhart, IN), snap-frozen, and cut into sections for comparison with paraffin-embedded cell sections (3) FFPE Cell Blocks Six cell pellets were fixed in 10% neutral buffered formalin immediately after harvest, at room temperature for 6,12,24h, 3,7, and 30 days, respectively. For further comparison with the cell model system, recently collected sample of human breast cancer tissues were processed by OCT-embedding and snap-freezing the corresponding routine FFPE block that was obtained from the Norris Cancer Hospital and Research Institute at the University of Southern California Keck School of Medicine (USC). This tissue block was processed routinely (formalin-fixed 24h and processed by automatic equipment). 

The extension of the cell model to multicomponent systems of spherical molecules of similar size, carried out initially by Prigogine and Garikian1 in 1950 and subsequently continued by several authors,2-5 was an important step in the development of the statistical theory of mixtures. Not only could the excess free energy be calculated from this model in terms of molecular interactions, but also all other excess properties such as enthalpy, entropy, and volume could be calculated, a goal which had not been reached before by the theories of regular solutions developed by Hildebrand and Scott8 and Guggenheim.7... 

The greatest success of the cell model of solutions was probably the qualitative prediction that mixtures of spherical molecules of the same size may simultaneously show a positive excess free... 

From the quantitative point of view, the success of the cell model of solutions was more limited. For example, a detailed analysis of the excess functions of seven binary mixtures by Prigogine and Bellemans5 only showed a very rough agreement between theory and experiment. One should of course realize here that besides the use of the cell model itself, several supplementary assumptions had to be made in order to obtain numerical estimates of the excess functions. For example, it was assumed that two molecules of species and fi interact following the 6-12 potential of Lennard-Jones ... 

Concerning point (b), a generalized theory was developed independently by Prigogine and his co-workers10 11 and by Scott.18 The main idea was to combine the concept of average potential involved in the cell model with the theorem of corresponding states for pure compounds, in such a way that ... 

The second basis of the APM is the cell model, which is required in order to define the average potential acting on a molecule in a mixture A + B. The particular way this average potential is constructed from the pair potentials eAA(r), e ir), and eBB(r) leads to the different versions of the APM. 

All these expressions clearly reduce to the theorem of corresponding states for a one-component system (cf. Eqs. (8) and (10)). The problem is now to attribute values to the reduced volumes and for A and B molecules in their respective mean fields in other words how is the available volume V shared between the molecules A and B We recover here a typical problem of the cell model. Three different assumptions on , (vBy have been proposed11 leading to slightly different versions of the APM ... 

In the simplest version of the transformation, the state of occupancy of a given bond lattice point is independent of the states of occupancy of other bond lattice points this corresponds to neglecting interaction between hydrogen bonds. The calculation of the distribution of possible occupation states of the bond lattice replaces the enumeration of occupancy states of the basic lattice section in the cell model, but in the simplest model the bond lattice occupancy distribution only accounts for a subset of possible basic lattice section occupancies. 

The random network bond lattice transformation very clearly displays some aspects of the relationship between the cell model and broken bond (multistate) models. We have already remarked on the analogies between states of occupancy... 

SOL. 15.1. Prigogine et V. Mathot, Application of the cell model to the statistical thermodynamics of solutions, J. Chem. Phys. 20, 49-57 (1952). 

MSE.IO. I. Prigogine et L. Saraga, Sur la tension superficielle et le modele ceUulaire de I etat liquide (On the surface tension and the cell model of liquid state), J. Chim. Phys. 49, 399-407 (1952). 

In the dense system the convergence of the virtual expansion is doubtful. In any case the higher coefficients are hard lo calculate here approximate theories have been developed, of which the cell model is an example,... 

Numerical calculations for the residual stresses in the anode-supported cells are carried out using ABAQUS. After modeling the geometry of the cell of the electro-lyte/anode bi-layer, the residual thermal stresses at room temperature are calculated. The cell model is divided into 10 by 10 meshes in the in-plane direction and 20 submeshes in the out-plane direction. In the calculation, it is assumed that both the electrolyte and anode are constrained each other below 1400°C and that the origin of the residual stresses in the cell is only due to the mismatch of TEC between the electrolyte and anode. The model geometry is 50 mm x 50 mm x 2 mm. The mechanical properties and cell size used for the stress calculation are listed in Table 10.5. 

Much of the modeling work to date has dealt with the separation of binary mixtures composed of a carrier and an impurity Such separation can be readily treated using the method of characteristics. While in some situations (Knaebel and Hill, (8), for instance) the same method can be used for binary mixtures of arbitrary composition, the cell model is readily and generally useful for this situation It was this... 

Fig. 2. Two variants of constructing the cell model of the closed circuit milling system.

The electrostatic effects are influenced by the micelle concentration. This effect can be viewed as a micelle-micelle interaction mediated by counterions. The most direct way for modelling the finite micelle concentration is to confine the volume per micelle by an outer radius if of finite size298 300). This is called the cell model. 

Recently, the stiff-chain polyelectrolytes termed PPP-1 (Schemel) and PPP-2 (Scheme2) have been the subject of a number of investigations that are reviewed in this chapter. The central question to be discussed here is the correlation of the counterions with the highly charged macroion. These correlations can be detected directly by experiments that probe the activity of the counterions and their spatial distribution around the macroion. Due to the cylindrical symmetry and the well-defined conformation these polyelectrolytes present the most simple system for which the correlation of the counterions to the macroion can be treated by analytical approaches. As a consequence, a comparison of theoretical predictions with experimental results obtained in solution will provide a stringent test of our current model of polyelectrolytes. Moreover, the results obtained on PPP-1 and PPP-2 allow a refined discussion of the concept of counterion condensation introduced more than thirty years ago by Manning and Oosawa [22, 23]. In particular, we can compare the predictions of the Poisson-Boltzmann mean-field theory applied to the cylindrical cell model and the results of Molecular dynamics (MD) simulations of the cell model obtained within the restricted primitive model (RPM) of electrolytes very accurately with experimental data. This allows an estimate when and in which frame this simple theory is applicable, and in which directions the theory needs to be improved. 

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]. 

Another attempt to go beyond the cell model proceeds with the Debye-Hiickel-Bjerrum theory [38]. The linearized PB equation is used as a starting point, however ion association is inserted by hand to correct for the non-linear couplings. This approach incorporates rod-rod interactions and should thus account for full solution properties. For the case of added salt the theory predicts an osmotic coefficient below the Manning limiting value, which is much too low. The same is true for a simplified version of the salt free case. 

Another viable method to compare experiments and theories are simulations of either the cell model with one or more infinite rods present or to take a solution of finite semi-flexible polyelectrolytes. These will of course capture all correlations and ionic finite size effects on the basis of the RPM, and are therefore a good method to check how far simple potentials will suffice to reproduce experimental results. In Sect. 4.2, we shall in particular compare simulations and results obtained with the DHHC local density functional theory to osmotic pressure data. This comparison will demonstrate to what extent the PB cell model, and furthermore the whole coarse grained RPM approach can be expected to hold, and on which level one starts to see solvation effects and other molecular details present under experimental conditions. 

The osmotic coefficient obtained experimentally from polyelectrolyte PPP-1 having monovalent counterions compares favorably with the prediction of the PB cell model [58]. The residual differences can be explained only partially by the shortcomings of the PB-theory but must back also to specific interactions between the macroions and the counterions [59]. SAXS and ASAXS applied to PPP-2 demonstrate that the radial distribution n(r) of the cell model provides a sufficiently good description of experimental data. 

Abstract In this chapter we review recent advances which have been achieved in the theoretical description and understanding of polyelectrolyte solutions. We will discuss an improved density functional approach to go beyond mean-field theory for the cell model and an integral equation approach to describe stiff and flexible polyelectrolytes in good solvents and compare some of the results to computer simulations. Then we review some recent theoretical and numerical advances in the theory of poor solvent polyelectrolytes. At the end we show how to describe annealed polyelectrolytes in the bulk and discuss their adsorption properties. 

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