Lattice models for solutions

Figure 3.3 Schematic representation of quasiciystalline lattice model for solution. (a) Mixture of molecules of equal size, (b) Mixture of solvent molecules with a polymer molecule showing the connectivity of polymer segments.

Figure 3.2 Schematic representation of quasicrystalline lattice model for solutions, (a) Simple solution mixture of molecules of equal size, white circles representing the solvent molecules and filled circles the solute molecules. It is assumed that solvent molecules can exchange sites with solute molecules. This results in an increase in the number of ways they can be arranged, and hence in an increase in entropy Jb) Polymer solution mixture of solvent molecules (unfilled circles) With a polymer molecule composed of chain segments (each segment represented by a filled circle) tied with chemical bonds. It is assumed that solvent molecules can exchange sites with polymer chain segments. This results in an increase in entropy. (After Flory, 1953.)...

Helfand E (1976) Theory of inhomograieous polymers. Lattice model for solution interfaces. Macromolecules 9 307 310... 

Figure 9.1 Two-dimensional lattice model for a solution of two different atoms of similar radius.

Figure 9.5 Two-dimensional illustration of the lattice model for polymer solutions. Black sites are occupied by the polymer chain, white sites by solvent monomers.

Applying the lattice model for polymer solutions J. S. Mackie and... 

A lattice model for an electrolyte solution is proposed, which assumes that the hydrated ion occupies ti (i = 1, 2) sites on a water lattice. A lattice site is available to an ion i only if it is free (it is occupied by a water molecule, which does not hydrate an ion) and has also at least (i, - 1) first-neighbors free. The model accounts for the correlations between the probabilities of occupancy of adjacent sites and is used to calculate the excluded volume (lattice site exclusion) effect on the double layer interactions. It is shown that at high surface potentials the thickness of the double layer generated near a charged surface is increased, when compared to that predicted by the Poisson-Boltzmann treatment. However, at low surface potentials, the diffuse double layer can be slightly compressed, if the hydrated co-ions are larger than the hydrated counterions. The finite sizes of the ions can lead to either an increase or even a small decrease of the double layer repulsion. The effect can be strongly dependent on the hydration numbers of the two species of ions. 

Assuming for the moment a lattice model for liquid structure, we can calculate the rate at which a molecule A encounters new neighbors in the solution as the rate at which it jumps to a new lattice site times Z/2, since it will change half its neighbors at each j ump. The frequency is... 

The influence of the interaction in binary solvents on AG r ions was analyzed by Y. Marcus [261], who assumed a quasi-lattice model for the electrolyte in such solutions. Free energies of transfer of various ions were collected and discussed [75, 76]. Ion solvation including mixed solvent media has been reviewed by several authors [45, 76, 262-265]. 

Flory ) and Huggins ) derived a now classical mean-field expression for the configurational entropy and energy of mixing, using a lattice model. The solution, containing moles of solvent and n moles of polymer, is described as a lattice of sites, of which are occupied by solvent and N

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In the lattice model for polymer solutions which assumes that there is no change in volume on mixing, volume fractions (pi and concentrations c are related as... 

We note that the necessary and sufficient condition, Aab = 0, for SI was derived here and is valid for mixtures at constant P, T. The condition (5.38) is very general for SI solutions. It should be recognized that this condition does not depend on any model assumption for the solution. For instance, within the lattice models of solutions we find a sufficient condition for SI solutions of the form (Guggenheim 1952)... 

Note that when b = a, y = 1/48, as it should for a circle. Equation [27] also applies to unidirectional flow between parallel flat plates, and can be used as a model for solute dispersion in rock fractures. Setting in Eq. [5] equal to the separation distance between two plates, it can be shown that y = 1/210 (Aris, 1959a Wooding, 1960). This result has been confirmed by numerical simulations (Koplik et al 1993) and lattice-gas automata (Perea-Reeves Stockman, 1997). 

In 1925 Ising [14] suggested (but solved only for the relatively trivial case of one dimension) a lattice model for magnetism in solids that has proved to have applicability to a wide variety of other, but similar, situations. The mathematical solutions, or rather attempts at solution, have made the Ising model one of the most famous problems in classical statistical mechanics. 

The micelle has too small an aggregation number to be considered as a phase in the usual sense, and yet normally contains too many surfactant molecules to be considered as a chemical species. It is this dichotomy that makes an exact theory of solubilization by micelles difficult. The primary theoretical approaches to the problem are based on either a pseudophase model, mass action model, multiple equilibrium model, or the thermodynamics of small systems [191-196]. Technically, bulk thermodynamics should not apply to solute partitioning into small aggregates, since these solvents are interfacial phases with large surface-to-volume ratios. In contrast to a bulk phase, whose properties are invariant with position, the properties of small aggregates are expected to vary with distance from the interface [195]. The lattice model of solute partitioning concludes that virtually all types of solutes should favor the interface over the interior of a spherical micelle. While for cylindrical micelles, the internal distribution of solutes... 

It should be noted that the general idea of splitting any thermodynamic quantity of solution into two terms as in (3.1.1) and (3.1.2) can be formally made exact. However, Eley also used a lattice model for water and assumed that water contained a fixed number of cavities, or holes. He was able to give a qualitative interpretation of the difference in AH and AS for an inert solute s in water and in other liquids. Today, the lattice type of model used by Eley and the assumption of a fixed number of holes are not acceptable. 

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