Solution, concentrated lattice model

To find the relationship between the vapor pressure pb of the B molecules and the solution concentration Xb, use Equation (11.50), pe(gas) = IcTInpe/pB.jnt-for the chemical potential of B in the vapor phase. For the chemical potential of B in the solution, use lattice model Equation (15.15), pB(liquid) = kTlnx + zwbbI2 + kTx,4fl(l - Xb). Substitute these equations into the equilibrium Equation (16.1) and exponentiate to get... 

The randomly occupied lattice model of a polymer solution used in the Rory-Huggins theory is not a good model of a real polymer solution, particularly at low concentration. In reality, such a solution must consist of regions of pure solvent interspersed with locally concentrated domains of solvated polymer. 

The addition of water to solutions of PBT dissolved in a strong acid (MSA) causes phase separation in qualitative accord with that predicted by the lattice model of Flory (17). In particular, with the addition of a sufficient amount of water the phase separation produces a state that appears to be a mixture of a concentrated ordered phase and a dilute disordered phase. If the amount of water has not led to deprotonation (marked by a color change) then the birefringent ordered phase may be reversibly transformed to an isotropic disordered phase by increased temperature. This behavior is in accord with phase separation in the wide biphasic gap predicted theoretically (e.g., see Figure 8). The phase separation appears to occur spinodally, with the formation of an ordered, concentrated phase that would exist with a fibrillar morphology. This tendency may be related to the appearance of fibrillar morphology in fibers and films of such polymers prepared by solution processing. 

A partitioning function for a system of rigid rod-like particles with partial orientation around an axis is derived from the use of a modified lattice model. The free energy of mixing is shown as a function of the mole numbers, the axis ratio of the solute particles and a disorientation parameter this function passes through a minimum with increase in the disorientation parameter. The chemical potentials display discontinuities at the concentration at which the minimum appears and then separation into an isotropic phase and a somewhat more concentrated anisotropic phase arises. The critical concentration, v, is given in the form 13) ... 

It is important to note that the theoretical results obtained by Flory about the dependence of the critical concentrations v and vj on x are in good agreement with experimental data. It is sufficient to remember, as an example, the results obtained by Flory for PBLG solutions in dioxane (Fig. 4). The discrepancy between the experimental results (solid curves) and theoretical calculations (dashed curves) looks quite natural on the account of a number of assumptions made when deriving the equation for the free energy on the basis of the lattice model (see also ). 

Physical interpretation of the X species employed in ion-exchange half reactions is governed by the molecular model used to describe the ion-exchange process. If the ion exchanger is viewed as a porous gel, for example, in which counterions are distributed within the charged solid lattice, the ion concentrations in the exchanger phase may be related to the corresponding bulk solution concentrations by (32)... 

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... 

Few studies have been made on transport processes involving concentrated solutions. In the concentrated solutions, in the range of dehydrated melt formation, incompletely hydrated melts and anhydrous salt melts, various structural models are described to define their properties, i.e. the free-volume model, the lattice-model and the quasi-crystalline model. Measured and calculated transport phenomena do not always represent simple ion migration of individual particles, but instead we sometimes find them to be complicated cooperative effects (27). 

The Onsager and Flory theories are both statistical theories on rigid rod liquid crystalline polymers, but the former is a virial approximation while the latter is a lattice model. The first is more applicable to dilute solutions while the latter works especially well for high concentrations and a highly ordered phase. These theories with experiments, especially critical volume fractions 4>i and critical order parameter Sc at nematic-isotropic transition are made below. 

The structure function of a homogeneous solution is related to the density-density fluctuations of monomers in the solution. However, the definition of the concentration is model-dependent. On a lattice, the monomer concentration is the number cp of monomer per site. For the standard continuous model, it is expressed as an area per unit volume, which is denoted by < . For a Kuhnian chain, the quantity = CA 1/V represents the monomer concentration. However, the definition of the structure function should not really depend on the model under consideration, and therefore we shall define this quantity in an intrinsic manner. 

In order to describe the collapse of a long-chain polymer in a poor solvent, Flory developed a nice and simple theory in terms of entropy and enthalpy of a solution of the polymer in water [14]. In order to obtain these two competing thermodynamic functions, he employed a lattice model which can be justified by the much larger size of the polymer than the solvent molecules. The polymer chains are represented as random walks on a lattice, each site being occupied either by one chain monomer or by a solvent molecule, as shown in Figure 15.8. The fraction of sites occupied by monomers of the polymer can be denoted as 0, which is related to the concentration c, i.e., the number of monomers per cm by 0 = ca, where is the volume of the unit cell in the cubic lattice. Though the lattice model is rather abstract, the essential features of the problem are largely preserved here. This theory provides a convenient framework to describe solutions of all concentrations. 

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