Chemical potential Flory model

The first qualitatively correct attempt to model the relevant chemical potentials in a polymer solution was made independently by Huggins (4, ) and Flory [6). Their models, which are similar except for nomenclature, are now usually called the Flory-Huggins model ( ). 

The solid curves in Fig. 5 are the results of a thermodynamic analysis based on a Flory-Huggins model for the interaction between the monomer and polymer and a phenomenological curvature energy model to describe the chemical potential of the monomer in the micelle [30]. 

Phase equilibrium may be dealt with through the equality of chemical potentials of each component in all phases while phase stability may be studied through the appropriate spino-dal conditions and the conditions for the critical points(13). The present LF model uses an "entropic" correction term in the expression for the chemical potential entirely analogous to the corresponding correction term of the Equation-of-State theory of Flory and coworkers (14) and which is equal to -rj for a binary mixture i-j. The "entropic parameter q j is a unitless adjustable binary parameter.This correction term does not appear in the expressions for the excess volume and the excess enthalpy of the mixture. 

The model appears to describe accurately sorption isotherms when the equation of state parameters of both polymer and penetrant are determined. Like the Flory-Huggins modef, the Sanchez-Lacombe model assumes that the different components mix randomly in a lattice. Unlike the Flory-Huggins model, the Sanchez-Lacombe model permits some lattice sites to be empty, which allows holes or free volmne in the fluid. The addition of free volume to the lattice permits volume changes upon mixing components. The amount of absorbed penetrant in the polymer is determined by equating the chemical potential of the penetrant and the chemical potential of the penetrant in the mixture and by satisfying the equation of state of the pure penetrant phase and of the polymer-penetrant mixture. At fixed temperature and pressure, these conditions are met by equations 5-7. 

Above we have introduced a special component in the system that was termed a vacancy. In our system the number of vacancies is countable and from this point of view it is natural to assign a chemical potential to this component. As a consequence there is no pV term in the thermodynamic analysis and we have five molecular components. Alternatively, one could choose to convert the chemical potential of the vacancies into a pressure and renormalize the chemical potentials of all other components accordingly. In this latter case there are only four molecular components, and in addition a —p Vterm occurs in the thermodynamic analysis. The first approach is the classical incompressible (lattice) solution and the second approach is known as a lattice gas (even when we model a condensed liquid solution). In Table 5.2 we have introduced Flory-Huggins interactions also for the vacancies... 

Use the Flory-Huggins expression for the solvent chemical potential in Equation 3.8 to show that in the semi-dilute regime, Vj 1 and therefore ln(l-V2) -V2-v /2, this model would predict that for high-molecular-weight polymers the osmotic pressure would scale as vf. Compare with the scaling law result. To what can you attribute the difference ... 

The characteristic ratio Coo characterizes chain flexibility. It depends on the 6 and torsional potential and is determined by the chemical structure of the monomers [20]. The rotational isomeric state (RIS) model, introduced by P.J. Flory [20] is essentially an adaptation of the one-dimensional Ising model of statistical physics to chain conformations. This model restricts each torsion angle to a discrete set of states (e.g., trans, gauche , gauche"), usually defined around the minima of the torsional potential of a single bond, V((f>) (see Fig. 2d). This discretization, coupled with the locality of interactions, permits calculations of the conformational partition... 

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