Pyun-Fixman model

Caroline and co-workers have recently reported measurements of translational diffusion coefficients in solutions of PS in two mixed-solvent systems at or near theta conditions. In the solvent CCb-methanol (85), they observed the diffusion theta state, defined when the coefficient y of Equation 41 equals 0.5, to occur at 25°C and a volume fraction of CCI4, (fyCCU = 0.8025. In this system there is strong preferential adsorption of the polymer for CCI4, and it is not possible to define a true theta state such that y = a = V2 and A2 = 0 simultaneously. Under diffusion theta conditions, the concentration dependence of Dt apparently is closely described by the Pyun-Fixman hard-sphere model. In the mixed solvent benzene—2 propanol, polystyrene exhibits a true theta condition at T = 25.5°C and (benzene) = 0.04. Frost and Caroline confirmed that y = 0.5 within experimental error in this system (86) and report that values of the parameter fcf are scattered between the extreme values corresponding to the predictions of Yamakawa (and Imai) and the soft-sphere model of Pyun-Fixman (or the Freed theory). 

Equation 73 satisfactorily explains frictional coeflBcient data for PS in a variety of solvents (87). The thermodynamic contributions to kjy are modelled quite well using the two-parameter theory of polymer solutions (87). Recent QLLS experiments on PS/tetrahydrofuran solutions are also in accord with the Pyun-Fixman theory (37,44), However, other studies of the frictional coeflBcient of poly-a-methylstyrene in trans-decalin and toluene indicate that the Pyun-Fixman theory but not the Yama-kawa theory fits the data at small excluded volumes (88). Neither theory works well at large excluded volumes (88). 

The idea of Kirkwood (25) is combined with the Rouse model by Pyun and Fixman (14). The theory allows a uniform expansion of the bond length by a factor a such as introduced by Flory. The nondiagonal term of the Oseen tensor is considered but only to the first order by a perturbation method. Otherwise, their theory is identical to Zimm s theory in Hearst s version in the treatment of the integral equation (14). 

In classical analysis, the concentration dependence of Dt has been described by Equations 35-37. For the simplest hydrodynamic model, that of impermeable neutral hard spheres, Pyun and Fixman (75) derived the result ... 

Corresponding theoretical analysis of concentration dependence of Dt in solutions of flexible coil macromolecules have been carried out exclusively using the classical theory embodied in Equation 37 and have also turned out to be very sensitive to details of the particular model. The original treatment of Pyun and Fixman (75) based on a model of soft interpenetrable spheres of uniform segment density has been most commonly applied to these data. Their result is ... 

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