Liquid–liquid phase separation in multicomponent polymer solutions. IX. Concentration‐dependent pair interaction parameter from critical miscibility data on the system polystyrene–cyclohexane

Abstract

Critical miscibility data obtained from measurements of phase‐volume ratios have been used to calculate the concentration dependence of the pair interaction parameter for the system polystyrene–cyclohexane. The measured temperature and concentration ranges are 11–30°C and 4–18% polymer by weight, respectively. With the Gibbs free energy of mixing expressed in polymer segment mole fractions, x*, the pair interaction parameter is g(x*, T) = 0.4961 + 71.92/T + 0.2312x* + 0.0750x*2. In a polymer volume fraction formulation the parameter is g(φ, T) = 0.4099 + 90.65/T + 0.2064 φ + 0.0518 φ2, which approximates to χ(φ, T) = 0.2035 + 90.65/T + 0.3092 φ + 0.1554 φ2. Comparison of the temperature and concentration dependence with that obtained by other authors shows very good agreement, even when extensive extrapolations in temperature and concentration are applied. The present function is believed to be the most accurate. Solutions of mixtures of two narrow‐distribution polystyrenes in cyclohexane show separation into three liquid phases under the exact conditions predicted by theoretical calculation with the present pair‐interaction function.

Phase equilibria and phase separation dynamics in a polymer composite containing a main‐chain liquid crystalline polymer

“…1,13 While the corresponding state, the equation of state or the lattice fluid theories are capable of satisfactorily explaining the possible coexistence of both UCST and LCST, an alternative approach in describing the phase diagrams of real polymer systems is to treat the interaction parameter to be a more general function of both temperature and concentration, v(T,/). Hence, the FH free energy of mixing may be expressed as:…”

“…v(T,/) is the generalized interaction parameter, but it cannot be determined explicitly. Hence, it has been expressed empirically as a product of two functions: temperature-dependent term and concentration-dependent term 13 as…”

“…The dependence of the v interaction parameter on polymer chain structures (e.g., molecular weight, shape, and rigidity), concentration, and temperature has been postulated by modern theories, [6][7][8][9][10][11][12] notably integral equation methods by Schweizer and et al, 7,8 the lattice cluster theory of Freed and coworkers, 9,10 and the continuum chain theory by Fredrickson et al 11,12 An inevitable consequence of any refined molecular theory is that the simplicity of the analytical expression afforded by the FH theory may be lost due to increasing arbitrary parameters. An alternative approach is to adopt the empirical scheme introduced by Koningsveld et al 13 that incorporates temperature and concentration dependence of the interaction parameter, and hence the unique features such as simplicity and predictability of the FH theory can be preserved.…”

Liquid-liquid equilibria for solutions of polydisperse polymers. Continuous thermodynamics for the close-packed lattice model

“…Koningsveld, Kleintjens and Shultz [24] reported high-quality experimental liquid-liquidequilibrium data for cyclohexane-polydisperse polystyrene systems. Data reported by Koningsveld et a1 include critical coordinates for nine samples with different number-average, weight-average and z-average molar masses, M,., Mw and Mz shown in Table 1.…”

Coexistence curve of polystyrene in methylcyclohexane. IX. Pressure dependence of tricritical point

“…For ternary systems of two polymer homolog with molecular weights M 1 and M 2 (ϾM 1 ) in a solvent, analysis of the original FH equation predicts three-phase separation, when the molecular weight ratio, rϭM 2 /M 1 , exceeds a critical value rϷ10. 2,3 For bimodal polyethylene 4 and bimodal polystyrene solutions 5 the three-phase separation was observed in the temperature range indicated by analysis of the FH equation with the empirically determined parameter g. A ternary system which has a critical value of r is a tricritical system in which the upper critical end point ͑UCEP͒ and lower critical end point ͑LCEP͒ coincide, to yield a tricritical point ͑TCP͒.…”