SUMMARY:A method is presented which allows the calculation of phase diagrams (spinodal, binodal and tie lines) on the basis of the Gibbs energy of mixing AG. No derivatives of AG with respect to the composition variables are required. This method is particularly useful in cases where the composition dependence of AG is very complex and no analytical representation of the derivatives can be given. The method is applied to a ternary mixture of two homopolymers with a copolymer consisting of the same monomers. The sequence distribution of the copolymer is kept constant between random and purely alternating, and phase diagrams are calculated for different chemical compositions of the copolymer. The complex phase separation behavior resulting for a 1 : 1 copolymer becomes much simpler as one monomeric unit starts to predominate in the copolymer.
Crystallization of molten polymers involves a liquid-solid transition. In its first stage, the solidification process may be viewed as a physical gelation, which can be monitored with small-amplitude oscillatory shear experiments. A threshold crystallinity, X cg, is required for the sample to reach the critical gel state (gel point). Samples of a metallocene random copolymer of ethene and 11 mol % 1-butene were quenched from the melt to a crystallization temperature TX and then held at that temperature for isothermal crystallization. The objective was to find the highest value of TX at which the polymer would crystallize only to the threshold crystallinity Xcg ∞ and no further. The result would be a stable critical gel at a temperature TX ) Tcg ∞ . We start at large degrees of supercooling where the critical gel is only a transient state which is passed through as the sample solidifies to greater degrees of crystallinity. The gel time increases with temperature, obeying a power law, and can be extrapolated to infinity at 68.8°C. The crystallinity at the gel point is about Xcg ) 1%, almost independent of T. The rheological experiments are inherently difficult since the characteristic transition behavior, where tan δ is independent of frequency, occurs only in the terminal frequency region which is at very low frequencies for the sample of this study.
The present calculations were performed on the basis of the Sanchez-Lacombe-Balasz lattice fluid theory. The two system specific parameters ε12* and δε* required for that purpose have been obtained from the spinodal temperatures measured (SANS) for mixtures of poly(vinyl methyl ether) (PVME) and deuterated polystyrenes (d-PS) by Schwahn and coworkers. The experimental data reported for atmospheric pressure and six representatives of the present system are well described theoretically, where ε12* does not depend on molar mass and δε* decreases only slightly as the chain length of d-PS is raised. The measured pressure influences on the spinodal conditions correspond to an approximately linear reduction of δε* with increasing P; this observation should reflect the volume changes associated with the formation of specific interactions. According to the present calculations the critical composition shifts markedly towards pure PVME as P is raised. Since experimental data are commonly expressed in terms of the Flory-Huggins theory, the current results were also translated into Flory-Huggins interaction parameters and evaluated with respect to the contributions of enthalpy and of entropy. The agreement between experimental information and that calculated from the Sanchez-Lacombe-Balasz lattice fluid theory is reasonable.
The phase diagrams of a blend of two homopolymers exhibiting a lower critical solution temperature (LCST) have been calculated on the basis of the generalized Gibbs energy of mixing (sum of the Gibbs energy of mixing of the stagnant blend and the energy the mixture can store during stationary flow) for various shear rates y. In the present paper the temperature dependence of the energy stored in blends was taken into account. The calculations yield a complex dependence: With increasing y the heterogeneous region of the phase diagram is first reduced (shear dissolution), then enlarged (shear demixing), and finally reduced again. So, with varying y, the influences of shear change sign twice; i.e., two inversions of the effects are observed. For very high y values the system behaves like in the quiescent state. Closed miscibility gaps can occur within two ranges of y: Within the first range the islands show up below Tc (the critical temperature of the system) and merge into the original miscibility gap as y is raised; the first inversion point is located in this area. The second y range of islands (entirely located above Tc) is observed within the regime of the second shear dissolution. The occurrence of islands can be suppressed for an appropriate choice of parameters and the previously published simpler behavior is regained.
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