Polymer Phase Separation
The thermodynamic phase diagrams and their implications to the kinetics of phase separation in the polymer-based miscible multi-component systems have been introduced. In addition, the microphase separation in diblock copolymer systems attracts specific t
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Polymer Phase Separation
9.1
Thermodynamics of Phase Separation
Phase separation is a spontaneous process for polymer chains to segregate from a mixture into a more concentrated phase with clear boundaries. The decrease of the system free energy after mixing two components leads to stable homogeneous polymer mixture. The Flory-Huggins equation for the mixing free energy of polymer-based mixtures shows that, the mixing entropy is always positive and favors mixing. Therefore, the total mixing free energy is mainly determined by the sign and magnitude of the mixing heat. For non-polar polymers, the mixing heat is always positive, as described by (4.9) (the Scatchard-Hildebrand equation) in Sect. 4.2.1. The mixing heat can be so large that the polymer solution becomes thermodynamically unstable, and spontaneously transforms into two coexisting phases: a polymer-rich phase, and a polymer-poor phase. The condition for a thermodynamic equilibrium between two coexisting phases is the equivalence of chemical potentials with respect to each component. The mixing free energy changes with concentrations in the homogeneous mixing states, as illustrated in Fig. 9.1a. Assuming a lattice polymer blend with the total volume N ¼ r1N1 þ r2N2, where two polymers with separate molecular weights r1 and r2 are blended with corresponding molecular numbers N1 and N2. The free energy density Dfm ¼ DFm/N. If we draw a tangent line from a given point on the curve of Dfm versus the volume fraction f2, its intercepts at f2 ¼ 0 and f2 ¼ 1 separately correspond to the chemical potentials Dm1 and Dm2 for two components, as defined by Dm1
@Dfm @f1
(9.1)
Dm2
@Dfm @f2
(9.2)
W. Hu, Polymer Physics, DOI 10.1007/978-3-7091-0670-9_9, # Springer-Verlag Wien 2013
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9 Polymer Phase Separation
Fig. 9.1 Illustration of mixing free energy as a function of polymer volume fractions. (a) The mixing state are stable over all f2; (b) f2 are stable only at points A and B and the regions outside of these two points
When the phase separation occurs, the curve of Dfm versus f2 exhibits a common tangent line at two points of A and B, as illustrated in Fig. 9.1b. This implies that at A and B states, Dm1A ¼ Dm1B (9.3) Dm2A ¼ Dm2B
(9.4)
The common tangent rule above is the thermodynamic condition for the equilibrium between A and B phases. The temperature dependence of the concentrations at A and B states outlines the phase coexistence curve, which is called the binodal line. When temperature is high enough, A and B points will merge at the critical point of phase separation. The thermodynamic condition for the critical point is that partial derivatives of both the first and the second orders for the free energy with respect to the concentration are equal to zero. From f1 þ f2 ¼ 1
(9.5)
@ 3 Dfm 1 1 ¼ kTð Þ¼0 3 2 @f1 r2 f2 r1 f1 2
(9.6)
@ 2 Dfm 1 1 ¼ kTð þ 2wÞ ¼ 0 2 r1 f1 r2 f2 @f1
(9.7)
and
Solve the above three simultaneous equations, we can obtain pffiffiffiffi r1 f2c ¼ pffiffiffiffi pffiffiffiffi r1 þ r2
(9.8)
9.1 Thermodynamics of Phase Separation
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