Polymer Physics

porous membranes and foam materials via thermal stimulating process to control
their microscopic structures (Graham and McHugh 1998 ). The porous membranes
made of polypropylene and poly(vinylidene fluoride) have been widely applied as
filters in water purification.
Polymer phase separation and crystallization, as introduced separately in the
previous two chapters, have different molecular driving forces that can be simulta-
neously expressed by the use of the lattice model. Adjusting the corresponding
driving forces, the mean-field theory can predict the phase diagrams, and at the
meanwhile molecular simulations can demonstrate the complex phase transition
behaviors of polymers in the multi-component miscible systems.
Let us open the discussion on the thermodynamic interplay. In a 32^3 cubic lattice
space of polymer solutions, the mean-field statistical thermodynamic theory can
predict the complete set of phase diagrams combining both liquid-liquid binodal
curves and liquid–solid coexistence curves of polymer chains with length
32 monomers, as illustrated in Fig.11.1a. Specifically, the binodal curve can be
calculated from the mixing free energy (10.11) with the chemical potential equiva-
lence between the dilute and concentrated phases (9.3) and (9.4). The coexistence
curve can be calculated from the absolute free energy (10.9) with the chemical
potential equilibrium between the solution and the crystalline phases. By the use of
the conformation energyEcas a reference, the mixing interaction parameterBand
the crystallization interaction parameterEpcan be reduced intoB/EcandEp/Ec,
respectively. One may see that, whenEp/Ecswitches from zero to one, the binodal
curve only shifts up slightly; but whenB/Ecchanges from 0.25 to 0.1, one order of
magnitude smaller thanEp/Ec, the binodal curve shifts down significantly. This
result implies that the mixing interaction parameter dominates the binodal curve,
and the crystallization interaction parameter does not play a significant role.
Figure11.1bshows the simulation results in parallel to Fig.11.1a. The simulation
results are roughly consistent with the theoretical phase diagrams under the same

12

10

8

6

4

2

0.0 0.2 0.4 0.6 0.8 1.0

Tl(

Ec

/k)

Tl(

Ec

/k)

T Td(1,0.25)

d(1,0.25)

Td(0,0.25)

Td(1,0.1)

Tm(1,0.1)

Td(0,0.25)

Td(1,0.1)

Tm(1,0.1)

12

10

8

6

4

2

0.0 0.2 0.4 0.6 0.8 1.0

a b

ff

Fig. 11.1(a) Theoretical phase diagrams of temperature versus polymer volume fraction in
polymer solutions with the chain length 32 monomers. (b) Molecular simulation results under
parallel conditions. The reduced interaction parameter sets are labeled asTd(Ep/Ec,B/Ec) and the
liquid–solid coexistence curveTm(Ep/Ec,B/Ec) (Hu et al.2003a) (Reprinted with permission)

224 11 Interplay Between Phase Separation and Polymer Crystallization