Linearized Analysis of Spinodal Decomposition

These results can be used to calculate the time evolution of structure factors as measured by small-angle light scattering or X-ray scattering. The structure factor for concentration is defined by 

We first consider the behavior oftheSD with lo = OinEq. (2.77). When lo = 0,we obtain b(q) = d(q) = 0 and then the kinetic equations (2.82) and (2.83) have no cross term between the gradients of the concentration and orientation order parameters. Such behaviors of SD in PDLC have been discussed theoretically (73, 119-122]. The time evolution of the structure factor for the concentration fluctuations falls into the Cahn-Hilliard classical category [102,103] and Eq. (2.84) results in the Cahn theory of SD for isotropic solutions. The structure factor for concentration in Eq. (2.92) is given by 

When/Kj, 0, from Eq. (2.88), the amplitude of any concentration fluctuation decreases with time because a(q) 0 and so the system is stable. If 0, concentration fluctuations are unstable for the wave vector in the range 0 q qo = and the amplitude of the corresponding modes grows 

When Lo has nonzero values, we have a coupling between concentration and orientation. This cross term plays a significant role in phase separation kinetics of solutions containing liquid crystals or liquid crystalline polymers [112,113,116]. The structure factors are affected by this cross term even in the linear regime of the SD. 

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