Dfðr;tÞ¼f<f>exp½Dh^2 ðf^00 þ 2 kh^2 Þt (9.35)wheretis the time,Dis the effective diffusion coefficient,his the wave number for the
concentration modulation (scattering vector),f^00 ¼∂^2 Dfm/∂f^2. One can see that
the concentration fluctuations will be amplified with an exponential function, and the
amplification factor
RðhÞ¼Dh^2 ðf^00 2 kh^2 Þ (9.36)Whenf^00 <0 (the system becomes unstable),R(h)can be larger than zero. The
critical condition ish<hc¼(f^00 /2k)1/2(the fluctuation size must be large
enough), and the maximum occurs athmax¼(f^00 /4k)1/2. The scattering factor
measured by experiments isS(h)¼<Df^2 >. Therefore
Sðh;tÞexp½ 2 RðhÞt (9.37)The maximum of the scattering intensity measured in experiments first increases
with time evolution athmax, and then shifts to smaller values ofh(corresponding to
larger sizes of new phases), as illustrated in Fig.9.7.
An optimized bi-continuous periodic structure occurs at the early stage of
spinodal decomposition. The small domains coalescence with each other at the
later stage, in order to minimize the total interfacial area and thus the total free
energy of the system. The structural evolution at the later stage is calledOstwald
ripening(Ostwald 1896 ). According to the Porod law,
Sðh!1Þ¼O
h^4(9.38)
The scattering experiment provides the interfacial densityO(the interfacial area
per unit of volume), and thus observes the decay ofOwitht^1. Correspondingly,
the linear domain size of the new phase increases as
LðtÞðDstÞ^1 =^3 (9.39)whereDis the diffusion coefficient,sis the free energy density of interfaces.
Equation (9.39) is also called theLifshitz-Slyozov law(Lifshitz and Slyozov 1961 ).
In the nucleation and growth mechanism of phase separation, large amplitude of
concentration fluctuation is necessary. Since each component diffuses towards its
low-concentration region (downhill diffusion), the interfaces between the separate
regions of the low and the high concentrations will be sharp. The interfacial
contribution to the free energy change is directly calculated with the interfacial
free energy densitys, instead of using a concentration-gradient function as in the
case of spinodal decomposition. In the present case, not only a large enough
concentration gradient, but also a large enough new phase domain are required.
176 9 Polymer Phase Separation