Dynamic structure factor of crosslinked polymer blends

In this paper, we are interested in the critical behavior of the dynamic structure factor of a crosslinked polymer blend made of two chemically incompatible polymer A and B  , when it is suddenly cooled down from a high initial temperature towards a final one very close to the spinodal point. Since the critical fluctuations occur over distances smaller than the mesh size (microdomains size), ξ*ξ*, the dynamic structure factor should be governed by a short-time behavior we want to determine. We demonstrate that the final time, t*t*, necessary to the appearance of microdomains alternatively rich in A and B  -polymers, scales as t*∼ξ*zt*∼ξ*z, with z a dynamic critical exponent. The investigation of the dynamic structure factor is first achieved using a mean-field approach, based on an extended Van Hove theory, and second by a scaling argument. The Van Hove theory is valid as long as the fluctuations of composition can be underestimated. Within the framework of this model, we determine an exact   form for the dynamic structure factor, and in particular, we find that the corresponding dynamic exponent is z0=6z0=6. Second, the study is extended to the case of crosslinked polymer blends of low-molecular-weight, where fluctuations of composition are strong enough near the spinodal temperature. Using a scaling argument, we prove that the scaling law for the dynamic structure factor is S(q,t)=qη-2f(qR(t))S(q,t)=qη-2f(qR(t)), with ηη the standard Ising critical exponent, f(x)f(x) a universal   scaling function, and R(t)∼t1/zR(t)∼t1/z some time-characteristic length. The latter can be interpreted as the size of instabilities domains at time t   , and it becomes of the order of ξ*ξ* at time t*t*. The growing process of instabilities is then stopped at the final time t*t*. We show that the dynamic exponent z   is not trivial and has as three-dimensional value z≃5.969±0.001z≃5.969±0.001. In dimension 2, we find an exact   value for this exponent that is z=234.