  We consider the swelling kinetics of layered structures. We focus on the case
of triblock copolymer mesogels, although our results are applicable to other
layered structures including clays. We assume the mesogels are swollen by a
solvent that is good for the bridging block but poor for the non-bridging
block. At long times the penetration front moves as in ordinary diffusion,
i.e., as $t^{1/2}$. At short times, however, the bending elasticity of the
non-bridging layers becomes important. This bending elasticity leads to a
$t^{1/6}$ relaxation of the penetration front at early times. The crossover
length between these two regimes is approximately the width of a single layer.
However, for a large number of lamellae there is a cooperative effect which
leads to a large enhancement of this crossover length.
