Kinetics of depolymerization.

Abstract

An approach to finding exact solutions to the binary fragmentation equation is presented. This approach is used to solve a general class of exact solutions with a fragmentation rate F(x,y)=(x+y${)}^{\mathrm{\ensuremath{\alpha}}}$\ensuremath{\delta}(x-y). This fragmentation rate describes a type of depolymerization in which the polymer chains always split in the middle at different rates that depend on the length of the polymer chain and the homogeneity index \ensuremath{\alpha}. For \ensuremath{\alpha}g0, corresponding to the case where larger sized fragments are more likely to split into two equally sized pieces, the asymptotic form of the scaled cluster size distribution \ensuremath{\Phi}(\ensuremath{\xi}) decays as ${\ensuremath{\xi}}^{\mathrm{\ensuremath{-}}2}$ exp(-${\ensuremath{\xi}}^{\mathrm{\ensuremath{\alpha}}}$/2\ensuremath{\omega}\ensuremath{\alpha}) as \ensuremath{\xi}\ensuremath{\rightarrow}\ensuremath{\infty} and exp[-\ensuremath{\alpha}(ln\ensuremath{\xi}${)}^{2}$/2 ln2] as \ensuremath{\xi}\ensuremath{\rightarrow}0, where \ensuremath{\xi} is the scaled mass. For \ensuremath{\alpha}0, we get a ``shattering'' transition. In this case, the scaled cluster size distribution has an asymptotic form that depends on the initial conditions. Finally, our approach is compared and contrasted with other approaches currently used to find exact solutions to the binary fragmentation equation.