Transition from random to ordered fractals in fragmentation of particles in an open system

We consider the fragmentation process with mass loss and discuss self-similar properties of the arising structure both in time and space, focusing on dimensional analysis. This exhibits a spectrum of mass exponents $\theta$, whose exact numerical values are given for which $x^{-\theta}$ or $t^{\theta z}$ has the dimension of particle size distribution function $\psi(x,t)$, where $z$ is the kinetic exponent. We obtained conditions for which the scaling and fragmentation process altogether break down and give explicit scaling solution for special case. Finally, we identify a new class of fractals ranging from random to non-random and show that the fractal dimension increases with increasing order and a transition to strictly self-similar pattern occurs when randomness completely ceases.