role in the first production of small grains, does not occur otherwise. We will surely need to combine our simulation with direct dust motion calculations [e.g. @2016MNRAS.456.4174H; @2018MNRAS.478.2851M] to check if the velocity levels assumed here are actually achieved. Because of the huge computational cost required, we leave this for a future work.]{}

For shattering, the total mass of the fragments is estimated following [@2010Icar..206..735K]. We consider a collision of two dust grains with masses $m_1$ and $m_2$. The total mass of the fragments ejected from $m_1$ is estimated as $$\begin{aligned}
m_\mathrm{ej}=\frac{\phi}{1+\phi}m_1,\end{aligned}$$ where $\phi\equiv{E_\mathrm{imp}}/(m_1Q_\mathrm{D}^\star)$ \[$E_\mathrm{imp}=\frac{1}{2}{m_1m_2}v_\mathrm{1,2}^2/(m_1+m_2)$ is the impact energy and $Q_\mathrm{D}^\star$ is the specific impact energy that causes the catastrophic disruption (i.e. the disrupted mass is $m_1/2$)\]. We adopt $Q_\mathrm{D}^\star =4.3\times 10^{10}$ cm$^2$ s$^{-2}$ (valid for silicate).[^3] Now we set the grain size distribution of shattered fragments. The maximum and minimum grain masses of the fragments are assumed to be $m_\mathrm{f,max}=0.02m_\mathrm{ej}$ and $m_\mathrm{f,min}=10^{-6}m_\mathrm{f,max}$, respectively . We adopt the following fragment mass distribution including the remnant of mass $m_1-m_\mathrm{ej}$ as $$\begin{aligned}
\mu_\mathrm{shat}(m,\, m_1,\, m_2) &=
\frac{(4-\alpha_\mathrm{f})m_\mathrm{ej}m^{(-\alpha_\mathrm{f}+1)/3}}{3\left[
m_\mathrm{f,max}^\frac{4-\alpha_\mathrm{f}}{3}-
m_\mathrm{f,min}^\frac{4-\alpha_\mathrm{f}}{3}\right]}\,
\Phi (m;\, m_\mathrm{f,min},\, m_\mathrm{f,mmax})\nonumber\\
&+ (m_1-m_\mathrm{ej})\delta (m-m_1+m_\mathrm{ej}),\label{eq:frag}\end{aligned}$$ where $\Phi (m;\, m_\mathrm{f,min},\, m_\mathrm{f,mmax})=1$ if $m_\mathrm{f,min}<m<m_\mathrm{f,mmax}$, and 0 otherwise, $\delta $ is Dirac’s delta function, 