$(1-\xi )$ is condensed into the dust phase\]. Therefore, $$\begin{aligned}
  \xi (t) \equiv
  \begin{cases}
    1-\mathcal{D}_{\rm tot}\slash Z\,\,&{(\rm accretion)},\\
    -1\,\, &{(\rm destruction)},
\end{cases}\end{aligned}$$ where $Z$ is the metallicity.

The destruction time-scale $\tau_\mathrm{dest}(m)$ is estimated based on the sweeping time-scale divided by the grain-mass-dependent destruction efficiency as [e.g. @1989IAUS..135..431M] $$\begin{aligned}
\tau_\mathrm{dest}(m)=
\frac{M_\mathrm{gas}}{\epsilon_\mathrm{dest}(m)M_\mathrm{s}\gamma},
\label{eq:tau_dest}\end{aligned}$$ where $M_\mathrm{gas}$ is the mass of the gas particle of interest, $M_\mathrm{s}=6800$ M$_{\sun}$ is the gas mass swept by a single SN blast, $\gamma$ is the rate of SNe sweeping the gas, $\epsilon_{\rm dest} (m)$ is the dust destruction efficiency as a function of grain mass. We adopt the following empirical expression for the destruction efficiency (described as a function of $a$ instead of $m$):[^2] $$\begin{aligned}
\epsilon_{\rm dest}(a)=1-\exp\left[-0.1\left(\frac{a}{0.1\,\micron}\right)^{-1}\right] .
\label{eq:eps}\end{aligned}$$

For accretion, the growth time-scale is estimated as (we fixed the sticking efficiency $S=0.3$ in HA19) $$\begin{aligned}
\tau_\mathrm{acc}(m) &= \frac{1}{3} \tau^\prime_\mathrm{0,acc}\left(\frac{a}{0.1~\micron}
\right)\left(\frac{Z}{\mathrm{Z}_{\sun}}\right)^{-1}
\left(\frac{n_\mathrm{H}}{10^3\,\mathrm{cm}^{-3}}
\right)^{-1}\left(\frac{T_\mathrm{gas}}{10\,\mathrm{K}}
\right)^{-1/2},
\label{eq:tau_acc}\end{aligned}$$ where $\tau^\prime_\mathrm{0,acc}$ is a constant [we adopt $\tau^\prime_\mathrm{0,acc}=1.61\times 10^8$yr appropriate for silicate, but the value for graphite has little difference; @2012MNRAS.422.1263H], and $T_\mathrm{gas}$ is the gas temperature.

### Shattering and coagulation {#subsubsec:shattering}

Shattering and coagulation are described by grain–grain collisions followed by the redistribution of fragments or coagulated grains [@1994ApJ...433..797J; @1996ApJ...469..740J; @2009MNRAS.394.1061H]. Shattering 