and $\alpha_{\rm f}$ is the power-law index of the fragment size distribution ($\alpha_{\rm f} = 3.3$; @1996ApJ...469..740J). Grains whose radii are smaller than the minimum grain radius ($3\times 10^{-4}~\micron$) are removed.

For coagulation, we adopt $$\begin{aligned}
\mu_\mathrm{coag}=m_1\delta (m-m_1-m_2)\end{aligned}$$ in equation (\[eq:shat\]).

Calculation of extinction curves {#subsec:ext}
--------------------------------

Extinction curves describe the wavelength dependence of the optical depth of dust. Extinction curves have been useful in constraining the grain size distribution [e.g. @2001ApJ...548..296W]. Here, we calculate extinction curves based on the grain size distributions, $n_{i}(a)$, where $i$ indicates the composition of the dust grains (we consider multiple compositions as we explain below). The extinction at wavelength $\lambda $ in units of magnitude is written as $$\begin{aligned}
A_{\lambda}=(2.5 \log_{10}\mathrm{e})L\displaystyle\sum_{i}\displaystyle\int_{0}^{\infty}
n_{i}(a)\,\upi a^{2}Q_{\rm ext}(a, \lambda),\end{aligned}$$ where $L$ is the path length, $Q_{\rm ext}(a, \lambda)$ is the extinction efficiency factor, which is evaluated by using the Mie theory [@1983asls.book.....B] with the same optical constants for silicate and carbonaceous dust (graphite) as in [@2001ApJ...548..296W]. In this paper, the mass fractions of silicate and carbonaceous dust are fixed to 0.54 and 0.46, respectively (number ratio 0.43:0.57) with both grain species having the same grain size distribution [@2009MNRAS.394.1061H]. Because this fraction is valid only for the Milky Way, 