Computational methods

The aggregate model and the numerical method to compute the optical properties of coagulated particles are adopted from HS. The aggregates are assumed to be in the form of PCA (50%) and CCA (50%) particles consisting of $0.01 \mu$m spherical subgrains. The spectral representation of inhomogeneous media (Bergman [1978]) and the discrete multipole method (DMM) are elaborated to calculate effective dielectric functions of aggregates (Stognienko et al. [1995]). At first, we use the DMM to calculate the spectral function of the aggregated particles of a special topology, when their subgrains touch each other only at one point. Then, we account for the interaction strength between the subgrains (percolation), which varies with the size and type of the aggregates, by an analytical expression (see Equation $13$ in HS). Finally, the effective dielectric functions of the aggregates are obtained by the spectral representation (HS, Eq. $5$). In the case of the composite aggregates, we compute the optical constants of the composite material by the Bruggeman effective medium theory (Bruggeman [1935]), generalized to many components by Ossenkopf ([1991]). The optical properties of the dust aggregates are calculated with the usual Mie theory. It should be noted that this numerical approach is valid in the static limit only, which means that the scale of inhomogeneities within the particles must be small compared to the wavelength. Given the $0.01 \mu$m size of the subgrains and the shortest wavelength of $0.1 \mu$m we considered, this condition is fulfilled.

We use the numerical approach of VM to model the (porous) composite and multishell dust particles. In this method, the composite grains are represented as spheres with many concentric shells, where each shell includes several layers of randomly distributed dust materials. The multishell grains are modelled exactly as the composite ones but each shell includes only one layer of a dust constituent. Then a generalized multilayered Mie theory can be applied to calculate their optical properties. As it has been shown by VM, a convergence in the optical behavior of the multishell particles is achieved if the number of shells exceeds $3$ and dust materials within shells are randomly ordered. In our calculations, we found that this number must be at least $20$, because highly absorbing materials, like troilite and iron, are used, which induces interference within the shells and prevents fast convergence. Thus, in our case, a typical composite grain is modelled as a spherical particle with about hundred shells. On the contrary, a multishell grain is represented by a spherical particle with only a few shells.

We modify somewhat the dust model for the case of multishell and composite spherical particles. We mixed the silicates and iron into one material with the Bruggeman mixing rule. A similar mixture of silicates, sulphides, and metals (GEMS, Glass with Embedded Metals and Sulphur) is found to be a common component of interplanetary dust particles (Rietmeijer & Nuth [2000]). However, the main reason for this change is the convergence failure of the applied numerical method for the case of multishell grains with iron layers. This is due to numerical uncertainty which arises in the calculation procedure for the Mie scattering coefficients in the case of highly absorbing materials, like iron (for further explanation, see Gurvich, Shiloah, & Kleiman [2001]).

We assume that each dust component has a total volume fraction in a particle according to its mass fraction and density, as specified in Table 1. For example, for the first temperature region and in the case of IPS silicate mineralogy, the mixture of iron and silicates occupies $8.9$%, troilite - $1.6$%, refractory organics - $23.4$%, volatile organics - $6$%, and water ice - $60$% of the entire particle volume, respectively. These values are similar in the case of NRM and IRS models. Thus, if the temperature is low, organics and ice are the dominant components of the dust grains.

Unlike to the composite particles, it is assumed in the case of multishell spherical grains that the distribution of dust materials is not random but follows their evaporation sequence. Thus, for the first temperature region in the protoplanetary disc, multishell spherical grains consist of a refractory core made of a mixture of silicates and iron and subsequent shells of troilite, refractory organics, volatile organics, and water ice. For higher temperature ranges, the number of shells is smaller since some materials are evaporated. In total, the number of shells in the case of multishell spherical particles varies from $2$ to $5$. For the fifth temperature range ($T>700$ K), where all troilite is converted to solid iron, we let this iron form an additional layer on the grain surface. It is an extreme and probably physically unjustified case, but this allows us to study the influence of the distribution of highly absorbing materials, like iron, within the grains on resulting opacities.

The porosity of particles is treated in a simple manner by the addition of $50\%$ of vacuum (by volume) inside. For the case of the porous composite spheres, we consider vacuum to be one of the grain constituents, which forms additional "empty" layers. In the case of the multishell grain model, we mix each shell with vacuum in the same way which is applied to create the composite spherical particles. That is, we subdivide each individual shell in many layers and fill some of them with vacuum according to the requested porosity degree.

With the two computational approaches described above, we calculated the ensemble averaged absorption and scattering cross section as well as albedo and the mean cosine of the scattering angle for all kinds of the grains. Applying Eqs. $1$-$5$ from Pollack et al. ([1985]), the dust monochromatic opacity and, consequently, the Rosseland and Planck mean opacities were obtained for temperatures below roughly $1\,500$ K and a density range between $10^{-18}$ gcm$^{-3}$ and $10^{-7}$ gcm$^{-3}$. A convenient analytical representation of the Rosseland and Planck mean opacities for every temperature region is provided as a $5$-order polynomial fit. This representation allows to calculate the Rosseland and Planck mean dust opacities accurately ($\sim 1$%) and quickly for any given temperature and density values within the model applicability range, which is important for computationally extensive hydrodynamical simulations. The corresponding fit coefficients can be found on the web page.