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# Appendix A: Equations for porosity evolution during growth

## A.1. Hit & stick regime

In the hit & stick regime, the mass of the grain doubles during a collision, and a volume of void $V_{void}$ is formed. According to Okuzumi et al. (2009), this captured volume $V_{void}$ corresponds to 99% of the initial volume $V_i$, thus adding to the initial volumes of the two grains before the collision. The filling factor $\phi \propto m/V$ is therefore multiplied by a factor of 2/2.99. For an arbitrary number of collisions $n$, there is a geometric relationship between the initial filling factor $\phi_0$ and the final $\phi_f$, given by $\phi_f = (2/2.99)^n\phi_0$ (Garcia & Gonzalez 2020). Moreover, the mass of a grain after the same number of collisions has grown by $m_f = 2^n m_0$, $m_0$ being the monomer mass. By isolating $n$ and combining the two expressions, we obtained the filling factor in the hit & stick regime (assuming $E_{kin} \ll E_{roll}$, Garcia & Gonzalez 2020):

$$ \phi_{h\&s} = \left(\frac{m}{m_0}\right)^{\psi}, \qquad (A.1) $$

with $\psi = \ln(2/2.99) / \ln 2 \approx -0.58$, where for the monomer $\phi_0 = 1$.

## A.2. Collisional compression regime

As both the grain mass and the relative velocity $v_{rel}$ increase, the kinetic energy increases. Since $v_{rel}$ depends on the Stokes number (Sect. 2.1), which in turn depends on the drag regime, there is an expression for each case (Garcia & Gonzalez 2020). Suyama et al. (2012) provide the volume of a grain after collision $V_f$ in terms of the volume of a grain before collision $V_i$. This relation can be transformed to incorporate filling factors using the fact that $\phi_f = m_f/V_f\rho_s = 2m_i/V_f\rho_s$ (Garcia & Gonzalez 2020):

$$ \phi_f = \frac{2m_i}{\rho_s} \left( \frac{(3/5)^5 (E_{\text{Kin}} - 3b E_{\text{roll}})}{N_{\text{tot}}^5 b E_{\text{roll}} V_0^{10/3}} + \frac{V_i^{-10/3}}{2^4} \right)^{3/10} . \qquad (A.2) $$

Here, $N_{tot}$ is the total number of monomers, which is $2m_i/m_0$, and $V_0$ is the volume of a monomer. This model is recursive, meaning that the evolution of porosity is tracked after each collision. However, in the case of global simulations, the integration time step rarely aligns with the time interval between collisions $\tau_{coll}$. Therefore, Garcia & Gonzalez (2020) transformed this discrete model into a continuous one to depend solely on the grain mass and quantities related to $r$, enabling its implementation in codes for global simulations. To achieve this, they assumed the approximation $E_{kin} \gg E_{roll}$.

In the Epstein and Stokes regimes for $St < 1$, one obtains

$$ \phi_{\text{Ep,St}<1} = \left( \frac{A}{2(2^{3/40} - 1)} \right)^{3/8} \left( \frac{m}{m_0} \right)^{-1/8}, \qquad (A.3) $$

$$ \phi_{\text{St,St}<1} = \left( \frac{A c_g a_0}{9 \nu_{\text{mol}} (2^{1/5} - 1)} \right)^{1/3}. \qquad (A.4) $$

$$ \nu_{\text{mol}} = \frac{5 \sqrt{\pi} m_{\text{mol}} c_g}{64 \rho_g \sigma_{\text{mol}}} \qquad (A.5) $$

is the gas molecular kinematic viscosity, where $m_{mol}$ and $\sigma_{mol}$ are the mass and molecular cross section of the H₂ molecule,

and A is a factor used to simplify the expressions and is given by

$$ A = \frac{243 \sqrt{2\pi} \text{Ro} \alpha a_0^4 \rho_s^2 c_g \Omega_K}{15625 \rho b E_{\text{roll}}} \qquad (A.6) $$

where $\rho = \rho_g + \rho_d$ is the total density. It is observed that in the Stokes regime with $St < 1$, the filling factor depends only on the local disc conditions. For regimes with $St > 1$, collisional compression is less efficient due to the decrease in $v_{rel}$. By neglecting $E_{kin}$, it can be derived from Okuzumi et al. (2012) and Garcia & Gonzalez (2020) that $\phi_f \propto m_f^{-1/5}$ for $St > 1$. Therefore, the filling factors for $St > 1$ can be expressed as

$$ \phi_{\text{Ep,St}>1} = \phi_{\text{Ep,St}<1}(M_4) \left(\frac{m}{M_4}\right)^{-1/5}, \qquad (A.7) $$

$$ \phi_{\text{St,St}>1} = \phi_{\text{St,St}<1}(M_5) \left(\frac{m}{M_5}\right)^{-1/5}, \qquad (A.8) $$

where $M_4$ and $M_5$ are the transition masses between the Epstein and Stokes regimes for $St < 1$ and $St > 1$. The transition masses are given in Appendix B. The equations giving $\phi$ in the various regimes can also be rewritten as a function of the size $s$ instead of the mass $m$, they are listed in Appendix C. However, the natural variable to describe dust evolution is $m$.

The filling factor resulting from the hit & stick and collisional compression regimes is noted $\phi_{coll}$.

## A.3. Static compression regime

In addition to undergoing compression due to collisions, aggregates can also be statically compacted either by the surrounding gas or by their own gravity. Kataoka et al. (2013) and Garcia & Gonzalez (2020) provided the relationship between static compression pressure and the filling factor:

$$ \phi = \left( \frac{a_0^3 P_{\text{comp}}}{E_{\text{roll}}} \right)^{1/3}, \qquad (A.9) $$

which is only valid for small filling factors $\phi < 0.1$. In the case of compression by gas, the pressure is given by

$$ P_{\text{comp}} = \frac{F_D}{\pi s^2} = \frac{m \Delta v \Omega_K}{\pi s^2 \text{St}}, \qquad (A.10) $$

which allows one to obtain the expression for the filling factor $\phi_{gas}$ (Garcia & Gonzalez 2020):

$$ \phi_{\text{gas}} = \left( \frac{m_0 a_0 \Delta v \Omega_K}{\pi E_{\text{roll}} \text{St}} \right)^{3/7} \left( \frac{m}{m_0} \right)^{1/7}. \qquad (A.11) $$

For self-gravity, the exerted pressure is given by

$$ P_{\text{comp}} = \frac{Gm^2}{\pi s^4}, \qquad (A.12) $$

and the expression for the filling factor $\phi_{grav}$ is (Garcia & Gonzalez 2020)

$$ \phi_{\text{grav}} = \left( \frac{Gm_0^2}{\pi a_0 E_{\text{roll}}} \right)^{3/5} \left( \frac{m}{m_0} \right)^{2/5}, \qquad (A.13) $$

where G is the universal gravitational constant. To obtain the final filling factor of a grain with mass $m$ at a distance $r$ from the
# Appendix A: Equations for porosity evolution during growth

## Appendix A.1: Hit & stick regime

In the hit & stick regime, the mass of the grain doubles during a collision, and a volume of void $V_{void}$ is formed. According to Okuzumi et al. (2009), this captured volume $V_{void}$ corresponds to 99% of the initial volume $V_i$, thus adding to the initial volumes of the two grains before the collision. The filling factor $\phi \propto m/V$ is therefore multiplied by a factor of 2/2.99. For an arbitrary number of collisions $n$, there is a geometric relationship between the initial filling factor $\phi_0$ and the final $\phi_f$, given by $\phi_f = (2/2.99)^n\phi_0$ (Garcia & Gonzalez 2020). Moreover, the mass of a grain after the same number of collisions has grown by $m_f = 2^n m_0$, $m_0$ being the monomer mass. By isolating $n$ and combining the two expressions, we obtained the filling factor in the hit & stick regime (assuming $E_{kin} \ll E_{roll}$, Garcia & Gonzalez 2020):

$$ \phi_{h\&s} = \left(\frac{m}{m_0}\right)^{\psi}, \qquad (A.1) $$

with $\psi = \ln(2/2.99)/\ln 2 \approx -0.58$, where for the monomer $\phi_0 = 1$.

## Appendix A.2: Collisional compression regime

As both the grain mass and the relative velocity $v_{rel}$ increase, the kinetic energy increases. Since $v_{rel}$ depends on the Stokes number (Sect. 2.1), which in turn depends on the drag regime, there is an expression for each case (Garcia & Gonzalez 2020). Suyama et al. (2012) provide the volume of a grain after collision $V_f$ in terms of the volume of a grain before collision $V_i$. This relation can be transformed to incorporate filling factors using the fact that $\phi_f = m_f/V_f\rho_s = 2m_i/V_f\rho_s$ (Garcia & Gonzalez 2020):

$$ \phi_f = \frac{2m_i}{\rho_s} \left( \frac{(3/5)^5 (E_{Kin} - 3b E_{roll})}{N_{tot}^5 b E_{roll} V_0^{10/3}} + \frac{V_i^{-10/3}}{2^4} \right)^{3/10} . \qquad (A.2) $$

Here, $N_{tot}$ is the total number of monomers, which is $2m_i/m_0$, and $V_0$ is the volume of a monomer. This model is recursive, meaning that the evolution of porosity is tracked after each collision. However, in the case of global simulations, the integration time step rarely aligns with the time interval between collisions $\tau_{coll}$. Therefore, Garcia & Gonzalez (2020) transformed this discrete model into a continuous one to depend solely on the grain mass and quantities related to $r$, enabling its implementation in codes for global simulations. To achieve this, they assumed the approximation $E_{kin} \gg E_{roll}$.

In the Epstein and Stokes regimes for $St < 1$, one obtains

$$ \phi_{Ep,St<1} = \left( \frac{A}{2(2^{3/40} - 1)} \right)^{3/8} \left( \frac{m}{m_0} \right)^{-1/8}, \qquad (A.3) $$

$$ \phi_{St,St<1} = \left( \frac{Ac_g a_0}{9v_{mol}(2^{1/5} - 1)} \right)^{1/3}. \qquad (A.4) $$

$$ v_{mol} = \frac{5\sqrt{\pi} m_{mol} c_g}{64 \rho_g \sigma_{mol}} \qquad (A.5) $$

is the gas molecular kinematic viscosity, where $m_{mol}$ and $\sigma_{mol}$ are the mass and molecular cross section of the H₂ molecule,

and A is a factor used to simplify the expressions and is given by

$$ A = \frac{243 \sqrt{2\pi} \text{Ro} \alpha a_0^4 \rho_s^2 c_g \Omega_K}{15625 \rho b E_{roll}}, \qquad (A.6) $$

where $\rho = \rho_g + \rho_d$ is the total density. It is observed that in the Stokes regime with $St < 1$, the filling factor depends only on the local disc conditions. For regimes with $St > 1$, collisional compression is less efficient due to the decrease in $v_{rel}$. By neglecting $E_{kin}$, it can be derived from Okuzumi et al. (2012) and Garcia & Gonzalez (2020) that $\phi_f \propto m_f^{-1/5}$ for $St > 1$. Therefore, the filling factors for $St > 1$ can be expressed as

$$ \phi_{Ep,St>1} = \phi_{Ep,St<1}(M_4) \left(\frac{m}{M_4}\right)^{-1/5}, \qquad (A.7) $$

$$ \phi_{St,St>1} = \phi_{St,St<1}(M_5) \left(\frac{m}{M_5}\right)^{-1/5}, \qquad (A.8) $$

where $M_4$ and $M_5$ are the transition masses between the Epstein and Stokes regimes for $St < 1$ and $St > 1$. The transition masses are given in Appendix B. The equations giving $\phi$ in the various regimes can also be rewritten as a function of the size $s$ instead of the mass $m$, they are listed in Appendix C. However, the natural variable to describe dust evolution is $m$.

The filling factor resulting from the hit & stick and collisional compression regimes is noted $\phi_{coll}$.

## Appendix A.3: Static compression regime

In addition to undergoing compression due to collisions, aggregates can also be statically compacted either by the surrounding gas or by their own gravity. Kataoka et al. (2013b) and Garcia & Gonzalez (2020) provided the relationship between static compression pressure and the filling factor:

$$ \phi = \left( \frac{a_0^3 P_{\text{comp}}}{E_{\text{roll}}} \right)^{1/3}, \qquad (A.9) $$

which is only valid for small filling factors $\phi < 0.1$. In the case of compression by gas, the pressure is given by

$$ P_{\text{comp}} = \frac{F_D}{\pi s^2} = \frac{m \Delta v \Omega_K}{\pi s^2 \text{St}}, \qquad (A.10) $$

which allows one to obtain the expression for the filling factor $\phi_{gas}$ (Garcia & Gonzalez 2020):

$$ \phi_{\text{gas}} = \left( \frac{m_0 a_0 \Delta v \Omega_K}{\pi E_{\text{roll}} \text{St}} \right)^{3/7} \left( \frac{m}{m_0} \right)^{1/7}. \qquad (A.11) $$

For self-gravity, the exerted pressure is given by

$$ P_{\text{comp}} = \frac{Gm^2}{\pi s^4}, \qquad (A.12) $$

and the expression for the filling factor $\phi_{grav}$ is (Garcia & Gonzalez 2020):

$$ \phi_{\text{grav}} = \left( \frac{Gm_0^2}{\pi a_0 E_{\text{roll}}} \right)^{3/5} \left( \frac{m}{m_0} \right)^{2/5}, \qquad (A.13) $$

where G is the universal gravitational constant. To obtain the final filling factor of a grain with mass $m$ at a distance $r$ from the
<figure><img src="image_20.png" /><figcaption>**Fig. B.1.** Transition masses for our standard disc model and for silicate grains.</figcaption></figure>

star, we take the value of $\phi$ given by the phenomenon that com-pacts the grain the most, which is the largest value among static compression by gas, static compression by gravity, and compres-sion by collision: $\phi_{min} = \max(\phi_{coll}, \phi_{gas}, \phi_{grav})$. All these regimes are presented in Fig. 1 in the $(s,\phi)$ plane for different distances to the star.

### A.4. Growth from above the compression limit

The minimum filling factor must be $\phi_{min}$. However, there are cases where the evolution of a grain (e.g. via bouncing) can pro- duce a filling factor larger than $\phi_{min}$, placing it in the space above the curves showed in Fig. 1. To take into account the increase of porosity of grains during growth above the limit, we can com- pute the filling factor as a function of the initial filling factor $\phi_i$, and the final and initial masses $m_f$ and $m_i$ in the case where $v_{rel} < v_{frag}$.

$$ \phi_{\text{grow}} = \phi_i \left( \frac{m_f}{m_i} \right)^{\psi} \quad \text{if } E_{\text{kin}} < 3bE_{\text{roll}}, \qquad (\text{A.14}) $$

$$ \phi_{\text{grow}} = \phi_i \left( \frac{m_f}{m_i} \right)^{-1/5} \quad \text{otherwise.} \qquad (\text{A.15}) $$

The resulting filling factor is then the maximum of $\phi_{grow}$ and $\phi_{min}$.

## Appendix B: Transition masses

The transition masses are the masses at which the behaviour of a grain changes. The different expressions of $\phi$ found for each regime in Sect. 2.2 are independent, the transition masses allowed us to relate them. For the transition masses $M_1$ and $M_2$ that connect the hit & stick regime to the Epstein and Stokes regimes with $St < 1$, we find

$$ \frac{M_1}{m_0} = \left( \frac{A}{2(2^{3/40} - 1)} \right)^{\frac{3/8}{\psi+1/8}}, \qquad (\text{B.1}) $$

$$ \frac{M_2}{m_0} = \left( \frac{A c_g a_0}{9 \nu_{\text{mol}} (2^{1/5} - 1)} \right)^{1/3\psi}. \qquad (\text{B.2}) $$

The transition mass $M_3$ corresponds to the transition between the Epstein and Stokes regimes

$$ \frac{M_3}{m_0} = \left( \frac{A}{2(2^{3/40} - 1)} \right)^3 \left( \frac{A c_g a_0}{9 \nu_{\text{mol}} (2^{1/5} - 1)} \right)^{-8/3}, \qquad (\text{B.3}) $$

$$ = \left( \frac{M_1}{m_0} \right)^{8\psi+1} \left( \frac{M_2}{m_0} \right)^{-8\psi} \qquad (\text{B.4}) $$

Finally, the expressions for $M_4$ and $M_5$ are obtained by equating the expression of $\phi$ for the Epstein or Stokes regimes with $St < 1$ to the expression of $\phi$ obtained by setting $St = 1$.

$$ \frac{M_4}{m_0} = \left( \frac{\rho c_g}{\rho_s a_0 \Omega_K} \right)^4 \left( \frac{A}{2(2^{3/40} - 1)} \right)^{-1}, \qquad (\text{B.5}) $$

$$ \frac{M_5}{m_0} = \left( \frac{9 \nu_{\text{mol}} \rho}{2 \rho_s a_0^2 \Omega_K} \right)^{3/2} \left( \frac{A c_g a_0}{9 \nu_{\text{mol}} (2^{1/5} - 1)} \right)^{-1/6}. \qquad (\text{B.6}) $$

Here, we have not considered the other Stokes regimes that are reached for even larger sizes than the linear regime as static grain compression is always dominant compared to these regimes. The various transition masses as well as the differ- ent expansion and compression regimes are shown in Fig. B.1, inspired by Fig. A1 of Garcia & Gonzalez (2020). We note that the transition mass $M_2$ is always larger than mass $M_1$. The Stokes regimes are reached in the inner regions of a disc, at a distance less than 10 au.

## Appendix C: Filling factor as a function of size

The equations giving the filling factor $\phi$ as a function of mass $m$ in Appendix A can be rewritten as a function of the size $s$ as

$$ \phi_{\text{h\&s}} = \left( \frac{s}{a_0} \right)^{\frac{3\psi}{1-\psi}}, \qquad (\text{C.1}) $$

$$ \phi_{\text{Ep-St}<1} = \left( \frac{A}{2(2^{3/40} - 1)} \right)^{1/3} \left( \frac{s}{a_0} \right)^{-1/3}, \qquad (\text{C.2}) $$

$$ \phi_{\text{St-St}<1} = \left( \frac{A c_g a_0}{9 \nu (2^{1/5} - 1)} \right)^{1/3}, \qquad (\text{C.3}) $$

$$ \phi_{\text{Ep-St}>1} = \left( \frac{\rho c_g}{\rho_s \Omega_K a_0} \frac{A}{2^{3/40} - 1} \right)^{1/4} \left( \frac{s}{a_0} \right)^{-1/2}, \qquad (\text{C.4}) $$

$$ \phi_{\text{St-St}>1} = \left( \frac{\rho c_g a_0}{2 \rho_s \Omega_K a_0^2} \frac{A}{2^{1/5} - 1} \right)^{1/4} \left( \frac{s}{a_0} \right)^{-1/2}. \qquad (\text{C.5}) $$

## Appendix D: Co-evolution of grain size and filling factor in PHANTOM simulations with and without compaction

Figure D.1 shows the joint evolution of grain size and filling factor in PHANTOM simulations without (top) and with (bottom) compaction during fragmentation and bouncing, with the same order of simulations as in Fig. 14. Without compaction (top panel), the trajectories for all simulations are very similar
<figure><img src="image_20.png" /><figcaption>Fig. B.1. Transition masses for our standard disc model and for silicate grains.</figcaption></figure>

star, we take the value of $\phi$ given by the phenomenon that com-pacts the grain the most, which is the largest value among static compression by gas, static compression by gravity, and compres-sion by collision: $\phi_{min} = \max(\phi_{coll}, \phi_{gas}, \phi_{grav})$. All these regimes are presented in Fig. 1 in the $(s, \phi)$ plane for different distances to the star.

## Appendix A.4: Growth from above the compression limit

The minimum filling factor must be $\phi_{min}$. However, there are cases where the evolution of a grain (e.g. via bouncing) can pro- duce a filling factor larger than $\phi_{min}$, placing it in the space above the curves showed in Fig. 1. To take into account the increase of porosity of grains during growth above the limit, we can com- pute the filling factor as a function of the initial filling factor $\phi_i$, and the final and initial masses $m_f$ and $m_i$ in the case where $v_{rel} < v_{frag}$.

$$ \phi_{\text{grow}} = \phi_i \left(\frac{m_f}{m_i}\right)^{\psi} \quad \text{if } E_{\text{kin}} < 3bE_{\text{roll}}, \qquad (\text{A.14}) $$

$$ \phi_{\text{grow}} = \phi_i \left(\frac{m_f}{m_i}\right)^{-1/5} \quad \text{otherwise.} \qquad (\text{A.15}) $$

The resulting filling factor is then the maximum of $\phi_{grow}$ and $\phi_{min}$.

## Appendix B: Transition masses

The transition masses are the masses at which the behaviour of a grain changes. The different expressions of $\phi$ found for each regime in Sect. 2.2 are independent, the transition masses al- lowed us to relate them. For the transition masses $M_1$ and $M_2$ that connect the hit & stick regime to the Epstein and Stokes regimes with $St < 1$, we find

$$ \frac{M_1}{m_0} = \left( \frac{A}{2(2^{3/40} - 1)} \right)^{\frac{3/8}{\psi+1/8}}, \qquad (\text{B.1}) $$

$$ \frac{M_2}{m_0} = \left( \frac{A c_g a_0}{9 \nu_{\text{mol}} (2^{1/5} - 1)} \right)^{1/3 \psi}. \qquad (\text{B.2}) $$

The transition mass $M_3$ corresponds to the transition between the Epstein and Stokes regimes

$$ \frac{M_3}{m_0} = \left( \frac{A}{2(2^{3/40} - 1)} \right)^3 \left( \frac{A c_g a_0}{9 \nu_{\text{mol}} (2^{1/5} - 1)} \right)^{-8/3}, \qquad (\text{B.3}) $$

$$ = \left(\frac{M_1}{m_0}\right)^{8\psi+1} \left(\frac{M_2}{m_0}\right)^{-8\psi} \qquad (\text{B.4}) $$

Finally, the expressions for $M_4$ and $M_5$ are obtained by equating the expression of $\phi$ for the Epstein or Stokes regimes with $St < 1$ to the expression of $\phi$ obtained by setting $St = 1$.

$$ \frac{M_4}{m_0} = \left( \frac{\rho c_g}{\rho_s a_0 \Omega_K} \right)^4 \left( \frac{A}{2(2^{3/40} - 1)} \right)^{-1}, \qquad (\text{B.5}) $$

$$ \frac{M_5}{m_0} = \left( \frac{9 \nu_{\text{mol}} \rho}{2 \rho_s a_0^2 \Omega_K} \right)^{3/2} \left( \frac{A c_g a_0}{9 \nu_{\text{mol}} (2^{1/5} - 1)} \right)^{-1/6}. \qquad (\text{B.6}) $$

Here, we have not considered the other Stokes regimes that are reached for even larger sizes than the linear regime as static grain compression is always dominant compared to these regimes. The various transition masses as well as the different expansion and compression regimes are shown in Fig. B.1, in- spired by Fig. A1 of Garcia & Gonzalez (2020). We note that the transition mass $M_2$ is always larger than mass $M_1$. The Stokes regimes are reached in the inner regions of a disc, at a distance less than 10 au.

## Appendix C: Filling factor as a function of size

The equations giving the filling factor $\phi$ as a function of mass $m$ in Appendix A can be rewritten as a function of the size $s$ as

$$ \phi_{\text{h\&s}} = \left( \frac{s}{a_0} \right)^{\frac{3\psi}{1-\psi}}, \qquad (\text{C.1}) $$

$$ \phi_{\text{Ep-St}<1} = \left( \frac{A}{2(2^{3/40} - 1)} \right)^{1/3} \left( \frac{s}{a_0} \right)^{-1/3}, \qquad (\text{C.2}) $$

$$ \phi_{\text{St-St}<1} = \left( \frac{A c_g a_0}{9 \nu (2^{1/5} - 1)} \right)^{1/3}, \qquad (\text{C.3}) $$

$$ \phi_{\text{Ep-St}>1} = \left( \frac{\rho c_g}{\rho_s \Omega_K a_0} \frac{A}{2^{3/40} - 1} \right)^{1/4} \left( \frac{s}{a_0} \right)^{-1/2}, \qquad (\text{C.4}) $$

$$ \phi_{\text{St-St}>1} = \left( \frac{\rho c_g a_0}{2 \rho_s \Omega_K a_0^2} \frac{A}{2^{1/5} - 1} \right)^{1/4} \left( \frac{s}{a_0} \right)^{-1/2}. \qquad (\text{C.5}) $$

## Appendix D: Co-evolution of grain size and filling factor in PHANTOM simulations with and without compaction

Figure D.1 shows the joint evolution of grain size and filling factor in PHANTOM simulations without (top) and with (bottom) compaction during fragmentation and bouncing, with the same order of simulations as in Fig. 14. Without compaction (top panel), the trajectories for all simulations are very similar to the 1D
<figure><img src="image_21.png" /><figcaption>**Fig. D.1.** Same as Fig. 14 but showing the filling factor $\phi$ plotted against grain size, with colour indicating the distance $r$.</figcaption></figure>

to the 1D case (Fig. 12), with a spread corresponding to the disc’s vertical extent. When compaction is considered (bottom panels), the compaction trajectory is more vertical compared to that obtained with PAMDEAS, and the bouncing plateau at $\phi \sim 0.4$, primarily populated with grains very close to the disc inner edge, is more clearly visible. The overall evolution appears more complex and the $s-\phi$ plane is more densely populated. A large spread of filling factors can be reached for a given grain size and at a given distance, grains of various $s$ and $\phi$ can be found, indicating the coexistence of porous and compacted dust.

Simulations with silicate grains and $v_{\text{frag, Si}} = 10$ and $20 \text{ m s}^{-1}$ (first two rows) are again quite similar, with grains reaching larger sizes and porosities before being fragmented and compacted for the latter value. The main difference is that at $t = 300 \text{ kyr}$ almost all grains are compacted or undergoing compaction for $v_{\text{frag, Si}} = 20 \text{ m s}^{-1}$, with very few left on the growth branch, while for $10 \text{ m s}^{-1}$ a large fraction of grains are still

growing, mainly in the outer disc. Despite the fact that simulations GF-Si-a02-Vf40 and GBFc-Si-a02-Vf40 (third rows) are not valid, it is still interesting to note that, due to larger sizes being reached, the gas and self-gravity compaction regimes are visible for solids larger than a few m (see Fig. 1). Again, the water ice simulations (bottom rows) are similar to that with silicate grains and $v_{\text{frag, Si}} = 20 \text{ m s}^{-1}$, with larger $s$ and smaller $\phi$, except for a fraction of still growing grains at large $r$.

## Appendix E: Influence of rotational disruption

Michoulier & Gonzalez (2022a) studied the effect of rotational disruption on the evolution of porous grains using 1D simulations with PAMDEAS. 3D disc simulations taking into account the evolution of gas and dust are necessary to understand in more detail when and where rotational disruption influences dust evolution. To that effect, we performed simulations modelling this
<figure><img src="image_21.png" /><figcaption>**Fig. D.1.** Same as Fig. 14 but showing the filling factor $\phi$ plotted against grain size, with colour indicating the distance $r$.</figcaption></figure>

case (Fig. 12), with a spread corresponding to the disc’s vertical extent. When compaction is considered (bottom panels), the compaction trajectory is more vertical compared to that obtained with *PAMDEAS*, and the bouncing plateau at $\phi \sim 0.4$, primarily populated with grains very close to the disc inner edge, is more clearly visible. The overall evolution appears more complex and the $s-\phi$ plane is more densely populated. A large spread of filling factors can be reached for a given grain size and at a given distance, grains of various $s$ and $\phi$ can be found, indicating the coexistence of porous and compacted dust.

Simulations with silicate grains and $v_{\text{frag, Si}} = 10$ and $20 \text{ m s}^{-1}$ (first two rows) are again quite similar, with grains reaching larger sizes and porosities before being fragmented and compacted for the latter value. The main difference is that at $t = 300$ kyr almost all grains are compacted or undergoing compaction for $v_{\text{frag, Si}} = 20 \text{ m s}^{-1}$, with very few left on the growth branch, while for $10 \text{ m s}^{-1}$ a large fraction of grains are still

growing, mainly in the outer disc. Despite the fact that simulations GF-Si-a02-Vf40 and GBFc-Si-a02-Vf40 (third rows) are not valid, it is still interesting to note that, due to larger sizes being reached, the gas and self-gravity compaction regimes are visible for solids larger than a few m (see Fig. 1). Again, the water ice simulations (bottom rows) are similar to that with silicate grains and $v_{\text{frag, Si}} = 20 \text{ m s}^{-1}$, with larger $s$ and smaller $\phi$, except for a fraction of still growing grains at large $r$.

## Appendix E: Influence of rotational disruption

Michoulier & Gonzalez (2022a) studied the effect of rotational disruption on the evolution of porous grains using 1D simulations with *PAMDEAS*. 3D disc simulations taking into account the evolution of gas and dust are necessary to understand in more detail when and where rotational disruption influences dust evolution. To that effect, we performed simulations modelling this
<figure><img src="image_22.png" /><figcaption>**Fig. E.1.** Comparison between PHANTOM simulations with growth, bouncing, fragmentation with compaction, and without (GBFc-*-a02-*-top panel) and with (GBFcD-*-a02-*-bottom panel) rotational disruption showing the radial grain distribution, with colour representing the dust filling factor. Simulation parameters are indicated in each row.</figcaption></figure>

phenomenon for silicates with $v_{frag, Si} = 10$ and $20 \text{ m s}^{-1}$, and for water ice with $v_{frag, Si} = 15 \text{ m s}^{-1}$, listed in the last three rows of Table 1. The grain properties were somewhat different compared to Michoulier & Gonzalez (2022a): the monomer size was set to 0.2 instead of 0.1 µm, and $\gamma_s = 0.2 \text{ J m}^{-2}$ for silicates. In fact, with these new parameters, the tensile strength $S_{max}$ should be larger by a factor of two.

Simulations with PAMDEAS showed signs of rotational disruption only for very small values of the viscosity parameter $\alpha < 5 \times 10^{-4}$, lower than that used with PHANTOM. The conditions necessary for its appearance in the midplane (which PAMDEAS simulates) are therefore challenging to achieve.

Figure E.1 compares 3D simulations including growth, bouncing and fragmentation with compaction, without (top) and with (bottom) rotational disruption. It shows the radial grain distribution, with colour representing the dust filling factor $\phi$. For simulations GBFc-Si-a02-Vf10 and GBFcD-Si-a02-Vf10 (top rows in both panels), very few differences are observed, which is expected since the fragmentation threshold is low enough for grains to fragment before they can be rotationally disrupted (which only happens at higher relative velocities and therefore larger sizes). For simulations GBFc-Si-a02-Vf20 and GBFcD-Si-a02-Vf20 (middle rows), rotational disruption is able to destroy grains during their fall towards the midplane,

which slows down their settling and limits dust enrichment in the midplane. Since the dust density is slightly lower, grains do not grow as much as in the case without disruption. This is mostly visible for grains outside of 50 au at $t = 100$ kyr. After 300 kyr, the differences have been largely erased by the subsequent dust evolution. Finally, for simulations GBFc-H2O-a02-Vf15 and GBFcD-H2O-a02-Vf15 (bottom rows), disruption has a minimal effect on the size and porosity of dust grains, only visible outside of 150 au. Indeed, the growth of water ice grains is very rapid and largely counteracts rotational disruption. It is at $t = 300$ kyr that the effect becomes more visible since, unlike the simulation without disruption, many small grains are observed between 200 and 500 au.

In general, disruption has little effect on the size and porosity of grains and their evolution in the disc, with fragmentation and compaction largely dominating grain growth and destruction.
<figure><img src="image_22.png" /><figcaption>**Fig. E.1.** Comparison between PHANTOM simulations with growth, bouncing, fragmentation with compaction, and without (GBFc-*-a02-*- top panel) and with (GBFcD-*-a02-*- bottom panel) rotational disruption showing the radial grain distribution, with colour representing the dust filling factor. Simulation parameters are indicated in each row.</figcaption></figure>

phenomenon for silicates with $v_{frag, Si} = 10$ and $20 \text{ m s}^{-1}$, and for water ice with $v_{frag, Si} = 15 \text{ m s}^{-1}$, listed in the last three rows of Table 1. The grain properties were somewhat different compared to Michoulier & Gonzalez (2022a): the monomer size was set to 0.2 instead of 0.1 µm, and $\gamma_s = 0.2 \text{ J m}^{-2}$ for silicates. In fact, with these new parameters, the tensile strength $S_{max}$ should be larger by a factor of two.

Simulations with PAMDEAS showed signs of rotational disruption only for very small values of the viscosity parameter $\alpha < 5 \times 10^{-4}$, lower than that used with PHANTOM. The conditions necessary for its appearance in the midplane (which PAMDEAS simulates) are therefore challenging to achieve.

Figure E.1 compares 3D simulations including growth, bouncing and fragmentation with compaction, without (top) and with (bottom) rotational disruption. It shows the radial grain distribution, with colour representing the dust filling factor $\phi$. For simulations GBFc-Si-a02-Vf10 and GBFcD-Si-a02-Vf10 (top rows in both panels), very few differences are observed, which is expected since the fragmentation threshold is low enough for grains to fragment before they can be rotationally disrupted (which only happens at higher relative velocities and therefore larger sizes). For simulations GBFc-Si-a02-Vf20 and GBFcD-Si-a02-Vf20 (middle rows), rotational disruption is able to destroy grains during their fall towards the midplane, which slows down their settling and limits dust enrichment in the midplane.

Since the dust density is slightly lower, grains do not grow as much as in the case without disruption. This is mostly visible for grains outside of 50 au at $t = 100$ kyr. After 300 kyr, the differences have been largely erased by the subsequent dust evolution. Finally, for simulations GBFc-H2O-a02-Vf15 and GBFcD-H2O-a02-Vf15 (bottom rows), disruption has a minimal effect on the size and porosity of dust grains, only visible outside of 150 au. Indeed, the growth of water ice grains is very rapid and largely counteracts rotational disruption. It is at $t = 300$ kyr that the effect becomes more visible since, unlike the simulation without disruption, many small grains are observed between 200 and 500 au.

In general, disruption has little effect on the size and porosity of grains and their evolution in the disc, with fragmentation and compaction largely dominating grain growth and destruction.