Issue 
A&A
Volume 670, February 2023



Article Number  L21  
Number of page(s)  10  
Section  Letters to the Editor  
DOI  https://doi.org/10.1051/00046361/202345887  
Published online  22 February 2023 
Letter to the Editor
Threshold velocity for the collisional growth of porous dust aggregates consisting of cohesive frictionless spheres
^{1}
Japan Agency for MarineEarth Science and Technology, 317325, Showamachi, Kanazawaku, Yokohama 2360001, Japan
email: arakawas@jamstec.go.jp
^{2}
Astronomical Institute, Graduate School of Science, Tohoku University, 63 Aramaki, Aobaku, Sendai 9808578, Japan
^{3}
National Astronomical Observatory of Japan, 2211, Osawa, Mitaka, Tokyo 1818588, Japan
Received:
11
January
2023
Accepted:
31
January
2023
Understanding the collisional outcomes of dust aggregates and their dependence on the material properties of the constituting particles is of great importance for understanding planet formation. Recent numerical simulations have revealed that interparticle tangential friction plays a crucial role in energy dissipation during collisions between porous dust aggregates, but the importance of friction for the collisional growth of dust aggregates remains poorly understood. Here we demonstrate the effects of interparticle tangential friction on the collisional growth of dust aggregates. We performed numerical simulations of collisions between equalmass porous dust aggregates consisting of cohesive and frictionless spheres. We changed the collision velocity and impact angle systematically and calculated the collisional growth efficiency as a function of the collision velocity. We found that the threshold velocity for collisional growth decreases when dust aggregates are made of frictionless spheres compared to frictional spheres. Our results highlight the importance of tangential interactions for the collisional behavior of dust aggregates and indicate that the predictive equation for the threshold velocity should be reconstructed.
Key words: planets and satellites: formation / protoplanetary disks
© The Authors 2023
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1. Introduction
Collisional growth of dust aggregates consisting of micro or nanosized grains is ubiquitous in the Universe. For example, collisional growth of soot nanoparticles produced by combustion processes is important in environmental science and engineering (e.g., Haynes & Wagner 1981). Atmospheric hazes on the early Earth and other planetary bodies are also dust aggregates consisting of micro or nanosized particles, and their collisional growth affects the atmospheric structure (e.g., Trainer et al. 2006; Zhang et al. 2017; Ohno et al. 2021). The first step of planet formation in circumstellar disks is collisional growth of micro or nanosized interstellar dust particles (e.g., Hayashi et al. 1985; Drazkowska et al. 2022), and dust growth might also trigger the formation of planetary objects around supermassive black holes in a galactic center (e.g., Wada et al. 2021).
The condition for collisional growth of dust aggregates has been studied with laboratory experiments (e.g., Blum & Wurm 2008; Güttler et al. 2010; Fritscher & Teiser 2021; Schräpler et al. 2022) and numerical simulations (e.g., Wada et al. 2009; Seizinger et al. 2013; Hasegawa et al. 2021; Osinsky & Brilliantov 2022). Previous studies found that the conditions for growth or fragmentation depend on the strength of interparticle forces that act on constituent particles that are in contact with each other.
Wada et al. (2009) performed a large number of numerical simulations of collisions between two equalmass dust aggregates consisting of submicronsized spherical ice particles. They varied the collision velocities and impact angles and obtained the collisional growth efficiency. Their numerical simulations revealed that the threshold velocity for collisional growth or fragmentation of dust aggregates, v_{fra}, is approximately 60 m s^{−1} when the radius of the constituting particles is 0.1 μm and the initial aggregates before collisions are prepared by ballistic particle–cluster aggregation (BPCA; Mukai et al. 1992). The authors also predicted that v_{fra} would be proportional to the square root of the energy required for breaking one interparticle contact, E_{break}, which is related to the connection or disconnection of particles due to interparticle normal motion (e.g., Dominik & Tielens 1997; Wada et al. 2007).
Arakawa et al. (2022b) also performed numerical simulations of collisions between dust aggregates and investigated the dependence of the threshold velocity on the strength of the viscous dissipation. They found that the main energy dissipation mechanism is not the normal interaction (i.e., viscous dissipation and connection or disconnection of particles), but the rolling friction between particles that are in contact.
Then Arakawa et al. (2022a) investigated the dependence of the threshold velocity on the strength of rolling friction. They revealed that the threshold velocity only very weakly depends on the strength of rolling friction, however. In their simulations, frictions arising from two other tangential motions (sliding and twisting) complemented the change in energy dissipation due to rolling friction.
Energy dissipation is essentially required for the collisional growth of dust aggregates. The results of Arakawa et al. (2022a,b) indicate that the threshold velocity for collisional growth or fragmentation are affected by the presence or absence of interparticle tangential frictions when the motions cannot complement one another.
Here, we report numerical simulations of collisions between two equalmass dust aggregates consisting of submicronsized spherical ice particles, with and without interparticle friction. We also systematically varied the collision velocities and impact angles, and we calculated the collisional growth efficiency by averaging over impact angles. We confirm that the collisional growth efficiency of dust aggregates clearly depends on the presence or absence of interparticle frictions associated with tangential motions. Our results indicate that the dependence of the collisional growth efficiency on the material properties of constituent particles is more complex than previously assumed.
2. Model
We performed threedimensional numerical simulations of collisions between two equalmass dust aggregates. Our numerical code was originally developed by Wada et al. (2007), and Arakawa et al. (2022b) introduced a viscous drag force for normal motion (e.g., Krijt et al. 2013). We prepared two initial aggregates before collisions by BPCA as in previous studies (Wada et al. 2009; Arakawa et al. 2022b,a). We set the number of particles in the target aggregate, N_{tar}, equal to that for the projectile aggregate, N_{pro}, and the total number of particles in a simulation, N_{tot} = N_{tar} + N_{pro}, was 100 000. The constituent dust particles were made of water ice, and the particle radius was r_{1} = 0.1 μm. The particle interaction model is briefly described in Appendix A.
The strength of interparticle normal dissipation is controlled by a parameter called the viscoelastic timescale, T_{vis} (e.g., Arakawa et al. 2022b). We performed numerical simulations with two models for interparticle normal dissipation: with dissipation (T_{vis} = 6 ps), and without dissipation (T_{vis} = 0 ps; see Appendix A.1).
The strengths of the interparticle tangential friction are characterized by the spring constants for rolling (k_{r}), sliding (k_{s}), and twisting (k_{t}). In this study, we also performed numerical simulations with two models for interparticle tangential motions: frictional and frictionless models. In the frictional model, we considered the interparticle tangential interactions as modeled by Wada et al. (2007), and we used the same values of k_{r}, k_{s}, and k_{t} as assumed in Wada et al. (2007). In the frictionless model, in contrast, we did not consider the interparticle tangential frictions; in other words, we set k_{r} = k_{s} = k_{t} = 0 (see Appendix A.2).
3. Results
Figure 1 shows snapshots of the collisional outcomes. We show the numerical results for the frictionless model (k_{r} = k_{s} = k_{t} = 0) without normal dissipation (T_{vis} = 0 ps), and we set v_{col} = 39.8 m s^{−1}, where v_{col} is the collision velocity of two dust aggregates. We found that the collisional behavior strongly depends on the impact angles. Comparing Fig. 1c with Fig. 2 of Arakawa et al. (2022a), we can visually understand that the collisional behavior also depends on the interparticle interaction model. Arakawa et al. (2022a) presented the results for the same collision velocity and impact angle, but used the frictional model with normal dissipation. In contrast, Fig. 1c of this study shows the results for the frictionless model without normal dissipation. We found that the number of small fragments significantly increases compared to the results of Arakawa et al. (2022a).
Fig. 1. Snapshots of the collisional outcomes. We show the numerical results for the frictionless model (k_{r} = k_{s} = k_{t} = 0) without normal dissipation (T_{vis} = 0 ps), and we set v_{col} = 39.8 m s^{−1}. Panels (a)–(d) are the time series of snapshots for , 3/12, 6/12, and 9/12, respectively. The time interval for each snapshot is 0.40 μs. 
In this study, we use the normalized impact parameter, B_{off}, to quantify the offset of oblique collisions. We defined B_{off} as B_{off} = b_{off}/b_{max}, where b_{off} is the impact parameter and b_{max} is the sum of the radii of the target and projectile aggregates (see Arakawa et al. 2022b). The radius of the dust aggregates was set equal to the characteristic radius, r_{c}, which is given by , where r_{g} is the gyration radius (e.g., Mukai et al. 1992; Wada et al. 2009). It is clear that for headon collisions, and ranges from 0 to 1.
When collisions in space are considered, the impact parameter should vary with each collision event. The average value of a variable A weighted over B_{off},
is useful to measure the average outcome of the collision.
3.1. Collisional growth efficiency and the threshold velocity for collisional growth or fragmentation
Figure 2 shows the B_{off}weighted collisional growth efficiency, ⟨f_{gro}⟩, as a function of v_{col} for different particle interaction models. The growth efficiency, f_{gro}, was defined for each simulation, and it is given by f_{gro} = (N_{lar} − N_{tar})/N_{pro}, where N_{lar} denotes the number of constituent particles in the largest remnant (see Hasegawa et al. 2021). The collisional outcome of each simulation is summarized in Fig. B.1.
Fig. 2. B_{off}weighted collisional growth efficiency, ⟨f_{gro}⟩. We note that the results for frictional models (gray lines) are identical to those presented in Fig. 3 of Arakawa et al. (2022b) for v_{col} ≥ 20.0 m s^{−1}. 
We found that ⟨f_{gro}⟩ clearly depends on the choice of particle interaction models. In our simulation of the frictional model without normal dissipation (solid gray line), ⟨f_{gro}⟩ ≃ 0 at approximately v_{col} = 60 m s^{−1}. Wada et al. (2009) also performed simulations with the same particle interaction model, and their result is in excellent agreement with ours. We defined the threshold velocity for collisional growth or fragmentation, v_{fra}, as the collision velocity that satisfies ⟨f_{gro}⟩ = 0.
In the frictional model with normal dissipation (dashed gray line), we found that v_{fra} ≃ 50 m s^{−1}, which is close to the value for the frictional model without normal dissipation. The dependence of ⟨f_{gro}⟩ on v_{col} in frictional models with and without normal dissipation is similar; their difference in ⟨f_{gro}⟩ is typically smaller than 0.1 in the range of 10 m s^{−1} ≤ v_{col} ≤ 100 m s^{−1}. Arakawa et al. (2022b) concluded that this independence of ⟨f_{gro}⟩ on the presence or absence of normal dissipation is consistent with the fact that the main energy dissipation mechanism is not normal dissipation, but interparticle tangential friction when frictional models are used for particle interaction.
In frictionless models, however, we found that v_{fra} is significantly lower than that of frictional models, for example, v_{fra} ≃ 30 m s^{−1} in the frictionless model without normal dissipation (solid red line). In addition, ⟨f_{gro}⟩ also decreases compared to the frictional models. For example, 0.1 ≲ ⟨f_{gro}⟩ ≲ 0.2 at v_{col} = 15.8 m s^{−1} in frictionless models, while 0.8 ≲ ⟨f_{gro}⟩ ≲ 0.9 at the same v_{col} in frictional models. These differences would be related to the difference in energy dissipation processes (see Appendix C).
We note that v_{fra} depends on the presence or absence of normal dissipation when tangential frictions are absent. In the frictionless model with normal dissipation (dashed red line), our numerical results show that v_{fra} ≃ 45 m s^{−1}, which is 1.5 times higher than that of the frictionless model without normal dissipation. Figure 2 shows that ⟨f_{gro}⟩ for the frictionless model with normal dissipation is significantly higher than that of the frictionless model without normal dissipation in the ranges of v_{col} ≪ 20 m s^{−1} and v_{col} ≫ 60 m s^{−1}, although their difference in ⟨f_{gro}⟩ is roughly within 0.1 in the intermediate range of v_{col}.
Based on the results of Wada et al. (2009), Wada et al. (2013) proposed an empirical formula to estimate v_{fra} as a function of the particle radius and material properties of constituting particles. The empirical formula is , where m_{1} is the mass of each particle. As both E_{break} and m_{1} are independent of the spring constants for tangential motions (k_{r}, k_{s}, and k_{t}), this equation cannot express the effects of tangential interactions on v_{fra}. Our numerical results, however, highlight the impact of tangential interactions on v_{fra}. Thus we need to modify the prediction formula for v_{fra}.
Our results indicate that v_{fra} depends not only on E_{break}, but also on interparticle energies associated with tangential motions. It is important to note that their dependences on the particle radius and material properties are different from each other (see Wada et al. 2007). For example, E_{break} is proportional to (see Appendix A.1), while the energy needed to slide a particle by π/2 radian around its contact point, E_{slide}, is proportional to . The energy needed to twist over π/2 radian, E_{twist}, is proportional to (see Appendix A.2). Although it seems an extreme and unrealistic assumption, v_{fra} might be proportional to when it is proportional to , or v_{fra} might be proportional to when it is proportional to .
Blum & Wurm (2008) reviewed laboratory experiments of collisions of dust aggregates and reported that the threshold velocity for the sticking of dust aggregates is proportional to , with x ≳ 1 for 0.1 μm ≲ r_{1} ≲ 1 μm and x ≲ 1 for 1 μm ≲ r_{1} ≲ 10 μm. Although these experimental results are for headon collisions and cannot be directly compared with our B_{off}weighted numerical results, we can speculate that the dependence of v_{fra} on r_{1} might not be given by a simple powerlaw relation.
3.2. Size distribution of fragments
As shown in Fig. 1, collisions of dust aggregates consisting of frictionless particles without normal dissipation (k_{r} = k_{s} = k_{t} = 0 and T_{vis} = 0 ps) produce a large number of small fragments. In circumstellr disks, the strength of turbulence driven by magnetorotational instability is a function of the gas ionization degree and depends on the number of small dust aggregates (e.g., Okuzumi & Hirose 2012). Thus, the number of small dust aggregates might also be the key parameter for planet formation via collisional growth of dust aggregates. Here, we quantify the size distribution of fragments in our simulations (see also Arakawa et al. 2022b; Hasegawa et al. 2022; Osinsky & Brilliantov 2022).
Arakawa et al. (2022b) defined N_{cum}(≤N) as the cumulative number of particles that are constituents of fragments that contain not more than N particles. Figure 3 shows the B_{off}weighted average of N_{cum}(≤N), ⟨N_{cum}(≤N)⟩. We found that ⟨N_{cum}(≤N)⟩ significantly differs among particle interaction models.
Fig. 3. B_{off}weighted average of N_{cum}(≤N), ⟨N_{cum}(≤N)⟩. (a) For the case of v_{col} = 20 m s^{−1}. (b) For the case of v_{col} = 39.8 m s^{−1}. 
As shown in Fig. 3, the size distributions of fragments for frictional models with and without normal dissipations (dashed and solid gray lines, respectively) are similar, particularly for larger fragments (N > 10 for v_{col} = 20 m s^{−1} and N > 10^{3} for v_{col} = 39.8 m s^{−1}). This trend is consistent with the results of Arakawa et al. (2022b).
We found that the number of small fragments strongly depends on the presence or absence of interparticle tangential friction. When we apply the frictionless model without normal dissipation for constituting particles (solid red line), the number of small fragments with N < 10^{2} was orders of magnitude larger than that of the frictional models. This difference might also imply the importance of interparticle tangential frictions on the energy dissipation and deformation of dust aggregates during collision.
We note that for frictionless models, the size distribution of fragments clearly depends on the strength of normal dissipation (T_{vis}). When we consider v_{col} = 20 m s^{−1} as an example, the difference in ⟨f_{gro}⟩ is small (approximately 0.1; Fig. 2), but the difference in ⟨N_{cum}(≤N)⟩ is approximately two orders of magnitude for N < 10^{2}. These simulation results might indicate that the mechanisms for aggregatewide deformation and the ejection of small fragments are different. We will test this hypothesis in future studies.
4. Conclusions
Understanding the collisional behavior of dust aggregates consisting of micro or nanosized grains is essential for understanding planet formation. The threshold velocity for collisional growth or fragmentation, v_{fra}, is one of the most important parameters that controls the size of dust aggregates in circumstellar disks (e.g., Okuzumi & Tazaki 2019; Arakawa et al. 2021), but the dependence of v_{fra} on the particle radius and the material properties of constituting particles is still under debate.
Arakawa et al. (2022a,b) revealed that the main energy dissipation mechanism for oblique collisions of dust aggregates is not the interparticle normal interaction (i.e., connection and disconnection of particles), but the tangential friction between particles in contact with each other. Thus, we expect that the collisional outcomes of dust aggregates could depend on the strength of interparticle frictions.
In this study, we demonstrated that v_{fra} depends on the strength of tangential interactions using numerical simulations of collisions between dust aggregates. We tested 2 × 2 = 4 types of particle interaction models in this study. In these models, the presence or absence of the interparticle tangential frictions is captured by differences in the spring constants (k_{r}, k_{s}, and k_{t}), and the presence or absence of the normal dissipation is captured by differences in the viscoelastic timescale (T_{vis}).
In the frictional model without normal dissipation, we found that v_{fra} ≃ 60 m s^{−1}, which is consistent with what has been reported in previous studies (e.g., Wada et al. 2009; Hasegawa et al. 2021). In contrast, in the frictionless model without normal dissipation, we found that v_{fra} ≃ 30 m s^{−1}, which is notably lower than for the frictional model (see Fig. 2). We also found that v_{fra} depends on the presence or absence of the normal dissipation when tangential frictions are absent, while v_{fra} is nearly independent of T_{vis} for frictional models. Our results further indicate that the dependence of v_{fra} on the particle radius and material properties cannot be described by a simple powerlaw relation (see Blum & Wurm 2008; Wada et al. 2009, 2013). Future studies of this point are essential, although a large number of numerical simulations are needed to construct a better fitting formula to predict v_{fra} as a function of particle radius and material properties.
The size distribution of fragments also depends on the choice of particle interaction models (see Fig. 3). As shown in Fig. 1, collisions of dust aggregates consisting of frictionless particles without normal dissipation produce a large number of small fragments. We also found that for frictionless models, the size distribution of fragments significantly depends on the strength of normal dissipation, even if the difference in ⟨f_{gro}⟩ is small. Our results might indicate that the mechanisms for aggregatewide deformation and the ejection of small fragments are different, although we need to investigate this hypothesis in future studies.
Acknowledgments
The anonymous reviewer provided a constructive review that improved this paper. Numerical computations were carried out on PC cluster at CfCA, NAOJ. H.T. and E.K. were supported by JSPS KAKENHI grant No. 18H05438. We thank American Journal Experts (AJE) for English language editing.
References
 Arakawa, S., & Krijt, S. 2021, ApJ, 910, 130 [NASA ADS] [CrossRef] [Google Scholar]
 Arakawa, S., Matsumoto, Y., & Honda, M. 2021, ApJ, 920, 27 [NASA ADS] [CrossRef] [Google Scholar]
 Arakawa, S., Tanaka, H., & Kokubo, E. 2022a, ApJ, 939, 100 [NASA ADS] [CrossRef] [Google Scholar]
 Arakawa, S., Tanaka, H., & Kokubo, E. 2022b, ApJ, 933, 144 [NASA ADS] [CrossRef] [Google Scholar]
 Blum, J., & Wurm, G. 2008, ARA&A, 46, 21 [Google Scholar]
 Dominik, C., & Tielens, A. G. G. M. 1997, ApJ, 480, 647 [Google Scholar]
 Dominik, C., & Tielens, A. G. G. M. 1995, Phil. Mag. Part A, 72, 783 [CrossRef] [Google Scholar]
 Dominik, C., & Tielens, A. G. G. M. 1996, Phil. Mag. Part A, 73, 1279 [CrossRef] [Google Scholar]
 Drazkowska, J., Bitsch, B., Lambrechts, M., et al. 2022, ArXiv eprints [arXiv:2203.09759] [Google Scholar]
 Fritscher, M., & Teiser, J. 2021, ApJ, 923, 134 [NASA ADS] [CrossRef] [Google Scholar]
 Gundlach, B., & Blum, J. 2015, ApJ, 798, 34 [Google Scholar]
 Güttler, C., Blum, J., Zsom, A., Ormel, C. W., & Dullemond, C. P. 2010, A&A, 513, A56 [Google Scholar]
 Hasegawa, Y., Suzuki, T. K., Tanaka, H., Kobayashi, H., & Wada, K. 2021, ApJ, 915, 22 [NASA ADS] [CrossRef] [Google Scholar]
 Hasegawa, Y., Suzuki, T. K., Tanaka, H., Kobayashi, H., & Wada, K. 2022, ApJ, accepted [arXiv:2212.10796] [Google Scholar]
 Hayashi, C., Nakazawa, K., & Nakagawa, Y. 1985, in Protostars and Planets II, eds. D. C. Black, & M. S. Matthews, 1100 [Google Scholar]
 Haynes, B. S., & Wagner, H. G. 1981, Progr. Energy. Combust. Sci., 7, 229 [NASA ADS] [CrossRef] [Google Scholar]
 Johnson, K. L., Kendall, K., & Roberts, A. D. 1971, Proc. Royal Soc. London Ser. A, 324, 301 [NASA ADS] [Google Scholar]
 Krijt, S., Güttler, C., Heißelmann, D., Dominik, C., & Tielens, A. G. G. M. 2013, J. Phys. D Appl. Phys., 46, 435303 [NASA ADS] [CrossRef] [Google Scholar]
 Mukai, T., Ishimoto, H., Kozasa, T., Blum, J., & Greenberg, J. M. 1992, A&A, 262, 315 [NASA ADS] [Google Scholar]
 Musiolik, G., Teiser, J., Jankowski, T., & Wurm, G. 2016, ApJ, 827, 63 [NASA ADS] [CrossRef] [Google Scholar]
 Ohno, K., Zhang, X., Tazaki, R., & Okuzumi, S. 2021, ApJ, 912, 37 [NASA ADS] [CrossRef] [Google Scholar]
 Okuzumi, S., & Hirose, S. 2012, ApJ, 753, L8 [CrossRef] [Google Scholar]
 Okuzumi, S., & Tazaki, R. 2019, ApJ, 878, 132 [Google Scholar]
 Osinsky, A., & Brilliantov, N. 2022, Phys. A: Stat. Mech. Appl., 603, 127785 [Google Scholar]
 Schräpler, R. R., Landeck, W. A., & Blum, J. 2022, MNRAS, 509, 5641 [Google Scholar]
 Seizinger, A., Krijt, S., & Kley, W. 2013, A&A, 560, A45 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Trainer, M. G., Pavlov, A. A., Dewitt, H. L., et al. 2006, Proc. Natil. Acad. Sci., 103, 18035 [CrossRef] [Google Scholar]
 Wada, K., Tanaka, H., Suyama, T., Kimura, H., & Yamamoto, T. 2007, ApJ, 661, 320 [NASA ADS] [CrossRef] [Google Scholar]
 Wada, K., Tanaka, H., Suyama, T., Kimura, H., & Yamamoto, T. 2009, ApJ, 702, 1490 [Google Scholar]
 Wada, K., Tanaka, H., Okuzumi, S., et al. 2013, A&A, 559, A62 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Wada, K., Tsukamoto, Y., & Kokubo, E. 2021, ApJ, 909, 96 [NASA ADS] [CrossRef] [Google Scholar]
 Zhang, X., Strobel, D. F., & Imanaka, H. 2017, Nature, 551, 352 [NASA ADS] [CrossRef] [Google Scholar]
Appendix A: Particle interaction model
Here, we briefly explain the particle interaction model used in this study. Details are described in Wada et al. (2007) and Arakawa et al. (2022b).
A.1. Normal motion
We assume that the normal force acting between two particles, F, is given by the sum of the two terms
where F_{E} denotes the force arising from the elastic deformation of particles, and F_{D} is the force related to the viscous dissipation.
When two particles in contact are elastic spheres with a surface energy, the elastic term is given by the following equation (Johnson et al. 1971):
where a is the contact radius, a_{0} is the contact radius at equilibrium, F_{c} = 3πγR is the maximum force needed to separate the two particles in contact, γ is the surface energy, and R = r_{1}/2 is the reduced particle radius. At the equilibrium state, a_{0} is given by
where ℰ^{*} is the reduced Young modulus.
The contact radius is a function of the compression length between two particles in contact, δ, and vice versa,
where is the equilibrium compression length at a = a_{0}. Two particles in contact separate when the compression length reaches a critical length, δ = −δ_{c}, where δ_{c} = (9/16)^{1/3}δ_{0}.
The viscous drag force is given by
where ν is Poisson’s ratio, v_{rel} is the normal component of the relative velocity of the two particles, and T_{vis} is the viscoelastic timescale (see Krijt et al. 2013).
In this study, we considered two cases: T_{vis} = 0 ps and T_{vis} = 6 ps. We set T_{vis} = 6 ps in our previous studies (e.g., Arakawa et al. 2022a), and the choice of T_{vis} = 6 ps was motivated by extrapolation of laboratory experiments (see Figure 5 of Arakawa & Krijt 2021). Krijt et al. (2013) predicted that T_{vis} would be approximately proportional to R, and Arakawa & Krijt (2021) confirmed this relation using experimental results of Gundlach & Blum (2015) and Musiolik et al. (2016).
The potential energy for the normal motion of the two particles in contact, U_{n}, is given by
The energy needed to break a contact in equilibrium by a quasistatic process (i.e., v_{rel} → 0 and F_{D} → 0), E_{break}, is
and E_{break} is proportional to .
A.2. Tangential motion
The tangential motion of two particles in contact is the combination of three motions: rolling, sliding, and twisting. The displacements corresponding to these motions are described as the rotation of two particles in contact. Wada et al. (2007) provided the particle interaction model for these tangential motions (see also Dominik & Tielens 1995, 1996), which is equivalent to the linear spring model with critical displacements to their elastic limits. The concept of the tangential interaction model is summarized in Figures 2 and 3 of Wada et al. (2007).
A.2.1. Rolling motion
The spring constant for the rolling displacement, ξ, is k_{r}. For the frictional model, we set
and we set k_{r} = 0 for the frictionless model. The energy needed to rotate a particle by π/2 radian around its contact point, E_{roll}, is useful for interpreting the collisional outcomes of dust aggregates from energetics (e.g., Dominik & Tielens 1997). Wada et al. (2007) derived that E_{roll} is given by
where ξ_{crit} is the critical rolling displacement. In this study, we set ξ_{crit} = 0.8 nm for water ice particles of r_{1} = 0.1 μm. The dependence of ξ_{crit} on r_{1} is poorly understood, however (see Arakawa et al. 2022a, and references therin).
A.2.2. Sliding motion
The spring constant for the sliding displacement, ζ, is k_{s}. For the frictional model, we set
where 𝒢^{⋆} = 𝒢/[2(2 − ν)], and 𝒢 is the shear modulus. We set k_{s} = 0 for the frictionless model, as is the case for k_{r}. The critical sliding displacement, ζ_{crit}, is given by ζ_{crit} = [(2 − ν)/(16π)]a_{0} (Wada et al. 2007). The energy needed to slide a particle by π/2 radian around its contact point, E_{slide}, is given by
and E_{slide} is proportional to .
A.2.3. Twisting motion
The spring constant for the twisting displacement, ϕ, is k_{t}. For the frictional model, we set
where 𝒢′ = 𝒢/2 is the reduced shear modulus. We set k_{t} = 0 for the frictionless model, as is the case for k_{r} and k_{s}. The critical angle for twisting, ϕ_{crit}, is set to ϕ_{crit} = 1/(16π) (Wada et al. 2007). The energy needed to twist over π/2 radian, E_{twist}, is given by
and E_{twist} is proportional to .
Appendix B: Collisional growth efficiency
Figure B.1 shows the collisional growth efficiency, f_{gro}. The square of the normalized impact parameter ranges from to 1 with an interval of 1/12. The collision velocity was set to 10^{(0.1i)} m s^{−1}, where i = 10, 11, ..., 20. Each panel shows f_{gro} for different particle interaction models. The gray lines in Figure B.1 are the B_{off}weighted collisional growth efficiency, ⟨f_{gro}⟩, and they are identical to those shown in Figure 2.
Fig. B.1. Collisional growth efficiency, f_{gro}, for different settings of B_{off} and v_{col}. (a) For the frictional model without normal dissipation. (b) For the frictional model with normal dissipation. (c) For the frictionless model without normal dissipation. (d) For the frictionless model with normal dissipation. 
Appendix C: Energy dissipation and interparticle connection or disconnection
We briefly check the energy dissipation in our simulations (Appendix C.1). In addition, we also show the numbers of connection and disconnection events (Appendix C.2).
C.1. Energy dissipation
Figure C.1 shows the total energy dissipation due to particle interactions from the start to the end of the simulations, E_{dis, tot} (see also Figure 8 of Arakawa et al. 2022a). For frictionless models, E_{dis, tot} is given by E_{dis, tot} = E_{dis, c} + E_{dis, v}, where E_{dis, c} is the energy dissipation due to the connection and disconnection of particles and E_{dis, v} is the energy dissipation due to the viscous drag force. The dashed gray line denotes the initial kinetic energy.
Fig. C.1. Total energy dissipation due to particle interactions from the start to the end of the simulations, E_{dis, tot}, as a function of v_{rel} and B_{off}. (a) For the frictional model without normal dissipation. (b) For the frictional model with normal dissipation. (c) For the frictionless model without normal dissipation. (d) For the frictionless model with normal dissipation. The dashed gray line denotes the initial kinetic energy (see Arakawa et al. 2022a). 
The relation between E_{dis, tot} and f_{gro} is complex. At approximately v_{col} = 10 m s^{−1}, E_{dis, tot} for frictionless models is larger than that for frictional models, but ⟨f_{gro}⟩ for frictionless models is lower than that for frictional models (see Figure 2). For and v_{col} = 100 m s^{−1}, E_{dis, tot} is nearly equal to the initial kinetic energy in Figures C.1(a), C.1(b), and C.1(d), and f_{gro} ∼ 1 in these cases (see Figure B.1). In contrast, E_{dis, tot} is significantly lower than the initial kinetic energy in Figure C.1(c), and the corresponding growth efficiency is f_{gro} ∼ −1 in this case.
Figure C.2 shows E_{dis, c} as a function of v_{rel} and B_{off}. We note that E_{dis, c} is identical to E_{dis, tot} for the frictionless model without normal dissipation (Figure C.2(c)). It is clear that E_{dis, c} for frictionless models is higher than that for frictional models in the range of 10 m s^{−1} ≤ v_{col} ≤ 100 m s^{−1}. For frictional models, the fraction of energy dissipation due to the connection and disconnection of particles is notably small: E_{dis, c}/E_{dis, tot} ≪ 1. This is because the main energy dissipation mechanism for collisions of dust aggregates is the tangential friction between particles that are in contact for frictional models (Arakawa et al. 2022b,a).
Fig. C.2. Energy dissipation due to connection and disconnection of particles, E_{dis, c}, as a function of v_{rel} and B_{off}. (a) For the frictional model without normal dissipation. (b) For the frictional model with normal dissipation. (c) For the frictionless model without normal dissipation. (d) For the frictionless model with normal dissipation. The dashed gray line denotes the initial kinetic energy. 
C.2. Interparticle connection and disconnection
We also verified the numbers of connection and disconnection events, as they are directly related to E_{dis, c}. Figures C.3 and C.4 show the numbers of connection and disconnection events in a collision between dust aggregates, N_{con} and N_{cut}, respectively (see also Figure 15 of Arakawa et al. 2022b).
Fig. C.3. Number of connection events in a collision between dust aggregates, N_{con}, as a function of v_{rel} and B_{off}. (a) For the frictional model without normal dissipation. (b) For the frictional model with normal dissipation. (c) For the frictionless model without normal dissipation. (d) For the frictionless model with normal dissipation. 
Fig. C.4. Number of disconnection events in a collision between dust aggregates, N_{cut}, as a function of v_{rel} and B_{off}. (a) For the frictional model without normal dissipation. (b) For the frictional model with normal dissipation. (c) For the frictionless model without normal dissipation. (d) For the frictionless model with normal dissipation. 
For frictionless models, N_{con} is larger than N_{tot} in the range of 10 m s^{−1} ≤ v_{col} ≤ 100 m s^{−1} (Figures C.3(c) and C.3(d)). In contrast, N_{con} is smaller than N_{tot} for frictionless models at approximately v_{col} = 10 m s^{−1} (Figures C.3(a) and C.3(b)). The large difference in N_{con} would be related to the difference in the degree of deformation of aggregates after collisions.
For the frictionless model without normal dissipation, not only N_{con} but also N_{cut} exceeds N_{tot} in the entire range of 10 m s^{−1} ≤ v_{col} ≤ 100 m s^{−1} (Figure C.4(c)). For other models, N_{cut} is significantly smaller than N_{tot} at approximately v_{col} = 10 m s^{−1} (Figures C.4(a), C.4(b), and C.4(d)). As shown in Figure 2, ⟨f_{gro}⟩ for the frictionless model without normal dissipation is notably lower than for the other three models. We therefore hypothesize that the dependence of ⟨f_{gro}⟩ on v_{col} might be the key to understanding the large difference in v_{fra} among particle interaction models.
All Figures
Fig. 1. Snapshots of the collisional outcomes. We show the numerical results for the frictionless model (k_{r} = k_{s} = k_{t} = 0) without normal dissipation (T_{vis} = 0 ps), and we set v_{col} = 39.8 m s^{−1}. Panels (a)–(d) are the time series of snapshots for , 3/12, 6/12, and 9/12, respectively. The time interval for each snapshot is 0.40 μs. 

In the text 
Fig. 2. B_{off}weighted collisional growth efficiency, ⟨f_{gro}⟩. We note that the results for frictional models (gray lines) are identical to those presented in Fig. 3 of Arakawa et al. (2022b) for v_{col} ≥ 20.0 m s^{−1}. 

In the text 
Fig. 3. B_{off}weighted average of N_{cum}(≤N), ⟨N_{cum}(≤N)⟩. (a) For the case of v_{col} = 20 m s^{−1}. (b) For the case of v_{col} = 39.8 m s^{−1}. 

In the text 
Fig. B.1. Collisional growth efficiency, f_{gro}, for different settings of B_{off} and v_{col}. (a) For the frictional model without normal dissipation. (b) For the frictional model with normal dissipation. (c) For the frictionless model without normal dissipation. (d) For the frictionless model with normal dissipation. 

In the text 
Fig. C.1. Total energy dissipation due to particle interactions from the start to the end of the simulations, E_{dis, tot}, as a function of v_{rel} and B_{off}. (a) For the frictional model without normal dissipation. (b) For the frictional model with normal dissipation. (c) For the frictionless model without normal dissipation. (d) For the frictionless model with normal dissipation. The dashed gray line denotes the initial kinetic energy (see Arakawa et al. 2022a). 

In the text 
Fig. C.2. Energy dissipation due to connection and disconnection of particles, E_{dis, c}, as a function of v_{rel} and B_{off}. (a) For the frictional model without normal dissipation. (b) For the frictional model with normal dissipation. (c) For the frictionless model without normal dissipation. (d) For the frictionless model with normal dissipation. The dashed gray line denotes the initial kinetic energy. 

In the text 
Fig. C.3. Number of connection events in a collision between dust aggregates, N_{con}, as a function of v_{rel} and B_{off}. (a) For the frictional model without normal dissipation. (b) For the frictional model with normal dissipation. (c) For the frictionless model without normal dissipation. (d) For the frictionless model with normal dissipation. 

In the text 
Fig. C.4. Number of disconnection events in a collision between dust aggregates, N_{cut}, as a function of v_{rel} and B_{off}. (a) For the frictional model without normal dissipation. (b) For the frictional model with normal dissipation. (c) For the frictionless model without normal dissipation. (d) For the frictionless model with normal dissipation. 

In the text 
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