Collisional Growth Efficiency of Dust Aggregates and Its Independence of the Strength of Interparticle Rolling Friction

The first step of planet formation is the pairwise collisional growth of dust aggregates consisting of submicron-sized grains (e.g., Wurm & Teiser 2021). Therefore, understanding the condition for collisional growth of dust aggregates is essential.

The collisional behavior of dust aggregates has been studied extensively, by using both laboratory experiments (e.g., Blum & Wurm 2008; Fritscher & Teiser 2021; Schräpler et al. 2022) and numerical simulations (e.g., Wada et al. 2009; Hasegawa et al. 2021; Umstätter & Urbassek 2021a). It is known that the threshold collision velocity for the fragmentation of dust aggregates, v_fra, depends on the strength of interparticle forces acting on constituent particles in contact.

The stickiness of dust particles has been investigated in a large number of studies (e.g., Güttler et al. 2012; Tanaka et al. 2012; Gundlach & Blum 2015). These studies found that the threshold velocity for the sticking in a head-on collision of individual dust particles is several times higher than that predicted for perfectly elastic spheres. Arakawa & Krijt (2021) concluded that the viscous dissipation for the normal motion would play a critical role in the collision between micron-sized water-ice particles.

Recently, Arakawa et al. (2022) performed numerical simulations of collisions between two equal-mass dust aggregates consisting of submicron-sized spherical ice particles. Surprisingly, they found that the main energy dissipation mechanism was not the viscous dissipation but the friction between particles in contact. Interparticle frictions are caused by tangential motions, and there are three types of tangential motions: rolling, sliding, and twisting (see Figure 2 of Wada et al. 2007). Therefore, we can imagine that the collisional behavior of dust aggregates would be affected by the strength of interparticle frictions.

The rolling friction is one of the strongest energy dissipation mechanisms. The interparticle rolling motion behaves elastically (i.e., without energy dissipation) when the length of the rolling displacement, ξ, is smaller than the critical rolling displacement, ξ_crit (e.g., Dominik & Tielens 1995; Wada et al. 2007). The energy dissipation starts when ξ exceeds ξ_crit.

As the interparticle rolling friction is the dominant mechanism for the energy dissipation within colliding dust aggregates, we can expect that the collisional outcome of dust aggregates must depend on ξ_crit; however, it is poorly understood how large ξ_crit is for micron-sized spheres. Dominik & Tielens (1995) constructed the theoretical model of rolling based on the contact theory developed by Johnson et al. (1971). They mentioned that ξ_crit should be of the order of the distance between atoms in the materials, and it is typically 0.2 nm. In contrast, Heim et al. (1999) measured the rolling friction torques between individual silica microspheres by using an atomic force microscope. They found, however, that the critical rolling displacement for silica microspheres with radii between 0.5 and 2.5 μm would be ξ_crit = 3.2 nm, which is 16 times larger than Dominik & Tielens (1995) assumed. We note that the upper limit of ξ_crit would be given by the contact radius of spheres at the equilibrium state, a₀, and ξ_crit = 3.2 nm is still smaller than a₀.

Krijt et al. (2014) modified the analytical theory for the rolling friction based on the concept of adhesion hysteresis, and they reported that ξ_crit would be given by ξ_crit ∼ a₀/24 for silica microspheres when the crack propagation rate is 1–10 μm s⁻¹. Recently, Umstätter & Urbassek (2021b) performed molecular dynamics simulations of amorphous Lennard-Jones grains and confirmed that ξ_crit depends on both the material parameters and the angular velocity for rolling.

Although the rolling friction is one of the main energy dissipation mechanisms for oblique collisions between dust aggregates, dependence of the collisional growth condition on ξ_crit is poorly understood. Wada et al. (2008) performed numerical simulations of head-on collisions between dust aggregates with various ξ_crit. They used highly porous dust aggregates prepared by ballistic cluster–cluster aggregation in their simulations, and they investigated the condition for collisional compression and fragmentation. They reported that both the onset of collisional compression and the maximum compression linearly depend on ξ_crit, but the size of the largest fragment formed after a collision is nearly independent of ξ_crit. These results would be naturally explained by the findings that the collisional compression is caused by interparticle rolling motions while the energy dissipation is primarily caused by the connection and the disconnection of particles (Wada et al. 2009). We note, however, that they focused on head-on collisions and did not study collisional behavior for oblique collisions. Suyama et al. (2008, 2012) also investigated the collisional compression of dust aggregates by sequential collision simulations, but they also focused on head-on collisions.

In this study, we report the results from numerical simulations of collisions between two equal-mass dust aggregates with various collision velocity and impact parameters. We changed the strength of the interparticle rolling friction systematically, which is controlled by ξ_crit. Surprisingly, we found that the threshold collision velocity for the fragmentation of dust aggregates is nearly independent of ξ_crit when we consider oblique collisions.

We performed three-dimensional numerical simulations of collisions between two equal-mass dust aggregates. The numerical code used in this study is identical to that developed by Arakawa et al. (2022), which considered the viscous drag for normal motion and also considered friction torques for tangential motions.

2.1. Material Parameters

We assumed that the dust particles constituting dust aggregates are made of water ice, and all particles have the same radius of r₁ = 0.1 μm. The material parameters of water-ice particles used in this study are listed in Table 1. The particle interaction model is identical to that used in Arakawa et al.(2022).

Table 1. List of Material Parameters Used in This Study (see Arakawa et al. 2022)

Parameter	Symbol	Value
Particle radius (μm)	r1	0.1
Material density (kg m−3)	ρ	1000
Surface energy (mJ m−2)	γ	100
Young's modulus (GPa)	${ \mathcal E }$	7
Poisson's ratio	ν	0.25
Viscoelastic timescale (ps)	Tvis	6

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2.2. Interparticle Rolling Motion

In this study, we regarded ξ_crit as a parameter. We applied a linear spring model with a critical displacement to the interparticle rolling motion (Wada et al. 2007). When the length of the rolling displacement is ξ, the potential energy stored by the rolling displacement is given by

$$\begin{eqnarray}&&{U}_{{\rm{r}}}=\displaystyle \frac{1}{2}{k}_{{\rm{r}}}{\xi }^{2},\end{eqnarray} \tag{ 1 }$$

and the spring constant, k_r, is given by

$$\begin{eqnarray}&&{k}_{{\rm{r}}}=\displaystyle \frac{4{F}_{{\rm{c}}}}{R}.\end{eqnarray} \tag{ 2 }$$

Here F_c = 3π γ R is the maximum force needed to separate the two particles in contact, and R = r₁/2 is the reduced particle radius. The critical energy required to start rolling, E_r,crit, is given by

$$\begin{eqnarray}\begin{array}{rcl}{E}_{{\rm{r}},\mathrm{crit}} & = & \displaystyle \frac{1}{2}{k}_{{\rm{r}}}{{\xi }_{\mathrm{crit}}}^{2}\\ & = & 6\pi \gamma {{\xi }_{\mathrm{crit}}}^{2}.\end{array}\end{eqnarray} \tag{ 3 }$$

Wada et al. (2007) introduced E_roll as the energy needed to rotate a particle by a π/2 radian around its contact point, which is given by

$$\begin{eqnarray}\begin{array}{rcl}{E}_{\mathrm{roll}} & = & {k}_{{\rm{r}}}{\xi }_{\mathrm{crit}}\pi R\\ & = & 12{\pi }^{2}\gamma R{\xi }_{\mathrm{crit}}.\end{array}\end{eqnarray} \tag{ 4 }$$

Figure 1 shows the dependence of E_r,crit and E_roll on ξ_crit. We normalized E_r,crit and E_roll by F_c δ_c, where δ_c is the critical stretching length for particle separation, which is given by

$$\begin{eqnarray}&&{\delta }_{{\rm{c}}}={\left(\displaystyle \frac{9}{16}\right)}^{1/3}\displaystyle \frac{{{a}_{0}}^{2}}{3R}.\end{eqnarray} \tag{ 5 }$$

The contact radius of spheres at the equilibrium state, a₀, is given by

$$\begin{eqnarray}\begin{array}{rcl}{a}_{0} & = & {\left(\displaystyle \frac{9\pi \gamma {R}^{2}}{{{ \mathcal E }}^{* }}\right)}^{1/3}\\ & = & 12.4\,\,\mathrm{nm},\end{array}\end{eqnarray} \tag{ 6 }$$

where ${{ \mathcal E }}^{* }={ \mathcal E }/[2(1-{\nu }^{2})]$ is the reduced Young's modulus.

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The energy needed to break a contact in the equilibrium by the quasistatic process, E_break, is given by

$$\begin{eqnarray}\begin{array}{rcl}{E}_{\mathrm{break}} & = & \left(\displaystyle \frac{4}{45}+\displaystyle \frac{4}{5}\times {6}^{1/3}\right){F}_{{\rm{c}}}{\delta }_{{\rm{c}}}\\ & = & 1.54{F}_{{\rm{c}}}{\delta }_{{\rm{c}}}.\end{array}\end{eqnarray} \tag{ 7 }$$

The blue horizontal line shown in Figure 1 represents E_break. We found that E_break > E_r,crit when ξ_crit ≤ 3.2 nm. Not only the potential energy related to the normal compression but also the potential energies stored by the tangential displacements dissipate when two particles separate. However, the contribution of the rolling displacement is negligible for ξ_crit ≤ 3.2 nm. In addition, Arakawa et al. (2022) confirmed that contributions of sliding and twisting displacements also play a minor role; they are one order of magnitude smaller than E_break.

The collision velocity of two dust aggregates is v_col, and the initial kinetic energy per one particle with the velocity of v_col/2 is

$$\begin{eqnarray}\begin{array}{rcl}{E}_{\mathrm{kin},1} & = & \displaystyle \frac{1}{2}{m}_{1}{\left(\displaystyle \frac{{v}_{\mathrm{col}}}{2}\right)}^{2}\\ & = & 33{\left(\displaystyle \frac{{v}_{\mathrm{col}}}{50\,\,{\rm{m}}.{{\rm{s}}}^{-1}}\right)}^{2}{F}_{{\rm{c}}}{\delta }_{{\rm{c}}},\end{array}\end{eqnarray} \tag{ 8 }$$

where ${m}_{1}=4\pi \rho {{r}_{1}}^{3}/3$ is the mass of each particle. Therefore, E_roll ≫ E_kin,1 for ξ_crit ≫ 3.2 nm and v_col ≪ 50 m s⁻¹, and the collisional deformation of dust aggregates due to rolling motion would be suppressed for large ξ_crit and/or small v_col cases.

2.3. Initial Condition

Figure 2 shows the snapshots of the collisional outcomes. Here we show the results for v_col = 39.8 m s⁻¹ and ${{B}_{\mathrm{off}}}^{2}=6/12$. We found that the collisional behavior is similar among different settings of ξ_crit as shown in Figure 2. We also analyzed the number of constituent particles in each fragment at the end of the simulation, t = 7.99 μs. The numbers of constituent particles in the three largest remnants are (N_lar, N_2nd, N_3rd) = (34042, 33299, 29827) for ξ_crit = 0.2 nm, (34170, 32457, 30940) for ξ_crit = 0.8 nm, (32571, 31598, 29338) for ξ_crit = 3.2 nm, and (48491, 31900, 19003) for ξ_crit = 12.8 nm, respectively. For these cases, most of the particles are included in the three largest remnants.

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We discuss the parameter dependence of the largest fragment formed after a collision. The collisional growth efficiency, f_gro, is defined as follows (e.g., Wada et al. 2013):

$$\begin{eqnarray}&&{f}_{\mathrm{gro}}=\displaystyle \frac{{N}_{\mathrm{lar}}-{N}_{\mathrm{tar}}}{{N}_{\mathrm{pro}}}.\end{eqnarray} \tag{ 9 }$$

Figure 3 shows the collisional growth efficiency for different settings of ξ_crit, B_off, and v_col. Panels (a)–(d) show the results for ξ_crit = 0.2 nm, 0.8 nm, 3.2 nm, and 12.8 nm, respectively. For head-on collisions with ${{B}_{\mathrm{off}}}^{2}=0/12$, f_gro ≃ 1 when 20 m s⁻¹ ≤ v_col ≤ 100 m s⁻¹, and for grazing collisions with ${{B}_{\mathrm{off}}}^{2}=12/12$, f_gro ≃ 0. These trends are consistent with those observed in previous studies (e.g., Hasegawa et al. 2021; Arakawa et al. 2022).

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We note that Schräpler et al. (2022) proposed a model for collisions between porous dust aggregates based on their experimental results. In their model, head-on collisions of porous aggregates result in bouncing when the collision velocity exceeds the sticking threshold, and the sticking threshold decreases with increasing ξ_crit. However, we have never seen bouncing behavior for head-on collisions even though ξ_crit is varied. Although the real reason is unclear, this discrepancy might originate from the difference in the radius and/or the structure of aggregates; the radius of dust aggregates used in this study is approximately 7 μm, while Schrapler et al. (2022) used centimeter-sized aggregates.

The impact parameter must vary with each collision event in protoplanetary disks. Then the average value of f_gro weighted over B_off would be useful to investigate whether dust aggregates can grow into larger aggregates in protoplanetary disks. The average value of f_gro weighted over B_off is given by

$$\begin{eqnarray}&&\langle {f}_{\mathrm{gro}}\rangle \equiv \displaystyle \frac{1}{\pi }{\int }_{0}^{1}{\rm{d}}{B}_{\mathrm{off}}\,2\pi {B}_{\mathrm{off}}{f}_{\mathrm{gro}}.\end{eqnarray} \tag{ 10 }$$

The gray lines in Figure 3 are the B_off-weighted collisional growth efficiency, 〈f_gro〉. As 〈f_gro〉 decreases with increasing v_col, the threshold collision velocity for the fragmentation of dust aggregates, v_fra, is defined as the velocity where 〈f_gro〉 = 0. We found that v_fra is nearly independent of ξ_crit; 50 m s⁻¹ ≤v_fra ≤ 60 m s⁻¹ for 0.2 nm ≤ ξ_crit ≤ 12.8 nm.

Figure 4(a) shows the B_off-weighted collisional growth efficiency. Although 〈f_gro〉 slightly depends on ξ_crit when we set 20 m s⁻¹ ≤ v_col ≤ 30 m s⁻¹, v_fra is nearly independent of ξ_crit in our simulations. Figures 4(b)–4(d) show the collisional growth efficiency for oblique collisions. As shown in Figures 4(c) and 4(d), for grazing collisions with ${{B}_{\mathrm{off}}}^{2}\geqslant 6/12$, f_gro is nearly independent of v_col around v_fra. In contrast, Figure 4(b) exhibits a clear dependence of f_gro on v_col around v_fra if we focus on oblique collisions with ${{B}_{\mathrm{off}}}^{2}\simeq 3/12$. Therefore, we can imagine that the energy dissipation for oblique collisions with ${{B}_{\mathrm{off}}}^{2}\simeq 3/12$ and its dependence on v_col would be the key to unveiling the ξ_crit dependence of 〈f_gro〉 and v_fra.

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We check the energy dissipation due to particle interactions from the start to the end of the simulations. Figure 5 shows the energy dissipation due to rolling friction (E_dis,r), sliding friction (E_dis,s), twisting friction (E_dis,t), connection and disconnection of particles (E_dis,c), and viscous drag force (E_dis,v). We also show the total energy dissipation due to all particle interactions, E_dis,tot ≡ E_dis,r + E_dis,s + E_dis,t + E_dis,c + E_dis,v. Here we set ${{B}_{\mathrm{off}}}^{2}=3/12$ and investigated the dependence on ξ_crit and v_col. Results for ${{B}_{\mathrm{off}}}^{2}=6/12$ and 9/12 are shown in the Appendix as a reference (see Figures 6 and 7). The particle interaction energies for interparticle motions (e.g., E_r,crit and E_roll) are summarized in Table 2 of Arakawa et al. (2022).

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We found that E_dis,r does not monotonically increase with ξ_crit; there is a maximum value of E_dis,r. It is clear that E_dis,r → 0 when ξ_crit → 0 because E_roll becomes zero. It is also natural that E_dis,r → 0 when ξ_crit → ∞ ; this is because the critical energy required to start rolling becomes E_r,crit → ∞ . Although it depends on v_col, E_dis,r takes the maximum around ξ_crit ≃ 3.2 nm for v_col ∼ v_fra. As the kinetic energy per one particle, E_kin,1, cannot overcome E_roll for v_col ∼ v_fra and ξ_crit ≫ 3.2 nm, the decrease of E_dis,r for the large ξ_crit would be consistent with the order of magnitude estimated in Section 2.2.

In contrast to E_dis,r, E_dis,s and E_dis,t take large values for both the lower and the upper limits of ξ_crit (i.e., 0.2 nm and 12.8 nm, respectively). For low-speed collisions with v_col ≲ v_fra, E_dis,t plays the major role in energy dissipation when ξ_crit is set to be close to the lower or the upper limits. For high-speed collisions with v_col ≳ v_fra, E_dis,s plays the major role in the energy dissipation when ξ_crit is set to be close to the lower or the upper limits. Conversely, E_dis,r is the strongest energy dissipation mechanism when we choose a moderate ξ_crit in the range of 0.8–3.2 nm for submicron-sized ice particles.

The strong dependence of E_dis,s on v_col is discussed in Section 5.2 of Arakawa et al. (2022). They claimed that the strong dependence could be originated from the magnitude relation between E_slide and E_kin,1, where E_slide is the the energy needed to slide a particle by a π/2 radian around its contact point.

Interestingly, the total energy dissipation due to all particle interactions, E_dis,tot, is nearly independent of ξ_crit. ⁴ In other words, our numerical results suggest that three mechanisms of tangential frictions could complement one another. The weak dependence of E_dis,tot on ξ_crit might be the reason why 〈f_gro〉 and v_fra are not strong functions of ξ_crit. As we prepared the initial dust aggregates via sequential hit-and-stick collisions between an aggregate and multiple single particles (e.g., Mukai et al. 1992), they are highly porous and have only two interparticle contacts per particle on average. Therefore, we can expect that there are higher degrees of freedom for the deformation of dust aggregates, and all three tangential motions (rolling, sliding, and twisting) can potentially play a prominent role in the energy dissipation if the strengths of the springs and their critical displacements have moderate values. We note that 〈f_gro〉 and v_fra considerably decrease when we perform numerical simulations of collisions between aggregates made of cohesive but frictionless spheres (S. Arakawa et al., in preparation). Although what controls the balance of energy dissipation via tangential frictions is still unclear, our results highlight the importance of three tangential motions on the collisional outcomes of porous dust aggregates made of submicron-sized particles.

The pairwise collisional growth of dust aggregates consisting of submicron-sized grains is the first step of planet formation, and understanding the collisional behavior of dust aggregates is therefore essential. It is known that the main energy dissipation mechanisms are the tangential frictions between particles in contact, namely, rolling, sliding, and twisting (e.g., Arakawa et al. 2022). However, there is great uncertainty for the strength of rolling friction, and the dependence of the collisional growth condition on the strength of rolling friction was poorly understood.

In this study, we systematically performed numerical simulations of collisions between two equal-mass dust aggregates made of submicron-sized ice spheres. We changed the strength of interparticle rolling friction, which is controlled by the critical rolling displacement, ξ_crit. As a result, we found that the threshold collision velocity for the fragmentation of dust aggregates is nearly independent of ξ_crit when we consider oblique collisions (Figures 3 and 4).

We checked the total energy dissipation due to particle interactions from the start to the end of the simulation (Figure 5). We found that the total energy dissipation due to all particle interactions, E_dis,tot, is nearly independent of ξ_crit. The energy dissipation due to rolling friction, E_dis,r, takes the maximum when we chose a moderate ξ_crit, while E_dis,r becomes small if ξ_crit is set to be close to the lower or the upper limits (0.2 nm and 12.8 nm, respectively). Conversely, the energy dissipations due to sliding friction, E_dis,s, and twisting friction, E_dis,t, become large at both the lower and the upper limits of ξ_crit. Therefore, our numerical results indicate that three mechanisms of tangential frictions could complement one another.

Numerical simulations of granular matter are performed in many fields of science and engineering. We note, however, that the particle interaction models differ among numerical codes. Although the majority of these codes do not include the particle interaction for the twisting motion, our simulations showed that twisting friction plays an important role when the collision velocity of dust aggregates, v_col, is not significantly larger than the threshold collision velocity for the fragmentation, v_fra. Therefore, the details of particle interaction models would be crucial for investigating the collisional behavior of dust aggregates from numerical simulations.

We acknowledge that plastic deformation, melting, and fragmentation of constituent particles are not considered in this study. These effects might be important for high-speed collisions as shown in molecular dynamics simulations (e.g., Nietiadi et al. 2020, 2022). The angular velocity dependence of the strength of tangential frictions is also ignored in our simulations. Although our results revealed that v_fra does not strongly depend on the strength of rolling friction alone, the collisional behavior might be affected when the strengths of all three tangential frictions increase/decrease simultaneously. Future studies on the particle interactions from laboratory experiments and molecular dynamics simulations would be essential.

Numerical computations were carried out on PC cluster at CfCA, NAOJ. H.T. and E.K. were supported by JSPS KAKENHI grant No. 18H05438. This work was supported by the Publications Committee of NAOJ.

Figures 6 and 7 show the total energy dissipation due to particle interactions from the start to the end of the simulations. We set ${{B}_{\mathrm{off}}}^{2}=6/12$ and 9/12 for Figures 6 and 7, respectively. The total energy dissipation due to all particle interactions, E_dis,tot, is nearly independent of ξ_crit not only for ${{B}_{\mathrm{off}}}^{2}=3/12$ but also for ${{B}_{\mathrm{off}}}^{2}=6/12$ and 9/12.

We note that E_dis,tot is smaller than the initial kinetic energy, N_tot E_kin,1. Figure 8 shows E_dis,tot as a function of ${{B}_{\mathrm{off}}}^{2}$ and v_col. Here we set ξ_crit = 0.8 nm, although E_dis,tot is nearly independent of ξ_crit. The reason why E_dis,tot is smaller than N_tot E_kin,1 is clear; fragments have kinetic energies after collisions as shown in Figure 2.

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