Collisions between Sintered Icy Aggregates

The shape of a pair of sintered grains is approximated by a circle of radius x smoothly touching each sphere (Figure 1(b)). The relation between the neck radius a and x is given by

$$\begin{eqnarray}&&x=\displaystyle \frac{{a}^{2}}{2(R-a)}.\end{eqnarray} \tag{ 1 }$$

The thickness of the neck, d, is the distance between two contact points on the grain surface (Figure 1(b)), which is given by

$$\begin{eqnarray}&&d=\displaystyle \frac{2{Rx}}{R+x}.\end{eqnarray} \tag{ 2 }$$

Mechanical interactions between the two sintered grains are approximated by those for an elastic disk (Figure 1(c)) with a radius of a. It is assumed that only the disk is deformed elastically and that the other part of the grain is rigid. The mechanical interactions of this model are linear, and plastic deformation is not taken into account.

If the disk is stretched along the direction connecting the two grain centers by the amount of ${u}_{\mathrm{ss}}$, the stress σ appearing in the disk is given by $\sigma ={u}_{\mathrm{ss}}E/d$, where E is the Young's modulus of H₂O ice. Then, the total force required for stretching is $\pi {a}^{2}\sigma =\pi {a}^{2}{u}_{\mathrm{ss}}E/d$. From this formula, the spring constant ${k}_{\mathrm{ss}}$ for stretching motion is given by

$$\begin{eqnarray}&&{k}_{\mathrm{ss}}=\displaystyle \frac{\pi {a}^{2}E}{d}.\end{eqnarray} \tag{ 3 }$$

If a force is applied parallel to the disk surface and a displacement of ${u}_{\mathrm{st}}$ is attained, the shear stress that appeared in the disk is $\sigma ={u}_{\mathrm{st}}G/d$, where G is the shear modulus. By the same argument for the stretching case, the spring constant for tangential (sliding) motion, ${k}_{\mathrm{st}}$, is given by

$$\begin{eqnarray}&&{k}_{\mathrm{st}}=\displaystyle \frac{\pi {a}^{2}G}{d}.\end{eqnarray} \tag{ 4 }$$

Suppose the upper surface of the disk is tilted by an amount of θ because of an external moment M with respect to the axis on the center of the disk surface. In this case, the net vertical force is zero. At a distance l from the axis, the vertical strain is given as $\theta l/d$ (Figure 3). The stress accompanied by this strain is $\theta {lE}/d$. By integrating this stress throughout the surface, the moment M around the axis on the surface is obtained as

$$\begin{eqnarray}M & = & 2{\displaystyle \int }_{-a}^{a}\displaystyle \frac{\theta {lE}}{d}\sqrt{{a}^{2}-{l}^{2}}{ldl}\\ & = & \displaystyle \frac{\pi {a}^{4}E}{4d}\theta ={k}_{\mathrm{sr}}\theta .\end{eqnarray} \tag{ 5 }$$

The elastic energy stored is written as ${k}_{\mathrm{sr}}{\theta }^{2}/2$.

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The neck is broken when a large force is applied. Fracturing takes place when the stress inside a solid material exceeds its tensile strength, which strongly depends on the number and size of internal cracks. A crack grows by breaking interatomic or intermolecular bonds at the crack edge. The elastic energy stored around a crack edge is consumed to break the bonds. When torque is applied to rotate a grain relative to an adjacent grain, the stress reaches tensile strength T at the edge of a neck when $\theta {aE}/d=T$. To complete fracturing, the elastic energy stored should be larger than the surface energy, $\pi {a}^{2}\gamma$. If the elastic energy is much larger than the surface energy, cracks propagate quickly and the material breaks at once. If the elastic energy is small, fragmentation stops after breaking some bonds. In this case, the elastic energy is insufficient to break the neck completely at the beginning of fracture. For simplicity, instantaneous breaking of the neck is assumed in this study. This point should be addressed by laboratory experiments.

Under the assumption that the neck fully breaks at once, the neck is broken when the displacement u (θ for rolling) reaches ${k}_{i}{u}^{2}/2=\pi {a}^{2}\gamma$ (i is rolling, stretching, or sliding). From this condition, critical displacements for stretching ${u}_{\mathrm{ss},{\rm{c}}}$, for sliding ${u}_{\mathrm{st},{\rm{c}}}$, and for rolling ${\theta }_{{\rm{c}}}$, are, respectively, given by

$$\begin{eqnarray}&&{u}_{\mathrm{ss},{\rm{c}}}=\sqrt{\displaystyle \frac{2\pi {a}^{2}\gamma }{{k}_{\mathrm{ss}}}},\quad {u}_{\mathrm{st},{\rm{c}}}=\sqrt{\displaystyle \frac{2\pi {a}^{2}\gamma }{{k}_{\mathrm{st}}}},\quad {\theta }_{{\rm{c}}}=\sqrt{\displaystyle \frac{2\pi {a}^{2}\gamma }{{k}_{\mathrm{sr}}}}.\end{eqnarray} \tag{ 6 }$$

When two sintered aggregates collide, some of the necks might break and new contacts between grains might form. Because the temperature increase associated with a collision is very low, a newly formed contact is a non-sintered one. The mechanical responses of non-sintered necks were described by Johnson (1987) and Dominik & Tielens (1997). We adopted the same equations used in those studies. Recently, a model taking account of adhesion hysteresis (Krijt et al. 2013a, 2013b) has been developed. A simulation using the model is desirable to determine the effect of adhesion hysteresis for non-sintered aggregates.

The radius of a non-sintered neck at the equilibrium is

$$\begin{eqnarray}&&{a}_{0}={\left(\displaystyle \frac{9\pi \gamma {R}^{* 2}}{{E}^{* }}\right)}^{1/3},\end{eqnarray} \tag{ 7 }$$

where ${E}^{* }=E/2(1-{\nu }^{2})$ (ν is Poisson's ratio). It should be noted that ${R}^{* }$ is the reduced radius of two contact grains, given by ${R}_{1}{R}_{2}/({R}_{1}+{R}_{2})$, with radii of R₁ and R₂. In this study, we focus on a monodispersed system and ${R}^{* }=R/2$.

When a force F is applied along the stretching direction, the neck radius changes. The relation between F and a can be written by

$$\begin{eqnarray}&&\displaystyle \frac{F}{{F}_{{\rm{c}}}}=4\left[{\left(\displaystyle \frac{a}{{a}_{0}}\right)}^{3}-{\left(\displaystyle \frac{a}{{a}_{0}}\right)}^{3/2}\right],\end{eqnarray} \tag{ 8 }$$

where ${F}_{{\rm{c}}}=3\pi \gamma {R}^{* }$ is the critical force required to break a non-sintered contact. On the other hand, the compression length (the shift of the distance between two centers of spheres) δ can be written as a function of a as

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

Combining Equations (8) and (9), we can obtain the relation between F and δ implicitly.

For rolling and sliding motions, simple elastic responses are assumed until the forces reach the critical values. The elastic energy stored as rolling deformation, ${E}_{\mathrm{nr}}$, is written using the spring constant, ${k}_{\mathrm{nr}}$, for rolling motion as

$$\begin{eqnarray}&&{E}_{\mathrm{nr}}=\displaystyle \frac{1}{2}{k}_{\mathrm{nr}}{\xi }^{2}=\displaystyle \frac{1}{2}\displaystyle \frac{4{F}_{{\rm{c}}}}{{R}^{* }}{\xi }^{2},\end{eqnarray} \tag{ 10 }$$

where the rolling displacement is $\xi =R\theta /2$ (see Figure 2(c) of Wada et al. 2007). When ξ reaches ${\xi }_{{\rm{c}}}$, the spring is cut and the stored elastic energy dissipates. This corresponds to the rolling friction due to the breaking of intermolecular bonds. In the simulation, the spring was newly formed immediately after the breaking. The estimated values of ${\xi }_{{\rm{c}}}$ varied from 2 Å (Dominik & Tielens 1995) to 32 Å (Heim et al. 1999). We adopted ${\xi }_{{\rm{c}}}=2$ Å and 30 Å in the simulation.

The critical force for tangential motion, ${F}_{\mathrm{nt}}$, is given by

$$\begin{eqnarray}&&{F}_{\mathrm{nt}}=\displaystyle \frac{{{Ga}}_{0}^{2}}{2\pi },\end{eqnarray} \tag{ 11 }$$

and this spring is broken when the displacement reaches

$$\begin{eqnarray}&&{u}_{\mathrm{nt}}=\displaystyle \frac{2-\nu }{16\pi }{a}_{0}.\end{eqnarray} \tag{ 12 }$$

As with the rolling motion, the spring immediately reformed after breaking.

As shown in Figure 1(a), the main part of a spherical grain shrinks during sintering. A sintered contact is broken mainly through rolling motion, as shown later. If the degree of sintering is low, the main part of the grain retains the original shape. In this case, a non-sintered contact is likely to form because the distance between the centers of two grains is slightly shorter than the sum of the two radii ($-4.2\times {10}^{-4}\,\mu {\rm{m}}$ for two 0.1 μm radius spherical grains). On the other hand, if the degree of sintering is high, the radius decreases substantially ($-0.091\,\mu {\rm{m}}$ for a 0.1 μm radius spherical grain when β increases to 0.7 provided that the radius of grains is uniform). The shrinkage depends on the size distribution of grains, which is determined by the thermal history of the grains (Kuroiwa & Sirono 2011). This point should be addressed separately from this study.

This shrinkage should affect the formation of a new contact immediately after the breaking of a sintered neck. In this study, the grain radius R decreased to $R^{\prime}$, as shown in Figure 4 during sintering. A new contact between sintered grains forms when the distance between the two grains is less than $2R^{\prime}$. The radius of the contact area, a₀, and other quantities in Section 2.2 were calculated using the decreased $R^{\prime}$. It should be noted that Figure 4 is the result for a straight chain of spherical grains. If the number of contacts per grain is more than two, the decrease in R should be larger than that shown in Figure 4. When taking shrinkage into account, a new contact formation is less probable than in the case without shrinkage.

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Typical examples of collisional outcomes of BCCA aggregates are shown in Figure 7. Non-sintered BCCA aggregates stick perfectly at a collision velocity of ${V}_{{\rm{c}}}=10\,{\rm{m}}\,{{\rm{s}}}^{-1}$. Even at a higher velocity of ${V}_{{\rm{c}}}=30\,{\rm{m}}\,{{\rm{s}}}^{-1}$, the sticking efficiency (defined as the ratio of maximum aggregate mass after collision to the total mass) is close to unity.

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On the other hand, three panels from the bottom in Figure 7 show the typical examples of the collisional outcomes of sintered aggregates. Two sintered aggregates stick at ${V}_{{\rm{c}}}=3.2\,{\rm{m}}\,{{\rm{s}}}^{-1}$, bounce at ${V}_{{\rm{c}}}=10\,{\rm{m}}\,{{\rm{s}}}^{-1}$, and totally rupture at ${V}_{{\rm{c}}}=30\,{\rm{m}}\,{{\rm{s}}}^{-1}$. Except at low velocities, these collisional outcomes differ substantially from those without sintering. Because sintering increases the elasticity and the strength of the neck between grains, energy dissipation through rolling motion cannot proceed. This is in contrast to the non-sintered cases, where rolling friction dissipates kinetic energy. As a result, bouncing is observed at ${V}_{{\rm{c}}}=10\,{\rm{m}}\,{{\rm{s}}}^{-1}$. For low collisional velocities, two sintered aggregates stick together without any notable deformation. At high collision velocities, the aggregates extensively disrupt to small fragments.

It might be considered that the high energy required to break a sintered contact leads to more efficient sticking as compared to non-sintered aggregates because of higher energy dissipation. This is not the case because only a limited number of contacts where stress is concentrated can break. Most of the contacts remain intact and do not contribute to energy dissipation. These are common phenomena observed in the breaking of solid material. Brittle disruption occurs through the propagation of cracks, and the breaking of intermolecular bonds occurs only at the edges of cracks where stress is concentrated. Thus, a substantial fraction of the initial kinetic energy remains even after the collision.

This is a totally different deformation mode from that in non-sintered aggregates. In a collision of non-sintered aggregates, the aggregate deforms through rolling of the grains. Because the critical force to roll is quite small, rolling deformation occurs throughout an aggregate. This deformation mode is similar to that of liquid, where molecules smoothly rotate around adjacent ones.

Figure 8 (left) shows the sticking efficiency, defined as the ratio of the maximum mass after collision to the total mass, as a function of collision velocity. It can be seen that non-sintered aggregates can grow (sticking efficiency more than 0.5) at ${V}_{{\rm{c}}}=50\,{\rm{m}}\,{{\rm{s}}}^{-1}$. This is consistent with previous studies (Wada et al. 2007, 2009). On the other hand, substantial disruption is observed even at ${V}_{{\rm{c}}}=15\,{\rm{m}}\,{{\rm{s}}}^{-1}$ for the $\beta =0.7$ sintered aggregates. A $\beta =0.7$ aggregate bounces at ${V}_{{\rm{c}}}=10\,{\rm{m}}\,{{\rm{s}}}^{-1}$. For velocities lower than ${V}_{{\rm{c}}}=3.2\,{\rm{m}}\,{{\rm{s}}}^{-1}$, the $\beta =0.7$ aggregate sticks.

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The collisional outcomes of $\beta =0.2$ aggregates are almost the same as those of non-sintered aggregates. This is because the neck is not strong enough to sustain the stress induced during a collision. Moreover, if a sintered neck is broken, a new neck without sintering is formed because the shrinkage of a grain does not proceed at low β (see Figure 4).

Figure 8 (right) shows the gyration radius of the largest aggregate after collision. The gyration radius is a measure of the aggregate size defined by

$$\begin{eqnarray}&&{R}_{{\rm{g}}}=\sqrt{\displaystyle \frac{1}{N}\displaystyle \sum _{i}{({{\boldsymbol{r}}}_{i}-{{\boldsymbol{r}}}_{0})}^{2}},\end{eqnarray} \tag{ 13 }$$

where ${\boldsymbol{r}}-{{\boldsymbol{r}}}_{0}$ is the positional vector measured from the center of mass ${{\boldsymbol{r}}}_{0}$. The gyration radius decreases monotonically as the collision velocity increases because of compaction and fragmentation. Below $10\,{\rm{m}}\,{{\rm{s}}}^{-1}$, the gyration radius of the non-sintered aggregate is the smallest among the three types of aggregates. This shows that compaction through the rolling motion of the grains proceeds efficiently in non-sintered aggregates. Compaction does not proceed in the $\beta =0.7$ aggregate below $10\,{\rm{m}}\,{{\rm{s}}}^{-1}$. The evolution of porosity (Kataoka et al. 2013) would thus be affected by sintering. The sintered aggregate has much less density than the non-sintered aggregate. Above $10\,{\rm{m}}\,{{\rm{s}}}^{-1}$, the gyration radius of the $\beta =0.7$ aggregate is the smallest compared to the other types of aggregates because of fragmentation.

Next, we conducted the collision simulations using compacted aggregates with $\beta =0.7$. Snapshots during collisions are shown in Figure 9. The aggregate grown with ${V}_{{\rm{g}}}=3\,{\rm{m}}\,{{\rm{s}}}^{-1}$ has a compacted structure compared to the BCCA aggregate. Then the compacted aggregates collided with a collision velocity of ${V}_{{\rm{c}}}=3\,{\rm{m}}\,{{\rm{s}}}^{-1}$. The collisional outcome for the non-sintered case is shown in Figure 9 (top). Even when an aggregate is compacted, the non-sintered aggregates stick perfectly. On the other hand, the sintered aggregates cannot stick; instead, they bounce. This is simply because a compacted aggregate is stiffer than a BCCA aggregate. For BCCA aggregates, perfect sticking occurs when ${V}_{{\rm{c}}}\leqslant 3.2\,{\rm{m}}\,{{\rm{s}}}^{-1}$ irrespective of the sintering degree (see Figure 8, left).

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Figure 10 compares the sticking efficiencies for compacted aggregates as a function of collision velocity. Data for $0\,{\rm{m}}\,{{\rm{s}}}^{-1}$ correspond to BCCA aggregates with $\beta =0.7;$ the results for $0\,{\rm{m}}\,{{\rm{s}}}^{-1}$ are identical to (triangles) those shown in Figure 8 (left). When the growth velocity is low (0 and $1\,{\rm{m}}\,{{\rm{s}}}^{-1}$), perfect sticking is realized at collision velocities less than $3\,{\rm{m}}\,{{\rm{s}}}^{-1}$. Above $10\,{\rm{m}}\,{{\rm{s}}}^{-1}$, substantial fragmentation takes place and sticking efficiency is approximately 0.2.

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For aggregates with growth velocities of $3\,{\rm{m}}\,{{\rm{s}}}^{-1}$ and $10\,{\rm{m}}\,{{\rm{s}}}^{-1}$, the collisional outcome is bouncing. Above $30\,{\rm{m}}\,{{\rm{s}}}^{-1}$, fragmentation occurs.

We have shown the results of sintered BCCA aggregate collisions with $\beta =0.2$ and 0.7. We also investigated the β dependence of collisional outcomes. Figure 11 shows the sticking efficiency of collisions with BCCA aggregates of various sintering degrees as a function of β with ${\xi }_{{\rm{c}}}=2\,\mathring{\rm{A}}$ (left) and 30 Å (right). The collisional velocities were 10 and $30\,{\rm{m}}\,{{\rm{s}}}^{-1}$. It can be seen that the sticking efficiency decreases as β increases in both cases. If the collisional velocity is $10\,{\rm{m}}\,{{\rm{s}}}^{-1}$, growth is possible for all values of β. This has already been suggested in Figure 8 (left), where the sticking efficiency of the $\beta =0.7$ aggregate with a collision velocity of $10\,{\rm{m}}\,{{\rm{s}}}^{-1}$ is larger than unity.

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On the other hand, qualitative changes were observed for ${V}_{{\rm{c}}}=30\,{\rm{m}}\,{{\rm{s}}}^{-1}$ collisions. Collisional growth took place only for $\beta \leqslant 0.4$ aggregates. Above $\beta =0.5$, bouncing and fragmentation occurred and collisional growth was impossible. It should be noted that the sticking efficiency decreased between $\beta =0.4$ and 0.5 for both collision velocities. For the BCCA aggregates, the sintering degree of $\beta =0.5$ was critical in the sense that the collisional growth was affected above this value. It should be noted that we have shown that the volume required for $\beta =0.5$ is 2% of the volume of a grain (Figure 2).

The mass of fragments produced during a collision is presented in Figure 12. A fragment is defined as an aggregate whose mass is less than one-fourth of the total mass. The fragment mass of ${V}_{{\rm{c}}}=30\,{\rm{m}}\,{{\rm{s}}}^{-1}$ collisions monotonically increases as β increases. On the other hand, the fragment mass of ${V}_{{\rm{c}}}=10\,{\rm{m}}\,{{\rm{s}}}^{-1}$ collisions has a peak at $\beta =0.5$. Aggregates of $\beta =0.7$ produced only a small mass of fragments. This is because an aggregate is hard, and thus few necks were broken. On the other hand, if β is low, an aggregate is too weak to sustain the stress, and plastic deformation through rolling occurs as in non-sintered aggregates. In these cases, almost perfect sticking takes place and the amount of fragments is negligible.

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Compacted aggregates are produced through successive collisions with a collision velocity of ${V}_{{\rm{g}}}$ using non-sintered aggregates. The compacted aggregates drift to the central star and may enter a sintering zone. It is thus interesting to investigate the case in which ${V}_{{\rm{c}}}={V}_{{\rm{g}}}$ for sintered aggregates. Figure 13 shows the β dependence of the sticking efficiency for aggregates grown at ${V}_{{\rm{g}}}$ colliding with ${V}_{{\rm{c}}}={V}_{{\rm{g}}}$ for ${\xi }_{{\rm{c}}}=2\,\mathring{\rm{A}}$ (left) and 30 Å (right). If ${V}_{{\rm{c}}}=1\,{\rm{m}}\,{{\rm{s}}}^{-1}$, the collisional outcome is perfect sticking, irrespective of β in both cases. If ${\xi }_{{\rm{c}}}=2\,\mathring{\rm{A}}$, the fraction of bouncing increases as β increases when ${V}_{{\rm{c}}}=2\,{\rm{m}}\,{{\rm{s}}}^{-1}$. For $3\leqslant {V}_{{\rm{c}}}\leqslant 5\,{\rm{m}}\,{{\rm{s}}}^{-1}$, only bouncing is observed except when $\beta =0.2$. At $\beta =0.2$, the fraction of bouncing increases as ${V}_{{\rm{g}}}$ increases. Bouncing at $\beta =0.2$ occurs when ${V}_{{\rm{c}}}=7.5\,{\rm{m}}\,{{\rm{s}}}^{-1}$. For ${\xi }_{{\rm{c}}}=30\,\mathring{\rm{A}}$, the probability of sticking is slightly higher than ${\xi }_{{\rm{c}}}=2\,\mathring{\rm{A}}$ cases.

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From these results, it is suggested that bouncing is one of the main collisional outcomes of compacted aggregates. It is important to understand the conditions of bouncing in order to clarify the collisional evolution of icy grain aggregates. The conditions will be discussed in Section 5.4.

Figure 14 shows the partition of energy after the collision of BCCA aggregates with ${V}_{{\rm{c}}}=10\,{\rm{m}}\,{{\rm{s}}}^{-1}$ as a function of β. The initial kinetic energy dissipates through breaking of the sintered bonds, sliding and rolling friction at non-sintered contacts, and elastic vibration. Some of the kinetic energy remains after the collision. In Figure 14, the leftmost points are non-sintered aggregates. Approximately 80% of the initial kinetic energy dissipates through rolling friction, which causes the smooth ductile deformation of the aggregates. The other 20% is dissipated by breaking non-sintered necks through stretching. The remaining kinetic energy is small, representing perfect sticking.

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At $\beta =0.2$, 20% of the kinetic energy dissipates through breaking sintered necks with rolling deformation. As β increases, the amount of energy dissipation through the breaking by rolling decreases. Energy dissipation through rolling friction decreases accordingly. This is because the number of newly formed non-sintered contacts decreases as the number of broken sintered contacts decreases. In turn, the elastic and remaining kinetic energies increase as β increases. From Figure 11, it can be seen that bouncing starts from $\beta =0.5$. At this value, 40% of the initial kinetic energy does not dissipate but remains as kinetic and elastic energies. As a result, the efficiency of energy dissipation decreases and the collisional outcome becomes bouncing.

As shown above, the number of sintered necks broken through rolling motion is an important quantity in collision. Figure 15 (left) shows the number of sintered bonds broken in a collision normalized by the total number of grains, ${N}_{\mathrm{cut}}/{N}_{{\rm{t}}}$, as a function of the initial kinetic energy per unit grain, ${E}_{{\rm{k}}}={{mV}}_{{\rm{c}}}^{2}/4$, normalized by the breaking energy of a sintered neck, ${E}_{\mathrm{sint}}=\pi \gamma {(\beta R)}^{2}$. Five types of aggregates are included: BCCA with $\beta =0.2$ and 0.7, and compacted aggregates with ${V}_{{\rm{g}}}=1$, 3, and 10 m s⁻¹. Although all modes of motion are included (rolling, sliding, stretching) as a cause of breaking a sintered neck in Figure 15 (left), more than 90% of the bonds were broken through rolling motion. It should be noted that data for all types of aggregates showed the same dependence except for BCCA with $\beta =0.2$. Least-squares fitting except $\beta =0.2$ gave $3.7\times {10}^{-2}{({E}_{{\rm{k}}}/{E}_{\mathrm{sint}})}^{0.97}$, indicating that the number is determined by the ratio ${E}_{{\rm{k}}}/{E}_{\mathrm{sint}}$. The number of broken bonds thus depends only on β, but not on the degree of compaction.

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On the contrary, the fraction of mass in fragments compared to the total mass depends on the degree of compaction, as seen in Figure 15 (right). The fragment fraction was the largest for BCCA with $\beta =0.7$ and for compacted aggregates with ${V}_{{\rm{g}}}=1\,{\rm{m}}\,{{\rm{s}}}^{-1}$. These two types of aggregates show almost the same dependence. For high collision velocities, these aggregates are disrupted totally, as shown in Figure 8 (left). On the other hand, more compacted aggregates and BCCA with $\beta =0.2$ produced fewer fragments. The collisional outcomes of BCCA aggregates with $\beta =0.2$ are close to perfect sticking at collision velocities of less than $32\,{\rm{m}}\,{{\rm{s}}}^{-1}$, where ${E}_{{\rm{k}}}/{E}_{\mathrm{sint}}=8.3$, which is almost identical to the value for non-sintered aggregates, as seen before (Section 5.1). For compacted aggregates of ${V}_{{\rm{g}}}=3\,{\rm{m}}\,{{\rm{s}}}^{-1}$ and $10\,{\rm{m}}\,{{\rm{s}}}^{-1}$, the collisional outcome is bouncing. The mass of fragments of these aggregates is between those of the other two types of aggregates. It can be seen that the fragment mass is different for aggregates with the same β. Less-compacted aggregates produced more fragments even when the number of broken necks was the same (Figure 15, upper right). Correspondingly, the sticking efficiency (the mass of the largest aggregate) differs between the aggregate types (Figure 15, bottom). This suggests that the fragments stick again to the main part of the aggregate during collision if the aggregate is compacted. Fragments are highly likely to collide with a nearby fragment because the packing fraction is high. In contrast, fragments produced inside a porous aggregate have a chance to escape from the other fragments. Thus, grain arrangement is a critical parameter that determines the final mass of the largest aggregate.

The collisional outcomes can be categorized as sticking, bouncing, and fragmentation. It is helpful to clarify the conditions of collisional outcomes.

In a non-sintered case, the condition given by Dominik & Tielens (1997) describes the collisional outcomes well. For example, maximum compression of an aggregate occurs when ${E}_{\mathrm{roll}}\simeq {E}_{{\rm{k}}}$, where ${E}_{{\rm{k}}}={{mV}}_{\mathrm{c}}^{2}/4$ is the kinetic energy of a grain and ${E}_{\mathrm{roll}}$ is the energy required to roll the grain 90° around an adjacent grain. Catastrophic disruption occurs when ${E}_{{\rm{k}}}\simeq 10{E}_{\mathrm{break}}$, where ${E}_{\mathrm{break}}$ is the energy required to break a non-sintered contact. These considerations based on energetics were successful because a non-sintered aggregate smoothly deforms like liquid. This behavior comes from the fact that the force required for rolling motion is substantially smaller than those for other motions including sliding and stretching. As a result, smooth ductile deformation of a non-sintered aggregate is possible. Energy thus plays a key role in the non-sintered case.

We showed that the number of broken bonds clearly depends on the ratio between the initial kinetic energy and the breaking energy of a sintered contact in Figure 15. A substantial fraction of sintered contacts are broken when ${E}_{{\rm{k}}}\simeq 10{E}_{\mathrm{sint}}$. Critical energies play a key role in fragmentation accordingly, as in the non-sintered case. However, the final mass of the largest aggregate depends on the packing fraction of an aggregate (Figure 15 bottom). The arrangement of grains determines the mass of the maximum aggregate.

The situation is different for bouncing, because bouncing occurs if the force at a newly formed non-sintered contact is higher then the critical force for breaking a non-sintered contact. We investigated how forces are induced in an aggregate in order to clarify bouncing, adopting a model given by Sirono & Greenberg (2000), where an aggregate is represented by chains connected periodically. If we assume the shape of a chain is triangular with a base length of L and a height $L/2$ (see Figure 16(a)), the number of grains in one chain, N, is given by

$$\begin{eqnarray}&&N=\left[\left(\displaystyle \frac{L}{2R-1}\right)+\left(\displaystyle \frac{L}{2R}\right)\right]+1.\end{eqnarray} \tag{ 14 }$$

Suppose the chains are connected to form a regular triangular lattice (Figure 16(b)). It should be noted that the two end grains in one chain are shared with six triangular cells. $N-2$ in-between grains are sheared with the two neighboring cells. In one cell, there are three end grains and $3\times (N-2)$ in-between grains. Thus, the number of grains contained in a unit cell is

$$\begin{eqnarray}&&{N}_{\mathrm{cell}}=3\times \displaystyle \frac{1}{6}+3\times \displaystyle \frac{N-2}{2}=\displaystyle \frac{3N-5}{2}.\end{eqnarray} \tag{ 15 }$$

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An end grain of a chain is shared with six chains. There are thus six contact points on an end grain. In one cell, there are three end grains with six contact points shared by six cells. There are three contact points contributed by end grains effectively in one cell. An in-between grain has two contact points. In one triangular cell, in-between grains contribute $2\times (N-2)/2\,\times \,3\,=\,3(N-2)$ contact points. In total, there are $3N-3$ contact points for each cell. Thus, the average number of contact points on a grain is given by

$$\begin{eqnarray}&&{N}_{\mathrm{con}}=\displaystyle \frac{6N-6}{3N-5}.\end{eqnarray} \tag{ 16 }$$

The average number approaches 2 as N increases.

When aggregates collide, forces are induced at the contacts. The maximum force induced in an aggregate is determined by the balance of kinetic energy and elastic energy stored in the chains. If the total number of chains in an aggregate is ${N}_{\mathrm{chain}}$, the balance can be written by ${N}_{\mathrm{chain}}{k}_{\mathrm{chain}}{u}_{\mathrm{chain}}^{2}/2={N}_{{\rm{g}}}{V}_{{\rm{c}}}^{2}/4$, where ${N}_{{\rm{g}}}$ is the number of grains in the aggregate, and ${u}_{\mathrm{chain}}$ is the maximum shrinkage of the chain. The effective spring constant of the triangular chain ${k}_{\mathrm{chain}}$ is given by Sirono & Greenberg (2000) as

$$\begin{eqnarray}&&{k}_{\mathrm{chain}}=\displaystyle \frac{12(N-1){k}_{\mathrm{sr}}}{(N-3)(N+1)}{\left(\displaystyle \frac{L}{2R-1}\right)}^{-2},\end{eqnarray} \tag{ 17 }$$

where the term ${k}_{\mathrm{sr}}=\pi {(\beta R)}^{4}E/4d$ appeared in Equation (5). Because ${N}_{{\rm{g}}}/{N}_{\mathrm{chain}}\simeq N$, the balance gives a maximum force ${F}_{\max }$ as

$$\begin{eqnarray}&&{F}_{\max }={V}_{{\rm{c}}}\sqrt{{k}_{\mathrm{chain}}{mN}/2}.\end{eqnarray} \tag{ 18 }$$

When two aggregates collide, the maximum compressive force given by Equation (18) is induced inside an aggregate. At the same time, non-sintered new contacts are formed during the collision. An aggregate bounces if the tensile force at the new contact is larger than the critical force for breaking a non-sintered contact. Because the maximum compressive and tensile forces are comparable, if ${F}_{\max }$ is larger than the critical force for breaking, ${F}_{{\rm{c}}}=3\pi \gamma {R}^{* }$, bouncing occurs. This condition can be written as

$$\begin{eqnarray}&&\displaystyle \frac{{F}_{{\rm{c}}}}{{F}_{\max }}\lt 1.\end{eqnarray} \tag{ 19 }$$

Figure 17 shows the ratio given by Equation (19) corresponding to compact aggregates produced with ${\xi }_{{\rm{c}}}=2\,\mathring{\rm{A}}$ (left) and BCCA aggregates (right). To calculate ${F}_{\max }$, we have to adopt N into Equation (18). For BCCA aggregates, we determined N from $L=100R$ from the gyration radius shown in Figure 8 (right). For compacted aggregates, N was determined from Equation (16) and the average number of contacts shown in Figure 6 (${\xi }_{{\rm{c}}}=2$ Å). In Figure 17 (left), the ratio was calculated for collisions of compacted aggregates with ${V}_{{\rm{c}}}={V}_{{\rm{g}}}$.

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From Figure 17 (left), the model predicts that aggregates compacted with ${V}_{{\rm{g}}}=2\,{\rm{m}}\,{{\rm{s}}}^{-1}$ colliding at the same velocity bounce if β is larger than 0.4. Aggregates of ${V}_{{\rm{g}}}=1\,{\rm{m}}\,{{\rm{s}}}^{-1}$ stick, irrespective of β, and aggregates of ${V}_{{\rm{g}}}=3\,{\rm{m}}\,{{\rm{s}}}^{-1}$ bounce if $\beta \geqslant 0.2$. These results explain the collisional outcomes shown in Figure 13 (left) well (${\xi }_{{\rm{c}}}=2$ Å).

For BCCA aggregates (Figure 17, right), the model predicts sticking for a wide range of collision velocities in contrast to the results shown in Figure 8 (left), where $\beta =0.7$ aggregates bounce at ${V}_{{\rm{c}}}=10\,{\rm{m}}\,{{\rm{s}}}^{-1}$. It should be noted that there are no chain loops inside a BCCA aggregate because of its formation mechanism i.e., the motion of the colliding aggregate stops when a new contact is formed.

When sintered BCCA aggregates collide, they elastically deform substantially (Figure 7). A chain of grains bends and new contacts are formed around the impact point. The length of the chains decreases and larger forces can appear as the elasticity of the aggregate increases. We identified loops consisting of grain chains and checked the the number of grains belonging to the loops. The average number of the grains was 44. We therefore adopted $N=44/2=22$ in Equation (18). The square in Figure 17 (right) is the ratio based on the calculated number of grains belonging to the loops. The ratio is almost unity in this case, well explaining the results obtained by numerical simulation (sticking efficiency of 0.6, corresponding to two bounces and three stickings in Figure 8, left).

As seen above, the threshold between bouncing and sticking is determined by forces induced in an aggregate and the sticking force needed to separate grains. This is in contrast to the collisions between non-sintered aggregates, in which the collisional outcomes are well classified based on energies associated with the motions of the grains.

A critical displacement for the irreversible rolling motion of ${\xi }_{{\rm{c}}}=2$ Å was estimated by Dominik & Tielens (1997). On the other hand, Heim et al. (1999) obtained ${\xi }_{{\rm{c}}}=32$ Å experimentally. Figure 13 displays the effect of ${\xi }_{{\rm{c}}}$ on the collisional outcomes of compacted aggregates. As seen in Figure 6, the number of contact points decreases as ${\xi }_{{\rm{c}}}$ increases. The number of contacts at ${V}_{{\rm{g}}}=3\,{\rm{m}}\,{{\rm{s}}}^{-1}$ with ${\xi }_{{\rm{c}}}=2$ Å is almost the same as that at ${V}_{{\rm{g}}}=10\,{\rm{m}}\,{{\rm{s}}}^{-1}$ with ${\xi }_{{\rm{c}}}=30$ Å. The results of ${V}_{{\rm{g}}}=3\,{\rm{m}}\,{{\rm{s}}}^{-1}$ with ${\xi }_{{\rm{c}}}=2$ Å in Figure 13 (left) and ${V}_{{\rm{g}}}=10\,{\rm{m}}\,{{\rm{s}}}^{-1}$ with ${\xi }_{{\rm{c}}}=30$ Å in Figure 13 (right), coincides well. This comparison shows that the number of contacts is a critical parameter, which in turn depends on ${\xi }_{{\rm{c}}}$.

On the other hand, Figure 11 (right) shows the sticking efficiencies of BCCA aggregates with ${\xi }_{{\rm{c}}}=30$ Å. Comparing Figure 11's left and right panels, no significant change is observed. This is because the number of contacts in both aggregates is the same. Although new contacts that form during a collision are non-sintered ones and their rolling motion depends on ${\xi }_{{\rm{c}}}$, BCCA aggregates at high collisional velocities break up totally and the dependence on ${\xi }_{{\rm{c}}}$ is not observed.

In the non-sintering zone of a protoplanetary nebula, an aggregate efficiently grows through collisional sticking, and the collision velocity increases as the aggregate size increases. At the same time, the aggregates are compacted. The compacted aggregates drift to the central star owing to gas drag. If an aggregate that has grown in a non-sintered zone drifts into a sintering zone, sintering occurs inside the aggregate and its mechanical properties change. Figure 13 suggests that compacted aggregates that grow with a velocity of ${V}_{{\rm{g}}}=3\,{\rm{m}}\,{{\rm{s}}}^{-1}$ cannot grow further because of bouncing if β is high enough.

On the other hand, aggregates that form inside a sintering zone are less compacted than those in a non-sintered zone (Figure 8 (right)). Figure 8 (left) shows that porous sintered aggregates break at a collision velocity of $20\,{\rm{m}}\,{{\rm{s}}}^{-1}$. In this case, the size of the aggregate is limited by fragmentation, not by bouncing, as discussed for compacted aggregates.

These two effects induce heterogeneity in the growth of icy dust aggregates in a protoplanetary nebula. In a non-sintered zone, perfect sticking of aggregates does not produce fragments in contrast to the sintered zone. This contrast can explain the symmetric pattern observed at HL Tau (ALMA Partnership et al. 2015). Okuzumi et al. (2016) reproduced the observational pattern assuming substantial fragment production at collisional velocities higher than 20 m s⁻¹. This velocity corresponds to the critical velocity for the growth of sintered BCCA aggregates. The effect of bouncing collisions and associated fragment production (Figure 12) should be included in future studies.

The simulations conducted in this study were 2D as a first step. Of course, 3D simulations are also required. However, important aspects of sintered aggregate collisions are common in 2D and 3D.

Figure 17 nicely explains the threshold between sticking and bouncing after a collision. This figure is based on a comparison between the critical pull-off force, ${F}_{{\rm{c}}}=3\pi \gamma {R}^{* }$, and the maximum force ${F}_{\max }$ (Equation (18)) attained in a collision. The pull-off force is common in 2D and 3D. How about the maximum force? The discussion in Section 5.4 is based on a comparison between the initial kinetic energy and the total elastic energy stored in an aggregate, which is independent of dimensionality. Thus, the conditions obtained in this study would be applicable also to 3D cases.

Moreover, we found that the number of broken sintered necks only depended on the initial kinetic energy (Figure 15, left). This point is also independent of dimensionality. Thus, the critical velocity at which fragmentation starts should be the same in both 2D and 3D cases. However, the mass of fragments might depend on the dimension because the mass depends on the arrangement of grains in the aggregate (Figure 15, right).

Another important point is the number of grains in an aggregate. In this study, aggregates composed of 1024 grains were used. This number is quite small for simulating actual aggregates. However, the discussion in Section 5.4 suggests that the collisional outcome, especially sticking or bouncing, is determined by the length of the chains composing the aggregates. In other words, a critical parameter is the packing fraction of an aggregate. The geometrical meaning of the packing fraction is quite different between 2D and 3D. The probability of a fragment escaping would be higher in 3D than in 2D. Thus, it is possible that the mass of fragments shown in Figure 15 (upper right) would be higher in 3D. This point should be addressed by 3D simulations.

As shown in Figure 15, sintered necks are broken during collisions. The shape of a broken neck is not smooth; instead, it has edges. The collisional outcomes of sintered aggregates highly depend on the stickiness of fragments. A fragment is produced through breaking necks by rolling motion and the fragment collides with other fragments. If the fragment sticks, mass loss due to fragmentation is small. If the fragment bounces, the mass loss is large.

One criterion of bouncing is ${F}_{{\rm{c}}}/{F}_{\max }\lt 1$, where ${F}_{{\rm{c}}}$ is the pull-off force, given by $3\pi \gamma {R}^{* }$. If an aggregate is highly porous, the shape of a fragment is a chain. When a fragment chain collides, it is highly probable that the end of the chain is the first to come into contact with the other chain. Because the end of a chain is produced by breaking a neck, the contacting surface is likely rough.

Although this effect is not included in this study, we can discuss a consequence of this effect. Suppose the rough surface has a surface curvature of $0.1R$. The lines shown in Figure 17 (right) would shift downward by a factor of 10. The region of bouncing is significantly reduced, and the collisional outcome tends to bouncing (or fragmentation).

Another important effect not included in this study is the motion of the grain during rolling. After a neck is broken through rolling, a grain rolls around the axis at the edge of the neck (see Figure 3). During rolling, the center of the rolling grain shifts upward because the distance between the axis and the center is larger than that between the contact surface and the center. As a result, the distance between two grain centers increases. Clearly this effect inhibits reconnection of the grains. This effect should be included in a future simulation.