THE STICKINESS OF MICROMETER-SIZED WATER-ICE PARTICLES

Water ice is a fascinating material, which is of utmost importance for the physical and chemical evolution of dense molecular-cloud cores and the outer reaches of protoplanetary disks, because it is one of the most abundant species in these objects (see the review by Dishoek et al. 2014). Its occurrence as solid material as part of the interstellar matter is typically in (sub-)μm-sized grains, which are often termed as "dust," together with other, less volatile, solids.

Dust particles in general are ubiquitous in space. Wherever dust grains are present, they give rise to exciting physical processes. Their interaction with electromagnetic radiation (absorption, emission, scattering) determines the appearance and the spectral energy distribution of many astronomical objects. When subjected to high-energy particle fluxes, sputtering can reduce the sizes of dust grains, enhance the abundances of gaseous species, and deliver ions into the gas phase. On the other hand, dust grains act as condensation nuclei in regions with high densities and low temperatures. Mutual dust collisions can lead to sticking (coagulation) for low impact velocities and to shattering (fragmentation) if the collision speeds are high enough. For intermediate velocities, cratering or erosion can occur (Blum & Wurm 2008). Depending on the temperature and the harshness of the environment, dust particles consist of highly refractory materials (e.g., oxides, metals, silicates) or exhibit a more volatile composition (e.g., organic material, ices). Interstellar dust particles in cold regions possibly possess a core-mantle-type morphology, with a refractory silicate core, an inner organic mantle, and an outer ice mantle (Li & Greenberg 1997).

Dust coagulation plays an important role in the densest interstellar regions, i.e., in molecular clouds (Ossenkopf 1993; Weidenschilling & Ruzmaikina 1994; Ormel et al. 2009) and protoplanetary disks (Blum & Wurm 2008; Zsom et al. 2010; Testi et al. 2014). Collision velocities in these regions are sufficiently low to allow two dust grains (or two small dust aggregates) to stick to one another after a collision. Evidence for dust growth in molecular clouds (e.g., Steinacker et al. 2010; Pagani et al. 2010) and in protoplanetary disks (Wilner et al. 2005; Rodmann et al. 2006; Testi et al. 2014) is available. In both environments, the dominating dust materials are silicates, carbonaceous materials, and ices. In contrast to silicates, for which empirical evidence for the low-velocity collision behavior exists (Blum & Wurm 2008; Güttler et al. 2010), the collision properties of carbonaceous materials and ices (and, in particular, of the most abundant water ice) are mainly unknown. Coagulation models (Wada et al. 2007, 2008, 2009; Wettlaufer 2010) assume a higher upper energy limit for sticking of ice particles than for silicates (for the latter, see Poppe et al. 2000), but empirical evidence for this conjecture is scarce. The only supporting fact for an enhanced stickiness of water ice with respect to silicates is the at least tenfold higher surface energy of water ice (0.19 J m⁻²; Gundlach et al. 2011) in comparison to silicates (0.02 J m⁻²; Blum & Wurm 2000; Gundlach et al. 2011). In addition, the work by Aumatell & Wurm (2014) suggests an even higher specific surface energy of water ice of 0.37 J m⁻².

Dust collisions can also lead to fragmentation of dust aggregates in molecular clouds (Ormel et al. 2009) if the collision velocities exceed the fragmentation threshold for aggregates, which is expected for velocities in the range of ∼1–100 m s⁻¹, depending on grain size and material. As in the case of coagulation, this effect has been investigated in the laboratory for silicates (Blum & Wurm 2008; Güttler et al. 2010; Blum et al. 2014) but not for ices. At very high collision velocities, a variety of impact experiments of (small) projectiles into (large) targets exist, even for ice at cryogenic temperatures (see, e.g., Koschny & Grün 2001, and references therein). At these impact velocities, crater formation with an excessive outflow of ejecta is generally observed.

As outlined above, we are still lacking quantitative data on the coagulation or fragmentation behavior of microscopic water-ice particles and aggregates thereof under the conditions found in molecular cloud cores and protoplanetary disks. For macroscopic ice particles (centimeter to decimeter sizes), laboratory experiments have been performed by Hatzes et al. (1988, 1991), Supulver et al. (1997), Heißelmann et al. (2010), and Hill et al. (2014). Their findings agree inasmuch that particles consisting of water ice do not stick to one another, even at velocities below 0.1 mm s⁻¹. However, a frost layer increases the sticking threshold by some sort of "Velcro" effect. It is, however, questionable whether these experiments can be used to derive the collision properties of microscopically small water-ice particles.

In this work, we present a novel experimental setup, which is used to grow ice aggregates composed of μm-sized water-ice particles at low temperatures (see Section 2). With this experiment, we are able to determine the sticking and the erosion thresholds of μm-sized water-ice particles at different temperatures. In Section 3, we characterize our μm-sized water-ice particles by discussing their size distribution, the specific surface energy of water ice and the temperatures of the particles before impact. The obtained experimental results are then presented in Section 4 and compared with a collision model in Section 5. This comparison is used to deduce the specific surface energy and the viscous relaxation time of the μm-sized water-ice particles. In Section 6, the obtained results are discussed and a short outlook on future experiments is given. Finally, Section 7 concludes this manuscript by reviewing the main results of this work.

2.1. Experimental Setup

We have constructed an experimental setup (see Figure 1 for a sketch and Figure 2 for a photograph of the experimental setup) to study the collision properties of μm-sized water-ice particles in the velocity range between ∼1 m s⁻¹ and ∼150 m s⁻¹. The μm-sized water-ice particles are produced by spraying distilled water with an inhalator (labeled 1 in Figure 2) into a sedimentation chamber (2). The sedimentation chamber is actively cooled by liquid nitrogen to temperatures between 110 K and 198 K. Thus, freezing of the water droplets occurs during sedimentation inside the cold sedimentation chamber (see Section 3.3), leading to crystalline, spherical water-ice particles. The produced particles are then extracted (3) by the suction provided by two connected vacuum chambers (4 and 5). The first vacuum chamber (4; typical pressure: 50 mbar–200 mbar) is used to separate the ice particles from the gas by a nozzle-skimmer arrangement. In the second vacuum chamber (5; typical pressure: 0.1–50 mbar), a collimated ice-particle jet is formed. The ice-particle jet is directed onto a cryogenically cooled target (temperature: ∼80 K). The collisions between the ice particles and the target are observed by a long-distance microscope (6) using an illumination by a pulsed laser beam in forward scattering (7). To prevent the laser light from entering the microscope, it is equipped with a light trap at the center of its entry lens. Thus, only the forward-scattered light of the particles within a scattering-angle range between 43 and 132 can reach the microscope camera.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The inset in Figure 2 shows the nozzle (8) used to form the ice-particle jet and the cryogenically cooled target (9) seen from top of the second vacuum chamber through a glass window. The distance between the nozzle and the cryogenically cooled target is 2 cm. The nozzle is positioned inside a cooling system (10) to ensure cryogenic temperatures. The cooling system (see Figure 3) allows active cooling to temperatures between ∼100 K and ∼300 K using liquid nitrogen. Heating cartridges are also included in the setup, which enable us to warm up the cooling system from cryogenic temperatures to room temperature within approximately 10 minutes.

Zoom In Zoom Out Reset image size

The temperature sensors (H in Figure 2 and IV in Figure 3) are used to monitor all relevant temperatures of the experimental setup, i.e., the cooling system, the nozzle, the pipes, and the sedimentation chamber, respectively. During all experimental runs, the cooling system, the first vacuum chamber, and the sedimentation chamber were actively cooled by liquid nitrogen, which provided cryogenic temperatures. Table 1 summarizes the measured temperatures during the experimental runs. The intervals of the measured temperatures are due to temperature fluctuations during the various experimental runs.

Table 1. Measured Temperatures during the Experimental Runs

Experimental Run	Tcooling system	Tsedimentation chamber	Tnozzle
	[K]	[K]	[K]
1	128–133	189–198	250–260
2	123–140	150–154	235–243
3	79–82	153–193	198–208
4	80–83	174–179	150–180
5	81–82	128–135	138–148
6	80–84	110–116	114–120

Note. The temperatures T_{cooling system}, T_{sedimentation chamber}, and T_nozzle are those of the cooling system (which is identical to the temperature of the target), the sedimentation chamber, and the nozzle 2 cm away from the target, respectively.

Download table as:  ASCIITypeset image

As mentioned above, the ice particles are accelerated by the pressure difference between the vacuum chambers. For typical pressures of 50 mbar to 200 mbar in the first vacuum chamber and 0.1 mbar to 50 mbar in the second vacuum chamber, the emerging particles are accelerated to velocities between 1 m s⁻¹ and 150 m s⁻¹. Figure 4 exemplarily shows μm-sized water-ice particles illuminated by the pulsed laser beam and observed with the long-distance microscope (dashed lines).

Zoom In Zoom Out Reset image size

2.2. Experimental Procedure

3.1. Size Distribution

3.2. Specific Surface Energy of Water Ice

3.3. Temperature of the μm-sized Water-ice Particles during the Passage of the Experimental Setup

3.4. Temperature of the Grown Ice Aggregates Composed of the μm-sized Water-ice Particles

The grown ice aggregates possess porous structures composed of μm-sized water-ice particles. We assume that the grown ice aggregates obtain a porosity of approximately 0.5. The heat conductivity of porous aggregates in vacuum is generally very low (in the order of 10⁻² W m⁻¹ K⁻¹; Krause et al. 2011; Gundlach & Blum 2012). Thus, the temperature of the region of the ice aggregates where the collisions with the impinging μm-sized water-ice particles occur (see Figure 5) is in thermal equilibrium with the ambient temperature provided by the copper cylinder (I in Figure 3) of the cooling system. Additionally, the temperature of the cold plate (II in Figure 3) is equal to the temperature of the copper cylinder during the experimental runs, which implies that the temperature gradient inside the sample is very small. Thus, the temperature of the grown ice aggregates is in equilibrium with the temperature of the cooling system (see Table 1).

Zoom In Zoom Out Reset image size

4.1. Growth and Erosion of Ice Aggregates Composed of μm-sized Water-ice Particles

With the experimental setup described above (see Section 2.1), we performed impact experiments with the μm-sized water-ice particles onto the cryogenically cooled target. As explained in Sections 3.4 and 3.3, the temperature of the impinging water-ice particles is approximately given by the temperature of the nozzle and the temperature of the target water-ice agglomerate is that of the cooling system. Both temperatures are presented in Table 1 for the six experimental runs. As long as the impact velocities were sufficiently small, porous ice aggregates formed on the target by direct sticking of the impinging μm-sized water-ice particles. The growing ice aggregates were observed by the long-distance microscope and the laser setup (see Figure 2). Figure 5 shows two examples of the growth of ice aggregates. The growing ice aggregates were also observed through the top window of the vacuum chamber utilizing a standard CCD camera (see Figure 6(a)). For comparison, we also collided uncooled water micro-droplets with the cryogenically cooled target (see Figure 6(b)) and sprayed water vapor onto the cooled target (see Figure 6(c)). We performed these different experiments in order to demonstrate the influence of the projectile state (solid, liquid, vapor) on the visual appearance of the growing ice targets.

Zoom In Zoom Out Reset image size

At higher velocities, the ice aggregates get eroded by the impinging μm-sized water-ice particles. Figure 7 exemplarily shows the erosion of two different ice aggregates. For velocities higher than the sticking threshold and lower than the erosion threshold, bouncing of the μm-sized water-ice particles occurred. In this case, no growth and no erosion of the ice aggregates was observed (see Section 4.2).

Zoom In Zoom Out Reset image size

4.2. Derivation of the Sticking and Erosion Thresholds of μm-sized Water-ice Particles

In order to measure the threshold velocities for sticking and erosion of the μm-sized water-ice particles, the growth and erosion behavior of the ice aggregates was observed using the long-distance microscope and the laser-illumination setup (see Figures 5 and 7). For each growth (erosion) sequence, the growth (erosion) rate of the ice aggregate (measured in units of μm growth length per second exposure of the target to the impinging projectile flux) and the corresponding velocities of the impinging μm-sized water-ice particles was measured. Additionally, sequences in which no growth and no erosion occurred were identified as dominated by bouncing of the water-ice particles. Also in these sequences, the velocities of the impinging water-ice particles were measured and the corresponding growth rate of the ice aggregate was set to zero.

The results of this analysis are summarized in Figure 8 for the different experimental runs, which were performed at different temperature (see Table 1 and Sections 3.4 and 3.3). Each panel of Figure 8 shows an individual experimental run during which the temperature of the nozzle and, therewith, the temperature of the impinging μm-sized water-ice particles were kept constant (see Table 1). In Figure 8 the velocities obtained during the growth, bouncing, and erosion sequences are denoted by the triangles, pluses, and squares, respectively. For each of these sequences, the mean velocity (dashed lines) of the μm-sized water-ice particles and the standard error σ of the mean velocity (dotted lines) are shown. It is important to mention that the shown growth rates are absolute values and are not normalized, because it was not possible to determine the number of impinging particles per unit area and time during the experiments.

Zoom In Zoom Out Reset image size

In the following, we will explain how the sticking threshold of the μm-sized water-ice particles was derived from the measured data. In order to be independent from model assumptions, we decided to determine a lower and an upper boundary to the sticking threshold for each temperature, which provides a maximum range within which the sticking threshold velocity can be found.

The lower boundary of the sticking threshold was derived from the lowest growth rates observed during the different experimental runs. We assume that in this case, all collisions with velocities lower than the mean velocities minus the corresponding 1σ deviations of the mean velocities led to a growth of the ice aggregates. These lower boundaries of the sticking threshold are shown as dash-dotted lines in Figure 8. The upper boundary of the sticking threshold was calculated by the assumption that in the case of bouncing, all impinging particles with velocities lower than the mean velocity caused growth of the ice aggregate whereas all water-ice particles impinging above the mean velocity cause erosion so that both processes just cancel. Thus, the upper boundaries of the sticking threshold are given by the mean values of the velocities measured during the bouncing sequences (solid lines in Figure 8).

Figure 9 shows the derived lower and upper boundaries of the sticking threshold (squares and diamonds, respectively) as a function of the temperature of the impinging μm-sized water-ice particles. For temperatures below ∼210 K, the sticking threshold of the μm-sized water-ice particles is within the measurement uncertainties constant at 9.6 m s⁻¹ (dashed-dotted line; this value is derived by calculating the arithmetic mean of the lower and upper boundaries obtained for each of the four experimental runs performed at temperatures below ∼210 K). For higher temperatures, the sticking threshold increases with increasing temperature to values of ∼18 m s⁻¹ for T = (239 ± 4) K and ∼40 m s⁻¹ for T = (255 ± 5) K.

Zoom In Zoom Out Reset image size

The measured sticking threshold of the μm-sized water-ice particles at low temperatures is approximately 10 times higher than the sticking threshold derived for μm-sized silica particles. The latter was measured by Poppe et al. (2000), who found a sticking threshold of 1.2 m s⁻¹ for silica spheres with radii of 0.6 μm (dotted line in Figure 9). The higher sticking threshold of the μm-sized water-ice particles in comparison to the μm-sized silica particles is mainly due to the difference of the specific surface energies of the two materials.

Between ∼210 K and ∼230 K, the sticking threshold changes from a temperature-independent value to an increasing function with increasing temperature. This transition is observed in the same temperature region where Higa et al. (1998) proposed a change of the brittle states of water ice. Whether a transition of the brittle states of water ice can increase the stickiness of water ice at higher temperatures is, however, speculative.

In addition to the derivation of the sticking threshold, we also investigated the erosion threshold of the μm-sized water-ice particles when impinging an ice-particle target (see solid and dash-triple-dotted lines in Figure 8). To derive the lower and the upper boundary of the erosion threshold, the same algorithm as in the case of the sticking threshold was used. This means that the lower boundary of the erosion threshold is given by the mean velocity of the bouncing sequence (which is, thus, equal to the upper threshold for sticking). The upper boundary of the erosion threshold was derived from the lowest erosion rates observed during the different experimental runs. In these cases, we assume that all collisions with velocities higher than the mean velocities plus the corresponding 1σ-deviations led to erosion of the ice aggregates.

In Figure 10, the lower and upper boundaries for the erosion threshold are shown by the diamonds and triangles as a function of temperature of the impinging water-ice particles. The erosion threshold at low temperatures (below ∼210 K) is within the uncertainties of our measurements constant at 15.3 m s⁻¹ and increases to ∼50 m s⁻¹ for higher temperatures. For comparison, the dashed line indicates the erosion threshold of ∼2 m s⁻¹ derived for spherical silica particles with radii of 0.75 μm and for unpassivated aggregate targets (Schräpler & Blum 2011). A very similar value of the erosion threshold was also found by Seizinger et al. (2013) who used computer simulations in order to determine the erosion rate of silica aggregates. Schräpler & Blum (2011) noted that passivation of dust aggregates occurs when more than 2 mg cm⁻² of dust collides with an aggregate. We used this critical value to check whether the water ice aggregates are being passivated by the impinging water-ice particles before the erosion of the ice aggregates is started. In our experiments, the ice aggregates are grown with a maximum speed of ∼100 μm s⁻¹ and we assume that the ice aggregates posses a porosity of 0.5. With the knowledge of the growth speed and the assumed porosity it is possible to derive the required time until passivation of the ice aggregates occurs (i.e., when the critical value of 2 mg cm⁻² is reached), which reads t ⩾ 400 s. In our experiments, the time interval between growth and erosion of the ice aggregates was always less than 60 s, which implies that the ice aggregates are not passivated before erosion.

Zoom In Zoom Out Reset image size

A comparison between the mean sticking threshold (see Figure 9) and the mean erosion threshold (see Figure 10) for the two materials shows that the ratio of the threshold velocities for erosion and sticking is ∼1.5 in the case of the μm-sized water-ice particles and ∼2 in the case of the μm-sized silica particles. These values are in reasonable agreement with the calculation performed by Dominik & Tielens (1997). They showed that the critical energy for sticking (their Equation (6)) is 4.8 times smaller than the energy required to break an existing contact between the particles (their Equation (8)). In terms of velocity (their Equation (9)), this means that the sticking threshold is $\sqrt{4.8} \, = \, 2.2$ times lower than the erosion threshold. However, we would like to remind the reader that the sticking threshold estimated from our experiments is derived for particle-aggregate collisions, whereas the calculations performed by Dominik & Tielens (1997) are considering particle-particle collisions.

Additionally it should be noted that our μm-sized water-ice particles possess a rather broad size distribution, which also influences the sticking threshold, because the smaller particles stick at higher velocities than larger ice grains. Thus, the values derived in this work are to be considered as averages over the size distribution.

In this section, we compare our results with the outcome of the collision model developed by Krijt et al. (2013). This collision model describes the energy dissipation in head-on collisions between spherical particles by taking into account adhesion, visco-elasticity and plastic deformation. The only material parameters in this model are the viscous relaxation time τ_vis and the specific surface energy of the material γ.

We used this model to calculate the coefficient of restitution after the collision of two equal-sized water-ice particles with radii of r = 1.5 μm for different collision velocities. For velocities lower than the sticking threshold, the coefficient of restitution remains zero because sticking occurs, i.e., the velocity after the collision is zero. At higher velocities, the coefficient of restitution reaches values between zero and unity depending on the collision speed, particle size, and material properties. Thus, the highest velocity for which the coefficient of restitution is zero can be regarded as the sticking threshold.

We performed different runs of the collision model to calculate the sticking threshold of the water-ice particles for various values of the viscous relaxation time and the specific surface energy. Both parameters were varied systematically and independently until the model matched the sticking threshold of 9.6 m s⁻¹ that was experimentally derived for temperatures below ∼210 K. We allowed for an uncertainty of ±0.3 m s⁻¹ for the sticking threshold calculated with the collision model. In order to take collisions between ice particles and ice aggregates into account, we changed the mass of the second collision partner to m_agg = 5 m₀, where m₀ is the mass of a water-ice particle with a radius of 1.5 μm (see Schräpler & Blum 2011, and Section 6).

Figure 11 shows those combinations of the viscous relaxation time and the specific surface energy that result in a sticking threshold of (9.6 ± 0.3) m s⁻¹ (squares). For the model calculations, a Young's modulus of 7 GPa, a Poisson ratio of 0.25, and a water-ice mass density of 930 kgm⁻³, respectively, were used. One run of the collision model shown in Figure 11 was performed by assuming a specific surface energy of γ = 0.19 J m⁻² (asterisk), a value that was derived from previous experiments performed with the same μm-sized water-ice particles as in this study (see Section 3.2; Gundlach et al. 2011). In this case, the viscous relaxation time becomes 1 × 10⁻¹⁰ s (dotted lines) and matches the range of values 1 × 10⁻¹² < τ_vis < 3 × 10⁻¹⁰ acquired by fitting other experiments, which were also performed with μm-sized particles (see Figure 8(a) in Krijt et al. 2013).

Zoom In Zoom Out Reset image size

In the previous sections, we presented a novel experimental setup with which we grew aggregates consisting of μm-sized water-ice particles under astrophysically relevant conditions. We measured the threshold velocities for sticking and erosion in collisions between individual μm-sized water-ice particles and aggregates thereof at different temperatures between 114 K and 260 K (see Section 2). The experiments showed that the sticking threshold is constant at 9.6 m s⁻¹ for temperatures below ∼210 K. At higher temperatures the sticking threshold increases to values between ∼30 m s⁻¹ and ∼50 m s⁻¹ (see Figure 9). This change in the sticking threshold with increasing temperature is probably caused by the transition from brittle state I to brittle state II at a temperature of 229 K (Higa et al. 1998). In comparison to μm-sized silica particles (Poppe et al. 2000), the sticking threshold of water-ice particles is approximately tenfold higher. This dependency was expected due to the increased specific surface energy of water-ice in comparison to silica (see Section 3.2). The estimated sticking threshold was derived for particle-aggregate collisions, which is the most probable case in protoplanetary disks.

A comparison of our experimental data with the collision model by Krijt et al. (2013) provides the opportunity to derive important material properties of μm-sized water-ice particles at cryogenic temperatures, i.e., the specific surface energy and the viscous relaxation time (see Section 5). Without any restrictions, it is only possible to provide possible combinations of the two material properties (see Figure 11). However, as the specific surface energy has been determined in previous laboratory experiments (Gundlach et al. 2011) to be γ = 0.19 J m⁻², we derive a viscous relaxation time of 10 × 10⁻¹⁰ s.

In addition to the sticking threshold, we also estimated the erosion threshold of the μm-sized water-ice particles. The experiments showed that the erosion threshold is also temperature independent at low temperatures (below ∼210 K) and reads 15.3 m s⁻¹ (see Figure 10). For comparison, μm-sized silica particles possess an erosion threshold of ∼2 m s⁻¹ (Schräpler & Blum 2011; Seizinger et al. 2013). This result shows that ice aggregates are more resistant to erosion than silica aggregates. For higher temperatures, the erosion threshold increases to a value of ∼50 m s⁻¹ at a temperature of (255  ±  5) K.

As shown in Section 5, our experimental results can be fitted with the collision model by Krijt et al. (2013) with reasonable assumptions for the surface energy and the viscous relaxation time (see Figure 11). We used this model to predict the threshold velocity for sticking between spherical water-ice particles. For this, we assumed a constant (temperature and particle-size independent) surface energy of γ = 0.19 J m⁻² and a viscous relaxation time that is grain-size dependent, as shown in Krijt et al. (2013; their Figure 8(b)), and reads

$$\begin{equation} \tau _{\rm vis} \, = \, \tau _{\rm vis,\ exp} \, \left(\,\frac{r \, \mathrm{[\mu m]}}{1.5\, \mathrm{\mu m}}\,\right)^{1.11} \,, \end{equation} \tag{ 1 }$$

where r is the particle radius and τ_vis, exp = 10 × 10⁻¹⁰ s is the viscous relaxation time derived from the experiments. The results of these calculations are shown in Figure 12, where we plot the threshold velocity for sticking as a function of grain radius for radii between 0.1 μm and 10 μm. The three curves show the case of grain–grain collisions (squares), grain–aggregate collisions (asterisks), and grain–grain collisions for core-mantle particles (pluses), respectively. Grain–aggregate collisions were approximated by assigning one of the two colliding grains an effective mass of five times the monomer mass. As Schräpler & Blum (2011) showed by comparing their experimental results on grain–aggregate collisions with model predictions by Konstandopoulos (2000), the target aggregate can be replaced by a single monomer with an effective mass of

$$\begin{equation} m_{\rm eff} \, = \, m \, \left(\,1\, +\ C\,n\,\right), \end{equation} \tag{ 2 }$$

with m, C = 1, and n = 4 being the monomer mass, a rigidity parameter, and the local coordination number. Thus, we get m_eff = 5 m. For the core-mantle particles (particle-particle collisions), all characteristic material parameters were kept constant with the exception of the mass density of both grains, which was raised from ρ = 930 kg m⁻¹ for pure water ice to ρ = 2000 kg m⁻¹ for core-mantle particles, assuming approximately equal volumes for the water-ice mantle and the silicate core.

Zoom In Zoom Out Reset image size

Following the collision models by Dominik & Tielens (1997) and Krijt et al. (2013), we can derive the grain size below which collisions are in the hit-and-stick regime, i.e., grains collide without any compaction, and above which growth is expected to occur as compact aggregates. The hit-and-stick threshold is derived by Dominik & Tielens (1997) to be

$$\begin{equation} v_{\mathrm{hit\hbox{-}and\hbox{-}stick}} = 5 E_{\mathrm{roll}}, \end{equation} \tag{ 3 }$$

where E_roll is the rolling-friction energy. In case of equal-size particles, the rolling-friction energy is given by (Dominik & Tielens 1997)

$$\begin{equation} E_{\mathrm{roll}} = 3 \pi ^2 \gamma r \xi, \end{equation} \tag{ 4 }$$

with ξ being the critical displacement before rolling starts. Following Krijt et al. (2013), this critical displacement is given by

$$\begin{equation} \xi = \frac{a_{\mathrm{eq}}}{12} \frac{\Delta \gamma }{\gamma }, \end{equation} \tag{ 5 }$$

with a_eq being the equilibrium contact radius between the ice grains and (Δγ/γ) ≈ 1. The contact radius for equal-size spheres is given by

$$\begin{equation} a_{\mathrm{eq}} = \left(\frac{9 \pi \gamma r^2}{4 E^*} \right)^{1/3}, \end{equation} \tag{ 6 }$$

with E* = 1.5 × 10¹⁰ Pa being the combined elastic modulus of water ice. Thus, the critical displacement scales as ξ∝r^−2/3 and we can calculate below which velocity sticking occurs in the hit-and-stick regime, which yields

$$\begin{equation} v_{\mathrm{hit\hbox{-}and\hbox{-}stick}} < \mathrm{min}(v_{\mathrm{stick}},v_{\mathrm{roll}}), \end{equation} \tag{ 7 }$$

with v_roll = (2 E_roll/m)^1/2 and m being the reduced mass of the projectile and the target particle. This velocity is also shown in Figure 12 (dashed line). As can be seen, the threshold velocity for sticking v_stick is always higher than v_roll in the monomer-size range shown in Figure 12 and for monomer grains as projectiles. This is not necessarily the case when two ice-aggregates collide, which is, however, not the subject of this study. Thus, water-ice grains grow to highly porous aggregates for velocities below v_roll (dashed line in Figure 12) and to rather compact aggregates above v_roll and below v_stick.

The relatively high sticking threshold of water-ice particles found in our experiments is not in contradiction to the idea that icy planetesimals form by direct sticking of sub-μm-sized water-ice particles as suggested by Wada et al. (2008, 2009), Suyama et al. (2008, 2012), Okuzumi et al. (2012), and Kataoka et al. (2013). In this formation scenario, 0.1 μm-sized water-ice particles are assumed to stick at velocities up to 60 m s⁻¹, which leads to the formation of very porous (with volume filling factors down to 10⁻⁵) ice aggregates. Due to the high porosity, the aggregates are not severely affected by radial drift in the protoplanetary disk and, thus, possess a rather long lifetime before they evaporate. After the formation of the very porous aggregates, gas compression and finally self-gravitational compression lead to a compaction of the planetesimals. So far, this formation scenario was lacking the experimental proof that sub-μm-sized water-ice particles do stick at the assumed relatively high velocities. With our measured sticking threshold of v_st = 9.6 m s⁻¹ for water-ice particles with a mean radius of r = 1.5 μm at low temperatures and applications of the model by Krijt et al. (2013), we can now support this formation scenario under the assumption of very small (sub-μm) ice particles. As can be seen from Figure 12, a critical sticking velocity of 60 m s⁻¹ is reached for water-ice particle radii of ∼0.7 μm in particle-particle and ∼0.2 μm for particle-aggregate collisions, respectively. In the case of core-mantle particles, the maximum particle sizes are even smaller. Thus, for maximum particle sizes as indicated above, the assumption of high stickiness of water-ice particles made by Wada et al. (2008, 2009), Suyama et al. (2008, 2012), Okuzumi et al. (2012), and Kataoka et al. (2013) seem to be in agreement with our experimental results and the collision model by Krijt et al. (2013). However, it is at least questionable whether the assumed small particle radii match the ice-particle sizes in protoplanetary disks. Thus, it would be interesting to repeat the model calculations for particles with radii of ∼1 μm.

In the future, we plan to perform further laboratory collision experiments with μm-sized water-ice particles. Next, we will focus on measuring the coefficient of restitution as a function of ice-particle size and temperature. We intend to derive the individual particle sizes of the impinging water-ice particles from the forward-scattered light that enters our long-distance microscope, using Mie scattering theory. These measurements will then allow us to disentangle the specific surface energy and the viscous relaxation time obtained by fitting the experimental data with the collision model developed by Krijt et al. (2013). Performing these future experiments at different temperatures will yield the temperature dependency of the specific surface energy of water ice for temperatures between ∼100 K and ∼250 K. In addition, experiments with core-mantle particles (water-ice particles containing a silicate core) will be performed in order to determine the influence of the silicate cores on the collision properties of μm-sized water-ice particles.

In this paper, we described our experimental work on the deduction of the collision properties of spherical, micrometer-sized water-ice particles under astrophysically relevant conditions, i.e., at low pressures and temperatures. The main results of this work are as follows.

• 1.  
  Spherical water-ice particles with a mean radius of 1.5 μm stick in collisions with aggregates of water-ice particles and under astrophysically relevant conditions as long as their impact velocities are below ∼10 m s⁻¹ (see Section 4.2 and Figures 5, 6, 8, and 9). The threshold velocity for sticking is about tenfold higher than that of spherical SiO₂ particles with comparable size (Poppe et al. 2000), which can be explained by a tenfold higher specific surface energy of water ice.
• 2.  
  The threshold velocity for sticking of 1.5 μm water-ice particles is temperature independent for T < 210 K and increases with increasing temperature for T > 210 K (see Figure 9). As all astrophysically relevant water-ice temperatures in molecular clouds and protoplanetary disks are well below 210 K (Bergin & Tafalla 2007; Lecar et al. 2006), laboratory experiments on the collision and sticking behavior of water-ice particles should be done at temperatures below 210 K to avoid increased stickiness, or sintering effects.
• 3.  
  The threshold velocity for sticking of the 1.5 μm water-ice particles can be reconstructed with the collision model of Krijt et al. (2013) with reasonable assumptions for the specific surface energy of water ice of γ = 0.19 J m⁻² (Gundlach et al. 2011) and the viscous relaxation time of τ_vis = 1 × 10⁻¹⁰ s, respectively. Applying the model by Krijt et al. (2013) allows the deduction of a threshold velocity for sticking in monomer-monomer collisions of ∼25 m s⁻¹ (see Figure 12). Varying the size of water-ice monomers in monomer-aggregate collisions gives sticking thresholds between ∼150 m s⁻¹ for grain radii of 0.1 μm and ∼2 m s⁻¹ for grain radii of 10 μm (see Figure 12). Core-mantle particles possess threshold velocities of 60%–70% of the values of pure ice particles (particle-particle collisions; see Figure 12).
• 4.  
  For impact velocities exceeding the sticking threshold, the water-ice aggregates become eroded (see Figure 7). The onset of erosion occurs typically for velocities above 1.5 times the upper limit for sticking (see Figures 9 and 10). This finding is in agreement with predictions by the collision model of Dominik & Tielens (1997).

This project was supported through the DFG priority program "Physics of the Interstellar Medium" under grant BL 298/19-1. We thank Sebastiaan Krijt, Carsten Güttler, and Daniel Heißelmann for programing their collision model in IDL and Florian Heimbach for providing the sketch of the experimental setup. Furthermore, we thank the anonymous reviewer for detailed and helpful comments, which helped improving this manuscript considerably.