Dust Collisions in the Protoplanetary Disk: Atomic Simulations to Better Understand the Surface Free Energy

Introduction: The formation of planetesimals within a protoplanetary disk is strongly influenced by the collision and eventual aggregation of nano- and micro-scale dust particles (1). This granular collision process is most often modelled using Johnson-Kendall-Roberts (JKR) contact models, a well-established theoretical approach that determines the sticking, bouncing and aggregation of colliding grains. The surface free energy (SFE) is a crucial physical input to this model, governing the properties of dust grains at their interfaces. The SFE is a measure of the energy difference between a free surface and the bulk of material.  Particle surfaces with high SFE  are more likely to adhere strongly to each other, promoting  particle coagulation and facilitating the growth of larger bodies. In contrast, particle surfaces with low SFE adhere weakly, making it more difficult for particles to aggregate.

While this value is a key input into JKR models, it is not well understood in the complete protoplanetary disk regime. Previous experimental studies on SFEs for silica have measured values ranging from 0.02–2.5 J m^-2, over two orders of magnitude in difference (2). These results are typically limited to specific temperatures and do not consider the complex minerals beyond SiO₂ likely to be found in natural dust particles during collisions. As an alternative, molecular dynamics (MD) modelling can be used to study these surfaces on the atomic scale, allowing for energy differences between bulk and surfaces to be calculated as a function of temperature for different minerals. While research has been conducted using MD for SFEs, results are needed for the temperatures, background gas properties, and mineral types important in protoplanetary disks. These mineral-specific values are essential for models of planet formation and disk evolution, while also influencing our understanding of the chemical processes occurring in the early solar system.

Methodology: In this study, we use MD simulations to study the SFEs of silica, albite, and anorthite. Silica is chosen as it is a commonly studied mineral both theoretically and experimentally. Albite and anorthite were chosen as they are the two endmembers in the plagioclase feldspar family, thought to be abundant on many celestial bodies. We use a ReaxFF potential to model all interactions, allowing for dynamic bond formation and, for future studies, chemical reactions. For each mineral we first calculate the potential energy of the bulk periodic sample. We then open a surface to vacuum, equilibrate the sample at the desired temperature, and calculate the subsequent (PE) of the surface. The SFE is then calculated as the energy difference per unit area of a periodic bulk sample and its corresponding surface divided by the area of the surface. We simulate SFEs for 30K, 100K, 300K, and 700K, a range of temperatures expected to be found at various radii within protoplanetary disks.

Results:  Table 1 displays the equilibrated SFE for each mineral type at the different simulated temperatures.

	Surface Free Energy (J/m2)
Temp (K)	Silica	Albite	Anorthite
30K	3.32	2.24	2.88
100K	3.04	2.21	1.70
300K	2.40	2.11	1.47
700K	2.24	1.71	1.06

 

For all temperatures silica has the highest SFE, followed by either albite or anorthite, suggesting that approximating grains as a simple silica structure may overestimate the SFE. In all cases the SFE decreases with temperature, potentially due to the increased atomic disorder with temperature reducing the energy needed to form a surface. For example, as temperature is increased from 30K to 700K the silica SFE is reduced by a factor of ~1.5. It is therefore important to account for specific temperatures within the disk when modelling grain interactions. These results suggest that a distribution of SFEs with temperature may be a more accurate approach. We can incorporate these different SFEs into various JKR relationships to show the effect on contact area, threshold velocity and pulling-off force.

In general, the SFEs at all temperatures are 1–2 orders of magnitude higher than commonly reported experimental values, which typically range from 0.02-0.03 J m^-2 (3-4). Results do however agree with limited MD simulations for the SFE of SiO₂ using different interatomic potentials (1). This discrepancy between experiment and theory may occur because experimental substrates are often studied at room temperature as surfaces that, when exposed to the ambient terrestrial environment, are likely to be hydroxylated and/or water passivated. This passivation can terminate free O bonds on the surface, potentially significantly influencing the energy of the surface. Therefore, if grains in protoplanetary disks are clean, as could be found at higher temperatures closer to the star, they will collide with a significantly different behavior than would be predicted using these experimental values. As such, SFEs need to be better quantified as a for both clean and covered surfaces as function of mineral type and temperature.  Here we provide a comprehensive set of simulations for clean surfaces in a true vaccum.

These novel results highlight the key role MD simulations can play in better understanding SFEs for grain contact modelling. SFEs are shown to be temperature and mineral specific, and likely highly sensitive to H or H₂O in the environment . These “clean surface” MD-derived values may be more appropriate than the comparatively lower experimental values when collisions occur in regions of the disk where water/H is less likely to adsorb.  Further work is required to quantify the influence of coverage along with the effects of amorphization.

References:

1. Kimura, K. Wada, H. Senshu and H. Kobayashi, Astrophys. J., 2015, 812, 67.

2. Blum, Res. Astron. Astrophys., 2010, 10, 1199.

3. Blum and G. Wurm, Icarus, 2000, 143(1), 138-146.

4. Kendall, N. M. Alford, and J. D. Birchall, Nature, 1987(London) 352, 794.