Collisional Growth Research Articles

The Dynamical Complexity of Surface Mass Shedding from a Top-shaped Asteroid Near the Critical Spin Limit

Abstract The regolith transport near the surface of an asteroid is inherently sensitive to the local topography. In this paper, conditions of surface mass shedding and the subsequent evolution of the shedding material are studied for the primary of 65803 Didymos, serving as a representative for a large group of top-shaped asteroids that rotate near their critical spin limits. We considered the influences of an asymmetric shape and a non-spherical gravity, and demonstrate that these asymmetries play a significant role in the shedding process as well as in the subsequent orbital motion. The mass shedding conditions are given as a function of the geological coordinates, and show a clear-cut dependency on the local topographic features. We find that at different stages of the Yarkovsky–O’Keefe–Radzievskii–Paddack spin-up, the bulged areas exhibit a uniform superior advantage of enabling mass shedding over the depressed areas. “Dead zones” free from mass shedding are found around the polar sites. Numerical simulations show that the orbital motion of the shedding material experiences a drastic change as the spin rate is approaching the critical limit. The “mass leaking” effect is reinforced as the spin rate increases; the lower spin rates correspond to a higher capability of trapping the lofted particles in the vicinity of the asteroid, which statistically improves the probability of collisional growth in orbit. We also find that the topological transition of the equilibrium point can in practice lead to rapid clearance of the shedding material and transport of their orbits to larger distances from the surface.

Bridging the condensation–collision size gap: a direct numerical simulation of continuous droplet growth in turbulent clouds

Abstract. In most previous direct numerical simulation (DNS) studies on droplet growth in turbulence, condensational growth and collisional growth were treated separately. Studies in recent decades have postulated that small-scale turbulence may accelerate droplet collisions when droplets are still small when condensational growth is effective. This implies that both processes should be considered simultaneously to unveil the full history of droplet growth and rain formation. This paper introduces the first direct numerical simulation approach to explicitly study the continuous droplet growth by condensation and collisions inside an adiabatic ascending cloud parcel. Results from the condensation-only, collision-only, and condensation–collision experiments are compared to examine the contribution to the broadening of droplet size distribution (DSD) by the individual process and by the combined processes. Simulations of different turbulent intensities are conducted to investigate the impact of turbulence on each process and on the condensation-induced collisions. The results show that the condensational process promotes the collisions in a turbulent environment and reduces the collisions when in still air, indicating a positive impact of condensation on turbulent collisions. This work suggests the necessity of including both processes simultaneously when studying droplet–turbulence interaction to quantify the turbulence effect on the evolution of cloud droplet spectrum and rain formation.

Effect of Turbulence on Collisional Growth of Cloud Droplets

We investigate the effect of turbulence on the collisional growth of micrometer-sized droplets through high-resolution numerical simulations with well-resolved Kolmogorov scales, assuming a collision and coalescence efficiency of unity. The droplet dynamics and collisions are approximated using a superparticle approach. In the absence of gravity, we show that the time evolution of the shape of the droplet-size distribution due to turbulence-induced collisions depends strongly on the turbulent energy-dissipation rate [Formula: see text], but only weakly on the Reynolds number. This can be explained through the [Formula: see text] dependence of the mean collision rate described by the Saffman–Turner collision model. Consistent with the Saffman–Turner collision model and its extensions, the collision rate increases as [Formula: see text] even when coalescence is invoked. The size distribution exhibits power-law behavior with a slope of −3.7 from a maximum at approximately 10 up to about 40 μm. When gravity is invoked, turbulence is found to dominate the time evolution of an initially monodisperse droplet distribution at early times. At later times, however, gravity takes over and dominates the collisional growth. We find that the formation of large droplets is very sensitive to the turbulent energy dissipation rate. This is because turbulence enhances the collisional growth between similar-sized droplets at the early stage of raindrop formation. The mean collision rate grows exponentially, which is consistent with the theoretical prediction of the continuous collisional growth even when turbulence-generated collisions are invoked. This consistency only reflects the mean effect of turbulence on collisional growth.

Illustration of microphysical processes in Amazonian deep convective clouds in the gamma phase space: introduction and potential applications

Abstract. The behavior of tropical clouds remains a major open scientific question, resulting in poor representation by models. One challenge is to realistically reproduce cloud droplet size distributions (DSDs) and their evolution over time and space. Many applications, not limited to models, use the gamma function to represent DSDs. However, even though the statistical characteristics of the gamma parameters have been widely studied, there is almost no study dedicated to understanding the phase space of this function and the associated physics. This phase space can be defined by the three parameters that define the DSD intercept, shape, and curvature. Gamma phase space may provide a common framework for parameterizations and intercomparisons. Here, we introduce the phase space approach and its characteristics, focusing on warm-phase microphysical cloud properties and the transition to the mixed-phase layer. We show that trajectories in this phase space can represent DSD evolution and can be related to growth processes. Condensational and collisional growth may be interpreted as pseudo-forces that induce displacements in opposite directions within the phase space. The actually observed movements in the phase space are a result of the combination of such pseudo-forces. Additionally, aerosol effects can be evaluated given their significant impact on DSDs. The DSDs associated with liquid droplets that favor cloud glaciation can be delimited in the phase space, which can help models to adequately predict the transition to the mixed phase. We also consider possible ways to constrain the DSD in two-moment bulk microphysics schemes, in which the relative dispersion parameter of the DSD can play a significant role. Overall, the gamma phase space approach can be an invaluable tool for studying cloud microphysical evolution and can be readily applied in many scenarios that rely on gamma DSDs.

Stochastic coalescence in Lagrangian cloud microphysics

Abstract. Stochasticity of the collisional growth of cloud droplets is studied using the super-droplet method (SDM) of Shima et al.(2009). Statistics are calculated from ensembles of simulations of collision–coalescence in a single well-mixed cell. The SDM is compared with direct numerical simulations and the master equation. It is argued that SDM simulations in which one computational droplet represents one real droplet are at the same level of precision as the master equation. Such simulations are used to study fluctuations in the autoconversion time, the sol–gel transition and the growth rate of lucky droplets, which is compared with a theoretical prediction. The size of the coalescence cell is found to strongly affect system behavior. In small cells, correlations in droplet sizes and droplet depletion slow down rain formation. In large cells, collisions between raindrops are more frequent and this can also slow down rain formation. The increase in the rate of collision between raindrops may be an artifact caused by assuming an overly large well-mixed volume. The highest ratio of rain water to cloud water is found in cells of intermediate sizes. Next, we use these precise simulations to determine the validity of more approximate methods: the Smoluchowski equation and the SDM with multiplicities greater than 1. In the latter, we determine how many computational droplets are necessary to correctly model the expected number and the standard deviation of the autoconversion time. The maximal size of a volume that is turbulently well mixed with respect to coalescence is estimated at Vmix = 1.5 × 10−2 cm3. The Smoluchowski equation is not valid in such small volumes. It is argued that larger volumes can be considered approximately well mixed, but such approximation needs to be supported by a comparison with fine-grid simulations that resolve droplet motion.

Estimating collision–coalescence rates from in situ observations of marine stratocumulus

Precipitation forms in warm clouds via collision–coalescence. This process is difficult to observe directly in situ and its implementation in numerical models is uncertain. We use aircraft observations of the drop‐size distribution (DSD) near marine stratocumulus tops to estimate collision–coalescence rates. Marine stratocumulus is a useful system to study collisional growth because it is initiated near the cloud top and the clouds evolve slowly enough to obtain statistically useful data from aircraft. We compare rate constants estimated from observations with reference rate constants derived from a collision–coalescence box model, the result of which is termed the enhancement factor (EF). We evaluate two hydrodynamic collision–coalescence kernels, one quiescent and one including the effects of small‐scale turbulence. Due to sampling volume limitations, DSDs must be averaged over length‐scales much greater than those relevant to the underlying physics, such that we also examine the role of averaging length‐scale with respect to process representation. Averaging length‐scales of 1.5 and 30 km are used, corresponding roughly to the horizontal grid lengths of cloud‐resolving models and high‐resolution climate models, respectively. EF values range from 0.1 to 40, with the greatest EFs associated with small mode diameter cases and a generally decreasing trend with drop size. For any given drop size or averaging length‐scale, there is about an order of magnitude variability in EFs. These results suggest that spatial variability on length‐scales smaller than 1.5 km prevents accurate retrieval of rate constants from large‐scale average DSDs. Large‐scale models must therefore account for small‐scale variability to represent cloud microphysical processes accurately. The turbulent kernel reduces EFs for all drop sizes, but can only account for at most half of the calculated EFs in marine stratocumulus.

Satellitesimal Formation via Collisional Dust Growth in Steady Circumplanetary Disks

Abstract The icy satellites around Jupiter are considered to have formed in a circumplanetary disk. While previous models have focused on the formation of the satellites starting from satellitesimals, the question of how satellitesimals themselves form from smaller dust particles has not yet been addressed. In this work, we study the possibility that satellitesimals form in situ in a circumplanetary disk. We calculate the radial distribution of the surface density and representative size of icy dust particles that grow by colliding with each other and drift toward the central planet in a steady circumplanetary disk with a continuous supply of gas and dust from the parent protoplanetary disk. The radial drift barrier is overcome if the ratio of the dust-to-gas accretion rates onto the circumplanetary disk, , is high and the strength of turbulence, α, is not too low. The collision velocity is lower than the critical velocity of fragmentation when α is low. Taken together, we find that the conditions for satellitesimal formation via dust coagulation are given by and . The former condition is generally difficult to achieve, suggesting that the in situ satellitesimal formation via particle sticking is viable only under extreme conditions. We also show that neither satellitesimal formation via the collisional growth of porous aggregates nor via streaming instability is viable as long as is low.

The Route to Raindrop Formation in a Shallow Cumulus Cloud Simulated by a Lagrangian Cloud Model

Abstract The mechanism of raindrop formation in a shallow cumulus cloud is investigated using a Lagrangian cloud model (LCM). The analysis is focused on how and under which conditions a cloud droplet grows to a raindrop by tracking the history of individual Lagrangian droplets. It is found that the rapid collisional growth, leading to raindrop formation, is triggered when single droplets with a radius of 20 μm appear in the region near the cloud top, characterized by large liquid water content, strong turbulence, large mean droplet size, broad drop size distribution (DSD), and high supersaturations. Raindrop formation easily occurs when turbulence-induced collision enhancement (TICE) is considered, with or without any extra broadening of the DSD by another mechanism (such as entrainment and mixing). In contrast, when TICE is not considered, raindrop formation is severely delayed if no other broadening mechanism is active. The reason for the difference is clarified by the additional analysis of idealized box simulations of the collisional growth process for different DSDs in varied turbulent environments. It is found that TICE does not accelerate the timing of the raindrop formation for individual droplets, but it enhances the collisional growth rate significantly afterward by providing a greater number of large droplets for collision. Higher droplet concentrations increase the time for raindrop formation and decrease precipitation but intensify the effect of TICE.

Collisions between Sintered Icy Aggregates

Abstract Collisions between sintered icy dust aggregates are numerically simulated. If the temperature of an icy aggregate is sufficiently high, sintering promotes molecular transport and a neck between adjacent grains grows. This growth changes the mechanical responses of the neck. We included this effect in a simulation code, and conducted collisional simulations. For porous aggregates, the critical velocity for growth, below which the mass of an aggregate increases, decreased from 50 m s−1 for the non-sintered case to 20 m s−1. For compacted aggregates, the main collisional outcome is bouncing. These results come from the fact that the strength of the neck is increased by sintering. The numerical results suggest that the collisional growth of icy grain aggregates is strongly affected by sintering.

Planetesimal Formation in the Warm, Inner Disk: Experiments with Tempered Dust

Abstract It is an open question how elevated temperatures in the inner parts of protoplanetary disks influence the formation of planetesimals. We approach this problem here by studying the tensile strength of granular beds with dust samples tempered at different temperatures. We find via laboratory experiments that tempering at increasing temperatures is correlated with an increase in cohesive forces. We studied dust samples of palagonite (JSC Mars-1a) which were tempered for up to 200 hr at temperatures between 600 and 1200 K, and measured the relative tensile strengths of highly porous dust layers once the samples cooled to room temperature. Tempering increases the tensile strength from 800 K upwards. This change is accompanied by mineral transformations, the formation of iron oxide crystallites as analyzed by Mössbauer spectroscopy, changes in the number size distribution, and the morphology of the surface visible as cracks in larger grains. These results suggest a difference in the collisional evolution toward larger bodies with increasing temperature as collisional growth is fundamentally based on cohesion. While high temperatures might also increase sticking (not studied here), compositional evolution will already enhance the cohesion and the possibility of growing larger aggregates on the way toward planetesimals. This might lead to a preferred in situ formation of inner planets and explain the observed presence of dense inner planetary systems.

Examination of an improved quasi‐stochastic model for the collisional growth of drops

AbstractThe evolution of cloud drop size distribution due to the collision‐coalescence process is generally described by a quasi‐stochastic model that solves the stochastic collection equation in a deterministic way. In this study, an improved quasi‐stochastic (IQS) model, which is derived by rigorously considering a finite model time step, is examined in the context of comparison with the normal quasi‐stochastic (NQS) model. The IQS model allows a large collector drop to collide with a small collected drop more than one time in a model time step even if the collision probability is small. The number distribution of collector drops then follows the Poisson distribution with respect to the number of collisions. Using a box model that takes turbulence‐induced collision enhancement into account, it is found that large drops in the IQS model tend to have larger sizes than those in the NQS model and that the IQS model accelerates large‐drop formation by a few minutes compared to the NQS model. The effects of the IQS model depend on the model time step and the shape of initial drop size distribution. The IQS model is incorporated into a detailed bin microphysics scheme that is coupled with the Weather Research and Forecasting model, and a single warm cloud is simulated under idealized environmental conditions. It is found that the onset of surface precipitation is accelerated in the IQS model.

Angle-Dependent Light Scattering Study of Silica Aggregate Growth in 1-D Methane/Air Flames with Hexamethyldisiloxane Admixture: Effects of Siloxane Concentration, Flame Temperature, and Equivalence Ratio

ABSTRACTSilica aggregate formation was studied in 1D premixed methane/hexamethyldisiloxane/air flames by angle-dependent light scattering measurements for various siloxane concentrations, flame temperatures, and equivalence ratios, using Guinier analysis to interpret the experimental data. A sublinear dependence of the aggregate radii of gyration of generated silica particles on residence time, and non-monotonic dependence on flame temperature with a maximum around 2000 K have been observed, with radii of gyration in the range of 10 to 120 nm. Furthermore, a lean flame environment appears to foster aggregate growth compared to rich and stoichiometric flames, in which growth is very similar. When fixing the initial conditions at the residence time corresponding to the first measurement point, a simple model describing particle evolution as a result of collisional growth and sintering predicts well the functional dependence of the growth of particle radii.

Stochasticity and predictability in terrestrial planet formation

Terrestrial planets are thought to be the result of a vast number of gravitational interactions and collisions between smaller bodies. We use numerical simulations to show that practically identical initial conditions result in a wide array of final planetary configurations. This is a result of the chaotic evolution of trajectories which are highly sensitive to minuscule displacements. We determine that differences between systems evolved from virtually identical initial conditions can be larger than the differences between systems evolved from very different initial conditions. This implies that individual simulations lack predictive power. For example, there is not a reproducible mapping between the initial and final surface density profiles. However, some key global properties can still be extracted if the statistical spread across many simulations is considered. Based on these spreads, we explore the collisional growth and orbital properties of terrestrial planets which assemble from different initial conditions (we vary the initial planetesimal distribution, planetesimal masses, and giant planet orbits). Confirming past work, we find that the resulting planetary systems are sculpted by sweeping secular resonances. Configurations with giant planets on eccentric orbits produce fewer and more massive terrestrial planets on tighter orbits than those with giants on circular orbits. This is further enhanced if the initial mass distribution is biased to the inner regions. In all cases, the outer edge of the system is set by the final location of the $\nu_6$ resonance and we find that the mass distribution peaks at the $\nu_5$ resonance. Using existing observations, we find that extrasolar systems follow similar trends. Although differences between our numerical modelling and exoplanetary systems remain, we suggest that CoRoT-7, HD 20003, and HD 20781 may host undetected giant planets.

FROM PLANETESIMALS TO PLANETS IN TURBULENT PROTOPLANETARY DISKS. I. ONSET OF RUNAWAY GROWTH

ABSTRACT When planetesimals grow via collisions in a turbulent disk, stirring through density fluctuation caused by turbulence effectively increases the relative velocities between planetesimals, which suppresses the onset of runaway growth. We investigate the onset of runaway growth in a turbulent disk through simulations that calculate the mass and velocity evolution of planetesimals. When planetesimals are small, the average relative velocity between planetesimals, v r ?> , is much greater than their surface escape velocity, v esc ?> , so that runaway growth does not occur. As planetesimals become large via collisional growth, v r ?> approaches v esc ?> . When v r ≈ 1.5 v esc ?> , runaway growth of the planetesimals occurs. During the oligarchic growth subsequent to runaway growth, a small number of planetary embryos produced via runaway growth become massive through collisions with planetesimals with radii of that at the onset of runaway growth, r p,run ?> . We analytically derive r p,run ?> as a function of the turbulent strength. Growing ∼ 10 M ⊕ ?> embryos that are suitable to become the cores of Jupiter and Saturn requires r p,run ∼ 100 ?> km, which is similar to the proposed fossil feature in the size distribution of main belt asteroids. In contrast, the formation of Mars as quickly as suggested from Hf-W isotope studies requires small planetesimals at the onset of runaway growth. Thus, the conditions required to form Mars, Jupiter, and Saturn and the size distribution of the main-belt asteroids indicate that the turbulence increased in amplitude relative to the sound speed with increasing distance from the young Sun.

ELECTRON HEATING IN MAGNETOROTATIONAL INSTABILITY: IMPLICATIONS FOR TURBULENCE STRENGTH IN THE OUTER REGIONS OF PROTOPLANETARY DISKS

ABSTRACT The magnetorotational instability (MRI) drives vigorous turbulence in a region of protoplanetary disks where the ionization fraction is sufficiently high. It has recently been shown that the electric field induced by the MRI can heat up electrons and thereby affect the ionization balance in the gas. In particular, in a disk where abundant dust grains are present, the electron heating causes a reduction of the electron abundance, thereby preventing further growth of the MRI. By using the nonlinear Ohm's law that takes into account electron heating, we investigate where in protoplanetary disks this negative feedback between the MRI and ionization chemistry becomes important. We find that the “e-heating zone,” the region where the electron heating limits the saturation of the MRI, extends out up to 80 AU in the minimum-mass solar nebula with abundant submicron-sized grains. This region is considerably larger than the conventional dead zone whose radial extent is ∼20 AU in the same disk model. Scaling arguments show that the MRI turbulence in the e-heating zone should have a significantly lower saturation level. Submicron-sized grains in the e-heating zone are so negatively charged that their collisional growth is unlikely to occur. Our present model neglects ambipolar and Hall diffusion, but our estimate shows that ambipolar diffusion would also affect the MRI in the e-heating zone.

The Role of the Collisions on the Weibel Instability Growth Rate in the Fast Ignition Scenario

In this paper, the effects of Coulomb elctron-ion collisions and plasma density gradient, $$\eta$$ , on the Weibel instability in inertial confinement fusion are investigated. The results are indicative that the corrected collision of Weibel instability growth rate of the relativistic region near the corona, $$\eta >0.3$$ , increases with increasing relativistic parameter, $$\upgamma$$ . Also, near the fuel core as $$\eta$$ goes down, the corrected collisional growth rate decreases with increasing $$\upgamma$$ for $$\mathrm{{\upgamma }}<6$$ and and increases with increasing $$\upgamma$$ for $$\upgamma >6$$ . Therefore, for $$\upgamma <6$$ , the effect of collision and fuel density gradient tend to stabilize the Weibel instability in fuel core, with the Weibel growth rate below the collisionless value. Also deposition condition of relativistic electron beam energy can be shifted to the fuel core for the suitable ignition.

Lagrangian Tracking Simulation of Droplet Growth in Turbulence–Turbulence Enhancement of Autoconversion Rate

The authors describe the Lagrangian cloud simulator (LCS), which simulates droplet growth in air turbulence. The LCS adopts the Euler–Lagrangian framework and can provide reference data for cloud microphysical models by tracking the growth of particles individually. The collisional growth in a stagnant flow is calculated by the LCS and also by solving the stochastic collision–coalescence equation (SCE). Good agreement is obtained between the LCS and SCE simulations. Comparisons between the results for stagnant and turbulent flows confirm that in-cloud turbulence enhances collisional growth. The enhancement is well predicted by the SCE method if a proper collision model is employed. To quantify the enhancement, the paper defines the time scale of the autoconversion process, in which cloud droplets grow into raindrops through collisions, as the time taken for 10% of the cloud to become rain (t10%). The authors then define the turbulence enhancement factor Eturb as [Formula: see text], where the overbar denotes the mean value of the LCS runs and the subscripts NoT and T indicate stagnant (nonturbulent) flow and turbulent flow simulations, respectively. It was found that the enhancement factor increases linearly with the energy dissipation rate, while it does not show a consistent dependence on the Reynolds number. The levels of statistical fluctuations in the autoconversion time scales were directly obtained for the first time. It is shown that the relative standard deviation of t10% simply follows the power law that the binomial distribution theory predicts, independently of the flow conditions.

Orbital evolution of planetesimals in gaseous disks

Planets are formed from collisional growth of small bodies in a protoplanetary disk. Bodies much larger than approximately 1 m are mainly controlled by the gravity of the host star and experience weak gas drag; their orbits are mainly expressed by orbital elements: semimajor axes a, eccentricities e, and inclinations i, which are modulated by gas drag. In a previous study, $\dot a$ , $\dot e$ , and $\dot i$ were analytically derived for e≪1 and i≪H/a, where H is the scale height of the disk. Their formulae are valid in the early stage of planet formation. However, once massive planets are formed, e and i increase greatly. Indeed, some small bodies in the solar system have very large e and i. Therefore, in this paper, I analytically derive formulae for $\dot a$ , $\dot e$ , and $\dot i$ for 1−e 2≪1 and i≪H/a and for i≫H/a. The formulae combined from these limited equations will represent the results of orbital integration unless e≥1 or i>π−H/a. Since the derived formulae are applicable for bodies not only in a protoplanetary disk but also in a circumplanetary disk, I discuss the possibility of the capture of satellites in a circumplanetary disk using the formulae.

Distributional correspondence of 94-GHz radar reflectivity with the variation in water cloud properties over the northwestern Pacific and China

This paper studied the behavior of 94-GHz radar reflectivity (Ze) with variation in the properties of low-level water clouds, such as the effective droplet radius (re), geometrical thickness (Dcld), and liquid water path (LWP), over the northwest Pacific and China. The changes in the distribution of maxZe (the largest Ze within a cloud layer) were examined in terms of variation in the cloud parameters such as small, mid and large categories, while maxZe had monomodal distributions regarding variation in re and Dcld, that appeared bimodal in the small category of LWP. It was confirmed that the small category of LWP contained both non-precipitating clouds in the incipient stage and raining clouds in the dissipating stage. Next, optically measured particle size was combined with LWP derived from the microwave measurement to classify the precipitation type. Applying maxZe and Dcld to the analysis of classified precipitation types corroborated the importance of Dcld for examining the occurrence of precipitation. Finally, the position of maxZe relative to the cloud top was investigated using a measure of the probability of precipitation (POP) according to variation in re. The results showed that the Pacific and China had ‘bow’ and ‘funnel’ shapes, respectively. The emergence of these shapes according to the variation in re was interpreted as the enhancement of Ze due to droplet collisional growth and the attenuation of Ze by the presence of large particles. Furthermore, a detailed analysis of smaller particles (<10μm in radius) reinforced the idea of rapid, efficient particle growth in the lower part of the cloud.

TURBULENCE-INDUCED RELATIVE VELOCITY OF DUST PARTICLES. III. THE PROBABILITY DISTRIBUTION

Motivated by its important role in the collisional growth of dust particles in protoplanetary disks, we investigate the probability distribution function (PDF) of the relative velocity of inertial particles suspended in turbulent flows. Using the simulation from our previous work, we compute the relative velocity PDF as a function of the friction timescales, tau_p1 and tau_p2, of two particles of arbitrary sizes. The friction time of particles included in the simulation ranges from 0.1 tau_eta to 54T_L, with tau_eta and T_L the Kolmogorov time and the Lagrangian correlation time of the flow, respectively. The relative velocity PDF is generically non-Gaussian, exhibiting fat tails. For a fixed value of tau_p1, the PDF is the fattest for equal-size particles (tau_p2~tau_p1), and becomes thinner at both tau_p2<tau_ p1 and tau_p2>tau_p1. Defining f as the friction time ratio of the smaller particle to the larger one, we find that, at a given f in 1/2<f<1, the PDF fatness first increases with the friction time, tau_ph, of the larger particle, peaks at tau_ph~tau_eta, and then decreases as tau_ph increases further. For 0<f<1/4, the PDF shape becomes continuously thinner with increasing tau_ph. The PDF is nearly Gaussian only if tau_ph is sufficiently large (>>T_L). These features are successfully explained by the Pan & Padoan model. Using our simulation data and some simplifying assumptions, we estimated the fractions of collisions resulting in sticking, bouncing, and fragmentation as a function of the dust size in protoplanerary disks, and argued that accounting for non-Gaussianity of the collision velocity may help further alleviate the bouncing barrier problem.