tokens: on the Reynolds number. We select the dependence appropriate for spherical grains. For better comparison of the coupling between dust particles with a certain size and the disk gas, the dimensionless Stokes number is defined asParticles with Stokes number of unity come to match the gas velocity in one local orbital period. For a wide range of gas densities, submicrometer dust grains have and hence come quickly to rest in the gas reference frame. The dust scale height ( h d ) in a turbulent protoplanetary nebula can be calculated following Dubrulle et al. ( 1995 ) and Youdin Lithwick ( 2007 ):For tightly coupled particles with , h d ≈ h g , and we approximate the dust scale height by that of the gas at t = 0. To model the dust size distribution at t = 0, we adopt a grain-size distribution with an MRN (Mathis et al. 1977 ) power-law index, , where N ( a ) da is the number of dust particles of radii between . We consider that dust grains already grow beyond the ISM size in the molecular cloud phase (Suttner Yorke 2001 ) and adopt the maximum and minimum sizes of the dust size distribution at t = 0 as a max = 1.0 μ m and a min = 0.1 μ m. To begin our grain growth and settling simulation, we make two assumptions: 1. Gas and dust of all sizes are dynamically coupled and well mixed at t = 0, with (this will not be true at later times); 2. The dust to gas mass ratio is η = 0.01, similar to the ISM. 3. Dust Evolution Model The main objective of this work is to obtain the disk temperature profile, which is controlled by the dust opacity. To simulate dust evolution, we introduce a hybrid model in which the collisional dust growth is implemented through a weighted Monte Carlo method along with a Lagrangian Monte Carlo prescription for vertical settling and diffusion. 3.1. Collision Model The outcome of a modeled collision between two dust particles in a protoplanetary environment has many possibilities according to laboratory experiments. On the experimental side, Güttler et al. ( 2010 ) presented 19 possible collisional outcomes for particles with various mass ratios, speeds, and porosities. However, it is prohibitively computationally expensive to include all possibilities in a global disk model. For simplicity, we adopt a collisional model that includes only sticking and fragmentation. We treat collisions as a binary process, identifying the smaller mass as the projectile and the bigger mass as the target . The collision outcome is determined by the relative velocity (see Section 3.2 for a description of our velocity computation). If the particles collide with a velocity less than a threshold velocity v frag , they stick and form a new particle with mass . When , the collision results in fragmentation. As particles grow by sticking, their eddy-crossing times drop, leading to lower coupling with the gas and higher collision speeds (Ormel Cuzzi 2007 ). When the collision speed reaches v frag , instead of sticking, both particles fragment. For such an
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