PHYSICAL REVIEW E 95, 013103 (2017) Friction factor for aerosol fractal aggregates over the entire Knudsen range James Corson, 1 George W. Mulholland, 1 and Michael R. Zachariah 1, 2, * 1 Department of Chemical and Biomolecular Engineering, University of Maryland, College Park, Maryland 20742, USA 2 Department of Chemistry and Biochemistry, University of Maryland, College Park, Maryland 20742, USA (Received 29 September 2016; published 4 January 2017) We develop an approach for computing the hydrodynamic friction tensor and scalar friction coefficient for an aerosol fractal aggregate in the transition regime. Our approach involves solving the Bhatnagar-Gross-Krook equation for the velocity field around a sphere and using the velocity field to calculate the force on each primary sphere in the aggregate due to the presence of the other spheres. It is essentially an extension of Kirkwood-Riseman theory from the continuum flow regime to the entire Knudsen range (Knudsen number from 0.01 to 100 based on the primary sphere radius). Our results compare well to published direct simulation Monte Carlo results, and they converge to the correct continuum and free molecule limits. Our calculations for clusters with up to 100 spheres support the theory that aggregate slip correction factors collapse to a single curve when plotted as a function of an appropriate aggregate Knudsen number. This self-consistent-field approach calculates the friction coefficient very quickly, so the approach is well-suited for testing existing scaling laws in the field of aerosol science and technology, as we demonstrate for the adjusted sphere scaling method. DOI: 10.1103/PhysRevE.95.013103 I. INTRODUCTION Aerosol fractal aggregates formed from the coagulation of smaller, spherical primary particles are found in many natural and industrial settings. Understanding the forces on these aggregates is important in a number of science and engineering disciplines, including combustion, fire safety, atmospheric and environmental sciences, materials engineering [1], and nuclear reactor safety [2]. The translational drag force for a particle moving slowly relative to the surrounding fluid—given by F =−ζ U 0 , where U 0 is the particle’s relative velocity and ζ is the orientation-averaged scalar friction factor—is particularly important because it influences the transport properties of the particle, including its diffusion coefficient and electrical mobility. In many practical applications, the primary sphere radius a is significantly less than the mean free path of the surrounding gas (λ ≈ 65 nm at standard temperature and pressure and an order of magnitude higher near a flame), so that the primary sphere is in or near the free molecule flow regime. At the same time, the radius of gyration R g for the agglomerate may be comparable to or larger than the mean free path, so that the aggregate is in the transition flow regime. As one example, for carbonaceous soot a ≈ 5–30 nm and R g ≈ 30–1000 nm. There are a number of theories and techniques for com- puting the translational friction factor of macromolecules and particle aggregates in the continuum regime, including Kirkwood-Riseman (KR) theory [3] and its extensions by Rotne and Prager [4], Yamakawa [5], and Chen et al. [6], as well as algorithms that use the Hubbard and Douglas analogy between the electrostatic capacitance and the friction factor [7–9]. Likewise, there are established methods for computing ζ in the free molecule regime that simulate the ballistic nature of interactions between gas molecules and aggregates [10–13]. In contrast, there are few approaches for the transition regime. Melas et al. [14] estimated the friction coefficient * mrz@umd.edu in the near-continuum regime by solving the Laplace equation with a slip boundary condition at the surface of the particle. In a followup paper, the authors determined that their collision rate method is valid for Knudsen numbers less than 2 [15]. Dahneke [16] developed the adjusted sphere method for the transition regime, which applies a slip correction factor to the continuum friction factor. The key to this development is the identification of an aggregate Knudsen number that reduces a problem involving two length scales (primary radius and aggregate radius of gyration) to a single dimensionless length. Dahneke’s approach is similar to the approach used to calculate the drag on a sphere in the transition regime, but the adjusted sphere method uses an adjusted Knudsen number based on geometric descriptions of the particle in the continuum (hydrodynamic radius, R H ) and free molecular [projected area (PA)] regimes. Through scaling analysis, Zhang et al. [17] developed an approach analogous to the adjusted sphere method and demonstrated that the approach yields friction factors com- parable to direct simulation Monte Carlo (DSMC) results for the aggregates they studied (spheres, dimers, and dense and open 20-particle aggregates). However, it requires knowledge of the hydrodynamic radius and the projected area of the particle, which may take tens of minutes to a few hours to obtain computationally for a single particle. Obtaining R H and PA experimentally is possible, but it requires painstaking transmission electron microscopy (TEM) measurements [18]. More rigorous computational techniques for calculating the transition regime friction factor—such as DSMC or molecular dynamics—are time-consuming: for instance, the reported DSMC calculation times in Ref. [17] were on the order of one CPU week for a given Knudsen number and a given aggregate. Thus, a self-consistent-field theory method for quickly estimating the scalar friction factor of an aggregate across the Knudsen range is highly desirable. In this paper, we present an approach for computing the hydrodynamic friction tensor H and the scalar friction coefficient ζ for fractal aggregates across the entire Knudsen range. Our approach involves solving for the velocity field 2470-0045/2017/95(1)/013103(6) 013103-1 ©2017 American Physical Society