The Single-Fiber Collision Rate and Filtration Efficiency for Nanoparticles I: The First-Passage Time Calculation Approach

We describe an approach to filtration-efficiency calculations as an alternative to the traditional depth filtration theory. The new approach involves linking the single-fiber efficiency to the collision rate coefficient/kernel between nanoparticles and fibers, and correspondingly inferring the collision kernel via dimensionless mean first-passage time (MFPT) calculations. This method has the advantage of easily incorporating the influences of particle diffusion, inertia, and particle size; therefore, all filtration mechanisms can be considered simultaneously. Through non-dimensionalization of the equation of motion for a particle in MFPT calculations (the Langevin equation), it is shown that both the single-fiber efficiency E_f and dimensionless particle-fiber collision kernel, H, are functions of the ratio of particle radius to filter-fiber radius, R, the solid volume fraction in the filter, V_f, the ratio of particle persistence distance to the particle-filter collision distance, Kn_D (the diffusive Knudsen number), and the ratio of the particle translational kinetic energy to the thermal energy χ_f. Using a Kuwabara flow-cell model to define the geometry and flow field, MFPT calculations are used to determine H and E_f for nanoparticles in atmospheric pressure systems, i.e., when particle inertia is negligible but when diffusion and interception act in tandem to collect particles. From MFPT results, regression equations for both H and E_f are developed. A comparison is made between MFPT results and commonly invoked depth-filtration single-fiber efficiency relationships, experimentally measured values, and H equations derived from Sherwood number correlations based upon measurements of heat transfer from a fluid flowing perpendicular to an array of cylinders. Good agreement is found with both measurements and previously developed equations over a wide range of parameter space.

Copyright 2014 American Association for Aerosol Research     

Radical polymerization of methyl methacrylate (MMA) initiated by KHSO5 – BTBAC system - a kinetic study

Kinetics of free radical polymerization of methyl methacrylate using potassium peroxomonosulfate as initiator in the presence of benzyltributylammonium chloride (BTBAC) as phase transfer catalyst was studied. The polymerization reactions were carried out under nitrogen atmosphere and unstirred conditions at a constant temperature of 60°C in ethyl acetate/water bi-phase system. The role of concentrations of monomer, initiator, catalyst, temperature, acid and ionic strength on the rate of polymerization (R_p) was ascertained. The orders with respect to monomer, initiator and phase transfer catalyst were found to be 1.5, 0.5 and 0.5 respectively. The rate of polymerization (R_p) is independent of ionic strength and pH. Based on the kinetic results, a suitable mechanism is proposed.     

Langevin Simulation of Aggregate Formation in the Transition Regime

We study the aggregation of monomers in an aerosol via non-dimensional Langevin simulations, in which particles remain in point contact upon collision, and report the hydrodynamic radii and projected areas of the formed aggregates with less than 300 primary particles. Unique from prior studies, in examining aggregation we monitor the evolution of the distributions of two Knudsen numbers: the traditional Knudsen number (Kn) and the diffusive Knudsen number (Kn_D), which both shift to smaller mean values as aggregation proceeds. As Kn transitions from large to small values, momentum transfer changes from a free molecular to a continuum process; analogously, as Kn_D decreases, aggregation is altered from occurring ballistically to diffusively in a dilute system. During simulations, the change in drag coefficient with both changing Kn and changing aggregate structure is accounted for. We find that as compared to completely coalescing particles (spheres), non-coalescing aggregates with the same initial Kn and Kn_D have Kn_D values, which decrease more rapidly due to aggregation; hence, aggregates are more likely to collide with one another diffusively when compared with their spherical counterparts of the same Kn distribution. Further, we find that aggregation with evolving Knudsen numbers does not lead to strong scaling between the number of monomers in a formed aggregate and the aggregate radius of gyration for aggregates composed of 300 or fewer primary particles. In spite of this, aggregate hydrodynamic radii and orientationally averaged projected areas are found to scale well with the number of monomers per aggregate.

Copyright 2015 American Association for Aerosol Research     

Determination of the Scalar Friction Factor for Nonspherical Particles and Aggregates Across the Entire Knudsen Number Range by Direct Simulation Monte Carlo (DSMC)

The friction factor of an aerosol particle depends upon the Knudsen number (Kn), as gas molecule–particle momentum transfer occurs in the transition regime. For spheres, the friction factor can be calculated using the Stokes–Millikan equation (with the slip correction factor). However, a suitable friction factor relationship remains sought-after for nonspherical particles. We use direct simulation Monte Carlo (DSMC) to evaluate an algebraic expression for the transition regime friction factor that is intended for application to arbitrarily shaped particles. The tested friction factor expression is derived from dimensional analysis and is analogous to Dahneke's adjusted sphere expression. In applying this expression to nonspherical objects, we argue for the use of two previously developed drag approximations in the continuum (Kn → 0) and free molecular (Kn → ∞) regimes: the Hubbard–Douglas approximation and the projected area (PA) approximation, respectively. These approximations lead to two calculable geometric parameters for any particle: the Smoluchowski radius, R _S, and the projected area, PA. Dimensional analysis reveals that Kn should be calculated with PA/πR _S as the normalizing length scale, and with Kn defined in this manner, traditional relationships for the slip correction factor should apply for arbitrarily shaped particles. Furthermore, with this expression, Kn-dependent parameters, such as the dynamic shape factor, are readily calculable for nonspherical objects. DSMC calculations of the orientationally averaged drag on spheres and test aggregates (dimers, and open and dense 20-mers) in the range Kn = 0.05–10 provide strong support for the proposed method for friction factor calculation in the transition regime. Experimental measurements of the drag on aggregates composed of 2–5 primary particles further agree well with DSMC results, with differences of less than 10% typically between theoretical predictions, numerical calculations, and experimental measurements.

Copyright 2012 American Association for Aerosol Research     

Determination of the Lateral Dimension of Graphene Oxide Nanosheets Using Analytical Ultracentrifugation

In this paper, a method to determine the lateral dimensions of 2D nanosheets directly in suspension by analytical ultracentrifugation (AUC) is shown. The basis for this study is a well‐characterized and stable dispersion of graphene oxide (GO) monolayers in water. A methodology is developed to correlate the sedimentation coefficient distribution measured by AUC with the lateral size distribution of the 2D GO nanosheets obtained from atomic force microscopy (AFM). A very high accuracy can be obtained by virtue of counting several thousand sheets, thereby minimizing any coating effects or statistical uncertainties. The AFM statistics are further used to fit the lateral size distribution obtained from the AUC to determine the unknown hydrodynamic sheet thickness or density. It is found that AUC can derive nanosheet diameter distributions with a relative error of the mean sheet diameter of just 0.25% as compared to the AFM analysis for 90 mass% of the particles in the distribution. The standard deviation of the size‐dependent error for the total distribution is found to be 3.25%. Based on these considerations, an expression is given to calculate the cut size of 2D nanosheets in preparative centrifugation experiments.     

The Single-Fiber Collision Rate and Filtration Efficiency for Nanoparticles II: Extension to Arbitrary-Shaped Particles

We extend the equations for the dimensionless collision kernel and filtration efficiency, attained previously via mean first-passage time (MFPT) calculations, to particles of arbitrary shape. Specifically, we show that the regression equations for the dimensionless collision rate found considering particle-fiber collisions driven by simultaneous diffusion and interception remain valid for non-spherical particles, provided that an appropriate collision length scale for the non-spherical particle (L) is defined and incorporated into the definitions of the dimensionless collision rate (H) and the diffusive Knudsen number (Kn_D). Regression equations are provided to calculate this length scale for quasifractal aggregates of varying fractal dimension, as well as cylinders. MFPT calculations reveal that, over ～5 orders of magnitude in H, these regression equations for the collision length are valid. Furthermore, using the previously attained proportionality between the predicted dimensionless collision rate and the single-fiber efficiency, comparison is made between the equations presented here and measurements of the penetration of both multiwalled carbon nanotubes and quasifractal aggregates through fibrous filters. Reasonable agreement is found between measured and predicted single-fiber efficiencies in both circumstances, supporting the use of the single-fiber efficiency calculation approach we developed.

Copyright 2014 American Association for Aerosol Research