## Abstract

Brownian particles transported in the vicinity of solid walls experience short-range dispersion forces which must be incorporated into the particle transport equatin written at the microscale (i.e., describing phenomena on a length scale capable of resolving distances comparable with the effective range of such forces). However, the majority of problems concerned with particle transport and deposition on the wall surfaces rather describe these processes from the macroscale or coarse-scale viewpoint, on which length scale the dispersion forces are irresolvable, and hence explcitly absent from the governing equations. Yet, the latter forces affect particle deposition rates in an indirect manner, i.e., implicitly in the form of a "fictitious" boundary condition imposed upon the macroscale particle concentration at the wall. This paper develops a rigorous singular perturbation solution scheme for deriving this "apparent" boundary condition. The scheme utilizes the fact that the exact, microscale particle transport problem is characterized by two disparate length scales: one, l, associated with the effective range of action of the specific dispersion force considered; the other, L, beign the system's macroscopic linear dimension (typically characterizing the size of the external boundaries circumscribing the system). As such, the analysis employs a matched asymptotic expansion solution scheme for the particle micro- and macroscale concentration in terms of the small parameter δ = l/L ≤ 1. The form of the effective macroscale boundary condition is shown to depend not only upon the parameter δ but also upon the magnitude of the depth of the primary minimum (made dimensionless with kT) exhibited by the potential of the specific dispersion force considered. Accordingly, the form of this effective boundary condition was determined for three distinct cases, these being respectively characterized by "shallow," "eep,"; and "very deep" potential energy minima. In the case of a deep minimum the effective macroscale boundary condition is shown to possess the form of a force-free particle surfce-transport equation formulated in terms of a surface-excess particle concentration, the latter being related to the bulk concentration at the wall by a linear adsorption isotherm. In the case of aerosol particles transported in the vicinity of a solid collector wall the interaction is mainly of the London-van der Waals type, which produces a very deep potential energy minimum. In this case the effective boundary condition imposed upon the macroscale particle concentration at the wall possesses the form of a "perfect-sink" boundary condition, one which is conventionally employed in the problem of aerosol filtration by porous filters. Finally, the case of hydrosol particles was studied, involving the existence of potential energy barriers in addition to the preceding potential energy minima. Here, the effective "first-order, irreversible, surface reaction" boundary condition at the wall, previously obtained by E. Ruckenstein and D. C. Prieve (J. Chem. Soc. Faraday II 69, 1522 (1973)) and Spielman and Friedlander (J. Colloid Interface Sci. 46, 22 (1974)), is rigorously derived and its range of validity is carefully delineated.

Original language | English |
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Pages (from-to) | 552-573 |

Number of pages | 22 |

Journal | Journal of Colloid and Interface Science |

Volume | 136 |

Issue number | 2 |

DOIs | |

State | Published - May 1990 |

Externally published | Yes |