Boundary Homogenization for Periodic Arrays of Absorbers

Abstract

We introduce a homogenization procedure for reaction-diffusion equations in domains whose boundary consists of small alternating regions with prescribed Dirichlet and Neumann data of comparable areas. The homogenized problem is shown to satisfy an effective Dirichlet boundary condition which depends on the geometry of the small-scale boundary structure. This problem is also related to finding the effective trapping rate for a Brownian particle next to a surface with a periodic array of perfect absorbers. We use the method of optimal geometric grids to numerically solve the unit cell problem of homogenization. The geometric homogenization factor is obtained for a number of cell geometries (stripes, square and hexagonal arrays of disk-shaped absorbers or emitters) as a function of the surface area fraction occupied by the absorbers. Empirical analytical expressions that give excellent fits to data for the entire range of area fractions and correct asymptotic behaviors in the limits of small and large absorber area fractions are proposed.

First Passage Statistics for the Capture of a Brownian Particle by a Structured Spherical Target with Multiple Surface Traps

“…The leading term in (1.11) agrees with the Berg-Purcell result (1.10b), while the 8d 1 √ f/π term theoretically explains one of the parameter fittings in [3,34]. A further discussion of this result and a comparison to other results in the literature, notably those in [2,3,34,51], are discussed in section 4.…”

“…We now compare our scaling law for κ in (4.11) with some other results in the literature [4,3,34,51] derived from either physical considerations or parameter fittings. In dimensionless form, the Berg-Purcell result (cf.…”

“…A further discussion of this result and a comparison to other results in the literature, notably those in [2,3,34,51], are discussed in section 4. Finally, in section 7 we briefly summarize our main results and discuss a few open problems worthy of further study.…”

“…More detailed studies [45,51,2,3,34] and fittings of the leakage parameter κ in a boundary homogenization procedure for nanotraps on either the sphere or on a flat plane are discussed in section 4. By taking the limit N 1 in our expression for the capacitance C 0 for a uniformly distributed configuration of nanotraps of a common radius σ, our result in section 4 for the low surface nanotrap coverage limit f = O(−σ 2 log σ) 1, and when cast in dimensional form, predicts that…”

“…For a periodic arrangement of nanotraps on the surface of an infinite planar boundary, in [3,34] the dimensionless leakage κ for the corresponding half-space version of (1.5) was closely fitted to full numerical results by (4.14)…”