In this paper we analyze the narrow capture problem for a single Brownian particle
diffusing in a three-dimensional (3D) bounded domain containing a set of small, spherical traps. The
boundary surface of each trap is taken to be a semipermeable membrane. That is, the continuous flux
across the interface is proportional to an associated jump discontinuity in the probability density.
The constant of proportionality is identified with the permeability \kappa . In addition, we allow for
discontinuities in the diffusivity and chemical potential across each interface; the latter introduces a
directional bias. We also assume that the particle can be absorbed (captured) within the interior of
each trap at some Poisson rate \gamma . In the small-trap limit, we use matched asymptotics and Green's
function methods to calculate the splitting probabilities and unconditional mean first passage time
(MFPT) to be absorbed by one of the traps. However, the details of the analysis depend on how
various parameters scale with the characteristic trap radius \epsilon . Under the scalings \gamma = O(1/\epsilon 2
) and
\kappa = O(1/\epsilon ), we show that the semipermeable membrane reduces the effective capacitance \scrC of each
spherical trap compared to the standard example of totally absorbing traps. The latter case is
recovered in the dual limits \gamma \rightarrow \infty and \kappa \rightarrow \infty , with \scrC equal to the intrinsic capacitance of a sphere,
namely, the radius. We also illustrate how the asymptotic expansions are modified when \gamma = O(1/\epsilon )
(slow absorption) or \kappa = O(1) (low permeability). Finally, we consider the unidirectional limit in
which each interface only allows particles to flow into a trap. The traps then act as partially absorbing
surfaces with a constant reaction rate \kappa . Combining asymptotic analysis with the encounter-based
formulation of partially reactive surfaces, we show how a generalized surface absorption mechanism
(non-Markovian) can be analyzed in terms of the capacitances \scrC . We thus establish that a wide
range of narrow capture problems can be characterized in terms of the effective capacitances of the
traps.