  The size of the smallest detectable sample feature in an x-ray imaging system
is usually restricted by the spatial resolution of the system. This limitation
can now be overcome using the diffusive dark-field signal, which is generated
by unresolved phase effects or the ultra-small-angle x-ray scattering from
unresolved sample microstructures. A quantitative measure of this dark-field
signal can be useful in revealing the microstructure size or material for
medical diagnosis, security screening and materials science. Recently, we
derived a new method to quantify the diffusive dark-field signal in terms of a
scattering angle using a single-exposure grid-based approach. In this
manuscript, we look at the problem of quantifying the sample microstructure
size from this single-exposure dark-field signal. We do this by quantifying the
diffusive dark-field signal produced by 5 different sizes of polystyrene
microspheres, ranging from 1.0 $\mu$m to 10.8 $\mu$m, to investigate how the
strength of the dark-field signal changes with the sample microstructure size,
$S$. We also explore the feasibility of performing single-exposure dark-field
imaging with a simple equation for the optimal propagation distance given
microstructure with a specific size and thickness, and successfully verify this
equation with experimental data. Our theoretical model predicts that the
dark-field scattering angle is inversely proportional to $\sqrt{S}$, which is
consistent with our experimental data.
