  We study the exciton localization and resulting optical response for
disordered tubular aggregates of optically active molecules. It has been shown
previously that such tubular structures allow for excitons delocalized over
more than a thousand molecules, owing to the combined effects of long-range
dipole-dipole interactions and the higher-dimensional (not truly
one-dimensional) nature of the aggregate. Such large delocalization sizes
prompt the question to what extent in experimental systems the delocalization
may still be determined by the aggregate size (diameter and length) and how
this affects the aggregate's optical response and dynamics. We perform a
systematic study of the size effects on the localization properties, using
numerical simulations of the exciton states in a cylindrical model structure
inspired by the previously derived geometry of a cylindrical aggregate of
cyanine dye molecules (C8S3). To characterize the exciton localization, we
calculate the participation ratio and the autocorrelation function of the
exciton wave function. Also, we calculate the density of states and absorption
spectrum. We find strong effects of the tube's radius on the localization and
optical properties in the range of parameters relevant to experiment. In
addition, surprisingly, we find that even for tubes as long as 750 nm, the
localization size is limited by the tube's length for disorder values that are
relevant to experimental circumstances, while observable effects of the tube's
length in the absorption spectrum still occur for tube lengths up to about 150
nm. The latter may explain changes in the optical spectra observed during the
aging process of bromine-substituted C8S3 aggregates. For weak disorder, the
exciton wave functions exhibit a scattered, fractal-like nature, similar to the
quasi-particles in two-dimensional disordered systems.
