  Homogeneous suspensions of red blood cells (RBCs or erythrocytes) in blood
plasma are unstable in the absence of driving forces and form elongated stacks,
called rouleau. These erythrocyte aggregates are often branched porous networks
-- a feature that existing red blood cell aggregation models and simulations
fail to predict exactly. Here we establish that alignment-dependent attractive
forces in a system of dimers can precisely generate branched structures similar
to RBC aggregates observed under a microscope. Our simulations consistently
predict that the growth rate of typical mean rouleau size remains sub-linear --
a hallmark from past studies -- which we also confirm by deriving a reaction
kernel taking into account appropriate collision cross-section, approach
velocities, and an area-dependent sticking probability. The system exhibits
unique features such as the existence of percolated and/or single giant cluster
states, multiple coexisting mass-size scalings, and transition to a branched
phase upon fine-tuning of model parameters. Upon decreasing the depletion
thickness we find that the percolation threshold increases and the morphology
of the structures opens up towards an increased degree of branching. Remarkably
the system self-organizes to produce a universal power-law size distribution
scaling irrespective of the model parameters.
