Length segregation in mixtures of spherocylinders induced by imposed topological defects

We explore length segregation in binary mixtures of spherocylinders of lengths $L_1$ and $L_2$ with the same diameter $D$ which are tangentially confined on a spherical surface of radius $R$. The orientation of spherocylinders is constrained along an externally imposed direction field on the sphere which is either along the longitude or the latitude lines of the sphere. In both situations, integer orientational defects at the poles are imposed. We show that these topological defects induce a complex segregation picture also depending on the length ratio factor $\gamma$=$L_2/L_1$ and the total packing fraction $\eta$ of the spherocylinders. When the binary mixture is aligned along longitudinal lines of the sphere, shorter rods tend to accumulate at the topological defects of the polar caps whereas longer rods occupy central equatorial area of the spherical surface. In the reverse case of latitude ordering, a state can emerge where longer rods are predominantly both in the cap and in the equatorial areas and shorter rods are localized in between. As a reference situation, we consider a defect-free situation in the flat plane and do not find any length segregation there at similar $\gamma$ and $\eta$, hence the segregation is purely induced by the imposed topological defects. It is also revealed that the shorter rods at $\gamma$=4 and $\eta \ge$0.5 act as obstacles to the rotational relaxation of the longer rods when all orientational constraints are released.