We theoretically study the dynamic polarization of lattice nuclear spins in
GaAs double quantum dots containing two electrons. In our prior work [Phys.
Rev. Lett. 104, 226807 (2010)] we identified three regimes of long-term
dynamics, including the build up of a large difference in the Overhauser fields
across the dots, the saturation of the nuclear polarization process associated
with formation of so-called "dark states," and the elimination of the
difference field. In particular, when the dots are different sizes we found
that the Overhauser field becomes larger in the smaller dot. Here we present a
detailed theoretical analysis of these problems including a model of the
polarization dynamics and the development of a new numerical method to
efficiently simulate semiclassical central-spin problems. When nuclear spin
noise is included, the results agree with our prior work indicating that large
difference fields and dark states are stable configurations, while the
elimination of the difference field is unstable; however, in the absence of
noise we find all three steady states are achieved depending on parameters.
These results are in good agreement with dynamic nuclear polarization
experiments in double quantum dots.