  Colloidal patchy particles with divalent attractive interaction can
self-assemble into linear polymer chains. Their equilibrium properties in 2D
and 3D are well described by Wertheim's thermodynamic perturbation theory which
predicts a well-defined exponentially decaying equilibrium chain length
distribution. In experimental realizations, due to gravity, particles sediment
to the bottom of the suspension forming a monolayer of particles with a
gravitational height smaller than the particle diameter. In accordance with
experiments, an anomalously high monomer concentration is observed in
simulations which is not well understood. To account for this observation, we
interpret the polymerization as taking place in a highly confined quasi-2D
plane and extend the Wertheim thermodynamic perturbation theory by defining
addition reactions constants as functions of the chain length. We derive the
theory, test it on simple square well potentials, and apply it to the
experimental case of synthetic colloidal patchy particles immersed in a binary
liquid mixture that are described by an accurate effective critical Casimir
patchy particle potential. The important interaction parameters entering the
theory are explicitly computed using the integral method in combination with
Monte Carlo sampling. Without any adjustable parameter, the predictions of the
chain length distribution are in excellent agreement with explicit simulations
of self-assembling particles. We discuss generality of the approach, and its
application range.
