  The steepest-entropy-ascent quantum thermodynamic framework is used to
investigate the effectiveness of multi-chain
polyethyleneimine-methylenephosphonic acid in sequestering rare-earth ions
(Eu$^{+3}$) from aqueous solutions. The framework applies a thermodynamic
equation of motion to a discrete energy eigenstructure to model the binding
kinetics of europium ions to reactive sites of the polymer chains. The energy
eigenstructure is generated using a non-Markovian Monte Carlo model that
estimates energy level degeneracies. The equation of motion is used to
determine the occupation probability of each energy level, describing the
unique path through thermodynamic state space by which the polymer system
sequesters rare-earth ions from solution. A second Monte Carlo simulation is
conducted to relate the kinetic path in state space to physical descriptors
associated with the polymer, including the radius of gyration, tortuosity, and
Eu-neighbor distribution functions. These descriptors are used to visualize the
evolution of the polymer during the sequestration process. The fraction of
sequestered Eu$^{+3}$ ions depends upon the total energy of the system, with
lower energy resulting in higher sequestration. The kinetics of the overall
sequestration are dependent on the steepest-entropy-ascent principle used by
the equation of motion to generate a unique kinetic path from an initial
non-equilibrium state.
