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During fast diffusion-influenced polymerization, nonequilibrium behavior of the polymer chains and the surrounding reactive monomers has been reported recently. Based on the laws of thermodynamics, the emerging nonequilibrium structures should be characterisable by some "extra free energy" (excess over the equilibrium Helmholtz free energy). Here, we study the nonequilibrium thermodynamics of chain-growth polymerization of ideal chains in a dispersion of free reactive monomers, using off-lattice, reactive Brownian Dynamics (R-BD) computer simulations in conjunction with approximative statistical mechanics and relative entropy (Gibbs-Shannon and Kullback-Leibler) concepts. In case of fast growing polymers, we indeed report increased nonequilibrium free energies of several kBT compared to equilibrium and near-equilibrium, slowly growing chains. Interestingly, the extra free energy is a non-monotonic function of the degree of polymerization and thus also of time. Our decomposition of the thermodynamic contributions shows that the initial dominant extra free energy is stored in the nonequilibrium inhomogeneous density profiles of the free monomer gas (showing density depletion and wakes) in the vicinity of the active center at the propagating polymer end. At later stages of the polymerization process, we report significant extra contributions stored in the nonequilibrium polymer conformations. Finally, our study implies a nontrivial relaxation kinetics and restoring of the extra free energy during the equilibration process after polymerization.