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Two-dimensional phase field simulations of stress-free ferroelectric nanoparticles with different long-range (LR) electrostatic interactions are conducted based on the time-dependent Ginzburg-Landau equation. Polarization patterns and the toroidal moment of polarization are found to be dependent on the LR electrostatic interaction and the size of the simulated nanoparticle. Phase field simulations exhibit vortex patterns with purely toroidal moments of polarization and negligible macroscopicdoi:10.1063/1.3043576 fatcat:fcy5kj7mpze7hho6cy27vniejm