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A model describing the growth of embryonic dust grains on account of accretion of neutral atoms and positively charged ionic species in a complex plasma has been developed. In deference to the recent emphasis on the character of the openness in complex plasma, the investigation is based on the balance of the number density and energy of electrons, ions, and neutral particles as well as the energy balance of the dust particles and the charge neutrality condition. To discuss the kinetics of the growth of the size of dust the processes of accretion of electrons, ions, and neutral species on the dust particles, the ionization of neutral atoms, and recombination of electrons and ions, and the elastic collisions between the constituent species of the dusty plasma have been considered; the energy exchange associated with these processes has also been taken into account. The dependence of the growth of dust particles and other relevant parameters on number density of embryonic dust grains has, in particular, been explored.