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Rapid solidification occurs under large driving force of transformation from the metastable undercooled liquid phase to the stable crystalline state. Using a formalism of extended irreversible thermodynamics, a phase-field model of rapid solidification in binary systems is derived. An entropy approach together with a grand potential density of a binary system is used to obtain the main governing equations of the model. Special attention is paid to equations of a rapidly solidifying binary system which are accompanied by essential deviations from local equilibrium in the transport of the conservative variables (such as inner energy and mass) and in the dynamics of non-conservative variables (such as phase field). The obtained equations are analyzed and compared with recent models and outcomes based on the grand potential approach to solidification. [ABSTRACT FROM AUTHOR]