A simple derivation of phase-field model for pure materials based on entropy functional is provided and then solved by adaptive finite element method (AFEM) to simulate the free dendritic growth from undercooled nickel melt. To investigate the evolutions of the dendrite and reproduce the real physical process，the modeling is performed in a larger domain and thinner interface with the highly computationally efficient and accurate AFEM. The simulated results show that the secondary arms grow in an unsymmetrical mode and their development is controlled by the thermal diffusion and affected by noises which are arbitrarily introduced in the phase-field governing equation. As the latent heat released during the migration of solid-liquid interface is accumulated sufficiently，it prompted initiation of other secondary arms at the same side of the primary arm. As the computation proceeds，the secondary arms become coarsened apparently through four different modes.