To investigate the local properties of heterogeneous nuclei on substrates, a phase-field model is extended to incorporate volume constraints and a third order line tension in the gradient free energy density formulation. The new model is applied to sessile drop simulations of Cu nuclei on Ni substrates to precisely analyse 3D equilibrium shapes and diffusion processes across the phase boundaries. In particular, the formalism with higher order potentials is used to investigate the length-scale dependent effect of the line tension on Young's force balance at triple lines in 3D. The employment of parallel and adaptive simulation techniques is essential for three-dimensional numerical computations. Early stage solidification microstructures of cubic Ni crystals are simulated by scale-bridging molecular dynamics (MD) and phase-field (PF) simulations. The domain of the PF computations is initialized by transferring MD data of the atomic positions and of the shape of the nuclei. The combined approach can be used to study the responses of microstructures upon nucleation.