The multi-scale and multi-physics simulations are carried out for the nano-emitters consisting of FCC (Al, Cu and Au), BCC (V, Mo and W) and HCP (Ti, Zn and Zr) metals, using the hybrid electrodynamics coupled with molecular dynamics-particle in cell simulations (PIC-ED-MD). We show that the tilting of nano-emitter at low temperature and small electric field (E-field) is mainly caused either by the movement of partial dislocations at the apex of nanotip or by the elastic local distortions of atomic registries away from their ideal lattice sites (FCC/BCC/HCP). At high E-field, the intense resistive heating due to the strong electron emission leads to the directly melting of the apex of nano-emitters. For nano-emitters consisting of low melting point metals such as Al, Zn and Au, the thermal runaway is driven by the elongation, thinning, and necking of the molten region. Meanwhile, the elongation, thinning, and sharpening produce the nano-protrusion at the apex of metal nano-emitters, and the detaching of atoms or atomic cluster from nano-protrusion mainly contribute to the thermal runaway event for refractor metals such as Ti, Zr, Mo and W. The critical E-field strength of metal nano-emitters is found to be strongly correlated with structural parameters (atomic coordination number of liquid and equilibrium lattice constant), thermodynamic quantities (cohesive energy and enthalpy of evaporation) and phase transition temperatures (melting point and boiling point). Those correlations enable us to establish either the single-variable linear fitting models or multi-variable linear regression models to predict the critical E-field value for metal nano-emitters with good credibility.