Fig. 1: The thermodynamics-triggered competitive growth (TTCG) model.

a Schematic illustration of \({\varepsilon }_{i,i+1}\) and \({\lambda }_{i}\) in the Eq. (1) and the representations in nonlayered and layered materials. \({\varepsilon }_{i,i+1}\) and \({\lambda }_{i}\) refer to the interaction energy per unit area between subunit i and i + 1, and the edge energy per unit area of subunit i, respectively. b Sketch map of vertical and lateral growth when a new cluster (yellow) grows with the n stacking subunits (lilac). The free energy change of vertical growth mainly comes from the interaction of subunit at the upper interface, while edge energy penalty is dominant for lateral growth. ∆E_ver and ∆E_lat represent the free energy change of vertical and lateral growth. \({\varepsilon }_{n,n+1}\) denotes the interaction energy between subunit n and n + 1. \({\lambda }_{n}\) and \({\lambda }_{B}\) represent the average edge energies of the initial structure and the new cluster, respectively. \({A}_{{s}_{B}}\) and \({A}_{{l}_{B}}\) are the basal and lateral contact areas. c The competition between \(\varepsilon\) term and \(\lambda\) term. Smaller \(\varepsilon\) term will promote lateral growth and smaller \(\lambda\) term leads to vertical growth. \({\varepsilon }_{c}\) is the circumstance correction term of the interaction energy and \({\lambda }_{c}\) is the circumstance correction term of the edge energy. d A schematic for the growth process of 2D Fe₃O₄ nanoflakes along the [111] direction guided by the TTCG model. The intrinsic binding energy difference is not negative enough. The facilitation of H₂O passivation (large \({\varepsilon }_{c}\)) and mica substrate (small \({\lambda }_{c}\)) lead to smaller ∆E, and therefore promote the synthesis of ultrathin Fe₃O₄ nanoflakes. e Every calculated value in the model for Fe₃O₄ with different orientations. f Total ∆E with and without circumstance correction terms for Fe₃O₄ with different orientations. g Optimized structures of Fe₃O₄ along the [111] direction without and with the H₂O adsorption. h Total ∆E of γ-, ε-, α-Fe₂O₃ in different orientations with and without correction terms. Red triangles indicate the lowest energy. i Calculated values in the growth model of ZnO, MoS₂, SmOCl, EuOCl, Fe₇S₈, and Cr₅Te₈.