Fig. 1 a Number of papers on the phase-field model all over the world during the year of 1900-2022; b top 20 rank of countries in publishing papers on the phase-field model

Fig. 2 a Number of papers in China during the year of 1900-2022; b number of papers from the communication of materials science in China during the year of 1900-2022

Fig. 3 Typical numerical discrete methods of PF equations [12]

Fig. 4 Main applications and keywords of the phase-field method

Fig. 5 A Dendrite structures when the equilibrium volume fractions of the liquid phase are about 35% (left) and 19% (right) in the Ni-Al-Nb ternary system. Points A-C on the left are remelting, coalescing, and smoothing, respectively. Points A-E on the right are coalescence, smoothing, Rayleigh instability, rounding, and shrinking away, respectively [77]. B Two representative morphologies of the solid-liquid interface, are the seaweed growth (left) and the dendritic growths (right) [18]. C Typical microstructures of the competitive growth in 2D (top) or 3D (bottom) [98,99]. D The role of interfacial energy anisotropy on dendrite morphology [106]. E Comparison of experimental and simulation results in the interdendritic zone [117]

Fig. 6 A In situ synchrotron XRD and phase-field simulation of phase transformations in LA147. a XRD datasets of as-quenched LA147 during natural aging. b Phase-field simulated and experimentally observed data of the spinodal of wavelength. c Simulated structure order parameter within Al-rich zones as a function of natural aging time and d corresponding microstructural evolution [153]. B Rafted microstructures from phase-field with a elastic and b viscoplastic [162]. C Microstructure with a bimodal initial particle size distribution: a-b simulation results. c-d SEM images of the KSN microcrystalline powders [173]

Fig. 7 A The microstructural evolution during the precipitation process. a Configuration of a pure edge dislocation loop; b–g microstructures of $\alpha $ precipitates at t = 1 × 10−3 s, 3 × 10−3 s and 5 × 10−3 s viewed from the $+ z$ direction b–d at $ T = {\text{1070 K }}$[195]. B Evolution of morphology of the premelting domain and of dislocation configuration in softening crystal phase domains. a $\varepsilon = { 0}{\text{.0570}}$; b $ \varepsilon = {0}{\text{.0600}}$; c $\varepsilon = {0}{\text{.0636}}$; d $\varepsilon = {0}{\text{.0642}}$ [31]. C Phase-field-crystal method guides the design of strength and ductility synergistic mechanism of Cu92Al5Ni3 (wt%) [200]. a Schematic diagram of microstructure under different treatment methods. b ${\text{II}}_{{1}} $ yellow dislocation pair slip mechanism. c ${\text{II}}_{{1}} { }$ yellow dislocation pair decomposition process. d ${\text{V}}_{{3}} $ and ${\text{ VI}}_{{3}}$ dislocation annihilation mechanism. e ${\text{II}}_{{1}}$ and ${\text{III}}_{{1}} $ dislocation annihilation mechanism. D Atomic arrangement of order domain boundary in B2-FeAl alloy. a $\text{(002)}\Vert \text{(002)}\updownarrow\text{1/2[010]}$ interface, b ${\text{(110)}\Vert \text{(110)}}_{\text{Fe}}\updownarrow\text{1/2 [1}\stackrel{\mathrm{-}}{1}\text{0]}$ interface, (c) $\text{(110)}\Vert \text{(110)}\updownarrow\text{1/2 [1}\stackrel{\mathrm{-}}{1}\text{0]}$ interface [212]. E 3D simulation results of Fe-15 at.% Cr alloy show the distribution of a′ precipitates under irradiation and (001) plane different dislocation densities [152]

Fig. 8 Elastic interaction energies between an H-phase particle and 12 individual variants of the B19’ martensite with contour value a1 − 0.267 kJ/mol, a2 − 0.240 kJ/mol. Concentration field (at.%) of b1 Ni and b2 Hf around a growing H-phase particle (Aging at 550 °C for 1296 s) b3 Ni and b4 Hf around an H-phase particle at equilibrium (aging at 550 °C for over 3 h) viewed from the same [110] direction for comparison [217]

Fig. 9 Typical microstructures due to the 3D phase-field simulations performed in a domain with 48 × 48 × 48 nm3 for thin films deposited with three different deposition rates (denoted as V) corresponding to the different gas-solid transition velocities and incident vapor rates at the deposition time of 10 min. (Ref. [233])

Fig. 10 Microstructure evolution of agglomeration for a 30-nm NiSi film on a monocrystal Si substrate annealing at 600 °C: a-b simulation results with orientation field and phase-field, c in situ SEM results (Ref. [235])

Fig. 11 Temporal evolution of domain structure formation in a two-dimensional model. a 40 steps; b 80 steps; c 120 steps; d 240 steps [236]. e Schematic of the ferroelectric thin film with 180° domain structures as solid-state refrigerators [240]. f [001] and [111] butterfly loops for Pb (Zr0.8Ti0.2) O3 [241]. g Nonlinear dielectric constant versus applied equiaxial strain [243]

Fig. 12 Strain distributions and spin textures in patterned La0.67Sr0.33MnO3 (LSMO) wires. a Cross-sectional view (shown in the black dotted circle in schematics on top of (b)) of the LSMO lithographically fabricated sample modeled color scale map showing the relaxation of the strain as a function of width. b Calculated strain profile across the LSMO fabricated samples as a function of the normalized position across the wires of different widths. c, f Topography of 0.5 and 1.0 μm wide LSMO wires taken at 4 K using contact mode atomic force microscope (AFM). d Magnetic force microscopy (MFM) image of 0.5 μm wide wire recorded at 4 K after zero-field cooling (ZFC) at the same place where the topographic image was taken. g MFM image of 1.0 μm wide wire recorded at 4 K in zero fields after ZFC. e, h Phase-field modeling of domains in 0.5 and 1.0 μm wide wires [249]

Fig. 13 Topology of the crack in asymmetric notched three-point bending test based on spectral decomposition [251]

Fig. 14 Topology of crack propagation in different materials. A Polymer material [269]; B fiber-reinforced composite laminate [283]; C geomaterials [288]; D cantilevered beam [297]

Fig. 15 A Phase-field of dynamic fracture of a pressurized cylinder [299]. B Force-displacement curves and the phase-field at some instant during cyclic loading [305]. C Different crack patterns under impact loading for different values of plastic work threshold with and without the triaxiality [312]

Fig. 16 Domestic phase-field conferences: a regional distribution, b age distribution