We present a fully implicit, finite-element phase-field framework for simulating ice–water solidification in both two and three dimensions. Our model couples a modified heat conduction equation with a regularized Ginzburg–Landau phase-field evolution law, explicitly accounting for anisotropic diffusion and temperature-dependent constitutive properties (density, specific heat and thermal conductivity). After verifying accuracy on a classical 2D undercooling benchmark, we carry out systematic parametric studies to assess the effects of time-step size and mesh resolution on interface dynamics. Three demonstrative cases—two in 2D and one in 3D—show that dendritic morphologies emerge naturally on uniform meshes, without any special adaptive remeshing. We further quantify how temperature-dependent parameters accelerate mid-stage growth kinetics. The proposed approach combines robustness, versatility and computational efficiency, and is released alongside documented Mathematica/AceGen source code to support reproducibility and further extensions.
KrivilyovMDMesarovicSDSekulicDP. Phase-field model of interface migration and powder consolidation in additive manufacturing of metals. J Mater Sci2017; 52: 4155–4163.
2.
TakakiT. Phase-field modeling and simulations of dendrite growth. ISIJ Inter2014; 54: 437–444.
3.
ShaoLChenBYangW, et al.Numerical modeling of ice crystal dendritic growth with humidity by using an anisotropic phase field model. Cryst Growth Des2024; 24: 1400–1409.
4.
DemangeGZapolskyHPatteR, et al.Growth kinetics and morphology of snowflakes in supersaturated atmosphere using a three-dimensional phase-field model. Phys Rev E2017; 96: 022803.
5.
HuangT-HHuangT-HLinYS, et al.Phase-field modeling of microstructural evolution by freeze-casting. Adv Eng Mater2018; 20: 1700343.
6.
LuoZTangGRavanbakhshH, et al.Vertical extrusion cryo(bio)printing for anisotropic tissue manufacturing. Adv Mater2022; 34: 2108931.
7.
DiazFForsythNBoccacciniAR. Aligned ice templated biomaterial strategies for the musculoskeletal system. Adv Healthc Mater2023; 12: 2203205.
8.
CriscioneARoismanIVJakirlićS, et al.Towards modelling of initial and final stages of supercooled water solidification. Int J Thermal Sci2015; 92: 150–161.
9.
LibbrechtKG. The physics of snow crystals. Rep Prog Phys2005; 68: 855–895.
10.
DemangeGZapolskyHPatteR, et al.A phase field model for snow crystal growth in three dimensions. NPJ Comput Mater2017; 3: 1–7.
11.
XuYMcDonoughJMTagaviKA, et al.Two-dimensional phase-field model applied to freezing into supercooled melt. Cell Preserv Technol2004; 2: 113–124.
12.
GlicksmanMELupulescuAO. Dendtritic crystal growth in pure materials. J Cryst Growth2004; 264: 541–549.
13.
CollinsJBLevineH. Diffuse interface model of diffusion-limited crystal growth. Phys Rev B: Rapid Commun1985; 31: 6119–6122.
14.
SakaneSTakakiTAokiT. Parallel-GPU-accelerated adaptive mesh refinement for three-dimensional phase-field simulation of dendritic growth during solidification of binary alloy. Mater Theory2022; 6: 1–19.
15.
GránásyLTóthGIWarrenJA, et al.Phase-field modeling of crystal nucleation in undercooled liquids – a review. Prog Mater Sci2019; 106: 100569.
16.
KarmaARappelW-J. Quantitative phase-field modeling of dendritic growth in two and three dimensions. Phys Rev E1998; 57: 4323–4349.
17.
MaoSCaoYChenW, et al.An anisotropic lattice boltzmann - phase field model for dendrite growth and movement in rapid solidification of binary alloys. NPJ Comput Mater2024; 10: 1–16.
18.
KobayashiR. Modeling and numerical simulations of dendritic crystal growth. Phys D: Nonl Phenom1993; 63: 410–423.
19.
KarmaARappelW-J. Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics. Phys Rev E1996; 53: 3017–3020.
20.
ProvatasNGoldenfeldNDantzigJ. Efficient computation of dendritic microstructures using adaptive mesh refinement. Phys Rev Lett1998; 80: 3308–3311.
21.
GuoZXiongSM. On solving the 3-D phase field equations by employing a parallel-adaptive mesh refinement (para-AMR) algorithm. Comput Phys Commun2015; 190: 89–97.
22.
ChenWHouHZhangY, et al.Thermal and solute diffusion in -Mg dendtrite growth of Mg-5wt.Zn alloy: A phase-field study. J Mater Res Technol2023; 24: 8401–8413.
23.
ZhaoYLiuKHouH, et al.Role of interfacial energy anisotropy in dendrite orientation in al-zn alloys: A phase field study. Mater Des2022; 216: 110555.
24.
TanQHosseiniSA. Modeling ice crystal growth using the lattice boltzmann method. Phys Fluids2022; 34: 013311.
25.
LampronOTherriaultDLévesqueM. An efficient and robust monolithic approach to phase-field quasi-static brittle fracture using a modified newton method. Comput Methods Appl Mech Eng2021; 386: 114091.
26.
JinTLiZChenK. A novel phase-field monolithic scheme for brittle crack propagation based on the limited-memory BFGS method with adaptive mesh refinement. Int Numer Methods Eng2024; 125: e7572.
27.
DziukGElliottCM. Finite elements on evolving surfaces. IMA J Numer Anal2007; 27: 262–292.
U.S. Secretary of Commerce on behalf of the United States of America. NIST chemistry webbook. thermophysical properties of fluid systems, 2023 last updated. (accessed on 30 January 2025).
30.
HarveyA. Properties of Ice and Supercooled Water. Boca Raton, FL: CRC Handbook of Chemistry and Physics, CRC Press, 2019.
31.
HughesTJR. The finite element method, linear static and dynamic finite element analyses. Englewood Cliffs, NJ: Prentice-Hall, 1987.
32.
Inc. Wolfram Research. Mathematica, version 14.2. Champaign, IL, 2024.
