Phase-field crystal (PFC) models have emerged as a powerful computational tool for simulating the growth of graphene on metal substrates, bridging the gap between atomistic simulations and continuum approaches. These models capture atomic-scale details while operating at experimental timescales, making them particularly suitable for studying graphene nucleation, domain evolution, and defect formation during chemical vapor deposition (CVD). The PFC framework describes the system through a periodic density field that represents atomic positions, enabling the study of crystalline order, elastic deformations, and diffusive dynamics over micron-scale areas and seconds-long durations.

The PFC model for graphene growth incorporates free energy functionals that account for both the graphene lattice and the metal substrate interactions. The atomic registry between graphene and the substrate is modeled through coupling terms that reflect the energetics of different stacking configurations. For example, on copper (111) surfaces, the PFC model reproduces the weak van der Waals interactions that lead to incommensurate growth, while on nickel (111), stronger covalent interactions produce commensurate structures. The model parameters are typically derived from density functional theory (DFT) calculations of adhesion energies, interfacial strain, and diffusion barriers, ensuring quantitative accuracy in the simulations.

Temperature effects are incorporated through thermal noise terms and temperature-dependent mobility coefficients in the dynamical equations. The PFC model captures the Arrhenius behavior of adatom diffusion and attachment kinetics, which governs nucleation density and growth rates. Simulations show that graphene nucleation on copper follows a temperature-dependent trend, with nucleation densities ranging from 10^2 to 10^4 nuclei per square micron for typical CVD temperatures between 900 and 1100°C. These values align with experimental measurements from low-energy electron microscopy (LEEM) studies.

Defect formation during growth arises naturally in PFC simulations through the interplay between deposition flux, surface diffusion, and lattice mismatch. The model reveals how point vacancies, Stone-Wales defects, and grain boundaries form during island coalescence. On copper substrates, simulations show that rotational disorder between graphene domains is prevalent, with misorientation angles distributed around 15° and 30°, consistent with Raman spectroscopy observations of polycrystalline films. The PFC approach quantifies the excess energy associated with grain boundaries, typically ranging from 0.5 to 1.5 eV/nm depending on the misorientation angle.

Grain boundary dynamics during growth are particularly well-captured by the PFC model. Simulations demonstrate how mobile boundaries can annihilate defects through grain coarsening, especially at higher temperatures. The model predicts grain growth exponents that match experimental findings, explaining how initial high nucleation densities evolve into larger domains under optimized growth conditions. For instance, simulations reproduce the transition from nanometer-scale to micrometer-scale grains when increasing the copper substrate temperature from 900°C to 1000°C.

Multilayer graphene formation is modeled by extending the PFC framework to include interlayer interactions. The simulations reveal different growth modes on various metal substrates. On copper, where carbon solubility is low, the model shows predominantly monolayer growth with occasional second-layer nucleation at defect sites. In contrast, on nickel with higher carbon solubility, the PFC simulations demonstrate subsurface carbon accumulation followed by precipitation, leading to multilayer formation. The thickness distribution from simulations matches experimental data, showing how substrate chemistry controls graphene layering.

Comparative studies of different metal substrates highlight the versatility of the PFC approach. On copper, the simulations show slow nucleation and large diffusion lengths, resulting in large graphene domains. Nickel substrates produce smaller domains due to higher nucleation density caused by stronger graphene-substrate bonding. Ruthenium and iridium substrates exhibit intermediate behavior, with the PFC model capturing the complex moiré patterns formed due to lattice mismatch. The simulations quantify how the interfacial energy landscape affects graphene orientation, with copper favoring random orientations and nickel inducing aligned growth.

Recent extensions of PFC models have enabled the study of heterostructure growth, such as hexagonal boron nitride-graphene lateral structures. The modified free energy functional includes composition-dependent terms that describe the phase separation dynamics during co-deposition. Simulations show how growth parameters control domain size and interface sharpness in these 2D heterostructures. Strain engineering applications have also been explored, with PFC models predicting how substrate thermal expansion mismatch induces controlled buckling and wrinkle formation in graphene.

Validation against experimental techniques has been crucial for establishing PFC model credibility. LEEM observations of graphene island shapes and growth velocities show quantitative agreement with simulations across different substrate orientations. Raman spectroscopy data on defect densities and strain distributions match the spatial variations predicted by PFC models. The simulations have successfully reproduced experimental trends in graphene quality as a function of growth parameters, including pressure, temperature, and gas flow rates.

The PFC approach has provided fundamental insights into graphene growth mechanisms that are difficult to obtain experimentally. Simulations have revealed how subtle changes in substrate step edges can dramatically alter nucleation behavior, explaining experimental observations of preferential growth at certain surface features. The models have also clarified the role of hydrogen in graphene growth, showing how it affects edge termination and domain morphology. These insights guide experimental efforts to optimize graphene quality and develop growth protocols for specific applications.

Future developments in PFC modeling will likely focus on incorporating additional physical effects, such as explicit gas-phase reactions and more detailed descriptions of substrate surface reconstructions. Coupling PFC models with machine learning techniques for parameter optimization represents another promising direction. As computational power increases, multi-scale PFC simulations covering entire wafer-scale growth processes may become feasible, further closing the gap between simulation and experiment in graphene research.

The success of PFC models in graphene growth simulations demonstrates their value as a predictive tool for 2D material synthesis. By capturing atomic-scale details at experimental scales, these models provide unique insights into the complex interplay between thermodynamics and kinetics during CVD growth. The quantitative agreement with characterization data validates the physical basis of the approach, while the ability to explore parameter space rapidly makes PFC modeling an essential complement to experimental studies in graphene research and development.