A method that calculates the time evolution of microstructure in the direction that lowers the free energy, treating an order parameter for the phase or crystal orientation as varying smoothly across interfaces. Used for solidification and precipitation.
Japanese name: フェーズフィールド法
Phase-field method appears in 2 practice questions across 1 topic, including Computational materials science.
When calculating microstructural change in solidification or precipitation, treating the interface as a surface of zero thickness requires tracking its position at every moment and imposing temperature or concentration conditions there (such as the Gibbs–Thomson effect). When shapes change in complex ways, as when dendrite arms branch or precipitates join, this tracking becomes very difficult. The phase-field method treats the interface as a region of finite width over which an order parameter changes smoothly and solves the same equations throughout space, so the interface position need not be tracked. The idea rests on Cahn and Hilliard's 1958 theory of the free energy of nonuniform systems.
The free energy is written as the sum of a function with two valleys corresponding to the two phases (a double well) and a term in the square of the gradient of the order parameter multiplied by a coefficient κ. With W the height of the barrier of the double well, the interfacial energy is proportional to √(κW) and the interface width to √(κ/W). A steep interface pays in the gradient term, while a gradual one pays by spreading the region sitting on the barrier, so the width is set by the balance between them. Real interfaces are on the order of nanometers wide, but calculations use much wider interfaces, with "thin-interface limit" formulations devised so that results do not depend on the width.
The reliability of phase-field results depends on the property values put in. For the chemical free energy of each phase, thermodynamic databases of the CALPHAD method are often used. Interfacial energy and its anisotropy, interface mobility and diffusion coefficients are obtained from experiments and from molecular dynamics or first-principles calculations. When a precipitate and the matrix have different lattice parameters, elastic strain energy is also added. The arrangement of cuboidal γ' in nickel-based superalloys, and its joining into plates under stress, can be reproduced only when this elastic term is included. When property values are uncertain, it is safer to use the results to read trends with conditions rather than absolute values.
Typical uses are solidification structures and segregation in welding and additive manufacturing, grain growth and coarsening of precipitates in heat treatment, spinodal decomposition and martensitic transformation. Recently it has also been used for fracture analysis, in which a crack is represented by an order parameter to calculate its path, and for phase separation as lithium enters and leaves battery electrodes. Because the interface width must be represented by several grid points, the grid spacing is set by the interface width, and the method mainly handles microstructures on scales from nanometers to micrometers. Three-dimensional calculations need very many grid points, so parallel computing with GPUs is widely used.
Answer: A. It uses a diffuse interface across which an order parameter changes smoothly and evolves the order parameter so as to lower the free energy, computing microstructural change without tracking the interface directly
In the phase-field method, an order parameter that takes, for example, 1 in the solid and 0 in the liquid is assigned to each point in space, and at the interface its value changes smoothly over a finite width. The total free energy is written as the sum of the chemical free energy at each point and an interfacial energy proportional to the gradient of the order parameter, and the system evolves in time in the direction that lowers it. Non-conserved quantities such as crystal orientation are handled by the Allen–Cahn equation, and conserved quantities such as concentration by the Cahn–Hilliard equation. Because the interface need not be tracked explicitly, microstructures with complex changes of shape, such as dendrite growth, grain growth and precipitate coarsening, can be calculated. Tracking interface points belongs to sharp-interface methods, and solving atomic equations of motion is molecular dynamics.
It is a continuous variable that indicates, at each location, which phase or grain the point belongs to. For example, it is 1 in the solid and 0 in the liquid and takes intermediate values at the interface. Sometimes a measurable quantity such as concentration is used, and sometimes a quantity introduced for the calculation, such as one distinguishing grains.
Usually not. Calculating with the real width would need too many grid points, so a much wider interface is used. It is therefore important to check that results do not change when the interface width is varied, or to use formulations devised to cancel the effect of the width.
The phase-field method solves partial differential equations based on free energy to calculate interface shapes in detail. The cellular automaton method updates the states of grid cells by simple rules; it is less accurate but computationally light, and suits large regions such as the grain structure of castings.
In the phase-field method, an order parameter that takes, for example, 1 in the solid and 0 in the liquid is assigned to each point in space, and at the interface its value changes smoothly over a finite width.
Which statement about the phase-field method is correct?The phase-field method treats microstructural change from nanometers to micrometers, and the finite element method treats stress and temperature at the scale of components as a continuum.
Which pairing of a materials computation method with the ra…Molecular dynamics Density functional theory Interatomic potential Finite element method CALPHAD method Materials informatics Monte Carlo method