diff --git a/_posts/2024-07-20-Solidification.md b/_posts/2024-07-20-Solidification.md index 99a1dda7..18829ca8 100644 --- a/_posts/2024-07-20-Solidification.md +++ b/_posts/2024-07-20-Solidification.md @@ -10,7 +10,7 @@ related_posts: false --- ## Phase-field equation for solidification processes -Liquid-solid phase transformations usually refer to solidification and melting. The formation of complex microstructures during the solidification from a liquid phase, such as formation of snow flakes or metallic alloys, and the accompanying diffusion and convection processes in the liquid and solid have been widely studied in the literature [1, 2, 3]. In particular, the evolution of microstructural scale of dendrites during the solidification process determines many physical and mechanical properties of metals, since almost every metallic system originates from the liquid state. +Liquid-solid phase transformations usually refer to solidification and melting. The formation of complex microstructures during the solidification from a liquid phase, such as formation of snow flakes or metallic alloys, and the accompanying diffusion and convection processes in the liquid and solid have been widely studied in the literature {% cite boettinger2002phase flemings1974solidification kuhn1986fundamentals --collection external_references %}. In particular, the evolution of microstructural scale of dendrites during the solidification process determines many physical and mechanical properties of metals, since almost every metallic system originates from the liquid state.
@@ -35,7 +35,7 @@ A phase-field formulation that describes the solidification process is as follow \label{2} \end{equation} -where $$\phi \in [-1,1]$$ represents the phase-field variable, which describes the location of different components (e.g., $$\phi = -1$$ in the solid and $$\phi = 1$$ in the liquid), $$\theta$$ is the temperature, $$\theta_m$$ is the melting temperature, $$H$$ represents the interfacial enthalpy per unit mass, $$\omega$$ is the kinetic undercooling coefficient, $$C_v$$ is the heat capacity per unit mass, $$\rho$$ is the density, $$l$$ is specific latent heat (energy per unit mass), $$k$$ is the thermal conductivity, $$\sigma$$ is the surface tension and $$h$$ is an interpolatory function that verifies $$h(+1) = 1, h(-1) = 0$$, e.g., $$h = \frac{1}{2} (1 + \phi)$$. The thermal conductivity is considered to be a function of the phase-field to account for different conductivity in the solid and liquid phases. We take $$k(\phi) = (1 + \phi) k_s + (1 - \phi) k_l$$, which satisfies that $$k(+1) = k_s$$, $$k(-1) = k_l$$. The function $$W(\phi)$$, also referred to as the double-well potential, is defined such that it has two local minima, which makes possible the coexistence of the different phases. Some important examples of double well functions can be found in [4]. In this work we will take the classical quartic potential $$W(\phi) = \frac{1}{4} (1 - \phi^2)^2$$. The function $$G(\phi)$$ vanishes in the pure phases. Depending on the form of the functional $$G(\phi)$$, the phase-field will converge faster or slower to the generalized Stefan problem. Here we will use the expression $$G(\phi) = (1 - \phi^2)^2$$. One key aspect to achieve good agreement with the reality of interest behind dendritic solidification is surface tension anisotropy. Herein, we introduce anisotropy by assuming that $$\sigma$$ depends on the unit normal to the liquid-solid interface, this is, +where $$\phi \in [-1,1]$$ represents the phase-field variable, which describes the location of different components (e.g., $$\phi = -1$$ in the solid and $$\phi = 1$$ in the liquid), $$\theta$$ is the temperature, $$\theta_m$$ is the melting temperature, $$H$$ represents the interfacial enthalpy per unit mass, $$\omega$$ is the kinetic undercooling coefficient, $$C_v$$ is the heat capacity per unit mass, $$\rho$$ is the density, $$l$$ is specific latent heat (energy per unit mass), $$k$$ is the thermal conductivity, $$\sigma$$ is the surface tension and $$h$$ is an interpolatory function that verifies $$h(+1) = 1, h(-1) = 0$$, e.g., $$h = \frac{1}{2} (1 + \phi)$$. The thermal conductivity is considered to be a function of the phase-field to account for different conductivity in the solid and liquid phases. We take $$k(\phi) = (1 + \phi) k_s + (1 - \phi) k_l$$, which satisfies that $$k(+1) = k_s$$, $$k(-1) = k_l$$. The function $$W(\phi)$$, also referred to as the double-well potential, is defined such that it has two local minima, which makes possible the coexistence of the different phases. Some important examples of double well functions can be found in {% cite gomez2017computational --collection external_references %}. In this work we will take the classical quartic potential $$W(\phi) = \frac{1}{4} (1 - \phi^2)^2$$. The function $$G(\phi)$$ vanishes in the pure phases. Depending on the form of the functional $$G(\phi)$$, the phase-field will converge faster or slower to the generalized Stefan problem. Here we will use the expression $$G(\phi) = (1 - \phi^2)^2$$. One key aspect to achieve good agreement with the reality of interest behind dendritic solidification is surface tension anisotropy. Herein, we introduce anisotropy by assuming that $$\sigma$$ depends on the unit normal to the liquid-solid interface, this is, \begin{equation} \sigma = \sigma_0(1 + \delta \cos(\alpha - \alpha_0)), @@ -68,7 +68,7 @@ The solutions $$\phi^h$$ and $$\theta^h$$ are defined as \end{equation} ## Results -Our computational domain is the square $$\Omega= \left[0,0.1 \right] \times \left[0,0.1 \right]$$, where the units are in cm. We employ a computational mesh comprised of 5122 $$\mathcal{C}^1$$-quadratic elements. We impose periodic boundary conditions for $$\phi$$ and $$\theta$$. We choose a suitable time step of $$\Delta t = 10e-6$$ s to ensure good numerical stability, specially at the beginning of the simulation. As an initial condition, we set a square-shaped seed located at the center of the domain. The reason for using a square-shaped crystal, instead of one with a smoother shape like a circle, is because the corners of the square will accelerate the formation of dendritic structures. The temperatures inside and outside the seed are 1700 K and 1500 K, respectively. The parameters used in all the simulations are summarized in Table 1. These parameters correspond to an Nickel-Copper alloy. In the following simulations, we will show the relations between the anisotropy modes and the shapes of crystals. +Our computational domain is the square $$\Omega= \left[0,0.1 \right] \times \left[0,0.1 \right]$$, where the units are in cm. We employ a computational mesh comprised of 5122 $$\mathcal{C}^1$$-quadratic elements. We impose periodic boundary conditions for $$\phi$$ and $$\theta$$. We choose a suitable time step of $$\Delta t = 10e-6$$ s to ensure good numerical stability, specially at the beginning of the simulation. As an initial condition, we set a square-shaped seed located at the center of the domain. The reason for using a square-shaped crystal, instead of one with a smoother shape like a circle, is because the corners of the square will accelerate the formation of dendritic structures. The temperatures inside and outside the seed are 1700 K and 1500 K, respectively. The parameters used in all the simulations are summarized in {% cite gomez2019review --collection external_references %}. These parameters correspond to an Nickel-Copper alloy. In the following simulations, we will show the relations between the anisotropy modes and the shapes of crystals. The Figures below show the evolution of the phase-field variable and the temperature field over time for different values of $$q$$. For all simulations nucleation starts at the center of the domain, and it triggers the solidification process. The phase change is achieved by undercooling the liquid phase below its melting temperature.