A two-dimensional (2D) quantitative phase-field model solved by adaptive finite element method is employed to investigate the effect of natural convection on equiaxed dendritic growth of Al-4 wt.%Cu alloy under continuous cooling condition. The simulated results are compared with diffusion-limited simulations as well as the experimental data obtained by means of in situ and real-time X-ray imaging technique. The results demonstrate that natural convection induced by solute gradients around the dendritic crystal has an obvious influence on the dendrite morphology and growth dynamics. Since the rejected solute cooper from solid is heavier than aluminum, it sinks down along the interface from the top arm tip to the bottom arm which results in the formation of a circulatory flow vortex on both sides of the dendrite. Hence, the convection promotes the top arm advancing into the melt progressively whereas it suppresses the growth of bottom severely. As the dendrite grows into a large size, the convection becomes more intense and the morphology shows distinguished asymmetric shape. When compared with experimental data, the growth velocity is found to agree substantially better with the simulation incorporating natural convection than the purely diffusive phase-field predictions.

Equiaxed dendritic crystal is one of the most common microstructures formed in the solidification process of materials, whose morphology, size, and composition distribution in castings are critical to the mechanical properties of the as-cast structural materials. Ever better understanding of its morphology evolution dynamics and related underlying physics are always important to obtain targeted grain features as well as deepen the knowledge of formation mechanism on such practical and theoretical important structure. Since many factors, including the characteristics of the alloy system (diffusion, anisotropy, melting point, etc.) and the external imposed conditions of solidification (composition, cooling rate, thermal gradient, undercooling, forced flow, etc.), exert strong influences on the growth shape and dynamics of equiaxed dendrites, it is of great importance to discriminate these factors on the growth behaviors [

As one of the factors impacting the dendrite growth dynamics, fluid flow, in particular the natural convection which is caused by the density variation in the melt, is one of the main driving forces to form various morphologies of individual dendrite in the real solidification conditions [

Besides experimental approaches, theoretical analysis and numerical computer simulations have also been exerted to understand the natural convection effects on dendritic crystal growth. Yet, analytic solutions which require lots of assumptions beforehand [

In this paper, the equiaxed dendrite growth from the isothermal melt cooled by a constant rate on Al-4 wt.%Cu alloy is simulated with the employment of a 2D quantitative phase-field model with incorporation of incompressible Navier-Stokes equations. The simulated natural convection and its evolution with time, as well as the dendritic growth dynamics, are characterized and analyzed in detail. And then the simulations were directly compared with diffusion-limited dendritic growth and the monitored data of the alloy by in situ and real-time X-ray radiography [

The phase-field model for directional solidification [

Then, the fluid flow is coupled using the method proposed by Beckermann and coworkers [

The governing equations of the model, (

The parameters used for phase-field simulation of equiaxed dendritic growth of Al-4 wt.%Cu alloy with natural convection.

Solute partition coefficient, | 0.14 |

Liquidus temperature, | 933.47 K |

Solute diffusion coefficient in solid, | 1.15 × 10^{−8} cm^{2}/s |

Solute diffusion coefficient in liquid, | 2.4 × 10^{−5} cm^{2}/s |

Gibbs-Thomson coefficient, | 2.36 × 10^{−5} cm⋅K |

Liquidus slope, | −3.5 K/wt.% |

Initial concentration, | 4 wt.% Cu |

Interface width parameter, | 54 |

Surface energy anisotropy strength, | 0.0106 |

Kinematic viscosity, | 5.0 × 10^{−3} cm^{2}/s |

Density, | 2.45 g/cm^{3} |

Thermal expansion coefficient, | 1.0 × 10^{−4} k^{−1} |

Solutal expansion coefficient, | 9.2 × 10^{−3 }(wt.%)^{−1} |

Gravitational acceleration, | −9.8 × 10^{2} cm/s^{2} |

Dimensionless friction coefficient, | 2.757 |

Cooling rate, | 0.5 K/min |

The simulation domain is rectangle with a size of 4304

Figure

Two snapshots of the simulated flow field and solutal field for

Figures

Isoconcentration contours around a dendrite with developed side branches. (a) The whole domain. (b) Close-up around the crystal.

The primary-dendrite arm length with time in the phase-field simulation.

The measured lengths in Figure

The growing dendrite leading to the increment of natural convection which in turn results in the asymmetry of the dendrite shape. (a) The variation of dendrite arm length ratio with time. (b) The relationship between the maximum flow strength and the dendrite arm length ratio.

To be more exact, as the dendrite grows, more solute will be rejected into the melt ahead of the solid/liquid interface. Meanwhile, the region of melt with different density caused by the rejected solute becomes wider and wider. In terms of the dimensionless Rayleigh number which indicates the strength of nature convection, the convective strength enhances notably by the expanded region of distribution of solute. Therefore, stronger convection forms around the dendrite. The pronounced influence of natural convection on solid growth dynamics can also be understood within the order of magnitude differences between the moving velocities of the tip front and the melt. At the end of the calculation, the maximum flow velocity is 342.71

Figure

Evolution of the dendrite tip velocities of three different tips in the phase-field simulation.

Furthermore, the evolution of the tip shape is examined of the three tips through measuring the radii as illustrated in Figure

Evolution of the dendrite tip radii of three different tips in the phase-field simulation.

The corresponding crystal shape to the variation of the tip radius of the upstream arm.

In order to examine the accuracy of the convective phase-field simulation, the calculated dendrite growth is compared with the in situ and real-time experimental observations on Al-4 wt.%Cu alloy [

One selected sequential image recorded by means of synchrotron X-ray radiography [

Figure

Comparison between experiment and phase-field simulation in the presence of natural convection. The phase-field simulation result without natural convection is also plotted.

First, a quantitative 2D phase-field model incorporating fluid flow dynamics is employed to investigate the effect of natural convection on the dendritic growth from the melt of Al-4 wt.%Cu alloy solidified by continuous cooling-down. The parameters of the model are adopted from the real synchrotron X-ray radiography experiment, and hence it is reasonable to link the phase-field simulation to the experiment. As expected, the simulation shows that natural convection induced by the gravity has an obvious effect on the morphology evolution of the equiaxed dendrite, leading to different tip velocities at different growth directions. The natural convection transports the solute rejected from the crystal downwards and redistributes the solute asymmetrically in the whole domain, which in turn causes the asymmetrical realistic undercooling at the four dendritic tips. As the dendrite grows, the impact of natural convection on dendrite evolution becomes more important due to the increment of convective strength. Finally, the quantitative comparison between experiments and simulations of convective and diffusive phase-field modeling reveals that natural convection plays a crucial role in the dendrite growth dynamics of metallic alloys on the earth. However, there are still some uncertainties required to be clarified, such as the difference between 2D and 3D simulations, the growth interaction between dendrites, and the flow pattern among many dendritic crystals driven by graded solute and temperature.

The authors declare that there are no competing interests regarding the publication of this paper.

This work is supported by National Natural Science Foundation for Young Scientists of China (Grant no. 51401223) and National Natural Science Foundation of China (Grant no. 51271184).