Advances in Materials Science and Engineering

Volume 2016 (2016), Article ID 5286168, 10 pages

http://dx.doi.org/10.1155/2016/5286168

## The Effect of Natural Convection on Equiaxed Dendritic Growth: Quantitative Phase-Field Simulation and Comparison with Synchrotron X-Ray Radiography Monitoring Data

Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Shenyang, Liaoning 110016, China

Received 18 May 2016; Accepted 30 August 2016

Academic Editor: Paolo Ferro

Copyright © 2016 Xin Bo Qi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1, 2].

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 [3, 4]. It alters the local distribution of heat and solute around the dendritic tip. Nevertheless, natural convection is unavoidable on the earth because of the inhomogeneous distribution of solute and heat that results in the expansion or compression of melt. Thus, the gravity drives the denser melt moving downwards and the lighter melt flowing upwards, provoking the convective flow of melt. Consensus has been already achieved that natural convection should be responsible for the deviation between experiments and theoretical predictions through plenty of experimental analyses on the dendritic growth of transparent organic alloys [5, 6]. For example, the comparison of the SCN crystals grown in the microgravity environment which was generated by the space shuttle Columbia [7–9] with that under terrestrial conditions performed by Glicksman et al. demonstrated that the gravity-driven convection is the main factor contributing to the deviation between the measured dendritic growth velocities and the classical diffusion-limited Ivantsov theory [10]. Despite the fact that some reliable conclusions have been drawn on the solidification of transparent organic alloys, there have been limited experiments on metallic systems. Moreover, it is well known that the solidification behavior of transparent alloys differs from that of metallic systems owing to their different thermophysical properties. And these previous experiments cannot illustrate the dynamic details of morphology evolution precisely, especially the distribution of solute. In recent years, synchrotron X-ray radiography has become an effective method to unveil the dynamical formation of solidification microstructures and grain structure in metallic alloys. Benefitting from this development, precise data by in situ and real-time observations are obtained to benchmark modeling and numerical simulations [11–17]. Bogno et al. [13, 16] have carried out a series of in situ and real-time observations of the equiaxed dendritic growth of Al-Cu alloys solidified by cooling-down at the European Synchrotron Radiation Facility (ESRF). They have carefully characterized and analyzed the dendritic growth velocity and the solute distribution among the dendritic grains. Nevertheless, flow effects on dendritic growth dynamics were not taken into account in their analysis because of the difficulty to estimate the natural convection. Through comparing the diffusion-based phase-field simulations with the measured characteristic growth parameters of equiaxed dendritic crystal on Al-4 wt.%Cu alloys, Chen et al. [16] have found that the gravity-driven melt convection plays an important role on the crystal growth in the in situ and real-time observed experiment. The transportation of solute by convection results in the pronounced discrepancy in the dynamics of equiaxed grain growth between experiment and quantitative simulations.

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 [18–23] are limited to incorporate the complex nonlinear effect of convection into the theoretical analyses of the dendritic growth. Additionally, these analytic solutions are not able to give the details of the morphology evolution of a dendrite and the flow pattern in the melt. In contrast, several numerical mathematical models, such as phase-field and front-tracking methods, have been extended to include fluid flow dynamics for dendritic growth, offering researchers a rather convenient way to investigate the melt convection effects [24–29]. Bänsch and Schmidt [30] presented a numerical algorithm based on a sharp-interface model to simulate the thermal convection with different boundary conditions such as Dirichlet and Neumann boundary conditions. Tönhardt and Amberg [31] simulated the natural convection on succinonitrile (SCN) using a two-dimensional phase-field model, which revealed that the influence of thermal natural convection increases with the growth of the dendrites. Zhao et al. [32] presented the simulation using a sharp-interface model and adaptive mesh technique with the data extracted from Tonhardt’s work and obtained the similar values and conclusions. Chen and Lan [33] used an efficient adaptive three-dimensional phase-field model to investigate the influence of thermal convection on SCN dendrite. Their simulation results are consistent with previous theories and experimental observations.

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 [16]. Within the comparison results, the effects of natural convection on equiaxed dendritic growth dynamics and morphology are thus clarified, and it is clearly demonstrated that natural convection is an important factor accounting for the discrepancy between predictions controlled by diffusion and experimental data.

#### 2. Phase-Field Model and Numerical Implementation

##### 2.1. Model Description

The phase-field model for directional solidification [34] is modified and adapted to simulate equiaxed dendritic growth under continuous cooling-down condition. This model is primarily proposed by Karma [35] where the solute antitrapping current is included in the solute conservation equation to eliminate the unphysical effects, such as the surface diffusion, surface stretching, and the jumping of chemical potential at the interface. In the employed model, the order parameter is defined as 1 in solid and −1 in liquid and changes smoothly at the interface layer. The equation for phase-field variable is given bywhere is the cooling rate, is the initial concentration of the alloy, and and are the interface width and relaxation time, respectively. is the coupling parameter between concentration field and phase field, expressed as . The chemical capillary length is with being the liquidus slope and being the solute partition coefficient, which are taken constants, and being the Gibbs-Thomson coefficient. As it has been demonstrated in the reference [16], the capillary length is taken as constant even though the temperature of the melt decreases with time. is the dimensionless solute concentration; , which is obtained by solving the solute conservation equation that is written including the terms describing the solute transport by convection:where is the solute partition coefficient and is the solute diffusion coefficient in liquid. and are the length scale and time scale, respectively.

Then, the fluid flow is coupled using the method proposed by Beckermann and coworkers [36]. A no-slip condition at the solid-liquid interface is imposed by adding a source term in the Navier-Stokes equations. Assuming the fluid is incompressible, the Navier-Stokes equations controlling the fluid motion are rewritten as follows:where is the liquid flow velocity, is the liquid kinematic viscosity, is the pressure in the melt, and is the density. The term is a dissipative interfacial force to constrain the flow velocity to be zero in the bulk solid phase. Physically, in the limit of a sharp interface, that is, , the liquid velocity has to approach 0 in order to generate a moderate drag force as illustrated in the expression of ; as a result, the usual no-slip condition at the solid/liquid interface is resumed. Then, the exact value of is determined from an asymptotic analysis to minimize the deviations between the diffuse-interface and sharp-interface solutions [37, 38]. The buoyancy force which depends on the differences of local temperature and composition is expressed using the Boussinesq approximation. The conservation equation for mass takes the following form:

##### 2.2. Numerical Simulation

The governing equations of the model, (1)–(4), are numerically solved to simulate equiaxed dendritic growth of Al-4 wt.%Cu alloy under continuous cooling-down condition. In order to increase computational efficiency, the adaptive finite element method is adopted, where the meshes are coarsened or refined according to the local error indicator [39]. The numerical implementation is based on the finite element package AFEPack [40]. The error indicator is based on the gradient jump of finite element solutions on the interface of adjacent elements. Here the error indicator is chosen aswhere is the finite element solution for the phase-field equation and is the boundary of the element . The parameters used in this study are listed in Table 1, and the cooling rate is set equal to that applied in the experiment.