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A new model of dendritic growth and solute distribution of Fe-0.04%C binary alloys was developed, which is based on the sharp interface model of dendritic growth. This innovative model improved the cellular automaton method, combined with the finite difference method, considered concentration field, temperature field and the shape of molten pool. This model simulated the growth morphologies of single equiaxial crystal, the relationship between tip growth velocity and time, multi-equiaxed crystals’ growth morphologies and solute distribution, the growth of columnar crystals, columnar-to-equiaxed transition after coupling temperature field, and compared with experimental results. The results indicate that crystallographic orientation has certain influence on dendritic morphologies, that the tip growth velocity tends to be stable with the extension of time in the end, that the shape of molten pool influences the growth morphologies of columnar crystals and that the solute mainly concentrates in dendritic roots and among grain boundaries. The simulated results are in accord with experimental results.

The dendritic microstructure determines material microstructure and mechanical properties during the process of metal solidification [

There are many detailed simulation models at present, such as the Phase Field (PF) method, Monte Carlo (MC) method, Cellular Automaton (CA) method, and Front Tracking method [

A description of a two-dimensional model is provided. The two-dimensional calculation domain was divided into 400 × 800 cells of uniform orthogonal arrangement. The grid size was

The weld molten pool is actually an irregular arc shape, and it continuously changes during solidification. In order to simplify the model, this work uses an ideal model by setting the radius of molten pool and assuming that the shape of molten pool remains the same during solidification. The interior of the molten pool distributes liquid alloy, and the outside is solid. If there is a crystal nucleus

The temperature field of molten pool was simulated by the FD method. There are two kinds of finite difference calculations about welding heat conduction. One is the steady-state heat conduction of finite difference; the other is transient heat conduction of finite difference. Because of the characteristics of welding heat source, including concentration, mobility and instantaneity, the transient heat conduction of finite difference equation is used to describe the temperature field of molten pool in this paper.

The boundary conditions are assumed as follows: the top of the molten pool is adiabatic; and, the wall of molten pool is heat-dissipation. Then the basic partial differential equation [

where

The explicit difference scheme is used to solve Equation (2) and the obtained

Here,

spectively time step and the length of the CA cell side.

In this paper, the quasi-continuous nucleation model is adopted, which is put forward by Rappaz et al. and Thevoz et al. [

where

The sharp interface model of dendritic growth established by Chen et al. [

here curvature undercooling is represented by

where

Solute redistribution is one of the most important steps in the process of dendritic growth. Assuming that the solid/liquid concentration at the interface satisfies the following equation:

where

where

The discharge of solute will inevitably lead to rising in liquid solute concentration around dendrites, forming a big concentration gradient, promoting the diffusion of solute, and being uniformly distributed to the adjacent liquid cells in the end. In fact, the diffusions of solute include diffusion in liquid phase, solid phase and interface. These three kinds of diffusions occur simultaneously and interact with each other. In order to simplify the model, the simulation assumes that the diffusion only occurs in liquid phase. Then the governing equation [

where

There are some restrictions about the selection of a time step. In this paper the explicit difference scheme of diffusion equation is adopted and obtains the time step according to the stability of discrete equation, meanwhile, the time step also meets the provisions of the CA method. So the time step can be determined by:

where

The numerical model of dendritic growth was established by researching and appropriately simplifying the theory of solidification. Fe-0.04%C binary alloys are selected as object to simulate the dendritic growth morphology of solidification process, solute distribution and temperature field in arc-shape molten pool. The physical parameters of Fe-0.04%C binary alloys are listed in

Figures 1(a)-(c) show the single dendrite morphology with different crystallographic orientations after 0.67 × 10^{−4} min. It is found that there is a big difference in secondary dendrites morphology of different crystallographic orientations by comparing the three dendritic morphologies. That is to say, the growth

Parameter | Value |
---|---|

Alloy composition | 0.04 |

Liquid slope | −80 |

Solute diffusion coefficient in the liquid phase | 3 × 10^{−9} |

Partition coefficient | 0.17 |

Anisotropy strength | 0.3 |

Gibbs-Thompson coefficient | 1.9 × 10^{−7} |

velocity of secondary dendrites with crystallographic orientation of 45˚ is slower than the other two orientations under the same growth time. The dendrite morphology with crystallographic orientation of 0˚ has the best symmetry. The difference of morphology is caused by mesh dependency, and this article will further improve CA growth algorithm to reduce the mesh dependency of simulations.

^{−4} min, the velocity fluctuation is less than 0.006 m/min. The reason is that dendrite rapidly grows at the beginning of solidification under the effect of supercooling. As the solidification proceeds, the discharged solute concentrates at the front of liquid/solid interface and restrains the growth of dendritic tip. Only when the rate of discharging solute in the interface is equal to the diffusive rate of solute from the interface to surrounding liquid phase, the growth velocity of dendritic tip will tend to a stable value.

Multi-equiaxed crystals with random crystallographic orientations of Fe-0.04%C binary alloys free growing from an undercooling melt after 0.50 × 10^{−4} min and 0.83 × 10^{−4} min were simulated. The time step was 0.17 × 10^{−}^{7} min. Assuming that the temperature in arc-shape molten pool was isothermal during the solidification. A continuous cooling condition with a constant cooling rate of 1200

K/min was imposed in the arc-shape molten pool.

^{−4} min, the primary trunks meet each other and almost stop growing as shown in

The simulated growth morphology of columnar crystals and the observed microstructure of the experiment are presented in ^{−}^{2} min. It can be seen that there are a large number of crystal nuclei on the molten pool wall, which rapidly grow and contact each other, thereby forming the fine equiaxed crystals. The unstable grains in front of the interface competitive grow as dendrites, owing to the solute diffusion and undercooling. Some of the dendrites in advantageous positions survive from competition and block other dendrites. During the process of gradually eliminating the dendrites in disadvantageous positions, the columnar crystals are formed. Due to the effect of

arc-shaped molten pool, another phenomenon should be noted that the columnar crystals on the bottom of molten pool are more concentrated than those on the edage of molten pool, and the dendritic arms of which are thinner.

In previous studies, for the simulation of dendrites freely growing from a constant undercooled melt, the calculation domain was maintained at uniform tem- perature field. However, the actual temperature field has a gradient transformation, thus affecting the growth morphology of dendrites. In order to accord with the actual solidification process of molten pool, the simulated CET was coupled with the changed temperature field in this model.

By comparing

Metal solidification theory holds that the columnar crystals’ spindles have strict crystallographic orientations as they begin to grow. The base metal grains are anisotropic on the molten pool wall of arc shape, and it is beneficial for columnar crystals to grow when the crystallographic orientations of columnar crystals’ spindles exactly match the base metal grains. However, the base metal grains are isotropic on the molten pool wall of a rectangle, and such an idealized model has a large difference with actuality.

The solid/liquid solute distributions are shown in

A newly-coupled two-dimensional CA-FD model was developed to quantitatively predict the dendritic growth and solute distribution during the solidification of Fe-0.04%C binary alloys in arc-shape molten pool. The following main results were gained:

1) The numerical simulations were performed for single equiaxial crystal and multi-equiaxed crystals with different crystallographic orientations. At the beginning of solidification, the tip growth velocity rapidly decreased, and tended to a stable value after about 0.17 × 10^{−}^{4} min, while the velocity fluctuation was less than 0.006 m/min. Dendritic morphology varied with the crystallographic orientation. The microsegregation of solute was mainly among grain boundaries or along the dendritic arms.

2) The growth morphology of columnar crystals after different times was simulated. Some of the dendrites in advantageous positions competitively grew by blocking other dendrites and become columnar crystals. Owing to the effect of molten pool shape, the columnar crystals on the bottom of molten pool were more compact compared with the columnar crystals on both sides of the molten pool. The CA simulated results for columnar crystals were found to be in agreement with theoretical prediction and experimental result.

3) The simulated dendrite morphology by CA-FD model which was coupled with a nonuniform temperature field was more consistent with solidification theory and closer to the actual solidification conditions than the original model of this article’s authors. The solute mainly enriched in the roots of dendrites, and the solute concentration of columnar crystals’ roots was higher than equiaxial crystals’ roots.

Zhang, M., Xue, Q., Li, L.L. and Li, J.H. (2017) A Two- Dimensional Computational Model for the Simulation of Dendritic Microstructure and Solute Distribution of Fe-0.04%C Alloys. World Journal of Engineering and Technology, 5, 175-187. https://doi.org/10.4236/wjet.2017.52014