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A phase-field model coupled to the multiphase/multiscale model is used to simulate the microstructural morphology and predict the CET during solidification. The considered mechanism for the CET is based on interactions of solute between the equiaxed grains and the advancing columnar front. The results for the solute concentration in liquid region, dendrite tip velocity, volume fraction of the liquid and solid are presented and discussed. The phase-field model is used to simulate the dendritic morphology of an alloy directionally solidified, by imposing a constant temperature gradient. The simulation of the equiaxed grains growth requires a further important element, the growth of grains with different crystallographic orientations. The grain orientations are generated randomly for each nucleus introduced in computational domain. Finally, the coupling results between the multiphase/multiscale model and phase-field are presented and discussed. For higher nuclei density present in the melt, a shorter distance between mold wall and the equiaxed zone in the solidification process can be observed. A solute concentration boundary layer exists in the liquid along the columnar grain contour. The concentrations in the solid indicate the presence of a microsegregation pattern. The simulated results show that the solidification features are consistent with those observed based on the metallographic examinations of cast microstructures reported in the literature.

Solidification is the main phenomenon taking place during casting. This, in turn, has long been known as a relatively inexpensive means for producing metal goods. Nowadays, a sizable portion of the concepts and methods developed over the years in support of the research into solidification phenomena can be successfully and economically translated to industrial scale. Noticeable improvement can thereby be achieved insofar as the quality of the pieces manufactured by solidification is concerned. For this reason, solidification studies are not just mandatory; they truly are a powerful industrial tool. For conventional technologies, thorough understanding and control of the solidification process open wide perspectives in terms of its economic potential, since it provides the shortest distance from metal input to final product. As a consequence, solidification is one of the most important specialties in Metallurgy and Materials Science. In-the-mold solidification of a metal, opposite to what may, at first, be surmised, is not a “passive” process in any way. On the contrary, the metal undergoes a liquid- to-solid transformation of a very dynamic nature. In its course, events take place―like nucleation and growth of dendritic structures―which, in the absence of a tight control, may compromise the final output or even halt the manufacturing process altogether. Such events can originate several types of material heterogeneities which drastically affect the metallurgical quality of the final product [

First of all, we present the governing equations the multiphase/multiscale model. This model is used to predict the CET. Then, equations of phase-field model are presented and discussed.

The governing equations are based directly as in [

where e_{s} is solid, e_{l} is the extradendritic liquid and e_{d}_{ }represents the interdendritic liquid. As shown in

The grain envelope volume fraction is defined as_{l}) is below_{l} can become greater than C_{0} (initial concentration). The presence of a solute profile in the solid in _{s}) is much smaller than that of the liquid (D_{l}), this results in incomplete solute diffusion in the solid. In fact, back-diffusion in the solid is neglected in the present study for the multiphase/multiscale model. For the one-dimensional system considered in the present study used to predict the CET during solidification of Al-Cu binary alloys, the interdendritic liquid solute and extradendritic liquid solute can be written, respectively, as in [

where k_{0} is the solute partition coefficient and S_{e} represents the envelope area of grains. The last term in equation (2) and (3) accounts for diffusion of solute from the dendritic growth into the undercooled extradendritic liquid. The d_{e} represents the diffusion length and it is discussed in greater details in [

where T_{f} is the melting point of the pure metal, and m_{l} is the liquidus line slope. Then, Equation (2) can be solved for the solid fraction, e_{s}, and Equation (3) provide the average solute concentration in the extradendritic liquid, C_{l}. The following equation is used to calculate the grain (or extradendritic liquid) volume fraction:

where V is the dendrite tip velocity [

where

The dimensionless undercooling (W) is defined as

Note that in Equation (8) the undercooling is defined relative to average solute concentration in the undercooled extradendritic liquid _{1}). The following expressions are used to obtain a first-order estimate of the characteristic half spacing (R_{f}) in the two types of growth:

equiaxed:

As proposed in [

The envelope diffusion length (d_{e}) is calculated as a function of the envelope Peclet number (P_{e}), from

where the instantaneous envelope radius (R_{e}) is given by.

The Equation (11) differs slightly from the one proposed in [_{0} instead of C_{l} in the dimensionless undercooling. Governing equations of phase-field model are presented in the following section.

In phase-field models, the state of the domain is customarily represented by a distribution of the single variable known as the “order parameter” or “phase-field variable,” here represented by the Greek letter f. In this present study, it is assumed that the solid state corresponds to a value of +1 for the order parameter, while, in the liquid region, f is taken to be 0. The region through which f decreases from +1 to 0 is defined as the solid/liquid interface [

The evolution of the solid region with time ^{2}Ñf is a diffusivity term. The next component of the equation, _{m}, the molar volume. The arguments to the natural logarithms, _{L}_{ }and C_{S}. The

In this equation D(f) is the solute diffusivity in the solid and in the liquid regions. The model carried out, takes into account solute diffusivity in the liquid and in the interface regions. The “smoothing” function h(f) and the function g(f), which models the surface tension effect around the interface, are defined as [

Equations (15) and (16) are widely employed in phase-field works. Notice

where d_{e} gauges the anisotropy. The value j controls the number of preferential growth directions. For example, with j = 0 we shall be looking at a perfectly isotropic case, while j = 4 is indicative of a dendrite with four preferential growth directions. Orientation of the maximum-anisotropy interface is identified by the q_{0} of Equation (17). Furthermore, e_{0} in that equation, and w in Equation (13) are model parameters associated with interface energy (s) and thickness (2l), respectively, according to the following expressions [

where m_{e} is the slope of the liquidus line at equilibrium, k_{e} is the equilibrium partition coefficient, D_{i} is the diffusion coefficient in the interface region, and the kinetic coefficient, b, is defined to be the inverse of the usual linear kinetic coefficient, m_{.k.}. For the binary alloy system, we use the same parameters shown in the [

with “r” a random number between −1 and +1. The “a” parameter is the noise amplitude. Maximum noise corresponds to f = 0.5, at the center of the interface, whereas at f_{ }=_{ }0 (liquid region) and f = 1 (solid region) there occurs no noise. That is to say, noise is generated at the interface. The Equations (13) and (14) were solved numerically. They were discretized on uniform grids using an explicit finite scheme.

The main objective of the present work is to propose a numerical simulation of microstructure evolution by phase-field method coupled to the solutal interaction mechanism for the columnar-to-equiaxed transition (CET). First of all, we show results of the CET model, and then, results obtained by phase-field method are presented, and finally the results of coupling between CET model and phase-field. Simulations in this present study are carried out disregarding the energy equation and instead imposing the following linear temperature profile:

where T_{0} is the initial temperature and

We begin by displaying the results obtained with help of the CET model for predictions of temporal evolution of the equiaxed grain volume fraction in the two cases. The solutal interactions are quantitatively examined in this section for purely equiaxed growth. Simulations are performed for a cooling rate of 0.005 K/s. The results are presented for two different equiaxed nuclei densities equal to 2.4 × 10^{+5} and 8.8 × 10^{+6} m^{?3}. According to Equa-

diffusivity in liquid region, D_{l} (m^{2}∙s^{−1}) | 3 × 10^{−9} |
---|---|

diffusivity in solid region, D_{s} (m^{2}∙s^{−1}) | 3 × 10^{−18} |

slope of liquidus line, m_{e} (K∙mol^{−1}) | 672 |

partition coefficient, k_{e} (−) | 0.14 |

liquidus temperature, T_{L} (K) | 923 |

Gibbs-Thomson coefficient, G (m∙K) | 2.41 × 10^{−7} |

molar volume, V_{m} (m^{3}∙mol^{−1}) | 1.095 × 10^{−}^{5} |

tion (9), these nuclei densities give the following characteristic half-spacings for the equiaxed grains, respectively: R_{f} = 10 and 3 mm. _{g}) in the that cases, together with the temperature variation. The present work and literature [^{+5} m^{?3}. One can see from _{e}) in Equation (10). Starting at 200 seconds, the grains undergo a period of rapid growth until the grain volume fraction (e_{g}) reaches values of about 0.95 at 350 seconds. This rapid increase in grain volume fraction (e_{g}) can be attributed to S_{e} in Equation (10) and to the solutal undercooling in Equation (8) for the dendrite tip velocity simultaneously reaching larger values. The time when the solutal undercooling (W) in Equation (8) reaches a negligibly small value and dendritic growth ceases is marked as an asterisk in ^{+6} m^{?3}, except that the undercooling dissipates earlier due to the somewhat larger values of S_{e} at early times. Also, the maximum equiaxed grain fraction, corresponding to the time when the undercooling vanishes, is about 0.7.

Through of numerical example (

^{+5} m^{−3}. On the other hand, when the nuclei density is 8.8 ´ 10^{+6} m^{−3}, the solutal under-cooling dissipates in about 260 seconds. When solutal under-cooling reaches a negligibly small value, this means that not only the equiaxed dendritic growth stops, but also the columnar growth, in this situation we can determine the columnar-to-equiaxed transition (CET) in solidification of Al-0.013 mol% Cu alloy. The main feature of the CET model is that both the equiaxed and columnar dendrite growth velocities are a function of a solutal under-cooling proportional to the difference between the local liquidus concentration and the local average solute concentration in the extradendritic liquid, in Equation (8). ^{t}^{+Δt}) and said growth velocity (V) during the solidification process.

The two different cases shown in ^{+6} m^{−3} case, the equiaxed grain density is relatively large, as would case if an inoculants (grain refiner) had been used, one can see from ^{+5} m^{−3} case, the curve profile is similar to the previous case, i.e., one can see for said case that the columnar front velocity is initially very slow until 40 seconds, after this, the velocity undergo a abrupt increase until reaches a maximum value of about 8.5 × 10^{−5} m/s at 320 seconds, then decreases to zero, it indicates that columnar grains stopped growing, and the CET occurred too. Both velocity and position curves are consistent; because when velocity decreases to zero the columnar front position becomes constant at the time. One can see, that at the time when the velocity is zero (at about 340 seconds), the columnar front position is constant. The

knowledge of the columnar front position is needed in order to decide which equation to use to calculate the characteristic half-spacing (R_{f}). The model results indicate that the CET occurred when the columnar front position has a maximum and the velocity of the columnar front is almost zero (^{+5} m^{−3} case. One can see that, until about 200 seconds the control volume is totally extradendritic liquid. As said liquid decreases both the solid and interdendritic liquid increases until about 350 seconds. At the 350 seconds, the equiaxed grains have enough time to reach a sufficiently high volume fraction to block the columnar grains growth. In this time, the extradendritic liquid fraction (l) becomes very small and the control volume is almost totally solid (s) and interdendritic liquid (d). Starting at 350 seconds, the solid fraction keeps the growth, while interdendritic liquid (d) decreases, i.e., from that time the interdendritic liquid (d) becomes solid. An interesting behavior concerning progress of solidification can be observed through results in

In this subsection, we present the results for numerical simulation of solutal interaction mechanism for the columnar-to-equiaxed transition. The results obtained by phase-field model are presented in the following subsection.

To calculate the governing equations, there are seven unknown values. Three of them are phase-field parameters. W and e are determined by solving Equations (18) and (19) simultaneously. Since the phase-field mobility is a function of temperature, it should be calculated with the temperature during the computation. The values of solute concentration in liquid, c_{L}, and solid, c_{S}, are also required; they are determined from Equation (14) depending on the values of f and c at each point and every time step. The governing Equations (13) and (14) above are solved numerically, using a finite-difference scheme. In the calculations, the system temperature is uniform and continuously decreased with a constant cooling rate from the initial temperature (T_{0}), which is slightly lower than the liquidus temperature of the Al-Cu alloy. We analyze columnar growth in a two-dimensional system. The phase field model is used to calculate the alloy directionally solidified from left to the right side of computational domain, by imposing a constant temperature gradient in an undercooled melts system. Solidification in the presence of walls is of great practical importance. In casting, solidification usually starts by heterogeneous crystal nucleation on the walls of the mold. Those crystals that grow from mold walls are known as columnar dendritic grains. The concentration field, growth and selection of columnar dendritic grains with constant cooling rate equal to 0.005 K/s for Al-0.013 mol% Cu are shown in

represents solute concentration, while the white represents the concentration of the segregated copper from solid to liquid region. The copper concentration in solid is much less than that of copper in liquid. All the columnar dendritic grains are growing from left surface but the shapes of the arms are different from each other due to the imposed noise by Equation (22). In this simulation, a ﬁne solid layer was added to left side of the domain boundaries.

During the calculations, the system temperature is uniform and continuously decreased with a constant cooling rate from the initial temperature of 922 K, which is slightly lower than the liquidus temperature of the Al-0.013 mol% Cu alloy. The advanced columnar grains in ^{?3} sec, 8.8 × 10^{?3} sec, and 2.38 × 10^{?2} sec are shown in Figures 7(a)-(c). The equiaxed grains grow faster along the crystallographic orientations. At the early stage, as shown in

The competitive growth of these grains during solidification process can be observed. Due to the solute redistribution from solid to liquid region, the interdendritic liquid just ahead of the interface always has a composition greater than liquid region distant of solid-liquid interface. While the copper concentration in solid region is the lowest, the highest concentration corresponds to the interdendritic liquid. The simulation patterns of dendritic growth shows typical equiaxed grains structure, and comparing these simulations (Figures 7(a)-(c)), one can see that the equiaxed grains structure is very sensitive to the competitive growth. The results of the coupled models are showed in the following subsection.

In spite of phase-field models being suitable for simulating solidification processes, they suffer from low computational efficiency. For example, for computation of a dendrite with side-branches, i.e. secondary and tertiary arms, the computational domain should be discretized into one millions points. Thus, the computational stability condition in an explicit finite scheme can be guaranteed only with a very small time step [^{+12} and 24.8 × 10^{+12} m^{−3}), in order reduce the solidification time and consequently the computer run time. With the increase in nuclei density in the bulk liquid, more equiaxed grains are nucleated. This, in turn, leads to a finer grain size in casting. Grain refinement by inoculation involves addition of particles which can act as substrates for heterogeneous nucleation. Inoculation is particularly widely practiced in the aluminium industry [_{g}) versus solidification time, in which the columnar-to-equiaxed transition (CET) can be determined. Despite of the profiles for the two cases are qualitatively similar (

For nuclei density equal to 8.4 × 10^{+12} m^{?3}, one can see that columnar-to-equiaxed transition takes place at 0.2 seconds. On the other hand, for a value equal to 24.8 × 10^{+12} m^{?3}, the columnar-to-equiaxed transition takes place earlier to the 0.13 seconds. This transition times are input data for phase-field model, the results are showed to following.

Figures 9(a)-(d) show the morphology of dendrites which are calculated for an initial concentration of 0.013 mole fraction. To simulate the chill effect near the mould wall, a fine solid layer was added at the bottom of

the computational domain before simulation. The fine solid layer becomes unstable in a supercooled domain, with branching advancing into the liquid region (^{+12} m^{−3}. For the said density, the columnar-to-equiaxed transition takes place at 0.13 seconds, which is determined by multiphase/multiscale model, as previously indicated in

Through comparison between

During the solidification process the extradendritic liquid (l) decreases, while both solid (s) and interdendritic liquid (d) increase gradually through that period. After the columnar-to-equiaxed transition (CET), the said interdendritic liquid begins to solidify, suggesting that solidification process approaches its ending (_{l}. In the [_{0}, to determine the undercooling for dendrite growth. In the [_{l} for equiaxed growth, but C_{0} for columnar growth. In this present study, C_{l} is used for both equiaxed and columnar growth. This difference constitutes the main new feature of the CET model. The dendritic morphology during the columnar-to-equiaxed transition (CET) in directional solidification is simulated using the phase-field model. The model relies on the solute conservation equation (Equation (14)) and a phase-field equation (Equation (13)) on the scale of the developing microstructure. A solute entrapment can be observed in the region next to the columnar grains and equiaxed (_{0}. The concentrations in the solid indicate the presence of a microsegregation pattern. It should be observed that the phase-field calculations physically reproduces competitive growth of the columnar grains from the perturbed interface (

A model based on coupling between the multiphase/multiscale model and phase-field was proposed in the present study. The multiphase/multiscale model proposed in the [_{0}). The columnar-to-equiaxed transition (CET) will occur, when the solute rejected from the equiaxed grains is sufficient to dissipate the solutal undercooling at

the columnar front, such that C_{l} has increased to

the columnar-to-equiaxed transition (CET) is strongly influenced by nuclei density (n), as suggested by previous studies [

Alexandre FurtadoFerreira,Ingrid Meirelles SalvinoTomaskewski,Késsia GomesParadela,Dimas Moraes daSilva,Roberto CarlosSales, (2015) Numerical Simulation of Microstructural Evolution via Phase-Field Model Coupled to the Solutal Interaction Mechanism. Materials Sciences and Applications,06,907-923. doi: 10.4236/msa.2015.610092