**M.Khomenko, V.Niziev, F.Mirzade**

Investigation of the effect of the dissolved substance on the recrystallization of the clad deposit during the laser fusion of the heat-resistant nickel alloy powders

In general, the nickel alloys contain a lot of impurities affecting the final microstructure. The aim of this work is the numerical investigation of the effect of the soluble impurity on the clad deposit microstructure during the LF with due regard for the impact of adjoint heat transfer processes and phase transformations (melting/crystallization). The model is based on the use of the surface mechanism of the crystal phase growth. It includes the non-linear equations of heat and mass transfer and kinetic equation for the conversion fields. The phase transformation, regarded as a non-equilibrium kinetic processes is associated with the emergence and growth of crystallites in a metastable system. The kinetics of phase transformation is described on the basis of the Kolmogorov-Johnson-Mehl-Avrami (KJMA) model [11—13] that is used for conditions of the inhomogeneous distribution of the molten pool temperature.

FUNDAMENTAL EQUATIONS

The self-consistent LF model used in the paper consideing the heat and mass transfer, phase transformations, motion of the impurity of the free surface of the molten pool is as follows:

The continuity equation:

. (1)

The heat conduction equation:

(2)

The momentum equation:

. (3)

The equation for diffusion concentration:

. (4)

The equation for the free surface:

, (5)

where t is the time, p – the pressure, T – temperature,

c – thermal capacity, ρ – density, λ – thermal conductivity, ui – fluid velocity along the corresponding coordinate i = x, y, z; µ – viscosity, C – the concentration of impurities, D – diffusion coefficient, ϕ – level function, F – velocity of the free surface motion.

The second term in the right part of equation (2) is responsible for the absorption of laser radiation. The third and fourth terms are the source terms associated with a phase transformation. In equation (3), the second term in the right part is used to reset the movement velocity in the interphase area, and the latter term is responsible for the Marangoni force that is the main driving force of the molten pool. The last term in the right part of equation (4) is responsible for the capture of impurities at high speed crystallization (kp – the impurity capture coefficient). The velocity of the free surface motion is determined by the powder фввшешщт, as well as еру hydrodynamic movement of fluid [10].

The phase transformation is described with the KJMA equation for the volume fraction of a new phase:

, (6)

where σ – the shape factor, r – the particle initial radius, G (τ, t) – the rate of their growth and – the nucleation rate.

All particles located in the cell grow at the same rate at each point in time. The growth rate is determined by the following equation:

, (7)

where d0 – the lattice distance, Ea – activation energy, – entropy of crystallization.

, (8)

where EG – the Gibbs energy that in the case of surface growth is equal to the following:

. (9)

The effect of the soluble impurities is considered by introducing an effective concentration-dependent melting temperature [14]:

. (10)

THE SOLUTION ALGORITHM

For calculations we have used еру spatially non-homogeneous grids. The equations (1)–(4) have been approximated by the finite volume method, and then solved by the preliminary bioconjugated gradient method using the open-ended library of C++ class OpenFoam. The results allow to calculate the effective thermal conductivity coefficient and melting point. Further, the equations (2), (5)–(6) have been approximated by the stabilizing correction method, and solved by the double-sweep method. As a result, we have obtained the consistent temperature history of the crystallization process that using the population approximation of the KJMA equation [15] allows to obtain the crystallite distribution by the size. The average particle radius can be calculated based on the particle distribution by the size. The simplest approach is to use the total concentration of particles of all sizes. In the case of surface growth:

. (11)

Due to the multimodal particle distribution by the size, it is important to know the average volume occupied by the particles. The volume weighted radius can be calculated on the basis of proportion occupied by the particles with a certain size:

. (12)

Typically, in order to normalize the particle size during the non-isothermal crystallization it is necessary to use the typical radius associated with a peak temperature [16]. Since, in this case, the peak temperature is changed for different spatial points, such a normalization does not allow to compare the radii with each other. In this paper in order to normalize the crystallite size we have used the given supercooling δT = 45K:

. (13)

For calculations we have used the cluster of the Institute of Laser and Information Technologies of the Russian Academy of Sciences with a сapacity of 3.3 teraflop operations per second [17]. The time per calculation is approximately 24 hours.

RESULTS

The obtained 3D distributions of the temperature and volume fraction of new phase in the clad deposit are shown in Fig. 1. The nickel alloys have higher density and latent heat of melting, so the molten pool is extended, despite the low scanning velocity. At the rear of the bead there is not practically any temperature gradient due to the allocation of latent heat of fusion in this field. The distribution of the liquid phase fields allow us to estimate the depth and breadth of fusion penetration, and the bead form – of the welding height for the used parameters of the welding process.

Transfer of the dissolved impurities is carried out by the hydrodynamic flow. Figure 2a shows the distribution of concentration on the initial stage of the welding process. As it can be seen from the picture, in the area where there are no melting vortex flows the impurity concentration remains at the initial values. Figure 2b shows the distribution of concentration at the stage when the welding process is becoming the stationary one. It can be seen that the hydrodynamic flows in the properly selected modes, can mix the impurity up to the almost uniform concentration.

By the time of 1500 ms from beginning of the welding process the temperature fields and the bead height are becoming stationary. Figure 3 shows the molten pool, and еру temperature fields for different concentrations of еру dissolved impurities under the same conditions of laser fusion. It can be seen that with the higher content of impurities the molten pool is extended, and the depth of fusion penetration is somewhat increased. In the case of equal energy input we obtain large molten pools, since it is believed that increase in the amount of impurities reduces the effective melting temperature.

For a detailed study of the microstructure of the clad deposits several areas along the bead depth can be used. Figure 4 shows the dynamics of the main variables in the three areas along the bead depth (381 mkm over the substrate surface, at the substrate level and 373 mkm below the substrate level). The figure shows the value of the average particle size in the area in question. Generation of the latent heat of crystallization occurs with increase in supercooling. In the areas remoted from the substrate the process is conducted in the virtually continuous supercooling conditions. Decrease in the effective melting temperature leads to reduction in the growth velocity. In turn, it slows down (delays) the metastable process. Thus, the greater supercooling can be reached that leads to increased nucleation velocity and decreased average crystallite size.

Figure 5 shows the spatial distribution of the average crystallite size in the cross section. The area that are almost adjacent to the substrate have a crystallite size less than the areas removed from it. This is due to the fact that relatively more effective heat removal into the substrate results in a rapid reduction in temperature.

In order to verify the model used for the microstructure calculation we have used the experimental measurements of the silumin microstructure deposited on an aluminum substrate by the laser fusion [18]. Figure 6 shows that the calculation gives the higher values of the crystallite size, however, the tendency relating to decrease in size depending on the depth from the bead surface is retained. The difference in values is evidently linked to inaccurate determination of the basic kinetic constants such as the activation energy and surface tension. Anyway, the calculation can be used as the evaluation method, as well as to determine the presence (or absence) of the microstructure gradient in the deposited bead.

SUMMARY AND CONCLUSION

• We have received the dynamics of temperature fields, the distribution of impurities and the profile of the deposited bead during the laser fusion with coaxial feed of the heat-resistant nickel powders.

• We have demonstrated how the concentration of the soluble impurities affects the supercooling temperature and alters the kinetics of the crystallization process that may lead to changes in the microstructural properties of the clad deposits.

• The high bead depositions may lead to the microstructural gradients connected with different cooling rate at the surface and in the depth of the bead (due to heat conduction in the substrate).

• The results obtained make it possible to improve the recommendations during the laser fusion of the heat-resistant powders for determination of the optimal exposure modes not only in terms of geometric parameters, but also in terms of the structure of the deposited bead.

We have determined the significant role of the soluble impurity in the process of rapid crystallization of the clad deposit that can be used for planning the experiments for deposition of the heat-resistant nickel-based alloys.

The authors enclose gratitude for Dubrov A.V. for the provided calculation program for the hydrodynamic flows of the molten pool. This paper has been prepared with the financial support from the state represented by the Ministry of Education and Science of Russia under the grant agreement No.14.626.21.0001 (the unique identifier: ПНИЭHRFMEFI162614X0001) according to the federal target-oriented program "Research and development by high-priority development areas of the scientific and technological complex of Russia for 2014–2020".