The mechanism of element inhomogeneity in TW-DED-arc fabricated γ-TiAl alloy

Abstract

The twin-wire directed energy deposition-arc (TW-DED-arc) method is a low-cost and efficient in situ alloying process for producing γ-TiAl alloy, a new generation material for aero-engine blades. Its characteristic of “twin-wire-one-drop” can successfully avoid the phenomenon of discordant melting and ineffective mixing. In this study, the mixing effect of “twin-wire-one-drop” was analysed, and droplets of different diameters were used for fabricating Ti-48Al walls. It was found that the mixing effect in the droplet was great, but there were still local unmixed areas, and a completely uniform Ti-48Al wall could be obtained by using small droplet mode. Meanwhile, incompletely mixing regions with composition difference greater than 5% appeared in many places on the sides of the Ti-48Al wall in huge droplet mode. A numerical model is established to simulate the mixing process after the droplet enters the molten pool. It is found that the secondary droplets generated in huge droplet mode are the main reason for the element inhomogeneity phenomenon. Therefore, keeping the droplet interval short and uniform is beneficial to the element in situ alloying.

Appendix: Governing equations and boundary conditions

1.1 Governing equations

• (1) Species (Ti and Al) conservation equation:

  $$\frac{\partial\left(\rho\varphi\right)}{\partial t}+\nabla\bullet\left(\rho v\varphi\right)=\varphi_s$$

  (A1)

  where ρ is the fluid density, which is substituted by the density of Ti-48Al in this model, φ is the weight percentage of alloy element. t is the time, v is a velocity vector, and φ_s is an alloy element source term caused by droplets. To save computing consumption, only φ_Al was calculated. As discussed in Sect. 3.1, the droplets enter the molten pool at intervals of 1.8 s or 0.225 s, respectively. The droplets are set to be positively spherical, and φ_Al of the 20% volume above the sphere is set to be 100% Al, φ_Al of the 80% volume below the sphere is set to be 39.11%, when φ_Al of the existing parts is set to be 48%. The difference between the Al content of the molten drop and the molten pool will cause the change of the distribution of Al content in the molten pool.

  According to Fick’s law, the diffusion behaviour of metal atoms in an alloy satisfies the following formula:

  $$J= -D\bullet \frac{dc}{dx}$$

  (A2)

  Sprengel et al. [40] measured interdiffusion in TiAl around the stoichiometry in single-phase couples Ti₅₀Al₅₀, given a D value at about 10⁻¹² m2/s. Given the assumption that the edge of the molten pool is 2 mm away from the molten pool center, and the average velocity in the molten pool is about 0.1 m/s, we can use Peclet number (Pe) to estimate the convective effect and diffusive effect.

  $${P}_{e}= \frac{{N}_{conv}}{{N}_{diff}}=\frac{LU}{D}\approx \frac{2\times {10}^{-3}\bullet 0.1}{{10}^{-12} }=2\times {10}^{8}$$

  (A3)

  Therefore, it is verified that the convective effect is several orders of magnitude larger than the diffusive effect [41], and the mentioned element inhomogeneity is mainly caused by macroscopic movement of liquid metal phases, so that the diffusive effect of alloy elements was ignored in our model [42, 43].

• (2) Mass conservation equation

  $$\frac{\partial \rho }{\partial t}+\nabla \bullet \left(\rho v\right)={m}_{s}$$

  (A4)

  where m_s is a mass source term mainly caused by droplets. Generally speaking, the increase in the mass of the entire computing domain is equal to the amount of wire feed feeding, that is, 18.6 mm³/s. Under different conditions, the droplets enter the molten pool at different frequencies, resulting in mass changes in the conservation equation. The movement of the existing workpiece also causes a change in mass. However, the volume entering the calculation domain is the same as the volume leaving the calculation domain, so the mass change of the calculation domain is mainly caused by the droplet.

• (3) Momentum conservation equation

  $$\frac{\partial \left(\rho v\right)}{\partial t}+\nabla \bullet \left(\rho vv\right)=-\nabla P+\nabla \bullet \tau +{m}_{s}v+{f}_{b}$$

  (A5)

  where P is the pressure, τ is a viscosity stress tensor, and f_b is the body force, which is referred to as gravity in this model. The gravity field is distributed along the negative Z-axis with a value of 9.8 m/s².

• (4) Energy conservation equation:

  $$\frac{\partial \left(\rho h\right)}{\partial t}+\nabla \bullet \left(\rho vh\right)=\nabla \bullet \left(K\nabla T\right)+{h}_{s}$$

  (A6)

  where h is the enthalpy, K is the thermal conductivity, T is the temperature, and h_s is an energy source. During the wire feeding process, the wires are heated and melted by the plasma arc, transferring energy to the molten pool. The droplets are set to appear at a temperature of 2200 K and enter the molten pool. During the simulation process, the upper surface of the liquid phase is always heated by the arc, and the distribution of the arc heat is discussed in detail in the following section.

• (5) VOF equation:

  $$\frac{\partial F}{\partial t}+\nabla \bullet \left(\rho vF\right)={F}_{s}$$

  (A7)

  where F is the volume fraction of the fluid in a cell and ${F}_{s}$ is a volume fraction source term caused by designed droplet.

1.2 Boundary conditions

The plasma arc heat input acted on the free surface was modelled as a fixed double-ellipse density function. The heat conduction and radiation were also applied on the free surface, while the evaporation heat loss is ignored owing to the low molten pool temperature. Considering that the arc heat was not applied to a flat workpiece, but to an existing wall with a certain shape, the height of the free surface was read and transformed in this model.

$$\begin{array}{c}{q}_{\left(x, y,z\right)}=\frac{{6\eta }_{q}{U}_{q}{I}_{q}}{\pi {\sigma }_{qx}{\sigma }_{qy}{\sigma }_{qz}}{\text{exp}}\left\{-\frac{{3\left(x-{x}_{0}\right)}^{2}}{{{\sigma }_{qx}}^{2}}-\frac{{3\left(y-{y}_{0}\right)}^{2}}{{{\sigma }_{qy}}^{2}}-\frac{{3\left({z}^{\mathrm{^{\prime}}}-{z}_{0}\right)}^{2}}{{{\sigma }_{qz}}^{2}}\right\}-{h}_{c} \left(T-{T}_{0}\right) -\sigma \varepsilon \left({T}^{4}-{T}_{0}^{4}\right)\\ \end{array}$$

(A8)

$$\sigma_{qx}=\begin{Bmatrix}\sigma_{qr}\;x\leq x_0\\\sigma_{qf}\;x>x_0\end{Bmatrix}$$

(A9)

$${z}{\prime}= \frac{z{z}_{0}}{{h}_{x,y}}$$

(A10)

where η_q is the plasma arc energy efficiency; U_q is the plasma arc voltage; I_q is the plasma arc current; σ_qx, σ_qy, and σ_qz are effective radii of the plasma arc heat in x, y, z directions; h_c is the heat transfer coefficient; T₀ is the ambient temperature (873 K); σ is the Stefan-Boltzmann constant; and ε is the surface emissivity. Specifically, σ_qr and σ_qf represent the distribution coefficients of the double ellipsoid model in the positive direction and the negative direction. h_x,y represents the z coordinate of the highest point of the molten pool at the same x and y coordinate position at this time. Due to the curved surface of the molten pool in the upper deposition, the z coordinate of the calculation element is deformed in this model, and the z coordinate value z′ is re-assigned according to its proportion to the current height of the molten pool surface.

Previous studies showed that the plasma arc energy efficiency was only 47 ~ 66% [44, 45]. Given that droplet formation takes up a portion of the arc energy, the plasma arc energy efficiency of 50% is adopted, which is lower than usual value.

The balance of plasma arc pressure and surface tension pressure on the free surface is expressed as:

$$-P+2\mu\frac{\partial v_n}{\partial n}={-P}_{arc}+\frac\gamma R$$

(A11)

$$\gamma ={\gamma }_{0}+\frac{\partial \gamma }{\partial T}\left(T-{T}_{m}\right)$$

(A12)

where μ is the fluid viscosity, v_n is a normal velocity vector, n is a surface normal vector, P_arc is the plasma arc pressure, R is the radius of the surface curvature, γ₀ is the surface tension of the fluid at the melting point, and T_m is the melting point.

The plasma arc pressure is approximated as a fixed gaussian distribution density function:

$$\begin{array}{c}{P_{arc}=P}_{max}\exp\left\{-\frac{{3\left(x-x_0\right)}^2}{\sigma_p^2}-\frac{{3(y-y_0)}^2}{\sigma_p^2}\right\}\\\end{array}$$

(A13)

P_max is the maximum plasma arc pressure and σ_p is the effective radii of the plasma arc pressure. The values are calculated and adjusted according to previous research [46] and high-speed camera records, where σ_p is 0.00351 and P_max is 600.

The Marangoni shear stress caused by the temperature gradient is applied on the free surface in the tangential direction:

$$-\mu \frac{\partial {{{v}}}_{{{t}}}}{\partial {{n}}}=\frac{\partial \gamma }{\partial T}\frac{\partial T}{\partial {{S}}}$$

(A14)

where μ is the fluid viscosity, exhibited in Fig. 3d, v_t is a tangential velocity vector, and S is a tangential vector.

The velocity is set as − 0.9 m/s at the x direction boundary. The value of φ_Al is also set as 0.3422 (48% in atomic ratio) on the wall boundary for initiation.

The details of the iteration scheme for numerically solving the governing equations with required equations can be found in our previous work [32, 46].