Effectiveness of Ni-based diffusion barriers in preventing hard zone formation in ferritic steel joints

Abstract

A numerical procedure based on finite difference method was used to simulate the formation of ‘hard’ and ‘soft’ zones, in dissimilar weldments of 9Cr–1Mo and 2¼Cr–1Mo steels during high temperature exposure. Kinetic analysis of the calculated diffusion profiles showed that the activation energy for carbon diffusion in Cr–Mo steels is marginally higher than that in Fe–C system. Calculations were extended to incorporate the effect of Ni-based interlayers between 2¼Cr–1Mo and 9Cr–1Mo ferritic steels. The presence of a diffusion barrier was found to reduce the propensity for formation of hard and soft zones, which is related to the interaction parameter $\varepsilon_{\rm C}^{\rm M}.$ Thickness of the interlayer required to suppress the formation of hard zone was optimized by the calculations. Transition joints of ferritic steels with Inconel 182 as the interlayer of thickness close to that predicted by the computations were fabricated and exposed to elevated temperature. Microstructural studies and hardness measurements further confirmed the effectiveness of Ni-based interlayers in preventing hard zone formation.

Appendix

Simulation of carbon diffusion profiles.

The sole objective of the model is to demonstrate the effectiveness of Ni-based interlayer in preventing the hard zone formation. For this purpose a simple single-phase model is considered, which does not take into account the precipitation of carbides.

The change in concentration of a diffusing element with respect to time can be obtained from the well-known Fick’s second law of diffusion,

$$ \frac{\partial C}{\partial t} = \frac{\partial }{\partial x}\left( {D\frac{\partial C}{\partial x}} \right) $$

(2)

where ‘D’ is the diffusion coefficient of the element in solid solution, ‘C’ is the concentration of the element in solution, ‘t’ is the time, and ‘x’ is the distance. The system under consideration is a ternary system of Fe–M–C type where ‘M’ indicates any major substitutional alloying element like Cr, Ni, or Mo. In the present case, the major alloying element considered is Cr. For ternary diffusion, Fick’s second law can be rewritten using Onsager’s equation [5] as

$$ \frac{{\partial C_{1} }}{\partial t} = \frac{\partial }{\partial x}D_{11} \frac{{\partial C_{1} }}{\partial x} + \frac{\partial }{\partial x}D_{12} \frac{{\partial C_{2} }}{\partial x} $$

(3)

$$ \frac{{\partial C_{2} }}{\partial t} = \frac{\partial }{\partial x}D_{21} \frac{{\partial C_{1} }}{\partial x} + \frac{\partial }{\partial x}D_{22} \frac{{\partial C_{2} }}{\partial x} $$

(4)

where C₁ and C₂ refer to carbon and chromium concentration in solution. D₁₁ and D₂₂ are the self-diffusion coefficients of carbon and chromium in solution, respectively. D₁₂ and D₂₁ are the cross diffusion coefficients, which take into consideration the effect of one element on the other. Expanding the R.H.S. of Eqs. 3 and 4 we get

$$ \frac{{\partial C_{1} }}{\partial t} = \frac{{\partial D_{11} }}{\partial x}\frac{{\partial C_{1} }}{\partial x} + D_{11} \frac{{\partial^{2} C_{1} }}{{\partial x^{2} }} + \frac{{\partial D_{12} }}{\partial x}\frac{{\partial C_{2} }}{\partial x} + D_{12} \frac{{\partial^{2} C_{2} }}{{\partial x^{2} }} $$

(5)

$$ \frac{{\partial C_{2} }}{\partial t} = \frac{{\partial D_{21} }}{\partial x}\frac{{\partial C_{1} }}{\partial x} + D_{21} \frac{{\partial^{2} C_{1} }}{{\partial x^{2} }} + \frac{{\partial D_{22} }}{\partial x}\frac{{\partial C_{2} }}{\partial x} + D_{22} \frac{{\partial^{2} C_{2} }}{{\partial x^{2} }} $$

(6)

Since the contribution due to $\frac{{\partial D_{11} }}{\partial x},$ $\frac{{\partial D_{12} }}{\partial x},$ $\frac{{\partial D_{22} }}{\partial x},$ and $\frac{{\partial D_{21} }}{\partial x}$ is zero, they are neglected. Then the above equations can be simplified as

$$ \frac{{\partial C_{1} }}{\partial t} = D_{11} \frac{{\partial^{2} C_{1} }}{{\partial x^{2} }} + D_{12} \frac{{\partial^{2} C_{2} }}{{\partial x^{2} }} $$

(7)

$$ \frac{{\partial C_{2} }}{\partial t} = D_{21} \frac{{\partial^{2} C_{1} }}{{\partial x^{2} }} + D_{22} \frac{{\partial^{2} C_{2} }}{{\partial x^{2} }} $$

(8)

Using Schmit method [24] the differential Eqs. 7 and 8 can be transformed to finite differential equations using a one-dimensional grid of mesh points in space x and time t. The mesh points are separated by a space increment of Δx and time increment Δt. For carbon, the second differential of concentration with respect to space can be written as

$$ \frac{{\partial^{2} C_{1} }}{{\partial x^{2} }} = \frac{{C_{1} \left[ {\left( {x + 1,t} \right)} \right] - 2C_{1} \left[ {x,t} \right] + C_{1} \left[ {\left( {x - 1,t} \right)} \right]}}{{\Updelta x^{2} }} $$

(9)

and the first differential of concentration with respect to time is

$$ \frac{{\partial C_{1} }}{\partial t} = \frac{{C_{1} [x,(t + \Updelta t)] - C_{1} [x,t]}}{\Updelta t} $$

(10)

Similarly for chromium,

$$ \frac{{\partial^{2} C_{2} }}{{\partial x^{2} }} = \frac{{C_{2} \left[ {\left( {x + 1,t} \right)} \right] - 2C_{2} \left[ {x,t} \right] + C_{2} \left[ {\left( {x - 1,t} \right)} \right]}}{{\Updelta x^{2} }} $$

(11)

$$ \frac{{\partial C_{2} }}{\partial t} = \frac{{C_{2} \left[ {x,(t + \Updelta t)} \right] - C_{2} \left[ {x,t} \right]}}{\Updelta t} $$

(12)

Substituting Eqs. 9, 10, and 11 in Eq. 7 and Eqs. 9, 11, and 12 in Eq. 8, the change in concentration with respect to time for a particular grid point is obtained as

$$ C_{1} \left[ {x,(t + \Updelta t)} \right] = C_{1} \left[ {x,t} \right] + A + B $$

(13)

$$ C_{2} \left[ {x,(t + \Updelta t)} \right] = C_{2} \left[ {x,t} \right] + C + D $$

(14)

where

$$ A = \frac{{D_{11} \Updelta t}}{{\Updelta x^{2} }}\left\{ {C_{1} \left[ {(x + 1),t} \right] - 2C_{1} \left[ {x,t} \right] + C_{1} \left[ {(x - 1),t} \right]} \right\} $$

(14a)

$$ B = \frac{{D_{12} \Updelta t}}{{\Updelta x^{2} }}\left\{ {C_{2} \left[ {(x + 1),t} \right] - 2C_{2} \left[ {x,t} \right] + C_{2} \left[ {(x - 1),t} \right]} \right\} $$

(14b)

$$ C = \frac{{D_{21} \Updelta t}}{{\Updelta x^{2} }}\left\{ {C_{1} \left[ {\left( {x + 1} \right),t} \right] - 2C_{1} \left[ {x,t} \right] + C_{1} \left[ {\left( {x - 1} \right),t} \right]} \right\} $$

(14c)

$$ D = \frac{{D_{22} \Updelta t}}{{\Updelta x^{2} }}\left\{ {C_{2} \left[ {(x + 1),t} \right] - 2C_{2} \left[ {x,t} \right] + C_{2} \left[ {(x - 1),t} \right]} \right\} $$

(14d)

To calculate the change in the values of C₁ and C₂ for each mesh element as a function of Δt from Eqs. 13 and 14, the values of D₁₁, D₁₂, D₂₁, and D₂₂ should be known. These values can be calculated as follows.

According to Birchenall and Mehl [25] flux of any diffusing element can be expressed in terms of its activity as

$$ J = - D_{\text{a}} \frac{{{\hbox{d}}a_{\text{C}} }}{{\hbox{d}}x} $$

(15)

where D_a is the activity diffusion coefficient of carbon. If it is assumed that component 1 (carbon) is a relatively dilute interstitial and component 2 (chromium in this case) is a dilute substitutional component of a ternary alloy, then

$$ \frac{{D_{12} }}{{D_{11} }} \cong \varepsilon_{12} X_{1} $$

(16)

where X₁ is the mole fraction of component 1. In terms of activity coefficient γ₁, the Wagner interaction parameter is given as

$$ \varepsilon_{12} = \left. {\frac{{\partial \left( {\ln \gamma_{1} } \right)}}{{\partial X_{2} }}} \right|_{{X_{1} ,X_{2} = 0}} $$

(17)

In such a case D_a can be assumed to be constant. This assumption has been used earlier by Buchmayr [9] and Kucera et al. [26] for simulating the carbon concentration profiles in dissimilar weldments. For ternary alloys, activity of carbon a_C is given by Eq. 1 [16] where the values of interaction parameter are taken as $\varepsilon_{\text{C}}^{\text{C}}$ = +1.33 and $\varepsilon_{\text{C}}^{\text{Cr}}$ = −72 [22] for Fe–Cr–C system and $\varepsilon_{\text{C}}^{\text{Ni}}$ = +2 [23] for Ni–Cr–C system. Upon substituting Eq. 1 in Eq. 15, the flux can be rewritten as

$$ J = - \left\{ {\left[ {D(0)(1 + \varepsilon_{\text{C}}^{\text{C}} C{}_{1})\exp (\varepsilon_{\text{C}}^{\text{C}} C_{1} + \varepsilon_{\text{C}}^{\text{Cr}} C_{2} )} \right]\frac{{\partial C_{1} }}{\partial x} + \left[ {D(0)C_{1} \varepsilon_{\text{C}}^{\text{Cr}} \exp (\varepsilon_{\text{C}}^{\text{C}} C_{1} + \varepsilon_{\text{C}}^{\text{Cr}} C_{2} )} \right]\frac{{\partial C_{2} }}{\partial x}} \right\} $$

(18)

Comparing the above equation with Fick’s first law

$$ J = - \left\{ {[D_{11} ]\frac{{\partial C_{1} }}{\partial x} + [D_{12} ]\frac{{\partial C_{2} }}{\partial x}} \right\} $$

(19)

the values of diffusion coefficients for carbon can be obtained as

$$ D_{11} = D_{C} (1 + \varepsilon_{\text{C}}^{\text{C}} C_{1} )\exp (\varepsilon_{\text{C}}^{\text{C}} C_{1} + \varepsilon_{\text{C}}^{\text{Cr}} C_{2} ) $$

(20)

and

$$ D_{12} = D_{11} (\varepsilon_{\text{C}}^{\text{Cr}} C_{2} ) $$

(21)

Similar analogy can be used to calculate diffusion coefficients for chromium as

$$ D_{22} = D_{\text{Cr}} (1 + \varepsilon_{\text{Cr}}^{\text{Cr}} C_{2} )\exp (\varepsilon_{\text{C}}^{\text{Cr}} C_{1} + \varepsilon_{\text{Cr}}^{\text{Cr}} C_{2} ) $$

(22)

and

$$ D_{21} = D_{22} (\varepsilon_{\text{C}}^{\text{Cr}} C_{1} ) $$

(23)

$\varepsilon_{\text{Cr}}^{\text{Cr}}$ is the Cr–Cr interaction parameter, which is taken as −5 [22]. D_C and D_Cr are concentration independent diffusion coefficients, D(0), for carbon and chromium, respectively. In the calculation, values of D_C and D_Cr are taken as 1.05 × 10⁻¹¹ m²/s and 2.51 × 10⁻¹⁷ m²/s, respectively, in ferritic steel and as 1.465 × 10⁻¹⁴ m²/s and 1.48 × 10⁻¹⁹ m²/s, respectively, in Ni-based interlayer [21].

The above-calculated values for diffusion coefficients are substituted in Eqs. 14a–14d. Knowing the values of C₁ and C₂ for the mesh points x, x + 1, and x − 1 for a particular time ‘t,’ C (x, t + Δt) can be calculated. The calculations were carried out on a one-dimensional finite difference mesh with 100 mesh points. To minimize the error, the width of each mesh point Δx is kept as 0.003 mm. Time increment is calculated from the criteria

$$ \Updelta t \le \frac{{\Updelta x^{2} }}{2D} $$

(24)

Detailed flow chart of the computer program that was developed to solve the numerical equations is given in Fig. 1.