Solidification of Ni-Re Peritectic Alloys

Abstract

Differential thermal analysis (DTA) and microstructural and microprobe measurements of DTA and as-cast Ni-Re alloys with compositions between 0.20 and 0.44 mass fraction Re provide information to resolve differences in previously published Ni-Re phase diagrams. This investigation determines that the peritectic invariant between liquid, Re-rich hexagonal close packed and Ni-rich face center cubic phases, L + HCP → FCC, occurs at 1561.1 °C ± 3.4 °C (1σ) with compositions of liquid, FCC and HCP phases of 0.283 ± 0.036, 0.436 ± 0.026, and 0.828 ± 0.037 mass fraction Re, respectively. Analysis of the microsegregation in FCC alloys yields a partition coefficient for solidification, k = 1.54 ± 0.09 (mass frac./mass frac.). A small deviation from Scheil behavior due to dendrite tip kinetics is documented in as-cast samples. No evidence of an intermetallic phase is observed.

Appendices

Appendices

1.1 Appendix A: Scheil for Composition Dependent Partition Coefficient

The Scheil equation in differential form is

$$ \frac{{df_{\text{S}} }}{{1 - f_{\text{S}} }} = \frac{{dC_{\text{L}} }}{{\left( {C_{\text{L}} - C_{\text{S}} } \right)}}. $$

(A1)

For C_S = (k₀ + k₁C_L)C_L, integration by partial fractions gives

$$ C_{\text{L}} (f_{\text{S}} ) = \frac{{C_{0} (1 - k_{0} )(1 - f_{\text{S}} )^{{k_{o} - 1}} }}{{1 - k_{0} - k_{1} \left( {1 - (1 - f_{\text{S}} )^{{k_{0} - 1}} } \right)}}, $$

(A2)

where C_L = C₀ for f_S = 0 has been used. An expression for C_S(f_S) is obtained as

$$ C_{\text{S}} (f_{\text{S}} ) = \left[ {k_{0} + k_{1} C_{\text{L}} (f_{\text{S}} )} \right]C_{\text{L}} (f_{\text{S}} ). $$

(A3)

1.2 Appendix B: Histogram (Probability Density Plot) for Microsegregation

The data obtained from the pixel by pixel EDS measurement technique can be used to construct a probability density plot, P(C_S)ΔC_S, which is the fraction of pixels with composition between C_S and C_S + ΔC_S. It is the probability that upon selecting a random location in the microstructure that the composition will lie between C_S and C_S + ΔC_S. Such a plot (Figure B1) gives a more sensitive measure of the microsegregation than Figure 6(c).

Fig. B1

Fraction of pixels P(C_S)ΔC_S measured with composition between C_S and C_S + ΔC_S for the Ni-20 pct Re as-cast alloy (ΔC_S = 0.01 mass fraction Re). Dashed red line shows Scheil behavior for k = 1.54

For microsegregation following the Scheil expression for constant k,

$$ C_{\text{S}} = kC_{0} (1 - f_{\text{S}} )^{k - 1} $$

(B1)

or

$$ 1 - f_{\text{S}} = \left( {\frac{{C_{\text{S}} }}{{kC_{0} }}} \right)^{{\frac{1}{k - 1}}} , $$

(B2)

P(C_S) is given by $- \frac{{df_{\text{S}} }}{{dC_{\text{S}} }}.$ Thus

$$ P(C_{\text{S}} )\Delta C_{\text{S}} = - \frac{{df_{\text{S}} }}{{dC_{\text{S}} }}\Delta C_{\text{S}} = \frac{1}{{k(k - 1)C_{0} }}\left( {\frac{{C_{\text{S}} }}{{kC_{0} }}} \right)^{{\frac{2 - k}{k - 1}}} \Delta C_{\text{S}} . $$

(B3)

Figure B1 shows the experiment probability plot for the Ni-20 pct Re as-cast materials and the corresponding Scheil histogram (C₀ = 0.1816, k = 1.54). The deviation from the Scheil prediction is evident. A large fraction of the sample, ≈ 15 pct of the pixels, have a narrow range of composition between 27 pct Re and 29 pct Re. This peak corresponds to the dendrite stalk and arm cores formed during fast solidification. The minimum near 25 pct Re leads to the observed abrupt change in backscatter contrast in images of the FCC dendritic structures in as-cast samples.

1.3 Appendix C: Dendrite Tip and Truncated Scheil

In the Scheil approach, the transport of solute in the liquid phase is assumed to be sufficiently rapid that the liquid can be treated as being uniform in composition at each temperature during cooling. First solidification begins at the dendrite tip. The diffusion in the liquid in front of each dendrite tip leads to the fact that solidification starts slightly below the liquidus temperature and with composition different than kC₀. Indeed the phase field method was used[24] to show the existence of relatively flat microsegregation profiles at the start of rapid dendritic solidification (See their Figure 12).

A simpler approach is given here. For a treatment of the kinetics of the dendrite tip in an alloy, the Ivantsov solution is obtained to determine the composition of solute at the dendrite tip. The marginal stability criterion is applied to determine the tip radius. The equations that govern the dendritic growth of binary alloy, C₀, at velocity, V, and temperature gradient, G, are now summarized.[25] The composition of liquid at the dendrite tip, C ^*_L , is given in terms of the Péclet Number, P = VR/2D; by

$$ C_{\text{L}}^{*} = \frac{{C_{\text{0}}^{{}} }}{1 - (1 - k)Iv(P)}, $$

(C1)

where R is the dendrite tip radius, k, is the partition coefficient, D is the liquid diffusion coefficient, and Iv(P) = Pexp(P)E₁(P), where E₁(P) is the first exponential integral. The tip radius is given by

$$ R = 2\pi \left[ {\frac{{T_{M} \varGamma }}{{m_{\text{L}} G_{\text{C}} \xi - G}}} \right]^{1/2} , $$

(C2)

where T_MΓ is the capillary constant and m_L is the liquidus slope. T_M is pure material melting point and Γ is the ratio of the liquid-solid surface energy and the enthalpy of fusion per unit volume. The parameter ξ is a small correction for large P given by

$$ \xi = 1 + \frac{2k}{{1 - 2k - \left( {1 + \left( {\frac{2\pi }{P}} \right)^{2} } \right)^{1/2} }}. $$

(C3)

The composition gradient in the liquid at the dendritic tip G ^*_C is obtained from the Ivanstov diffusion solution as

$$ G_{\text{C}}^{*} = - \frac{2P(1 - k)}{R}C_{\text{L}}^{*} . $$

(C4)

The tip temperature is then given by

$$ T^{*} = T_{\text{M}} + m_{\text{L}} C_{\text{L}}^{*} - T_{\text{M}} \varGamma /2R, $$

(C5)

where m_L is the liquid slope. For small temperature gradient, the dendrite tip temperature approaches the liquidus temperature as the dendrite tip velocity approaches zero. With increasing dendrite tip velocity, the dendrite tip temperature drops below the liquidus. Thus, first solidification occurs at a temperature below the liquidus.

The simplest approach to couple the dendrite tip analysis to the remainder of the solidification path is called the “Truncated Scheil” method originally proposed by Flood and Hunt[26] and modified to conserve solute by Boettinger.[25] First, the dendrite operating point (tip temperature and tip liquid composition neglecting curvature undercooling) is determined as described above. Second, a solute balance is performed using the liquid and solid compositions at the tip to determine the fraction of solid formed at the tip as

$$ C_{\text{L}}^{*} (1 - f_{\text{S}}^{\text{tip}} ) + kC_{\text{L}}^{*} f_{\text{S}}^{\text{tip}} = C_{\text{0}} $$

(C6)

$$ f_{\text{S}}^{\text{tip}} = \frac{{C_{\text{L}}^{\text{tip}} - C_{\text{0}} }}{{C_{\text{L}}^{\text{tip}} (1 - k)}}. $$

(C7)

Third, a Scheil calculation is performed for the remainder of the solidification path. The calculation for this last step of the procedure would be started with “initial” values for the liquid compositions equal to the tip liquid composition and the fraction of solid equal to f ^tip_S .

In summary,

$$ C_{\text{S}} (f_{\text{S}} ) = \left\{ {\begin{array}{*{20}l} {kC_{L}^{*} } & {0 \le f_{\text{S}} < f_{\text{S}}^{\text{tip}} } \\ {kC_{L}^{*} \left[ {\frac{{1 - f_{\text{S}} }}{{1 - f_{\text{S}}^{\text{tip}} }}} \right]^{k - 1} } & {f_{\text{S}}^{\text{tip}} \le f_{\text{S}} < 1} \\ \end{array} } \right.. $$

(C8)

Figure C1 shows three plots (a) dendrite tip solid composition vs dendrite tip velocity, (b) fraction solid at the tip vs velocity and (c) the microsegregation profile, solid composition vs fraction solid for the truncated Scheil model. Calculations are for the materials parameters in Table C1. Estimate of the solidification velocity is obtained using the expression, $V = \ell \dot{T}/(L/C),$ where ℓ is a characteristic size of the casting (3 mm), $\dot{T}$ is the estimated cooling rate (10 K/s) and (L/C) is the ratio of heat of fusion to the heat capacity of Ni (666 K). This estimate yields a velocity of ≈ 5 × 10⁻⁵ m/s, and gives a dendrite tip fraction solid of 0.11. This value roughly corresponds to the fraction solid width of the plateau seen at the start of the microsegregation profiles seen in Figure 6(c) and fraction solid width of the peak in Figure B1.

Fig. C1

Three plots: (a) dendrite tip solid composition vs dendrite tip velocity, (b) fraction solid at the tip vs velocity and (c) the microsegregation profile, solid composition vs fraction solid for the truncated Scheil model for V = 5 × 10⁻⁵ m/s. The arc connects common values of the velocity on the two graphs. Note the flat composition profile in (c) at small values of fraction solid

Table C1 Materials Parameters for Dendrite Tip Calculation