Decoupled growth mechanism of Fe₄₀Ni₄₀P₁₄B₆ eutectic alloy

Abstract

Decoupled growth often occurs in the nonfacetted-facetted eutectic systems. And it is generally considered that the nonfacetted solid solution acts as the leading phase in the decoupled growth. In this work, Fe₄₀Ni₄₀P₁₄B₆ eutectic alloys were systematically studied via solidification of undercooled melts and crystallization of amorphous alloys. Upon solidification of melts subjected to different undercoolings, as the undercooling increases, the growth mechanism develops from cooperative growth to decoupled growth. Upon crystallization of amorphous alloys, the partially crystallized sample consists only of strongly faulted intermetallic (Fe,Ni)₃(P,B) with chemical composition deviating from stoichiometry. Formation of supersaturated solid solution γ(Fe, Ni) in the solidification and supersaturated intermetallic (Fe,Ni)₃(P,B) in the amorphous crystallization indicates that decoupled growth results from solute trapping and disorder trapping in rapid growth of solid solution and intermetallic, respectively. Further application of rapidly quenched experiments and theoretical analysis declare that the decoupled growth results from a competition between the growth of γ(Fe, Ni) and (Fe,Ni)₃(P,B), which are controlled by solute trapping and disorder trapping, respectively.

Appendix—Theoretical Analysis of Phase Competition

Crystalline growth of an alloy shows much more complexity than that of a pure material. Besides the advancement of the interface, one also needs to consider the solute redistribution across the interface.12 When the interface conditions deviate significantly from the local equilibrium, one needs to consider a nonequilibrium partition coefficient, k. Several models have been developed to correlate the nonequilibrium partition coefficient, k, with the interface conditions, and the most widely used model for a dilute binary alloy is of Aziz12:

$$k\left( v \right) = \frac{{{X_{\text{S}}}}} {{{X_{\text{L}}}}} = \left( {{k_{\text{e}}} + \frac{v} {{{v_{\text{d}}}}}} \right)/\left( {1 + \frac{v} {{{v_{\text{d}}}}}} \right),$$

((A1))

where k is the ratio of the solute concentration in the growing solid (X_S) to that in the liquid (X_L) at the interface, k_e is the equilibrium value of k, v is the interface velocity, and v_d is the diffusive speed discussed above. From this equation, it is evident that when the interface velocity is much larger than the interface diffusion velocity, k approaches unity and "partitionless" solidification can occur.

For metallic alloys, a collision-limited relationship is assumed between velocity and driving force under all circumstances.

$$v\left( {T,{X_{\text{S}}},{X_{\text{L}}}} \right) = {v_{\text{S}}}\left[ {1 - \exp \left( {\Delta {G_{{\text{eff}}}}\left( {T,{X_{\text{S}}},{X_{\text{L}}}} \right)/RT} \right)} \right]\,$$

((A2))

where ΔG_eff is the entire free energy dissipated at the interface, R the universal gas constant, T the interfacial temperature and the kinetic prefactor, v_s is the rate of the forward flux alone, and corresponds to the hypothetical maximum growth velocity at infinite driving force.

Since the undercooling is relatively small, solute diffusivity in undercooled free solidification is treated to be independent of underccoling for simplicity, and the values of v_s and v_d are assumed to be constants (in Aziz’s model,12v_s is assumed to be the velocity of sound, much larger than v_d). So the S-L interface velocity is only thermodynamic-controlled and increases continuously with the undercooling. However, this deviates a lot from Wang’s experiment19 about the solidification of Fe-10% Sb system, where the growth velocity of α-Fe dendrite does not always increase with the undercooling but for extremely large undercooling, decreases with the undercooling. So, for large undercooling (that is the case for amorphous crystallization), solute diffusivity is highly dependent on temperature, and v_s and v_d could not be treated as constants. Here, the Arrhenius relationship is assumed for v_s and v_d:

$${v_{\text{s}}} = {v_{{\text{s0}}}}\exp \left( { - E/RT} \right),$$

((A3a))

$${v_{\text{d}}} = {v_{{\text{d0}}}}\exp \left( { - E/RT} \right),$$

((A3b))

where v_s0 and v_d0 are constants and E is the activation energy.

If the liquid and solid phases are sufficiently dilute that Henry’s law is valid, an expression for ΔG_eff can be obtained12 as

$$ {\frac{{\Delta {G_{{\text{eff}}}}}} {{RT}}} s\;\;\;\ = \;\;\;\ {{X_{\text{S}}}\ln \left( {\frac{k} {{{k_{\text{e}}}}}} \right) + \left( {{X_{\text{L}}} + X_{\text{S}}^{\text{e}} - {X_{\text{S}}} - X_{\text{L}}^{\text{e}}} \right)}\\ {} \;\;\;\ = \;\;\;\ {k{X_{\text{L}}}\ln \left( {\frac{k} {{{k_{\text{e}}}}}} \right) + \left( {1 - k} \right){X_{\text{L}}} + \left( {{k_{\text{e}}} - 1} \right)X_{\text{L}}^{\text{e}},}\\ $$

((A4))

where $X_{\text{S}}^{\text{e}}$ and $X_{\text{L}}^{\text{e}}$ are the equilibrium values of X_S and X_L, respectively. If the phase diagram liquidus and solidus boundaries are linear for the compositions of interest, they can be described by

$$T = {T_{\text{M}}} + m_{\text{L}}^{\text{e}}X_{\text{L}}^{\text{e}}\quad \left( {{\text{liquidus}}} \right),$$

((A5a))

$$T = {T_{\text{M}}} + m_{\text{S}}^{\text{e}}X_{\text{S}}^{\text{e}}\quad \left( {{\text{solidus}}} \right).$$

((A5b))

Eutectic system Fe₄₀Ni₄₀P₁₄B₆ consists of solid solution γ(Fe, Ni) and intermetallic (Fe,Ni)₃(P,B). On one hand, because the solubility of Fe and Ni in intermetallic (Fe,Ni)₃(P,B) is negligible compared to that of P and B in solid solution γ(Fe, Ni), the value of k_e$k_{\text{e}}^{{\text{II}}} >k_{\text{e}}^{\text{I}},X_{\text{L}}^{{\text{II}}} = X_{\text{L}}^{\text{I}}$ is smaller for intermetallic than for the solid solution. On the other hand, because the composition of eutectic point (Fe₄₀Ni₄₀P₁₄B₆) is much closer to that of the intermetallic (Fe,Ni)₃(P,B) than γ(Fe, Ni), the value of X_L is smaller for (Fe,Ni)₃(P,B) than for γ(Fe, Ni), assuming that X_L does not change during the transformation.

Free energy, ΔG_eff has significant impact on interface velocity. Larger absolute value of ΔG_eff/RT (ΔG_eff/RT is negative) suggests larger interface velocity. Take the derivative of Eq. (A4) with respect to k_e and X_L, respectively.

$$\frac{{\partial \frac{{\Delta {G_{{\text{eff}}}}}} {{RT}}}} {{\partial {k_{\text{e}}}}} = \frac{1} {{{k_{\text{e}}}}}\left( {X_{\text{S}}^{\text{e}} - k{X_{\text{L}}}} \right) < 0.$$

((A6a))

$$\frac{{\partial \frac{{\Delta {G_{{\text{eff}}}}}} {{RT}}}} {{\partial {X_{\text{L}}}}} = k\ln \left( {\frac{k} {{{k_{\text{e}}}}}} \right) + \left( {1 - k} \right) >0.$$

((A6b))

It can be seen for Eq. (A6) that the increase in k_e and the decrease in X_L make the free energy and the interface velocity increase. Take the derivative of Eqs. (A6a) and (A6b) with respect to k, respectively.

$$\frac{{{\partial ^2}\frac{{\Delta {G_{{\text{eff}}}}}} {{RT}}}} {{\partial {k_{\text{e}}}\partial k}} = - \frac{{{X_{\text{L}}}}} {{{k_{\text{e}}}}} < 0,$$

((A7a))

$$\frac{{{\partial ^2}\frac{{\Delta {G_{{\text{eff}}}}}} {{RT}}}} {{\partial {X_{\text{L}}}\partial k}} = \ln \left( {\frac{k} {{{k_{\text{e}}}}}} \right) >0.$$

((A7b))

It can be seen for Eq. (A7) that as k increases (induced by the increase in the growth velocity), the extent of influences of k_e and X_L on the growth velocity decreases and increases, respectively.

To further illustrate how k_e and X_L influence the growth velocity, the full numerical solution of the equation set [Eqs. (A1)-(A5)] is plotted in Fig. A1, which is interface temperature versus relative velocity (v/v_s)or k for fixed X_L. The parameters used in the calculation are listed in Table A1. Curve II and curve I correspond to the same X_L value but to different k_e values $k_{\text{e}}^{{\text{III}}} = k_{\text{e}}^{\text{I}},X_{\text{L}}^{{\text{III}}} >X_{\text{L}}^{\text{I}}$, and the growth velocity of curve II is larger than that of curve I at the same temperature (Fig. A1a); curve I and curve III correspond to the same k_e value but to different X_L values $k_{\text{e}}^{{\text{IV}}} >k_{\text{e}}^{\text{I}},X_{\text{L}}^{{\text{IV}}} >X_{\text{L}}^{\text{I}}$, and the growth velocity of curve I is larger than that of curve III at the same temperature (Fig. A1b); curve I and curve IV correspond to different k_e and X_L values $\left( {X_{\text{S}}^{\text{e}}/X_{\text{L}}^{\text{e}}} \right)$, and the growth velocity of curve IV is larger than that of curve I when k is relatively small, the result is just opposite when k is relatively large (Fig. A1c). For Fe₄₀Ni₄₀P₁₄B₆ eutectic system, the intermetallic (Fe,Ni)₃(P,B) has smaller values of k_e and X_L compared to the solid solution γ(Fe, Ni), which is like the case of curve I and curve IV (Fig. A1c). Therefore, as the undercooling increases (k increases), the leading phase in the decoupled growth of Fe₄₀Ni₄₀P₁₄B₆ eutectic system is γ(Fe, Ni) at first, and (Fe,Ni)₃(P,B) at last. Moreover, as the leading phase changes from γ(Fe, Ni) to (Fe,Ni)₃(P,B), the growth velocities of the two phases could probably become comparable with each other.

TABLE A1.

Parameters used in the full numerical solution of the equation set [Eqs. (A1)-(A5)].

FIG. A1.

Interface temperature versus velocity, numerical solution of Eqs. (A1)-(A5) for fixed X_L. To demonstrate how k_e and X_L influence the interface velocity, the different values of k_e and X_L are adopted: (a) $\left( {k_{\text{e}}^{{\text{II}}} >k_{\text{e}}^{\text{I}}} \right)$; (b) $\left( {X_{\text{L}}^{{\text{III}}} >X_{\text{L}}^{\text{I}}} \right)$; (c) $\left( {k_{\text{e}}^{{\text{IV}}} >k_{\text{e}}^{\text{I}},X_{\text{L}}^{{\text{IV}}} >X_{\text{L}}^{\text{I}}} \right)$.