Coarsening behavior of precipitate Al₃(Sc,Zr) in supersaturated Al-Sc-Zr alloy via melt spinning and extrusion

Abstract

Al-Sc-Zr alloy contains precipitate Al(Sc,Zr) shows good coarsening resistance due to the small diffusivities of Sc and Zr. Two supersaturated Al-0.4Sc-0.4Zr at% alloys were produced using melt-spinning and following extrusion. The coarsening behavior of supersaturated Al-0.4Sc-0.4Zr at% alloys was investigated using scanning transmission electron microscopy (STEM). Accelerated coarsening was observed in the extruded melt-spun ribbon with larger precipitate size and higher precipitate volume fraction than in melt-spun ribbon. The accelerated effective diffusivity after extrusion was explicitly linked to the initially refined extruded microstructure. Precipitate volume fraction and chemical composition in both melt-spun ribbon and extruded rod during annealing were quantified with the help of STEM and energy-dispersive x-ray spectroscopy. Precipitate size and number density were predicted using Umantsev-Olson-Kuehman-Voorhees model for ternary alloys, and the predictions showed good agreement with experimental results by considering changes in diffusivity and precipitate volume fraction.

Appendices

Appendix 1 TEM sample thickness determination using EELS

The relative sample thickness in the units of the local inelastic mean free path λ was determined by the ratio of zero-loss electrons intensity to the total transmitted intensity [56]:

$$t/\lambda = - \ln \left( {I_{0} /I_{t} } \right)$$

(9)

where $I_{0}$ is the zero-loss intensity and $I_{t}$ is the total intensity. The areas for the precipitate count and thickness measurement were chosen along the edge of the twin-jet polished holes. Since the thickness of the twin-jet polished samples increases quickly away from the edge of the hole, the sample thickness was measured at a series of points away from the edge (Fig. 

Figure 16

a An example of sample thickness measurement for the selected area along the polished hole. b Sample thickness was measured at five places away from the edge in the spectrum image. c Spectrum of spectrum image of place #1

16).The absolute thickness was determined using the mean free path length of Al due to the small number of solute atoms. The relative and absolute thickness of the measured points were obtained according to Eq. 9 (Table 6).

The sample microstructures include precipitate size, dislocation density, and precipitate number density. These characteristics were approximately measured within areas which are less than 200 nm away from the edge.

Several, but not all, areas of dislocation density analysis had the thickness characterized by EELS thickness measurement. Areas under constant imaging conditions (near < 001 > Al) with similar gray scale contrast were also utilized for the microstructure measurement. In order to verify that the thickness approximations were consistent, the individual dislocation segment lengths in the grains were also measured. That is, because the dislocation segment lengths are limited by surface intersection, significant variations in grain thickness from sample to sample would result in significant variations in dislocation segment lengths. Such a measurement is thus a check that the assumptions of similar thicknesses (and thickness distributions) is valid. As shown in Table

Table 7 Mean dislocation length in overaged melt-spun ribbon (error bars are 95% standard error of the mean)

7, the variation in segment length between the relevant samples is varies within the standard deviation of the measurement. The significance of the mean difference between two mean values is in the range of 50%–90% (sample size equals to 10) [57].

Determining the defect density (number/volume) using the projected density (number/area) in an electropolished foil is not straight forward. Our approach has been to measure (via EELS) or estimate the thickness variation of the sample and use the following approach involving an average thickness with a linear variation in thickness from the edge of the hole to the interior of the grain. The sample thickness, $y$, at position $x$ is defined as:

$$y = m \cdot x + b$$

(10)

where $m$ is the linear thickness change rate with increasing x, $b$ is the initial thickness at $x = 0$. The particle count depends on the sample thickness, with the defect count over a variation in thickness given by the numerator in Eq. 11 with the effective volume given by the denominator in Eq. 11, thus revealing the defect density. Note that the area term in Eq. 11 need not be specified as it algebraically cancels.

$$\frac{{{\text{Number}}\,{\text{of}}\,{\text{Defects}}}}{{{\text{Volume}}}} = \frac{{\frac{1}{L}\mathop \int \nolimits_{0}^{L} \rho \cdot \left( {m \cdot x + b} \right) \cdot {\text{area d}}x}}{{\frac{1}{L}\left[ {\mathop \int \nolimits_{0}^{L} \left( {m \cdot x + b} \right) \cdot {\text{area d}}x} \right]}} = \rho$$

(11)

To test this approximation, we examine the projected defect density as a function of distance from $x = 0$. For all overaged melt-spun ribbon and extruded rod samples, the defect counts of precipitate and dislocation were measured in the range of 0–100 nm and 0–200 nm from hole edge into grain interior, respectively. The defect density ratios from the different counts (0–200 nm/0–100 nm) were listed in Table

Table 8 Ratios of experimentally measured precipitate area number density and dislocation area density in the range of 0–200 nm to 0–100 nm from the edge to the grain, respectively

8. According to Table 6, the measured sample thickness was 31 nm, 52 nm and 75 nm at position $x = 0$ nm, 100 nm and 200 nm respectively.

Considering the twin-jet electropolished sample showed trapezoid-shape with linearly increasing thickness from the polished hole edge to grain (Table 6), the ratios of precipitate number and total dislocation line length are determined as:

$$\frac{{\mathop \int \nolimits_{0}^{2L} \rho \cdot \left( {m \cdot x + b} \right) \cdot {\text{area}}\,{\text{d}}x}}{{\mathop \int \nolimits_{0}^{L} \rho \cdot \left( {m \cdot x + b} \right) \cdot {\text{area}}\, {\text{d}}x}} = \frac{2 \cdot L \cdot m}{{2 \cdot b + L \cdot m}} + 2$$

(12)

where $L = 100{\text{nm}}$ for each measurement step. According to Table 6, the measured sample thickness was 31 nm, 52 nm and 75 nm with position away from the edge at 0 nm, 100 nm and 200 nm respectively. Thus values are determined as $m = 0.22$ and $b = 31{\text{nm}}$. The ratio has a value of ~ 2.5, similar to the values found in the Table 8.

Appendix 2 Increasing time exponent n as a function of evolving precipitate composition

The precipitate radius $R$ change described in traditional LSW model was defined as:

$$\frac{{{\text{d}}R}}{{{\text{d}}t}} = \frac{{2D\Omega^{2} C_{{{\text{eq}}}} \gamma }}{kTR} \cdot \left( {\frac{1}{R} - \frac{1}{R}} \right)$$

(13)

where D is the diffusivity of solute, $\gamma$ is the interfacial energy, $\Omega$ is the solute atomic volume, $C_{{{\text{eq}}}}$ is the equilibrium solubility determined by the phase diagram, $k$ is the boltzmann constant and T is the over aging temperature. The precipitate radius change depends on the size comparison between the actual precipitate size $\left\langle R \right\rangle$ and the mean precipitate size $R$. The time exponent of traditional LSW is 3 according to Eq. 13.

The composition of precipitates in melt-spun ribbon versus precipitate radius during annealing is plotted in Fig. 

Figure 17

Solute concentration of precipitate in melt-spun ribbon change with precipitate radius aged from 12 to 72 h at 400 °C (error bars are 95% standard error of the mean)

17. The concentration of Sc decreased from 19.5 to 18.9% and the concentration of Zr increased from 3.1 to 3.4% with precipitate radius coarsened from 7.5 to 12.1 nm. Note that the experimentally measured compositions of precipitate are the average composition of entire precipitate. The composition change of Sc and Zr might lead more significant impact than measured values due to the inhomogeneous distribution of elements.

The confidence in these reported solutes changes was estimated using the Thiel-Sen estimator which is the median of the slopes through all points (Table

Table 9 The confidence of solutes changes for precipitate in supersaturated and dilute alloys using Thiel-Sen estimator

9). Based on the confidence of solutes changes using data from both supersaturated, the decrease of Sc composition is poorly supported with less than 30% confidence, and the increase in Zr composition shows at least 70% confidence. As for the dilute alloys [50], the decrease in Sc composition in precipitate is found to be statistically significant with 95% confidence, and the increase of Zr composition shows 85% confidence. Thus, the composition of Sc in precipitate is thought decreased and Zr is increased with annealing.

Using the lattice mismatch difference between ${\text{Al}}_{3} {\text{Zr}}$ versus ${\text{Al}}_{3} {\text{Sc}}$ (1.32% for ${\text{Al}}_{3} {\text{Sc}},$ $0.75{\text{\% for Al}}_{3} {\text{Zr}}$) and the concentrations changes (Table 9), the lattice mismatch for an average size precipitate decreased 1.6% after overaging from 12 to 72 h. As the interfacial energy per unit area $\gamma$ decreases with lattice mismatch $\varepsilon$, the interfacial energy change could be estimated in terms of radius change using the chain rule as [58]:

$$\gamma = \gamma_{0} + \left( {\frac{{{\text{d}}\varepsilon }}{{{\text{d}}C_{Zr} }} \cdot \frac{{{\text{d}}C_{Zr} }}{{{\text{d}}R}} + \frac{{{\text{d}}\varepsilon }}{{{\text{d}}C_{Sc} }} \cdot \frac{{{\text{d}}C_{Sc} }}{{{\text{d}}R}}} \right) \cdot \frac{{{\text{d}}\gamma }}{{{\text{d}}\varepsilon }} \cdot \left( {R - R_{0} } \right) = \gamma_{0} + \alpha \cdot \left( {R - R_{0} } \right)$$

(14)

where $R_{0}$ and $\gamma_{0}$ are the initial precipitate radius and interfacial energy and $\alpha < 0$.Substituting Eq. 14 into Eq. 13:

$$\frac{{{\text{d}}R}}{{{\text{d}}t}} \cong \frac{{aD\Omega^{2} C_{{{\text{eq}}}} }}{kT} \cdot \left( {\gamma_{0} + \alpha \cdot \left( {R - R_{0} } \right)} \right) \cdot \left( {\frac{1}{\left\langle R \right\rangle } - \frac{1}{R}} \right) \cdot \frac{1}{R}$$

(15)

Compared to the traditional LSW theory, Eq. 15 introduces an extra term describing the change in interfacial energy with precipitate radius, $\alpha$ (due to the change in Sc/Zr concentration with precipitate radius). Examining the specific case for $R = 2\left\langle R \right\rangle$, which is the radius that has the maximum radial velocity in the LSW treatment:

$$\frac{{{\text{d}}R}}{{{\text{d}}t}} \cong \frac{{aD\Omega^{2} C_{{{\text{eq}}}} }}{kT} \cdot \left( {\gamma_{0} + \alpha \cdot \left( {2\left\langle R \right\rangle - R_{0} } \right)} \right) \cdot \left( {\frac{1}{{4\left\langle R \right\rangle^{2} }}} \right)$$

(16)

Integration of Eq. 16 gives a hypergeometric solution, which has a closed form Taylor series about $\alpha$ (linear) as:

$$8\left\langle R \right\rangle^{3} \cdot \left( {\frac{1}{{3\gamma_{0} }} - \frac{\alpha }{{\gamma_{0}^{2} }}\left( {\frac{\left\langle R \right\rangle }{2} - \frac{{R_{0} }}{3}} \right)} \right) = t \cdot \frac{kT}{{aD\Omega^{2} C_{{{\text{eq}}}} }} + {\text{constant}}$$

(17)

The precipitate radius $R$ coarsening at 400 °C described in Eq. 17 are plotted in Fig. 

Figure 18

Precipitate radius predicted using modified LSW versus traditional LSW from t = 0 s to t = 10000 s at 400 °C

18 from 0 to 10000 s.

As the average precipitate radius calculated based on the modified LSW (Eq. 17) for $\alpha < 0$ is found smaller than the average precipitate radius calculated based on traditional LSW at given annealing duration, the time exponent of radius for the traditional LSW was increased to larger than 3.

As the average precipitate radius calculated based on the modified LSW (Eq. 17) for $\alpha < 0$ is found smaller than the average precipitate radius calculated based on traditional LSW at given annealing duration, the time exponent of radius for the traditional LSW was increased to larger than 3. The increased time exponent was estimated from precipitate size change in modified LSW (Table

Table 10 Four precipitate sizes were chosen from modified LSW (Fig. 18)

10).

The time exponent was found increased to 3.3 for modified LSW plot (Fig.

Figure 19

Log–log plot of mean precipitate radii of modified LSW

19).

Appendix 3 Increasing time exponent n due to increasing precipitate volume fraction

Precipitate volume fractions were found to increase during overaging heat treatment of both the melt-spun ribbon and extruded rod. For a small increase in the total volume of the precipitates during coarsening, the traditional LSW approach can be modified into:

$$\frac{d}{{{\text{d}}t}}\mathop \sum \limits_{{{\text{part}}}} \frac{4\pi }{3}R^{3} = \delta_{1}$$

(18)

where $\delta_{1} = 0$ in the traditional LSW derivation, but is taken as a positive constant in this modification. Rewriting Eq. 13 as:

$$\mathop \sum \limits_{{{\text{part}}}} R^{2} \frac{{{\text{d}}R}}{{{\text{d}}t}} = \delta_{2}$$

(19)

where $\delta_{1}$ in Eq. 18 and $\delta_{2}$ in Eq. 19 are non-zero constants. The growth rate of a particle with radius $R$ is still considered as controlled by the mean field average concentration, $\left\langle c \right\rangle$:

$$\frac{{{\text{d}}R}}{{{\text{d}}t}} = - D\frac{{c^{{{\text{eq}}}} - \left\langle c \right\rangle }}{R}\Omega$$

(20)

where $D$ is the solute diffusivity, $\Omega$ is the atomic volume, $c^{{{\text{eq}}}}$ is the equilibrium solute concentration at the interface of a particle of radius, R, as determined by the Gibbs-Thompson effect.

Combining Eqs. 19 and 20 gives:

$$\mathop \sum \limits_{part} R\left( {c^{eq} - \left\langle c \right\rangle } \right) = - \delta$$

(21)

where $\delta = 3\delta_{1} /\left( {4\pi D\Omega } \right)$ is a constant. Substituting the Gibbs-Thompson expression $c^{eq} = c^{eq} \left( \infty \right) \cdot \left( {1 + \frac{2\gamma \Omega }{{kTR}}} \right)$ into Eq. 21:

$$\mathop \sum \limits_{{{\text{part}}}} R \cdot \left[ {\left\langle c \right\rangle - c^{{{\text{eq}}}} \left( \infty \right) \cdot \left( {1 + \frac{2\gamma \Omega }{{kTR}}} \right)} \right] = \delta$$

(22)

where $c^{{{\text{eq}}}} \left( \infty \right)$ is the solute concentration from the phase diagram, $\gamma$ is the interfacial energy, and $k$ is the Boltzmann constant. Rearranging Eq. 22 gives:

$$\left( {\left\langle c \right\rangle - c^{{{\text{eq}}}} \left( \infty \right)} \right) \cdot \mathop \sum \limits_{{{\text{part}}}} R + \frac{{2\gamma \Omega c^{{{\text{eq}}}} \left( \infty \right)}}{kT} \cdot \mathop \sum \limits_{{{\text{part}}}} 1 = \delta$$

(23)

Equation 23 could be further rearranged gives:

$$\left( {\left\langle c \right\rangle - c^{eq} \left( \infty \right)} \right) \cdot \left\langle R \right\rangle + \frac{{2\gamma \Omega c^{eq} \left( \infty \right)}}{kT} = \frac{\delta }{{N^{tot} }}$$

(24)

where $\left\langle R \right\rangle$ is the average particle radius, given by:

$$\left\langle R \right\rangle = \frac{{\mathop \sum \nolimits_{{{\text{part}}}} R}}{{N^{{{\text{tot}}}} }}$$

(25)

where $N^{{{\text{tot}}}} = \mathop \sum \limits_{{{\text{part}}}} 1$ is the total number of particles. Based on Eq. 24, the average concentration $\left\langle c \right\rangle$ is expressed as:

$$\left\langle c \right\rangle = \frac{\delta }{{N^{{{\text{tot}}}} \cdot \left\langle R \right\rangle }} + c^{{{\text{eq}}}} \left( \infty \right) \cdot \left( {1 + \frac{{2\gamma \Omega c^{{{\text{eq}}}} \left( \infty \right)}}{kT \cdot \left\langle R \right\rangle }} \right)$$

(26)

Substituting Eq. 26 into Eq. 20 gives:

$$\frac{{{\text{d}}R}}{{{\text{d}}t}} = \frac{D\Omega }{R} \cdot \left( {\frac{\delta }{{N^{{{\text{tot}}}} \left\langle R \right\rangle }} + \frac{{2\gamma \Omega c^{{{\text{eq}}}} \left( \infty \right)}}{kT\left\langle R \right\rangle } - \frac{{2\gamma \Omega c^{{{\text{eq}}}} \left( \infty \right)}}{kTR}} \right)$$

(27)

The total number of particles can be expressed in terms of equilibrium precipitate volume fraction $f_{v}^{{{\text{eq}}}}$ as:

$$N^{{{\text{tot}}}} = \frac{3}{{4\left\langle R \right\rangle^{3} }} \cdot f_{v}^{{{\text{eq}}}}$$

(28)

Substituting Eq. 28 into Eq. 27 gives:

$$\frac{{{\text{d}}R}}{{{\text{d}}t}} = \frac{D\Omega }{R} \cdot \left( {\frac{{4\delta \left\langle R \right\rangle^{2} }}{{3f_{v}^{{{\text{eq}}}} }} + \frac{{2\gamma \Omega c^{{{\text{eq}}}} \left( \infty \right)}}{kT\left\langle R \right\rangle } - \frac{{2\gamma \Omega c^{{{\text{eq}}}} \left( \infty \right)}}{kTR}} \right)$$

(29)

Solving for $R_{{{\text{max}}}}$ in terms of $R$:

$$\frac{d}{dt}\left( {\frac{dR}{{dt}}} \right) = 0 = - \frac{{D\Omega \cdot \left( {4\delta R\left\langle R \right\rangle^{3} kT - 12c^{eq} \left( \infty \right)f_{v}^{eq} \left\langle R \right\rangle \gamma \Omega + 6c^{eq} \left( \infty \right)f_{v}^{eq} R\gamma \Omega } \right)}}{{3f_{v}^{eq} R^{3} \left\langle R \right\rangle kT}}$$

(30)

Yields an $R_{{{\text{max}}}}$ of:

$$R_{{{\text{max}}}} = \frac{{6c^{{{\text{eq}}}} \left( \infty \right)f_{v}^{{{\text{eq}}}} \left\langle R \right\rangle \gamma \Omega }}{{2\delta \left\langle R \right\rangle^{3} kT + 3c^{{{\text{eq}}}} \left( \infty \right)f_{v}^{{{\text{eq}}}} \gamma \Omega }}$$

(31)

Equation 29 is arranged by substituting $R = R_{{{\text{max}}}}$:

$$\frac{{{\text{d}}\left\langle R \right\rangle }}{{{\text{d}}t}} = \frac{{D\Omega \cdot \left( {4\delta \left\langle R \right\rangle^{3} KT + 6c^{{{\text{eq}}}} \left( \infty \right)f_{v}^{{{\text{eq}}}} \gamma \Omega } \right)^{2} }}{{72c^{{{\text{eq}}}} \left( \infty \right)f_{v}^{{{\text{eq}}^{{2}} }} \left\langle R \right\rangle^{2} kT}} \cdot \left[ {\frac{{6c^{{{\text{eq}}}} \left( \infty \right)f_{v}^{{{\text{eq}}}} \gamma \Omega }}{{2\delta \left\langle R \right\rangle^{3} KT + 3c^{{{\text{eq}}}} \left( \infty \right)f_{v}^{{{\text{eq}}}} \gamma \Omega }} - \frac{{144c^{{{\text{eq}}}} \left( \infty \right)f_{v}^{{{\text{eq}}}} \gamma \Omega \left\langle R \right\rangle^{3} kT}}{{\left( {4\delta \left\langle R \right\rangle^{3} KT + 6c^{{{\text{eq}}}} \left( \infty \right)f_{v}^{{{\text{eq}}}} \gamma \Omega } \right)^{2} }}} \right]$$

(32)

Simplify the above equation by choosing the first order of $\delta$ gives:

$$\frac{{{\text{d}}\left\langle R \right\rangle }}{{{\text{d}}t}} = \frac{{D\Omega^{2} c^{{{\text{eq}}}} \left( \infty \right)\gamma }}{{4kT\left\langle R \right\rangle^{2} }} + \frac{{2D\Omega^{2} \left\langle R \right\rangle \gamma }}{{kTf_{v}^{{{\text{eq}}}} }}$$

(33)

Integrating Eq. 33 gives the evolution duration $t$:

$$t = \frac{{f_{v}^{{{\text{eq}}}} \cdot \ln \left( {\left\langle R \right\rangle^{3} + \frac{{c^{{{\text{eq}}}} \left( \infty \right)f_{v}^{{{\text{eq}}}} \gamma \Omega }}{4\delta kT}} \right)}}{3D\Omega \delta } + C$$

(34)

where $C$ is a constant that at $t = 0$ has a value of:

$$C = - \frac{{f_{v}^{{{\text{eq}}}} \cdot \ln \left( {\left\langle {R_{0} } \right\rangle^{3} + \frac{{c^{{{\text{eq}}}} \left( \infty \right)f_{v}^{{{\text{eq}}}} \gamma \Omega }}{4\delta kT}} \right)}}{3D\Omega \delta }$$

(35)

Rearranging Eq. 34 gives:

$$t = \frac{{f_{v}^{{{\text{eq}}}} \cdot \ln \left( {\left\langle R \right\rangle^{3} + \frac{{c^{{{\text{eq}}}} \left( \infty \right)f_{v}^{{{\text{eq}}}} \gamma \Omega }}{4\delta kT}} \right)}}{3D\Omega \delta } - \frac{{f_{v} \cdot \ln \left( {\left\langle {R_{0} } \right\rangle^{3} + \frac{{c^{{{\text{eq}}}} \left( \infty \right)f_{v}^{{{\text{eq}}}} \gamma \Omega }}{4\delta kT}} \right)}}{3D\Omega \delta }$$

(36)

Equation 36 is rearranged and simplified by choosing first order of $\delta$ to obtain:

$$t = - \frac{{4kT \cdot \left( {\left\langle R \right\rangle^{3} - \left\langle {R_{0} } \right\rangle^{3} } \right) \cdot \left( {2\delta \left\langle R \right\rangle^{3} kT + 2\delta \left\langle {R_{0} } \right\rangle^{3} kT - c^{{{\text{eq}}}} \left( \infty \right)f_{v}^{{{\text{eq}}}} \gamma \Omega } \right)}}{{3Dc^{{{\text{eq}}}} \left( \infty \right)^{2} f_{v}^{{{\text{eq}}}} \gamma^{2} \Omega^{2} }}$$

(37)

Equation 37 is changed into the general LSW when $\delta = 0$:

$$t = \frac{{4kT \cdot \left( {\left\langle R \right\rangle^{3} - \left\langle {R_{0}^{3} } \right\rangle } \right) \cdot c^{{{\text{eq}}}} \left( \infty \right)f_{v}^{{{\text{eq}}}} \gamma \Omega }}{{3Dc^{{{\text{eq}}}} \left( \infty \right)^{2} f_{v}^{{{\text{eq}}}} \gamma^{2} \Omega^{2} }}$$

(38)

An experimental value for $\delta = 3\delta_{1} /\left( {4\pi D\Omega } \right)$ is determined as $1.2 \times 10^{14} {\text{mol}}/{\text{m}}^{2}$. The precipitate radius $R$ coarsening at 400 °C described in Eqs. 37 and 38 are plotted in Fig.

Figure 20

Precipitate radius predicted using modified LSW versus traditional LSW from t = 0 s to t = 10000 s at 400 °C

20 from 0 to 10000 s.

The precipitate radius calculated by modifying LSW to consider increasing precipitate volume fraction is found to be larger than the general LSW, so the time exponent of radius for the modified LSW was decreased to smaller than 3. Based on the discussions of time exponent in "Appendixs 2and 3", the values of time exponent in melt-spun ribbon and extruded rod compromised to the changed precipitate composition and increased precipitate volume fraction during annealing.