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According to Figure 27 and Figure 28, the observed maximum yield strength for 1.8Zn0.05Ni and 1.7Zn0.07Zr are ~125 (250 ℃) and ~156 MPa (450 ℃), respectively, using a conversion factor of 1₃ between Vickers microhardness and yield strength[56]. These observed strengths are due to solid solution and precipitation strengthening mechanisms. Grain size reduction and strain hardening of the specimens were not considered since aging is not expected to have any significant effect on grain size and the specimens were not deformed. The yield strengths of the specimens have contributions from modulus mismatch, order, Orowan, and coherency strengthening mechanisms due to the spheroidal coherent precipitates formed during aging. The relative contributions of these mechanisms can be quantified as follows.

Modulus mismatch strengthening, ∆𝜎𝜎𝑚𝑚𝑠𝑠, results from the difference between the shear

moduli of the precipitate and matrix phases. It can be estimated by using[33, 57]: ∆𝜎𝜎𝑚𝑚𝑠𝑠 = 0.0055𝑀𝑀(∆𝐺𝐺)3�2(_𝐺𝐺2𝜙𝜙_{𝐴𝐴𝐴𝐴})1�2(〈𝑅𝑅〉_b)3𝑚𝑚� −12 (14)

where the Taylor factor[58] 𝑀𝑀 = 3.06 and ∆𝐺𝐺 is the difference in the shear modulus between the precipitate and matrix, which are approximated as 30.6 and 40 GPa for Al3- xZnxNi and Al3-xZnxZr, respectively. The shear moduli of L12-type Al3-xZnxNi, Al3-xZnxZr

and the matrix are taken as 𝐺𝐺Al3−xZnxNi = 56 GPa (same as the value for Al3Ni[59, 60]),

𝐺𝐺Al3−xZnxZr= 67.4 GPa (same as the value for Al3Zr[61-63]), and 𝐺𝐺𝐴𝐴𝑡𝑡 = 25.4 𝐺𝐺𝑀𝑀𝑀𝑀,

respectively. The volume fraction was approximated as 𝜙𝜙 ≅ 0.003 (estimated from the tie line of the Al3Zr phase diagram) for Al(3-x)ZnxZr. Since the added Zn is assumed to replace

the Al site, it is expected that Al3Zr and Al(3-x)ZnxZr have similar volume fractions[12].

This assumption is also confirmed by the heat treatment curves. Using the same approach, 𝜙𝜙 ≅ 0.002 for Al(3-x)ZnxNi. 〈𝑅𝑅〉 is the average particle radius of the specimens. Due to their

large number density and small interparticle spacing, most of the precipitation strengthening of 1.7Zn0.07Zr comes from the small size particles segregated in the dendritic centers. Hence, the values of 〈𝑅𝑅〉 used for the yield strength estimates were

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2.5 𝑇𝑇𝑝𝑝 (dendrite core) and 4.0 𝑇𝑇𝑝𝑝 for the precipitates present in 1.7Zn0.07Zr and 1.8Zn0.05Ni, respectively (Table 5). The Burgers vector magnitude of the matrix was[64, 65] 𝑏𝑏 = 0.286 𝑇𝑇𝑝𝑝, and m = 0.85 is a constant.

At peak strength, order strengthening ∆𝜎𝜎𝑡𝑡𝑠𝑠 (which is due to the formation of antiphase

boundaries (APBs) as matrix dislocations shear ordered particles) is given by[33, 57, 64]:

∆𝜎𝜎𝑡𝑡𝑠𝑠 = 0.81𝑀𝑀𝛾𝛾𝐴𝐴𝐴𝐴𝐴𝐴_2𝑏𝑏 (3𝜋𝜋𝜙𝜙₈ )1�2 (15)

where 𝛾𝛾𝐴𝐴𝐴𝐴𝐴𝐴 ~0.48 and 0.445 𝐽𝐽𝑝𝑝−2 are taken as the average APB energies for Al3-xZnxNi

and Al3-xZnxZr alloys based on several reported values for Al3Ni and Al3Zr for the (111)

plane[66-68]. APB energies are difficult to determine, and different methods tend to give varying values for the same intermetallic compound. Therefore, APB energies have only been reported for a limited number of compounds[69-71]. Recently, Rudy and Sauthoff reported a APB energy of ~0.4 𝐽𝐽𝑝𝑝−2 _{for NiAl[72]. A study has already reported the APB}

energy for Ni3Al ~0.195 𝐽𝐽𝑝𝑝−2[73].

During the over-aging, Orowan strengthening ∆𝜎𝜎𝑡𝑡𝑐𝑐 is given by[33, 57, 64]:

∆𝜎𝜎𝑡𝑡𝑐𝑐 = 0.4𝑀𝑀_{𝜋𝜋√(1−𝜈𝜈)} 𝐺𝐺𝐴𝐴𝐴𝐴𝑏𝑏 ln (

2𝑅𝑅 𝑏𝑏)

𝜆𝜆𝑐𝑐−𝑐𝑐 (16)

where the mean planar radius[64] is given by 𝑅𝑅 =𝜋𝜋₄〈𝑅𝑅〉, 𝜈𝜈 = 0.354 is the Poisson’s ratio for Al, and the inter-precipitate distance[64] 𝜆𝜆𝑐𝑐−𝑐𝑐 = ��_3𝜙𝜙2𝜋𝜋−𝜋𝜋₂� 〈𝑅𝑅〉.

Coherency strengthening, ∆𝜎𝜎𝑐𝑐𝑠𝑠, can be estimated using[33, 57, 64]:

∆𝜎𝜎𝑐𝑐𝑠𝑠 = 𝑀𝑀𝑀𝑀(𝜖𝜖𝐺𝐺𝐴𝐴𝑡𝑡)3�2(〈𝑅𝑅〉𝜙𝜙𝑏𝑏_Γ )1�2 (17)

where 𝑀𝑀 = 2.6 for fcc metals[64], 𝜖𝜖 is the mismatch parameter approximated by 2₃𝛿𝛿; 𝛿𝛿 = 1.23% and 0.75% is taken as the lattice parameter mismatch for the Al3-xZnxNi and Al3- xZnxZr precipitates estimated from the composition-dependent lattice parameter[30-32,

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Generally, precipitation hardening is governed by either the dislocation by-pass (Orowan- type) or dislocation shearing mechanisms. At peak strength, only the shearing mechanism was considered and the contributing factors in this mechanism are coherency strengthening (∆𝜎𝜎𝑐𝑐𝑠𝑠), modulus mismatch strengthening (∆𝜎𝜎𝑚𝑚𝑠𝑠), and order strengthening (∆𝜎𝜎𝑡𝑡𝑐𝑐)[33, 57].

Regardless of the unavailability of exact data and approximations made for some of the parameters used in the equations above, these models successfully predicted the maximum yield strength observed experimentally. Using equations 14-17 and the data in Table 5, the contribution of each strengthening mechanism to the yield strength of 1.7Zn0.07Zr, was estimated as ∆𝜎𝜎𝑡𝑡𝑐𝑐 ≈ 148 𝑀𝑀𝑀𝑀𝑀𝑀, ∆𝜎𝜎𝑡𝑡𝑠𝑠≈ 114 𝑀𝑀𝑀𝑀𝑀𝑀, ∆𝜎𝜎𝑐𝑐𝑠𝑠 ≈ 18 𝑀𝑀𝑀𝑀𝑀𝑀 and ∆𝜎𝜎𝑚𝑚𝑠𝑠 ≈

118 𝑀𝑀𝑀𝑀𝑀𝑀 for Orowan, order, coherency strain, and modulus mismatch strengthening, respectively. The larger value between ∆𝜎𝜎𝑐𝑐𝑠𝑠+ ∆𝜎𝜎𝑚𝑚𝑠𝑠 (~ 136 𝑀𝑀𝑀𝑀𝑀𝑀) and

∆𝜎𝜎𝑡𝑡𝑠𝑠 (~ 114 𝑀𝑀𝑀𝑀𝑀𝑀) determines the resultant yield strength increment due to the shearing

mechanism[33, 57]. As a result, the yield strength (∆𝜎𝜎𝑐𝑐𝑐𝑐𝑡𝑡) of 1.7Zn0.07Zr at 450 ℃, due

to precipitation hardening was estimated as 136 𝑀𝑀𝑀𝑀𝑀𝑀 from ∆𝜎𝜎𝑐𝑐𝑠𝑠+ ∆𝜎𝜎𝑚𝑚𝑠𝑠 (ignoring

Orowan strengthening). Therefore, the significant difference between the estimated and measured yield strength of 1.7Zn0.07Zr is 20 MPa, which could be attributed to the solid solution strengthening (not considered in the estimate) from the remaining Zn in the matrix. The difference between estimated and measured strength values could also be a result of some of the approximated parameters used in the estimation, due to a lack of exact theoretical or experimental data for the Al3-xZnxZr precipitate.

Table 5. Measured precipitate mean radii,〈𝑅𝑅〉, of L12 precipitates located at the dendritic

centers of 1.8Zn0.05Ni and 1.7Zn0.07Zr, after isochronal aging at 250 and 450 ℃ respectively.

Specimen Aging Temperature ℃ Mean precipitate

radius 〈𝑅𝑅〉 𝑇𝑇𝑝𝑝 Number of counted precipitates 1.8Zn0.05Ni 250 4.0 285 1.7Zn0.07Zr 450 2.5 201

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Similar to the estimation made for the 1.7Zn0.07Zr alloy, the yield strength due to precipitation (∆𝜎𝜎𝑐𝑐𝑐𝑐𝑡𝑡) of the 1.8Zn0.05Ni alloy at peak strength was estimated as 108 MPa

from ∆𝜎𝜎𝑐𝑐𝑠𝑠+ ∆𝜎𝜎𝑚𝑚𝑠𝑠. The measured peak strength for this specimen was 125 MPa. The

observed difference between the estimated and measured strength values is 17 MPa, which could also be attributed to the solid solution strengthening due to the remaining Zn left in the matrix and the gross approximation made for some of the parameters used.