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Relationship Between the Mechanical Properties of an Irradiated Metal Alloy and the Mechanical Properties of the Irradiated Elements of the Alloy

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16 June 2023

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19 June 2023

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Abstract
In the article, we are trying to find a general law that describes the physical properties of a complex compound through the same physical properties of compound elements. For metal alloys operated under extreme conditions - neutron and proton irradiations and thermal heating, the mechanical properties of these alloys and their elements have been studied. It turned out that the ultimate tensile strength of tantalum alloys used in the design of space reactors can be expressed in terms of the weighted sum of the ultimate tensile strengths of its elements in the range of neutron doses: (0 - 0.96) displacement per atom (dpa) for Ta–10W alloy and (1.4 - 1.7) dpa for T–111 alloy with a neutron energy of more than 0.1 MeV. For tungsten binary alloys used in the design of fusion reactors, the Vickers hardness can also be expressed in terms of the weighted sum of the Vickers hardnesses of the alloy elements in the range of proton irradiation doses: (0 - 0.05) dpa for W-1%Re alloy, (0.05 - 0.2) dpa for W-3%Re alloy and (0.05 - 1) dpa for W-1%Ta with a proton energy of 1 MeV.
Keywords: 
Subject: Physical Sciences  -   Condensed Matter Physics
From our previous results regarding thermal conductivity [1,2] and melting/solidification temperature [3] of nuclear fuel fragments, oxide nuclear fuel and other nuclear ceramics, we concluded that for an arbitrary material, we can express its arbitrary physical property through atomic weighted or mass weighted sum of the corresponding physical properties of the elements of this material. This conclusion is also confirmed by the result on the superconductivity transition temperatures of high-temperature Y- and Hg-cuprates [4]. We have shown that in limited ranges of temperatures [1,2] or mole fractions of mixtures of actinide oxides [3], such weighted sums are multiplied by a constant. In other ranges of temperatures or mole fractions, we describe the physical properties of nuclear materials in terms of a weighted sum multiplied by a variable that changes according to the change in lattice parameters with mole fractions of the elements or temperature [3].
In this paper, we also describe the mechanical properties of metal alloys in terms of the mass-weighted sum of the mechanical properties of the alloy elements, multiplied by a constant in certain ranges of irradiation doses. We cannot conclude how to describe the mechanical properties of these alloys in other ranges of irradiation doses, since we did not find data on the behavior of their lattice parameters with a change in the irradiation dose.
If we consider an arbitrary solid body represented as a chemical compound, or a mechanical mixture, or a solid solution AnBmCk…, then the best formula for describing an arbitrary physical property LAnBmCk… of this solid body through the physical properties of its elements is the following:
1 L A n B m C k ... ( Ω ) = 1 ( n ⋅ L A ( Ω ) + m ⋅ L B ( Ω ) + k ⋅ L C ( Ω ) + ... ) ⋅ [ 1 n ⋅ A + 1 m ⋅ B + ... + h ( Ω ) k ⋅ C ] ,
where LA(Ω) is a physical property of element A, etc., A is the atomic number of element A and so on. Ω is a set of external and internal parameters of the medium and the solid, respectively – temperature T, pressure p, radiation D, impurity concentration x or some elements of this solid, and so on: Ω = {T, p, D, x, …}, where T, p, D, x, … ∈ R (set of real numbers). h(Ω) is a structural parameter depending on the dependence of the lattice parameters of AnBmCk… on the elements Ω and h ∈ Q (set of rational numbers) [1,2,3]. h(Ω) is associated in (1) with the most electronegative element, in our case it is the symbol C. Presence of the atomic numbers of all elements of the given complex compound is necessary because charges of nuclei define electronic structure of the solid body.
It turned out that for some Ω′ ⊂ Ω the structure parameter h is a constant: h(Ω′) ≡ h [1], [2], [3]. In this case the physical observable LAnBmCk…(Ω′) can be represented in the next form:
L A n B m C k ... ( Ω ′ ) = ( n ⋅ L A ( Ω ′ ) + m ⋅ L B ( Ω ′ ) + k ⋅ L C ( Ω ′ ) + ... ) ⋅ b , w h e r e b = [ 1 n ⋅ A + 1 m ⋅ B + ... + h k ⋅ C ] − 1 ,
and Ω′ = {T′, p′, D′, x′, …}, where T′ ∈ [T1 ÷ T2] (some temperature range) and so on.
Consider a set of tantalum alloys used in the construction of space reactors. The first is the T–111 alloy with the following content (wt%) of elements: Ta- balk, Hf – 2%, W – 8.5% [5]. Samples T–111 were irradiated with fast neutrons with energy E > 0.1 MeV with doses of [0 ÷ 1.7] dpa and irradiation temperatures of [723 – 923] K (Table 2) [5]. Mechanical tests when measuring the yield strength σ0.2 and ultimate tensile strength σt of these samples were carried out at the same temperatures: Tirrad = Ttest. To apply formula (1), it is necessary to know the same mechanical values for all elements of the alloy (Ta, Hf and W) in the same radiation doses and temperature ranges as for the alloy. The authors of [5] found this out only for the Ta solvent. For monocrystalline and polycrystalline samples of tungsten W, we found the necessary data in [6]. But in [6], the data obtained using proton irradiation at higher doses (Table 1) and lower Tirr are given. Mechanical tests of samples W were carried out at room temperature RT (Table 1). In any case, we used these data when calculating according to formula (1), since we did not find another alternative.
Formula (1) for T-111 has the following form:
1 σ 0.2 T − 111 ( Ω T − 111 ) = 1 0.895 ⋅ σ 0.2 T a ( Ω T a ) + 0.085 ⋅ σ 0.2 W ( Ω W ) [ 1 0.895 ⋅ T a + 1 0.02 ⋅ H f + h ( Ω T − 111 ) 0.085 ⋅ W ] ,
where atomic numbers: Ta = 73, Hf = 72, W = 74. The electronegativities X of these elements are as follows: XTa = 1.5 [7], XHf = 1.3 [7], XW = 2.36 [8]. Therefore W is related to h in (1.1). The set of parameters Ωmaterial = {D, Tirr, Ttest}material is associated with a specific material and their values for its elements are presented in Table 1 and Table 2.
To calculate the ultimate tensile strength, it is necessary to replace σ0.2 with σt in formula (1.1). The result of using formula (1.1) is presented in Table 2. Since the parameters in the set Ω vary when moving from material to its elements, the mechanical values of the elements for calculation according to (1.1) were taken with the closest values of irradiation doses and temperatures. In the case of W, we only have a dose of 10.3 dpa. This is more than we need, but we took data for W at these irradiation doses, having no alternative. We neglected the influence of the mechanical properties of Hf in (1.1), but in any case there are no such data for irradiated Hf. From Table 2 it can be seen that in the absence of irradiation, the mechanical properties of T–111 can be expressed through (2) at fixed h for different Tirr and Ttest. In the presence of radiation, only σt is expressed through (2) with data for polycrystalline W and DT-111 = (1.4 – 1.7) dpa. Since we do not have data on the lattice parameters of T–111 at various temperatures and radiation doses, we cannot reveal the dependence of h(Ω) on the lattice parameters. Probably from [3] it should look like this: h(Ω) = g1⋅∂Da + g2⋅∂Tirra + g3⋅∂Ttesta, where a is the lattice parameter, gi are constants.
The next tantalum alloy is ASTAR–811C with the content (wt%): Ta-balk, Hf - 0.9%, Re - 1.39%, W - 8.2% [5]; the electronegativities: XTa = 1.5 [7], XHf = 1.3 [7], XW = 2.36 [8], XRe = 1.9 [7]. Formula (1) has the following form:
1 σ 0.2 A S T A R ( Ω A S T A R ) = 1 0.8951 ⋅ σ 0.2 T a ( Ω T a ) + 0.082 ⋅ σ 0.2 W ( Ω W ) [ 1 0.8951 ⋅ T a + 1 0.009 ⋅ H f + 1 0.0139 ⋅ Re + h ( Ω A S T A R ) 0.082 ⋅ W ] ,
where atomic nymbers: Ta = 73, Hf = 72, W = 74, Re = 75. For ultimate tensile strength, it is necessary to change σ0.2 to σt in (1.2). We neglected the influence of σ0.2 and σt for Hf and Re on the calculation results, in any case we did not find the necessary data for these elements. It can be seen from Table 2 that only for zero radiation it is possible to express σ0.2 for the ASTAR–811C alloy in the form (2) with a constant h as Tirr changes and with data for monocrystalline W. The behavior of h(Ω) with a change in the irradiation dose is difficult to describe unless data on the lattice parameters are available.
Two more words about ASTAR–811C. It contains 0.025% of carbon [5]. Taking the data for carbon in the state of diamond σ0.2 = 40 GPa [9], the product 0.00025⋅40 GPa gives a contribution comparable to the term for W in the weighted sum in (1.2). But we neglected this, although carbon has a decisive influence on the mechanical properties of this alloy, forming microstructural objects in the alloy [5]. Probably, this neglect leads to the absence of the constant h at different doses D for the ASTAR-811C data in Table 2. But again, we did not find suitable data for the irradiated diamond.
The last tantalum alloy is Ta–10W with the content (wt%): Ta- balk, W - 10.2% [5]. Formula (1) has the following form:
1 σ 0.2 T a − 10 W ( Ω T a − 10 W ) = 1 0.898 ⋅ σ 0.2 T a ( Ω T a ) + 0.102 ⋅ σ 0.2 W ( Ω W ) [ 1 0.898 ⋅ T a + h ( Ω T a − 10 W ) 0.102 ⋅ W ] ,
where again we need to change σ0.2 to σt for ultimate tensile strength calculations. The calculation data are given in Table 2. From Table 2 it can be seen that with an irradiation dose in the range (0 ÷ 0.96) dpa and irradiation and mechanical testing temperatures close to 700 K, the ultimate tensile strength of the Ta–10W alloy is described by the expression (2) where h is a constant value. There is no data for lattice parameters changing with dose radiation and temperatures of irradiation and mechanical test.
We have considered binary W alloys studied under proton irradiation in the dose range (0 ÷ 1) dpa and with proton energy E = 1 MeV [10]. Temperature of irradiation 800°C, Vickers hardness HV test was carried out at room temperature. The authors of [10] studied pure tungsten under the same conditions as alloys (Table 3). Formula (1) for W-1%Re and W-3%Re alloys is as follows:
1 Н V W − w t % R e ( Ω W − w t % R e ) = 1 ( 1 − w t ) ⋅ H V W − w t % R e ( Ω W ) ⋅ [ 1 w t ⋅ Re + h ( Ω W − w t % R e ) ( 1 − w t ) ⋅ W ] ,
where W = 74, Re = 75, wt = 0.01 for W-1%Re or 0.03 for W-3%Re. We neglected the Vickers hardness for Re due to the lack of relevant data for this element. Table 3 shows that HV for W-1%Re and W-3%Re alloys can be described by formula (2) in the dose ranges (0 ÷ 0.05) dpa and (0.05 ÷ 0.2) dpa, respectively. There are no data on the change in lattice parameters during irradiation for these alloys to clarify the form of h(Ω).
Formula (1) for W-1%Ta and W-3%Ta alloys is as follows:
1 Н V W − w t % T a ( Ω W − w t % T a ) = 1 ( 1 − w t ) ⋅ H V W − w t % T a ( Ω W ) ⋅ [ 1 w t ⋅ T a + h ( Ω W − w t % T a ) ) ( 1 − w t ) ⋅ W ] ,
where W = 74, Ta = 73, wt = 0.01 for W-1%Ta or 0.03 for W-3%Ta. We neglected the Vickers hardness of Ta due to the lack of relevant data for this element. The calculation results are given in Table 3. We see that for W-1%Ta in the dose range (0.05 ÷ 1) dpa, the structural parameter h is a constant, and therefore the Vickers hardness can be described by formula (2). There are no data on the change in lattice parameters during irradiation for these alloys to clarify the form of h(Ω).
As a result, we can conclude that the existence of ranges of radiation doses and temperatures within which the mechanical properties of various metal alloys can be described by a weighted sum of the mechanical properties of the alloy elements, multiplied by a constant, tells us that this is a manifestation of a general law that can similarly describe any physical property of a compound. In other ranges of radiation and temperatures, we can describe an arbitrary physical property of a complex compound as a weighted sum of the physical properties of the constituent elements, multiplied by a variable depending on the lattice parameters of the complex compound. Unfortunately, such a simple idea, understandable even intuitively, i.e. if you want to know the properties of a complex compound under some conditions, you need information about the properties of all the constituent elements of the compound under the same conditions, this simple idea is completely ignored by researchers at the present time.

References

  1. Kizka, V. Thermal Conductivity of ZrO2, ZrSiO4, (U,Zr)SiO4 and UO2: Numerical Approach. Preprints.org 2023, 2023020406. [Google Scholar] [CrossRef]
  2. Kizka, V. Thermal Conductivity of Carbides and Nitrides of Zr, Th and U: Numerical Approach. Preprints.org 2023, 2023030403. [Google Scholar] [CrossRef]
  3. Kizka, V. Numerical Description of Melting/Solidification Temperature of Oxide Nuclear Fuels. Preprints.org 2023, 2023050174. [Google Scholar] [CrossRef]
  4. Kizka, V.A. Numerical Approach in Superconductivity: An Advanced Study. Recent Advances in Mathematical Research and Computer Science 2022, 6, 56–63. [Google Scholar] [CrossRef]
  5. Grossbeck, M.L.; Wiffen, F.W. Swelling and tensile properties of EBR-II-irradiated tantalum alloys for space reactor applications. 1985. United States. 5740. Available online: https://www.osti.gov/servlets/purl/5740504.
  6. et al. Mechanical properties of pure tungsten and tantalum irradiated by protons and neutrons at the Swiss spallation-neutron source. Nuclear Materials and Energy 2023, 34, 101338. [Google Scholar] [CrossRef]
  7. James, E.; Huheey, Ellen, A. Keiter, and Richard L Keiter. Inorganic Chemistry: Principles of Structure and Reactivity, 4th ed.; HarperCollins College Publishers: New York, 1993. [Google Scholar]
  8. Allred, A.L. Electronegativity Values from Thermochemical Data. Journal of Inorganic and Nuclear Chemistry 1961, 17, 215–221. [Google Scholar] [CrossRef]
  9. Mil’man, Y.V.; Pidgornyuk, E.M.; Katrusha, A.N.; et al. Mechanical properties of type IIb synthetic diamond at a temperature of 900°C. J. Superhard Mater 2012, 34, 308–313. [Google Scholar] [CrossRef]
  10. Takeshi, Miyazawa; et al. Microstructural evolution in tungsten binary alloys under proton and self-ion irradiations at 800°C. Journal of Nuclear Materials 2023, 575, 154239. [Google Scholar] [CrossRef]
Table 1. Mechanical properties and conditions of irradiation of W-samples irradiated with protons with an energy of 580 MeV and unirradiated W-samples, irradiated with neutrons with energy E > 0.1 MeV and unirradiated Ta-samples, and tested for strength at room temperature RT: yield strength σ0.2, ultimate tensile strength σt. Data taken from [5] for Ta and from [6] for W.
Table 1. Mechanical properties and conditions of irradiation of W-samples irradiated with protons with an energy of 580 MeV and unirradiated W-samples, irradiated with neutrons with energy E > 0.1 MeV and unirradiated Ta-samples, and tested for strength at room temperature RT: yield strength σ0.2, ultimate tensile strength σt. Data taken from [5] for Ta and from [6] for W.
Material Tirrad(°K) Dose (dpa) σ0.2(MPa) σt(Mpa)
Wmonocrystal 300 0 417 464.5
405 10.2 670 685
Wpolycrystal 300 0 broken 555.3
408 10.2 broken 268
728 0 78.25 246.5
853 0 61.8 214
Ta 703 0.96 465 536
753 1.4 620 689
853 1.7 514 551
Table 2. Mechanical properties and conditions of neutron irradiation with neutron energy E > 0.1 MeV of irradiated and non-irradiated Ta alloys. The experimental data were taken from [5], the theoretical values were obtained according to (1.1) – (1.3). Fitting parameters h are also shown. Relative errors δ are also shown, they are calculated by the formula: δ = ∣1 – σtheory/σexperiment∣⋅100%. The use of data for polycrystalline or monocrystalline W in calculations is also displayed in the column headings for theoretical σt. Here Ttest = Tirr.
Table 2. Mechanical properties and conditions of neutron irradiation with neutron energy E > 0.1 MeV of irradiated and non-irradiated Ta alloys. The experimental data were taken from [5], the theoretical values were obtained according to (1.1) – (1.3). Fitting parameters h are also shown. Relative errors δ are also shown, they are calculated by the formula: δ = ∣1 – σtheory/σexperiment∣⋅100%. The use of data for polycrystalline or monocrystalline W in calculations is also displayed in the column headings for theoretical σt. Here Ttest = Tirr.
Material Dose (dpa) Tirr [°K] σ0.2(Mpa) experiment σt(Mpa) experiment h, σ0.2(Mpa) (δ%) theory h, σt(Mpa) (δ%) theory, monocrystal h, σt(Mpa) (δ%) theory, polycrystal
0 723 569 628.5 h = –3.3, 569.8(0.1%) h = –1.86, 628.2(0.05%) h = –1.78, 627.6(0.1%)
833 486 552.7 h = –3.3, 490.3(0.9%) h = –1.86, 557.9(0.9%) h = –1.78, 559.4(1.2%)
T–111 0.78 723 1370 1370 h = –2.3, 1375(0.4%) h = –2.0, 1373(0.2%) h = –2.2, 1395.9(1.9%)
1.4 803 1370 1370 h = –1.7, 1392.2(1.6%) h = –1.4, 1385.3(1.1%) h = –1.49, 1352.2(1.3%)
1.7 923 1050 1070 h = –1.4, 1061.2(1.1%) h = –1.2, 1062.4(0.7%) h = –1.49, 1091(2.0%)
763 710 855 h = –14.4, 720.6(1.5%) h = –13.42, 845.1(1.2%) h = –13.4, 860.2(0.6%)
0 873 586 745 h = –14.43, 579.2(1.2%) h = –13.42, 750.1(0.7%) h = –13.34, 742.4(0.3%)
ASTAR–811C 1.1 763 1385 1425 h = –13.2, 1376(0.6%) h = –13, 1427.8(0.2%) h = –13.13, 1417.6(0.5%)
1.7 873 1230 1270 h = –12.71, 1217.1(1.0%) h = –12.65, 1268.6(0.1%) h = –12.8, 1261.7(0.6%)
0 728 486 564 h = 1.65, 482.4(0.7%) h = 3.5, 561.1(0.5%) h = 3.6, 564.8(0.1%)
Ta–10W 803 447 521 h = 1.56, 441.7(1.4%) h = 3.4, 514.4(1.3%) h = 3.5, 519.5(0.3%)
0.96 703 1020 1030 h = 3.5, 1014.5(0.5%) h = 3.9, 1036.2(0.6%) h = 3.6, 1033.4(0.3%)
1.4 753 989.5 989.5 h = 4.7, 979.9(1.0%) h = 5.2, 977.9(1.2%) h = 4.8, 992.1(0.3%)
Table 3. Vickers hardness HV of semples of tungsten alloys at various irradiation doses with protons with an energy 1 MeV at an irradiation temperature of 800°C. Experimental data are taken from [10]. The relative error δ is also shown, it is calculated by the formula: δ = ?1 – HVtheory/HVexperiment?⋅100%. .
Table 3. Vickers hardness HV of semples of tungsten alloys at various irradiation doses with protons with an energy 1 MeV at an irradiation temperature of 800°C. Experimental data are taken from [10]. The relative error δ is also shown, it is calculated by the formula: δ = ?1 – HVtheory/HVexperiment?⋅100%. .
Dose (dpa) HV[HV] of pure W, experiment HV[HV] of W-1%Re, experiment h, HV[HV] (δ%) of W-1%Re, theory HV[HV] of W-3%Re, experiment h, HV[HV] (δ%) of W-3%Re, theory HV[HV] of W-1%Ta, experiment h, HV[HV] (δ%) of W-1%Ta, theory HV[HV] of W-3%Ta, experiment h, HV[HV] (δ%) of W-3%Ta, theory
0 408 369 h = –18,371.4(0.6%) 365 h = 45, 369.4(1.2%) 443 h = –34, 445.9(0.7%) 463 h = 29, 459.8(0.7%)
0.05 537 484 h = –18,488.8(1.0%) 449 h = 50, 456.5(1.7%) 536 h = –28,538.3(0.4%) 530 h = 38, 528.3(0.3%)
0.2 612 602 h = –24, 602.4(0.07%) 528 h = 50,520.3(1.5%) 609 h = –28,613.5(0.7%) 644 h = 33, 647.8(0.6%)
1 659 675 h = –27, 676.2(0.2%) 609 h = 45, 596.7(0.03%) 655 h = –28,660.6(0.8%) 716 h = 31, 719.4(0.5%)
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