Mechanisms of Neutron Irradiation Hardening in Impurity-Doped Ferritic Alloys

Abstract

Mechanisms of neutron irradiation hardening in phosphorus (P)-doped, sulfur (S)-doped, and copper (Cu)-doped ferritic alloys have been studied by applying a rate theory to the temperature dependence of the yield strength. Hardening behavior induced by neutron irradiation at various temperatures (473 to 711 K) is characterized in terms of the variations in athermal stress and activation energy for plasticity controlled by precipitation or solid solution, and kink-pair formation with the content and type of impurities. In P-doped alloys, neutron irradiation below 563 K brings about a remarkable increase in the athermal stress and activation energy, due to the dispersion of fine (∼1.7-nm) P-rich precipitates that is more extensive than that for the Cu-rich precipitates reported in irradiated steel. During neutron irradiation above 668 K, precipitation hardening occurs to some extent in Cu-doped and S-doped alloys, compared to small or negligible hardening in the P-doped alloys. In alloys with a low to moderate content of various dissolved impurities subjected to high-temperature irradiation, the formation of kink pairs becomes considerably difficult. Differing dynamic interactions of dissolved and precipitated impurities, i.e., P and Cu, with the nucleation and growth of dislocations are discussed, giving rise to irradiation hardening.

Appendices

Appendix I

1.1 Rate theory

A fundamental principle of rate controlling theory is that the shear strain rate (dγ/dt) is proportionally related to the average velocity of dislocations (V) and the density of mobile dislocations (ρ _m ):[15]

$$ d\gamma /dt = \rho _{m} {\mathbf{b}}V $$

(AI-1)

where b is the magnitude of the Burgers vector. Under steady state, the value of ρ _m parabolically increases with increasing shear flow stress (τ _s ); i.e.,

$$ \rho _{m} = \alpha {\left( {\tau _{s} /\mu {\mathbf{b}}} \right)}^{2} $$

(AI-2)

where μ is the shear elastic modulus and α is a dimensionless constant. The dislocation mobility is thermally activated under the shear flow stress and is appreciably changed depending on two types of barriers, such as discrete obstacles and lattice resistance. The rate of dislocation motion can be described in terms of the Gibbs free energy (ΔG(τ _s )) that depends on the flow stress

$$ V = \beta {\mathbf{b}}\nu {\text{ exp }}{\left[ { - \Delta G{\left( {\tau _{s} } \right)}{\text{/R}}T} \right]} $$

(AI-3)

where β is a dimensionless constant, R is the gas constant, ν is the Debye frequency, and T is the absolute temperature. The magnitude of activation energy is related to how the nucleation and growth of dislocations are influenced by discrete obstacles such as solute atoms, precipitates, and forest dislocations, and by lattice resistance. Although the activation area is theoretically introduced to describe the effect of flow stress, the Gibbs free energy is phenomenologically described in Eq. [AI-4], because of ambiguous physical meanings of the activation area indicating the temperature dependence:[15,16]

$$ \Delta G{\left( {\tau _{s} } \right)} = \Delta F^{i} {\left\{ {1 - {\left( {\tau _{s} /\tau ^{i}_{a} } \right)}^{p} } \right\}}^{q} $$

(AI-4)

where ΔF ⁱ is the Hermholtz free energy not influenced by the stress; $\tau ^{i}_{a}$ is the athermal shear stress at 0 K; p and q are empirical factors that would depend on the shape of dislocation barriers and the superscript of the activation energy and athermal stress; and subscript i = o and l, for DO- and LR-controlled plasticity. Using Eqs. [AI-1] through [AI-4], the rate equation for the dislocation motion is given by

$$ d\gamma /dt = \alpha \beta \nu {\left( {\tau _{s} /\mu } \right)}^{2} {\text{exp }}{\left[ { - {\left( {\Delta F^{i} /{\text{R}}T} \right)}{\left\{ {1 - {\left( {\tau _{s} /\tau ^{i}_{a} } \right)}^{p} } \right\}}^{q} } \right]} $$

(AI-5)

In the case of the DO-controlled glide, (τ _s /μ)², p and q can be unity because of the large value of ΔF ^o. The pre-exponential term [(dγ/dt) _o , = αβν] is empirically given by 10⁶ s⁻¹. Converting the shear flow stress and strain components into the tensile yield stress (σ _y ) with σ _y  = √3τ _s and ε = γ/√3, the rate equation for DO-controlled plasticity using the tensile stress and strain components is given by

$$ d\varepsilon /dt = {\left\{ {10^{6} /\surd 3} \right\}}{\text{exp }}{\left[ { - {\left( {\Delta F^{o} /{\text{R}}T} \right)}{\left\{ {1 - {\left( {\sigma _{y} /\sigma ^{o}_{a} } \right)}} \right\}}} \right]} $$

(AI-6)

For LR-controlled plasticity with low ΔF ^l, p = 3/4, q = 4/3, and (dγ/dt) _o  = 10¹¹ s⁻¹ are experimentally used, and the rate equation is

$$ d\varepsilon /dt = {\left\{ {{\left( {10^{{11}} /3\surd 3} \right)}} \right\}}{\left( {\sigma _{y} /\mu } \right)}^{{\text{2}}} {\text{exp }}{\left[ { - {\left( {\Delta F^{l} /{\text{R}}T} \right)}{\left\{ {1 - {\left( {\sigma _{y} /\sigma ^{l}_{a} } \right)}^{{3/4}} } \right\}}^{{4/3}} } \right]} $$

(AI-7)

The temperature dependence of the shear modulus of iron is given by

$$ \mu {\text{ (MPa)}} = 35.36{\left( {2050 - 0.81T} \right)} $$

(AI-8)

Plasticity controlled by obstacles and lattice resistance, which are affected by the type and quantity of impurities and neutron irradiation, can be characterized in terms of the athermal stress and activation energy, determined using Eqs. [AI-6] through [AI-8].

According to Reference 15, the dislocation mobility through discrete obstacles and lattice resistance can be categorized into three groups with different activation energies: (1) for weak obstacle strength (lattice resistance and solid solution), ΔF < 0.2 μ b ³ (120 kJ/mole); (2) for a medium strength (dislocations, defect clusters, and small or weak precipitates), 0.2 μ b ³ (120 kJ/mole) <ΔF < 2 μ b ³ (1200 kJ/mole); (3) for a strong strength (large or strong precipitates), ΔF > 2 μ b ³ (1200 kJ/mole).

The athermal stress is inversely related to the obstacle spacing (L) by

$$ \sigma _{a} = \kappa _{o} \mu {\mathbf{b}}/L $$

(AI-9)

where κ _o is a dimensionless dislocation-pinning factor depending on the type of obstacles, i.e., solute solution (κ _s ) and precipitate (κ _p ).[15,17,30] The inverse obstacle spacing (1/L) is given by χ√C _i /b and φ√f _v /d for solid solution and precipitation, where χ and φ are constant, C _i is the solute content, and f _v and d are the volume fraction and diameter of precipitates, respectively.[21] Then σ _a is given by

$$ \sigma _{a} = \kappa _{s} \chi \mu \surd C_{i} {\text{ for solid solution}} $$

(AI-10a)

$$ \sigma _{a} = \kappa _{p} \varphi \mu {\left( {{\mathbf{b}}{\text{/d}}} \right)}\surd f_{v} {\text{ for precipitation}} $$

(AI-10b)

The pinning strength changes as a function of the type and size of the medium-to-strong precipitates, that is, κ _p is proportionately related to the critical angle (ω) at which a dislocation can cut an obstacle:[18,36,37]

$$ \kappa _{p} \propto {\text{cos }}{\left( {\omega /2} \right)} = {\left\{ {1 - {\left( {\mu _{p} {\text{/}}\mu _{{{\text{Fe}}}} } \right)}^{{\text{2}}} } \right\}}^{{1/2}} $$

(AI-11a)

$$ \kappa _{p} \propto {\text{cos }}{\left( {\omega /2} \right)} = {\left\{ {{\text{ln (}}d{\text{)}} + 0.7} \right\}}/2\pi {\text{ when }}d\ll{\text{ }}L $$

(AI-11b)

Appendix II

1.1 Analysis of small-angle neutron scattering

The neutron scattering intensity can be given as a function of the scattering vector (q), which is related to the neutron wavelength (λ) and the neutron scattering angle (θ):

$$ {\mathbf{q}} = 4\pi {\text{ sin }}{\left( {\theta /2} \right)}/\lambda $$

(AII-1)

To analyze precipitates induced by neutron irradiation, the difference in the neutron scattering intensity between the irradiated and unirradiated alloys (ΔI(q)) was measured. The volume fraction of precipitates (f _v ) is related to the invariant of neutron scattering intensity (Q) that is given by ∫ q ²ΔI(q)d q and the difference in the scattering length density in the matrix and precipitate (Δρ) as given in[26,27]

$$ f_{v} = Q/2(\pi \Delta \rho )^{2} $$

(AII-2)

Assuming the atomic mass is constant, the Δρ can be given by superimposing the nuclear and magnetic scattering, is

$$ {\left( {\Delta \rho } \right)}^{{\text{2}}} = {\left\{ {{\left( {\rho _{n} } \right)}_{p} - {\left( {\rho _{n} } \right)}_{{{\text{Fe}}}} } \right\}}^{{\text{2}}} + {\left\{ {{\left( {\rho _{m} } \right)}_{p} - {\left( {\rho _{m} } \right)}_{{{\text{Fe}}}} } \right\}}^{{\text{2}}} {\text{ sin}}^{{\text{2}}} \phi $$

(AII-3)

where ϕ is the angle with respect to the direction of an applied magnetic field. Then

$$ {\left( {\Delta \rho } \right)}^{2}_{h} = {\left\{ {{\left( {\rho _{n} } \right)}_{p} - {\left( {\rho _{n} } \right)}_{{{\text{Fe}}}} } \right\}}^{{\text{2}}} {\text{ for the horizontal direction: }}\phi = 0{\text{ deg}} $$

(AII-4a)

$$ {\left( {\Delta \rho } \right)}^{2}_{v} = {\left\{ {{\left( {\rho _{n} } \right)}_{p} - {\left( {\rho _{n} } \right)}_{{{\text{Fe}}}} } \right\}}^{{\text{2}}} + {\left\{ {{\left( {\rho _{m} } \right)}_{p} - {\left( {\rho _{m} } \right)}_{{{\text{Fe}}}} } \right\}}^{{\text{2}}} {\text{ for the vertical direction: }}\phi = 90{\text{ deg}} $$

(AII-4b)

where the nuclear and magnetic scattering length density of the precipitate were estimated using (ρ _n ) _p  = Σc _i (ρ _n ) _i and (ρ _m ) _p  = Σc _i (ρ _m ) _i , where c _i is the atomic fraction of each element.[28] The average volume fraction of precipitates was determined by the invariants in the horizontal and vertical direction (Q _h and Q _v ), experimentally measured together with (Δρ) _h and (Δρ) _v . In addition, the average diameter of precipitates (d) is derived using

$$ \Delta I({\mathbf{q}}) \propto {\text{exp }}( - {\mathbf{q}}^{2} d^{2} /20) $$

(AII-5)