Stacking fault energy measurements in solid solution strengthened Ni–Cr–Fe alloys using synchrotron radiation

Abstract

The stacking fault energy (SFE) in a set of experimental Ni–Cr–Fe alloys was determined using line profile analysis on synchrotron X-ray diffraction measurements. The methodology used here is supported by the Warren–Averbach calculations and the relationships among the stacking fault probability (α) and the mean-square microstrain (<ε²_L>). These parameters were obtained experimentally from cold-worked and annealed specimens extracted from the set of studied Ni-alloys. The obtained results show that the SFE in these alloys is strongly influenced by the kind and quantity of addition elements. Different effects due to the action of carbide-forming elements and the solid solution hardening elements on the SFE are discussed here. The simultaneous addition of Nb, Hf, and, Mo, in the studied Ni–Cr–Fe alloys have generated the stronger decreasing of the SFE. The relationships between SFE and the contributions on electronic structure from each element of additions were established.

Introduction

The solid solution strengthened Ni–Cr–Fe alloys are an important group of face-centered-cubic (FCC) materials, which have an excellent combination of good mechanical properties and corrosion resistance. These kinds of Ni-alloys are widely used in applications involving operations at moderate temperatures, aggressive environments and high stress, such as in petrochemical, power and nuclear industries [1], [2], [3].

Ni–Cr–Fe alloys, generally, contain minor additions of C, N, Ti, Al and Nb, among others, with the aim to favor primary and secondary precipitations, increasing strength and to improve thermal properties behavior [3]. Some fabrication and joining processes used on this type of alloys, such as welding and hot forming involve simultaneous combination of heating and reheating cycles, high deformation and stress, and mechanical restrictions. During application of these processes, Ni–Cr–Fe alloys could present hot ductility reduction and, consequently, intergranular fracture [4], [5], [6], [7].

The relationships between SFE and the plastic deformation are broadly known [8]. SFE affects the ability of FCC-crystalline materials to distribute and to concentrate the strain in lattice during plastic deformation because, depending of SFE value, the cross slip process is made difficult, affecting the ductility [9]. In order to better understand the mechanism behind hot ductility reduction phenomenon this work aims to determine the SFE values. Likewise, it will establish the relationships among SFE and chemical composition, dislocation movements, and potential of recovery and recrystallization in a set of Ni–Cr–Fe alloys, which presents different susceptibility to ductility-dip cracking.

A stacking fault (SF) is a lattice planar defect, which consists in a local region in the crystal where the regular stacking of compact atomic planar sequence has been interrupted. In FCC crystals a SF is produced when an A-type atomic layer or any other layer occupies an alternative stacking site, changing the sequence of layers from ABCABC….to ABABABA…. In FCC crystals two types of SF are possible resulting from the removal (intrinsic) or introduction (extrinsic) of an extra layer [9].

Stacking fault energy (SFE) is a very important physical property of polycrystalline systems, because it is related to other important properties and phenomena, such as creep, strain deformation, annealing twins, formation of dislocations, swelling, stress corrosion cracking, phase transformation stability, and electron/vacancy density, among others. The SFE strongly influences processes of dislocation movement, such as cross slip and climbing, providing important information regarding the formation of defects during deformation and subsequent nucleation of new recrystallized grains. Generally, a low SFE value in FCC polycrystalline materials leads to a low mobility of dislocations, making difficult the climbing and cross slip processes. This condition leads to obtain a homogeneous distribution of dislocations and a high dislocation density inside grain compared to high SFE material [8], [9].

A dislocation in a FCC-lattice may dissociate into two partial dislocations separated by an intrinsic stacking fault as shown in Fig. 2. In the SF region the close packed atomic stacking sequence changes from an FCC sequence ABCABC… to an HCP sequence ABAB…. Then, the SFE is the energy per unit area associated with a stacking fault. The SFE also leads to the formation of localized HCP structures, such as the case of TRIP and TWIP steels [10], [11]. The stacking fault probability (α) is inversely proportional to SFE, therefore, when the stacking fault probability is high, and the partial dislocations separation (d) is large, then the SFE is low.

Several experimental methods have been used to determine the SFE with different levels of precision and accuracy. Transmission electron microscopy (TEM) is the most direct method for measurement of SFE, which involves the direct measurements of dislocation-node radii [12]. Measurement method by TEM is considered the most accurate method, especially for low SFE materials; however, this method is statistically poor. Methods based on X-ray measurements are considered as low precision, despite these have better statistical in the bulk material compared to TEM method. X-ray measurements include texture, third-stage single crystal work hardening rates and others.

Experimental measurements using X-ray diffraction, and calculation methods based on line profile analysis and peak shifting can be applied to a wide set of materials with the aim to obtain its SFE [13]. In the past there was some controversy when X-ray diffraction was used to determine SFE, because there was little clarity of relationships among stacking fault energy (SFE), stacking fault probability (α), and the mean-square microstrain, which is orthogonal to diffracting plane {hkl} and averaged over L distance (<ε²_L>_hkl). However, researches by Reed and Schramm have clarified this issue, establishing that the SFE can be expressed through Eq. (1), where it shows the relationships among stacking fault probability, and microstrain [13], [14].

$$SFE=\frac{K_{111}{\omega }_0{G}_{(111)}{a}_0}{\pi \sqrt{3}}\frac{<{\epsilon }_{50\AA }^2{>}_{111}}{\alpha }{A}^{-0,37}$$

Where K₁₁₁ and ω₀ are constants dependent of material; G₁₁₁ is the shear modulus along the (111) plane and is defined as 1/3(c₄₄+c₁₁– c₁₂), where c_ij are the elastic stiffness coefficients; a₀ is the lattice constant, α is the stacking fault probability, and <ε²_50 Å>₁₁₁ is the mean-square microstrain, which is measured orthogonal to diffracting plane {111} and averaged over 50 Å of distance. A is a constant term equivalent to the Zener anisotropy correction (A=2c₄₄/(c₁₁– c₁₂)) [13]. Using the partial Burgers vector b_p=a₀/√6, Eq. (1) can be re-arranged into Eq. (2).

$$\frac{SFE}{G_{(111)}\cdot b_p}=\frac{\sqrt{2}{K}_{111}{\omega }_0}{\pi }\frac{<{\epsilon }_{50A}^2{>}_{111}}{\alpha }{A}^{-0,37}$$

Usually, the constant term √2K₁₁₁ω₀/π is calculated using, both experimental and theoretical data. In this case, experimental data are lattice constant (a₀), mean-square microstrain (<ε²_50 Å>₁₁₁) and stacking fault probability (α) (the right side on Eq. (2)). The shear modulus G₍₁₁₁₎ and the SFE values, (the left side on Eq. (2)) are defined from literature information. It must be kept in mind that both the values of (<ε²_50 Å>₁₁₁) and α depend on the cold working degree; therefore, both measurements must be performed on the same specimen and at the same condition with the aim to guarantee the successful of SFE measurements [13], [14].

With the aim to measure the SFE of a set of materials, it is a common practice, first to establish the constant term of the Eq. (2) from well-known information available of a set of pure elements with FCC crystalline structure. It is necessary to perform the described procedure on, at least, three pure materials with the aim to obtain a good spectrum of SFE values, and consequently an acceptable linear fit. Once the constant term √2K₁₁₁ω₀/π has been defined, then it can be used to calculate the SFE of each experimental alloy, using the experimental values of mean-square microstrain, stacking fault probability, shear modulus, and Burgers vector. The mean-square microstrain values can be determined using Fourier coefficients analysis method applied on {111} peak signal, which must be free of instrumental effects [15]; meanwhile, stacking fault probability (α) can be calculated from Eq. (3) [16].

$$\mathrm{\Delta }2\theta ={(2{\theta }_{200}-2{\theta }_{111})}_{\mathrm{CW}}-{(2{\theta }_{200}-2{\theta }_{111})}_{\mathrm{ANN}}=-\frac{45\sqrt{3}}{{\pi }^2}\alpha \left[\tan {\theta }_{200}-\frac{1}{2}\tan {\theta }_{111}\right]$$

In Eq. (3), 2θ_hkl represents the {hkl} peak line position in degrees. The CW and ANN indices indicate cold-worked and annealed state of specimen, respectively.