In-situ neutron diffraction during stress relaxation of a single crystal nickel-base superalloy

The stress relaxation behaviour of a single crystal nickel-base superalloy has been quantiﬁed using time-of- ﬂight neutron diffraction analysis for a range of temperatures relevant to casting. A new iterative analysis methodology is described to isolate the lattice strain behaviour of the c matrix and c (cid:2) precipitate phases from data obtained suﬃciently rapidly to help elucidate the microscopic effect of macroscopic stress relaxation. The independent response of c and c (cid:2) is revealed, showing the temperature sensitivity of lattice strain relaxation. The c / c (cid:2) response is discussed in the context of thermo-mechanical conditions that may affect the propensity for recrystallisation.

Nickel-base superalloys are typically chosen for turbine blade applications due their excellent high temperature mechanical and environmental properties. To confer suitable creep performance, the material is cast into a single crystal via directional solidification, involving a controlled mould withdrawal from the furnace. During this process, macroscopic residual stresses are produced in the component due to localised plastic flow induced from the different thermal expansions of the metal and the ceramic mould during cooling. These are sufficient for the metal to experience visco-plasticity through creep and stress relaxation, affecting the resultant dislocation density. This has implications to recrystallisation during subsequent processing heat treatments. Such artifacts are unacceptable during turbine blade manufacture. It is therefore critical to quantify macromechanical and micromechanical strains, in addition to their sensitivity to processing conditions. An attractive method for quantifying micromechanical strains in nickel-base superalloys is diffraction via the evaluation of lattice strains. In particular, techniques that offer high angular resolution to separate the disordered A1 c matrix and ordered L1 2 structured c precipitates are desirable. Experiments are often complicated by the desire to obtain information at service or processing relevant stresses and/or temperatures. Laboratory X-ray sources can be equipped with in-situ testing capability, though their low flux and energy typically prohibits the measurement of the inherently weak c superlattice reflections. Experiments are instead commonly performed at either synchrotron or neutron sources. To date, experimenters have investigated dynamic behaviour such as tensile [1,2], creep [3][4][5], processing heat treatments & phase transformations [6][7][8][9] and stress relaxation [10,11]. In this study, an analysis methodology is developed that enables quantification of c and c lattice strains to investigate the stress relaxation behaviour in-situ during time-of-flight neutron diffraction measurements of a single crystal nickel-base superalloy. With a necessarily rapid data acquisition time to describe such phenomena, the new method enables analysis of diffraction data with a low signal to noise ratio that would be impossible using existing data fitting strategies.
Time-of-flight diffraction experiments were performed on the ENGIN-X neutron diffractometer [13], ISIS, UK. The experimental setup adopted during the acquisition of neutron diffraction data is shown in Fig. 1. The neutron signal was obtained from a probed volume measuring 8 × 4 × 4 mm 3 , controlled by the incident beam cross section and the collimator size. Samples were heated in an optical furnace at 10 • C min −1 to 800 • C then at 5 • C min −1 to the test temperatures 900 • C, 940 • C, 980 • C and 1000 • C measured with a K-type thermocouple, followed by a 20 min hold period before  loading. Using a tensile rig on the beam line, four specimens were loaded at a strain rate of 3.3 × 10 −5 s −1 to different initial stress levels (600 MPa, 480 MPa, 400 MPa and 380 MPa). Stresses were held at these magnitudes under load control for 9 min. Thereafter, relaxation was conducted under displacement control for 9 min. The load, hold and relaxation cycle was repeated for 6-7 cycles, with each cycle increasing the macroscopic stress by 10 MPa.
The measurement of c and c lattice parameters has been achieved by adopting a new data fitting strategy, developed from a previous methodology used for the analysis of X-ray diffraction data from polycrystalline nickel-base superalloys [9]. This method is tailored for time-of-flight neutron diffraction data from single crystal superalloys, incorporating the characteristic line profile asymmetry arising from the thermalisation process of neutrons, producing a time distribution of the neutron pulse [14]. The line profile shape used is a convolution product of a Voigt function and a trailing exponential function, where the former is defined as where d is d-spacing converted from time-of-flight and d c is the d-spacing coordinate of the peak centre of mass. The full fitting function, V exp (x), is given by where g is a constant and H(x) is the Heaviside step function. The line profile asymmetry from the ENGIN-X instrument was determined by performing single peak fitting (using V exp function) of a CeO 2 standard on all reflections in the d-spacing range 1.
To correctly fit the superalloy diffraction patterns, the parameters contributing to the intensity of individual reflections are considered. Following [15], the integrated intensity, I of a polychromatic incident neutron beam for a single Laue spot is where I 0 (k 0 ) is the intensity-wavelength (k 0 ) distribution, F hkl is the structure factor for reflection hkl, Dv is the diffracted volume,  small, (typically −5 × 10 −3 to 1 × 10 −3 [16,17]) Eq. (3) can be used to express the ratio of intensity between the two reflections: All absent terms from Eq. (3) are eliminated, being equivalent for c and c (including unit cell volumes where v c a ≈ v c a ). Whilst the above intensity expression has been simplified; neglecting absorption, extinction and thermal vibration effects is valid as their contributions are approximately cancelled in Eq. (3). For the (200) c reflection, the structure factor (simplified from general expressions, described elsewhere [9]) is where c Z is the composition and b Z is the bound coherent scattering length for each element, Z, with terms for the latter tabulated in [18]. For the ordered L1 2 structured c with a Ni 3 Al stoichiometry, the (200) fundamental reflection structure factor is where an element Z may be present on a 'Ni' site, A or 'Al' site, B. Their respective site occupancies are c Z A and c Z B . CMSX-4 possesses a strong elemental partitioning to a single c site, except Cr which has a composition dependent site preference [19]. The present study assumes Ni and Co occupy site A, whilst all remaining elements including Cr are occupant on the B site. Following a prediction of equilibrium temperature dependent phase compositions using the thermodynamic calculation tool, JMatPro [20], c compositions were used in Eq. (5) and c compositions were used to evaluate c site occupancies, and the terms c Z A and c Z B in Eq. (6). The c composition is not predicted nor measured experimentally [21] to change dramatically as a function of temperature, hence, Cr is not expected to switch site preference. Phase equilibria calculations provide c & c volume fractions, shown in Fig. 2 (d). These values along with calculated structure factor values (shown in Fig. 2 (d)) were used to evaluate the c/c intensity ratio in Eq. (4).
All analysis and fitting functions were written in Matlab using the following methodology: b G . Fit these parameters with respect to pattern number as shown in Fig. 2 (e). Repeat the fitting of all spectra using fitted b L & b G values. 8. The peak width scaling parameter a is now refined. The fitting error of the {200} c,c fundamental reflections are calculated for all patterns in a test using the w 2 Pearson's cumulative test statistic. A histogram of these errors summed Sw 2 was fitted with a normal distribution, as shown in Fig. 2 (f). Repeat steps 2-7, using different a values (0.6 to 1.6 used here). The selected value of a has the narrowest Sw 2 distribution close to zero. The variation of Sw 2 with a is shown in Fig. 2 (g).
Example raw data at room temperature with no stress applied is shown in Fig. 2 with fitted (100) and (300)   and background are free variables, similar to the method described in [16]. Method B uses a V exp (x) line profile shape, Method C adds the I c 200 /I c 200 ratio constraint (described in step 5 above), Method D adds a constraint to g for all reflections (step 1), Method E uses fitted values of b G & b L (step 7), and Method F includes the refined parameter a (step 8). Values of Sw 2 from I 200 indicate the success of the fitted reflection; each constraint added between Methods A to F improves the fit. Lattice parameters a c and a c are affected by the fitting constraints; converging to correct values as Sw 2 becomes small. Adopting Method F does not invalidate Method A, however, such procedures are only suitable for data with sufficiently high signal to noise ratio. For rapid data acquisition, as considered here, a more robust analysis routine, i.e. Method F, must be adopted.
The material response to the stress relaxation experiment for the range of tested temperatures as a function of time is shown in Fig. 4.   Considering recrystallisation, the temperature dependent relative strengths of c and c must play a role. At 1000 • C, c is considered undeformable [23], giving a high dislocation density in the c channels, and thus a high driving force for recrystallisation. However, during relaxation at 1000 • C in this study, 4 c (200) changes are significant and does not support the observation that c is undeformable. The magnitude of lattice strain change during relaxation at 1000 • C for c and c is greater than at a lower temperatures; this is significant considering the applied macroscopic stress was lower. The drops in the tensile direction lattice strains, for example, provides evidence that more stored elastic strain energy is released at increased temperatures. How the relaxation of c is coupled to the deformation of c, however, remains unclear. It is likely that the precipitates themselves are subjected to dislocation cutting [12] and/or experience elastic relaxation due to the plastic deformation of the surrounding c. This temperature dependent phase selectivity for deformation is likely to determine the propensity for recrystallisation and the future selection of turbine blade solidification processes.