# Quantification of Solid Solution Strengthening and Internal Stresses through Creep Testing of Ni-Containing Single Crystals at 980 °C

^{*}

## Abstract

**:**

## 1. Introduction

_{2}-ordered γ′ precipitates coherently embedded in a face-centered cubic γ-matrix [2]. Others have focused on the improvement of the matrix strength by alloying solid solution strengthening elements such as tungsten (W), molybdenum (Mo), rhenium (Re) [3], and ruthenium (Ru) [4]. The following subchapters will discuss different contributions to the stress state present in Ni-based superalloys during external creep load.

#### 1.1. Internal Back Stress

#### 1.2. Solid Solution Strengthening

#### 1.3. Dislocation Density Induced Internal Back Stress

#### 1.4. Misfit Stress

_{2}-ordered ${\gamma}^{\prime}$-precipitates in nickel-based superalloys are coherently embedded in the $\gamma $-matrix. Both phases are based on an fcc lattice structure with slightly different lattice parameters. This leads to a misfit stress ${\sigma}_{\mathrm{misfit}}$ [17,18], which can be roughly approximated by:

#### 1.5. Orowan Stress

#### 1.6. Cutting of γ′-Phase

#### 1.7. Summary Stress Contributions

- Dislocation density varies depending on the orientation of individual matrix channels with respect to external load direction.
- The misfit can vary locally from 0 to −4 × 10
^{−3}in interdendritic areas as compared to the dendrite center, respectively [19]. - The distance L between ${\gamma}^{\prime}$-precipitates is distributed between 10 and 150 nm, resulting in a change of the Orowan stress within a factor of 15. Additionally, coarsening occurs during long-time creep, resulting in an increase of L.
- The stresses can convert into each other, e.g., misfit-induced stress can turn into dislocation-induced stress.

^{−7}s

^{−1}was reached, leading to lifetimes of ≈12 days for 10% strain. This allows enough time for diffusional processes to take place, being similar for all alloys.

## 2. Materials and Methods

#### 2.1. Material

- (A)
- Pure Ni (single-phase fcc) with 99.9 at.% technical purity and configurational entropy equal to zero. No heat treatment was carried out. Solid solution strengthening can be excluded. For the sake of readability, this sample composition will also be referred to as an “alloy”, knowing that this might not be a precise denomination.
- (B)
- Equiatomic alloys with single-phase fcc microstructure: CrCoNi with 33.3 at.% for each element and the so-called Cantor alloy (CrMnFeCoNi with 20 at.% each) [26]. These two compositions exemplify a configurational entropy ${\mathrm{S}}_{\mathit{config}}$ of 1.10·R (generally called medium entropy alloy) and 1.61·R (high entropy alloy). No heat treatment was carried out on CrCoNi and Cantor alloy. Due to the evaporation of Mn during the SX casting procedure, the Mn content was adjusted before casting to obtain 20 ± 1 at.% in the cast single-crystal sample.
- (C)
**Single-phase**solid solution strengthened Ni-based alloys with fcc structure. The two alloys of this group have the composition of the matrix of the two alloys of Group (D). The compositions of the matrix phases were calculated with Thermo-Calc Software and database TTNi7 at 800 °C [27] and compared with experimentally determined matrix composition. A temperature of 800 °C was used to ensure the single-phase state at 980 °C. No precipitates were observed in these alloys; hence, they remain single-phase, see, for example, Figure 2a. Homogenization treatment was performed at 1290 °C for 48 h, see Table 3. The two alloys within this group have similar compositions, except for the element Re. Thereby, the contribution of solid solution strengthening of Re can be quantitatively determined. It should be noted that 3.1 atomic% of Re converts into 9.0 weight%.- (D)
- The commercially available two-phase Ni-based superalloys CMSX-3 (no Re, 1st generation) and CMSX-4 (1.0 at.% (3 wt.%) Re, 2nd generation) [28]. The exceptionally good mechanical properties of Ni-based superalloys at high temperatures are mainly due to the two-phase matrix/${\gamma}^{\prime}$-microstructure (Figure 2b). A specific heat treatment, see Table 3, was used after single-crystal casting in order to obtain such a microstructure, resulting in an average edge length of the cuboidal ${\gamma}^{\prime}$-particles of 410 ± 30 nm (CMSX-3) and 470 ± 50 nm (CMSX-4). The ${\gamma}^{\prime}$-volume fraction for the alloys is about 60% for CMSX-3 and about 70% for CMSX-4 at 980 °C.

#### 2.2. Sample Processing

^{−4}Pa or encapsulated in quartz glass tubes, using the heat treatment parameters listed in Table 3. Lower temperature heat treatment (annealing with <900 °C) was carried out in air since the sample surface was removed before creep testing. Thereafter, samples were cut into flat creep samples using wire electrical discharge machining (EDM). Out of one single-crystalline rod, more than 10 creep specimens could be eroded. The flat sides were sanded with a 2000 grid and etched in order to get rid of the EDM re-cast layer to finally achieve the creep samples.

#### 2.3. Creep Testing

^{−3}and 10

^{−9}s

^{−1}, resulting in lifetimes of several 10 min to more than 1 year (in theory, if the test was not aborted). Since the focus of this research is on the minimum or secondary creep rate, respectively, tests with strain rates below 10

^{−8}s

^{−1}were generally aborted after reaching a stationary state. Pure Ni and the high entropy Cantor alloy were tested in the range of 5–14 MPa, the single-phase medium entropy and matrix alloys of Group (C) were tested in between 15 and 75 MPa and Group (D), and the two-phase Ni-based superalloys were tested in between 170 and 300 MPa. This paper summarizes 25 creep tests with a minimum of three tests for each alloy in order to obtain a reliable Norton exponent for each alloy.

#### 2.4. Simulation of Two-Phase Steady-State Creep by Finite Element Calculations

## 3. Results and Discussion

#### 3.1. Experimental Evaluation of the Steady-State Creep Rate

#### 3.2. Creep Results at 980 °C over a Wide Range of Applied Stress

^{−7}s

^{−1}(horizontal dashed line in Figure 6), and the logarithm to base 10 of the extrapolate $\dot{{\epsilon}_{0}}$ at an applied stress σ = 100 MPa (vertical dashed lines in Figure 6) are listed in Table 5.

^{−7}s

^{−1}is reached for the single-crystal pure Ni samples and the high entropy Cantor alloy; then, the medium entropy alloy CrCoNi and the two single-phase matrix alloys follow. The stress needed to reach a steady-state strain rate of 10

^{−7}s

^{−1}in MatrixCMSX-4 compared to pure Ni is increased by a factor of 4 solely by solid solution strengthening. Finally, the two-phase Ni-based superalloys exhibit a wide gap, showing the strongest creep resistance. The sequence for the increase in Re content is the same: the matrix of CMSX-3 (no Re) has less creep resistance than the matrix of CMSX-4 (with 3.1 at.% or 9 wt.% Re), which is the same for the two-phase superalloys: CMSX-3 (no Re) is less creep-resistant than CMSX-4 (with 1.0 at.% or 3 wt.% Re).

^{−7}s

^{−1}by a factor of 12 and 9, respectively. Due to the strong solid solution strengthener Re in the matrix alloy of CMSX-4, the contribution of the precipitation hardening in CMSX-4 is smaller compared to CMSX-3 and its corresponding matrix. Both superalloys have similar composition except for Re, see Table 1. Thereby, the solid solution-strengthening potential of Re in the Ni-based superalloys is quantified. An addition of 1.0 at.% Re in CMSX-4 leads to a higher stress to reach a steady-state strain rate of 10

^{−7}s

^{−1}of a factor of 1.2 as compared to CMSX-3 with no Re content.

#### 3.3. Quantification of Solid Solution Strengthening and Internal Back Stresses by Precipitation Hardening

^{−7}s

^{−1}rises almost by a factor of 10. This is in very good agreement with our experimental observations. The presented FE model can quantitatively assess the magnitude of the precipitation hardening effect. Overall, the simulations predict a higher creep rate for CMSX-3 than for CMSX-4 by a factor of over 300. At a volume fraction of 70%, the different creep resistance of the matrix (see simulation input data in Table 4) leads to an increase in the simulated two-phase creep rate of a factor of 34. This is again in good agreement with the factor of 25 between the creep rates of the matrix of CMSX-3 and CMSX-4. The presented simulation data also show the quantitative influence of the precipitate volume fraction on the creep rate. The creep rate of CMSX-3 with a more realistic γ′ volume fraction of 60% is again a factor of 10 higher compared to the simulation with a volume fraction of 70% as in CMSX-4.

^{−7}s

^{−1}is reached (210 and 260 MPa, respectively). The internal stresses of the two-phase alloys are calculated via Equation (3) as

#### 3.4. Combined Effect of Various Strengthening Mechanism

## 4. Conclusions

^{−7}s

^{−1}, we can summarize:

- An influence of solely configurational entropy on creep strength is not observed. However, solid solution strengthening is of course of very high importance. Bearable stresses are four times higher for the matrix of CMSX-4 compared to pure Ni.
- We quantify the combined internal back stresses due to the various acting strengthening mechanisms in two-phase superalloys to be:
- –
- ${\sigma}_{0}^{\mathrm{CMSX}-4}=143\mathrm{MPa}$ for the Ni-based superalloy CMSX-4.
- –
- ${\sigma}_{0}^{\mathrm{CMSX}-3}=16\mathrm{MPa}$ for CMSX-3.

- Additions of Re: 1.0 at.% (3 wt.%, respectively) Re in the superalloys, corresponding to 3.1 at.% (9 wt.%, respectively) in the single-phase matrix, result in a strength increase of the matrix by a factor of 1.7 (1.2, respectively) in the two-phase material.
- The element Re influences the lattice misfit, the γ′ precipitate size, and morphology and thereby increases the internal back stress.
- Quantification of these effects is only possible by using single-crystal samples.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic Norton plot of the stationary creep rate over the applied stress for a material with an internal back stress ${\sigma}_{0}=100\text{}\mathrm{MPa}$ (green line) and a matrix response with n = 6 (orange line). For an applied stress in the range of 180–300 MPa, a stress exponent of around 12 is determined (blue line). The blue line is shifted vertically due to different $\dot{{\epsilon}_{0}}$.

**Figure 2.**Microstructure of the investigated (

**a**) single-phase matrix alloys and the (

**b**) two-phase Ni-based superalloy with cuboidal, dark γ′-precipitates (ordered) and thin, bright γ-matrix channels (face-centered cubic, fcc).

**Figure 3.**Process chain of SX-casting, heat treatment, specimen production, and creep testing at the chair of metals and alloys in Bayreuth.

**Figure 4.**Finite element model of the γ/γ′ microstructure with a γ′-volume fraction of 60%. A constant load ${\sigma}_{ext}$ was applied in [001]-direction, and periodic boundary conditions were applied to all surfaces.

**Figure 5.**Creep experiments of CMSX-3 (

**c**,

**d**) and the corresponding matrix alloy MatrixCMSX-3 (

**a**,

**b**). Strain over time (

**a**,

**c**) and strain rate over time (

**b**,

**d**) are shown. Minimum creep rate ${\dot{\epsilon}}_{min}$ and transition creep rate ${\dot{\epsilon}}_{T}$ are indicated.

**Figure 6.**Minimum/transition creep rates of seven different alloys, all of them as single crystals at 980 °C and 100 MPa. Creep parameters are listed in Table 5.

**Figure 7.**(

**a**) Simulated two-phase creep rates with CMSX-3 and CMSX-4 matrix and a γ′ volume content of 60% and 70%. (

**b**) Experimental two-phase creep curves shifted by the calculated internal back stresses ${\sigma}_{0}$ for every alloy. (

**c**) Reduced creep rate at 100 MPa for two-phase CMSX-3 and CMSX-4 compared to their respective matrix alloys.

**Table 1.**Nominal composition of the alloys in at.%. Measured composition deviates in all cases only slightly.

in at.% | Group | Al | Ti | Cr | Mn | Fe | Co | Ni | Mo | Hf | Ta | W | Re |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Ni | (A) | - | - | - | - | - | - | 100 | - | - | - | - | - |

CrCoNi | (B) | - | - | 33 | - | - | 33 | 33 | - | - | - | - | - |

Cantor | - | - | 20 | 20 | 20 | 20 | 20 | - | - | - | - | - | |

matrixCMSX-3 | (C) | 3.1 | 0.1 | 21.4 | - | - | 20.3 | 51.1 | 0.9 | - | 0.1 | 3.0 | - |

matrixCMSX-4 | 3.3 | 0.1 | 20.0 | - | - | 19.6 | 50.0 | 0.9 | - | 0.1 | 2.9 | 3.1 | |

CMSX-3 | (D) | 12.4 | 1.3 | 9.1 | - | - | 4.7 | 67.6 | 0.3 | 0.05 | 2.0 | 2.6 | - |

CMSX-4 | 12.6 | 1.3 | 7.7 | - | - | 9.9 | 62.8 | 0.4 | 0.05 | 2.2 | 2.1 | 1.0 |

**Table 2.**Phases present in the alloys of groups (A) to (D). The configurational entropy ${S}_{\mathit{config}}$ of the solid solution alloys after Equation (12) is also given. The two-phase alloys are marked with a * to indicate that the entropy is calculated from ${S}_{\mathit{config}}$ of the respective matrix phases multiplied by the volume content of the matrix phase (60% for CMSX-3 and 70% for CMSX-4).

Alloy | Group | Phases | ${\mathit{S}}_{\mathit{config}}\mathbf{in}\text{}\mathit{R}$ |
---|---|---|---|

Ni | (A) | single-phase fcc | 0.00 |

CrCoNi | (B) | 1.10 | |

Cantor | 1.61 | ||

matrixCMSX-3 | (C) | 1.27 | |

matrixCMSX-4 | 1.37 | ||

CMSX-3 | (D) | two-phase fcc + L1_{2} | 0.51 * |

CMSX-4 | 0.41 * |

Alloy | Group | Heat Treatment | Literature |
---|---|---|---|

Ni | (A) | - | |

CrCoNi | (B) | ||

Cantor | - | ||

MatrixCMSX-3 MatrixCMSX-4 | (C) | 1290 °C/48 h | this work |

CMSX-3 | (D) | 1293 °C/2 h + 1298 °C/3 h, cooling with > 100 °C/min, 1080 °C/4 h, 870 °C/20 h | [17] |

CMSX-4 | 1320 °C/2 h, cooling with > 100 °C/min, 1140 °C/6 h, cooling with > 100 °C/min, 870 °C/20 h | [29] |

Parameter | Value | |
---|---|---|

Elastic constants | γ-matrix | C_{11} = 210 GPaC _{12} = 160 GPaC _{44} = 90 GPa |

γ′-phase | C_{11} = 200 GPaC _{12} = 140 GPaC _{44} = 100 GPa | |

Lattice misfit | −2.2 × 10^{−3} | |

Creep parameters at ${\sigma}_{ref}$ = 100 MPa | matrix CMSX-3 | ${\epsilon}_{0}$= 2 × 10^{−3} s^{−1}n = 5.5 |

matrix CMSX-4 | ${\epsilon}_{0}$= 8 × 10^{−5} s^{−1}n = 5.5 | |

Mass density | 8900 kg m^{−3} | |

γ′ volume fraction | CMSX-3 | 60% |

CMSX-4 | 70% |

**Table 5.**Norton exponents, bearable stresses for a steady-state strain rate of ${10}^{-7}{\text{}\mathrm{s}}^{-1}$ and creep rates at 100 MPa of the tested elements/alloys at a temperature of 980 °C. Lifetimes at 980 °C and 100 MPa are extrapolated by assuming a strain to failure of 10% and a constant strain rate throughout the sample lifetime.

Alloy | Norton Exponent | Stress $\mathit{\sigma}$ in MPa to Reach a Steady-State Strain Rate of 10^{−7} s^{−1} | ${\mathbf{Log}}_{\mathbf{10}}\left(\dot{{\mathit{\epsilon}}_{\mathbf{0}}}\mathbf{in}\text{}{\mathbf{s}}^{-\mathbf{1}}\right)\text{}\mathbf{at}\text{}\mathbf{100}\text{}\mathbf{MPa}$ | Extrapolated Life Time |
---|---|---|---|---|

Ni | 5.2 | 7.5 | −1.2 | 2 s |

CrCoNi | 5.8 | 22 | −3.2 | 3 min |

Cantor | 5.1 | 8.4 | −1.5 | 3 s |

MatrixCMSX-3 | 6.1 | 18 | −2.5 | 32 s |

MatrixCMSX-4 | 5.6 | 30 | −4.1 | 21 min |

CMSX-3 | 6.6 | 210 | −9.1 | 4 years |

CMSX-4 | 12.5 | 260 | −12.3 | >6000 years |

**Table 6.**Different contributions to the internal back stress in a Ni-based superalloy with high volume fraction of γ′ phase. Estimations have been obtained with: α = 1, G = E = 100 GPa, $b={a}_{\langle 110\rangle}/2=250\text{}\mathrm{pm}$, $\rho =1014\text{}{\mathrm{m}}^{-2}$, L = 50 nm, $\mathsf{\delta}\text{}=-{2.2\times 10}^{-3}$ [36,37], ${E}_{\mathrm{APB}}=0.15{\text{}\mathrm{J}\xb7\mathrm{m}}^{-2}$ [41,42].

Stress Type | Estimated Internal Back Stress Contribution | Lateral Extension | Location | Remarks |
---|---|---|---|---|

solid solution strengthening | w, f_{0} unknown and r^{−3} dependence | very short range~r^{−3} | matrix | always active during plastic deformation as long as a continuous matrix exists |

dislocation density | 250 MPa | const. | matrix | varies strongly locally, in exchange with misfit stress |

misfit | 220 MPa | const. | matrix | varies strongly locally on two different length scales, in exchange with dislocation density, can be negative |

Orowan | 500 MPa | const. | matrix | reduced during creep if coarsening occurs, increases with increasing dislocation density (apparent smaller separation of γ′ phase) |

cutting | 600 MPa | const. | γ′ phase | constant within γ′ phase |

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Glatzel, U.; Schleifer, F.; Gadelmeier, C.; Krieg, F.; Müller, M.; Mosbacher, M.; Völkl, R. Quantification of Solid Solution Strengthening and Internal Stresses through Creep Testing of Ni-Containing Single Crystals at 980 °C. *Metals* **2021**, *11*, 1130.
https://doi.org/10.3390/met11071130

**AMA Style**

Glatzel U, Schleifer F, Gadelmeier C, Krieg F, Müller M, Mosbacher M, Völkl R. Quantification of Solid Solution Strengthening and Internal Stresses through Creep Testing of Ni-Containing Single Crystals at 980 °C. *Metals*. 2021; 11(7):1130.
https://doi.org/10.3390/met11071130

**Chicago/Turabian Style**

Glatzel, Uwe, Felix Schleifer, Christian Gadelmeier, Fabian Krieg, Moritz Müller, Mike Mosbacher, and Rainer Völkl. 2021. "Quantification of Solid Solution Strengthening and Internal Stresses through Creep Testing of Ni-Containing Single Crystals at 980 °C" *Metals* 11, no. 7: 1130.
https://doi.org/10.3390/met11071130