Molecular dynamics simulations of dopant effects on lattice trapping of cracks in Ni matrix*

Shulan Liu(劉淑蘭) and Huijing Yang(楊會靜)

School of Physical Science and Technology,Tangshan Normal University,Tangshan 063000,China

Keywords: alloying elements,doping,Ni crack,lattice trapping

1. Introduction

The outstanding high-temperature mechanical properties of Ni-based single crystal(SC)superalloys make them popular for applications related to advanced airplane engine blades and power-generation turbines.[1,2]Theγ-Ni phase is a popular host matrix for the Ni-based SC superalloys as it affects the overall mechanical properties of the resulting materials,[2,3]while the crack inγphase restricts its applications. Analysis of theγphase microstructure evolution was reported[4-7]earlier. However,the fracture in superalloys are complex,and the strengthening mechanism ofγ-Ni matrix is still not very clear.Thus,understandings on how to strengthen theγ-Ni matrix and minimizeγ-Ni matrix fracture has been becoming the important research.

The rupture leading to a complete part failure often results from cracks, which are formed because of the atomic bond breakage. The fracture stress is closely connected to the critical activation of cracks and their propagation,which,in turn,depends on the lattice trapping of the cracks.[8]If the lattice traps the crack,it does not further propagate or heals when the mechanical loads are above or below the Griffith load (KGth)values,which are also called upper and lower lattice trapping limits(K±IC),respectively. The Griffith load is defined as a material state,at which the total energy change is zero when the crack propagates in a specific lattice direction of a cell with the defined in-plane boundaries.[9]At the Griffith load,a finite lattice resistance barrier exists between the isoenergetic states,which could trap the crack tip at either state. These phenomena manifest themselves as the lattice trapping effects.

The lattice trapping of material can be affected by several factors. The lattice trapping effect depends on the kinks and the interaction range of the interatomic force field, as well as on the relative stiffness of the atomic bonds at the propagation front.[9-12]Moreover,the directions of the crack front and propagation, as well as the interatomic potential and the material’s structure components, also affect the lattice trapping limit and range.[13-17]Because of the industrial importance of the Ni-based SC superalloys, understanding their brittle fracture behavior upon cracking is essential. Thus, research on crack lattice trapping of Ni matrix is needed.

Mechanical properties of Ni-based SC superalloys depend on their chemical compositions. Experiments and simulations indicated that doping superalloys with Re,Ru,Co,and W improved their mechanical properties.[18-27]Our previous work reported the lattice trapping limit increased for the Ni or Ni3Al cracks when Re or W was added and that W incorporation yielded much stronger effects for the Ni3Al cracks.[4,13,15]Other reports indicated the effect of Co on the crack propagation along theγ/γ'interface was minimal.[28,29]In general,Ni matrix strengthening mechanisms involving solid solution formation (with Co, Mo, W, Re, and Ru dopants) are complex. Simultaneous consideration of (i) the influence of the dopants on the mechanical properties of industrial Ni-Al-X(X=Re,Ru,Co,or W)superalloys(which was assessed previously by the embedded-atom-method, EAM)[30-33]and (ii)the importance of crack lattice trapping by the Ni matrix,analysis of fracture,strengthening mechanisms,crack propagation and crack lattice trapping of the Ni matrix doped with Re,Ru,Co and/or W is needed to design the material with the best performance.

Because of the accuracy and cost concerns, experiments to obtain data for this analysis might be hard to conduct, especially at the atomic scale. However,this can be achieved by using the molecular dynamics (MD) method, a powerful and reliable approach (especially when combined with the EAM potentials) to solve the above problem and to predict the superalloy properties.[4,13,29,34-39]Re, Ru, and Co dopants predominantly accumulated in theγphase,while W was observed in bothγandγ'phases.[40,41]Therefore,this work focused on building a model with random substitution of Ni by 1 and 3 at.% ofX(X=Re, Ru, Co or W) in theγmatrix. We performed MD simulations to understand how these dopants affect Ni matrix crack lattice trapping limit and range at 300 K(the simulations showed that the Ni crack propagates in a brittle manner at 300 K). We also calculated the corresponding mechanical properties, surface energy, and atomic bonding strength of the Re,Ru,Co,or W dopedγ-Ni matrix.

2. Computation details

2.1. Calculation model

We constructed a model with a sharp crack in the(010)[001] orientation (with the crack plane and crack front oriented perpendicular to the [010] and parallel to the [001]directions)according to the anisotropic linear elastic displacement field[42](see Fig.1(a)).The atomic configurations for the cracks in the undoped and 3 at.%doped Ni matrix are shown in Figs. 1(b)-1(d). The model contained 120, 160, and 160 atomic layers along thex,y,andz([100],[010],and[001])directions, respectively, which accounts for the 1536000 atoms in total. To confirm the appropriate and feasible model size,two larger models with 3 at.%Re atoms were also constructed.One of these large models expanded the side lengths inxandydirections and contained 200, 200, and 160 atomic layers along thex,y,andzaxes,respectively. The other large model expanded the side length in thez-direction and contained 120,160,and 200 atomic layers along thex,y,andzaxes,respectively. The smaller-sized model adopted in this work was chosen because the configuration evolution and physical properties(such as surface energy and elastic constants)did not depend on the system size. The initial crack length was set as 1/3 of the crack propagation length.

The boundary conditions in thez-direction during the lattice trapping calculations were periodic. For thexandydirections, we used the fixed-displacement boundary conditions of the six outer atomic layers. Dopant atoms (at both 1 and 3 at.%levels)substituted Ni atoms randomly. Such setup showed reliable results in previous studies.[4,13,29,34]To further confirm this setup reliability, we studied three different random distributions of Re atoms at 3 at.%doping level. The deviations of the lattice and elastic constants as well as surface energy and lattice trapping limits for these distributions were negligible, which confirms the reliability of the current random doping scheme chosen in this work.

Fig.1. A diagram showing simulation model: (a)Geometrical presentation of the lattice trapping. (b)Crack tip configuration for undoped Ni before relaxation. (c)-(d)Crack tip atomic configuration for the doped Ni(at 3 at.%level)by Re,Ru,Co or W before relaxation. Blue balls are Ni atoms;red balls are dopant atoms.

2.2. Simulation details

The Newton equations of motion at 300 K were processed by the gear algorithm[43]using 5×10?15s time step. The Ni-Al-X(X=Re,Ru,Co or W)EAM potentials[30-33]of the doped matrix were used to describe the cracks. The EAM potentials were used to fit the parameters obtained from the first-principles calculations or experimental results, such as lattice constants,cohesive energies, elastic constants of Ni, Al,X(X=Re, Ru, Co or W) and their compounds. The EAM potentials typically provide reasonable results in describing the dislocations or/and crack systems of the Ni-based superalloys.[4,13,15,18,29,30,34]The MD simulations were performed by using the XMD program.[44]During the lattice trapping modeling, the cracks were firstly simulated under different loads, after which the atoms at the crack tips were fully relaxed by using the boundary condition described in Subsection 2.1. The initial loading or unloading was set at the stress intensity to fulfill the Griffith criterion,while the crack tip was always set as stable and not allowed to advance or heal. The starting structures were in their relaxed configurations, with the further loading or unloading being proportional to the boundary displacements. The upper and lower trapping limits()were determined based on the sudden decrease and increase of the average bond lengths,respectively,along the crack front at the crack tip (see Fig. 2). The loading increase and decrease increments were set as 1% of the Griffith load(). During the loading(or unloading),each relaxed step was set as 4000 for each time,while the number of the loadings(or unloadings)was 20 times of that. Thus,the total number of steps was equal to 80000. The formula for the lattice trapping range(?K),shown below,was taken from elsewhere:[4,13-15]

Fig.2. An example of the bond length contour maps of atom pairs across the crack surface at the crack tip. This specific model was constructed for the 1 at.%Re-doped Ni system during the loading process. Pink squares denote the Re atom positions. The bond length bar is shown on the right-hand side.

3. Results and discussion

3.1. Mechanical parameters

To examine the reliability of the EAM potentials,the lattice constanta, the elastic constantsCi j(includingC11,C12,andC44), the elastic modulus (including the bulk modulusB=(C11+2C12)/3,Young’s modulusE=9GB/(G+3B)and the shear modulusG)for the undoped and doped(at both 1 and 3 at.%levels)Ni matrix without cracks were calculated. The shear modulusGwas computed as the arithmetic Hill average of Voigt(GV)and Reuss(GR)bounds as follows:

whereS11,S12,andS44are the elastic compliances.[45]

All calculated mechanical parameters are listed in Table 1. The value ofaof the Ni matrix increased as it was doped with Re,Ru or W.However,the lattice constant of the Co-doped Ni barely changed,which agrees with the previous work.[29]The lattice parameter change upon doping can be explained by the size of the atomic radii of the dopants (which are equal to 1.37, 1.34, 1.25, and 1.39 ?A for Re, Ru, Co and W,respectively)relative to the atomic radius of the matrix Ni atoms(equal 1.25 ?A as well as by the interatomic interactions(discussed in Subsection 3.3). The strength of the Re-, Ruand W-doped Ni matrix was significantly higher at the 3 at.%doping level (than at the 1 at.% level), judging by the higher values of the elastic constantsC11,C12, andC44and the elastic moduliB,EandG(see Table 1). Specifically, the values of the bulk moduliBof the Re-, Ru- and W-doped Ni at the 1 ( 3 at.%) level were equal to 180.72 (184.57), 179.51(183.41), and 180.75 (184.94) GPa, respectively, which represents 1.08% (3.23%), 0.40 % (2.58%), and 1.10% (3.44%)increase in comparison to the undoped Ni matrix. The order of the matrix strengthening by the dopant addition can be described as W>Re>Ru.No difference was observed between the values ofa(orCij) of the undoped Ni matrix calculated using different Ni-Al-X(X=Re,Ru,Co or W)EAM potentials, which agrees with the elastic constant values calculated previously.[29,46,47]Thus, the application of ternary Ni-Al-X(X=Re,Ru,Co or W)EAM potentials to the doped and undoped Ni matrix is a valid and reliable approach.

Table 1. Lattice(a)and elastic(Cij)constants as well as bulk(B),shear(G),and Young’s(E)moduli for the Ni matrix doped with 1 and 3 at.%of Re,Ru,Co or W.

3.2. Crack lattice trapping

3.2.1. Crack lattice trapping limits and range

Fig. 3. Crack lattice trapping limits and ranges in undoped Ni (marked as“No”)and Ni matrix doped with 1 and 3 at.%of Re,Ru,Co and W.

The lattice trapping ranges(?K)were small for all doped Ni matrices(see Fig.3). Specifically,when 3 at.%of Re,Ru,Co or W was added,the corresponding ?Kvalues were equal to 0.044,0.045,0.022,and 0.035,respectively.Such small ?Kvalues were very likely because of the long-range interatomic potential,which,according to some studies,[4,10,12,13,15,48]often indicates negligible lattice trapping by the breaking bonds.Thus,the interatomic interactions of Ni matrix doped with Re,Ru,Co or W possessed long-range characteristics.

Fig.4. Upper and lower crack surface atomic configurations at the crack tip of Ni crack system doped with(a)1 at.%and(b)3 at.%of Re under loading condition. Ni and Re atoms are shown as blue and red balls,respectively.

3.2.2. Average atomic and surface energies

To further investigate the strengthening mechanisms of the doped Ni crack system,we used MD simulations to calculate the average atomic (Eeatom) and surface (γ) energies (see Fig.5),defined below:[4,7,13,29]

Small values ofEeatomindicate more stable systems. Figure 5 shows the calculatedEeatomvalues for each system.The addition of Re, Ru and W decreased theEeatomvalues, which were smaller for the 3 at.% doping levels than for the 1 at.% doping levels.Eeatomwere equal to?4.492(?4.586) eV/atom,?4.472 (?4.528) eV/atom, and?4.502(?4.615) eV/atom for the Ni matrix doped with 1 at.% (or 3 at.%)of Re,Ru,and W,respectively. Thus,the addition of Re,Ru,and W improved the Ni matrix stability. However,Eeatomdid not change when Co was added to the matrix.

Surface energyγincreased as the Re,Ru,and W elements were added at the 3 at.%level(see Fig.5). They were equal to 1626.92 J/m2, 1621.48 J/m2, and 1667.25 J/m2, respectively,which is 2.22%,1.88%,and 4.76%higher than theγof the undoped Ni matrix(which was equal to 1591.49 J/m2). The surface energy of the matrix did not change when Co was added.Combining this data with the average atomic and surface energies, we concluded that the influence on the strengthening could be placed in the order of W>Re>Ru>Co. The same order was obtained for the dopant effect on the lattice trapping limits discussed in Subsection 3.2.1.

Fig.5. The surface(γ)and the average atomic(Eeatom)energies for the undoped and doped(with 1 and 3 at.%of Re,Ru,Co or W)Ni crack systems.

3.2.3. Bonding strength between atoms

Earlier in this paper, we concluded that the Ni crack lattice trapping and the corresponding mechanical parameters of the Ni matrix were influenced by the dopant nature (Re, Ru,Co or W).Such behavior could be related to the interactions of the dopants with the matrix atoms. Thus,to better understand this phenomenon, we compared the strengths of the Ni-Ni,Ni-Re, Ni-Ru, Ni-Co, and Ni-W bonds by using the EAM potentials.[29,34,52]

Typically, the deeper the potential well of the EAM, the stronger the atomic bonding strength. Comparison of EAM potential wells of the Ni-Ni and Ni-X(X=Re,Ru,Co or W)bonds indicated that the Re-Ni,Ru-Ni and W-Ni interactions are stronger than the Ni-Ni ones(see Fig.6). For instance,the absolute values of the Ni-Ni,Ni-Re,Ni-Ru,Ni-Co,and Ni-W potentials were equal to 0.21,0.35,0.30,0.18,and 0.51 eV at the curve minima located atr= 2.58, 2.65, 2.61, 2.55,and 2.64 ?A, respectively. Thus, the interaction strength between Ni and these dopants could be described by the order of W>Re>Ru>Co, which agrees with the results discussed in Subsection 3.2.

Fig.6. Ni-Ni,Ni-Re,Ni-Ru,Ni-Co and Ni-W pair potentials calculated by EAM as a function of the atomic distance r.

The results discussed above helped us conclude that Ni doping with Re,Ru,and W created strong Ni-Re,Ni-Ru,and Ni-W bonds, which, in turn, improved the matrix mechanical properties and stability as well as prevented crack tip bond breakage and promoted the crack healing at 300 K. Doping with Co exhibited the opposite effect because of the formation of weaker Ni-Co bonds. The best Ni matrix strengthening effect and crack brittle fracture improvement were observed when 1 and 3 at.%of W were incorporated into the Ni matrix.Thus,out of the dopants tested in this work,W demonstrated the strongest effect.

4. Conclusion

This work demonstrated theoretical simulations by an MD method performed to understand how doping of Ni matrix with Re,Ru,Co,and W affect its lattice trapping at 300 K.We implemented a Ni crack model with randomly distributed Re, Ru, Co, and W, added at the 1 and 3 at.% levels. Out of all additives tested in this work,W addition into the Ni matrix demonstrated the most improved mechanical strength(namely,the elastic constants,the bulk modulus,the shear modulus and Young’s modulus)and stability as well as crack advancement prevention and healing. The crack lattice trapping ranges were small for undoped and doped Ni matrices because of the longrange Ni-Re, Ni-Ru, Ni-Co, and Ni-W interatomic interactions. The comparison of the pair potential well depths obtained by EAM showed that the bonding strengths between the dopant and Ni were in the order of W>Re>Ru>Co. The Ni-Re,Ni-Ru and Ni-W bonds were stronger than the Ni-Ni ones. Out of all the dopants, W demonstrated the strongest bond with the Ni matrix,while Co exhibited the weakest.

Acknowledgement

We are grateful to Professor T.Yu and Z.G.Liu for beneficial discussions.