Understanding creep in TiAl alloys on the nanosecond scale by molecular dynamics simulations

(cid:1) The applied stress and microstructure model address the caveats of atomistic creep. (cid:1) Physically afﬁne atomistic models reduce creep rates by three orders of magnitude. (cid:1) Dislocation gliding (strain bursts) and colony boundary sliding dominate the creep. (cid:1) Interlamellar interfaces inhibit both the dislocation nucleation and its mobility. the driving mechanisms of primary creep on the nanosecond scale that agree very well with recent experimental observations, thus contributing towards the overar-ching goal of bridging the gap between atomistic creep simulations and continuum-scale creep simula- tions for engineering applications.


Introduction
The aviation industry demands more fuel-efficient aero engines with significantly smaller CO 2  with four different microstructures, namely duplex, nearly gamma, nearly lamellar, and fully lamellar microstructure [3]. These four microstructures mechanical properties range from excellent high temperature creep resistance, fracture toughness, strength to tensile ductility. In particular, the fully lamellar microstructure excels for high-temperature applications [4]. This microstructure typically consists of two intermetallic phases for engineering applications: c-TiAl (ordered face-centered tetragonal -L1 0 ) and a 2 -TiAl (ordered hexagonal -DO 19 ). The constituent intermetallic phases form a platelet-like structure consisting of so-called lamella [1]. Several lamellae are stacked together in an identically oriented fashion to form so-called colonies. Accordingly, fully lamellar TiAl exhibits a complex microstructure with hierarchical interfaces across the length scales starting from colony interfaces (highest level) to single-lamellar interfaces (lowest level).
Creep refers to time-dependent inelastic (permanent) deformation under stresses below the yield stress. It typically accelerates with increasing temperature. Thus, in practice, creep resistance and structural stability of c(TiAl)-based alloys are limiting the service temperature in aero engines [5]. Continuous efforts have been made to improve the creep resistance of two-phase c(TiAl)-based alloys by tailoring the microstructure [6]. In parallel, continuummechanical modeling contributed to understanding their microstructure-property relationship [7] significantly. A recent study [8] pointed out the shortcomings of the existing creep models of fully lamellar TiAl alloys and concluded that for realistic creep simulations, not only bulk creep but also interfacial creep should be taken into account, e.g., by cohesive zone models [9][10][11].
While continuum mechanics has substantially contributed to our understanding of the creep of TiAl on the meso-and macroscale, it cannot unravel its atomistic and nanomechanical foundations. To this end, molecular dynamics (MD) simulations are required. While over the last years, a considerable number of MD studies of TiAl alloys have been published [12][13][14][15][16][17][18][19][20][21][22], only two recent studies specifically addressed creep [23,24], however, with several serious limitations like unrealistic TiAl atomistic models combined with heuristic stresses for creep simulation. This is particularly surprising, noting that over the last two decades MD simulations of creep in metallic materials have attracted rapidly increasing attention [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. They helped understand the relation between creep deformation mechanisms and thermo-mechanical conditions on the nanoscale. Moreover, they yielded atomistically informed parameters for creep calculations on larger length scales. So far, MD simulations of creep in metallic materials typically suffer from several important limitations. First, to keep the computational cost feasible, very small systems are often studied, resulting in models with a nanocrystalline structure rather than a physically rigorous microcrystalline one (e.g., TiAl alloys). Second, current MD models of metal creep usually assume very high stresses and temperatures. This automatically induces a high strain rate. Consequently, most of these studies observed the three main creep stages (primary, secondary and tertiary creep) within time scales accessible by MD simulations, ranging from a few hundred picoseconds to a few nanoseconds. Note that the rapid deformation resulting from such high applied stresses, some times above r y , violates creep conditions per definition.
These limitations are important because the high volume fraction of grain boundaries in nanocrystalline materials considerably decreases the creep resistance compared to microcrystalline structures, as observed typically in TiAl [33]. Moreover, the combination of high stress and temperature typically assumed activates either the dislocation or diffusion-driven mechanisms, constrained by the nanocrystalline topology and Hall-Petch effects, again limiting the scope that can be addressed with MD simulations. This way, the currently made modeling assumptions severely limit the transferability of atomistic insights from MD creep simulations to real TiAl.
In summary, four critical issues have to be revisited on the way to more realistic MD creep models: the magnitude of the applied stress, the type of microstructure model, the temperature level, and the strain rate. In this work, we study the influence of the first three issues on the fourth issue with a focus on TiAl alloys. To this end, we study herein altogether eight different atomistic models of TiAl creep, which differ from each other by their degree of realism. They range from highly simplified models under unphysically high assumed stresses -in many respects similar to the models studied so far in the literature -up to MD models with a largely realistic microstructure combined with realistic loading and service temperature. That is, the applied stress lies in the range r ¼ 0:3r y À 0:9r y (with yield stress r y ) and the applied temperatures in the range 0:55T M À 0:7T M (with melting temperature T M ). We demonstrate that such conditions reduce the creep strain rate by around three orders of magnitude compared to previous similar MD studies of creep in metallic materials, bringing our models much closer to a realistic regime. Such realistic atomistic models and optimal thermo-mechanical conditions enhance the physical reliability of the observed nanomechanisms during MD creep simulations, which are in agreement with several experimental works [5,6,[40][41][42][43][44][45]. Though the strain rates and time scale considered in our study are still much faster than under typical service conditions, these improvements enable us to gain new insights into the atomistic mechanisms of creep in c(TiAl)-based alloys.

Atomistic modeling of c-TiAl alloy
At the atomic scale, materials can be modeled as many-body systems consisting of a finite number of particles that represent atoms. MD can accurately capture the thermal, mechanical, chemical, and thermodynamic behavior of such systems, however, within the adequacy of the interatomic potential. Herein, we used the MD code LAMMPS [46] (Large-scale Atomic Molecular and Massively Parallel Simulator). To describe the interactions between atoms, we used the so-called embedded atom method (EAM) [47] type potential for the TiAl system developed by Zope and Mishin in [48]. This EAM potential has already been successfully used in several MD studies, including ours of c(TiAl)-based alloys [14,22,49]. It correctly predicts the lattice constants, cohesive energy, and elastic constants of a 2 -Ti 3 Al, even though this information was not explicitly part of the data used for fitting the parameters of the potential. c(TiAl)-based alloys for engineering applications usually consist of two intermetallic phases: c-TiAl (ordered face-centered tetragonal -L1 0 structure) and a 2 -Ti 3 Al (ordered hexagonal -DO 19 structure) [3]. In our TiAl models, we considered the lattice parameters a 0 = 3.998 Å and c=a 0 = 1.047 for the c-TiAl phase [48] along with the crystallographic X-direction ½1 1 0, Ydirection ½1 1 2, and Z-direction [1 1 1]. Similarly, we chose the lattice parameters for the a 2 -Ti 3 Al phase as a 0 = 5.7884 Å and c=a 0 = 0.820 along with the crystallographic X-direction ½1 1 0, Y-direction [1 1 0], and Z-direction [0 0 1].
These two phases are arranged in stacked lamellae [1]. Several lamellae with identical orientation form so-called colonies, as shown in the experimentally observed microstructure (Fig. 1).
The thickness of individual lamellae ranges on the order of 10 1 -10 3 nm and their aspect ratio can be as large as 10 3 . The size of colonies is on the order of micrometers.
Herein we model colonies with either one intermetallic phase (c-TiAl) or two such phases, namely c-TiAl and a 2 -Ti 3 Al. Accordingly, we refer to such models using the qualifier 'single-phase' or 'two-phase'. In general, there are both c/c and c=a 2 interlamellar interfaces [22]. The c=a 2 interface is generally aligned with the close-packed plane and direction, i.e., f1 1 1g c jjð0 0 0 1Þ a 2 [1]. The c=c interfaces always share the same [1 1 1] direction, and in real materials, the phases are rotated by a multiple of 60 around this axis relative to each other. In the simulated model, we only consider one particular c=c interface with a rotation of 60 , the socalled pseudo-twin (c=c PT ) interface.
Unlike the previous studies of MD creep in TiAl [23,24], we considered physically rigorous atomistic models and thermomechanical conditions expected in component's service scenario. Therefore, in this study, we focus on understanding the implications of both the microstructure model and the applied stress on the observed strain rate and associated nanomechanics.
Typically, a fully lamellar TiAl microstructure can be characterized by the following key parameters: thickness of c lamellae k c , thickness of a 2 lamellae k a 2 ; a 2 volume fraction u a 2 , separation distance of a 2 lamellae d a 2 , and characteristic colony size k C . Note that the experimentally reported lamellar thicknesses typically represent the average value of the thickness distribution [7,50].
Herein we consider eight different atomistic models M i ; i ¼ 1; . . . ; 8 divided into two categories. The models M 1 -M 4 in the first category are referred to as simplified atomistic models (SAMs), M 5 -M 8 in the second category are referred to as microstructure-informed atomistic models (MIAMs). The SAMs resemble the type of MD model that has been mostly used in the literature [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] to study creep at an acceptable computational cost. To this end, they [23,24] use a highly simplified microstructure whose characteristic lengths are typically below the ones observed in reality (see also Table 1 and Table 2). By contrast, the domain sizes of the MIAMs are much larger than the ones of the SAMs, allowing for much more realistic representations of the microstructure, where MIAM M 8 exhibits characteristic lengths that are largely realistic (with the main exception of the colony size k C , which is still much smaller than in reality). Details of the microstructure of the MIAMs and how it compares to experimental observations are presented in Table 1 and Table 2.

Simplified atomistic models -SAMs
The four SAMs studied herein are illustrated in Fig. 2. M 1 and M 2 exhibit a simplified regular microstructure, M 3 and M 4 a more realistic random microstructure. All SAMs except for M 3 consist of both c and a 2 lamellae, i.e., two-phase, whereas M 3 serves for studying the (rather fictitious) case of an absence of the a 2 phase. In Fig. 2, the different phases, interfaces, and colony (grain) boundaries can be visually distinguished by their specific centrosymmetry parameter (CSP) values. The CSP measures the local lattice disorder around an atom. It is thus sensitive to phase type, defects, surfaces, and interfaces. For atoms embedded in a defect-free c-TiAl lattice, the CSP is zero. By contrast, for one in a defect-free a 2 -TiAl lattice, it is 7. Even higher CSP values are reached in the neighborhood of dislocation cores, interfaces, colony boundaries, and surfaces.
Due to their simplicity, SAMs typically cannot resolve all relevant details of the microstructure. Rather they can only be used to study the role of specific microstructural features. The regular c þ a 2 mono-and poly-colony SAM M 1 and M 2 are used to study the role of interfaces. By contrast, the poly-colony SAM M 3 and M 4 were designed to study in particular topological effects observed in the presence of a random colony structure (nonuniform shape/size). Moreover, as the only difference between M 3 and M 4 is the presence of the a 2 phase in the latter, it is instructive to compare these two models, since in real materials, the creep resistance of pure c TiAl is not satisfactory.    grain shapes in SAM (M 3 ; M 4 ) result in thermodynamic forces that contribute to grain shape changes. To eliminate such so-called topological driving forces [26] as far as possible, the MIAMs rely on regular colony shapes [27]. Hence, M 5 -M 8 use uniform hexagonal colonies (grains) with a grain size of 40 nm (Fig. 3). The crystallographic orientations of colony 1 in X-, Y-, and Z-direction were ½1 1 0, ½1 1 2, and [1 1 1], respectively. The crystallographic orientations in the colonies 2-4 are rotated around the X-axis by 30°, 45°, and 60°or 20°. The only difference between M 5 and M 6 is the presence of the a 2 phase in the latter. As also for M 3 and M 4 , this difference allows us to examine the effect of the presence of an a 2 phase and c=a 2 interfaces. For MIAM M 7 and M 8 , all colonies consist of lamellae. In M 7 only c=c PT interfaces appear, resulting from the presence of a pseudo-twin (PT) c PT phase [22]. The difference between M 7 and M 8 is the additional presence of an a 2 phase in the latter. As apparent from Table 2, the microstructure of M 6 and M 8 are realistic with respect to the values of their characteristic parameters except for the colony size, which is unrealistically small (nanoscale rather than microscale) to keep the computational cost feasible. Compared to MIAM M 6 ; M 8 has a slightly higher u a 2 and d a 2 (Table 1) and the a 2 phases are distributed over two colonies. MIAM M 5 and M 6 represent non-lamellar nearly gamma and duplex microstructure models, whereas MIAM M 7 and M 8 resemble the fully lamellar microstructures typically used in engineering. Yet, compared to the MD models used so far in the literature to study creep in TiAl [23,24], the MIAM can be considered a substantial step towards physical reality. In particular, M 6 -M 8 allows us to study the contribution of lamellar c=a 2 and c=c PT interface creep in TiAl by MD simulations in a realistic setting.

Preprocessing
The preprocessing of our SAMs and MIAMs was performed in several steps. First, for each initial configuration, energy minimization was performed using FIRE and conjugate gradient algorithms in LAMMPS [46]. Then the minimized configuration was heated to 300 K with a subsequent constant temperature and pressure equilibration for 100 ps. Subsequently, the temperature in the models was increased at 2.57 K ps À1 up to the target temperature, where another equilibration was performed, first for 300 ps at constant pressure (to obtain the target volume) and then for another 300 ps at constant volume (to distribute the kinetic energy evenly within the model). The heating and equilibration simulations were realized using a Nosé-Hoover thermostat with a time step size of Table 2 Properties of real TiAl microstructures (left) and overview to which extent these are represented in the 8 atomistic TiAl models M1 -M8 studied herein.

Creep simulations
We applied isobaric isothermal (NrT) conditions using the Parrinello-Rahman technique [52] and periodic boundary conditions to study creep by two categories of simulations. The creep response will be discussed in terms of three stages: (1) primary creep -creep strain slowly increases, however, with a decreasing creep rate, (2) secondary creep -creep strain steadily increases at a constant creep rate, (3) tertiary creep -creep strain exponentially increases with accelerating creep rate.
Category I (unphysically high stress): in this category, we adopted the approach of many previous MD studies of creep (e.g., [23,24]) and assumed an unphysically high stress above yield stress to accelerate creep in our simulations and reduce thereby the computational cost. Our four SAMs were subjected to uniaxial tensile stresses of 1000 MPa, 1500 MPa, and 2000 MPa in Zdirection for 200 ps. In X-and Y-direction, we applied pressure of 0 bar. Time step size was 2 fs. As shown in Table 3, r y in our SAMs typically ranges between 1.28 GPa and 1.32 GPa so that the applied stresses in Z-direction of 1500 MPa, and 2000 MPa are unphysically high. In category I, we performed all simulations at the two temperatures 0:65T M ¼ 1073K and 0:71T M ¼ 1173K with the melting temperature of TiAl T M ¼ 1650K.
Category II (realistic stress): in this category, we considered the physically much more realistic stress range in Z-direction 0:3r y À 0:9r y with yield stress r y . To compute r y uniaxial tension tests were simulated at a temperature of 0:65T M and a strain rate of 10 9 =s. A study of the role of the strain rate would be instructive but was beyond the scope of our study. In category II, we studied the SAM M 3 and M 4 with a pressure of 0 bar in X-and Ydirection and a time step size of 2 fs. The MIAMs M 5 -M 8 were simulated with the (even slightly more realistic atmospheric) pressure of 1 bar in X-and Y-direction and a time step size of 1 fs to resemble reality as accurately as possible with these most detailed MD models.

Postprocessing
Postprocessing and visualization of results were performed using in-house scripts together with the visualization tool OVITO [53]. The following quantities were evaluated: Atomic displacements: d ¼ r t À r 0 , that is, the difference between the current position r t of an atom and its initial position r 0 .
Shear strain: the shear strain denotes the local (scalar) shear invariant (after [54]) of the Green-Lagrangian strain tensor E ¼ 1=2ðF T F À IÞ computed using the atomic strain modifier [53]. Here, F and I indicate deformation gradient and identity tensor. The scalar shear strain field helps to visualize and characterize defects in atomic systems under deformation.
Dislocation density: The evolution of the dislocation density was analyzed using the dislocation-identification algorithm DXB [55], calculating the dislocation density q Dis ¼ l Dis =X box as a ratio between the total dislocation length l Dis and the volume of the simulation domain X box .
Common neighbor analysis: The local structural environment of an atom, e.g., crystalline phase or defect, was characterized using the common neighbor analysis (CNA) algorithm. For the lowest stress of 1000 MPa, both the mono-and the polycolony SAM with regular microstructure, M 1 and M 2 , revealed only a primary creep stage. For the medium stress of 1500 MPa, additionally a secondary creep stage, and for the highest stress of 2000 MPa, even a tertiary creep stage are observed. There are two main differences between the regular mono-and polycolony SAM. First, insensitivity to temperature is observed for the former, suggesting mainly stress-driven deformation mechanisms such as mechanical twinning, stress-induced a 2 transformation, and dislocation motion. These mechanisms observed in MD simulations, e.g., mechanical twinning and stress-induced phase transformation, especially under high stresses, confirm previous experimental claims [1,42]. By contrast, the temperature dependence of creep is observed in the latter for the medium and highest stress. This suggests that the activation of temperature-driven deformation mechanisms there requires considerable mechanical driving forces. The second important difference between the mono-colony and poly-colony SAM is the much faster creep in the latter. As the main difference between the mono-colony SAM and the poly-colony SAM is the existence of colony (grain) boundaries, the faster creep in the poly-colony SAM can most likely be attributed to the presence of these boundaries, highlighting their key role in TiAl alloys creep.

Category I (unphysically high stress)
Note that the orientation of the lamellar interfaces is most likely not the reason for the increased creep rate in the case of the polycolony SAM M 2 . For there, the colonies 2-5 exhibit the same orientation of the lamellar interface w.r.t the loading direction as the mono-colony model (so-called soft mode/ 45 to loading direction), and the central colony 1 has a lamellar orientation perpendicular to the loading direction (so-called hard mode [1]), which should rather decrease than increase the creep rate. Fig. 5 illustrates the evolution of the atomic displacement in the Z-direction d z (relative to the initial configuration) and the microstructure evolution. Indeed colony 1 (hard-mode) exhibits the lowest creep displacements, as expected from its loading perpendicular to its lamellar interfaces.
It is instructive to examine the mechanisms responsible for the large creep deformation in the poly-colony SAM M 2 , particularly in colonies 3 and 5. To this end, we computed the atomic shear strains at 200 ps. As shown in Fig. 6, peak regions of atomic shear strains correspond to regions of twinning and considerable colony boundary sliding, identifying, in particular, the latter of these two mechanisms as a main nanomechanical driver of the deformation. In Fig. 6(a), green arrows indicate the direction of the twinned and parent region of the intermetallic c phase. Finally, note that the Table 3 ry identified from the uniaxial tension tests (Fig. 7)  Both random poly-colony SAM M 3 and M 4 in Fig. 4 show a similar magnitude and temperature dependence of the creep strain for each of the three applied stresses. As already for the regular monocolony and poly-colony SAM M 1 and M 2 , and likely for the same reasons, temperature effects are observed only for the medium and highest stress. We observe a primary creep stage at all stresses but only at the medium and highest stress, also a secondary and tertiary creep stage. In nanocrystalline materials, the bulk dislocation activation volume is typically larger than the grain size, ham-pering the associated contributions to inelastic deformation. Consequently, the high colony (grain) boundary volume fraction dominates inelastic deformation even at high temperatures. Like many previous studies [23,24] with MD models similar to our SAMs, our random poly-colony models are also dominated by colony boundary effects.

Discussion
Among the four different SAMs M 1 -M 4 studied, in category I, the regular c þ a 2 mono-colony lamellar SAM M 1 shows the highest creep resistance. This can be explained by the absence of topological creep contributions from colony boundaries. It is noteworthy that also in the SAM with c=a 2 interfaces, we observe substantial creep deformations due to dislocation-dominated mechanisms, although in a recent study [22] c=a 2 interlamellar interfaces were found to form barriers to incoming dislocations from the c phase under loading normal to the interface. While our computational study of different SAMs provided some insights into the nanomechanics of creep in TiAl, the SAMs suffer from severe limitations similar to previous MD studies [23,24], as pointed out above. In most key aspects, their microstructure is substantially different from real TiAl microstructures (Table 1, 2). The short simulation periods are a consequence of unphysically high applied stresses. However, these also lead to unphysical behavior such as dislocation-mediated inelastic deformation and damage within a few hundred ps. Indeed, stresses partially even above r y put into question to which extent the observed deformation can be classified as a classical creep. Generally, the unphysical magnitude of the applied stresses severely compromises the transferability of atomistically computed parameters, for example, to meso-or macro-scale continuum models.

Category II (realistic stress)
In category II, we considered the poly-colony SAM M 3 and M 4 and MIAMs M 5 -M 8 . The most important difference between both categories is the much larger domain size of the latter, allowing for resemblance to real microstructures in all key aspects except for the colony size. The yield stresses r y of the six models M 3 -M 8 were determined in simulated uniaxial tensile tests. Fig. 7 shows the stress r ZZ vs. strain e ZZ in these tests in Z-direction. The yield stress r y was identified in these tests by determining the stress r ZZ at the onset of substantial dislocation activation [31] causing irreversible plastic deformation. The identified values of r y are    Table 3. Note that the computed r y of the different MIAMs show the following hierarchy r y ðM 6 À DuplexÞ < r y ðM 5 À NGÞ 6 r y ðM 7 ; M 8 À FLÞ, where NG and FL refer to Nearly Gamma and Fully Lamellar microstructure, respectively. This hierarchy is in perfect agreement with the experiment [42], thus, physically reliable MD models.
The creep simulations in category II focus on the primary and secondary creep stage because these are most relevant during the lifetime of high-temperature engineering materials like TiAl. Tertiary creep mainly determines how such materials fail at the end of their lifetime. Nevertheless, it is worth noting that the simulations are spanning several nanoseconds, while only 200 ps were simulated in the previous category I.

SAM M 3 and M 4 : c and c þ a 2
Observations: In the low-stress range 0:3r y to 0:5r y , both random poly-colony SAM experience only very small creep strains with only a primary creep regime (Fig. 8 a, c). For 0:6r y , additionally, secondary creep is observed, and for 0:7r y and beyond show even tertiary creep. The c þ a 2 SAM M 4 shows a much higher creep resistance than the c SAM M 3 , at least for r P 0:6r y .
Discussion: Generally, the high grain boundary volume fraction in nanocrystalline materials restricts bulk dislocation activity and favors grain-boundary-dominated mechanisms [56]. For nanocrystalline Cu, Wang et al. [28] observed a transition from grain boundary diffusion as dominant creep mechanism to grain boundary sliding with increasing stress and eventually dislocation-driven creep. The SAM M 3 and M 4 are both nanocrystalline. This suggests that the acceleration of creep with increasing stress observed in Fig. 8 may be understood from a similar transition to more effective deformation mechanisms. To examine this aspect, we computed the evolution of the dislocation density q Dis during creep (Fig. 8 b, d). For 0:9r y and 0:8r y ; q Dis increases exponentially for both SAM M 3 and M 4 , suggesting dislocation-driven creep indeed at high stresses. For low stresses 6 0:7r y , both M 3 and M 4 exhibit a nearly constant q Dis over time, suggesting that in this regime creep is not dislocation-dominated, but grain boundary sliding. The stress-dependent role of dislocation-driven creep is also illustrated in Fig. 9, showing the CNA computed microstructure evolution of SAM M 3 in the stress range 0:7r y to 0:9r y . At high stress (0.9 r y ), we observe numerous planar faults traced by dislocations originated from colony boundaries traversing the colony within 1000 ps. This phenomenon significantly declines with the level of applied stress (even for a much longer time), in agreement with the idea of a transition to non-dislocation-dominated deformation mechanisms in the low-stress regime.
In SAM M 4 , the dislocation density at high stresses increases more slowly than in SAM M 3 (Fig. 8 b, d). Attributing this to the only difference between both models, the presence of an a 2 phase in M 4 underlines the role of c=a 2 interfaces in hampering the motion of dislocations. The hindrance stems from the a 2 phase crystallographic configurations with restricted and incompatible slip systems for incoming dislocations from the c phase [22].
Indeed, a similar phenomenon has been reported for nanotwinned metals [34].
To keep the computational cost low, many MD studies of creep have focused so far on nanocrystalline materials such as SAM M 3 and M 4 discussed in this section (TiAl [23,24] and others [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]). While such studies can certainly provide important insights, they also reveal that grain-boundary-based mechanisms govern the creep deformation at low stresses. This implies, however, that the high volume fraction of grain boundaries in nanocrystalline materials makes their creep resistance remarkably low, as pointed out already by [33]. As a consequence, the results of MD simulations of nanocrystalline systems can only to a very limited extent be translated to the technologically much more relevant microcrystalline systems. A step towards such systems will be done in the next section. In the c poly-colony MIAM M 5 we observe high creep strain from an avalanche of strain bursts (jump events) in the beginning, followed by slowly increasing strain for the applied stress 0:9r y .
Such high creep deformation is attributed to colonies with pure c phase and without any interlamellar interfaces that could hinder dislocation mobility. The magnitude of strain bursts reduces with decreasing applied stress. Generally, the pure c microstructure M 5 appears more susceptible to creep. The addition of the a 2 phase and the resulting c=a 2 interfaces appear to enhance the creep resistance relatively in M 6 . As shown in the insets in Fig. 10, the dislocation density does not change significantly during the simulations. This suggests that the dislocation creep observed in the SAM models M 3 and M 4 is a result of the much larger topological driving forces that arise due to randomized grain shapes.  the yield stress at temperature 0:65T M ¼ 1073K. Generally, the magnitude of strain bursts increases with temperature. The particularly high strain jump at 973 K forms an outlier that could not be explained within this study. In a previous study [22] we found that the yield stress r y decreases with increasing temperature. Combining those results and the observations here, the higher magnitude  of strain jumps with temperature may be explained by the fact that with increasing temperature the applied stress corresponds to an increasing fraction of yield stress (at that temperature).
To understand the role of the strain bursts driving primary creep, we compared the microstructural evolution at two different stresses between M 5 and M 6 ( Fig. 13). At stress 0:8r y ; M 5 shows dislocation nucleation from the colony boundary triple junctions 2/4/1 and 3/4/1 and the colony boundary 1/3 (yellow arrows). The dislocations traverse along the glide plane until encountering a colony boundary. There, their further mobility is hindered due to an insufficient driving force to cross the colony border and change their glide plane thereby. Consequently, the central colony 1 with hard orientations shows no considerable dislocation glide activities. By contrast, at the higher stress 0:9r y , the driving forces for dislocation motion are high enough to cause a host of concerted    dislocation glide activities also across colony boundaries resulting in an avalanche of strain bursts. For MIAM M 6 at the stress 0:8r y , a dislocation nucleated from colony boundary 3/4 glides (yellow arrow at time 1 ns). Also, a twinning dislocation (blue arrow at time 2 ns) from colony boundary 1/3 traverses colony 3, forming a twin boundary (1 1 1)½1 1 2. Note that, in M 6 even at the stress 0:9r y dislocation gliding appears considerably impeded (Fig. 13).
Discussion: Combining Fig. 10 and Fig. 13, we conclude that the atomistic origin of the strain bursts observed in MIAM M 5 and M 6 is the nucleation and gliding of dislocations. Note that both increases with stress, as does also the strain bursts driven creep. We underline that this interpretation is not in contradiction to the nearly constant dislocation density q Dis shown in the insets in Fig. 10. MIAM M 5 and M 6 exhibit already considerable initial dislocation densities due to misfit dislocations at colony boundaries. However, most of these dislocations remain sessile due to unfavorable slip systems and thus do not contribute to creeping. Note that creep is driven by the few dislocations gliding and the relatively small number of new dislocations nucleating. Nevertheless, the number of the latter is small because dislocation emission from colony boundaries requires a relatively high localized activation stress [29]. Thus, in our simulations, the number of nucleating dislocations remained much smaller than the number of initially existing dislocations so that no visible increase of the total dislocation density was observed over time.
Recently, [57] reported strain bursts for nano-creep experiments in semi-crystalline polymers and assumed the origin of these bursts in dislocation mobility. Our MD simulations support this interpretation. Thus, they not only unravel for the first time strain bursts as a nano-creep mechanism in TiAl but also contribute to our understanding of creep beyond the specific case of TiAl.
Noting that dislocation nucleation at colony boundaries plays an important role in the strain bursts driving creep, it is instructive to study its origin. Fig. 14 reveals the nucleation of two dislocations from the 1/2 colony boundary (white arrows in Fig. 14d). Interestingly, this nucleation directly follows substantial colony boundary sliding indicated by a high atomic shear strain (yellow arrows in Fig. 14b, c), suggesting that colony boundary sliding acts as an assisting precursor mechanism of dislocation nucleation. These nanomechanical insights from MD creep simulations validate the predictability of the unified creep formula based on dislocation glide and colony boundary sliding by Zhang and Deevi [42], who proposed following the careful analysis of several experimental works of various TiAl alloys in a broad temperature range 600-900°C. It is worth noting that no previous MD works reported such colony boundary sliding precursor mechanism in TiAl alloys.
Comparing M 5 and M 6 , the latter exhibits considerably higher creep resistance at the macroscale associated with considerably impeded nucleation and gliding of dislocations at the atomic scale (Fig. 13). Noting the difference between M 5 and M 6 is only the presence of a 2 lamellae in the latter, this simulation result is in excellent agreement with the experimentally well-known positive effect of the a 2 phase on the creep resistance of TiAl alloys [1].    In M 7 , dislocations nucleate predominantly from the junction of colony boundary and lamellar interface. Strain imbalance across the lamellar interface also contributes to a few dislocation nucleation events, as reported in experiments [58]. At the beginning (after 5 ps), interfacial gliding, i.e., dislocation gliding along the c=c PT lamellar interfaces, marks the onset of creep. Interestingly, interfacial gliding dislocations as the resulting creep mechanism from our MD simulations agree with the experimental work on fully lamellar TiAl [6]. As time progresses, there exist three dislocation propagation mechanisms: (1) interfacial gliding (white arrows), (2) gliding on planes parallel to the lamellar interface (blue arrows), and (3) gliding on planes oblique to the lamellar interface (yellow arrows). Generally, the dislocation gliding is impeded by c=c PT lamellar interfaces (hampering gliding on oblique planes), colony boundaries (hampering interfacial gliding and gliding on parallel planes), and dislocation-dislocation interactions. After 2 ns, the colony boundaries 1 and 3 undergo substantial sliding resulting in shear strain. Thus, colony boundaries act as both source and sink for dislocations.
Compared to M 7 , one observes in M 8 (Fig. 16) the same three dislocation propagation mechanisms. For dislocation nucleation, M 8 shows not only colony boundary junctions but also homogeneous colony boundaries between two c regions (blue arrows after 5 ps) and heterogeneous colony boundaries between c and a 2 regions (yellow arrow after 200 ps) as nucleation sites. In M 8 , the additional a 2 phase greatly reduces colony boundary sliding (elliptical markers), dislocation nucleation and propagation. For example, in colony 3, the a 2 phase suppresses interfacial glide along c=a 2 interfaces. Consequently, in colony 3, dislocation gliding is limited to c=c PT lamellar interfaces and planes parallel to c=a 2 interfaces in c regions.
Under the given thermo-mechanical conditions, in our MD simulations, dislocation gliding (arrows) and colony boundary sliding (elliptical markers) appears to dominate the primary creep (Fig. 16). Thus, any hindrance to these two mechanisms can be expected to reduce creep. Compared to the c phase, both the c PT and a 2 regions appear to contribute negligibly to the atomic mechanisms of creep. In M 7 and M 8 , the interfaces govern the creep deformation, whereas in M 5 and M 6 , also some bulk contribution exists. This observation agrees with the experimentally reported creep on nearly gamma TiAl microstructure (like our M 5 ), which suggests that primary creep is caused by dislocation gliding both on lamellar interfaces and c lamellae (bulk) [40].
Discussion: For the applied thermo-mechanical conditions, M 7 and M 8 show a slow creep strain evolution. Our MD MIAMs captured several experimentally reported nanomechanical events (e.g., dislocation gliding [40,41], colony boundary sliding [42], interfacial glide [6], and dislocation emission from interfaces [5,43,44]) during creep deformation. In Fig. 17, we comprehensively depicted the observed dislocation-based nanomechanical events from our MD creep simulations. Dislocation-based nanomechanical events categorized into nucleation, propagation, and hindrance govern the observed creep. Unlike in M 5 and M 6 , there were no significant strain jump events in MIAM M 7 and M 8 (except at 0:9r y ) due to the fully lamellar microstructure, which seems to lead to the absence of sequential gliding activities spanning several colonies and additionally a severely reduced dislocation gliding distance. In M 5 and M 6 , the dislocation gliding predominantly occurs in pure c colonies, where the gliding distance is proportional to the colony size k C . By contrast, in M 7 and M 8 , this distance reduces to the lamellar spacing k c . This nanomechanical insight from MD simulation holds in perfect agreement with the experimental works [41,45,59], which claim a decrease in primary creep strain with decreasing lamellar spacing in fully lamellar microstructure (like our M 7 and M 8 ).
Trapped dislocations at lamellar interfaces and colony boundaries hinder subsequent interfacial gliding and other oncoming dislocations [1,41]. Dislocation locking occurs when two cross-gliding dislocations encounter, providing additional resistance to further dislocations mobility. These MD results for MIAM M 7 and M 8 confirm the following experimental observations about creep mechanisms in fully lamellar TiAl [41,45]: lamellar interfaces may (1) inhibit dislocation gliding such that a fine lamellar spacing reduces the gliding distance; (2) trap dislocation segments, restricting dislocation motion.
Qualitatively, our MD simulations of dislocation propagation agree with the experiments reported in [6,40,41], which show dislocation motion at and along the c=c and c=a 2 interfaces. Hence, interface sliding contributes to the strain, especially in the primary creep regime. Interestingly, the microstructure of the TiAl in [6] had the lamellar thicknesses of k c ¼ 100 . . . 300nm (10 1 higher than in our simulations) and k a 2 ¼ 10 . . . 50nm (same order), which make the microstructure largely comparable to the one in our MD simulations. Note, however, that our MIAM M 7 and M 8 exhibit smooth colony boundaries, whereas the ones in [6] were serrated, which likely reduced colony boundary sliding due to interlocking. Interestingly, however, also our MD simulations showed only slight colony boundary sliding, which thus appears to be of no significant importance (at least on the length and time scale considered herein).
In our MD simulations, the central colony (ID 1) significantly assisted to dislocation gliding and creep in M 5 and M 6 but not in M 7 and M 8 (Fig. 16). The reason is the presence of the c=c PT and c=a 2 interfaces in the latter. These are subject to nearly normal loading in colony 1, resulting in minimal deformation (hard mode) [22]. Chen et al. [60] reported restricted dislocation mobility due to fine lamellar spacing in softly oriented grains and retarded shear in hard oriented grains. Also, well-interlocked lamellae along colony boundaries resist intergranular cracking, hampering dislocation nucleation. In our models, the absence of preexisting interfacial dislocations and the high applied stresses (although below r y ) enabled dislocation nucleation from colony boundaries and strain imbalance [58] near the lamellar interfaces resulting in gliding on interlamellar interfaces and on its parallel planes (Fig. 17). Thus, our simulations show dislocation nucleation from colony boundary triple junctions and heterogeneous regions. Fig. 18 compares the creep rate (computed as mean creep rate between the two points in time 1ns and 4ns) vs. applied stress for the SAM M 3 and M 4 with the three of the MIAMs M 6 -M 8 , whose microstructures are in many respects close to the ones used in reality for engineering applications. The creep rate of the c þ a 2 SAM M 4 is always slightly lower than the one of the pure c SAM M 3 , which can be understood from the creep-impeding presence of c=a 2 interfaces as discussed already above. Interestingly, [32] reported a similar observation for nanotwinned models with decreased twin boundary spacing. Compared to SAM M 3 and M 4 , the creep rate in MIAM M 6 is reduced by two orders of magnitude. We suggest that there are two main reasons for this. On the one hand, the MIAM like M 6 exhibit regular colony shapes. In contrast, the random colony shape in the SAM M 3 and M 4 can be expected to give rise to topological driving forces resulting in larger inelastic deformation. On the other hand, the colony (grain) size in the MIAM M 6 is considerably larger than in the SAM M 3 and M 4 , which can be expected to reduce grain boundary effects and thus also associated creep substantially. In fact, several studies on nanocrystalline models reported reducing creep rates for increasing grain size [29,33,35]. In the future, more comprehensive parameter studies separately varying the different microstructural parameters (k c ; k a 2 ; d   nanomechanics and the strain rate hierarchy for different microstructures considered herein, qualitatively.

Summary & conclusions
Over the last years an increasing number of MD studies of creep has been proposed for TiAl [23,24] and other metallic materials [26][27][28][29][30][31][32][33][34][35][36][37][38][39]. These studies typically focused on nanocrystalline and nanotwinned materials. The reported creep mechanisms include grain boundary diffusion (Coble and Nabarro-Herring creep), grain boundary sliding, and grain-boundary-assisted dislocation nucleation and motion. They can thus be broadly classified into diffusion-driven and dislocations-driven creep. Typically these studies considered very small simulation domains together with high stress (for dislocation-driven creep) and high temperature (for diffusion-driven creep) to observe creep-like phenomena within a time scale accessible to MD. However, as discussed herein on the basis of simplified atomistic models (SAMs) that resemble those previously used in MD creep studies, this leads to highly unphysical phenomena such as creep rates so high that tertiary creep occurs within a few hundred picoseconds. Such rapid deformation is rather the time scale of fracture than the one of creep. Hence, insights gained from such MD simulations may be misleading, in general not only quantitatively but even qualitatively. For such a short time scale may not allow studying deformation mechanisms whose activation takes considerable time but which may yet dominate in the long run. In particular, it is often not possible to observe the thermal activation of deformation mechanisms on very short time scales [64]. To this end, SAMs exposed the limitations of the previous MD creep studies. This paper's major objective is to point out how these caveats of current MD creep models can be overcome, at least to some extent. As a first step, we tested the effect of decreasing stress and temperature to a more realistic regime than most previous studies [23,24].
That is, we decreased stress clearly below yield stress r y to the regime 0:3r y -0:9r y and temperature clearly below melting temperature to the regime 0:55T M -0:7T M . As we demonstrated, this decreases the creep rate (for the same microstructure) by around two orders of magnitude, thus, improving the physical reliability of our MD models. However, decreasing stress and temperature alone is insufficient because also the nanocrystalline structure used in most previous simulations substantially differs from the microstructure used in typical TiAl alloys for engineering applications. This is important because length scale also plays a crucial role in creep via the activation volume [28,64]. To address this aspect, we designed the microstructure-informed atomistic models (MIAMs) with a microstructure whose characteristic geometric parameters largely matched the ones experimentally observed (with the exception of the colony size, which still had to be chosen smaller than in typical engineering alloys to keep the domain size and thus computational cost acceptable). This decreased the creep rate by another two to three orders of magnitude. Effectively, this allowed us to study creep in TiAl -to our best knowledge for the first time -in a largely realistic microstructural setting and on the nanosecond scale. Due to the high computational cost of our simulations, only the primary creep regime could be studied. However, for this regime, we were able to identify as the main driving mechanism of creep on the atomic scale the nucleation of dislocations from colony boundaries and their gliding. Apparently, this mechanism is assisted by colony boundary sliding as a precursor mechanism and leads to a sequence of characteristic strain bursts that macroscopically results in primary creep. Interestingly, this observation is in excellent agreement with recent experimental observations reported by [57] for nano-creep experiments in semi-crystalline polymers.
Our MD simulations also confirmed the experimentally already known creep-reducing role of a fully lamellar microstructure [41,45]. The simulations revealed that such microstructures reduce the creep rate by around one order of magnitude. In particular, they demonstrated that this effect could be attributed to the inhibiting effect of the interlamellar interfaces on the nucleation and mobility of dislocations, which corroborates our understanding of the atomic foundations of the creep resistance of TiAl.
Altogether, our results demonstrate that for MD studies of creep, it is essential to set both temperature and stress to realistic values and use a microstructure that resembles the real one close enough to capture at least the essential phenomena on the atomic scale properly. Unlike the previous studies of MD creep in TiAl [23,24], our study showed a longer primary creep stage of realistic MIAM models besides capturing different underlying nanomechanical events driving creep. Furthermore, these observed nanomechanical events from MD creep simulations reaffirm the findings of several experimental works [5,6,[40][41][42][43][44][45]. This way, our computational study could unravel the driving mechanism of -at least primary -creep on time and length scales not yet addressed so far by atomistic simulations in TiAl alloys. Such nanomechanical insights are an important step towards atomistically informed creep laws for mesoscale continuum models that are computationally efficient enough to capture the much longer time and length scales most relevant for engineering applications. We recommend incorporating our insights gained while investigating atomistic creep of other technological materials using MD simulations. Yet, we note that this paper is but a first step in this direction. A further extension of the time and length scale of our MD model will be required to bridge the gap to the mesoscale and continuum.

Data Availability
The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.