Understanding and designing mechanical twinning by ab initio modeling

Deformation of TWIP steels can be traced to a few elementary processes on the atomic scale. Ab initio-based materials design enables the identification of key control parameters that trigger certain microstructure features, such as twin nucleation (Figure 1).^{Reference Steinmetz, Jäpel, Wietbrock, Eisenlohr, Gutierrez-Urrutia, Saeed-Akbari, Hickel, Roters and Raabe8} The twin-boundary formation in austenite is characterized by local adaptation of the sequence of {111} atomic layers. Such adaption is only possible if the energy to create the change in the stacking sequence—the SFE—is sufficiently low. Recent advances in ab initio simulation tools provide access to the dependence of the SFE on chemical composition, stress state, or temperature with high accuracy.^{Reference Dick, Körmann, Hickel and Neugebauer25,Reference Hickel, Grabowski, Körmann and Neugebauer26}

For example, DFT allows for calculation of the SFE dependence on the C content in the Fe-C system. Simulations reveal an enormous increase in SFE with increasing C content.^{Reference Abbasi, Dick, Hickel and Neugebauer27} An underlying assumption behind this result is that C atoms are distributed homogenously. This assumption is justified for the undeformed state and would also hold after the formation of stacking faults, provided that C atoms remain trapped at octahedral sites and are displaced together with the lattice during the diffusionless rearrangement. This is realistic, since at room temperature, diffusion of C in fcc Fe is negligible.

To estimate local gradients in the C chemical potential induced by formation of an intrinsic stacking fault (ISF), we performed supercell calculations that contain such a defect.^{Reference Abbasi, Dick, Hickel and Neugebauer27} The results reveal there is a significant thermodynamic driving force to out-diffuse C from the local hexagonal close-packed region that is formed next to an ISF. Kinetic simulations indicate that such diffusion will become relevant if the sample is heated by several hundred K²⁸ (Figure 2). Since segregation of solutes out of the defect is opposite to the commonly observed decoration of dislocations (i.e., the Suzuki effect), we refer to it as the anti-Suzuki effect.

Figure 2.

Nonmagnetic ab initio calculation of the energetics of a single C atom along a diffusion path from the center of the stacking fault into the matrix. The jump between the (111) atomic layers is schematically shown in the lower part of the figure, where the colors of the atoms indicate the stacking sequence of these layers. The different shaded areas indicate the hexagonal close-packed (center of faults, two kinds of layers) and face-centered-cubic (fcc) stacking sequence (three kinds of layers). The green squared symbols in the upper part of the figure (connected by a dotted line as a guide for the eye) provide the C solution enthalpies in the interstitial sites, whereas the red solid line represents the energy barrier for the diffusion as determined in a nudged-elastic band (NEB) calculation. The black dashed line visualizes an energy profile obtained after renormalizing the ab initio values to match the experimental value in the bulk fcc region.^{Reference Hickel, Sandlöbes, Marceau, Dick, Bleskov, Neugebauer and Raabe28}

Ab initio-based mechanism maps for TWIP steels

For coarse-graining the ab initio-calculated SFEs into a wider composition-dependent phase space, a subregular solution thermodynamics model was applied for creating SFE maps for a range of steels.^{Reference Saeed-Akbari, Imlau, Prahl and Bleck29,Reference Song, Ingendahl and Bleck30}

Figure 3 shows three-dimensional (3D) deformation mechanism maps of high-Mn austenitic steels according to the calculation of the SFE, including TRIP (transformation-induced plasticity), TWIP, and MBIP (microband-induced plasticity) effects. A SFE value of 20 mJ/m² marks the borderline between TRIP and TWIP,^{Reference Saeed-Akbari, Imlau, Prahl and Bleck29–Reference Renard, Ryelandt and Jacques37} and a value of 50 mJ/m² is seen as an upper limit for TWIP.^{Reference Park, Kim, Sung, Lee, Hwang and Lee38} With the addition of Al, the SFE increases, and the κ-phase (Fe, Mn)₃AlC precipitates in Fe-Mn-Al-C steels.^{Reference Springer and Raabe39–Reference Gutierrez-Urrutia and Raabe42}Figure 3 reveals the regime where Kappa-carbide occurs (at 873 K) based on CALPHAD (CALculation of PHAse Diagrams) calculations, together with the 3D composition-dependent SFE map of an Fe-Mn-Al-C steel (at 300 K). The SFE maps predict the prevalent deformation mechanisms in Fe-Mn-C and Fe-Mn-Al-C steels; this enables the design of optimal strain hardening.^{Reference Bouaziz, Allain, Scott, Cugy and Barbier1,Reference Steinmetz, Jäpel, Wietbrock, Eisenlohr, Gutierrez-Urrutia, Saeed-Akbari, Hickel, Roters and Raabe8,Reference Raabe, Springer, Gutierrez-Urrutia, Roters, Bausch, Seol, Koyama, Choi and Tsuzaki9,Reference Pierce, Jimenez, Bentley, Raabe, Oskay and Wittig21,Reference Song, Ingendahl and Bleck30,Reference Herrera, Ponge and Raabe43–Reference Kuzmina, Herbig, Ponge, Sandlöbes and Raabe48}

Figure 3.

Three-dimensional composition-dependent SFE map for Fe-Mn-Al-C steel at 300 K. Note: MBIP, microband-induced plasticity; TRIP, transformation-induced plasticity; TWIP, twinning-induced plasticity; and SFE, stacking-fault energy.^{Reference Song, Ingendahl and Bleck30}

Combinatorial TWIP steel design

A combinatorial approach was used in translating the simulated trends into real materials. It combined co-sputtering of thin-film composition spreads with high-throughput characterization to define composition–structure–property relationships.^{Reference Springer and Raabe39} Here, we focus on the composition dependence of the elastic properties.

Figure 4 shows the experimental setup for the co-sputter deposition of Fe-Mn-X composition spreads from Fe-, Mn-, and transition-metal targets (X). Utilizing this experimental setup, Fe-Mn,^{Reference Gebhardt, Music, Takahashi and Schneider49–Reference Gebhardt, Music, Ekholm, Abrikosov, von Appen, Dronskowski, Wagner, Mayer and Schneider51} Fe-Mn-C,^{Reference Reeh, Music, Gebhardt, Kasprzak, Jäpel, Zaefferer, Raabe, Richter, Schwedt, Mayer, Wietbrock, Hirt and Schneider52} Fe-Mn-X, X = Al, Si,^{Reference Gebhardt, Music, Ekholm, Abrikosov, Vitos, Dick, Hickel, Neugebauer and Schneider53,Reference Gebhardt, Music, Kossmann, Ekholm, Abrikosov, Vitos and Schneider54} and Fe-Mn-X (X = {Cr, Co, Ni, Cu})^{Reference Reeh, Kasprzak, Klusmann, Stalf, Music, Ekholm, Abrikosov and Schneider55,Reference Reeh, Music, Ekholm, Abrikosov and Schneider56} composition spreads were synthesized.

Figure 4.

Schematic representation of the combinatorial approach: Fe-Mn-X composition spreads are grown by co-sputtering from three magnetrons where the geometric arrangement of the plasma sources results in concentration gradients. Young’s modulus and lattice parameter data obtained by x-ray diffraction (XRD) and nanoindentation experiments along the Co concentration gradient are presented and compared to ab initio simulations.^{Reference Reeh, Kasprzak, Klusmann, Stalf, Music, Ekholm, Abrikosov and Schneider55}

Characterization of the structures by x-ray diffraction and of elastic properties by nanoindentation along the concentration gradients allowed relating the effects of composition on the equilibrium volume and to the Young’s modulus. These results were compared to quantum mechanical predictions. This correlative strategy enables validation of structure and elastic properties as well as of the underlying magnetic effects.^{Reference Music, Takahashi, Vitos, Asker, Abrikosov and Schneider50–Reference Staunton, Gyorffy, Pindor, Stocks and Winter57} The trends revealed by the thin-film combinatorial results were subsequently used for rapid TWIP alloy screening by a bulk combinatorial approach that included high-throughput synthesis, thermomechanical processing, and mechanical testing of the alloy (i.e., a process sequence referred to as RAP: rapid alloy prototyping).^{Reference Springer and Raabe39} The chemical composition dependence of the Young’s modulus data served as input for the plasticity models is described next.^{Reference Steinmetz, Jäpel, Wietbrock, Eisenlohr, Gutierrez-Urrutia, Saeed-Akbari, Hickel, Roters and Raabe8,Reference Roters, Eisenlohr, Kords, Tjahjanto, Diehl and Raabe58–Reference Ma, Roters and Raabe60}

Constitutive modeling of strain hardening in TWIP steels

The observed dependence of the SFE and the elastic properties on composition were used to inform a constitutive deformation model.^{Reference Steinmetz, Jäpel, Wietbrock, Eisenlohr, Gutierrez-Urrutia, Saeed-Akbari, Hickel, Roters and Raabe8,Reference Roters, Eisenlohr, Kords, Tjahjanto, Diehl and Raabe58–Reference Ma, Roters and Raabe60} The equations related to the dislocation mechanisms are given in Reference Reference Roters61. Twinning is split into nucleation and growth. Nucleation is formulated based on the reaction of two dislocations to form a twin nucleus.^{Reference Karaman, Sehitoglu, Beaudoin, Chumlyakov, Maier and Tome4} For the nucleus to grow, a critical stress must be overcome that is dependent on the SFE.^{Reference Steinmetz, Jäpel, Wietbrock, Eisenlohr, Gutierrez-Urrutia, Saeed-Akbari, Hickel, Roters and Raabe8} The SFE is obtained by combined ab initio and thermodynamic modeling.^{Reference Dick, Körmann, Hickel and Neugebauer25–Reference Saeed-Akbari, Mosecker, Schwedt and Bleck32} Twin growth is instantaneous until an obstacle is encountered. Further details are given in References Reference Steinmetz, Jäpel, Wietbrock, Eisenlohr, Gutierrez-Urrutia, Saeed-Akbari, Hickel, Roters and Raabe8 and Reference Roters61.

The ab initio-informed constitutive model is thus capable of describing the strain-hardening behavior of the TWIP steel as a function of composition (through the SFE), temperature, and strain rate (Figure 1 and Figure 5).

Figure 5.

Strain-hardening behavior corresponding to the true stress–true strain curves shown in Figure 1 for an Fe-22Mn-0.6C twinning-induced plasticity steel (wt%) at different temperatures. One parameter set was used for the whole temperature range.^{Reference Steinmetz, Jäpel, Wietbrock, Eisenlohr, Gutierrez-Urrutia, Saeed-Akbari, Hickel, Roters and Raabe8}

TWIP steel microstructures revealed by ECCI

ECCI enables quantitative characterization of the deformation structures of TWIP steels for a wide field of view.^{Reference Gutierrez-Urrutia and Raabe2,Reference Steinmetz, Jäpel, Wietbrock, Eisenlohr, Gutierrez-Urrutia, Saeed-Akbari, Hickel, Roters and Raabe8,Reference Gutierrez-Urrutia, Zaefferer and Raabe10,Reference Gutierrez-Urrutia and Raabe11,Reference Gutierrez-Urrutia, Zaefferer and Raabe62–Reference Gutierrez-Urrutia and Raabe64} Figure 6 shows the evolution of the deformation substructure upon tensile deformation in an Fe-22Mn-0.6C (wt%) steel. The orientation dependence of the deformation structures is provided by the TA-IPFs (TA: tensile axis; IPF: inverse pole figure).^{Reference Gutierrez-Urrutia, Zaefferer and Raabe18} The ECC image of a dislocation pattern at optimum contrast is characterized by bright structures with a sharp boundary contrast over a dark background. This figure reveals the complex evolution of the dislocation patterns and their interaction with the evolving twin substructure, which is associated with the low SFE of TWIP steels (20–40 mJm^–2), high C content, and orientation dependence of the deformation mechanisms.^{Reference Gutierrez-Urrutia and Raabe2,Reference Steinmetz, Jäpel, Wietbrock, Eisenlohr, Gutierrez-Urrutia, Saeed-Akbari, Hickel, Roters and Raabe8–Reference Gutierrez-Urrutia and Raabe11,Reference Gutierrez-Urrutia, Zaefferer and Raabe18,Reference Gutierrez-Urrutia, Zaefferer and Raabe62–Reference Gutierrez-Urrutia and Raabe64}

Figure 6.

Evolution of the deformation substructure upon tensile deformation in an Fe-22Mn-0.6C (wt%) twinning-induced plasticity steel examined by the electron channeling contrast imaging technique and plotted in tensile axis-inverse pole figure (TA-IPF) diagrams.^{Reference Gutierrez-Urrutia and Raabe2,Reference Gutierrez-Urrutia, Zaefferer and Raabe18} The large blue arrow indicates the texture changes during deformation (HDDW, high-density dislocation wall; DC, dislocation cell).

At the early stage of deformation (0.1 true strain/380 MPa), the substructure is dominated by dislocation cell blocks (CBs) and dislocation cells (DCs). These patterns are associated with wavy and planar slip characteristics, respectively.^{Reference Gutierrez-Urrutia and Raabe2,Reference Gutierrez-Urrutia, Zaefferer and Raabe18} Crystallographic CBs are formed in grains containing a limited number of {111} active slip systems, such as in grains oriented close to 〈112〉//TA directions. Noncrystallographic CBs delimited by high-density dislocation walls (HDDWs) that can deviate by up to 10° from the {111} slip planes are attributed to multiple slips. These dislocation patterns are observed in grains oriented close to 〈111〉//TA directions. DCs are built up in grains when a high number of slip planes are activated and dislocation cross-slip is enabled (i.e., grains oriented close to 〈001〉//TA directions [red dots in the corresponding TA-IPF shown in Figure 6]).

At higher stress levels (720 MPa), twinning increases, leading to the development of a well-defined twin substructure. At this stage, we can distinguish between three types of deformation substructures, namely, Types I, II, and III. Type I is a twin-free structure that is formed by fine equiaxed DCs. This substructure forms in grains oriented close to 〈001〉//TA directions (red dots in the TA-IPF). Type II is characterized by a lamellar twin structure along the primary twinning system. This substructure is formed in grains oriented along the line between 〈001〉//TA and 〈111〉//TA directions (green dots in the TA-IPF). Type III is characterized by a well-defined twin substructure consisting of a primary twin system and one or two secondary twin systems. This substructure forms in crystals oriented close to 〈111〉//TA directions (blue dots in the TA-IPF).

TEM-based analysis of the SFE and defect arrangements

Analysis of the influence of SFE on the deformation mechanisms requires an experimental measurement of its value. Analysis of the partial dislocation spacing using weak-beam dark-field (WBDF) TEM techniques^{Reference Gutierrez-Urrutia, Zaefferer and Raabe62} has proven adequate for direct SFE determination.^{Reference Williams and Carter65} A quantitative value of the SFE can be determined from the separation distances of Shockley partial dislocation pairs.^{Reference Pierce, Jimenez, Bentley, Raabe, Oskay and Wittig21}Figure 7a shows a WBDF image of a partial dislocation pair exhibiting separation distances in the range of 8–10 nm in a sample after light deformation (1.5%). The fitted experimental data indicate a SFE of 21 ± 6 mJ/m² for the investigated Fe-14Cr-16Mn-0.3C-0.3N steel.^{Reference Mosecker, Pierce, Schwedt, Beighmohamadi, Mayer, Bleck and Wittig66}

Figure 7.

Transmission electron microscope investigations of an austenitic Fe-14Cr-16Mn-0.3C-0.3N alloy. (a) Measurement of Shockley partial-dislocation separation distances based on weak-beam dark-field imaging with a beam direction near [111] on defects in the (111) habit plane using 〈–220〉 type g-vectors. The images were acquired using g(3g) diffracting conditions. (b) Bright-field image of extended stacking faults (SFs) blocked at twin lamellae (inset: selected area diffraction pattern) and (c) 〈111〉 twin plane reflection showing dislocation pileups at planar defects at a true strain ε = 0.21.^{Reference Mosecker, Pierce, Schwedt, Beighmohamadi, Mayer, Bleck and Wittig66}

Further insight into strain hardening can be gained by conventional TEM analysis of strained samples. The primary mechanical twins contribute to the strain hardening by acting as obstacles to dislocations gliding on different slip systems, Figure 7b, which shows mechanical twins and intersecting stacking faults after 0.21 true strain. Dislocation accumulation at the twin boundaries is evident in Figure 7c, which is a higher magnification image of the twinned area in Figure 7a under diffraction conditions that make the dislocation arrangement visible.^{Reference Mosecker, Pierce, Schwedt, Beighmohamadi, Mayer, Bleck and Wittig66}

Conclusions and outlook

For alloys whose strain hardening is determined by mechanical twinning, ab initio-guided alloy design can be realized by revealing the composition dependence of the SFE and by considering these trends in internal variable plasticity models, including twinning, dislocation substructure evolution, and grain-size effects. We applied this concept to the development of Fe-Mn-C-based TWIP steels. Excellent agreement of the predictions was found with combinatorial synthesis and mechanical test data for an Fe-22Mn-0.6C TWIP steel (wt%) between 293 and 873 K using a single set of physically motivated parameters. Applying this design concept in the future and also to other alloy variants in the field of advanced high-strength steels requires including further deformation mechanisms (ε-martensite, α-martensite) and their interdependence. This is specifically important for the development of next-generation middle-Mn high-strength TRIP steels.