Abstract
Assessing the fingerprint of a material’s microstructure is key for supporting materials design. With the emergence of a wide range of 3D characterization techniques, it is critical to understand the main differences in fingerprints reconstructed from 2D and 3D datasets. To this end, we introduce a graph-based microstructure reconstruction framework that enables structural comparisons of twin domain networks in high purity Ti using 3D and 2D electron backscatter diffraction. Insights into the structure of the twin networks are facilitated by combining statistical analysis of twin crystallography with visual and graphical analysis of the novel graph abstractions of the twins. We demonstrate that compared to 3D reconstructions, conventional 2D views of twinning miss key aspects of the microstructure including the high interconnectivity of domains into networks that span the full reconstruction volume. The reduced cross-grain and in-grain twin connectivity typically observed in 2D has notable implications on our understanding of how twinning mediates the plastic response of microstructures and how twin networks evolve. It is thus clear that 3D characterization is critical for accurately inferring both twin network morphologies as well as the key unit processes facilitating network formation.
Introduction
Microstructure characterization is a foundational principle of physical metallurgy. The fingerprint of microstructures – including grain size, grain morphology, texture, phase fraction, porosity, and defect density – provides insights into the mechanisms mediating deformation including the Hall-Petch effect, work hardening, phase transformations, and dynamic recrystallization. Historically, the materials community has characterized microstructures through analysis of 2D images, however numerous 3D characterization techniques have recently emerged including serial-sectioned electron microscopy and electron backscatter diffraction1,2, 3D electron tomography3,4, atom probe tomography5,6, 3D high-energy X-ray diffraction microscopy7,8, and X-ray computed tomography9. As 3D reconstructions of complex microstructural features become more readily available, new questions arise in how to mesh conventional 2D data and novel 3D data to better understand the linkage between a material’s full 3D microscale fingerprint and its macroscopic mechanical response. In particular, it is necessary to understand (i) the biases embedded in 2D views of microstructures that cannot be corrected using existing quantitative stereology principals, and (ii) what additional quantitative information can only be extracted from 3D datasets.
These questions are critical as many engineering metals exhibit complex hierarchical microstructures and it is not easy to determine either a compact intelligible representation of the microstructure or how individual microstructural features (e.g., precipitates, lath boundaries, twin boundaries) condition plasticity. This is the case for a wide range of alloy systems including metals processed by additive manufacturing or welding, ferritic martensitic steels, and materials deforming via the activation of diffusionless transformations. Any bias in our analysis of the microstructure will necessarily bias our understanding of property-microstructure linkages.
In this context, this study will focus on diffusionless transformation (DTs). DTs are key phenomena observed to accommodate plasticity in a wide range of material systems including metals with low symmetry crystal structures10, face-centered cubic alloys with low stacking fault energies11,12, nanocrystalline metals13,14,15, high entropy alloys16,17, and twinning/transformation induced plasticity (TWIP/TRIP) steels18,19. DTs lead to the formation of 3D domains within the parent matrix through either reorientation of the parent lattice or reordering of regions of the lattice into a new phase. As the transformation process often carries a high amount of plastic strain, transformed domains induce stresses in the surrounding parent matrix and adjacent grains that can stimulate activation of additional DTs, termed transmission20. Conceptually, the percolation of these transformed domains could provide a pathway for easy plastic dissipation throughout the microstructure, both intrinsically due to the plastic strain carried by the domain and by additional plastic deformation in the transformed domains. As such, predicting the mechanical response of materials that deform through DTs is particularly complex due to activation of multiple concurrent deformation mechanisms. While the formation of DTs in some material systems such as TRIP and TWIP steels and nano-twinned Ti and Zr has been correlated to an increase in ductility21,22,23, other materials that exhibit extensive deformation twinning such as Mg display relatively low ductility and a susceptibility to crack propagation along chains of transformed domains24,25. Thus, the activation of DTs is not inherently a salient predictor of fracture and failure behavior. Instead, differences in the morphology and connectivity of the domains that evolve during deformation are expected to mediate the different mechanical responses observed and shed light on the unit processes involved with network growth. To better elucidate the influence of DTs on material properties, it is necessary to accurately characterize the full 3D fingerprint of the domain networks that form throughout microstructures, the characteristics of which are not well understood and have only recently become the subject of attention.
Over the past two decades, statistical analyses of materials exhibiting DTs – particularly deformation twinning – have investigated numerous characteristic features of the DT process including nucleation and growth of twin domains26,27,28,29,30,31,32,33, twin variant selection during nucleation32,34,35,36,37, and twin transmission across grain boundaries in a range of alloy systems20,38,39,40,41,42,43. However, these prior studies have primarily solely characterized 2D fingerprints of deformed microstructures. While 2D analyses can effectively capture twin volume fraction, twin thickness, and twin-parent orientation relationships, they risk improperly characterizing the dependence of DT activation on grain size, the full morphology of the transformed domains, and the overall domain network connectivity. These prior 2D studies have also predominately focused on the individual unit processes (e.g., nucleation and transmission) that grow the network rather than the structure of the network itself. In comparison to extensive 2D studies, full 3D twin networks have not been comprehensively characterized in part due to the complexity and time associated with 3D analysis. Recent investigations have begun to address this by combining mechanical serial sectioning of the microstructure with electron backscatter diffraction (EBSD), revealing complex 3D morphologies of twin domains44, twin nucleation stimulated by local stored energy density45, rotational fields induced by slip activity ahead of 3D twin tips46, variations in grain-scale twin activity at different 2D layers47, and long, tortuous twin chains with complex 3D paths48. Initial insights into full 3D domain networks have also observed higher numbers of in-grain and cross-grain contacts for individual twins than initially expected from 2D measurements as well as an increase in the interconnectivity of the network with increasing macroscopic strain48,49. However, the differences between the full 3D fingerprint and the sections of the network reconstructed only from 2D views of the microstructure are not well understood. It is therefore important to assess and better understand the additional insights that 3D datasets offer in comparison to traditional 2D analysis. As such, the present study investigates the following questions: (i) what biases are introduced when inferring the structures of 3D twin domain networks based solely on 2D information, and specifically to what degree do network features characterized in 3D differ compared to the same features characterized in 2D? (ii) How might conventional 2D only views of twin networks impact our understanding of how twin networks evolve and how they mediate deformation?
In the present work, we address this through a quantitative examination of 2D area reconstructions and 3D volume reconstructions of twin networks extracted from the same deformed microstructure. By combining conventional statistical analysis with novel graph analysis, our results reveal that 2D reconstructions of twin networks systemically underrepresent the degree of connectivity present in the network, particularly the number of cross-grain contacts that form for each twin. These undercounted twin contacts in 2D views of the microstructure suggest a more disconnected network composed of isolated twins and individual twin pairs compared to the densely interconnected fingerprint identified in the full 3D reconstructions. Further, it is more common to observe low-alignment connections indicative of incidental contacts between separate twin chains in the full 3D reconstructions than in 2D. These results imply that conventional 2D analyses of deformation twinning bias our understanding of both the mechanisms driving the growth of the network as well as the influence of the network on the mechanical response and failure modes of the microstructure. Crucially, the present work clearly details that 3D analysis is necessary to accurately capture the fingerprint of twin domain networks.
Results
2D and 3D measurements of deformation twin networks
In our study, the structure of a twin network formed through compression of high-purity Ti is investigated by concurrent 3D analysis of adjacent volumes and 2D analysis of adjacent areas in a deformed microstructure. The twin networks were formed via through-thickness compression (near parallel to the grain <c > -axes) to 2.6% macroscopic strain at ~80 K to induce a high density of compression twins along different concurrently active variants throughout the microstructure48. Three 3D cubes and three large 2D areas from the same cross-section of the microstructure were extracted as illustrated in Fig. 1. Full microstructure orientation maps for each 3D volume and 2D area are illustrated in Supplementary. Small-scale 2D slices were additionally extracted from the 3D volume of each cube along the through-thickness, rolling, and normal axes. The 3D cube and 2D twin networks were then reconstructed using the Microstructural Evaluation Tool for Interface Statistics (METIS3D)50 following the procedure detailed in ref. 48. The large-area 2D measurements comprised cross-sectional areas ranging from 2.2–3.2 × 105 µm2 containing nearly 12000 individual twins, and the 3D cubes comprised volumes of 3.3–4.4 × 105 µm3 containing around 2200 total twins. The EBSD datasets were collected with a step size of 0.2 µm in sections taken every 0.2 µm. The resolution was selected to balance detection of small twins and small twin contacts in the reconstruction volume with the overall measurement time. Twin segmentations errors were manually corrected in METIS3D as detailed in the methods section.
a Relative positions of 2D and 3D microstructure regions extracted from a cryogenically compressed (through-thickness) high-purity Ti test specimen, (b) 3D EBSD reconstructions from serial-sectioned cubic volumes of the same as-compressed cross section, (c) 2D slices extracted from 3D EBSD Cube 1, and (d) 2D EBSD reconstructions from planar slices of the as-compressed cross section (2D grain maps cropped to similar dimensions along ZL for visualization).
The topology of the twin network is complex and is characterized by many twin contacts which can be either intra- or trans-granular. Representative examples of morphologies observed for in-grain and cross-grain contacts in the 3D reconstructions are illustrated in Fig. 2a, b respectively. The in-grain contact in Fig. 2a consists of a long rectilinear contact facet with a plane oriented perpendicular to K1 in the blue twin and approximately 45° from K1 in the orange twin. These in-grain twin connections are observed frequently in both the 3D and 2D reconstructions and generally display a high contact area due to the limited relative orientations between twins on different variants. The cross-grain contact in Fig. 2b displays a well-aligned twin pair with a high shared contact area along a facet near-normal to the η1 direction in each of the two adjacent twins. Of note, unlike in-grain twin connections, the contact facets for the cross-grain twin pairs are dependent on the orientation of the parent grain boundaries and may vary even between different pairs of the same variants in two adjacent grains. Given the high number of twin contacts that form during network growth, characterization and analysis of the topology of twin networks can be facilitated by using graph abstractions of the networks.
a Twin contacts that form in-grain twin junctions through the intersections of separate twin variants in a single parent grain, and (b) twin contacts that form cross-grain twin pairs that connect at the shared parent grain boundary. The twins are illustrated by orange and blue point clouds of the exterior twin boundary and local reference frame axes detailing the K1, η1, and λ1 directions placed at each twin centroid. Grain boundaries are illustrated by a grey point cloud.
The use of graphs for twin network characterization was inspired by prior network analyses51 in structural biology52, the social sciences53, and microstructural analysis of dislocation structures54,55. In the present work, we extend the use of graph abstractions by developing a novel framework to succinctly visualize the fingerprint of the twin network present in the microstructure. As an example, different components of the twin fingerprint in 3D Cube 1 are visualized in Fig. 3. The connectivity of the twin network that evolves to accommodate plastic deformation comprises a variety of characteristic parameters that can be embedded in the graph, with nodes representing individual twins and edges representing contacts between twins. For Cube 1, the graph is initially generated by assigning nodes at the centroids of each twin – projected to a single plane for the 3D volumes – as shown in Fig. 3a with the node size scaled by the cube root volume of the twin as shown in Fig. 3b. The orientation of each twin variant is next encoded in each node and visualized by color mapping the node by the twin’s nominal Schmid factor (SF) as observed in Fig. 3c. While local stress states that activating twinning will vary from the macroscopic loading conditions, the differences between the twin SFs are detailed to highlight the differences between the alignment of the twins with respect to the macroscopic loading axis. The latter broadly provides a quantitative description of the difference between local and macroscopic stresses throughout the network. Finally, the connectivity in the twin network – arising from co-nucleation, cross-boundary transmission, intragranular junction formation, and incidental contact between separately nucleated isolated twins or twin chains – is also visualized by edges connecting the nodes. In-grain junctions and cross-grain contacts are differentiated by line style as shown in Fig. 3 (d-e), with intragranular twin contacts shown as dashed lines and intergranular twin contacts shown as solid lines. For twins that contact across grain boundaries, the geometric alignment of the twins is an important parameter that both reflects the mechanism by which the contact formed as well as the extent to which shear discontinuities are localized along the shared boundary. The cross-grain twin alignments in the graph are illustrated by color mapping the edges using the Luster and Morris56 m’ parameter:
where \({{{\bf{K}}}}_{{{\bf{1}}}}^{{{\bf{x}}}}\) and \({{{\boldsymbol{\eta }}}}_{{{\bf{1}}}}^{{{\bf{x}}}}\) are the twin plane normal and twin shear direction, respectively, for connecting twins \({{\rm{A}}}\) and \({{\rm{B}}}\) at each twin contact. We note m’ varies from −1 for perfectly anti-aligned twins to 1 for perfectly aligned twins. The cross-grain twin contact in Fig. 2b illustrates an example of a high m’ (well-aligned) twin pair. The edge colors are scaled from −0.5 to 1 as no adjoining twin pairs are observed in highly anti-aligned configurations due to the sample texture and loading conditions. A highly aligned twin pair indicates that the lattice shearing induced by a twin on one side of the grain boundary is near fully accommodated by the twin on the opposite side, generating minimal plastic discontinuities along the grain boundary interface. Misaligned twin pairs alternatively indicate possible sources of high shear localization, or hot spots for defect accumulation during continued deformation.
Graph abstraction of the reconstructed twin network in 3D Cube 1 following 2.6% macroscopic strain. a Scaled planar projection of each twin centroid, (b) projected centroids with node size proportional to twin volume1/3, (c) nodes color mapped by each twin’s macroscopic Schmid factor, (d) twins in intragranular contact connected by edges, (e) twins in intragranular and intergranular contact connected by edges, (f) intergranular edges color mapped by twin-to-twin \({{\rm{m}}}^{\prime}\).
The use of graph abstractions to visualize twinning offers a unique insight into the structure of the twin networks that evolve during deformation. The differences in both the spatial density and sizes of the nodes reflect heterogeneities in the path by which the network formed in different regions of the microstructure. In some volumes, local twinning is growth-dominated and the plastic shearing is carried by a few large twins. However, in other regions, higher densities of smaller twins are observed indicating a higher rate of nucleation or transmission processes during continued deformation. The extent to which each twin accommodates strain in the parent lattice also depends on the growth or thickening rate of the twinned lattice, influenced in part by the local resolved shear stresses along the coherent twin plane. This can be visualized using the nominal SF color mapping for each node.
As illustrated in Fig. 3f, the twin network in 3D Cube 1 displays a highly interconnected set of cross-grain contacts that includes nearly every twin in the reconstruction volume. Separate communities of dense cross-grain and in-grain connections are observed across the microstructure, and larger twins display notably higher numbers of twin-twin contacts than conventionally expected from prior 2D studies. While most twins form variants well oriented to accommodate the macroscopically imposed stresses (SF > 0.4), a notable fraction is also observed on lower SF variants broadly distributed throughout the microstructure. Further, high degrees of twin connectivity are observed both for twins with relatively high and low nominal SF. This is shown by two example twin hubs in the lower (-YL) half of the microstructure in Fig. 3f where twin TA (SF = 0.47) displays 10 cross-grain contacts and twin TB (SF = 0.34) displays 18 cross-grain contacts. Critically, one must note that not all contacts take place at either the same location along the twin’s exterior surface or at the same grain boundary.
The introduced graph framework for visualizing deformation twinning consequently provides a snapshot of the salient features of twin fingerprints and is applicable to any other DT domain network. The associated database contains all topologic and orientation information such that further identifiers can be added to node and edge parameters depending on the characteristics of interest for the network. For example, in microstructures with textures where both tensile and compression twin modes may activate depending on the orientation of the parent grains, the nodes and edges can be modified to highlight interactions between twins of the same mode or between twins in different modes. Alternatively, the intragranular edges could be modified to highlight co-variant contacts or secondary twinning, and the cross-grain edges could be modified to contrast contact at forward, lateral, or coherent twin boundary facets for each adjacent twin. Higher-order connectivity could also be introduced for graphs of materials with ordered domains, such as contact within or across prior β boundaries in Widmanstätten Ti microstructures.
Graph abstractions are also effective tools for comparing different networks, such as the evolution of twin connectivity at increasing amounts of deformation. As such, in the present study we leverage graphs to detail how full 3D reconstructions of nominally similar twin networks compare to reconstructions consisting of only 2D slices of the same microstructure.
Microstructural fingerprint of twin networks in 2D vs 3D: identifying the missing twin scaffold spanning the sample volume
In this section, the differences in network morphology identified in 2D and 3D microstructure reconstructions are explored through statistical and graph analysis. Graph abstractions for each of the three 3D volumes and three large-area 2D twin network reconstructions are illustrated in Fig. 4a–c, d–f respectively. To best illustrate the differences in the 2D and 3D networks, the node centroids are modified to use a force-directed placement (FDP) model for an improved visual comparison between the 3D and 2D networks. In the FDP model, the graph is considered as a distribution of nodes connected by springs (edges) of equivalent stiffness. The placement of the nodes along the 2D view plane is then calculated through reduction of forces within the network, resulting in near-equidistant node spacing with nearly equivalent edge lengths and minimized edge overlaps. This reduced edge overlap helps to visually differentiate different twin contacts. Graph abstractions using the twin centroid node projections are provided in Supplementary A. The twin connectivity parameters – normalized by the total number of twins in the reconstruction – are listed in Table 1. The twin connections include contacts between twins of any mode, with the distribution of identified twin modes in each sample detailed in Table S1.
Graph abstractions of (a–c) the reconstructed twin networks in 3D Cubes 1-3, and (d–f) 200 × 200 μm cropped regions from the center of the reconstructed twin networks in large-area 2D Areas 1-3. Node centroids are assigned through a force-directed placement (FDP) algorithm in NetworkX71.
A key feature of the fingerprint of twin networks is the connectivity (e.g., number of twin-to-twin contacts, interconnectedness of twin chains) at individual twin nodes throughout the microstructure. Twin connectivity indicates the density of the network that evolves during deformation and is driven by the formation of twin connections through transmission, co-nucleation, or incidental contacts between twins that initially nucleated through unrelated events. The reconstructions clearly exhibit large differences in connectivity, with nearly fully interconnected networks observed in the 3D reconstructions and disperse communities of cross-grain contacts evident in the 2D reconstructions. We note that for comparison, only twins and edges in the center 200 × 200 µm region of the 2D slices (containing a total number of twins closer to that of the 3D cubes) are visualized in Fig. 4d–f. The graph abstractions of the full large-area 2D twin networks are illustrated in Supplementary A. The network statistics also highlight these differences in twin connectivity. While the 3D cubic volumes display 0.74 in-grain and 0.70 cross-grain contacts on average for each twin, the 2D slices of the network show significantly less twin-to-twin contacts with only 0.35 in-grain and 0.17 cross-grain contacts present for each twin. As a result, a notably higher fraction of isolated (no adjacent twin contact) twin nodes and twin pairs are observed in the 2D reconstructions compared to the 3D reconstructions.
The graph abstractions further illustrate biases in the conclusions that might be inferred for the mechanisms by which the twin network evolves when analyzing either the 2D or 3D twin fingerprint. In 2D, twinning appears to occur primarily through isolated nucleation with occasional formation of cross-grain contacts, usually through transmission. Twin pairs, twin triplets, and small twin chains are prevalent throughout the 2D microstructure. The few high-contact twin hubs in 2D generally comprise a large number of in-grain twin junctions that are expected to form from a single large twin on one variant intersecting many twins of a different variant in the same grain. In a few cases, growth of a chain via sequential transmission into a third or fourth grain are also present. The high number of satellite isolated twins is also evident from the large number of nodes along the periphery of the graphs. This indicates that by 2.6% macroscopic strain, the twinning process is still dominated by the nucleation of individual twins, and the growth of individual twin domains will impact the macroscopic mechanical response more strongly than the interactions between twins. Alternatively, the high number of twin-to-twin interactions in the 3D reconstructions indicates that by 2.6% strain, twinning has fully percolated across the microstructure and twin formation is more strongly associated with transmission processes or co-nucleation than isolated nucleation. The 3D reconstructions further illustrate that the twin domains form a connected network structure within the microstructure by 2.6% strain, as opposed to the mostly dispersed domains observed in the 2D slices. The interconnectedness of the network is also apparent in the high degree of edge overlap in the 3D graphs compared to the 2D graphs. As increasingly complex arrangements of connected twins form, the graph spreading via force reduction – clearly evident in the 2D networks – is constrained in the 3D reconstructions by the additional twin contacts. The high-contact twin hubs in the 3D reconstructions also contain a far higher fraction of cross-grain connections than typically overserved in 2D views of twinning.
One might expect that by simply analyzing an increasing number of 2D slices, the statistics would converge to be more representative of the actual 3D network. We show that this does not hold true for the connectivity of twin networks. However, as twins are 3D defects that exhibit complex morphologies and growth characteristics and the investigated microstructure contains a strong basal texture, the connectivity statistics presented in Table 1 could be biased by the ND orientation of the 2D slice plane within the sample or by differences in twin activity in different regions of the microstructure. To test this, 2D reconstructions of the twin network in Cube 1 were generated by virtually slicing every 0.3 µm along the XL (through-thickness, 336 slices), YL (287 slices), and ZL (166 slices) axes of the cube for comparison to the full 3D network. Twins in each slice were reconstructed separately, and the distributions of the connectivity observed in each 2D layer are shown in Fig. 5c–e. The twin area fraction in each original serial-sectioned layer (ZL axis, 248 layers) was further calculated as shown in Fig. 5b, and the twin area fractions for the 2D slices are observed to be consistent with the total volume fraction for the full 3D cube (overlaid dashed line). Further, while the 2D slices display a considerable spread in contact statistics (expected due to the small number of twins in each slice), all slices contain a significantly lower number of twin-twin and twin-grain boundary contacts per twin compared to the full 3D reconstruction. The distributions of in-grain and cross-grain connectivity in the 2D cube slices also overlap with the statistics measured for the large-scale 2D scans. This confirms that the differences in connectivity between the 2D and 3D reconstructions in Table 1 do not simply arise from heterogeneities in the local twin activity or due to the texture of the twins. To verify this, 3D and 2D connectivity statistics were further calculated for synthetic microstructures containing overlapping cylindrical discs as detailed further in Supplementary. Four synthetic volumes containing randomly dispersed discs with either weak rotational axis textures, representative twin textures, or strongly aligned textures are illustrated in Fig. S5. The synthetic microstructures were each sliced along three orthogonal axes similar to Fig. 5, and the connectivity in the 2D slices were similarly consistently lower than the full 3D volume regardless of the domain texture, the domain size, or the slicing axis.
a Virtual 2D slices of the 3D twin network in Cube 1. Distributions of the (b) twin volume fraction in the serial sectioned (ZL) slices and the ratios of the number of (c) stopped contacts (twin-to-grain boundary contacts with no contact to adjoining twins) (d) in-grain contacts, and (e) cross-grain contacts to the total number of twins in each XL (purple, left column, 366 slices), YL (orange, middle column, 287 slices), and ZL (green, left column, 166 slices) 2D slice of the network in Cube 1, compared to the full 3D network shown by the overlaid dashed grey lines. The mean twin volume fraction in both the full 3D network in Cube 1 as well as all ZL slices of Cube 1 is 0.26. The mean stopped contacts, in-grain contacts, and cross-grain contacts per twin in the full network in Cube 1 is 1.79, 0.70, and 0.72 compared to only 0.59, 0.43, and 0.31 respectively for the mean of all XL, YL, and ZL 2D slices of Cube 1.
The impact of the differences in twin connectivity identified in 2D slices of the microstructure on the interconnectivity of individual twins and twin chains is further detailed in Fig. 6. The total number of both in-grain and cross-grain contacts for each twin node – the node degree in graph theory – is shown in Fig. 6a. These distributions highlight how each twin contributes to the structure of the full network, and the variations in twin-scale behavior match the differences in connectivity in the full networks. On average, each twin forms 3 twin-to-twin contacts in the 3D reconstructions but only 1 contact in the 2D reconstructions. The difference in the medians of the distributions is less pronounced but displays the same trend, with a median of 2 contacts per twin in the 3D reconstructions but only 1 in 2D. Around 35% of twins in the 2D reconstructions are isolated with no other twin contact compared to only 19% of twins in the 3D reconstructions. Further, only 1% of twins in 2D form a hub of five or more separate contacts, while in the 3D reconstruction 18% of twins display five or more contacts and 5% of twins form over ten separate connections. A small fraction of twins (2%) in the 3D reconstructions are large twin hubs with over 15 separate connections and are not displayed in Fig. 6a. Twins that appear as hubs in the graph generally do not display all twin contacts (edges) at the same location, as the edges are spread across different twin-to-grain boundary contact locations. These edges are primarily single paired twin contacts, although some triple junctions of twin contacts are also observed. As such, the plasticity mediated by these high-contact twins is very complex compared to simpler configurations of isolated twins or twins that only form a single adjoining twin contact.
Histograms detailing differences in (a) the total number of cross-grain and in-grain twin-to-twin contacts that each twin forms and (b) longest twin path lengths, for the combined data of each twin in all 3D twin network reconstructions (orange, left column, 2173 total twins) compared to the combined data for all twins in all 2D slices along the serial-sectioning (Z) axis of the 3D cubes (green, middle column, 26,907 total twins) and large-area 2D twin network reconstructions (purple, right column, 11,536 total twins). The median number of twin-to-twin contacts is 2 in the 3D reconstructions and 1 in both the 2D slices of the 3D cubes and in the large-area 2D reconstructions. The median longest path of twin contacts is 12 in the 3D reconstructions, 3 in the 2D slices of the 3D Cubes, and 2 in the large-area 2D reconstructions.
Another important feature of the fingerprint of twin networks is the length of the chains of twin contacts that evolve throughout the microstructure during deformation. Individual twin chains grow through sequential twin transmission events across grain boundaries, and separate chains may intersect either at grain boundaries or within grains. The total length of these twin paths, measured as the distance (number of edges) from each twin to the furthest away twin in each interconnected chain, is indicative of how the domain network spreads during deformation. The distribution of the longest path of connected twin contacts is shown in Fig. 6b. As individual twin chains that initially formed separately cannot be differentiated after deformation, the longest path of connected twin contacts for each twin is measured by calculating the shortest path length between each twin and all other interconnected twins in the graph. The maximum of these short path lengths is identified as the longest path for each twin. The median longest connected twin path length, a comparative metric for the length of twin chains, is 12 twin-to-twin contacts in the 3D reconstructions but only 2 in the 2D networks (4 if excluding all isolated twins). This indicates that the 2D slices of the network predominantly only identify single paired twins or short connected segments of twin chains. Note that the longest twin path lengths in the 3D reconstruction will be influenced by the size of the 3D volumes, and longer twin paths may be measured if the volume of the 3D reconstruction increases. For the 3D data, only 4% of twins form a short chain with a path length between two and five twins. The fractions of short chains and isolated twins in 3D is also likely an overestimate, as the distribution includes twins that contact the reconstruction boundary which may form twin contacts outside the reconstruction volume49. Thus, most of the twin domains in the 3D fingerprint comprise large, highly interconnected networks throughout the sample volume compared to the more fragmented twin networks identified in 2D.
Cross-grain contacts in 2D and 3D networks: signatures of twin transmission, co-nucleation, and incidental contact
The prior observations raise an additional question: if 2D reconstructions of the twinned microstructures systemically underrepresent the connectivity of the network, do the missed connections in 2D reflect a random sampling of all possible twin-to-twin contacts, or are certain types of contacts more commonly missed in 2D? To address these questions, we investigate the differences in alignment of the cross-grain twin contacts identified in 2D and 3D reconstructions of the network. Twin connectivity arising from cross-grain twin contacts has attracted extensive interest across a wide range of HCP alloy systems. As transmission across grain boundaries is a dominant pathway for the growth of twin chains, recent studies explored the applicability of geometric parameters, particularly m’ (Eq. (1)), to describe or predict the formation of adjoining cross-grain twin pairs during deformation39,42. Statistical analyses of 2D datasets revealed that m’ was insufficient as an individual parameter for predicting twin pair formation at grain boundaries, and that a non-negligible fraction of cross-grain twin pairs formed with relatively poor macroscopic alignment (m’ near 0). Leveraging the prior comparisons of twin connectivity, in this section the cross-grain contacts are analyzed to explore (i) the geometric alignment of the missed connections in the 2D data and (ii) how these missed connections bias the applicability of m’ for either describing or predicting the formation of adjoining twin pairs.
The distribution of m’ for all cross-grain contacts is shown in Fig. 7a. The bimodal profile is similar for each category of reconstructions with peaks near m’ = 1 and m’ = 0. However, the 3D reconstructions (top row) display a significantly higher fraction of misaligned cross-grain (m’ ≈ 0) contacts than the large-area 2D reconstructions (bottom row). Specifically, cross grain pairs with m’ < 0.5 comprise 47% of contacts in the 3D reconstructions but only 19% of contacts in the large-area 2D reconstructions. Thus, the undercounted cross-grain connections in 2D reconstructions are primarily the low-alignment twin pairs that are less likely to form through transmission. Consequently, the fraction of high m’ twin pairs is notably lower in the 3D reconstructions compared to the 2D reconstructions. While 61% of cross-grain pairs in the large-scale 2D slices display a high alignment of m’ > 0.8, only 36% of pairs display similar alignment in 3D.
a Histogram distributions of m’ for all observed cross-grain twin contacts. b Distribution of the ratios of observed m’ to maximum possible m’ between each incoming twin and all six possible outgoing twin variants for each cross-grain twin contact. Note that a m’/m’max ratio of 1 indicates that the observed cross-grain twin pair formed between the two variants with the highest possible geometric alignment, and the distributions only consider variants of the \(\left\{11\bar{2}2\right\}\) twin mode in each adjacent grain. c Propensity for forming an adjoining twin pair (ATP) for each twin-to-grain boundary contact, measured as the fraction of twin-boundary intersections that include a cross-grain contact to the total number of twin-boundary intersections at maximum possible m’ configurations ranging from 0 to 1. The cross-grain twin contact statistics are detailed for all combined 3D twin network reconstructions (orange, top row, 1517 cross-grain twin contacts) compared to all combined large-area 2D twin network reconstructions (purple, bottom row, 1938 cross-grain twin contacts). In the large-area 2D reconstructions, 40% of cross-grain twin contacts display m’ > 0.9 and 68% of twin-grain boundary contacts display m’/m’max > 0.9, compared to only 22% and 41% respectively for the 3D reconstructions. While the propensity for ATP formation increases with increasing m’max above 0.6 in both the 3D and large-area 2D reconstructions, the propensity for ATP formation for all twin-boundary contacts with a low maximum alignment of m’max < 0.6 is 0.32 in the 3D reconstructions compared to only 0.10 in the large-area 2D reconstructions.
The applicability of m’ for cross-grain contact formation further depends on the alignments of all possible twin variants that could form forward of each twin-to-grain boundary intersection. This can be tracked by calculating the ratio between m’ and m’max, the best possible alignment between the incoming twin and all outgoing twin variants for each twin-boundary, as shown in Fig. 7b. In the current analysis only outgoing variants of the dominant \(\left\{11\bar{2}2\right\}\) compression twin mode are considered and each of the two individual twin-boundary intersections are analyzed separately for existing cross-grain twin contacts. The cross-grain pairs in both the 2D and 3D reconstructions display a strong peak near 1, revealing that many twin pairs consist of the highest possible alignment variants in the two adjacent grains. However, 68% of cross-grain twin pairs in the large-area 2D slices form at m’/m’max > 0.9 compared to only 41% in the 3D reconstruction. Further, only 14% of cross-grain twin pairs in the large-area 2D reconstructions form at relatively low m’/m’max < 0.2 compared to 39% in the 3D reconstructions. Prior geometric analyses indicated that in cases where m’max between two adjacent grains is relatively low ( < 0.8), the variants that can accommodate the maximum plastic dissipation are often not the variants with the highest geometric alignment39. As such, our results suggest that the increase in low m’/m’max contacts observed in the full 3D network reflects either a higher propensity for twin transmission driven by local stress accommodation (instead of m’) than previously expected, or an increase in the number of incidental contacts. In addition, contacts forming from these mechanisms would generally have a lower contact surface area than the higher m’ connections that are less commonly missed in the 2D slices of the network.
The influence of twin alignment on cross-grain pair formation can also be measured by the propensity for adjoining twin pair (ATP) formation – the ratio of paired vs. total twin-grain boundary contacts for every twin – at increasing m’max as shown in Fig. 7c. The propensity for twin pair formation increases notably in the 2D reconstructions with increasing m’max above 0.6 but interestingly remains constant for all possible twin pair configurations with m’max below 0.6. In the 3D reconstructions, the propensity for twin pair formation is significantly higher than the 2D reconstructions for all possible twin pairs with m’max > 0.1 and increases monotonically with increasing m’max up to 1. As the 3D reconstructions identify all ATPs present in the measured volume, the discrepancy between the actual propensity for ATP formation and the observed propensity in the 2D data reflects the missing twin connections detailed in Figs. 3, 4. As the discrepancy gets increasingly larger with decreasing m’max, the size of the contact patches between adjacent twins must decrease with decreasing twin alignment.
Discussion
Twin network stereology: biases in 2D interpretations of 3D deformation twin networks
The structure of the twin network that forms during deformation introduces a radical change in the microstructure and is expected to strongly influence the mechanical response of the host grains. Further, accurate characterization of the morphology of the network, including the twinned volume fraction, the connections between twins and twin chains, and the tortuosity of twin transmission during chain growth is crucial for predicting both the influence of twins on local plasticity as well as damage accumulation and fracture behavior.
By comparing 2D and 3D reconstructions of twin networks from the same deformed test specimen, the blind spots of 2D slices are apparent. A qualitative comparison of the graph abstractions of the networks shows stark differences between the two reconstructions. While 3D graphs portray twin networks as near-fully interconnected within the microstructure, the 2D graphs depict the network as dispersed, isolated twins with some individual paired contacts and a few short chains. These differences in connectivity are akin to comparing fiber reinforced composites composed of long woven filaments to those with short, discontinuous chopped fiber segments. The origin of the difference is simply an extension of early work in the field of microstructural analysis. From fundamental stereology principles57,58, it is well known that morphological characteristics (i.e. size, length, surface area, volume fraction, particle spacing) of microstructural features may require corrections when extrapolating analyses from 2D slices to a full 3D volume. These also require assumptions regarding the 3D shape of both the microstructural features and the host grains.
In the case of twin networks, 2D slices of the microstructure can never count more twin-to-twin contacts than the true number of contacts in the full 3D volume. In the present results, there are distinct differences in the 2D and 3D distributions related to adjoining twin pairs that must be rationalized in terms of morphological differences in the different types of contacts (i.e., well-aligned transmission, less well-aligned transmission, co-nucleation, incidental contacts). The underrepresented cross-grain connections in 2D are more likely to be adjoining pairs of low-m’ twins (Fig. 7a), and the smaller fraction of these contacts must result from smaller overall surface area for the low-m’ facets compared to higher-m’ facets. For cross-grain twin contacts that exhibit a lower geometric compatibility (particularly a low \({{{\bf{K}}}}_{{{\bf{1}}}}^{{{\bf{A}}}}\cdot {{{\bf{K}}}}_{{{\bf{1}}}}^{{{\bf{B}}}}\) with a large twist angle along the boundary plane), the contact patch is expected to be small due to the misalignment of the twins and further twin thickening along any direction is expected to have a minor impact on the shared contact area. We postulate that incidental contacts from the intersection of two twins that formed through unrelated processes – indicative of contact between separate twin chains – are expected to have the smallest interaction areas.
Representative examples of twin pairs exhibiting these different contact morphologies are illustrated in Fig. 8. Note that these are illustrative examples but not a formal statistical analysis. In Fig. 8a, a contact between two well-aligned twins likely formed through transmission is shown, and the contact patch between the two twins is relatively large due to high overlap at the boundary. These high-area contacts are statistically more likely to be identified in a random slice of the microstructure, evident in the high peaks near m’ = 1 for both the 3D and 2D reconstructions in Fig. 8a. Alternatively, for twin contacts that form through co-nucleation, the twin alignment may be either high (m’ ≈ 1) or low (m’ ≈ 0) assuming that both twins are well-oriented to accommodate the macroscopic strain (SF close to 0.5, with the twin plane oriented near 45° from the loading axis). An example contact expected to have formed through the low-alignment co-nucleation configuration (due to high overlap at the boundary between the two twins and both twins terminating within their respective parent grains) is shown in Fig. 8b, with a high misorientation between the twins but a similarly high-area contact patch as observed for the well-aligned twin pair. While these co-nucleation events generate high contact patches that are more easily identified in random 2D slices of the microstructure, we speculate that they occur less frequently compared to transmission events. This would partially explain the secondary peak observed in the 2D m’ distribution around m’ = 0 in Fig. 7a. Finally, cross-grain contact between two randomly misoriented twins, indicative of incidental contact, is shown in Fig. 8c. This configuration, particularly at increasing twist misorientation angles, results in a small contact area that will be less commonly identified in a random 2D slice of a microstructure.
Cross-grain twin pair configurations resulting from a contact between highly aligned twin, (b) contact between low-m’ twins with a high λ1 alignment, and c contact between randomly misoriented twins. (Top row) 3D twin and parent grain morphologies extracted from 3D Cube 1, (bottom row) diagrams of cross-boundary contact illustrating the contact patches (green) between the twin pairs (orange, blue). Contacts (a, b) are representative of high-contact area cross-grain twin configurations, while contact (c) is representative of a low-contact area cross-grain twin configuration.
In the 3D reconstruction, all of these adjoining twin pairs are captured while in 2D analyses the identified contacts are instead proportional to the number density and contact areas of the different types of adjoining twin pairs. The identification of these incidental contacts in the 3D reconstruction, as well as the tortuous nature of twin chain propagation48 further explains why significantly longer twin paths are observed in the 3D networks compared to the 2D networks. However, due to the tortuous series of connections that comprise each individual chain, there is no straightforward way to scale the twin connectivity identified in 2D slices of the microstructure to the actual 3D network. As such, the full morphology of the domain networks can only be accurately captured through 3D characterization.
Consequences of 2D views of twin fingerprints on rationalizing deformation and network evolution
Several important aspects of plasticity and failure are not well represented by the twin network fingerprints identified in 2D analyses. Low-alignment cross-grain contacts that are underrepresented in 2D reconstructions are likely sources of shear localization during continued deformation. As such, we speculate that the slices of twinned microstructures highlighted by 2D analyses will provide an overly conservative estimate of the connectivity of the twin networks for input into microstructure-sensitive models. Further, as twin chain lengths are lower in 2D reconstructions, attempts to model the morphology of the twin network based solely on 2D analysis will also be biased. This will impact fracture models in materials exhibiting extensive deformation twinning.
In HCP materials, twin-grain boundary and twin-twin intersections have been shown to preferentially nucleate microcracks59,60,61,62,63,64 due to the mechanical incompatibilities that arise at shear discontinuities across twin interfaces49. Twin-to-matrix interfaces have further been observed as preferential paths for crack propagation, likely arising from stress concentrations that form along the forward and lateral edges of the twins65,66,67,68. As such, both the connectivity and the geometric alignment of the twin network is expected to influence the fracture toughness of the microstructure. For example, interconnected networks with well-aligned twin contacts may provide a minimal-energy pathway for microcracks to coalescence and rapidly progress to failure. In cases where interconnected networks contain a high density of misaligned contacts, cracks are instead expected to deflect at twin-twin intersections which can delay critical failure65. Networks containing isolated twins and twin pairs may also result in crack arrest at the end of the dispersed twin segments. This suggests several possible issues when generating fracture models based on 2D slices of twinned microstructures. First, if models are calibrated solely using 2D views of the network, they may underestimate the role of intergranular crack growth along twin boundaries if the network is assumed to have lower connectivity and shorter chains than actually present in the full microstructure. This may lead to increasingly inaccurate predictions if the calibrated model is extrapolated to other materials with higher or lower twin volume fractions. Further, qualitative insights about crack propagation may also be biased if only analyzing 2D views of the twinned microstructure. If the twins are assumed to be more dispersed and disconnected than in the actual network, rapid crack propagation along twin chains may be improperly attributed to low resistance of either grain boundaries or slip bands to crack growth.
The biases in 2D reconstructions of twin networks may also impact studies of network development during deformation. Depending on the alloy composition, texture, and loading conditions, twin formation in HCP materials generally occurs through a combination of isolated nucleation/co-nucleation at grain boundaries and transmission from existing neighboring twins. As such, the network of twin domains is conventionally expected to evolve by the growth of twin chains that form through sequential transmission processes originating at initial ‘seed’ isolated twins. The lower connectivity and increased fraction of isolated twins identified in 2D slices of the twin network tends to magnify the relative role played by nucleation in network growth69 relative to that of transmission. Alternatively, in the 3D reconstructions the true density and alignment of contacts in the full twin network portrays a different picture of network evolution. Instead of diffuse, disconnected twin domains and short chains of twins forming by 2.6% strain, the twin domains are observed to coalesce into a near fully interconnected network that contains approximately equivalent amounts of high (m’ > 0.5) and low (m’ < 0.5) alignment twin contacts. This indicates that the full 3D twin network does not primarily evolve through extensive isolated nucleation and occasional well-aligned transmission events, but rather by a mix of both highly aligned transmissions and highly mis-aligned cross-grain contacts. The implications of the current 3D twin characterization on modeling is that the role played by 3D transmission mechanisms should be emphasized, which will most likely attenuate the role played by twin nucleation at grain boundaries.
Prior analysis of 2D cross-grain connections also proposed that the formation of occasional mis-aligned contacts is guided by local stresses, as the transmitted variant with the highest plastic dissipation potential may not be the variant with the highest alignment (particularly at twin-grain boundary contacts with low m’max)39. As these low m’/m’max contacts are far more frequently observed in 3D reconstructions of the full twin network compared to 2D reconstructions (Fig. 7b), the present results suggest that local stresses may play an even greater role in determining variant selection during twin transmission than previously assumed. This also mirrors prior observations that the twin chains that form the backbone of the network likely display more tortuous growth paths than would be typically expected from 2D views of the microstructure48. As such, the interconnectedness in the twin network is expected to evolve during continued deformation through interweaving of separate twin communities that aggregate together as the density of incidental twin contacts increases. This is further reflected in comparisons of 3D cross-grain contacts in the present study to 2D cross-grain contacts at differing strain levels in prior 2D studies. It was previously observed that the applicability of geometric compatibility to describe cross-grain twin contact formation decreased with increasing macroscopic strain39 due to an increase in the relative fraction of low-m’ contacts. The current results suggest that this decrease in the descriptive accuracy of m’ at increasing strains in prior 2D studies arises due to an increase in the formation of low-alignment, low-contact-area incidental twin connections as the network evolves during continued deformation.
In conclusion, this study presents a novel statistical and graph-based comparison of the microstructural fingerprint of deformation twin domain networks extracted from 2D and 3D network reconstructions. In comparison to full 3D reconstructions, conventional 2D views of twinning miss key features of the twin network, particularly that the twin domains exhibit high interconnectivity and form a scaffold that spans nearly the entire microstructure. The diffuse and disconnected twin network identified in 2D reconstructions reflects an underrepresentation of the number of twin-to-twin contacts actually present in the full network, particularly low-alignment cross-grain twin connections that form through the incidental contact of separate twin chains. The reduced twin connectivity observed in 2D reconstructions has notable implications on our understanding of both how the network evolves – overemphasizing the role of twin nucleation compared to twin transmission – as well as how diffusionless transformations mediate plastic deformation in a wide range of alloy systems. In particular, the results of the study emphasize that 3D characterization is critical for accurately identifying the fingerprint of highly twinned microstructures.
Methods
Experimental
Titanium with a high purity of 99.999% was sourced from Fine Metals, provided as a 12.5 mm plate in a hot-rolled state. The titanium plate was cold-rolled to achieve a 25% thickness reduction perpendicular to its original rolling direction and then recrystallized at 650 °C in a high vacuum for 15 min, resulting in an average grain size of approximately 25 μm. EBSD mapping revealed a resultant microstructure with a rolling texture characterized by high basal (0002) pole intensities within 30° of the plate’s through-thickness direction, as observed in the pole figures in Fig. 9.
Pole figures detailing the texture of the pre-compression high-purity Ti microstructure along the rolled through-thickness (\({{{\rm{X}}}}^{{{\rm{L}}}}\)) direction.
A compression cube of dimensions 8 mm × 10 mm x 10 mm was sectioned from the heat-treated plate. The cube was compressed at 87 K along the through-thickness (TT) direction to 2.6% macroscopic strain, and the deformation conditions were specifically selected to induce a high density of twins in the microstructure. A slice was then taken from the center of the cube perpendicular to the TT direction, mechanically ground, and polished using 0.4 μm colloidal silica mixed with 20% hydrogen peroxide for the final polish. Three EBSD maps ranging in size between 0.41 mm × 0.53–0.73 mm were collected with a step size of 0.2 µm at a working distance of 25 mm and accelerating voltage of 20 kV in different regions on the same surface (Fig. 1 (a)) using an EDAX Velocity Super Camera. Note that no statistical method was used to predetermine sample size. Next, a Thermo Fisher Scientific G4 Plasma Focused Ion Beam (PFIB) was used to fabricate and lift out cubes for 3D-EBSD slicing. The samples were milled using 30 keV Xe+ at 2.5 µA for regular cross-section cutting and 1 µA for undercutting. The bottom edge of each cube was machined with a triangular surface extending from the cube size (i.e., inverted house geometry) to aid in layer alignment. The milled 3D geometries of exterior dimensions ~110 × 110 × 130 µm were then lifted and mounted on the edge of Si wafers for data collection. Three 3D-EBSD data cubes of volumes between 3.3–4.4 × 105 µm3 were collected from the same surface using both an in-plane and inter-plane step size of 0.2 µm and a working distance of 4 mm and accelerating voltage of 20 kV. To minimize curtaining, each slicing step consisted of two rocking-milling steps ( ± 8° rotation and ±1.5° counter tilt). Note that while the experimental framework cannot identify initial twin nuclei that are smaller than the step size or beam interaction volume, due to the rapid kinetics involved in twin propagation and growth, it is expected that these small nuclei contribute negligibly to both the mechanical response of the microstructure and the structure of the twin network. Further, as twins thinner than the step size may be present in both the 2D and the 3D reconstructions, excluding these twins from the analysis is expected to have a negligible impact on comparisons of 2D and 3D network statistics.
While the 3D cubes (and consequently the 2D slices of the 3D cubes) were polished via PFIB, the final polish step for the large-scale 2D datasets was a 0.4 μm colloidal silica-peroxide mixture. As such, the EBSD pattern quality varies slightly between the datasets. A schematic of the sample highlighting the relative locations of the 2D-EBSD and 3D-EBSD regions is shown in Fig. 1a. Note that it is experimentally challenging to scale up the data collection volume. Serial sectioning via plasma focused ion beam with EBSD is often used because it provides fine spacing between the slices. However, this technique limits the collection volume to ~2 × 106 µm3 due to the milling rate and data collection time. Other serial sectioning techniques such as tri-beam tomography and mechanical polishing allow for sampling of larger volumes but the layer spacing is too coarse ( ~ 1 µm) to capture fine deformation twins ( ~ 100–200 nm). Consequently, due to the limited size, the 3D cubes were selected in the cross-sectional region exhibiting high twin fractions, rationalizing the minor differences in the total twin volume fraction between the 2D and 3D reconstructions detailed in Table 1.
Data reconstruction
The EBSD scans were aligned using a two-step procedure. First, a rough alignment was performed through cross-correlation of the edges of the inverted house sample geometry, followed by a second fine tuning alignment step using cross-correlation of segmented parent grain boundaries from the 2D EBSD reconstructions of each adjacent layer. The alignment procedure assumes near-equiaxed parent grain morphologies, which is accurate in the present study. We note that the alignment procedure only consists of a rigid translation of the voxels in each layer. Registration inaccuracies from differences in tilt from layer-to-layer are expected to be negligible in the current experimental framework.
The twins in each 2D and 3D dataset were identified and reconstructed using METIS3D48. Separate orientation fragments (either parent grains or twins) are first identified in each 2D layer by grouping adjacent voxels with a misorientation below a threshold of 5 degrees. The average domain statistics (e.g., size, orientation, centroid) are calculated for each fragment, and the 2D domain connectivity is then determined by identifying adjacent voxels belonging to different fragments. Next, grains are reconstructed by comparing the disorientation between adjacent fragments to the expected disorientations generated by activation of any \(\{11\bar{2}2\}\), \(\{10\bar{1}1\}\), or \(\{11\bar{2}4\}\) compressive twin variant or any \(\{10\bar{1}2\}\) or \(\{11\bar{2}1\}\) extension twin variant. Parent-twin connections are assigned if any of the twin disorientations are below a threshold of 7 degrees, and grains are reconstructed by finding all sequentially adjacent fragments with positive parent-twin relationships (e.g., adjacent parent domain – twin domain – parent domain are grouped into a single grain). We note that if a twin spans the full length of the grain in a 2D layer, the fragments of the parent grain may be disconnected. To identify the parent phase and reconnect parent fragments, connections are then added between all fragments within each grain and a misorientation threshold of 5 degrees is assigned to determine the total areas of each domain in the grain with similar orientations. The parent domain is identified as the combined domain with the largest total area, and the parent fragments are combined into a single fragment. A common error when processing EBSD measurements of twinned microstructure is the twin splitting into separate disconnected fragments, particularly for thin cross-sections of twins. To address this, ellipsoids are fit to each twin fragment and twin fragments in each grain are merged together into a single twin strip if the following criteria are met: (i) the misorientation between two twin domains is less than 5 degrees, (ii) the orientation of the long axis of the ellipsoid and the orientation of the line connecting the two twin centroids is within 5 degrees, and (iii) the sum of the half lengths for each twin is less than 20% of the distance between their centroids. For the 3D EBSD reconstructions, inter-layer connections are then assigned between fragments if the misorientation between the two fragments is less than a given threshold and either (i) the in-plane distance between the two fragment centroids is less than the square root of the larger fragments area, or (ii) the in-plane coordinates of the voxels in one fragment are within the borders of the in-plane coordinates of the second fragment. We note that due to the potential for systemic differences in orientations between different layers, an offset misorientation is assigned as the minimum misorientation between any two fragments in adjacent layers. The misorientation threshold for matching inter-layer fragments is thus assigned as 5 degrees + this offset.
Erroneously segmented regions of the reconstructions were then fixed using the manual editing mode in the METIS3D software48,50. First, improper twin-parent connections were fixed in each 2D layer. In cases where the disorientation between two adjacent grains or a grain and a twin in an adjacent grain is less than 7 degrees from one of the twin disorientation relationships, all fragments in the two separate grains will be erroneously combined. These can be clearly identified visually by viewing only twinned fragments, as all parent fragments of the smaller grain are assigned as twin fragments of the larger grain. These improper twin-parent connections are then manually turned off in the METIS3D interface. Next, twin fragments that are improperly merged while generating 3D inter-layer connections are fixed. Edges are drawn between each 2D twin fragment in each 3D merged twin, and each 3D twin is visually checked layer-by-layer to verify that two closely spaced twins of the same variant are not erroneously merged together. Note that twins that appear separate in some 2D layers may either merge or display a boundary distance below the in-plane step size in other layers, such that the apparently disparate 2D fragments actually belong to a single twin. We further note that all automated reconstruction steps and manual edits in METIS3D were run on a standard 10-core desktop workstation with 64 GB of memory, and the reconstruction process only requires sufficient memory to load a minimum of three adjacent EBSD layers. The processing execution time during the reconstruction is negligible for the size of the EBSD datasets analyzed in the present study.
Finally, because small, disconnected fragments of twins reconstructed during EBSD indexing may be improperly identified as separate twins (if they do not meet the criteria for twin strip merging in 2D) and bias the statistical analyses of the networks, the twin reconstructions were then thresholded during post-reconstruction analysis by removing twin fragments fully contained within the interior of grains with a volume of less than 1 µm3 (3D Cubes) or area less than 1 µm2 (2D Areas). On average, the thesholded twins comprised 7% of the total number of identified twins but less than 1% of the total twinned volume in both the 3D Cubes the 2D Areas.
The 2D and 3D orientation maps in Fig. 1 and Supplementary A are color mapped by assigning an RGB triplet to each voxel according to the lattice plane normal parallel to the loading axis. The color legend for the HCP fundamental zone is detailed in each figure. All 2D EBSD images are generated using EDAX OIM Analysis, and all 3D EBSD images are generated using Paraview70. Note that the 3D rendering model in Paraview may slightly influence the luminance of the colors.
Data availability
The source data used in this study are available in the Materials Commons repository at https://doi.org/10.13011/m3-66pj-k548 (registration required to download the data).
Code availability
The source code for METIS3D cannot be released at this time, as the authors are in the process of filing for a commercial license.
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Acknowledgements
This work was fully funded by the US. Department of Energy, Office Basic Energy Sciences Project FWP 06SCPE401. The 3D data was obtained at the Electron Microscopy Laboratory at Los Alamos and Los Alamos National Laboratory, an affirmative action equal opportunity employer, is managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA, under contract 89233218CNA000001.
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D.A.G.: Investigation, data curation, formal analysis, writing-original draft; H.T.V.: conceptualization, investigation, data curation, formal analysis, writing-original draft; R.J.M.: data curation, investigation, writing – review & editing; M.M.S.: investigation, data curation; C.N.T.: conceptualization, investigation, writing – review & editing; L.C.: investigation, writing – review & editing, funding acquisition.
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Greeley, D.A., Vo, H.T., McCabe, R.J. et al. A quantitative comparison of the fingerprint of twinned microstructures through surface and three-dimensional techniques. Nat Commun 16, 9773 (2025). https://doi.org/10.1038/s41467-025-64740-0
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DOI: https://doi.org/10.1038/s41467-025-64740-0
