Three-dimensional character of the deformation twin in magnesium

Deformation twins are three-dimensional domains, traditionally viewed as ellipsoids because of their two-dimensional lenticular sections. In this work, we performed statistical analysis of twin shapes viewing along three orthogonal directions: the ‘dark side’ (DS) view along the twin shear direction (η1), the twinning plane normal (TPN) view (k1) and the ‘bright side’ (BS) view along the direction λ(=k1 × η1). Our electron back-scatter diffraction results show that twins in the DS and BS views normally exhibit a lenticular shape, whereas they show an irregular shape in the TPN view. Moreover, the findings in the TPN view revealed that twins grow faster along λ the lateral direction than along η1 the forward propagation direction at the initial stages of twin growth. These twin sections are irregular, indicating that growth is locally controlled and the overall shape is not perfectly ellipsoidal. We explain these findings using atomistic models, and ascribe them to differences in the mobility of the edge and screw components of the twinning dislocations.


Overall comment
"In this paper the authors are trying to answer the question if the kinetics of twin growth in BS and DS directions are comparable. To do so, they characterized the twin shape from three orthogonal directions, especially in the twin plane normal direction. Their Mg plate specimen was compressed by 1%. They drew their conclusions based on their EBSD characterization of many twins in this condition. The EBSD characterization was made on the surface of their specimens. It is a 2D observation." Answer: We thank the reviewer for the interpretation of our work. We want to highlight what we believe is the main result from our study: statistical analysis of hundreds of twin shapes in the early stages of formation allows us to infer the mechanisms of forward and lateral propagation. We conclude that in the first stage twins have a tendency to expand laterally and, past a critical 'width', forward propagation is favored. This result is novel and we complement it with MD simulations that justify such behavior in terms of twin-facet mobility and formation of interface dislocations. Statistical analysis plays an important role since twin shapes are neither perfect nor identical, and driving stresses are likely to vary from grain to grain. Our analysis of 2D EBSD sections has the advantage over 3D twin reconstruction that a much larger number of twins can be analyzed, and that the statistical analysis of their sections reveals more microscopic information about the shape of the twin domain. We will explain this in the following point-bypoint response in further detail.

Comment #1
"One question to the authors is why not do some 3D EBSD work to directly establish the real shape of twins. 3D EBSD can be done quite easily nowadays. I think the best way is to use 3D EBSD to show the 3D shape of twins, rather than making deductions from 2D intersections. From the 3D EBSD, the real shape of a single twin can be worked out more accurately." Answer: The assertion that '3D EBSD can be done quite easily today' is misleading. True that the software DREAM-3D is available for reconstructing sections measured with EBSD. This technique has been applied mainly to reconstruct aggregates and investigate grain geometry [T.
Liu et al, 2017, 2018] and for reconstituting the shape of three 'well developed' twins in Mg [Fernandez et al, 2013]. However, the technique by serial sectioning with a focused ion beam (FIB) has limitations for been applied to our problem, where early stages of twinning are addressed.
Specifically, the resolution required to reconstruct incipient twins is at least 0.5 micron. At such scale, damage and heat induced by FIB introduce uncertainties larger than 1 micron [Gutierrez, 2017, andMingard et al, 2018]. Moreover, the number of twins that can be characterized with this technique is relatively small.
In addition, there is the issue of automated twin-matrix orientation and twin features recognition (such as dimensions, inclination of habit plane with respect to the section, etc). Finally, once all twin features are collected, specific software is required for analyzing correlations. In our DOE-BES Program we are currently in the process of developing such 3D software. The bottom line is that analysis of twin structures and their correlations using FIB, 3D EBSD, and automated image recognition software is still in its infancy. Furthermore, an advantage of using a statistical approach to analyze 2D sections (in our case this was done manually via exhaustive microscopy work) is that one can obtain distributions of shapes and general trends associated with the twinparameters studied. In addition to being extremely time intensive, 3D-EBSD reconstruction technique will not provide sufficient number of twins to overcome variabilities and derive a meaningful conclusion in this study. As an example, the statistics on twin aspect ratios shown in Figure R1, for twins that intersect none, one or two grain boundaries, indicate that in all three cases there is a threshold length that divides the early faster lateral growth from the later faster forward growth. The main text has been revised accordingly. Figure R1 Statistical analysis showing that for all three types of twin shapes, they all have critical "width" to divide the early faster laterally growth from the later faster forward growth.  ▪ "Since the TD has step character, the meaning of 'planar core' should be clarified."

Answer：
(a) A revision has been made accordingly, as "The segment of a TD loop which is parallel to the Burgers vector is also parallel to and has pure screw character" (b) A revision has been made accordingly, as "For a general dislocation with planar core, a screw dislocation has higher mobility than an edge dislocation". Also described Answer: We thank the reviewer to point out these typos. A revision has been made accordingly.
Even if the authors feel that 3D EBSD is beyond their capability, they should at least try to analyse their 2D data points by taking the truncation effect into account. The values of twin dimensions that they provided in several figures will change when the truncation correction is made, or when the 3D approach is used.
The essence of this paper is the statement made by the authors: a deformation twin grows faster in the DS direction than the BS direction in the initial stages of twin growth. What is the definition for initial stages of twin growth? For the twins that were shown in their EBSD maps, they are already several microns or even tens of microns. These sizes are far beyond initial stages of twin growth, they are also far beyond the sizes of twins in their MD simulations. The definition for the "initial stages" is quite arbitrary, so is the critical size at which the preference of growth occurs. These vague definitions do not help reader to fully understand their work.
The expectation from the classical theory is that growth along the shear direction (DS in authors' terminology) should be faster, unless the growth front develops facets of much lower interfacial energy. It would add value to this work if the authors could provide some experimental information on the structures of the facets of twins before and after the growth preference is changed.

Reviewer #1 (Remarks to the Author):
Even if the authors feel that 3D EBSD is beyond their capability, they should at least try to analyse their 2D data points by taking the truncation effect into account. The values of twin dimensions that they provided in several figures will change when the truncation correction is made, or when the 3D approach is used.
The essence of this paper is the statement made by the authors: a deformation twin grows faster in the DS direction than the BS direction in the initial stages of twin growth. What is the definition for initial stages of twin growth? For the twins that were shown in their EBSD maps, they are already several microns or even tens of microns. These sizes are far beyond initial stages of twin growth, they are also far beyond the sizes of twins in their MD simulations. The definition for the "initial stages" is quite arbitrary, so is the critical size at which the preference of growth occurs.
These vague definitions do not help reader to fully understand their work.
The expectation from the classical theory is that growth along the shear direction (DS in authors' terminology) should be faster, unless the growth front develops facets of much lower interfacial energy. It would add value to this work if the authors could provide some experimental information on the structures of the facets of twins before and after the growth preference is changed.

Response to reviewer
Overall response: The authors appreciate the comprehensive feedback and challenges provided by the reviewer. As a result of it we have added new experimental characterization, revised our interpretation of results and some of our conclusions, and eliminated parts that were deemed more speculative. We believe that the revised manuscript is clearer than before and more impactful.
The most relevant and novel aspect of this paper is the characterization of twin morphology by analyzing non-standard sections, namely, viewing them along the twin propagation direction and along the twin plane normal. In addition, we added an analysis of the typical twin sections showing the twin profile along the coherent twin plane and the forward propagation direction.

Comments #1
1. Even if the authors feel that 3D EBSD is beyond their capability, they should at least try to analyse their 2D data points by taking the truncation effect into account. The values of twin dimensions that they provided in several figures will change when the truncation correction is made, or when the 3D approach is used.
Answer: If by 'truncation effect' the reviewer refers to sectioning the twin with a plane nonparallel to the coherent twin boundary (what we call TPN view) or not perpendicular to the CTB (DS view), the answer is: we establish the relative orientation between the twin planes and the section plane and do not consider twins that exceed 5 degree misorientation. The procedure is explained in main text (page 4) and Fig. 3 (here is Fig. R1). In addition, when measuring the main directions along long axes λ and η1, and along short axis along k1, we correct the length accordingly. A schematic of BS section is shown below. On the other hand, if the reviewer refers to 'where' the cut intersects the twin, we present below an explanation. We classified twins in three types based on their junctions at GBs as they appear in the EBSD sections. They are Type 0 if neither side of the twin is connected to GBs, Type 1 if only one side is connected to a GB and Type 2 if both sides are connected to GBs. Since twins in annealed HCP nearly always propagate from grain boundaries, the schematic below attempts to capture such situation as shown in Fig. R2: (a) represents our typical picture of forward propagation after one twin nucleation, that showing type 1 case from BS view ( and ) and type 0 case from DS view ( and ); (b) represents another scenario when twin propagation start laterally towards , that showing type 0 case from BS view ( and ) and type 1 case from DS view ( and ).
Moreover, in our main text (page 4), we found: "As shown in Fig. 4a, the number of BS twins ( -) that qualify is relevant for the 3 types, with prevalence of type 2; twins in DS view ( -) are overwhelmingly type 2, and practically zero number of type 0 is observed; in what concerns TPN twins, they are exclusively type 1 and type 2 in about equal numbers, indicating that propagation always start at a GB." This evidence suggests that the lateral expansion of twins is relatively easy, since most of the DS view ( -) extend from side to side of the grain (type 2), compared to the BS view ( -).

Comments #2
2. The essence of this paper is the statement made by the authors: a deformation twin grows faster in the DS direction than the BS direction in the initial stages of twin growth. What is the definition for initial stages of twin growth? For the twins that were shown in their EBSD maps, they are already several microns or even tens of microns. These sizes are far beyond initial stages of twin growth, they are also far beyond the sizes of twins in their MD simulations. The definition for the "initial stages" is quite arbitrary, so is the critical size at which the preference of growth occurs.
These vague definitions do not help reader to fully understand their work.

Answer:
We agree with the reviewer that 'initial stage' may convey different length scales to different readers. We have added clarifications in the introduction: "Twinning involves three sequential processes: nucleation, propagation and growth, which are associated with formation and migration of twin boundaries (TBs). This work focuses on the twin morphology at the propagation stage, defined as the twin having partially or fully traversed the grain, but before it starts growing in thickness after being arrested at the opposite grain boundary", and "Experimental results also show that twins grow faster in the direction (associated with migration of lateral TBs) than in the direction (associated with migration of forward TBs) at the initial stages of twin growth, before intersection with grain boundaries that arrest the forward and lateral propagation".

Comment #3
The expectation from the classical theory is that growth along the shear direction (DS view in authors' terminology) should be faster, unless the growth front develops facets of much lower interfacial energy. It would add value to this work if the authors could provide some experimental information on the structures of the facets of twins before and after the growth preference is changed.
First a clarification: in our nomenclature the propagation along the shear direction would be apparent in the Bright Side (BS) view of the twin.

Answer:
We thank the reviewer's comment. As shown in Fig. R5, we provide additional facet analysis that may suggesting possible to form {426 ̅ 7} facet from TPN view (Fig.1e) Figure R5. The EBSD maps showing the possible facets as shown in Figure 1e.
We are not aware of a classical theory result about the 3D motion of twins. Classical crack or twin analysis tends to look always at the 2D forward stress field, and in a few of those calculations (mostly finite element method) also to propagate. Even in such 2D case the problem is complex because dislocation emission relaxes the forward stress, and the analysis should include plasticity.
We do not recall the lateral stress field and associated relaxation being discussed in the literature.
As a consequence, we asked one of our collaborators (Dr. M. Arul Kumar) to calculate the 3D stress field around a flat twin domain (10:10:1 aspect ratio). He used the Fast plane (plotted below), is positive on the forward and the lateral side of the twin (Fig. R3). Allowing for plasticity makes a qualitative difference in the calculated stress field. A profile of Resolved Shear along the twin direction is included below, plot along forward and laterally oriented lines.
Observe that inside the twin there is a negative back-shear induced by the twin transformation, but the positive shear along the rim will favor both, edge forward twin dislocations and screw lateral twin dislocations. Since the forward rim and lateral rim shear has about the same value, and since screw TDs have a lower activation barrier, it is to be expected that this will favor the lateral expansion of the twin. This result can be regarded as valid for a continuum micrometers scale. In what concerns the atomic scale, we provide simulation results in Supplementary Fig. S6 which also show a rapid lateral propagation of the twin (Fig. R4). Under loading, the twin nucleus grows faster on the lateral side than on the forward side.
In sum, the reviewer rightly questions the idea of a transition, and requests experimental evidence of micron sized facets of twins (which we do not have). We revised our speculation that there is a transition from lateral to forward propagation based on a change in mechanisms. At the scale of the twins that we measure (well over one micron, as the reviewer points out) it is unlikely. Instead, we attribute the transition to the fact that the expanding twin is arrested by barriers that have a separation close to the ~40-micron transition revealed by Fig. 4. The path between barriers is consistent with the grain size or other twins in the grain. We added such argument in the analysis of Fig. 4.
"Such behavior holds up to ~ 40 m, past this length, the length appears to be systematically larger than . However, the scarcity and dispersion of the results does not guarantee a firm conclusion, and we speculate that at this point the propagation of the expanding twin is affected by grain boundaries or by other twins."