Geometry of Butterfly Martensite in Fe-18Ni-0.7Cr-0.5C Alloy

Abstract

To clarify the geometry of the junction of martensite which forming butterfly-type martensite, electron microscopy, theoretical analysis using the phenomenological theory of martensite crystallography and rank-1 connection were carried out on Fe-18Ni-0.7Cr-0.5C (mass%) alloy. Martensite plates in this alloy exhibit {252}_γ habit plane and K-S OR and are in good agreement with the double shear PTMC by Ross and Crocker. The frequency of formation of V1/V2 and V1/V16 (butterflies) pairs was 20% each, with these two pairs alone accounting for 40% of the total. Theoretical analysis using rank-1 connection of variants revealed that butterfly-pairing is not a geometrically compatible morphology, even for alloys in which butterfly martensite forms frequently. We have also revealed theoretically that the junction plane of the butterfly-pairing keeps (100)_γ independent of the mode of lattice invariant shear (i.e. morphology of martensite plate). Preferential formation mechanism for butterfly-pairing is discussed based on the common lattice invariant deformation between V1 and V16.

1. Introduction

1.1. Variant Pairing

Martensitic transformation is a diffusionless transformation achieved by cooperative and shear displacement of atoms¹⁾ and is one of the crucial phase transformations in steel and shape memory alloys. A characteristic martensitic microstructure is formed when crystallographically equivalent martensitic variants are connected with each other.^2,3,4,5,6) The combinations of these martensitic variants are not random, but specific variant pairs are connected at high frequencies.^{4,5,7,8,9,10,11,12)} In this paper, such junction of two variants is called variant pairing. Understanding the dominant factors of variant pairing tendency provides a better understanding of martensitic transformation and a guideline for designing the martensitic microstructure to improve its mechanical properties.^13,14,15)

In α’ martensite (body-centered cubic/bcc or body-centered tetragonal/bct) of steel, 24 habit plane variants can be produced from a single prior γ grain.¹⁶⁾ α’ martensitic microstructures exhibit four morphologies depending on the alloy composition:²⁾ lath, butterfly, lenticular, and thin-plate. It should be noted that lath, lenticular, and thin-plate are names for the shape of a single variant plate, whereas butterfly martensite is a name for a form of two variants connected each other. According to previous reports,^3,4,8) butterfly-type variant pairing appears even in alloys where lath, lenticular, and thin-plate martensite are known to occur. Butterfly morphology^4,17) corresponds to V1/V16 pair according to the notation proposed by Morito et al.,⁶⁾ which is based on the Kurdjumov-Sachs (K-S) orientation relationship (OR). Hereinafter, the variant pairing that is equivalent to V1/V16 pair is referred to as “butterfly-pairing”. Variant pairing tendency depends on the martensitic morphology,^2,3,4,5,6,8) and the martensitic microstructure in which V1/V16 pairs occur frequently is called butterfly martensite. Although there have been fewer crystallographic studies of butterfly martensite than the others, butterfly-pairing is universally present in the martensitic microstructure in steel, so further study of butterfly-pairing is useful for understanding variant selection.

Evaluating the crystallography of martensite theoretically helps to understand variant pairing. The phenomenological theory of martensite crystallography (PTMC) rigorously describes the geometry of the martensite when the interface between the prior austenite and martensite (i.e. habit plane) is an invariant plane.^{3,4,18,19,20)} According to the PTMC, the shape strain (deformation gradient) matrix U of the martensitic transformation is expressed as follows:^18,19,21)

  

$$\mathbf{U}=\mathbf{R}\mathbf{B}{\mathbf{P}}_2{\mathbf{P}}_1$$(1)

where R is the rigid body rotation, B is the Bain deformation and P₁ and P₂ are the lattice invariant deformations. U can be rewritten more generally as an invariant plane deformation, as shown in the following equation:

  

$$\mathbf{U}=I+g\left(\mathbf{d}\otimes \mathbf{p}\right)$$(2)

where I is the identity matrix, g is the magnitude of the shape strain, d is the shape deformation vector and p is the habit plane normal. The PTMC well explains the crystallography of martensite such as habit planes and austenite-martensite OR for lenticular and thin-plate martensite.^3,4,20) Even for lath martensite, the PTMC explains the experimental results when the lattice invariant deformation is appropriately selected.^18,19,21) The following subsections provide an overview of the variant pairing in steel as revealed by experiments and the PTMC theoretical analysis, and show the motivation for studying butterfly martensite crystallography in this work.

1.2. Self-accommodation in Variant Pairing

The phenomenon in which multiple habit plane variants combine to reduce the transformation strain in average during transformations is called self-accommodation.^3,17) Okamoto et al.³⁾ calculated the shape strain of thin-plate martensite using the PTMC and reported that the arithmetic mean of the shape strain of four variants (habit plane group, HP group) such that the habit plane normal around {110}_γ is closest to I, causing self-accommodation. Hereafter subscripts “γ” and “α′” represent austenite and martensite, respectively. Morito et al.⁶⁾ performed similar calculations for lath martensite and showed that self-accommodation is caused by forming six variants (close-packed plane group, CP group) that form a packet. It has also been reported that the development of packets is effective on self-accomodation.⁶⁾ These results well explain the observed microstructure in terms of self-accommodation.

In contrast, there are no reports that self-accommodation occurs in butterfly-pairing. Umemoto et al.⁵⁾ reported that two variants forming butterfly-pairing belong to the same Bain group, based on transmission electron microscopy (TEM) observations in an Fe-20Ni-1.3Cr-0.5C alloy. The pairs observed corresponded to V1/V16. Niho et al.²²⁾ investigated variant pairing tendency in an Fe-0.56C-0.20Si-0.71Mn alloy and reported that more than 80% of the pairs observed were V1/V16. As V1/V16 pairs belong to the same Bain group,¹⁷⁾ the formation of butterfly-pairing cannot be explained by self-accommodation.

1.3. Incompatibility in Variant Pairing

Recently, it has been shown that the continuity of deformation between two habit plane variants significantly influences the frequency of variant pairing in shape memory alloys.^{9,10,11,12,23)} Several groups, including us, have investigated variant pairing in lenticular martensitic microstructures based on the rank-1 connection.^4,17) The necessary and sufficient condition for the continuity of deformation between two habit plane variants at the junction plane is expressed by the following equation:²⁴⁾

  

$$\mathbf{Q}{\mathbf{U}}_i-{\mathbf{U}}_j=\mathbf{b}\otimes \mathbf{m}$$(3)

where U_i, U_j (i, j are variant number) are the total shape strain matrices of the habit plane variants, b is the vector associated with the discontinuity of shape deformation at the junction plane, m is the junction plane normal, Q (rotation angle θ) is the rigid body rotation to maintain the continuity of deformation at the junction plane. Equation (3) is referred to as the rank-1 connection, Kinematic Compatibility condition and Hadamard jump condition. Equation (2) is a special case of Eq. (3). The larger θ makes higher strain energy during the formation of the pairing, because the rotation Q necessarily breaks the invariant plane condition at the habit plane in exchange for the coherence at the junction plane. It is, therefore, expected that the variant pairs with smaller θ are preferentially formed. In lenticular martensite, V1/V6, V1/V16 and V1/V17 pairs are frequently formed, accounting for 75% of the total.⁴⁾ θ of V1/V6, V1/V16 and V1/V17 are 0.55°, 4.95° and 0.01° respectively, and these three pairs are the combinations with the smallest θ. Similar results have been obtained for thin-plate martensite.²⁵⁾

The θ of V1/V6 and V1/V17 are 0.55° and 0.01° whereas that of V1/V16 (butterfly-pairing) is 4.95°, more than one order of magnitude larger. Based on this result, one would expect that few butterfly-pairing forms just below the martensitic transformation start temperature (M_s) because a larger driving force is required for the formation of butterfly-pairing. However, the authors have found that formation frequency of V1/V16 pair just below the M_s is higher than that at 77 K in lenticular martensite.⁴⁾ Despite their unfavourable geometry, V1/V16 pairs occur with high frequency in the temperature range where a small driving forces is available. This fact suggests that origin of butterfly-pairing is differ from the other pairs.

There are also interesting experimental facts about V1/V16 pairing in lath martensite. The dominant variant pairing are V1/V2, V1/V4, and V1/V6.^6,8) The rotation Q for butterfly-pairing is about 5° in lath so that energy cost of butterfly-pairing is much higher. Butterfly-pairing is, therefore, considered to form in the late stage of the transformation where large driving force is available. However, similar to the experimental results for lenticular martensite, it has been reported that butterfly-pairing are frequently formed in the early stage of the transformation in lath as well. Hayashi et al.²⁶⁾ found that the size of martensitic plates forming butterfly-pairing is larger than that of lath. They have proposed a model in which butterfly-pairing is formed in the early stage of the transformation. Kohne et al.⁷⁾ investigated the change in preferential variant pairing tendency upon the progress of the transformation in high-carbon steels. They reported that V1/V16 pairs are preferentially formed in the early stage of the transformation. These results are consistent with those reported by Hayashi et al. As regards for lath martensite, the transformation process upon cooling in a low carbon steel was investigated by Nambu et al.²⁷⁾ using in-situ observation. Their results also suggest that V1/V16 pairs are formed in the early stage of the transformation. These results suggest that the geometrically unfavorable butterfly-pairing can frequently form also in lath martensite even when the driving force is low.

1.4. The Objective of this Study

Given the above backgrounds, the question arises as to whether small Q is really achieved in V1/V16 pair in alloys where butterfly martensite is said to occur predominantly, and whether butterfly-pairing is geometrically advantageous? In this study, the butterfly martensite microstructure in an Fe-18Ni-0.7Cr-0.5C (mass%) alloy was analyzed by electron microscopy and theoretical analysis using the PTMC and rank-1 connection.

2. Experimental Procedures

An Fe-18Ni-0.7Cr-0.5C alloy was prepared by vacuum melting followed by hot-rolling. Table 1 shows the results of the chemical composition analysis of the ingots. The hot-rolled ingots were sealed in quartz tubes under an Ar atmosphere, annealed at 1273 K for 3.6 ks and then quenched by breaking the quartz tubes in iced water. The martensitic transformation start temperature (M_s) was determined to be 233 K by differential scanning calorimetry (DSC). Martensitic transformation was thermally induced by sub-zero treatment in a temperature-controlled ethanol bath just below the M_s for 1 min, resulting in a microstructure composed of martensite and retained austenite.

Table 1. Chemical composition of the alloy.

C	Cr	Ni	P	S	N
0.5±0.005	0.70±0.01	18.00±0.02	<0.01	<0.001	<0.003

Specimens for observations were cut from the center of the sub-zero treated ingot to avoid the surface martensite. The surfaces of the specimens were mechanically polished with emery papers followed by polishing with colloidal silica. The lattice parameters were determined by X-ray diffractometry of the polished samples at room temperature to be a_γ = 359.13 ± 0.05 pm and a_M = c_M = 287.69 ± 0.21 pm. Electron backscattered diffraction (EBSD) analysis were performed at an acceleration voltage of 15 kV using a field-emission gun-type scanning electron microscope (FE-SEM, HITACHI SU5000). The step size in EBSD measurement was set to 0.5 μm. Crystallographic orientation maps were analyzed using TSL OIM Analysis 7.3 software. Variants in the maps were identified by comparing the martensite/austenite ORs based to the theoretical K-S OR which was calculated from the orientation of the retained austenite. Variants notation used is the KS variants shown in Table 2. Backscattered electron (BSE) observations were performed at an acceleration voltage of 15 kV to observe the morphological features of the martensite, using the BSE detector equipped with the FE-SEM.

Table 2. Notation of variants based on K-S ORs.⁶⁾

Variant	Plane parallel	Direction parallel	Bain group
V1	${\left(111\right)}_{\gamma }\parallel {\left(011\right)}_{{\alpha }^{\prime }}$	${\left[\overline{1}01\right]}_{\gamma }\parallel {\left[\overline{1}\overline{1}1\right]}_{{\alpha }^{\prime }}$	B1
V2		${\left[\overline{1}01\right]}_{\gamma }\parallel {\left[\overline{1}1\overline{1}\right]}_{{\alpha }^{\prime }}$	B2
V3		${\left[01\overline{1}\right]}_{\gamma }\parallel {\left[\overline{1}\overline{1}1\right]}_{{\alpha }^{\prime }}$	B3
V4		${\left[01\overline{1}\right]}_{\gamma }\parallel {\left[\overline{1}1\overline{1}\right]}_{{\alpha }^{\prime }}$	B1
V5		${\left[1\overline{1}0\right]}_{\gamma }\parallel {\left[\overline{1}\overline{1}1\right]}_{{\alpha }^{\prime }}$	B2
V6		${\left[1\overline{1}0\right]}_{\gamma }\parallel {\left[\overline{1}1\overline{1}\right]}_{{\alpha }^{\prime }}$	B3
V7	${\left(1\overline{1}1\right)}_{\gamma }\parallel {\left(011\right)}_{{\alpha }^{\prime }}$	${\left[10\overline{1}\right]}_{\gamma }\parallel {\left[\overline{1}\overline{1}1\right]}_{{\alpha }^{\prime }}$	B2
V8		${\left[10\overline{1}\right]}_{\gamma }\parallel {\left[\overline{1}1\overline{1}\right]}_{{\alpha }^{\prime }}$	B1
V9		${\left[\overline{1}\overline{1}0\right]}_{\gamma }\parallel {\left[\overline{1}\overline{1}1\right]}_{{\alpha }^{\prime }}$	B3
V10		${\left[\overline{1}\overline{1}0\right]}_{\gamma }\parallel {\left[\overline{1}1\overline{1}\right]}_{{\alpha }^{\prime }}$	B2
V11		${\left[011\right]}_{\gamma }\parallel {\left[\overline{1}\overline{1}1\right]}_{{\alpha }^{\prime }}$	B1
V12		${\left[011\right]}_{\gamma }\parallel {\left[\overline{1}1\overline{1}\right]}_{{\alpha }^{\prime }}$	B3
V13	${\left(\overline{1}11\right)}_{\gamma }\parallel {\left(011\right)}_{{\alpha }^{\prime }}$	${\left[0\overline{1}\overline{1}\right]}_{\gamma }\parallel {\left[\overline{1}\overline{1}1\right]}_{{\alpha }^{\prime }}$	B1
V14		${\left[0\overline{1}\overline{1}\right]}_{\gamma }\parallel {\left[\overline{1}1\overline{1}\right]}_{{\alpha }^{\prime }}$	B3
V15		${\left[\overline{1}0\overline{1}\right]}_{\gamma }\parallel {\left[\overline{1}\overline{1}1\right]}_{{\alpha }^{\prime }}$	B2
V16		${\left[\overline{1}0\overline{1}\right]}_{\gamma }\parallel {\left[\overline{1}1\overline{1}\right]}_{{\alpha }^{\prime }}$	B1
V17		${\left[110\right]}_{\gamma }\parallel {\left[\overline{1}\overline{1}1\right]}_{{\alpha }^{\prime }}$	B3
V18		${\left[110\right]}_{\gamma }\parallel {\left[\overline{1}1\overline{1}\right]}_{{\alpha }^{\prime }}$	B2
V19	${\left(11\overline{1}\right)}_{\gamma }\parallel {\left(011\right)}_{{\alpha }^{\prime }}$	${\left[\overline{1}10\right]}_{\gamma }\parallel {\left[\overline{1}\overline{1}1\right]}_{{\alpha }^{\prime }}$	B3
V20		${\left[\overline{1}10\right]}_{\gamma }\parallel {\left[\overline{1}1\overline{1}\right]}_{{\alpha }^{\prime }}$	B2
V21		${\left[0\overline{1}\overline{1}\right]}_{\gamma }\parallel {\left[\overline{1}\overline{1}1\right]}_{{\alpha }^{\prime }}$	B1
V22		${\left[0\overline{1}\overline{1}\right]}_{\gamma }\parallel {\left[\overline{1}1\overline{1}\right]}_{{\alpha }^{\prime }}$	B3
V23		${\left[101\right]}_{\gamma }\parallel {\left[\overline{1}\overline{1}1\right]}_{{\alpha }^{\prime }}$	B2
V24		${\left[101\right]}_{\gamma }\parallel {\left[\overline{1}1\overline{1}\right]}_{{\alpha }^{\prime }}$	B1

The formation frequency of variant pairing was evaluated by counting the number of variant boundaries. The reconstructed boundaries function in the OIM Analysis is used to define variant boundaries. If multiple variant boundaries were detected from a single variant pair, the longest one was analyzed. The detected variant pairs (Vi/ Vj) were converted into their equivalent V1/Vk pairs (i, j, k = 1–24).

The average normal of the habit and junction planes were determined by a method proposed by Okamoto et al.²⁸⁾ This method uses the results of single trace analysis for some variants and the least-squares method. Traces with lengths of less than 1 μm were excluded from the analysis because the direction of the trace could not be determined.

3. Results

3.1. Variant Map

Figures 1(a)–1(c) show variant maps in the [001]_γ‖ND, [011]_γ‖ND, [111]_γ‖ND prior γ grains. Retained austenite is shown in black on the variant maps. Insets show {001}_α′ pole figures obtained by EBSD and calculated position of {001}_α′ poles of K-S OR which were calculated from the retained austenite. The area of each measurement is (a) 163 μm × 262 μm, (b) 163 μm × 194 μm and (c) 303 μm × 310 μm. The experimentally and theoretically obtained {001}_α′ poles are in good agreement for all grains. Therefore, we judged that each variant holds K-S OR, and the variant maps were constructed as in Figs. 1(a)–1(c). The tolerance angle from K-S OR was defined within 5°. All 24 variants were confirmed in all the prior γ grains. The total numbers of detected martensite plates are 2399, 749 and 1734 in Figs. 1(a), 1(b) and 1(c), respectively. The prior γ grain sizes are (a) 200 μm, (b) 129 μm and (c) 274 μm respectively.

Fig. 1. Variant maps in (a) [001]_γ‖ND, (b) [011]_γ‖ND, (c) [111]_γ‖ND prior γ grains. Insets show {001}_α′ pole figures obtained by EBSD and calculated positions of {001}_α′ poles of K-S OR. Retained austenite is shown in black on the variant maps. The arrows in map indicate the typical butterfly-pairing.

V-shaped variant pairs were observed in Fig. 1 as indicated by the arrows; they are characteristic of butterfly-pairing. The formation frequency of variant pairing are shown in Fig. 2. The total numbers of the analyzed variant pairs are 2633, 650 and 1553 in Figs. 2(a), 2(b) and 2(c), respectively. All pairing are converted to V1/Vi pairs as explained in the section 2. V1/V2 and V1/V16 pairs formed frequently in all prior γ grains. Each V1/V2 and V1/V16 accounts for about 20% of all observed pairs (40% in total). V1/V2 pairs are known as a typical variant pair in lath martensite. The third most frequently formed pair was V1/V6 pair (about 10%).

Fig. 2. Formation frequency of variant pairing in (a) [001]_γ‖ND, (b) [011]_γ‖ND, (c) [111]_γ‖ND grains.

Figure 3 shows a BSE-SEM image of a typical butterfly-pairing and its schematic. The plane orientations in Fig. 3(b) show the plane traces calculated from the prior γ crystal orientation. The traces of (252)_γ, (252)_γ and (100)_γ were in good agreement with traces of the habit planes of V1 and V16, and their (100)_γ junction plane. These results are morphologically and crystallographically consistent with the previous report.⁵⁾

Fig. 3. (a) a BSE-SEM image of a typical butterfly-pairing and (b) its schematic.

3.2. Theoretical Analysis of Variant Pairing

The deformation gradient of the habit plane variant is required to analyze the geometry of variant pairing. In this study, lattice invariant deformations are assumed that reproduce the experimentally determined habit plane normal and K-S OR. Figure 4 shows the results of the habit plane determination in the experiments, which shows that the habit plane normal is well defined. The deviation angle Δθ between the experimental habit plane normal and the (252)_γ was evaluated. The values of Δθ for each prior γ grain were different, with a maximum value of Δθ = 6.7° and a minimum value of Δθ = 1.8°. From this result, it is considered that the habit plane in this study is consistent with {252}_γ, previously reported as the habit plane of butterfly martensite.⁵⁾

Fig. 4. Determination of the habit plane in experiment. (252)_γ (blue) and the average habit plane normal determined by single trace analysis (red) in (a) [001]_γ‖ND, (b) [011]_γ‖ND, (c) [111]_γ‖ND grains.

3.3. Deformation Gradient of the Habit Plane Variant

To evaluate the shape strain matrix reproducing the {252}_γ habit plane normal and the K-S OR, the double lattice invariant shear model proposed by Ross and Crocker¹⁸⁾ and by Kelly¹⁹⁾ were applied. Table 3 shows the parameters used in the calculations. A comparison of the theoretically reproduced and experimentally determined habit plane normal is shown in Fig. 5. The habit plane normal reproduced by using the Ross and Crocker model agrees with the experimental habit plane normal. This result indicates that the Ross and Crocker model better represents the shape strain of butterfly martensite than the Kelly model. Therefore, the Ross and Crocker model was adopted in the following analysis. Lattice invariant deformations were not identified in this study. However, as will become apparent in the later discussion, the geometry of the butterfly-pairing is insensitive to the lattice invariant deformation as long as the shape deformation is an invariant plane deformation.

Table 3. Crystallographic parameters of V1 derived from PTMC assuming double lattice invariant shear in the Fe-18Ni-0.7Cr-0.5C alloy.

Input	Lattice parameter (pm)	aγ = 359.14
		aα′ = cα′ = 287.17
		Kelly	Ross and Crocker
	Shear system of P1	$\left(112\right){\left[\overline{1}\overline{1}1\right]}_{{\alpha }^{\prime }}$	$\left(\overline{1}12\right){\left[1\overline{1}1\right]}_{{\alpha }^{\prime }}$
		$\left(=\left(101\right){\left[\overline{1}01\right]}_{\gamma }\right)$	$\left(=\left(011\right){\left[0\overline{1}1\right]}_{\gamma }\right)$
	Shear system of P2	$\left(110\right){\left[\overline{1}1\overline{1}\right]}_{{\alpha }^{\prime }}$	$\left(112\right){\left[\overline{1}\overline{1}1\right]}_{{\alpha }^{\prime }}$
		$\left(=\left(100\right){\left[01\overline{1}\right]}_{\gamma }\right)$	$\left(=\left(101\right){\left[\overline{1}01\right]}_{\gamma }\right)$
	Magnitude of P2	0.0832	0.2866
Output	Magnitude of P1	0.2726	0.0853
	Habit plane normal	(0.486, 0.726, 0.486)γ	(0.416, 0.808, 0.417)γ
	Shape deformation direction	[0.222, 0.720, 0.658]γ	[0.444, 0.710, 0.547]γ
	Magnitude of shape strain	0.238	0.140
	Misorientation angle from exact K-S OR	3.12°	2.30°

Fig. 5. (001)_γ standard stereographic projection showing habit plane normal by the PTMC assuming Kelly model (black) and Ross and Crocker (RC) model (gray), and experiment (white) for (a) [001]_γ‖ND (b) [011]_γ‖ND (c) [111]_γ‖ND grains.

3.4. Rank-1 Connection of Variant Pair

The incompatibility of the variant pairs was evaluated based on the rank-1 connection using the shape strain matrix obtained in the section 3.3. The magnitude of rotation by Q (θ), the rotation axis r, and the junction plane m required to satisfy the rank-1 connection for V1/Vk pairs are shown in Table 4. Pairs with no solution were denoted as “No sol.”. It should be noted that there are two solutions for a given variant pair.²⁹⁾ It was found that solutions are categorized into nine types (V1/V2, V4, V6, V7, V8, V16, V17, V21, and V24). Comparing θ of each type of solution, the pairs with the first and the second smallest θ are V1/V2 (θ = 0.01°) and V1/V17 (θ = 0.91°). Interestingly, the butterfly-pairing (V1/V16) observed as frequently as V1/V2 pairing has a larger θ of 6.15° in the present alloy. This large θ in the present alloy indicates that butterfly-pairing is not geometrically advantageous, even for alloys where butterfly martensite forms frequently.

Table 4. Nine types of solutions of rank-1 connection between V1 and the 23 other variants.

Ui	Uj	Rotation angle θ	Rotation axis r	Junction plane m
V1	V2	0.01°	${\left[\overline{0.503},\mathrm{\hspace{0.17em} }\overline{0.703},\mathrm{\hspace{0.17em} }\overline{0.503}\right]}_{\gamma }$	${\left(\overline{0.420},\mathrm{\hspace{0.17em} }\overline{0.805},\mathrm{\hspace{0.17em} }\overline{0.420}\right)}_{\gamma }$
		1.15°	${\left[\overline{0.571},\mathrm{\hspace{0.17em} }0.589,\mathrm{\hspace{0.17em} }\overline{0.571}\right]}_{\gamma }$	${\left(\overline{1}01\right)}_{\gamma }$
	V4	2.49°	${\left[\overline{0.669},\mathrm{\hspace{0.17em} }\overline{0.669},\mathrm{\hspace{0.17em} }\overline{0.325}\right]}_{\gamma }$	(0.685, 0.685, 0.250)γ
		12.65°	${\left[\overline{0.307},\mathrm{\hspace{0.17em} }\overline{0.307},\mathrm{\hspace{0.17em} }0.901\right]}_{\gamma }$	${\left(\overline{1}10\right)}_{\gamma }$
	V6	1.97°	[0.251, 0.685, 0.685]γ	${\left(\overline{0.301},\mathrm{\hspace{0.17em} }\overline{0.674},\mathrm{\hspace{0.17em} }\overline{0.674}\right)}_{\gamma }$
		13.82°	${\left[\overline{0.901},\mathrm{\hspace{0.17em} }0.306,\mathrm{\hspace{0.17em} }0.306\right]}_{\gamma }$	${\left(0\overline{1}1\right)}_{\gamma }$
	V7	7.16°	${\left[\overline{0.703,}\mathrm{\hspace{0.17em} }0.103,\mathrm{\hspace{0.17em} }0.703\right]}_{\gamma }$	${\left(0.333,\mathrm{\hspace{0.17em} }\overline{0.882},\mathrm{\hspace{0.17em} }\overline{0.333}\right)}_{\gamma }$
		8.40°	${\left[\overline{0.707},\mathrm{\hspace{0.17em} }\overline{0.001},\mathrm{\hspace{0.17em} }0.707\right]}_{\gamma }$	(101)γ
	V8	7.12°	${\left[\overline{0.708,}\mathrm{\hspace{0.17em} }0.000,\mathrm{\hspace{0.17em} }0.707\right]}_{\gamma }$	${\left(0\overline{1}0\right)}_{\gamma }$
		8.45°	${\left[\overline{0.777},\mathrm{\hspace{0.17em} }0.000,\mathrm{\hspace{0.17em} }0.630\right]}_{\gamma }$	(0.158, 0.000, 0.987)γ
	V16	6.15°	[0.000, 0.611, 0.792]γ	(0.000, 0.041, 0.999)γ
		6.18°	${\left[0.000,\mathrm{\hspace{0.17em} }\overline{0.459,}\mathrm{\hspace{0.17em} }0.889\right]}_{\gamma }$	${\left(\overline{1}00\right)}_{\gamma }$
	V17	0.91°	${\left[0.553,\mathrm{\hspace{0.17em} }\overline{0.589},\mathrm{\hspace{0.17em} }0.589\right]}_{\gamma }$	(011)γ
		13.57°	${\left[\overline{0.895},\mathrm{\hspace{0.17em} }\overline{0.316},\mathrm{\hspace{0.17em} }0.316\right]}_{\gamma }$	${\left(0.324,\mathrm{\hspace{0.17em} }\overline{0.669},\mathrm{\hspace{0.17em} }0.669\right)}_{\gamma }$
	V21	1.51°	${\left[\overline{0.589},\mathrm{\hspace{0.17em} }0.589,\mathrm{\hspace{0.17em} }\overline{0.553}\right]}_{\gamma }$	(110)γ
		13.29°	${\left[\overline{0.394},\mathrm{\hspace{0.17em} }0.394,\mathrm{\hspace{0.17em} }0.830\right]}_{\gamma }$	${\left(0.648,\mathrm{\hspace{0.17em} }\overline{0.648},\mathrm{\hspace{0.17em} }0.399\right)}_{\gamma }$
	V24	5.79°	${\left[\overline{0.848},\mathrm{\hspace{0.17em} }\overline{0.530},\mathrm{\hspace{0.17em} }0.000\right]}_{\gamma }$	${\left(\overline{0.969},\mathrm{\hspace{0.17em} }\overline{0.247},\mathrm{\hspace{0.17em} }0.000\right)}_{\gamma }$
		7.57°	${\left[\overline{0.889},\mathrm{\hspace{0.17em} }0.458,\mathrm{\hspace{0.17em} }0.000\right]}_{\gamma }$	(001)γ
	Other variants	No sol.	No sol.	No sol.

3.5. Comparison with Rank-1 Connection at the Junction Plane

A comparison between the experimental and theoretical junction planes was made to confirm the validity of the theoretical analysis. The junction plane normal of the preferentially formed V1/V2, V1/V6 and V1/V16 pairs were determined by single trace analysis. Figure 6 shows the results for [001]_γ‖ND grain shown in Fig. 1(a). The red and light blue circles indicate the junction plane normal corresponding to the two solutions of θ. The deviation angle Δθ between the experimental and theoretical junction planes was calculated using single trace analysis and is summarized in Table 5. One of these theoretical junction planes agrees with the experimental junction plane with an error of at the most 6.4°. This result reiterates that the habit plane variant observed in this study can be considered to have lattice invariant deformation and shape strain following Ross and Crocker’s model, and also indicates that the junction planes of variants maintain the rank-1 connection. In contrast, the theoretical junction plane corresponding to the other solution is perpendicular to the experimental one. This means that only one of the geometrically possible variant pairing appears in the actual phase transformation for the butterfly-pairing. This trend is reported also in shape memory alloys and explained by the elastic strain energy associated with pairing.^30,31)

Fig. 6. Determination of the junction plane in experiment. Theoretical junction plane normal (red and light blue, corresponding to solutions with large and small rotation angles θ, respectively) and the average junction plane normal determined by single trace analysis (yellow) in [001]_γ‖ND grain for (a) V1/V2, (b) V1/V6, (c) V1/V16.

Table 5. Deviation angle Δθ between the experimental and theoretical junction planes.

Ui	Uj	Rotation angle θ	Rotation axis r	Deviation angle Δθ between the experimental and theoretical junction planes
				[001]γ‖ND	[011]γ‖ND	[111]γ‖ND
V1	V2	0.01°	${\left[\overline{0.503},\mathrm{\hspace{0.17em} }\overline{0.703},\mathrm{\hspace{0.17em} }\overline{0.503}\right]}_{\gamma }$	2.4°	0.5°	2.9°
		1.15°	${\left[\overline{0.571},\mathrm{\hspace{0.17em} }0.589,\mathrm{\hspace{0.17em} }\overline{0.571}\right]}_{\gamma }$	90.0°	90.0°	90.0°
	V6	1.97°	[0.251, 0.685, 0.685]γ	6.4°	1.2°	3.5°
		13.82°	${\left[\overline{0.901},\mathrm{\hspace{0.17em} }0.306,\mathrm{\hspace{0.17em} }0.306\right]}_{\gamma }$	90.0°	90.0°	90.0°
	V16	6.15°	[0, 0.611, 0.792]γ	90.0°	90.0°	90.0°
		6.18°	${\left[0,\mathrm{\hspace{0.17em} }\overline{0.459},\mathrm{\hspace{0.17em} }0.889\right]}_{\gamma }$	0.0°	0.0°	0.0°

3.6. Morphologies of Variant Pair

Morphologies of V1/V2 and V1/V16 pair satisfying the rank-1 connection are schematically shown in Fig. 7. Shear planes and shear directions of the first and the second lattice invariant shear, parallel close-packed planes, parallel close-packed directions are shown in the light blue plane and black lines in Fig. 7, respectively. The junction plane of V1/V2 pair has a near-parallel relationship with the habit plane, the parallel close-packed direction and the second shear direction, respectively. All crystallographic features of V1/V16 pair have mirror-symmetry across the junction plane. In particular, two variants have the same first shear plane and first shear direction.

Fig. 7. Schematic diagram of morphologies for (a) V1/V2 and (b) V1/V16. Shear planes and shear directions of the first and the second shear, parallel close-packed plane, parallel close-packed direction are shown in the light blue planes and black lines. V1 and V16 have the same first shear plane and direction, (011)[0$\overline{1}$1]_γ.

4. Discussion

4.1. Junction Plane Normal of Butterfly-pairing

The junction plane normal in butterfly-pairing (V1/V16) is (100)_γ as shown in Fig. 6(c); it is consistent with the previous report.⁵⁾ Generally, the total shape strain changes as the martensite morphology changes due to the change in the mode of lattice invariant deformation. It is natural that the junction plane normal, which should satisfy the rank-1 connection, also changes as the shape strain changes. However, the junction plane normal of V1/V16 pair is (100)_γ in both lenticular and thin-plate martensite.^3,4) This fact implies that (100)_γ junction plane of V1/V16 pair is universal independent of the total shape strain (i.e. lattice invariant deformation). Therefore, we discuss below why the junction plane normal of V1/V16 pair is always (100)_γ in terms of the geometry at the junction plane.

According to Eq. (2), the shape strain matrix for V1, U₁, is expressed as:

  

$${\mathbf{U}}_1=\mathbf{I}+g\left(\mathbf{d}\otimes \mathbf{p}\right)=\left(\begin{array}{ccc}1+g{d}_1{p}_1& g{d}_1{p}_2& g{d}_1{p}_3\\ g{d}_2{p}_1& 1+g{d}_2{p}_2& g{d}_2{p}_3\\ g{d}_3{p}_1& g{d}_3{p}_2& 1+g{d}_3{p}_3\end{array}\right)$$(4)

where d=(d₁,d₂,d₃)^T and p=(p₁,p₂,p₃)^T.

Based on the symmetry between V1 and V16, the d and p components of V16 are expressed as d=(d₁,−d₂,−d₃)^T and p=(p₁,−p₂,−p₃)^T, hence the shape strain matrix for V16, U₁₆, is expressed as follows:

  

$${\mathbf{U}}_{16}=\left(\begin{array}{ccc}1+g{d}_1{p}_1& -g{d}_1{p}_2& -g{d}_1{p}_3\\ -g{d}_2{p}_1& 1+g{d}_2{p}_2& g{d}_2{p}_3\\ -g{d}_3{p}_1& g{d}_3{p}_2& 1+g{d}_3{p}_3\end{array}\right)$$(5)

In summary, U₁₆ is easily expressed as U₁₆=CU₁C where C is the 180° rotation around ê=(100)^T. Then, the rank-1 connection of V1 and V16 are expressed as follows using a rotation matrix Q∈SO(3) and vectors b∈${ℝ}^3$ and ê following the proposition 2.2 by Bhattacharya:³²⁾

  

$$\mathbf{Q}{\mathbf{U}}_{16}-{\mathbf{U}}_1=\mathbf{b}\otimes \hat{\mathbf{e}}$$(6)

Therefore, if the shape strain is an invariant plane deformation, then V1/V16 pair always has a solution (Q, b, m) for the rank-1 connection, and one of the two solutions has the junction plane normal of (100)_γ. In other words, the junction plane of butterfly-pairing is always (100)_γ even if the lattice invariant deformation changes. It is a geometrical consequence that the junction plane of butterfly-pairing is (100)_γ independent of the alloy composition.

The rotation Q can be obtained analytically as follows. As a special case, Q = I if

  

$$d_1=0$$(7)

or

  

$$p_2=p_3=0$$(8)

In general, Q ≠ I and the rotation axis r and rotation angle θ of Q are given follows.

  

$$\mathbf{r}=\frac{1}{\sqrt{p_2^2+p_3^2}}{\left(0,p_3,-p_2\right)}^{\text{T}}$$(9)

  

$$cos\theta =1-\frac{2g^2{d}_1^2\left(p_2^2+p_3^2\right)}{{\left(1+g\left(d_2{p}_2+d_3{p}_3\right)\right)}^2+g^2{d}_1^2\left(p_2^2+p_3^2\right)}$$(10)

  

$$sin\theta =\frac{2g{d}_1\left(1+g\left(d_2{p}_2+d_3{p}_3\right)\right)\sqrt{p_2^2+p_3^2}}{{\left(1+g\left(d_2{p}_2+d_3{p}_3\right)\right)}^2+g^2{d}_1^2\left(p_2^2+p_3^2\right)}$$(11)

These results show that the rotation of variants to form butterfly-pairing depends on the total shape strain, whereas the junction plane is always (100)_γ.

4.2. Geometric Perspective on Variant Pairing

A comparison of the frequency of variant pairing in the present alloy and lenticular martensite is shown in Fig. 8. The angles in Fig. 8 indicate θ for the pairing. In the present alloy, V1/V16 (butterfly-pairing) and V1/V2 (typical pair in lath martensite) were frequently observed.

Fig. 8. Frequency of variant pairing observed in (a) butterfly martensite (butterfly M) and (b) lenticular martensite (lenticular M).⁴⁾

V1/V2 pairs were rarely observed in lenticular martensite, whereas they were observed with the highest frequency in butterfly martensite. The θ of V1/V2 pair in the present alloy is 0.01°, which is very small compared to that of lenticular martensite (θ = 7.8°). It is deduced that the smaller the incompatibility at the junction plane, the less energy is required to form a pair. Therefore, pairs with smaller rotation Q (i.e., V1/V2) are preferentially formed in the present alloy. The θ of V1/V2 pair in the present alloy is smaller than that of in the lenticular martensite due to the difference in the mode of lattice invariant deformation.¹⁷⁾

The second most frequent pairing in the present alloy is V1/V16. However, the θ of V1/V16 pair is more than five orders larger than that of V1/V2 pair. The reason for V1/V16 formation in this alloy cannot be explained in terms of incompatibility at the junction plane. And also, V1/V16 pairing is not an advantageous pair to form in terms of self-accommodation because V1 and V16 belong to the same Bain group. Why are V1/V16 pairs nevertheless preferentially formed?

The first possibility is that the lattice invariant deformations adopted in the theory differ from actually occurred ones. The θ of V1/V16 pair can be obtained from Eq. (10) or Eq. (11). Table 6 shows the rotation angle θ calculated from several PTMC models which can explain the habit plane of butterfly martensite (Single shear, Ross and Crocker, Kelly), and experimental measurement using an Fe-7.9Cr-1.11C alloy which exhibits {252}_γ habit plane.³³⁾ The θ of V1/V16 in all cases show values in the vicinity of 5°. Therefore, the incompatibility at the junction plane of V1/V16 is always large, and the reason for the preferential formation of V1/V16 pairs cannot be explained even if another lattice invariant deformation is applied to the present analysis.

Table 6. Rotation angle θ of V1/V16 calculated from shape strain, habit plane, shape deformation direction.

	Lattice invariant shear system		Magnitude of shape strain	Habit plane	Shape deformation direction	Rotation angle θ
	P1	P2
Single shear	$\left(112\right){\left[\overline{1}\overline{1}1\right]}_{{\alpha }^{\prime }}$	–	0.227	(0.181, 0.777, 0.602)γ	${\left[\overline{0.205},\mathrm{\hspace{0.17em} }0.703,\mathrm{\hspace{0.17em} }\overline{0.681}\right]}_{\gamma }$	5.1°
Ross and Crocker	$\left(\overline{1}12\right){\left[\overline{1}\overline{1}1\right]}_{{\alpha }^{\prime }}$	$\left(112\right){\left[\overline{1}\overline{1}1\right]}_{{\alpha }^{\prime }}$	0.140	(0.416, 0.808, 0.417)γ	${\left[\overline{0.444},\mathrm{\hspace{0.17em} }0.710,\mathrm{\hspace{0.17em} }\overline{0.547}\right]}_{\gamma }$	6.2°
Kelly	$\left(112\right){\left[\overline{1}\overline{1}1\right]}_{{\alpha }^{\prime }}$	$\left(110\right){\left[\overline{1}1\overline{1}\right]}_{{\alpha }^{\prime }}$	0.238	(0.486, 0.726, 0.486)γ	${\left[\overline{0.222},\mathrm{\hspace{0.17em} }0.720,\mathrm{\hspace{0.17em} }\overline{0.658}\right]}_{\gamma }$	5.1°
	$\left(112\right){\left[\overline{1}\overline{1}1\right]}_{{\alpha }^{\prime }}$	$\left(21\overline{1}\right){\left[\overline{1}1\overline{1}\right]}_{{\alpha }^{\prime }}$	0.205	(0.470, 0.712, 0.470)γ	${\left[\overline{0.240},\mathrm{\hspace{0.17em} }0.712,\mathrm{\hspace{0.17em} }\overline{0.660}\right]}_{\gamma }$	4.8°
	$\left(112\right){\left[\overline{1}\overline{1}1\right]}_{{\alpha }^{\prime }}$	$\left(121\right){\left[\overline{1}1\overline{1}\right]}_{{\alpha }^{\prime }}$	0.306	(0.508, 0.695, 0.508)γ	${\left[\overline{0.198},\mathrm{\hspace{0.17em} }0.729,\mathrm{\hspace{0.17em} }\overline{0.655}\right]}_{\gamma }$	5.7°
Fe-7.9Cr-1.11C32)	–	–	0.184	(0.358, 0.853, 0.379)γ	${\left[\overline{0.264},\mathrm{\hspace{0.17em} }0.684,\mathrm{\hspace{0.17em} }\overline{0.680}\right]}_{\gamma }$	4.9°

4.3. Formation Processes of Butterfly-pairing

Finally, we present a possible reason why V1/V16 pair is formed for the case where martensite plate follows Ross-Crocker model as supported by the present experimental results. As summarized in Fig. 7, V1 and V16 have the same P₁ so that V16 would be easily branched from V1 if the P₁ of V1 occurs in the austenite at the growing edge of V1. In cases where P₁ has already occurred in the austenite at the tip of V1, the energy required for nucleation of V16 is reduced because V16 can form without causing P₁ itself. A schematic of this V1/V16 formation is shown in Fig. 9. Similar consideration has been given by Takayama et al.³⁴⁾ and Iwashita et al.³⁵⁾ for V1/V4 pair in lath martensite.

Fig. 9. Schematic illustration of the formation of V1/V16 pair. (a) V1 plate grow and the first lattice invariant shear P₁ occurs in austenite around the growth tip. (b) V16 plate grow from the tip area with P₁. The solid arrows indicate the direction of the growth of martensite plate.

5. Conclusion

The variant-pairs in an Fe-18Ni-0.7Cr-0.5C alloy were investigated experimentally and theoretically to reveal the nature of the butterfly-type martensite and following conclusions were obtained.

(1) Martensite plates in this alloy exhibits {252}_γ habit plane and K-S OR and are in good agreement with the double shear PTMC by Ross and Crocker.

(2) The frequency of formation of V1/V2 and V1/V16 (butterflies) pairs was 20% each, with these two pairs alone accounting for 40% of the total. The third most preferential pair was V1/V6 (about 10%).

(3) The experimentally determined junction planes of the variant pairs formed preferentially (V1/V2, V1/V6 and V1/V16) were consistent with the theoretical ones derived from the rank-1 connection.

(4) θ (magnitude of rotation Q) for V1/V2, V1/V6 and V1/V16 pair were 0.01°, 1.97° and 6.15°, respectively. Butterfly-pairing was not a geometrically advantageous morphology, even for alloys in which butterfly martensite forms frequently.

(5) V1/V16 pair always has a solution (Q, b, m) for the rank-1 connection when the total shape strain of martensite is an invariant plane deformation. One of the junction planes of V1/V16 is always (100)_γ regardless of the mode of the lattice invariant deformation. It is therefore considered that even if the martensite morphology changes, the junction plane of the butterfly-pairing keeps (100)_γ.

(6) V16 can be easily nucleated by branching from the growing tips of V1 because one of the lattice invariant shear (P₁) is common in V1 and V16.

Acknowledgements

The alloy ingot was provided by the Nippon Steel Corporation. The authors thank Dr. Yoshihiro Suwa (Nippon Steel Corporation) for his valuable advice and discussion. This work was supported by Nippon Steel Corporation, Proterial Materials Science Foundation, JST SPRING (Grant Number JPMJSP2106), and JSPS KAKENHI (Grant Number 20K15046, 21H04613).

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