Shape-Memory Effect and the Topology of Minimal Surfaces

Abstract

Martensitic transformations, viewed as continuous mappings between triply periodic minimal surfaces (TPMSs), as suggested by Hyde and Andersson (Z. Kristallogr. 1986, 174, 225–236), are extended to include paths between the initial and final phases. Reversible transformations, which correspond to shape-memory materials, occur only if lattice points remain at flat points on a TPMS throughout a continuous transformation. For the shape-memory material NiTi, the density functional calculations by Hatcher et al. [Phys. Rev. B 2009, 80, 144203] yield irreversible and reversible paths with and without energy barriers, respectively, in agreement with our theory. Although there are TPMSs for face-centered and body-centered cubic crystals for iron, the deformation between them is not reversible, in agreement with the non-vanishing energy barriers obtained from the density functional calculations of Zhang et al. (RSC Advances 2021, 11, 3043–3048).

1. Introduction

Martensitic transformations are diffusionless solid-to-solid transformations triggered by the rapid collective movement of atoms across distances smaller than a lattice spacing [1,2,3]. This transformation is abrupt, showing a structural discontinuity at a particular temperature, and it is often accompanied by changes in physical properties, which can be exploited for applications. Materials exhibiting martensitic transformations include metals and alloys [2], ceramics [4], and biological systems [5].

The shape-memory effect (SME) [3,6], where a material returns to its original shape when heated after a plastic deformation, accompanies some martensitic transformations. Shape-memory materials have been known since the 1930s [7], but studies of this effect began in earnest only in the 1960s, with the discovery of NiTi-based materials [8,9]. Ni-Ti alloys are the most widely used shape-memory and superelastic alloys, combining a pronounced SME with superelasticity, corrosion resistance, biocompatibility, and superior engineering properties. Shape-memory alloys have various practical applications [10]. For example, the shape-memory alloy NiTi is used in pipe coupling [6], orthodontic treatment devices [11], in the fabrication of self-expandable vascular stents [12,13], and various venous filters [14].

Here, we study martensitic transformations from a geometric and topological perspective. The martensitic transformation is interpreted as a continuous deformation between minimal surfaces, and it includes information about the relevant symmetry and topological properties of the two end states. We build on the approach of Hyde and Andersson [15], to develop a theory of martensitic transformations that encompasses not just the end points, as in their original work, but also the path between the initial and transformed states. Transformation paths that not only maintain the topological properties of the end states, but remain as triply periodic minimal surfaces over the path, are reversible and are, therefore, expected to exhibit the shape-memory effect. Our approach is consistent with available density functional calculations along martensitic transformation paths and provides a topological distinction between reversible and irreversible transformations.

2. Minimal Surfaces

Surfaces that minimize their area for given boundary conditions are called minimal. Determining the surface with the least area that spans a boundary curve was first formulated by Lagrange as a variational problem. He showed that the equation for a minimal surface must satisfy what is now known as the Euler–Lagrange equation. In geometrical terms, the local shape near a point of a three-dimensional smooth surface is determined by the principal curvatures $k_1$ and $k_2$ measuring the maximum and minimum bending near that point. The surface is characterized by the mean curvature $M=-{\displaystyle \frac{1}{2}}(k_1+k_2)$ and the Gaussian curvature $K=k_1{k}_2$ [16,17]. A surface with $K=0$ everywhere is flat, e.g., a plane. A surface with $M=0$ everywhere is locally area-minimizing and is called minimal. The catenoid (the surface of revolution of a catenary), the (circular) helicoid (whose boundary is a circular helix), and the plane are standard examples of minimal surfaces.

The Bonnet transformation [18] establishes a mapping between minimal surfaces. This transformation is continuous, preserves the mean and Gaussian curvatures at every point on the surface, and is an isometry of the surface: that is, all distances within the surface are preserved, as there is no stretching or wrinkling. The Bonnet transformation is the weighted sum of two minimal surfaces parametrized by the Bonnet rotation angle. As the rotation angle increases, a continuous family of minimal surfaces is generated.

Minimal surfaces with repeating units in three independent directions are called triply periodic minimal surfaces (TPMSs). Minimal surfaces can have points where $K=0$; these are called flat points, and they will play a central role in what follows. The flat points on the minimal surfaces considered here are isolated. The symmetry of each TPMS is a space group, just as for ordinary crystals [19]. The first TPMSs were discovered by Schwarz [20], including the primitive (P), diamond (D), and hexagonal (H) surfaces (Figure 1). Schoen [21,22] discovered the gyroid (G), which is obtained when deforming P to D surfaces along the Bonnet path and is locally isometric to both the P and D surfaces [21,23]. Chen and Weber [24] discovered that the P surface can deform into the H surface within a space of TPMSs; the intermediate surfaces are in the $oPa$ and $oPb$ families. Surfaces in the $oPa$ family can be obtained by stretching or compressing the P surface along its three necks [Figure 2a,b], while those in the $oPb$ family involve displacements of necks and holes on the surface [Figure 2c,d]:

Since the point group of a lattice is a (proper or improper) subgroup of the point group of its TPMS [25], we require the point group of the crystal to be a subgroup (proper or improper) of that of the TPMS for the two end states. There is also a group–subgroup relation between the point group of the crystal and its corresponding Bravais lattice. Thus, the TPMS can be understood as the Bravais lattice that gives the basic symmetry and structural information of this crystal. Now, a topological transformation between two surfaces is bicontinuous (continuous in both directions), one-to-one, and reversible. Some properties are preserved under topological transformations and others are not. The genus of a surface is its topological type. Roughly speaking, the genus is the number of holes in a surface. Thus, surfaces with the same genus are topologically equivalent to each other, but surfaces with a different genus are not topologically equivalent.

The transformation between body-centered cubic (BCC) and face-centered cubic (FCC) lattices is not a topologically permitted martensitic phase transformation, because the genus of the corresponding TPMS is 4 for BCC but is 6 for FCC [26]. Hence, there cannot be a topological transformation between these lattices. However, the hexagonal close-packed (HCP)–BCC, HCP–hexagonal transformation in Zr, the BCC–HCP transformation in Na, and the B2–B19′ transformation in NiTi are topologically permitted to be martensitic.

3. Deformation Paths of Martensitic Transformations

Explanations for BCC–FCC (and reverse) transformation include the celebrated Bain tetragonal distortion [27], the Kurdjumow–Sachs (KS) [28] and Nishiyama–Wassermann (NW) [29,30] shear models, together with the Bogers–Burgers (BB) [31] and the Olson–Cohen (OC) [32] hard-sphere models. The geometries for these paths are shown in Figure 3. The Bain path [Figure 3a,b] is based on a deformation along $\langle 001\rangle$, which yields a body-centered tetragonal (BCT) structure. The BCT unit cell is compressed by about 21% along the z-direction and expanded by about 12% along the x- and y-directions [33].

Despite its simplicity and intuitive appeal, the orientation relationship between the parent (FCC) and daughter (BCC) phases of the Bain path has not been observed in experiments. The NW path [Figure 3c,d] begins with a shear that produces a monoclinic unit cell. Along this path, the unit cell changes from orthogonal to monoclinic and back to orthogonal. Computational studies [33,34] indicate this path is favored over the Bain path, though there is still a barrier for the transformation. The KS shear model [Figure 3e,f], similar to the NW model, starts with a shear that produces a monoclinic cell, but has different orientational relationships. The BB (OC) hard-sphere model indicates the FCC–BCC lattice transformation is realized by a shear of type $\langle 112\rangle \left\{111\right\}$ [31,32].

In non-magnetic materials, these deformation paths are not topologically continuous. Take the Bain deformation as an example [Figure 4a]. Each lattice point in a BCC lattice has 8 nearest neighbors while in an FCC lattice each lattice point has 12 nearest neighbors. Here, the lattice points are equivalent points in a Bravais lattice. The crystal structure of an elemental or compound material is obtained by assigning a basis of one or more atoms to each lattice point. For instance, each lattice point in BCC Fe is conventionally chosen to be occupied by a Fe atom. However, for BCC NiTi, the basis of each lattice point consists of one Ni atom and one Ti atom. In an FCC lattice, the length of the green bond line ($l_{\mathrm{g}}$) and the length of the cyan bond line ($l_{\mathrm{c}}$) are equal, while in a BCC lattice they are not. When lattice parameters satisfy $a=b>c$, we have $l_{\mathrm{g}}>l_{\mathrm{c}}$ and some bond lines connecting the lattice point A will “break”, which leads to a decrease in the number of bond lines connecting one lattice point to its nearest neighbors from 12 (FCC) to 8 (BCC). Similar arguments can be made for the KS, NW, and BB (OC) paths [Figure 4b–d].

4. Density Functional Calculations of Martensitic Transformation Paths

Table 1. Energy barriers calculated along different transformation paths for the given materials. All shears are given in the coordinate system of the B2 phase of NiTi. $\Delta E$ is expressed in units of meV/atom. No magnetism but structure transformation was considered in these calculations.

Material	Initial	Path	Final	$\mathit{\Delta }\mathit{E}$
NiTi [35]	B2	$\langle 100\rangle \left\{011\right\}$ basal shear	B19′	∼37
	B2	$\langle 100\rangle \left\{011\right\}$ bilayer shear	B19′	∼22
	B2	109°–B19′	B19′	0
NiTi [36]	B2	[111] elongation	R	0
NiTi [35,37,38,39]	B2	$\langle 100\rangle \left\{011\right\}$ alternate bilayer shear	B33	0
NiTi [40]	B2	$\langle \overline{1}10\rangle \left\{110\right\}$ basal shear	B19	∼13
Ti [41]	HCP($\alpha$)	${\left[0001\right]}_{\alpha }\left|\right|{\left[01\overline{2}0\right]}_{\omega }$,${\left\{11\overline{2}0\right\}}_{\alpha }\left|\right|{\left\{0001\right\}}_{\omega }$	Hexagonal ($\omega$)	∼9
Ti [42]	HCP($\alpha$)	${\left[11\overline{2}0\right]}_{\alpha }\left|\right|{\left[01\overline{1}1\right]}_{\omega }$,${\left(0001\right)}_{\alpha }\left|\right|{\left(0\overline{1}11\right)}_{\omega }$	Hexagonal ($\omega$)	∼9.5
Zr [43]	HCP($\alpha$)	${\left[11\overline{2}0\right]}_{\alpha }\left|\right|{\left[01\overline{1}1\right]}_{\omega }$,${\left(0001\right)}_{\alpha }\left|\right|{\left(0\overline{1}11\right)}_{\omega }$	Hexagonal ($\omega$)	22
Na [44]	BCC	Bain	FCC	∼0.79
Fe [34,45]	$\gamma$-Fe	Bain (FCT)	$\alpha$-Fe	∼20
	$\gamma$–Fe	Bain	$\alpha$–Fe	∼45
Fe [34]	$\gamma$–Fe	NW (FCT)	$\alpha$–Fe	∼37
Fe [46]	$\gamma$–Fe	BB	$\alpha$–Fe	∼25
Cu [47]	BCT(BCC)	Bain	FCC	1
Cu [48]	BCT(BCC)	Bain	FCC	1.3
Cu [49]	FCC	BB	BCC	∼700
Al [50]	BCT(BCC)	Bain	FCC	∼30
Al [50]	FCC	BB	BCC	∼100

compiles energy barriers calculated with the density functional theory (DFT) of materials undergoing martensitic transformations along several paths. The calculations for Fe did not account for magnetism. We can see that paths involving Bain/NW/BB deformations, which are not topologically continuous, have a non-vanishing energy barrier. Only shape-memory alloys have barrierless paths that reside in a space of TPMSs. However, the austenite (B2) phase of NiTi has the space group $Pm\overline{3}m$. After a rapid quench, NiTi transforms to martensite. We can locate the Bravais lattice of B2 NiTi on a P surface, as shown in Figure 5.

DFT calculations [35] also show that energy barriers vary with the paths of the B2→B19′ NiTi martensitic transformation, with the energy barrier going to zero when a path meets our geometrical criteria. The paths involving only the monolayer (resp., bilayer) $\langle 100\rangle \left\{011\right\}$ shear/shuffle have an energy barrier up to 37 meV/atom (resp., 22 meV/atom), while the path that admits the intermediate phase 109°–B19′, allowing both shear and structural relaxation, has no barrier, with the energy per atom decreasing throughout the transformation [35]. As shown in Figure 6b, a single-layer $\langle 100\rangle \left\{011\right\}$ shear requires $l_1/l_2=1.40$ and $\gamma ={98}^{\circ }$.

We will report elsewhere that the end phases of a martensitic phase transformation yield Bravais lattices whose lattice sites are located at flat points on a TPMS, such that (i) the lattice structure and periodicity are preserved, (ii) the point group of the crystal is a (proper or improper) subgroup of the point group of the TPMS. The lattice parameters used to find the TPMS corresponding to NiTi in different phases are given in Appendix B. The TPMS corresponding to the phase under a single-layer $\langle 100\rangle \left\{011\right\}$ shear closest to the requirements given by Figure 5b is shown in Figure 6d, in which $l_1^{\prime }/l_2^{\prime }=1.40$ and $\gamma ={98}^{\circ }$. However, in Figure 6b, ${\gamma }_2=141.{89}^{\circ }$ and ${\gamma }_3={98}^{\circ }$, while in Figure 6d, ${\gamma }_2=125.{09}^{\circ }$ and ${\gamma }_3=112.{86}^{\circ }$. Similarly, for the bilayer $\langle 100\rangle \left\{011\right\}$ shear, as shown in Figure 6c, $l_1/l_2=0.70$ and $gamma={98}^{\circ }$. The TPMS closest to this requirement is given in Figure 6e, with $l_1^{\prime }/l_2^{\prime }=0.70$ and $\gamma ={\gamma }_3={98}^{\circ }$. However, ${\gamma }_2=112.{86}^{\circ }$, which disagrees with the requirement given by Figure 6d that ${\gamma }_2=109.{47}^{\circ }$. Thus, the TPMS given in Figure 6e cannot describe the crystal lattice shown in Figure 6d. For another choice of lattice points, as shown in Figure 6f, the conditions $l_1^{\prime }/l_2^{\prime }=0.7$ and $\gamma ={98}^{\circ }$ cannot be satisfied simultaneously. Therefore, none of the single-layer or bilayer $\langle 100\rangle \left\{011\right\}$ shears can reside on a TPMS.

However, for the path that allows the occurrence of both the bilayer and single-layer $\langle 100\rangle \left\{011\right\}$ shear with an intermediate 109°–B19′ phase, as shown in Figure 7a, we can find a TPMS belonging to the oPb family [Figure 7c] on which this phase can reside. Both the 109–B19′ phase and the B19′ phase [Figure 7b,d] can reside on a surface belonging to the $oPb$ family, and the shear mechanism of this path is compatible with the continuous deformation within a space of TPMSs formed by surfaces belonging to the $oPb$ family. According to Table 1, this differs from the two paths mentioned above, so this path is barrierless.

We see similar arguments can be made for the B2–R phase transformation via the ${\left[111\right]}_{\mathrm{B}2}$ elongation. As shown in Figure 8, we can find the Bravais lattice of the R phase of NiTi described by an H surface. We can see the ${\left[111\right]}_{\mathrm{B}2}$ elongation can be viewed as a combination as an extension of the lattice along ${\left[110\right]}_{\mathrm{B}2}$ and an extension along ${\left[001\right]}_{\mathrm{B}2}$. This deformation conforms to the deformation from the P surface to the H surface within a space of the $oPb$ family. This path is also barrierless. The B2–B33 transformation via an alternate bilayer $\langle 100\rangle \left[011\right]$ shear and the B2–B19 transformation via a $\langle \overline{1}10\rangle \left\{110\right\}$ basal shear are shown in Figure 9 and Figure 10. It can be seen that the B2–B33 path conforms to the surface deformation within the $oPa$ family while the B2–B19 path does not, and we see the B2–B33 path is barrierless while the B2–B19 path is not.

The details of the atomic-scale mechanisms responsible for martensitic transformations are important for a quantitative understanding of the transformation processes, including the calculation of energy barriers. Our approach, based on the transformations between minimal surfaces, provides a conceptually simple description based on geometrical and topological principles. As with transformation paths, such as proposed by Bain and NW, and according to the group–subgroup relation [51], no attempt is made to model the temperature-driven nucleation process, just to establish a relationship between the initial and final phases and, in our case, the intermediate states as well. We are looking at the change of crystal structures during the stress-driven diffusionless transformations that are determined by the energy barrier calculated by DFT.

5. Summary and Conclusions

We have studied the martensitic transformation and the shape-memory effect from the perspective of the topology and differential geometry of TPMSs. We propose that martensitic transformations can be understood as continuous deformations between TPMSs, with the Bravais lattice positions of two end phases being at flat points. If a martensitic transformation continues within a space of TPMSs and all the lattice points remain at flat points throughout the process then we expect the occurrence of the SME. Our method does not account for the nucleation process, nor for the effects of temperature in such a stress-driven diffusionless process. Nevertheless, our method is relatively simple, general, and far less time-consuming in comparison to DFT and, thus, can provide a fast screening method for the discovery of more novel shape-memory alloys.