Shape memory alloys that produce and recover from large deformation driven by martensitic transformation are widely exploited in biomedical devices and micro-actuators. Generally their actuation work degrades significantly within first a few cycles, and is reduced at smaller dimensions. Further, alloys exhibiting unprecedented reversibility have relatively small superelastic strain, 0.7%. These raise the questions of whether high reversibility is necessarily accompanied by small work and strain, and whether high work and strain is necessarily diminished at small scale. Here we conclusively demonstrate that these are not true by showing that Au_(30)Cu_(25)Zn_(45) pillars exhibit 12 MJ m^(−3) work and 3.5% superelastic strain even after 100,000 phase transformation cycles. Our findings confirm that the lattice compatibility dominates themechanical behavior of phase-changing materials at nano to micron scales, and points a way for smart micro-actuators design having the mutual benefits of high actuation work and long lifetime

Bhattacharya, Kaushik

Chen, Xian

Greer, Julia R.

James, Richard D.

Ni, Xiaoyue

Caltech Authors - Main

Supporting Information:
Exceptional resilience of small-scale Au30Cu25Zn45 under cyclic
stress-induced phase transformation
Xiaoyue Ni1, Julia R. Greer1, Kaushik Bhattacharya1, Richard D. James2, and Xian Chen *3
1Division of Engineering and Applied Science, California Institute of Technology, Pasadena CA
91125
2Department of Aerospace Engineering andMechanics, University of Minnesota, Minneapolis,
Minnesota 55455
3Department of Mechanical and Aerospace Engineering, Hong Kong University of Science and
Technology, Hong Kong
TABLE OF CONTENTS
I. Information of supporting movies · · · · · · · · · · · · · · · · · · · · · · 1
II. Geometrically nonlinear theory of martensite for the formation of microstructure un-
der uniaxial compression · · · · · · · · · · · · · · · · · · · · · · · · · · · 2
II.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
II.2 Crystallography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
II.3 Energy minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
II.4 Deformation by forming (1¯01) type I twin . . . . . . . . . . . . . . . . . . . . . . . . 6
III. Reference figure for the re-entrant corner of martensite microstructure · · · · · · 7
I. Information of supporting movies
Movie S1: Evolution of microstructure of Au30Cu25Zn45 1µm-diameter pillar during in-situ
compression-induced martensitic transformation
Movie S2: Evolution of microstructure of Au30Cu25Zn45 2µm-diameter pillar during in-situ
compression-induced martensitic transformation
The movies record the morphology change and the corresponding stress-strain response
(converted from the force-displacement data) of the Au30Cu25Zn45 pillars within a complete
*To whom all correspondences should be addressed. Email: xianchen@ust.hk
cycle of phase transformation. The nanomechanical experiments were conducted by a
custom-made in-situ nanoindenter module (PI 85 PicoIndentor, Hystron, Inc) inside a
scanning electron microscope (Versa 3D, FEI).
II. Geometrically nonlinear theory of martensite for the formation of
microstructure under uniaxial compression
In this section of the SI, we will give a mathematical statement for determination of martensite
variant that will form in favor of the uniaxial compressive loading condition.
II.1 Kinematics
Consider a Au30Cu25Zn45 pillar defined in Figure S1 subjected to w : h = 1 : 3 as caved from a
single grain of austenite before loading. Upon loading, the cubic to monoclinic phase
transformation will be induced by the uniaxial compressive stress. Such a symmetry-breaking
transformation results in a set of 12 distinct martensite variants13: M = {U1, ...,U12} yielding
the following order,
U1 =
α ² 0² δ 0
0 0 γ
 ,U2 =
 α −² 0−² δ 0
0 0 γ
 ,U3 =
δ ² 0² α 0
0 0 γ
 ,U4 =
γ 0 00 δ −²
0 −² α
 (S1)
U5 =
γ 0 00 δ ²
0 ² α
 ,U6 =
 α 0 −²0 γ 0
−² 0 δ
 ,U7 =
α 0 ²0 γ 0
² 0 δ
 ,U8 =
δ 0 ²0 γ 0
² 0 α
 (S2)
U9 =
γ 0 00 α ²
0 ² δ
 ,U10 =
γ 0 00 α −²
0 −² δ
 ,U11 =
 δ 0 −²0 γ 0
−² 0 α
 ,U12 =
 δ −² 0−² α 0
0 0 γ
 . (S3)
The values α = 1.0591, ² = 0.0073, δ = 1.0015 and γ = 0.9363 are determined by the lattice
parameters of austenite and martensite, which were measured by the X-ray diffraction
experiment12. The blue domain ΩM in Figure S1 is considered as a mixture of deformations of
martensite due to the structural phase transformation. Let f ∈ [0,1] represent the average
volume fraction for twinning, the average deformation gradient within the martensite region
can be interpreted as
F f = Rˆ f [(1− f )U+ f Uˆ], (S4)
for U, Uˆ ∈M related by Uˆ = (−I+ 2e⊗ e)U(−I+ 2e⊗ e) where e is one of the two-fold axes of
austenite, |e| = 1. Rˆ f ∈ SO(3) is some rigid rotation depending on f . The austenite and
martensite interface m (also called the habit plane by metallurgists), can be determined from
the crystallographic equation27
F f − I= b⊗m, (S5)
for some vector b ∈ R3. This equation is not generally solvable due to the lack of compatibility
between the two lattices. Once the cofactor conditions are satisfied, we can find a pair of solu-
tions to the Eq. (S5) for every f ∈ [0,1]13. In particular, f = 0 and f = 1 correspond to the habit
plane between a single martensite variant and austenite.
Ω ΩA
ΩA
ΩM
F f
m f
-Nˆ
Figure S1: Kinematic model for microstructure evolution from austenite to martensite subjected to uni-
axial compression.
II.2 Crystallography
We utilized the X-ray Laue microdiffraction to characterize the crystallographic orientation of
the Au30Cu25Zn45 pillars used in our nanomechanical experiments. This measurement was
conducted at the Advanced Light Source Beamline 12.3.2. An area of 3×4 mm2 was scanned
by the polychromatic synchrotron X-rays from which we collected a sequence of Laue patterns
to construct the orientation mapping for the sample surface by the method outlined in
reference15. Figure S2(a) shows the spatial distribution of angle between Z-axis and c-axis
where X-Y-Z and a-b-c represent the sample and crystal coordinates respectively as illustrated
in Figure S2(c). It shows an angular variation of 2 degree from 10.6 to 12.4 within the scanned
austenite region. The region chosen for FIBing is marked as the black box in Figure S2(a),
which corresponds to the SEM image in Figure S2(b). The Laue pattern of austenite within this
region is shown in Figure S2(d) and the angle between the normal of sample surface and the
c-axis is 11.25 degree. The outer normal vector of these micro-pillars is characterized as
Nˆ= 1
ρN
OA[001]L21 = (0.150,−0.125,0.981) for some scalar ρN , (S6)
in which the orientation tensor
OA =

1.6039 −0.05659 0.2431
0.08753 1.6081 −0.2032
−0.2338 0.21104 1.59199
 . (S7)
Figure S2: Orientation information of the austenite domain from which the micro-pillars were fabri-
cated. (a) Orientation mapping of the area scanned by the polychromatic synchrotron X-ray. (b) The
overview of SEM image of pillars within the region marked in (a). (c) The orientation of the sample coor-
dinate, X-Y-Z, relative to the crystal coordinate of austenite, a-b-c, corresponding to (d) Laue pattern.
II.3 Energy minimization
In the scenario that the reference configuration in austenite phase is deformed by a uniaxial
loading, the total free energy E depends on the deformation gradient ∇y, where y :Ω→R3,
E =
∫
Ω
(φ(∇y)−P :∇y)dx. (S8)
The first term of integrand is the Helmholtz free energy density. P is the first Piola-Kirchhoff
stress defined as
P=σNˆ⊗ Nˆ (S9)
where σ< 0 for compression and Nˆ is the outer normal of top surface of the pillar, measured in
previous section. We need to find a deformation gradient F ∈ R3×3 that minimizes φ(F)−P : F,
and the minimizing deformation will be given by y= Fx for x ∈Ω.
For the martensitic transformation, we have the martensite energy wellsWM = {SO(3)U : U ∈
M } such that φ(F)<φ(A) for all A ∈R3×3 if F ∈WM . Thus, we can find a martensite variant from
the setM that minimizes Eq. (S8) with the application of the load given in Eq. (S9) by solving
the following maximization problem:
max
U∈M
(
max
R∈SO(3)
σUNˆ ·RT Nˆ). (S10)
Since σ< 0, this maximization problem can be converted to a minimization problem28:
min
U∈M
(√
Nˆ ·U2Nˆ), (S11)
subjected to some constraint on rotation matrix R to avoid the case that RNˆ = −Nˆ. By direct
calculation, (S11) is minimized by four martensite variants and they are U1, U2, U3 and U12
listed in (S1)-(S3). The uniaxial recoverable strain is calculated by
²N =
√
Nˆ ·U2Nˆ−1=−0.047 (S12)
and the shear strain is determined by
²S = tan
(
arccos
Nˆ ·FNˆ
|FNˆ|
)
, (S13)
for F = RˆU that solves the crystallographic equation (S5). In the case of U1, there exist two
conjugate solutions for Rˆ, vector b and the habit plane m listed in Table S1, which result in two
shear strains, 0.0699 and 0.0395.
Rˆ b m
Sol. I

0.9985 −0.000182 0.05428
−0.000182 0.99998 0.006705
−0.05428 −0.006705 0.9985


−0.07758
−0.00958
0.07758


0.7416
0.0916
0.6645

Sol. II

0.9985 −0.000182 −0.05428
−0.000182 0.99998 −0.006705
0.05428 0.006705 0.9985


0.07758
0.00958
0.07758


0.7416
0.0916
−0.6645

Table S1: Two solutions of the crystallographic equation (S5) for variant U1.
II.4 Deformation by forming (1¯01) type I twin
We construct a set of twin lamellae by the variant U1 and variant U5 listed in (S1) and (S2),
which satisfy the cofactor conditions for (1¯01) type I twin closely (verified in reference15). A
deformation sequence is proposed as
y( f ;x)=

F f x+a
∫ (x·n)
0
χ( f ; s)ds, if g ( f ;x)> 0
x, else.
(S14)
The deformation gradient F f is calculated from Eq. (S4) by substituting U = U1 and
e = 1p
2
(1¯01) for the volume fraction f ∈ {0.0,0.45,0.05,0.46,0.15,0.47,0.15,0.05,0.05,1.0}. The
resultant normal strain is -0.03207 by Eq. (S12) and the shear strain is 0.01885 by Eq. (S13). The
vectors a = [0.1624,0.02,0.1455] and n = e are the type I twinning parameters. χ( f ; s) is a
piecewise function defined as,
χ( f ; s)=

(1− f )(s−bsc), bsc ≤ s < bsc+ f ;
− f (s−dse), bsc+ f ≤ s < dse.
(S15)
This function ensures a rank-1 interface between the two martensite variants13. The function
g ( f ;x) defines the austenite and twinned martensite interfaces consisting of alternative
normals m0 = (0.7416,0.0916,−0.6645) and m1 = (−0.6645,0.0916,0.7416),
g ( f ;x)=
 (x−p f ) ·m1, bx ·nc ≤ x ·n< bx ·nc+ f(x−p f ) ·m0, bx ·nc+ f ≤ x ·n< dx ·ne. (S16)
The vector p f is the vertex of a triple junction at which the martensite twin pair meet austenite
as shown in Figure S4.
Figure S3: Compatible interface between austenite and type I twinned martensite at volume fraction
f = 0.4. The red region are the austenite (undeformed phase), while the blue/green lamellae are the twin
pair corresponding to (U1,U5). Either variant can form a exact interface with austenite lattice. Austenite
and two martensite variants meet at the triple junction p f .
III. Reference figure for the re-entrant corner of martensite microstructure
Figure S4: Formation of the type II twin boundary in Cu-Al-Ni single crystal (in the paper by Ichinose et al
1985 25). (a) - (c) shows the evolution of the twin boundary during stress-induced phase transformation.
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Exceptional resilience of small-scale Au_(30)Cu_(25)Zn_(45) under cyclic stress-induced phase transformation

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Exceptional resilience of small-scale Au_(30)Cu_(25)Zn_(45) under cyclic stress-induced phase transformation

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