Xian Chen, chen1561@umn.edu
Co-authors: Yintao Song, Richard D. James
From compatibility conditions to stress-free microstructure for martensitic transformation
The cofactor conditions (CC) are the conditions of super compatibility between phases for martensitic transformation. By
satisfying CC, austenite and variants of martensite can fit together without elastic transition layers for any twinning volume
fraction between 0 and 1. Here we discuss different forms of CC in Type I/II, Compound twins and domains, followed by the
prediction of their possible microstructures. Then we calculate the geometric linear case of CC. Finally, we show real examples
whose lattice parameters were tuned to satisfy CC closely for both Type I and II twin system.
Materials undergoing reversible phase transformations are desirable for applications in medical devices, microelectronics and
energy conversion devices [1]. The conditions of elastic compatibility of the phase-transforming materials have profound
impact on these applications. Because they dominate the hysteresis, the reversibility and the resistance to functional degradation
upon cyclic operation [2, 3]. It has been proven experimentally that the degree of hysteresis and the cycling ability of the
materials can be optimized [4] by making the lattice parameters satisfy a set of geometric compatibility conditions
λ2 = 1
a · Ucof(U2 − I)n = 0
trU2 − det U2 − |a||n|
4 ≥2
(1)
(2)
(3)
where λ2 is the middle eigenvalue of the transformation stretch tensor U, the vector a, n are the twinning parameters. Microscopically, the satisfaction of these conditions means a perfect lattice points match at the interface between the initial and final
phase, which results in stress-free microstructure in the material. Therefore, it allows for materials transforming back and forth
freely without paying internal elastic penalty. This is why the materials satisfying CC exhibit tremendous reversibility.
References
[1] Srivastava, V., Song, Y. Bhatti, K. and James, R. D. Adv. Eng. Mater. 1, 97-104 (2011)
[2] Ball, J. M and James, R. D. Arch. Ration. Mech. Anal. 100, 12-52 (1987)
[3] Chen, X., Srivastava, V., Dabade, V. and James, R. D. J. Mech. Phys. Solids, 61, 2566-2587 (2013)
[4] Song, Y., Chen, X., Dabade, V., Shield, T.W. and James, R. D. Nature, 502, 85 (2013)
[5] Chen, X., Song, Y. and James, R. D. in prep.
From compatibility to stress-free
microstructure for martensitic materials
Xian Chen!
!
Aerospace Engineering and Mechanics
University of Minnesota
Support by MURI project (AFOSR)
and NSF-PIRE
Martensitic transformation
Applications
Aerospace engineering
Microelectronics
Medical device
Flexinol®!
actuator!
wire
geometry adjustable Chevron
stent
Durability
Hysteresis
Reversibility
Materials undergoing phase transformation
Irreversible
Tin pest
Reversible
A material made by!
satisfying some mathematic conditions
Structural transformation
em
3
initial structure!
(austenite)
final structure!
(martensite)
F
for some
em
2
ea3#
ea2#
=
I
ea1
•
U
stretch matrix
depends on lattice parameters
of initial and final lattices.
•
For martensitic transformation
em
1
c
U
Ø
a
b
Û
Û
determines
the variants of martensite
b
c
a
i.e. cubic to monoclinic, there are 12 variants.
Ø
Ø
c
b
a
Structural transformation
em
3
initial structure!
(austenite)
final structure!
(martensite)
F = RU !
for some R∈SO(3)
em
2
ea3#
ea2#
=
I
ea1
•
U is the transformation
stretch matrix that only
depends on lattice parameters
of initial and final lattices.
•
For martensitic transformation
em
1
c
U
Ø
a
b
Û
Û
determines
the variants of martensite
b
c
a
i.e. cubic to monoclinic, there are 12 variants.
Ø
Ø
c
b
a
Structural transformation
em
3
initial structure!
(austenite)
final structure!
(martensite)
F = RU !
for some R∈SO(3)
em
2
ea3#
ea2#
=
I
ea1
•
U is the transformation
stretch matrix that only
depends on lattice parameters
of initial and final lattices.
•
For martensitic transformation
em
1
c
U
Ø
a
Ûi = Qi U
b
Ûj = Qj U QjT
QiT
b
the symmetry-related
Û = Q U QT, for Q ∈ 𝒫a
determines
the variants of martensite
c
a
i.e. cubic to monoclinic, there are 12 variants.
Ø
Ø
c
b
a
In the case of Tin pest
Initial phase: tetragonal!
a0 = 5.83!
c0 = 3.18
.
=
Final phase: cubic!
a = 6.48
.
.
.
=
=
.
.
I
.
.
U1 or U2
I
.
U3
condition for a compatible interface
Deformation gradients are differed by a rank1 matrix
A-B=b⊗m
A
B
for some b, m ∈ ℝ3
Conditions of compatibility in general case
twinned martensite
Let U and Û be symmetry related variants
austenite
and they satisfy
QÛ - U = a ⊗ n !
for some Q ∈ SO(3) f
n
1-f
n
a
U
QÛ
U
Favg = Rf [f U + (1 - f )QÛ]
200μm
mf
( +
)
=
has solutions for some f ∈ [0, 1] iff the
C(f) = (U + f n ⊗ a) (U + f a ⊗ n) has middle
eigenvalue 1.
J. Ball and R. D. James, Rat. Mech. Arch. 1987"
RV. Kohn and S. Muller, Phil. Mag., 1992
If C(f) has middle eigenvalue 1 for some f
twinned martensite
austenite
type I twin with twinning plane
n = (0, 1, -1)
f = 0.208
f = 0.792
n
QÛ - U = a ⊗ n
( +
)
m = (0.52,0.01,-0.85)
m = (0.52,-0.85,0.01)
m = (-0.52,0.01,-0.85)
m = (-0.52,-0.85, 0.01)
=
There are 4 solutions
per twin system (U, a, n)
{,
{,
{
,
{
,
+
+
,
,
+
,
,
+
+
}
}
+
}
}
J. Ball and R. D. James, Rat. Mech. Arch. 1987
A degeneracy case: if λ2 = 1
where λ2 is the middle eigenvalue of U
a perfect interface for f = 0 "
( +
)
=
R0U - I = b0 ⊗ m0
or a perfect interface for f = 1"
R1U - I = b1 ⊗ m1
Hysteresis and λ2=1
80
TiNiPt
TiNiCu
ZnAuCu
72
TiNiPd
TiNiAu
mf
Hystersis (degree Celcius)
64
56
48
40
# of strains reduces
32
24
austenite
16
single variant!
martensite
8
0
0.950.960.970.980.99 1 1.011.021.031.041.051.061.071.08
middle principle stretch of transformatio matrix
J. Cui et al., Nature Mater., 2006"
Z. Zhang et al., Acta Mater., 2009"
X. Chen et al., JMPS, 2013"
Y. Song et al., Nature, 2013
Further degeneracy for (U, a, n)
"
Cofactor Conditions (CC)
Let U = UT and detU > 0, and define Û = (-I + 2 e ⊗ e) U (-I + 2 e ⊗ e) for some
|e| = 1, so that Q Û - U = a ⊗ n for some Q ∈ SO(3) and a, n ∈ ℝ3, then the
equations of crystallography theory of martensite
( +
)
=
has solutions {Rf , bf ⌦ mf } for every f ∈ [0, 1] if and only if
▪ (CC)1 λ2 =1
Click to play
▪ (CC)2
▪ (CC)3
·
(
) =
| | | |
There are infinitely many strains !
cofactor conditions in different twin systems
Let U and Û = (-I + 2 e ⊗ e) U (-I + 2 e ⊗ e) define a twin system.
physically, e, |e|=1 is a 2-fold symmetry axis of the austenite.
(CC)
(CC)
(CC)
X
Type I
satisfied simultaneously
Type II
Compound
λ
X
e ⋅ v 2 = 0,
e ⋅ v2 = 0
⊥
(2 1− (e ⋅ v1)2 (e ⋅ v3 )2 +1) detU2 ≥ 2
where XI = |U-­‐1e|, XII = |U e| denote the inverse and forward stretches along the 2-­‐fold axis e. X.Chen et. al., JMPS, 2013
microstructure by
satisfying CC
compound twin: plane strain
Compound twin
•
λ2=1"
•
e⋅v2=e⊥⋅v2=0"
•
inequality condition
for every f ∈ [0, 1]"
•
Rf (U + f a ⊗ n) - I = bf ⊗ mf "
•
mf ⋅ v2 = 0
A general case "
of transition layer
v2
mf
e
e⊥
type I twin: stress-free triple-junction
•
λ 2 = XI = 1
Rf (U + f a ⊗ n) - I = bf ⊗ mf!
Rf is the same for every f ∈ [0, 1]
for half of the solutions
All rank-1 connected triple-junctions"
RU - I = b0 ⊗ m0"
RQÛ - I = b1 ⊗ m1"
RQÛ - RU = Ra ⊗ n
n
I
RU
Perfect interface
transition layer
m0
m1
RQÛ
X. Chen et.al, JMPS, 2013
Diamond nucleation of austenite
The austenite (red lattice) in a diamond shape can grow from a slim band of M
variant (blue lattice) in another M variant (green lattice) by satisfying the Cofactor
Condition for Type I twin
This nucleation process occurs
at zero elastic energy.
Au
ste
nit
e
M2
M1
X. Chen et.al, JMPS, 2013
type II twin: parallel phase boundaries
•
λ2 = XII = 1"
➞ Twinning plane || habit plane for every f ∈ [0, 1]
mf
nII
X. Chen et.al, JMPS, 2013
CC satisfied for both Type I and Type II twins
simultaneously
•
λ2 = XI = XII = 1
Six rank-1 connections
=
Triple junction
Quad junction
=
=ˆ
)=ˆ
(
)=ˆ
(
(
a
te
s
u
M1
e
nit
M3
M1
M2
M2
)=ˆ
…result in six perfect interfaces!
M4
X. Chen et.al, JMPS, 2013"
Y. Song et.al, Nature, 2013
Combination of tri-quad junctions
with varying the volume fractions
continuously
au
s
ite
n
te
M1
M2
austenite
marte
M1
M2
nsite
M3
M4
X. Chen et.al, JMPS, 2013"
Y. Song et.al, Nature, 2013
the CC material has infinitely many
flexibility to form zero energy
microstructure
"
these are not just
math…
A family of alloys satisfies CC closely
Differential Scanning Calorimetry "
for an alloy family Zn45AuxCu55-x
3.1℃
Trends of hys.
2.4℃
6
Hysteresis =
0.5℃
0.8℃
4.2℃
4.8
5.1℃
Hysteresis ℃
2.6℃
3.6
2.4
1.2
0
24
26
28
30
Au at%
32
Structural determination by XRD
Austenite
17℃
Martensite
-50℃
L21
M18R"
(P2)
Y. Song et.al, Nature, 2013
Lattice parameters and Cofactor Conditions
(CC)2 for Type I twin
(CC)2 for Type II twin
Y. Song et.al, Nature, 2013
1
Cycling the phase transformation
~16,000
Shift of trans. temps. °C
0
-2
-4
-6
Au30
-8 Au27
Au25
As Af Ms Mf
64
12
8
25
6
51
2
16
32
8
4
2
1
-10
Number of cycles
Y. Song et.al, Nature, 2013
210 211 212 213 214
Irreproducible microstructure from cycle to cycle
Zn45Au30Cu25 λ2 = 1.0006"
XI = 1.0081"
XII = 0.9996
Click to watch the movie of reversible phase transformation
Y. Song et.al, Nature, 2013
Ongoing work
Theoretical prediction
Optical microscopy
Ongoing work
Theoretical prediction
Quantitative
verification of the
theory
Optical microscopy
SEM
EBSD
Ongoing work
Theoretical prediction
Quantitative
verification of the
theory
Optical microscopy
SEM
EBSD
Summary
•
formulation of the cofactor conditions (CC) for martensitic phase
transformation
•
geometric interoperation of CC in type I, type II and compound
twin systems
•
possible stress-free microstructure by satisfying CC in different
twin systems
•
discovery of materials that satisfy CC closely, which show
enhanced reversibility and unusual microstructure
Thank you
This work is done by University of Minnesota group lead by Prof. Richard James
and ongoing work is collaborating with CMU group lead by Prof. Marc de Graef
and LBNL advanced light source.
"
Supported by ASFOR MURI project: Managing the Mosaic of Microstructure and
NSF-PIRE.