Thermal response of infinitely extended layered Nickel-Titanium
Shape memory alloy thin film with variable material properties
Abhijit Bhattacharyya1,a, Mehmet Mete Ozturk1,b
1
Smart Materials and MEMS laboratory, Department of Systems Engineering, Donaghey College
of Engineering and Information Technology, University of Arkansas at Little Rock
Little Rock, AR, USA 72204
a
[email protected], [email protected]
ABSTRACT
This paper reports a study of the thermal response of an infinitely extended shape memory alloy thin film. Motivated
by experiments reported in the literature about SMA thin films on a silicon substrate, the thin film is taken to have
three layers from the bottom to the top – an amorphous layer, a non-transforming austenitic layer and a transforming
SMA layer. The boundary conditions are taken to be adiabatic and convective at the bottom of the film and the top
respectively. The material properties of the transforming layer (thermal conductivity, electrical resistivity and
specific heat) are taken to evolve hysteretically with temperature, commencing from an initial room temperature
state of martensite. All the results are presented in non-dimensional form. The steady state results are compared with
an analytical solution. The computations of the transient response are carried out with ANSYS. The thermal
response of the 3-layer model is compared with that of a 1-layer model (where the entire film is a SMA
transforming layer) and it is seen that the the temperature of the top surface for the 3-layer model is higher than that
of the 1-layer model. It is also seen that the evolution of the specific heat has the least effect whereas the evolution
of the electrical resistivity has the most effect on the thermal response of the 3-layer model. The thermal response of
the infinitely extended films provides a benchmark against which the response of finite sized films can be assessed.
Keywords: Heat Transfer, Shape Memory Alloy, Thin Film.
1.Introduction
A Shape Memory Alloy (SMA) thin film is an important candidate for the fabrication of a microsensor
and/or a microactuator. Its shape memory effect (SME) is due to a solid–solid phase transformation between a high
temperature phase of austenite and a low temperature phase of martensite. The phase transformation is also
accompanied by a temperature dependent change in material properties such as the thermal conductivity, electrical
resistivity and heat capacity [1]. Fabrication usually results in thin fims that are inhomogeneous. Thus, for example,
it has been reported that a SMA thin film fabricated by magnetron sputter deposition has three layers – an
amorphous layer, a non-transforming austenite layer and a transforming SMA layer – with the amorphous layer
being adjacent to the substrate [2]. Furthermore, thermal modeling of the thin films has been studied and published
by various researchers [3-8]. Thermal modeling of thin films has been studied in literature from different aspects, for
instance a thermal model of NiTi thin film is studied by Favelukis et al. [3], Ramadan et al. [4] investigated thermal
wave reflection in a multilayer slab using the dual phase lag model, Liu & Cheng [5] studied numerical analysis in
layered films by a hybrid method of laplace transform and control volume, Cheah et al. [6] modeled the microscale
heat conduction effect in electric package numerically for different thermal boundary conditions, Al-Nimr et al. [7]
studied thermal behavior of a multi layered thin slab by a periodical signal. Aviles et al. [8] discussed the dynamical
behavior of the thin metallic film-substrate system, theoretically and experimentally. They tried to obtain a physical
picture of distribution of temperature on thin film and substrate and proposed a dynamic thermal model. While
thermal studies of thin films have a rich history in the literature, such studies for shape memory alloys with variable
material properties and layered structure are sparse and need to be carried out in order to have an accurate
understanding of the thermal response of SMA thin films.
In this study, an infinitely extended, 3-layered NiTi SMA thin film is modeled and analyzed with
commercially available finite-element software ANSYS. The temperature variation of the 3 layered thin film is
investigated under the following conditions: convective cooling on the free surface, adiabatic boundary condition on
the bottom surface (at the interface with the substrate) and uniform heat source. The obtained results are presented
for both steady state and transient conditions. The steady state response is validated by comparison with analytical
results. The thermal response of the film during heating is also compared with the thermal response of a single
layered thin film undergoing phase transformation.
2. Definition of the boundary value problem
The microstructure of a thin film is usually complicated and inhomogeneous. Thus, for example,
microstructural study of a NiTiPd thin film by Rabei et al. [2] has shown that the SMA thin film has three layers –
an amorphous layer (referred sometimes as Layer 1 subsequently), a non-transforming austenite layer (Layer 2) and
a top transforming SMA layer (Layer 3) – from the bottom (substrate) to the top. While the details of the
microstructure may certainly vary from one film to another, for this computational study of a NiTi thin film, we
consider an infinitely extended, 3-layered SMA thin film as shown in the schematic of Figure 1. We assume an
adiabatic boundary at the bottom and a top surface across which there is convective heat transfer. While, in practice,
thin films are certainly of finite extent, the boundary value problem addressed in this paper serves as a benchmark
problem against which the response of thin films of finite dimensions may be assessed. The conservation equations
as well as the boundary conditions are given below in dimensional as well as non-dimensional form.
2.1 Dimensional conservation equations and boundary conditions
The energy conservation equation can be written as [1]
k T
ρ T J
C
,
T
Hp T
(i=1,2,3) ,
(1)
where k T , ρ T and C , T are the temperature dependent thermal conductivity, electrical resistivity and the
heat capacity of the ith layer respectively. The term, Hp T , represents the temperature dependent evolution of
latent heat where “H” is its magnitude. In particular, the parameters for the three layers are defined as
k T
k
, k T
k
and
k T
k
ξ T
k
k
ρ T
ρ
,
ρ T
ρ
and
ρ T
ρ
ξ T
ρ
ρ
C
,
T
p T
C
,
0, p T
,
C
,
T
C
,
0 and p T
and C
,
.
T
C
,
ξ T
C
,
,
,
C
,
,
(2)
The parameters used in Eq.2 are the following: kAmor , Amor and Cv,Amor pertain to the amorphous layer, kAus , Aus
and Cv,Aus pertain to the SMA in a purely austenite state and kMar , Mar and Cv,Mar pertain to the SMA in a purely
martensitic state. The parameter , is defined as the martensite volume fraction. It ranges between zero (pure
austenite) and one (pure martensite) and is taken as
1
ξ T
ξ
0,A
T
A ,
ξ
0, M
T
M ,
/
1
/
(3)
where we have used a sigmoidal function to represent the evolution of the martensite volume fraction. Finally, As,
Af, Ms and Mf are the austenite start & finish temperature, martensite start & finish temperature respectively. With
Eq.3, the last of Eq.2 can be used to show that p 3 (T)  ξ(1  ξ) .
We assume an adiabatic boundary at x = 0 and a convective surface at x = L. Note that L is the total
thickness of the film, defined as L ∑ L . The interfaces between the layers will be taken to be thermally perfect,
i.e. the heat flux and temperature are taken to be continuous. The initial condition, boundary conditions and interface
conditions are written as
T x, 0
T ,0
∂T
0, t
∂x
k
L
k
L ,t
L ,t
k
L, t
L,
0,
L , t , T L , t
k
L , t , T
L
-k
x
h T L, t
L
T L , t ,
L ,t
T
L
L , t ,
T .
(4)
In the above equations, “h” is the convection coefficient and “T0” is the ambient temperature.
2.2 Non-Dimensional conservation equations and boundary conditions
The governing equations given in the previous section are rewritten in non-dimensional form below. We
propose a non-dimensional temperature and length as given below
T
,x
,
(5)
using which the governing equations (Eq.1) are non-dimensionalized and given below
∂T
∂
k Ti
∂x
∂x
ρ Ti J̅
C
Ti
,
∂Ti
∂t̅
Hp Ti
∂Ti
,
∂t̅
(6)
whereas the boundary conditions, interface and initial conditions (Eq.4) are written as
T
,0
0,
T1
̅
0, ̅
0
0,
1,
, ̅
̅
, ̅
̅
, ̅
, ̅ ,
̅
-
, ̅ ,
̅
̅
1, ̅
, ̅ ,
, ̅
, ̅ ,
1, ̅ .
(7)
The process of non-dimensionalization has resulted in the identification of the following non-dimensional
parameters. These are
t̅
k
LC
J̅
t,
,
ρ
Tk
L
J ,
H
1
C
T
,
H,
h
L
h,
k
where
p Ti
k
k
k T1
ρ T1
C
,
T1
C ,
C ,
k T2
,
,
C
k
k
p Ti T ,
, ρ T2
,
C
C
T2
,
,
k
T3
andρ T3
andC
,
ξ T3 k
k
k
andk T3
,
C
T3
ξ T3 C
C ,
,
,
,
C
,
.
(8)
2.3 A Steady state Analytical Solution
A steady state analytical solution of Eq.6 is given below. In order to generate this solution, we assume that
the properties of Layer 1 and Layer 2 are constant, the heat source in those layers is set to zero, and the heat source
in Layer 3 is assumed to be a constant,
(its non-dimensional counterpart is
). For the purpose of the
analytical solution, the thermal conductivity of Layer 3 is taken to depend linearly on temperature such that
k 3 (T) k Mar 1 α T T0  where  is a constant parameter. The nondimensional steady state temperature is given as
∝
T x, ∞
∝
T x, ∞
T x, ∞
where
α  αT0 .
qL
k
k
∝
qx
2k
∝
∝
L
q L
2k
T
L
T
L ,∞ ,
∝
L
L ,∞ ,
,
(9)
3. Analytical & Numerical Results
Most of the results are presented in non-dimensional form. However, in order to calculate the results,
material properties of the austenite and martensite phase are needed. While the literature does report some of these
properties (electrical resistivity and heat capacity) for thin films, an accurate determination for all properties
(including thermal conductivity) as part of the same study on thin films is not available. For the purpose of this
computational study, we have used the material properties determined for NiTi wires and reported in one of our
=8.371 x 10-4Ω.mm and C v,Aus =5.92 x 106 J/Km3 for the
earlier publications. These are kAus =28 W/mK ,
=9.603 x 10-4Ω.mm and C v,Mar =4.50 x 106 J/Km3 for the martensite
austenite phase and kMar =14 W/mK ,
phase [1]. The transformation temperatures have been taken as
A f  355
0
C
. The ambient temperature has been taken as
T0  293
0
C
M f 323 0 C
,
0
M s  333 C ,
0
A s  345 C and
.
As far as the amorphous layer is concerned, we have not been able to locate any of the material properties
in the literature. Usually, thermal conductivity and electrical conductivity are lower (or electrical resistivity is
higher) for amorphous materials as compared to their crystalline counterparts with identical stoichiometry. These
films were obtained from the Ramirez Laboratory1 in Yale University. The tests reveal that the electrical resistivity
of the amorphous thin film is 46% higher than the crystalline thin film (or conversely, the electrical conductivity is
68% lower). In absence of thermal conductivity data for the amorphous film, we shall assume that the thermal
conductivity is also 68% lower for the amorphous film as compared to the value of the crystalline film at room
temperature. As for the specific heat, in absence of reliable data, we take it to be the same value as that of the
crystalline thin film. Usually, the specific heat has a more modest impact on the thermal response of the SMA as
compared to the latent heat during the phase transformation. Thus, we shall assume that the specific heat of the
amorphous layer is identical to the specific heat of the crystalline layer at room temperature. The properties of the
crystalline layer are now needed. We have done temperature dependent x-ray diffraction of the crystalline thin film
at the University of Arkansas Institute for Nanoscience and Nanoengineering and based on direct comparison
method, we have determined the phase fractions to be 98.7%. austenite and 1.3% martensite at room temperature.
Assuming a rule of mixtures of the material properties, the thermal conductivity, the electrical resistivity and the
heat capacity of the crystalline thin film are found to be 27.818 W/mK, 8.38706 x 10-4Ω.mm and 5.901618 x 106
J/Km3 respectively based on the properties of the individual phases determined by us for NiTi wires[8]. Thus, the
=12.24508 x 10-4Ω.mm and C
material properties for the amorphous layer are taken as kAmor =8.90176 W/mK,
6
3
v,Amor =5.901618 x 10 J/Km . Finally, where we have given results pertaining to a 3-layer SMA, we have taken the
thicknesses of Layers 1, 2 and 3 to be 0.2 microns, 0.5 microns and 1 micron respectively [2]. Where needed, the
normalized convection coefficient is taken to be 2.43 (this corresponds to a dimensional convection coefficient of
2107 W/m 2 ).
3.1 Steady State Analytical results
In order to partially validate the finite element computations, the temperature distribution in the SMA thin
film is studied along its thickness under steady state conditions. While the numerical results have been obtained
using the finite element software ANSYS, the analytical results are calculated using the equations given in Section
2.3. One-dimensional LINK32 elements are used as well as LINK34 elements at the surface where the film is
exposed to convective heat transfer. The parameter, , in the analytical solution is taken to be 0.1. The
nonimensional steady state temperature vs. the nondimensional distance is given in Figures 2(a) and 2(b). The first
figure pertains to the three layer model and the second figure pertains to results corresponding to a single crystalline
transforming layer spanning the entire thickness of the film. In either case, comparison of the finite element solution
is made with the analytical solution; as is quite obvious, the comparison is excellent. The Figure 2(a) also contains
the results of the finite element calculations where the total number of elements, N, have been taken as 30, 40 , 50
and 60. In the first case, each layer has 10 elements of equal thickness within the layer. In the second case, Layers 1,
1
We are grateful to Dr.A.G.Ramirez and Dr.X.Huang. Their addresses are: Dr.A.G.Ramirez, Department of
Mechanical and Materials Engineering, Yale University, New Haven, CT 06520, X.Huang, Memry Corporation, 3
Berkshire Blvd, Bethel, CT 06801.
2 and 3 have 10, 10 and 20 elements; in the third case, 10, 20 and 20 elements have been taken. Finally, the last case
has 20, 20 and 20 elements respectively. It is clear from the steady state results in Figure 2(a) that there is no
distinguishable difference between the different cases. The rest of the figures uses the material properties of the
shape memory alloy as discussed in the previous paragraph.
The Figures 3(a)-(d) give the evolution of the non-dimensional temperature with respect to non-dimentional
time for four different locations: interface between the substrate and Layer 1 ( x  0 ), Layer 1 and Layer 2 (
x  L1 ), Layer 2 and Layer 3 ( x  L1  L2 ), and top of Layer 3 ( x  1 ). This information is given for four
different levels of spatial discretization, all at a normalized time increment of 0.005. We see that there is virtually no
difference in the computational response at the four different locations for the various levels of discretization.
Finally, we provide the non-dimensional temperature – time response at x  1 for three different time increments,
Δt = 0.05, 0.005 and 0.0005 in Figure 4. The difference between the three cases is noticeable but marginal. In light
of these studies (Figures 2, 3 and 4), in the rest of the calculations, we have taken 10 elements in each layer, with
equal element thicknesses in each layer and Δt =0.005. As well, the normalized current density has been taken to be
J  0.55677 . The Figures 5(a) and (b) compare the heating response of the 3-layer model and the 1-layer model. The
Figure 5(a) gives the temperature evolution at the interface between the substrate and film, while the Figure 5(b)
gives the temperature evolution at the surface exposed to the environment. In all these plots, and more prominently
for the 1-layer model response, the typical “knee” during the phase transformation while heating is clearly obvious.
The effect of the phase transformation accompanied by the exchange of latent heat is effectively equivalent to an
increase in the heat capacity, leading to a slower change in temperature than would otherwise have happened
(leading to the “knee”). Clearly, while the “knee” is present for the 3-layer model (this will be more apparent in a
subsequent figure), it is clearly not as prominent as the 1-layer model, and we attribute this difference to the
presence of an amorphous layer and a non-transforming SMA layer in the 3-layer model. Also notice the difference
between the response of the two models is clearly higher at the interface of the SMA with the substrate as compared
to the top surface of the thin film suggesting that the effects of Layers 1 and 2 will further diminish if the thickness
of the transforming layer was thicker (Layer 3). While this is to be expected, the computational study can offer some
guidance to experimenters on the range of SMA film thicknesses for which the interfacial layers may cease to have
an impact on the thermal response of the SMA layer. We point out another interesting feature – the “knee” has
appeared in the response of the 1-layer model at its top surface (Figure 5(b)). This location sees a maximum
temperature that does not exceed 0.2. This value is lower than the normalalized austenitic finish temperature, which
is 0.21. This means that while the top surface in the 1-layer model starts to undergo the martensite to austenite
transformation, the temperatures are not high enough to finish the transformation. This is in contrast to the bottom of
the film (solid line in Figure 5(a)) where the normalized temperature for the 1-layer far exceeds 0.21. Such is not the
case for either location in the 3-layer model, where the temperatures are higher. This is probably to be expected
because the presence of a less thermally conductive amorphous layer at the bottom (in case of the 3-layer model)
results in less heat loss to the substrate and therefore more efficient heating at a given current density. More detailed
studies will be offered in the future. The effect of evolving material properties is studied in the last figure – Figure 6
for the 3-layer model. In this figure, the solid curve is the thermal response of the 3-layer model. Notice that this
curve is slightly lower than the dotted curve which was generated by setting the latent heat to zero but allowing all
other properties to change as for the solid curve. The dashed curve (short dashes) is calculated by allowing the
thermal conductivity not to change with temperature, in addition to setting the latent heat to zero. The dashed curve
is somewhat different from the dotted curve, underpredicting the temperature in the initial transient response and
then overpredicting the temperature when the steady state is being approached. The top two curves (which almost
overlay on each other; more about this soon) are quite a bit higher than the remaining curves. In these top curves, the
electrical resistivity is allowed to remain at its initial martensitic value and not allowed to change with phase
transformation to the lower value of austenite (ofcourse, the latent heat continues to be zero in these two curves);
thus, the attained temperature is higher. Finally, the difference between the top two curves is that the specific heat is
not allowed to change with temperature in one case; yet, the difference is marginal. Thus, in conclusion, it appears
that the specific heat has an insignificant effect while electrical resistivity has a significant effect on the thermal
response of the top surface of the SMA.
4. Conclusions
In this paper, the thermal response of infinitely extended Nickel-Titanium SMA thin films is studied. The thin film
has a three-layered structure that includes an amorphous layer right above the substrate, followed by a nontransforming crystalline austenite layer and a transforming crystalline SMA layer. The response of this thin film is
compared with the response of a SMA thin film with identical overall thickness and with just one crystalline SMA
transforming layer. The structures are studied in a context of 1D boundary value problem modeled by using
Galerkin finite element method in which the heat is generated electrically and the ends of the thin film have
adiabatic and convection boundary conditions respectively. The steady state response is validated by comparing with
a known analytical solution as well. Parametric studies have been performed by including the latent heat and by
excluding it to represent the effect of phase transformation on the temperature during heating of an initially
martensitic thin film. For the given set of parametric studies, it appears that the temperature dependence of the
specific heat is of least importance whereas the temperature dependence of the electrical resisitivity is most
important.
5. References
[1] Faulkner M.G., Almaraj J.J. & Bhattacharyya A., ‘Experimental determination of thermal and electrical
properties of NiTi shape memory wires‘, Smart Materials & Structures, 9, 632-639 (2000).
[2] Lee J.W., Thomas B. And Rabiei A., ‘Microstructural study of titanium-palladium-nickel base thin film shape
memory alloys’, Thin Solid Films, 500, 309-315 (2006).
[3] Favelius J.E., Lavine A.S. and Carman G.P., ‘An experimentally validated thermal model of thin film NiTi’,
Proc. SPIE 3686, 617-629 (1999).
[4] Ramadan K. and Al Nimr M., ‘Thermal wave reflection and transmission in a multilayer slab with imperfect
contact using the dual-phase-lag’, Heat transfer Engineering, 30(8), 677-687 (2009).
[5] Liu K. and Cheng P., ‘Numerical analysis for dual-phase-lag heat conduction in layered films’, Numerical Heat
Transfer: Part A, 49, 589-606 (2006).
[6] Cheah T.S., Seetharamu K.N., Ghulam A.Q., Zainal Z A and Sundararjan T., ‘Numerical modeling of microscale
heat conduction effects in electronic package for different thermal boundary conditions’, Proc. IEEE Electronics
Packaging Technology Conference (2000).
[7] Al Nimr M.A., Naji M. and Abdallah R.I., ‘Thermal behavior of a multi-layered thin slab carrying periodic
signals under the effect of the dual-phase lag heat conduction model’, International Journal of Thermophysics,
25(3), 949-966 (2004).
[8] Aviles F., Oliva A.I. and Aznaraz J.A. ‚‘Dynamical thermal model for thin metallic film substrate system with
resistive heating‘, Applied Surface Science, 206, 336-344 (2003).
x
Convective Surface
L3
Transforming SMA layer
L2
Non Transforming Austenite Layer
L1
Amorphous layer
Adiabatic Surface
Figure 1. Schematic representation of the boundary value problem.
0.4
0.4
3 Layer
N=
0.3
1 Layer
30
40
50
60
N=30
0.3
̅
̅
0.2
0.2
0.1
0.1
analytical
numerical
analytical
numerical
0
0
0
0.2
0.4
̅
0.6
0.8
1
0
0.4
0.4
0.3
0.3
̅ =0,  ̅ =0.005
0.2
0.1
0.6
̅
̅
0.8
1
̅ = ̅ =0.11765,  ̅ =0.005
0.2
30
40
50
60
N=
0.4
Figure 2(b). The steady state temperature variation
with respect to thickness in the single layer SMA
thin film.
Figure 2(a). The steady state temperature variation
with respect to thickness in the three-layer SMA
thin film.
̅
0.2
0.1
0
30
40
50
60
N=
0
0
1
2
̅
(a)
3
4
5
0
1
2
̅
(b)
3
4
5
0.4
0.4
0.3
0.3
̅
N=
0.1
̅
̅ =0.2941,  ̅ =0.005
̅= ̅
0.2
0.2
30
40
50
60
̅ =1,  ̅ =0.005
0.1
0
30
40
50
60
N=
0
0
1
2
̅
3
4
5
0
1
(c)
2
3
̅
(d)
4
Figure 3(a)-(d): Temperature evolution in the three-layer thin film with respect to time at different locations, ̅ = 0, ̅
and 1 respectively.
N=30,
̅=
̅
5
̅
̅ =1
0.05
0.005
0.0005
̅
Figure 4. Temperature evolution in the three-layer thin film with respect to time for different time steps at ̅ =1.
̅
0.4
0.4
3 Layer
0.3
0.3
̅
̅
1 Layer
3 Layer
0.2
0.2
0.1
0.1
1 Layer
̅ =0, N=30,  ̅ =0.005
̅ =1, N=30,  ̅ =0.005
0
0
0
1
2
3
̅
4
0
5
1
2
(a)
̅
3
4
5
(b)
Figure 5. (a)Temperature evolution at the interface between film and substrate ( ̅ =0) for single layer and three-layer SMA film
during heating (b) Temperature evolution at the top surface ( ̅ =1) for single layer and three-layer SMA film during heating.
0.4
̅ =1, N=30,  ̅ =0.005
0.3
̅
0.2
̅
̅
̅
̅
̅
0.1
0
0
1
, ̅
, ̅
, ̅
, ̅
, ̅
2
, ̅
, ̅
, ̅
, ̅
, ̅
̅
, ̅
, ̅ =0
, ̅ =0
, ̅ =0
, ̅ =0
3
4
5
Figure 6. Effect on temperature evolution at the top surface ( ̅ =1) for the three-layer SMA film during heating
due to evolving material properties in the transforming layer (Layer 3).