Combining pattern instability and shape-memory hysteresis for phononic switching.

Abstract

We report a fully reversible and robust shape-memory effect in a two-dimensional nanoscale periodic structure composed of three steps, the elastic instability governing the transformation, the plasticity that locks in the transformed pattern as a result of an increase in glass transition temperature (T(g)), and the subsequent elastic recovery due to the vapor-induced decrease in T(g). Solvent swelling of a cross-linked epoxy/air cylinder structure induces an elastic instability that causes a reversible change in the shape of the void regions from circular to oval. The pattern symmetry changes from symmorphic p6mm to nonsymmorphic p2gg brought via the introduction of new glide symmetry elements and leads to a significant change in the phononic band structure, specifically in the opening of a new narrow-band gap due to anticrossing of bands, quite distinct from gaps originating from typical Bragg scattering. We also demonstrate that numerical simulations correctly capture the three steps of the shape-memory cycle observed experimentally.

Full text

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Combining Pattern Instability and
Shape-Memory Hysteresis for Phononic
Switching
Ji-Hyun Jang,†Cheong Yang Koh,†Katia Bertoldi,‡,§Mary C. Boyce,*,‡
and Edwin L. Thomas*,†
Institute for Soldier Nanotechnologies, Department of Materials Science and
Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139,
Institute for Soldier Nanotechnologies, Department of Mechanical Engineering,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, and Multi
Scale Mechanics, TS, CTW, UniVersiteit Twente, P.O. Box 217,
7500 AE Enschede, The Netherlands
Received February 25, 2009; Revised Manuscript Received April 10, 2009
ABSTRACT
Wereport afullyreversibleandrobust shape-memoryeffect inatwo-dimensional nanoscaleperiodicstructurecomposedof threesteps, the
elasticinstabilitygoverningthetransformation, theplasticitythatlocksinthetransformedpatternasaresultofanincreaseinglasstransition
temperature (Tg), and the subsequent elastic recovery due to the vapor-induced decrease in Tg. Solvent swelling of a cross-linked epoxy/air
cylinder structure induces an elastic instability that causes a reversible change in the shape of the void regions fromcircular to oval. The
pattern symmetry changes fromsymmorphic p6mmto nonsymmorphic p2gg brought via the introduction of new glide symmetry elements
and leads to a significant change in the phononic band structure, specifically in the opening of a newnarrow-band gap due to anticrossing
ofbands, quitedistinctfromgapsoriginatingfromtypical Braggscattering. Wealsodemonstratethatnumerical simulationscorrectlycapture
the three steps of the shape-memory cycle observed experimentally.
Structures with a variety of geometries with nanometer-scale
features and submicrometer periodicities have many applica-
tions in optics and acoustics,1-5and importantly their
properties can be quite sensitive to structural details. Revers-
ible control of feature shape in periodic structures via an
external triggering stimulus provides an opportunity for
tunable devices such as optical/sonic switches. Phononic
crystals offer new functionalities including negative effective
bulk modulus and effective mass density.6-8This is made
possible by the purposeful design of the pattern at the
requisite length scales to interact with the phonons. Most
work to date on phononic structures has concentrated on
obtaining large band gaps (∆ω/ωgap) and, to a lesser extent,
lensing effects.9The former still remains very much an open
challenge with no systematically superior way of searching
for optimal band structures.6,10In addition, dynamic tunability
of phononic properties would provide a significant step
toward the adoption of phononic crystals for practical device
applications such as a switch where the properties change
upon the activation of an external stimulus. Ideally, the small
triggering field allows the structure to remain in the “linear”
regime of operation with the cycling hysteresis to ensure
device repeatibility, long device lifetime, etc. In order to
induce useful effects within the same frequency window, the
pattern change should ideally occur without change in pattern
periodicity but at the same time create a significant change
in the pattern symmetry/geometry so as to significantly alter
the wave propagation behavior and hence phononic properties
of the periodic structure.
Recently it was shown that mechanical instability in
elastomeric structures can trigger a dramatic pattern trans-
formation11,12and can alter the theoretical phononic band
gaps.13However, the transformed pattern has new period-
icit(ies) and reverts to the initial shape upon elimination of
the stimulus (solvent swelling or applied mechanical
force).11,12,14-16
Hence there is a need to find a means to capture the shape
changes created by the mechanical instability and then
recover the original shape with the same triggering field
avoiding dependence on the duration of the triggering field.
* Corresponding authors: Edwin L. Thomas, telephone (617) 253-6901,
fax (617) 253-5859, e-mail elt@mit.edu; Mary C. Boyce, telephone (617)
253-2342, fax (617) 258-8752, e-mail mcboyce@mit.edu.
†Institute for Soldier Nanotechnologies, Department of Materials Science
and Engineering, Massachusetts Institute of Technology.
‡Institute for Soldier Nanotechnologies, Department of Mechanical
Engineering, Massachusetts Institute of Technology.
§Multi Scale Mechanics, TS, CTW, Universiteit Twente.
NANO
LETTERS
2009
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We utilize a mechanical instability to trigger an abrupt
change in the phononic properties. SU8, the material of
choice for the fabrication of complex two-dimensional (2D)
and three-dimensional structures in microelectromechanical
systems, is used as our matrix due to its excellent exposure-
contrast as well as good mechanical and thermal stability.17,18
High rotational symmetries usually promote the ability to
obtain a complete band gap by virtue of greater symmetry.
This is the case for the band gaps which occur at Brillouin
zone boundaries formed via the well-known Bragg type
scattering mechanisms. The underlying physical theory that
embodies the interpretation and indeed the design of a
phononic structure is steeped in the elegance of group
theory.19Phase transitions typically involve symmetry changes
from supergroups to subgroups and afford limited switching
capabilities. Another possibility to open a band gap is through
inducing anticrossing of bands that possess the same sym-
metries; their like-symmetries cause them to interact as they
cross each other in k-space, leading to level repulsion and,
hence, avoided crossing.
The requirements for a complete band gap rest upon
several different parameters, including the volume fractions
of the two components, the network topology, ratios of the
transverse and longitudinal velocities of each component,
and the ratios of these velocities between the respective
components. The hexagonal void cylinder/epoxy matrix
structure requires an air cylinder radius rcof >0.44a where
a is the lattice parameter for existence of a complete band
gap;10hence our starting p6mm structure with rc) 0.336a
(a ) 610nm, rc) 205 nm) and void volume fraction of 0.41
possesses only partial band gaps. The patterned SU8 film
fabricated by interference lithography20is 1.5 µm thick with
a 700 nm thick SU8 buffer layer adhering the film to a glass
substrate.
Figure 1a shows the theoretical band diagram for the p6mm
structure utilizing a primitive unit cell; note there is only a
partial band gap centered at 3.25 GHz (∆ω/ω ∼ 10%) along
the Γ-M direction. In order to make a comparison to the
p2gg structure which has two void regions per unit cell, we
recalculated the p6mm structure utilizing a supercell (Figure
1b). We focus on the change in symmetry of the 20th and
21st bands of the p6mm supercell. After the transformation,
an avoided crossing occurs in the p2gg structure, causing
the previous partial band gap to fully open and form a
complete gap between the 20th and 21st bands (Figure 1c).
The in-plane displacement fields of bands 20 and 21 of the
original p6mm are shown in Figures 1d and 1e and possess
antisymmetric and symmetric displacements about the mirror
line, which allows the bands to cross at the K point. Parts f
and g of Figure 1 show the displacement fields corresponding
to the same two bands in the p2gg structure which now
possess like symmetries. Their symmetry originates from the
glide symmetries present in the p2gg structure. A complete
discussion is beyond the scope of this Letter and will be
addressed in a future work (see Supporting Information for
the videos of the displacement fields accompanying mode
propagation which show the glide symmetry). The formation
of such flat, relatively dispersionless bands (see bands 21
and 22 in Figure 1c) is favored at higher frequencies where
the original bands are typically flatter and hence offer the
possibility for switchable narrowband filters.
Brillouin light scattering (BLS)21,22was employed for
experimental verification of the band dispersion relations
because of the high frequency (gigahertz) operating range
of these phononic crystals. BLS allows for a direct measure-
ment of the phonons present in the structure at a particular
scattering vector corresponding to the position in k-space;
hence the band diagram is obtained directly in this way. In
all of our experiments, the collection was done in the VV
geometry; hence only modes which are quasi-longitudinal
or mixed modes which have symmetries that can couple to
the incident radiation with the set polarization will be
detected.23In addition, due to the wings of the central
Rayleigh peak, frequencies below 1 GHz are not detectable.
The experimentally measured phononic band dispersion
superposed onto the theoretical band structure diagrams of
the original p6mm and the transformed p2gg structure are
shown in parts a-d of Figure 2. Measurements were taken
in the two principal directions (corresponding to [10] and
[11]) along the ΓP and ΓQ directions for the original and
transformed structure. We plot the experimental data ac-
cording to the actual scattering angle geometries and
superpose onto the theoretical band diagrams that are plotted
in the repeated zones scheme, which reflects the actual
physical scattering situation more accurately than the reduced
zone scheme. There is good agreement between the theoreti-
cal band dispersion diagram and the experimentally detected
phonon modes. In addition, we also detect two dispersionless
modes, which correspond to the glass substrate and the SU8
buffer layer present in our samples. This buffer layer signal
also serves as an internal reference for the detected intensity
signals. To experimentally illustrate the differences in the
phononic band properties of the two structures, parts e and
f of Figure 2 compare the frequency spectra taken at the
scattering angle q ) 0.00288 nm-1, in the [10] directions,
as indicated. The peaks in the spectra were fitted with a
multiple Lorentz oscillator model,6,23the insets in parts e
and f of Figure 2 providing evidence for the individual
(dashed lines) and combined (solid lines) fit to the experi-
mental spectra, showing the clear change in the phononic
properties as the structure is transformed from p6mm to p2gg.
It is important to note that, besides a change in the
frequencies of modes, the nature of the corresponding
displacement fields would be different as well.
We trigger the pattern instability/shape memory effect of
the 2D patterned SU-8 utilizing a good solvent to alter the
mechanical properties of SU-8 by shifting its Tg through
plasticizing the polymer while imparting stress fields through
concurrent swelling. This allows the shape transformation
to occur as a result of a bifurcation-type elastic instability,
which is only accessible in the rubbery regime, as shown in
Figure 1h. At relatively small strain fields (∼5%), the SU-8
undergoes a bifurcation elastic instability, inducing the
transformation from p6mm to a p2gg structure.
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Figure 1. Phononic band structure calculations before and after transformation. Theoretically calculated in-plane phononic band diagrams (a) for
the original p6mm and (b and c) for the original p6mm, using a supercell and the transformed p2gg structures. The supercell is utilized in (b) in
order to keep the same unit cell size as in (c), for comparison. The red arrow labeled in (b) points to the position in k space where the crossing
occurs in the p6mm, whereas the arrow in (c) points to that where anticrossing occurs, causing the gap to open. Note also that as a result of the
reduced symmetry, several high symmetry points (M, K points) in (c) have the degeneracies lifted in. (d and e) Displacement fields of corresponding
modes 20 and 21 in p6mm which cross (indicated in Figure 1b). (d) is antisymmetric about the mirror plane while (e) is symmetric, thus
crossing is allowed. (f and g) Displacement fields of corresponding modes 20 and 21 in p2gg which have interacted to avoid crossing, both
(f) and (g) have symmetry corresponding to the glide elements (refer to Supporting Information for attached videos to show clear dependence).
All displacement fields have the k vector along the [11] direction. (h) The strain energy of the structure as a function of the strain induced
by the swelling (εsw), showing the bifurcation which occurs at a critical swelling strain, favoring transformation from the p6mm (blue) to
the p2gg (red) structures.
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The sequence of proceeding steps for the cyclable shape-
memory effect in our 2D patterned SU8 material involves
three simple steps (see Figure S1 in Supporting Information).
In step I, the cross-linked network is swollen in a good
solvent (NMP (N-methylpyrrolindone)) leading to an elastic
instability at a certain swelling strain. An ESEM (environ-
mental scanning electron microscope) image of the NMP
swollen structure demonstrates the transformation occurring
during the swelling process (see Figure S2 in Supporting
Information).
In step II the sample is immersed in a large amount of
isopropyl alcohol (IPA), a nonsolvent. This causes the
outermost surface of the swollen epoxy to lose NMP much
faster than the interior. The outer surface regions retract to
increase the size of the solvent/nonsolvent filled voids as
the epoxy contracts (and the local Tgincreases). When this
near-surface region attains a Tgabove room temperature, it
ceases to deform, while the matrix around this “surface skin”
continues to contract as the solvent continues to diffuse out.
As the IPA/NMP solution is completely evaporated, the
Figure 2. Phononic band structure measurements and calculations. Superposed experimentally measured phononic dispersion modes and
the theoretical band dispersion relations in the ΓM and ΓK directions, respectively, for (a and b) the undeformed p6mm structure and (c and
d) the deformed p2gg structure. The solid red line is the theoretical line for the glass; the open red circles are the corresponding experimental
points. The dashed blue line is the theoretical line for the SU8 buffer layer present, and the solid blue triangles are the corresponding
experimental data points. The solid black lines correspond to the quasi-longitudinal modes, the dashed lines the transverse modes (which
cannot be detected), dotted lines correspond to mixed modes of various symmetries corresponding to the respective symmetry group of the
structure, with solid circles the respective experimental data points. (e and f) The experimental frequency spectra taken at q ) 0.00287
nm-1in the ΓM direction of the structures before and after transformation. The frequency spectra correspond to the dashed lines in parts
a and c of Figure 2, respectively, which are cuts of the dispersion relation at q ) 0.00287 nm-1. The solid lines are combined fits to the
experimental spectra while the dashed lines are fits for the individual peaks, illustrating the peak positions.
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sample shrinks in volume but retains the transformed pattern
due to the high activation barrier required to modify the rigid
epoxy surface regions. Because of the confinement of the
patterned SU8 by the highly cross-linked thin (700 nm) SU8
buffer film fixed to the rigid glass substrate, there is no
detectable change in the center to center spacing of the voids
between deformed and original patterns as indicated in fast
Fourier transform (FFT) patterns (Figure 2b).24
Step III is the recovery process to the original state (state
1) by a short time (5 min) exposure to NMP vapor. Upon
exposure, the deformed sample absorbs a sufficient amount
of NMP, to drop the Tg but not enough to retrigger the
swelling induced instability, facilitating relaxation of the
stresses and reversion to the original sample pattern (Figure
3c).
By reimmersion of recovered samples into NMP, the cycle
composed of sequential steps, from the transformation by
swelling-induced pattern instability to subsequent freezing
to fix the new hole shape, and finally the vapor recovery to
release the stress, can be repeated. Key to our process is the
hysteresis in the instability arising from the gradient in
solvent concentration (creating a gradient in polymer mobil-
ity) during the IPA quench step which causes the formation
of sufficiently rigid regions that arrest the full recovery to
the original structure. Only with subsequent exposure to the
solvent vapors is there sufficient simultaneous sample
mobility (yet with low swelling so as to avoid the instability
regime), to permit the shape change to bring the structure
back to its original state with complete solvent evaporation.
To better understand the mechanics of the instability and
shape-memory effects16,25,26observed, numerical simulations
were conducted. The stress-strain behavior of SU-8 is
captured using a two-mechanism constitutive model27(see
Supporting Information). The stress response is decomposed
into two mechanisms: the resistance due to stretching and
orientation of the molecular network (σN) and the resistance
arising from intermolecular interactions (σV). The shape
memory behavior is taken into account by taking σv to
depend on (T-Tg), where T is room temperature. The
intermolecular interactions are negligible when T > Tgand
the material is characterized by a rubbery behavior; however,
as the Tgmoves through room temperature, intermolecular
interactions increase and lock in the deformation. The
constitutive model is implemented into the commercial finite
element code ABAQUS enabling simulation of the swelling,
the instability, and the shape memory behaviors. Figure 4c
shows the normalized Von Mises stress (σVM/σVMMax) and
the ratio between the two axes of the holes (b/a) as a function
of swelling strain (εsw) or T-Tgshowing clear agreement
with the experimentally observed instability and shape-
memory behavior. The analysis consists of three steps as
indicated in Figure 4a (see Figure S4 in Supporting Informa-
tion for Tgat each step):
Figure 3. Pattern instability and shape-memory effect for phononic switching. (a) SEM image of original p6mm hexagonal lattice with
circular air cylinders and a periodicity of 610 nm. (b) SEM image of new (temporary) oval air hole shapes having p2gg symmetry due to
the liquid solvent swelling induced instability, followed by subsequent nonsolvent freezing of the oval hole shape. (c) SEM image of the
structure from (b) after relaxation by solvent vapor treatment. The transformed pattern reverts back to the original structure. (d) SEM image
of the temporary air hole shapes in the 10th transformation cycle demonstrating the reversibility of the process. The permanent and temporary
shapes (c), (d) in 10th cycle are very similar to the initial shapes (a), (b) at the beginning of the cycle. The insets are calculated FFT power
spectra from SEM images, demonstrating that the center to center spacing of the holes is constant during pattern transformation.
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Step I. Immersion in NMP and swelling. Tgdecreases to
∼ -10 °C, so that the σvvanishes. Bloch wave analysis
detects a mechanical instability occurring for a swelling strain
εsw) 0.04 leading to swelling-induced pattern transformation
(Figure 4a-B).
Step II. Immersion in IPA and subsequent drying. The
material Tg increases to ∼112 °C, and the intermolecular
interactions increases, making the material much stiffer and
providing retention of the transformed pattern (Figure 4a-C).
Step III. NMP vapor treatment and subsequent drying.
The material Tgdecreases to ∼4 °C so that the structure again
exhibits a rubbery behavior (σvvanishes again) and the initial
shape and pattern are elastically recovered (Figure 4a-A).
In conclusion, we have shown a simple way to provide a
cyclable shape-memory effect in nanoscale lattices fabricated
via IL that allows for potential property control in many
useful devices. The internal stress arising due to swelling of
cross-linked glassy SU8 structures in a good solvent followed
by quenching of the deformed structure in a poor solvent
creates a symmorphic to nonsymmorphic structural trans-
formation with the consequence of anticrossing of certain
phononic bands resulting in the opening of a complete band
gap. Removal of the stress by lowering the Tgvia exposure
to good solvent vapor returns the structure back to its original
state. Numerical simulations correctly capture the shape-
memory effect observed experimentally. The pattern instabil-
ity and shape-memory effects we have shown in the periodic
nanoscale structures suggest applications in tunable optical
and sonic applications.
Acknowledgment. We thank Steven E. Kooi for technical
assistance and Taras Gorishnyy for helpful discussion. This
work is supported in part by the Institute for Soldier
Nanotechnologies of the U.S. Army Research Office with
Contract No. W911NF-07-D-0004 (E.L.T., M.C.B.), the
National Science Foundation with Grants CMS-0556211
(E.L.T.) and DMR-0804449 (E.L.T.).
Supporting Information Available: The interference
lithographic setup, preparation of samples, basic character-
ization of the structures with transformed and recovered
patterns, details of the constitutive model, numerical simula-
tion of shape-memory effect, numerical calculation of
phononic band structure, and BLS measurement. This
material is available free of charge via the Internet at http://
pubs.acs.org.
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