One-Step Nanoscale Assembly of Complex Structures via Harnessing of an Elastic Instability.
ABSTRACT We report on a simple yet robust method to produce orientationally modulated two-dimensional patterns with sub-100 nm features over cm2 regions via a solvent-induced swelling instability of an elastomeric film with micrometer-scale perforations. The dramatic reduction of feature size ( approximately 10 times) is achieved in a single step, and the process is reversible and repeatable without the requirement of delicate surface preparation or chemistry. By suspending ferrous and other functional nanoparticles in the solvent, we have faithfully printed the emergent patterns onto flat and curved substrates. We model this elastic instability in terms of elastically interacting "dislocation dipoles" and find complete agreement between the theoretical ground-state and the observed pattern. Our understanding allows us to manipulate the structural details of the membrane to tailor the elastic distortions and generate a variety of nanostructures.
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ABSTRACT: We report the results of an experimental investigation into buckling in elastic and plastic cellular materials under dynamic compression. The buckling instabilities are in the form of a global pattern switch where the square array of circular holes is transformed into a set of orthogonal ellipses. Properties of the instabilities in the elastic and plastic cellular materials are compared and contrasted. The case of the elastic structure is considered as a delayed pitchfork bifurcation. On the other hand, the response of the plastic lattice is complex, and an irreversible global instability is only found above a critical compression rate.Applied Physics Letters 01/2013; 103(15):151909-151909-4. · 3.52 Impact Factor
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ABSTRACT: We report on the development of a PDMS-SMA composite whose surface micro wrinkles can be dynamically programmed by an electrical current supplied to the SMA wire. It is advantageous over other techniques for surface topographical modulation, including portability, real-time programmability, no requirement for specific surface chemistry, operability under ambient conditions, and relative ease of control. A simplified mechanical model is also developed to describe the force-deflection balance of the PDMS-SMA composite. The wavelengths and amplitudes of the wrinkles when different currents applied to the SMA are characterized, and the experimental results agree with the theoretical model. The developed composite device can be applied to programmable modulations of surface adhesion, friction, wettability, etc.Smart Materials and Structures 10/2014; 23(11). · 2.02 Impact Factor
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ABSTRACT: An approach to obtain analytical closed-form expressions for the macroscopic 'buckling strength' of various two-dimensional cellular structures is presented. The method is based on classical beam-column end-moment behaviour expressed in a matrix form. It is applied to sample honeycombs with square, triangular and hexagonal unit cells to determine their buckling strength under a general macroscopic in-plane stress state. The results were verified using finite-element Eigenvalue analysis.Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences 07/2014; 470(2167):20130856. · 2.38 Impact Factor
One-Step Nanoscale Assembly of
Complex Structures via Harnessing of
an Elastic Instability
Ying Zhang,†,‡Elisabetta A. Matsumoto,†,§Anna Peter,†,‡Pei-Chun Lin,†,‡,|
Randall D. Kamien,*,†,§, and Shu Yang*,†,‡
Laboratory for Research on the Structure of Matter, UniVersity of PennsylVania, 3231
Walnut Street, Philadelphia, PennsylVania 19104, Department of Materials Science
and Engineering, UniVersity of PennsylVania, 3231 Walnut Street, Philadelphia,
PennsylVania 19104, and Department of Physics and Astronomy, UniVersity of
PennsylVania, 209 South 33rd Street, Philadelphia, PennsylVania 19104
Received January 16, 2008; Revised Manuscript Received February 18, 2008
We report ona simple yet robust methodtoproduce orientationally modulatedtwo-dimensional patterns withsub-100 nmfeatures over cm2
regionsviaasolvent-inducedswellinginstabilityof anelastomericfilmwithmicrometer-scaleperforations. Thedramaticreductionof feature
or chemistry. By suspendingferrous andother functional nanoparticles inthe solvent, we have faithfully printedthe emergent patterns onto
to tailor the elastic distortions and generate a variety of nanostructures.
The demand for higher density, faster speed, and lighter
devices drives the need for developing inexpensive fabrica-
tion tools that create ever more complex patterns with smaller
features.1Many current fabrication techniques rely upon top-
down processes;2Nature, on the other hand, provides us with
examples of intrinsic, bottom-up effects from the phyllotactic
growth of plants, to animal stripes, and to fingerprints. In
those systems, instabilities, packing constraints, and simple
geometries drive the formation of delicate, detailed, and
beautiful patterns. Mechanical instabilities in soft materials,
precipitated by dewetting, swelling, and buckling, are often
viewed as failure mechanisms that can interfere with the
performance of devices. Recently these instabilities have been
exploited to assemble complex patterns,2–10to fabricate novel
devices such as stretchable electronics11and microlens
arrays,12–14and to provide a metrology for measuring elastic
moduli and the thickness of ultrathin films.15,16
* To whom correspondence should be addressed. E-mail: shuyang@
seas.upenn.edu (S.Y.); email@example.com (R.D.K.).
†Laboratory for Research on the Structure of Matter, University of
‡Department of Materials Science and Engineering, University of
§Department of Physics and Astronomy, University of Pennsylvania.
|Present address: Department of Mechanical Engineering, National
Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, 10617, Taiwan.
Figure 1. PDMS membrane with a square lattice of holes before
and after swelling by toluene. Optical images of (a) the original
PDMS membrane with hole diameter D ) 1 µm, pitch P ) 2 µm,
and depth H ) 9 µm, and (b) the swollen PDMS membrane with
diamond plate structures. (c-d) Schematic illustration of (a) and
Vol. 8, No. 4
10.1021/nl0801531 CCC: $40.75
Published on Web 03/12/2008
2008 American Chemical Society
Along these lines, we have harnessed an elastic instability
in a flexible poly(dimethylsiloxane) (PDMS) membrane with
a periodic arrangement of circular pores. When exposed to
a solvent, these pores elliptically deform and elastic interac-
tions between them generate long-range orientational order
of their axes into a “diamond plate” pattern. We lace the
solvent with nanoparticles and find that solvent diffusion and
evaporation drive the convective assembly17–19of the nano-
particles onto the PDMS surface along these distorted pores.
The resulting uniform film can be transferred onto a glass
substrate via contact printing. The features of the printed
nanoparticle pattern are up to 10 times sharper than those of
the original membrane. The membrane can be reused, and
thus our method of nanoprinting through elastic patterning
and convective assembly offers a one-step, repeatable process
without the requirement of delicate surface preparation or
chemistry. The resulting structural anisotropy of the nano-
particle films could be used to generate similar anisotropic
magnetic, photonic, phononic, and plasmonic properties. Our
technique could be implemented in other material systems
such as polymers and composites that undergo swelling and
brings a new design mechanism for device engineering.
PDMS is an elastomer that has been widely used in soft
lithography for low-cost fabrication of microdevices.20Here,
we have replica-molded a PDMS membrane with circular
pores from an array of 1 µm diameter silicon pillars spaced
2 µm apart on a square lattice.21When exposed to an organic
solvent, such as toluene, PDMS gels swell by as much as
130%.22As the osmotic pressure builds, the circular pores
in the PDMS deform and eventually snap shut to relieve the
stress (Figure 1a-d), much as the joints in railways and
bridges expand and contract to maintain structural integrity
in response to changes in moisture and temperature. The
resulting “diamond plate” pattern persists over large regions
of the sample. Because the elastic deformation of the PDMS
membrane is induced by solvent swelling, the diamond plate
pattern in PDMS is stable in the wet state and snaps back to
the original square lattice once the solvent evaporates (see
Figure S1 and the movie in the Supporting Information). To
capture the diamond plate before evaporation and, more
Figure 2. Convective assembly of Fe3O4nanoparticles in toluene on the PDMS membrane with a square lattice of holes (D ) 1 µm, P )
2 µm, and H ) 9 µm). (a-c) Schematic illustration of the self-assembly process. (a) A solution of nanoparticles suspended in an organic
solvent (e.g., toluene) is deposited onto the patterned PDMS membrane. (b) The organic solvent swells the PDMS membrane and deforms
the holes into the diamond plate structure. (c) Nanoparticles convectively assemble on the swollen PDMS membrane, capturing the deformed
structure. The nanoparticle film is then transferred to a glass substrate by microcontact printing. (d) Scanning electron micrograph (SEM)
of the original PDMS membrane. (e) SEM of the deformed PDMS membrane after casting a drop of nanoparticle/toluene solution on
PDMS, followed by heating in an oven at 65 °C for 2 h and removal of the nanoparticle film. The diamond plate structure can be clearly
seen in the PDMS membrane although the shape is somewhat relaxed compared to that of nanoparticle assembly due to the elastic nature
of the PDMS. (f) SEM of the nanoparticle pattern lifted off from the deformed PDMS membrane. Inset: higher magnification of an individual
elliptical nanoparticle deposit, which is 78 nm wide by 2.3 µm long. The ridge height is 392 nm according to atomic force microscopy
(AFM) section analysis. (g) A larger field of view of the nanoparticle pattern, illustrating high uniformity over a large area. (h) SEM of the
dislocation in the diamond plate pattern. Note that the overall symmetry of the sample is unaffected by the defect. (i) Optical photography
of the nanoparticle film that was transfer-printed on a glass capillary. The colorful reflection is due to the diffraction grating of the underlying
pattern, which is determined by the periodicity and refractive index of the film and the viewing angle.
Nano Lett., Vol. 8, No. 4, 2008 1193
importantly, to utilize this deformation for assembly of
complex functional structures, we suspended superparamag-
netic Fe3O4nanoparticles (∼10 nm in diameter) in toluene
and applied the solution to the PDMS membrane (Figure
2a). As the PDMS swells (Figure 2b), the convective
assembly of the nanoparticles follows (Figure 2c), which
faithfully replicates the deformed PDMS membrane (Figure
2e-f). Once dried, the elastic membrane returns to its
original state (Figure 2d) and can be reused. We find that
the nanoparticle film can be transfer-printed onto another
substrate, which can be either hydrophobic (e.g., Au and
polystyrene) or hydrophilic (e.g., glass).23,24The ability to
release our nanoparticle films from PDMS (i.e., the carrier)
onto a targeted substrate may be attributed to the difference
in particle-substrate adhesion,25,26which strongly depends
on both the interfacial energy and contact area. During solvent
evaporation, capillary forces hold the particles tightly together
on the PDMS surface; the low surface energy and compliance
of PDMS, on the other hand, maximizes the contact of nano-
particles on the hard substrate for subsequent transfer. Support-
ing this, we find that when air-dried for 2 h or dried in an oven
at 65 °C for 10 min, the nanoparticle films could no longer be
reduced film quality when transferred onto a hydrophobic
surface. This observation suggests that residual solvent may
play an important role in wetting, which increases the contact
between the nanoparticle film and the target substrate,
therefore increasing the adhesion and transfer yield.26
Scanning electron microscopy (SEM) (see Figure 2f inset)
reveals a dramatic reduction in feature size: the width of
each elliptic nanoparticle deposit (78 nm) is smaller than
1/10the diameter of the initial pore (1 µm). The separation
between neighboring ellipses remains 2 µm, demonstrating
that the deformation preserves the original lattice. Other
groups27report similar feature size reduction, from ∼1.6 µm
to ∼200 nm, however this comes about via several cycles
of compression and replication of the PDMS molds.
Previously, the elastic instabilities of a polymer matrix
have been harnessed to produce patterns exhibiting long-
range order; when a polymer gel network is swollen by a
solvent, the osmotic pressure competes with the elasticity
of the inner gel surface to produce beautiful and regular
wrinkle patterns with wavelengths on the order of tens to
hundreds of micrometers.3,7,16Our approach differs in that
we need not balance elastic forces with kinetic effects. Even
more fascinating, our nanopatterning method does not require
delicate template preparation or surface chemistry28,29and
we achieve a ground state with long-range order. As shown
in Figure 2g, this “diamond-plate” pattern persists over the
entire sample (up to cm2in our experiments depending on
the available size of the original Si master) with no random
defects; the “phase slip” dislocations (Figure 2h) do not alter
the overall symmetry of the system, thus preserving many
of the interesting physical properties of the film. For instance,
the brilliant red-orange image shown in Figure 2i is diffrac-
tion from our highly regular, optically anisotropic product.
To understand the response of the membrane under
swelling, we have used continuum elasticity theory to model
the local deformation of each pore as a small amount of
material inserted into an unstressed elastic sheet. Formally,
we may view this insertion as two equal and opposite edge
dislocations with macroscopic Burgers vectors forming an
effective “dislocation dipole” (as defined in Figure 3a).
According to linear elasticity theory,30the energy of a
deformation given in terms of the linearized strain tensor uij
) (∂iuj+ ∂jui)/2 is
where λ and µ are Lamé coefficients. However, to find the
ground-state configuration, we need only consider the energy
of interaction between two dislocation elements, as the self-
energy is independent of the orientation of the dislocation
dipole. The interaction energy per hole of two dislocation
dipoles is dependent only on their dislocation densities bb1(x b)
and bb2(x b)
where Y2 ) 4µ(λ + µ)/(λ + 2µ) is the two-dimensional
Young’s modulus. From this expression, we obtain the
interaction energy per hole
Figure 3. Theoretical calculation of minimum energy states of
square lattice of holes using dislocation dipoles. (a) Diagram of
the interaction between two dislocation dipoles separated by R b. The
Burgers vector’s density for a dislocation dipole of strength d
located at x b is bb(x b) ) z ˆ × dˆb[δ2(x b - db/2) - δ2(x b + db/2)]. (b)
Several local configurations of pairs of dipoles and their respective
energies per hole (offset by an angle-independent constant), all given
in units (Y2b2d2)/2πR2. (c) The minimal energy configuration for a
2 × 2 plaquette of dipoles. The angles
are θ2) θ3) –1/2arcsin(1/10) ≈ 0.05, θ1) θ4) θ2+ π/2. This
configuration has an energy of –2.55(Y2b2d2)/(2πR2). (d) Minimum
energy configurations 20 × 20 square lattice of holes obtained by
numeric minimization of the pairwise interactions of a 20 × 20
square lattice of dipoles.
Nano Lett., Vol. 8, No. 4, 2008
[cos(θ1+θ2) sin(θ1) sin(θ2)+1
where θ1 and θ2 are the angles each dipole makes with
respect to the vector separating them and R is the distance
between them (see Figure 3a). The long-range behavior of
the interaction is responsible for the long-range order. The
energies of several local configurations are shown in Figure
3b. The ground-state configuration for a given lattice is
determined by minimizing the sum of all possible pairwise
interactions over each angle. The ground state of a 2 × 2
plaquette of holes is a set of mutually perpendicular
dislocation dipoles (Figure 3c). Numeric minimization of the
energy for larger square lattices (Figure 3d) corroborates not
only the results of the 2 × 2 plaquette (Figure 3c) but also
those of our experiments (Figure 1b). Moreover, we see that
the larger lattices have a higher degree of alignment with
the lattice axes, confirming that interaction with distant
dipoles is essential and provides the long-range fidelity of
the patterns over centimeter length scales. Our model should
apply at all length scales: indeed, the resulting diamond plate
aligned along the two lattice directions is similar to the
deformation pattern recently seen in 10 × 10 arrays of
millimeter size holes in elastomeric cellular solids subjected
to uniaxial compression.31
Because the emergent pattern is the result of energy
minimization of the neighboring pores upon deformation,
we can tune the specific characteristics of each nanoparticle
film simply by changing the diameter and spacing of the
holes and the lattice symmetry in the PDMS membrane.
Figure S2 (Supporting Information) shows diamond plates
with different lengths and widths obtained from the square
lattice of the PDMS membranes with variable pore size and
spacing, demonstrating the transferability of our simple
process. The dipole interactions require that the swollen
regions around each pore overlap. Thus, the spacing of the
pores and the rate of diffusion can influence the diamond
plate pattern. In our experiments the diamond plate is
obtained in PDMS membranes with high aspect ratios (height
H/diameter D ) 2 to 18) and pitch P/diameter D ratios up
The analytic model posed here enables us to rationally
design new motifs via externally imposed elastic forces. By
mechanically stretching the perforated PDMS membrane with
square lattice along a lattice direction (Figure 4a),32we exert
an external force on the dislocation dipoles favoring align-
ment along the strain direction. This competes with the
internal stresses caused by swelling, and by varying the
strength of the external stress, we are able to create an even
richer library of morphologies. These patterns vary continu-
ously from slight distortions of the original pattern up to
10% strains, to a binary pattern of circles and lines from
30–50% strain, to a rectangular lattice of aligned ovals for
strains in excess of 50% (see Figure 4b-c).
We have demonstrated a one-step method for the rational
design of functional nanoscale motifs with adjustable feature
size and shape. By taking advantage of the elastic instability
Figure 4. Complex patterns of Fe3O4nanoparticle assembly obtained from PDMS membranes that are mechanically stretched at different
strain levels. (a) Schematic illustration of the mechanical stretching of the PDMS membrane. (b) Schematics of the lattice configurations
at different strain levels. (c) Corresponding SEM images of Fe3O4nanoparticle assemblies from the stretched PDMS membranes. The
original membrane has a square lattice of holes with D ) 750 nm, P ) 1.5 µm, and H ) 9 µm.
Nano Lett., Vol. 8, No. 4, 2008 1195
caused by swelling in a patterned elastomeric membrane,
we offer an elegant means to direct the formation of complex
ordered patterns while simultaneously capturing nanoparticles
into structured films via convective assembly. Simply chang-
ing parameters in the patterned PDMS membrane facilitates
transformation to complex morphologies. These structures
are easily transferred to both flat and curved surfaces as well
as superimposed to create multilayered devices. We anticipate
that the method presented here will be the basis for designing
new tools in nanoscale manufacturing and for dynamically
Experimental Section. Replica-Molding of Patterned
PDMS Membrane.21Silicon masters with a square lattice
of cylindrical pillars were fabricated by the conventional 248
nm photolithography/etching process. The pillars have
diameter D ranging from 350 nm to 2 µm, pitch P ranging
from 800 nm to 5 µm, and height H ranging from 4 to 9
µm, respectively. The silicon masters were cleaned by
sonication in deionized water, isopropyl alcohol, and acetone
for 10 min, respectively. They were then placed in the oxygen
plasma cleaner (PDC-001, Harrick Scientific Products, Inc.)
at 23 W for 10 min. Finally, they were silanized with release
vapor (Gelest Inc.) for 2 h under vacuum in a desiccator.
The PDMS prepolymer and its curing agent (RTV615, GE
Silicones) were mixed in a weight ratio of 10:1 and degassed
to remove air bubbles. The prepolymer mixture was poured
onto the silicon master and cured in a convection oven at
80 °C for two hours before removal by careful pealing.
Convective Assembly of Fe3O4Nanoparticles on PDMS
Membrane. Superparamagnetic Fe3O4nanoparticles (average
diameter of 10 nm) were extracted from a commercially
available ferrofluid (EMG911, Ferrotec Corporation, USA).
First, 5 mL of acetone was added into 1 mL of ferrofluid to
precipitate the nanoparticles, which were then collected using
a strong magnetic bar. The nanoparticles were washed with
acetone three times and redispersed into toluene with a
concentration of 2.5% w/v. A small amount of the Fe3O4
nanoparticle toluene solution was applied onto the PDMS
membrane using a cotton swab and blow-dried under N2for
20 s. The obtained thin nanoparticle film was then transfer-
printed onto a cover glass and lifted off from the PDMS
membrane for further study.
Convective Assembly of Fe3O4Nanoparticles on Me-
chanically Stretched PDMS Membranes. A customized jig
was constructed from a large acrylic base and two sliders
whose positions could be adjusted continuously by two long-
thread M4 wing screws. Small binder clips connected to each
of the two sliders were clamped to the edges of the PDMS
membrane (see Figure 4a). After stretching the PDMS
membrane,32the nanoparticle solution was applied using the
same method described above.
Acknowledgment. We thank J. Ashley Taylor (University
of Wisconsin) for providing some Si masters and Dinesh
Chandra for insightful discussion. This work was supported
by NSF MRSEC DMR05-20020. Y.Z., P.L., and S.Y. were
also supported in part by NSF CAREER award DMR-
0548070 and a 3M Nontenured Faculty Grant. E.M. and
R.D.K. were supported in part by gifts from L. J. Bernstein
and H. H. Coburn.
Supporting Information Available: In-situ optical and
AFM images of partially restored perforated PDMS mem-
brane during toluene evaporation shown in Figure 1 and
corresponding video (.qt). SEM images of Fe3O4nanoparticle
assemblies from different PDMS membranes with square
lattices. This material is available free of charge via the
Internet at http://pubs.acs.org.
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