Mechanical Self-Assembly of a Strain-Engineered Flexible Layer: Wrinkling, Rolling, and Twisting

Self-shaping of curved structures, especially those involving flexible thin layers, is attracting increasing attention because of their broad potential applications in, e.g., nanoelectromechanical andmicroelectromechanical systems, sensors, artificial skins, stretchable electronics, robotics, and drug delivery. Here, we provide an overview of recent experimental, theoretical, and computational studies on the mechanical self-assembly of strain-engineered thin layers, with an emphasis on systems in which the competition between bending and stretching energy gives rise to a variety of deformations, such as wrinkling, rolling, and twisting. We address the principle of mechanical instabilities, which is often manifested in wrinkling or multistability of strain-engineered thin layers. The principles of shape selection and transition in helical ribbons are also systematically examined. We hope that a more comprehensive understanding of the mechanical principles underlying these rich phenomena can foster the development of techniques for manufacturing functional three-dimensional structures on demand for a broad spectrum of engineering applications.

Figure 1

Schematics of a (a) released bilayer, (b) bent bilayer with inner radius $R$, and (c) wrinkled structure with deflection profile $\zeta (x,y)$, amplitude $A$, and wavelength $\lambda$. (d) $E_w$ (solid line) and energy of planar relaxation (dashed line) as a function of $h$. (e) Wavelength $\lambda$ (solid line, left axis) and amplitude $A$ (dashed line, right axis) as a function of length $h$, in the case of wrinkled structure. The vertical dotted-dashed line marks the $h_{\mathrm{cw}}$. Reprinted with permission from Ref. [52] (© 2009 by the American Physical Society).

Figure 2

Phase diagram of favorable shapes of a strained bilayer based on the energetic comparison between $E_w$ and $E_{\text{bent}}$. The solid curve indicates the boundary separating the bent and wrinkled shapes. $R_{\mathrm{eq}}$ is shown for the bent structure and wavelength $\lambda$ for the wrinkled structure. The dashed curve is the phase boundary curve for $\overline{ϵ}=-1.0\%$. Reprinted with permission from Ref. [52] (© 2009 by the American Physical Society).

Figure 3

Morphology evolution for increasing etching time. SEM images are shown for (a) $h=3.4\mu \mathrm{m}$ (top view); (b) $h=5.7\mu \mathrm{m}$ (top view); (c) $h=5.7\mu \mathrm{m}$ (side view) and (d) $h=10.3\mu \mathrm{m}$ (side view). Adapted from Ref. [64] with permission of The Royal Society of Chemistry.

Figure 4

(a) A schematic illustrating the formation of the wrinkles perpendicular to the etching front. The subsequent bond-back effect leads to the formation of a nanochannel network. Adapted with permission from Ref. [68] (© 2007). (b) Optical microscopy images showing a linear nanochannel network with single-sided (upper panel) and double-sided branch channels (middle panel). A corresponding SEM image (bird view) of a single-sided linear nanochannel network is given in the lower panel. Adapted with permission from Ref. [68]. (c) Si ribbon structures formed on a PDMS substrate prestrained to 50%. Reprinted with permission from Macmillan Publishers Ltd: Nature Nanotechnology, Ref. [69] (© 2006).

Figure 5

(a) Schematic of the unetched (strained QW), bonded-back (partially relaxed QW), and wrinkled (bent QW) structures. (b) Band diagram and quantized energy levels corresponding to the structure shown in (a). Light gray lines are guides to the eye. (c) Transition energy of the bent QW as a function of the bending curvature. Three bent QW models are presented. Reprinted with permission from Ref. [87] (© 2007 American Chemical Society).

Figure 6

Micro-Raman mapping measurements of the wrinkled Si layer, and the corresponding section analysis: (a),(c) 514-nm laser; (b),(d) 325-nm laser. Reprinted with permission from Ref. [71] (© 2013, AIP Publishing LLC).

Figure 7

Theoretical model for the strain distribution in the wrinkled silicon layer. Reprinted with permission from Ref. [71] (© 2013, AIP Publishing LLC).

Figure 8

When a bilayer is freed, its top layer contracts, and its bottom layer expands, causing the bilayer to roll up. From Ref. [96], reprinted with permission from AAAS.

Figure 9

Optical images of rolled-up microtubes made out of (a) Pt, (b) $\mathrm{Pd}/\mathrm{Fe}/\mathrm{Pd}$, (c) ${\mathrm{TiO}}_2$, (d) ZnO, (e) ${\mathrm{Al}}_2{\mathrm{O}}_3$, (f) ${\mathrm{Si}}_x{\mathrm{N}}_y$, (g) ${\mathrm{Si}}_x{\mathrm{N}}_y/\mathrm{Ag}$, and (h) diamondlike carbon. Adapted with permission from Ref. [94] (© 2008).

Figure 10

A patterned wheel of anchored rectangular pads (a) before and (b) after lithography in SEM, with magnified pads displayed in (b). ${\mathrm{In}}_{0.3}{\mathrm{Ga}}_{0.7}\mathrm{As}/\mathrm{GaAs}$ film is grown on (100) GaAs. Reprinted with permission from Ref. [114] (© 2008 IEEE).

Figure 11

Two- and three-dimensional plots of the rolling direction as a function of the tube circumference ($c$), width ($a$), and rectangular layer length ($b$). Blue indicates long side rolling, red short side, and green for a combination. Adapted with permission from Ref. [63] (© 2010 American Chemical Society).

Figure 12

Schematic illustration of the shape evolution of chiral bilayer membranes (a) with the molecules tilted with respect to their neighbors (indicated by the arrows) twisted into a helical ribbon (b) and a closed tubule when the lipid molecules are deposited from the saturated solution. From Ref. [128], reprinted with permission from AAAS.

Figure 13

(a) $\mathrm{InGaAs}/\mathrm{GaAs}$ strain-induced curl; (b) releasing the strain generates helices; (c) bilayer fabrication; (d) helix pitch is a function of pattern orientation. Adapted with permission from Ref. [145] (© 2006 American Chemical Society).

Figure 14

SEM top view images of $\mathrm{SiGe}/\mathrm{Si}/\mathrm{Cr}$ helical nanoribbons (the layer thickness is $11/8/21\mathrm{nm}$). The inset in (h) does not have a Cr layer. The arrows in (a) denote the $⟨110⟩$ direction on the substrate. All the strips in (a)–(g) have a misorientation angle of 10° from $⟨110⟩$. The width decreases from 1.30 to $0.70\mu \mathrm{m}$ at an interval of 100 nm from (a) to (h). In (h), both nanoribbons have a misorientation of 5°. Adapted with permission from Ref. [10] (© 2006 American Chemical Society).

Figure 15

Tunable helical ribbons. (a) Illustration of a helical ribbon. The principal bending directions, ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$, are rotated from the geometric axes, ${\mathbf{d}}_x$ and ${\mathbf{d}}_y$, by a misorientation angle $\phi$. (b),(c) A piece of latex-rubber sheet is prestretched twice as much in the vertical direction than in the horizontal direction before a series of adhesive strips are attached to it along different misorientation angles ($\phi =0^{°}$, 15°, 30°, 45°, 60°, 75°, and 90°). The ribbon at the center is made of prestretched top and bottom layers with a misorientation angle of 30°. The released multilayer strips deform into coiled shapes with the pitch and helix angle depending on $\phi$. Reprinted with permission from Ref. [9] (© 2011, AIP Publishing LLC).

Figure 16

(a) Open Bauhinia pods. (b) Manufacturing a mechanical analog of a Bauhinia pod. Bonding two perpendicularly prestretched latex-rubber sheets and gluing them together forms a residually stressed compound sheet. A strip is cut from the sheet along a direction that forms an angle $\theta$ with either direction. (c)–(e) Narrow strips cut at angle $\theta =0°$, 15°, 30°, 45°, 60°, 75° and 90° from a latex sheet, Bauhinia pods, and theoretical predictions, respectively. (f) Dependence of the radius and pitch on angle θ. Symbols correspond to data points of the latex sheet, whereas lines are the theoretical predictions. (a)–(f) are from Ref. [15], reprinted with permission from AAAS. (g) The schematics for the method used to create paper models, which generates the coiling behavior that mimics plant structures. (h) The shapes of paper models for different combinations of the two control parameters: the dimensionless width and the misorientation angle. (g) and (h) are from Ref. [149], reprinted with permission from AAAS.

Figure 17

(a) Theoretical cross section of isosceles nanoribbons, where $X(s)$ is the path of the perimeter. The gray area is the imaginary mirror image. (b) Coiled nanoribbons immersed in water. (a) and (b) are adapted from Ref. [150] with permission (© 2013). (c) Left panel: A flat sheet made of the composite gel with 1-mm-wide patterned stripes of PNIPAm gel (PG) and PNIPAm/PAMPS gel (BG) that form an angle to the long axis of the sheet. Right panel: A helical ribbon formed when subjected to an external stimulus. (d) Helical ribbons generated after incubating the gel sheets for 24 h in a 1M NaCl solution. The right panels display conical helices generated in the NaCl solution by gradually changing the geometric parameters such as the ribbon width. (c) and (d) are reprinted by permission from Macmillan Publishers Ltd: Nature Communications, Ref. [151] (© 2013). (e) A 365-nm UV light is used to selectively cross-link a PNIPAM copolymer sheet. The photomask restricts the regions of the sheet that are exposed to light, while the edges of the sheet under the edges of the photomask receive an intensity gradient of light, giving rise to a cross-linking gradient. (f) The swelling constant $\mathrm{\Omega }$ is low in the center of the ribbons, indicating that the centers have a limited capacity to swell. The edges are able to swell much more, resulting in a helix with defined radius $R_e$ in a swollen state. (e)–(g) are reprinted from Ref. [152] (© 2014 with permission from Elsevier).

Figure 18

(a) Flowchart showing the halftone lithographic patterning method. A simple photomask is used to give the entire region a low dose of UV, and then a halftone photomask gives selective high doses of UV. (b) Enneper’s minimal surfaces generated from radially symmetric patterning. Surfaces with three to six nodes are generated. (c) Sphere generated from asymmetric patterning, with areas of high distortion excised. From Ref. [161], reprinted with permission from AAAS.

Figure 19

(a) Phase diagram showing the cutoff point for when polymer bilayer particles will form complete tubes and rings (above lines) versus arcs and coils (below lines). The $x$ axis is the ratio of layer thicknesses, the $y$ axis is the radius of curvature, and each line corresponds to a different biaxial modulus. (b) Examples of structure found above and below the lines. (a) and (b) are reprinted with permission from Ref. [162] (© 2008, AIP Publishing LLC).

Figure 20

Various structures formed by LC elastomers adhered to a polystyrene layer. Bends, folds, and twists are introduced, as well as a four-arm grabbing structure. Reproduced (adapted) from Ref. [164] with permission of The Royal Society of Chemistry.

Figure 21

(a) Thermal cycle of shape memory fiber-embedded elastomers. When stretched and cooled below the glass transition temperature of fibers, the composite bends on unloading. When reheated, it returns to a flat state. (b),(c) Self-folding box made from this technique, where the fibers are embedded in the folds. Reprinted with permission from Ref. [5] (© 2013, AIP Publishing LLC).

Figure 22

(a) Nanohelix structure formed from wavy silicon ribbon selectively bonded to an elastomer sheet. (b) A similar technique is used to make double-helix structures, where the second layer of helices stands on top of the first. This second layer is not bonded to the elastomer. From Ref. [166], reprinted with permission from AAAS.

Figure 23

(a) Schematic representation of cylindrical helical and purely twisted ribbons. The bottom shapes feature the locally cylindrical and saddlelike curvatures in these multilayered ribbons. Adapted from Ref. [127] with permission. (b) Side view of schematics of the direction of twisted-nematic-elastomer (TNE) ribbons. (c) A schematic diagram shows the top view of $L$ and $S$ geometry. (e) Temperature dependence of the inverses of pitch and diameter of the helices. The lines (both dashed and solid) represent theoretical predictions. The (red) circles and (blue) squares represent the data of the L geometry and the data of the $S$ geometry, respectively. (f) Cylindrical helical ribbons formed by the wide TNE where the thickness of the ribbons is $35.2\mu \mathrm{m}$. The $L$-geometry ribbon is left-handed at 374 K and right-handed at 336 K. Reprinted with permission from Macmillan Publishers Ltd: Nature, Ref. [127] (© 1999).

Figure 24

Monostable and bistable helices. (a) Monostable helical ribbons. An elastic strip is bonded to a prestretched latex-rubber sheet, with a misorientation angle $\phi$ ranging from 0 to 90° at a 30° interval. The prestretches are 0.24 and 0.12 in the vertical and horizontal directions, respectively. (b) Monostable helical ribbons. (c) Bistable helical ribbons. (d) The other stable shapes of the same ribbons in (c). The color is indexed according to the out-of-plane displacement. Reprinted with permission from Ref. [22] (© 2014, AIP Publishing LLC).

Figure 25

The stable helical configurations driven by misfit strains in bilayer ribbons modeled using finite-element simulations. The misfit strain in the top layer is ${\underset{\_}{\underset{\_}{ϵ}}}_t^0=({ϵ}_1{\cos }^2\varphi +{ϵ}_2{\sin }^2\varphi ){\underset{\_}{e}}_1\otimes {\underset{\_}{e}}_1+({ϵ}_1{\sin }^2\varphi +{ϵ}_2{\cos }^2\varphi ){\underset{\_}{e}}_2\otimes {\underset{\_}{e}}_2+({ϵ}_1+{ϵ}_2)\sin \varphi \cos \varphi ({\underset{\_}{e}}_1\otimes {\underset{\_}{e}}_2+{\underset{\_}{e}}_2\otimes {\underset{\_}{e}}_1)$, where (a) ${ϵ}_1=0.02$, ${ϵ}_2=-0.02$, $\varphi =\pi /4$, $L=0.4\mathrm{m}$, and $W=0.04\mathrm{m}$; (b) ${ϵ}_1=0.1$, ${ϵ}_2=-0.1$, $\varphi =\pi /4$, and $W=0.04\mathrm{m}$; (c) ${ϵ}_1=0.1$, ${ϵ}_2=-0.1$, $\varphi =\pi /4$, $L=0.4\mathrm{m}$, and $W=0.08\mathrm{m}$; and (d) ${ϵ}_1=0.5$, ${ϵ}_2=-0.5$, $\varphi =\pi /4$, $L=0.4\mathrm{m}$, and $W=0.08\mathrm{m}$. Transition from a purely twisted ribbon (c) to a spiral ribbon (d) occurs when the misfit strain is increased by 5 times. The color (in this figure and the following) is indexed according to the total displacement to help visualize the deformation involved. With kind permission from Springer Science+Business Media, Ref. [168] (© 2014).

Figure 26

(a) Saddle-shape structure formed from concentric ring-patterned sheet. (b) Icosahedron formed from projection-patterned sheet. (c) Multistage material that coils upon reaching 40° and uncoils upon reaching 65°. (d) Multistimulus material that curls into a long thin tube under high pH and a short fat tube under high salinity. Adapted with permission from Ref. [173] (© 2013 American Chemical Society).

Figure 27

(a),(b) LC helix curling, uncurling, and inversion under UV light exposure. Helices where the handedness is reinforced by the alignment director will curl when exposed to UV light, but those where handedness is opposite to alignment director will uncurl and eventually invert. (c)–(f) When selectively irradiated by UV light near the kink, the end of the helical ribbon constructed from liquid-crystal polymer moves away and off axis from the other end. Reprinted with permission from Macmillan Publishers Ltd: Nature Chemistry, Ref. [174] (© 2014). (g),(h) Diagram showing how alignment director of the initial LC network informs the handedness of the resulting helix. (g) and (h) are adapted with permission from Ref. [175] (© 2014 American Chemical Society).

Figure 28

(a) Diagram showing programmed shape transition under hydration. (b),(c) Dry and wet states of uniform curling and uncurling composites. (d),(e) Dry and wet states of twisting composites. Reprinted with permission from Macmillan Publishers Ltd: Nature Communications, Ref. [176] (© 2013).

Figure 29

(a) Fiber ribbon that overwinds around the perversion when pulled. (b) Twistless spring with low bending stiffness and high twisting stiffness that unwinds when pulled. From Ref. [11], reprinted with permission from AAAS. (c)–(e) are adapted from Ref. [12] with permission of The Royal Society of Chemistry. (f) Helices with decreasing cross-section height-to-width ratio. The structure moves from helix, to hemihelix, to hemihelix with multiple perversions. (g) Number of perversions as function of aspect ratio and prestrain. Adapted from Ref. [179] with permission.

Figure 30

(a)–(c) Thin stretched rubber sheets are stitched to unstretched rubber tubes to induce twisted patterns that resemble the looping guts in chick embryos in (c). (d),(e) Scaling laws for the loop shape, size, and number at three stages in the development of chick guts. Reprinted with permission from Macmillan Publishers Ltd: Nature, Ref. [17] (© 2011).

Figure 31

(a) Geometry of a symmetric mesa design. Both ends (in red) are fixed. (b) A strained nanoribbon with symmetric left-handed and right-handed segments (both ends being fixed). Here, ${\varnothing }_1={\varnothing }_2=75°$, and ${ϵ}_0=0.024$. (c) SEM image of a typical helical structure (diameter of $1.4\mu \mathrm{m}$) formed by a $\mathsf{V}$-shaped mesa with both ends fixed to the substrate. The two arms of the $\mathsf{V}$-shaped mesa form helices, with opposite chirality. The inset shows the mesa design and the rolling direction of the helix as indicated with a white arrow. (d) A helical nanoribbon with both left-handed and right-handed segments with only one fixed end. Here, ${\varnothing }_{\mathrm{L}}=50°$, ${\varnothing }_{\mathrm{U}}=40°$, and ${ϵ}_0=0.05$. In the upper segment (of length $0.8\mu \mathrm{m}$), the effective misfit strain tensor of the bottom layer is ${ϵ}_b={ϵ}_0{e}_{U2}\otimes e_{U2}$. In the lower segment (of length $2.4\mu \mathrm{m}$), the effective misfit strain tensor of the bottom layer is ${ϵ}_b={ϵ}_0{e}_{L2}\otimes e_{L2}$. The color indicates the total displacement. (a), (b), and (d) are adapted from Ref. [169] with permission of The Royal Society of Chemistry. (c) and the inset of (d) are adapted from Ref. [138] (© IOP Publishing, reproduced with permission, all rights reserved).

Figure 32

(a) Measurement of the strain field; (b) the closed leaf is cut to illustrate the natural principal curvatures; (c) mean curvature as $K_m$ a function of $K_{xn}$. (a)–(c) are reprinted with permission from Macmillan Publishers Ltd: Nature, Ref. [180] (© 2005). (d) Snapping surface of concave microlenses that mimics the Venus flytrap. Adapted from Ref. [181] with permission (© 2007). (e) A swelling-induced snapping microbot made of hydrogel shell with microfluidic channels embedded (a scanning electron microscope image embedded). Adapted from Ref. [182] with permission from The Royal Society of Chemistry.

Figure 33

(a) Blue and yellow latex sheets are perpendicularly prestretched and bonded to a thicker elastic strip. The multilayer sheet forms a doubly curved shape that conforms to a torus (b). Small (c) and large (d) thin squares form saddles, while thin strips (e), (f) form stable semicylinders. Reprinted with permission from Ref. [185] (© 2012 by the American Physical Society).

Figure 34

(a) Multistable shell design space, with bistability in red. (b)–(g) Total strain energy as a function of misfit axis orientation, for orientations $f_2/f_1=\sqrt{3}$, $f_2/f_1=1$, $f_2/f_1=1/\sqrt{3}$, $f_2/f_1=-1/\sqrt{3}$, $f_2/f_1=-1$, and $f_2/f_1=-\sqrt{3}$. (a)–(g) are reprinted with permission from Ref. [185] (© 2012 by the American Physical Society). (h) A variety of shapes of a prestressed shell structure with zero stiffness. The shell can be transformed from one configuration to another along either the clockwise or anticlockwise path, and each shape can be held merely by friction with the table’s surface. First published in Journal of Mechanics of Materials and Structure in 2011, from Ref. [186] (published by Mathematical Sciences Publishers © 2011). (i) is adapted from Ref. [187] with permission (© 2011 ASME).

Figure 35

(a) Images and schematic diagrams of a circular disk buckles axisymmetrically with principal curvatures ${\kappa }_1={\kappa }_2$ and bifurcates with two distinct positive curvatures. (b) Normalized curvature versus nonhomogeneously swollen time. (c) By minimizing the total strain energy, the relationship between normalized curvature and normalized time is obtained and is in good agreement with experimental data. (d) The circular disk relaxes back to its original flat state. Adapted from Ref. [16] with permission of The Royal Society of Chemistry.