![2 1 Structural Engineering of Flexible Electronics
1.2 Applications of Flexible Electronics
Flexible electronics that can apply on complex curvilinear surfaces have aroused
extensive interest and become one of the greatest concerned, cutting-edge interdis-
ciplines. They have broad innovative application prospects like wearable electronics
[1],epidermalelectronics[2],roboticskin[3],andaircraftsmartskin[4].Therequire-
ments of flexible electronics differ from application to application, from bendability
and rollability for easier handling of large area photovoltaics, to conformability onto
irregular shapes, and to hybrid of stretchability, twistability, foldability, and deforma-
bility for electronic skin. This section will present some representative application
scenarios, offering insights into the significance of structural engineering of flexible
electronics.
1.2.1 Wearable Human Healthcare
Flexible electronics can be built into our clothes and accessories, attached to our skin,
and even implanted in our bodies. The corresponding branches are wearable elec-
tronics [5, 6, 7], epidermal electronics [2, 8, 9], and implantable electronics [10–13],
respectively (Fig. 1.1). Such bioelectronics have been changing conventional medical
diagnoses by endowing them with combined features of wearability, comfortability,
remote operation, and timely feedback. Specifically, they can be utilized for contin-
uous, noninvasive, real-time, and comfortable monitoring of vital biometric signs,
which provide important clinically related information for disease diagnosis, preven-
tive healthcare, and rehabilitation care [14, 15]. In a word, flexible electronics offer
bright promise for next-generation healthcare and biomedical applications.
Various physical and physiological signals can be measured by wearable elec-
tronics, such as electrophysiological signals [16, 17], body motion [18], and muscle
movement [19] (e.g., walking, jogging, and eye blinking). Such real-time monitoring
of muscle activities is useful for clinical gait analysis and muscle fatigue evaluation,
which can improve exercise performance and even prevent unexpected catastrophic
situations.Withepidermalelectronics,vitalphysiologicalsignals,suchaspulse,heart
and respiration rate, body temperature, and skin and breath moisture, can be contin-
uously and long-timely tracked in a mechanically unfeelable manner [20, 21]. In this
way, chronic diseases like diabetes and glaucoma can be effectively diagnosed and
managed with noninvasive, transcutaneous therapeutic treatment. Moreover, with
the advent of biocompatible and bioresorbable materials, implantable electronics
have been developed for in vivo recording of internal conditions, such as intracranial
pressure [22], biochemical constituents (e.g., metabolites and electrolytes) [23], and
electrophysiological signals [24] (e.g., electrocardiogram, ECG; electromyogram,
EMG; electroencephalogram, EEG; electrooculogram, EOG; and electroglottogram,
EGG). Direct contact with the dynamic curvilinear tissues and organs can provide
accurate neural and physiological signals.](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-23-320.jpg)
![1.2 Applications of Flexible Electronics 3
Fig. 1.1 Flexible electronics can be used as wearable electronics [5–7], epidermal devices [2, 8,
9], and implantable electronics [12, 13]
One of the most challenges of flexible electronics for human applications is how
to maintain intimate and effective contact with the complex, time-dynamic surfaces
of skin, tissues and organs during service [25]. For example, organs such as the heart,
arteries, and alveoli undergo periodic areal strains of up to several tens of percent.
Structural engineering and materials engineering are two main strategies. Structural
engineering can circumvent the high threshold of intrinsically flexible/stretchable
materials. Moreover, rational structural designs can not only enable the stretcha-
bility of conventional stiff, non-stretchable, and even brittle materials, but realize the
programmable deformability of such materials to better adapt the targeted surfaces.
1.2.2 Robotics and Haptic Interface
The research of robotics has grown into a popular field in the past few years (Fig. 1.2a)
[26]. Electronic skin (E-skin) is the most intuitive application of flexible electronics
in robotics to enhance perceptibility and interactivity, which is emerging rapidly with](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-24-320.jpg)
![4 1 Structural Engineering of Flexible Electronics
the goal of matching or even surpassing the performance of human skin (Fig. 1.2b,
c) [27, 28]. Moreover, soft robots can integrate E-skin and soft actuators to achieve
integrated control of sensing and driving (Fig. 1.2d, e) [29–32], and programmability
and multifunction [33]. With the increasing demand, flexible electronics have broad
application prospects in the field of soft robots [34]. Mechanical compliance is a
prerequisite for E-skin to accommodate complex dynamic environments while main-
taining its multiple functionalities. In this regard, structural engineering of E-skin
has been developed for compliant designs, enabling the integration of imperceptible
sensing systems with great conformability to 3D surfaces.
More importantly, analogy to human skin, E-skin should exhibit extraordinary
sensing abilities for numerous tactile and thermal stimuli, object recognition, texture
discrimination, slip detection, sensory-motor feedback, etc. The flexible sensors are
Fig. 1.2 Flexible electronics in soft robotics. a Human–machine interaction [26]. b Electronic
skin [28]. c Prosthetic skin [27]. d Self-powered soft robot [31]. e Soft, autonomous robots [32]. f
Programmable, reprocessable multifunctional soft robots [33]](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-25-320.jpg)
![1.2 Applications of Flexible Electronics 5
the key to realizing this goal. Taking the flexible pressure sensors as examples,
microstructural engineering of the active layers is one of the most admirable strate-
gies to enhance the performance of flexible sensors like sensitivity, linearity, and
sensing range. Common microstructural designs include regular pyramids [29, 35],
hemispheres [30], and pillars [36, 37]. Moreover, bioinspired hierarchical structural
arrays [38], randomly distributed spinosum [39], and bioinspired cone arrays [40]
have been proposed successively. Recently, Bai et al. reported an iontronic pressure
sensor with ultra-broad-range high sensitivity based on graded intrafillable architec-
ture [41]. Ji et al. synergistically optimized the sensitivity and linearity of flexible
pressure sensors via double conductive layer and porous microdome array [42]. Not
long ago, Liu et al. introduced Origami designs to broaden the sensing range of flex-
iblesensors[43].Inthefuture,continuousbreakthroughsofstructuralengineeringfor
the applications of flexible electronics in soft robotics will appear, such as the intro-
duction of emerging mechanical metamaterials [44, 45], and the structural design of
a driving-sensing integrated strategy suitable for soft robots.
1.2.3 Smart Skin in Aircraft
Recently, the demands of flexible electronics applied in the field of aeronautics are
increasing such as the flexible smart sensing skin [4, 46–49]. The flexible smart
sensing skin can realize the in-situ measurement of aerodynamic parameters without
changing the structure attributes and flow field environment, making it a promising
candidate in wind tunnel test, unmanned driving, morphing aircraft, and other related
fields. Compared with the traditional method of pressure taps [50], the pressure-
sensitive paint [51], and the conventional smart skin [52], recent significant tech-
nical advances in flexible electronics have overcome many historic drawbacks, e.g.,
avoidance of the inevitable structural damage and significant increase of weight,
real-time and in-situ measurement, and high-density sensing network. However,
unlike the applications in human and robots, the extreme operating environments
of aircrafts impose strict limitations to the materials. The widely used intrinsi-
cally flexible/stretchable materials, such as PDMS and Ecoflex, are incompetent and
unsuitable for such situations due to insufficient strength, temperature sensitivity
(low glass-transition temperature), high-frequency fitration, and so on. Therefore,
the structural engineering of flexible electronics made of conventional stiff, stable,
and high-performace materials is natural and sagacious choice.
Furthermore, the intelligent flexible sensing (iFlexSense) skin is a key enabling
technology for the wind tunnel test and the future “Fly-by-Feel” control of morphing
aircraft (Fig. 1.3). It represents the next-generation skin of aircraft that can exhibit
more powerful sensing functions. The smart skin can “feel”, “think”, and “react”
in real time based on high-resolution state-sensing, awareness, and self-diagnostic
capabilities, endowing the aircraft with the ability to change shape according to
the fly states and structural health [4]. Undoubtedly, such advanced functions pose
great challenges to the mechanical properties of the flexible smart sensing skin,](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-26-320.jpg)
![6 1 Structural Engineering of Flexible Electronics
Fig. 1.3 Intelligent flexible sensing (iFlexSense) skin for aircraft [54]
especiallyforthedeformability.Itshouldholdextrafeatures:(i)capabilityofadaptive
deformation with the infrastructural skin, (ii) low membrane stiffness to allow small
actuation requirement, (iii) effectively sensing air loads while maintaining initial
surface geometry, and (iv) high elasticity and recovery to allow multiple cycles of
deformation [53]. Empirically and reasonably, the realization of these targets mainly
relies on the structural engineering of flexible electronics. Predictably, future aircrafts
with more controlled systems will lead to the increasing need for flexible electronics
applied on the aircraft surface.
1.3 Structural Strategies
1.3.1 Wavy Strategy
The wavy strategy was first proposed in the pioneering work of flexible electronics
in 2006, in which flexible devices made of single-crystal silicon were buckled into
microscale, periodic, and wavy shapes [55]. Thin silicon ribbons (20 ~ 320 nm) were
firstly fabricated by lithographic processing and then transferred to the surface of
a pre-strained elastomer substrate (e.g., PDMS). After releasing the pre-strain, the
silicon ribbons were compressed to wavy structures. Wavy silicon can be reversibly
stretched and compressed to large levels of strain without damage. The strains of
the whole system during deformation are accommodated mainly through changes
of the wavelengths and amplitudes of the wavy structures rather than the mechan-
ical properties of materials. The wavy silicon ribbons of 100 nm thickness can be
stretched up to 30% that overwhelms the 1% strain limit of silicon. Such strategy
renders the conventional stiff, nonstretchable, and even brittle materials sufficiently
deformable to act as flexible electronics. It not only allows large-scale stretchability](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-27-320.jpg)
![1.3 Structural Strategies 7
without degradation in electronic performances, but also realizes good conforma-
bility on complex surfaces. The wavy strategies pave the new way for the structural
engineering of flexible electronics.
Many mechanical models have been established to analyze the configuration of
wavy structures. In the beginning, under the small-deformation assumption (<5%)
and the principle of minimum potential energy, Khang et al. [55] and Huang et al. [56]
developed an energetic method to describe the buckling behaviors, in which the out-
of-plane displacement is assumed as a sinusoidal profile. Afterward, considering the
practical large deformations (>5%), Jiang et al. [57] and Song et al. [58] developed a
nonlinear finite-deformation model, by which the wavelength and amplitude of wavy
profile can be predicted.
Following this direction, the theoretical framework is gradually and continuously
improved,andmoreandmoresituationsareconsidered,includingpost-buckling[59],
finite-width effect [60], local versus global buckling [61], thermomechanical analysis
[62, 63], adhesion-governed buckling [64], rigid/soft wavy surfaces [65, 66], 1D/2D
wavy surfaces [67, 68], and moderately large-deflection theory [69]. Relying on the
underlying mechanisms, the wavy designs are widely applied in flexible electronics.
Initially, Khang et al. presented a stretchable single-crystal silicon p-n diode on a
PDMS substrate at −11%(top), 0% (middle), and 11% (bottom) applied strains [55].
Furthermore, Kim et al. demonstrated stretchable and foldable silicon integrated
circuits based on 2D wavy designs [70]. Ko et al. realized the wrapping of a silicon
membrane circuit on a golf ball (Fig. 1.4a) [71]. Noteworthily, except for the full
bonding method, the selective bonding method is also widely used to generate wavy
structures on the elastomeric substrates. For example, Sun et al. constructed metal–
semiconductor field-effect transistors based on buckled wavy GaAs ribbons [72].
Fig. 1.4 Various structure strategies in stretchable electronics: a Wavy [71], b serpentine [73], c
self-similar [74] strategies](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-28-320.jpg)
![8 1 Structural Engineering of Flexible Electronics
1.3.2 Island-Bridge Strategy
Originating from and then beyond the wavy strategies, the island-bridge strategies
were proposed to achieve higher levels of stretchability. The non-stretchable func-
tional components reside at the low-strain islands and the stretchable interconnects
form the bridges to accommodate the mechanical stretchability and electrical conduc-
tivity. Due to the much lower stiffness of the bridges than the islands, the islands
and adhered functional devices are mechanically isolated. In other words, the key
point of island-bridge strategies lies in the design of bridges. By now, there are three
representative successful cases, namely arc-shaped [75, 76], serpentine [77–80], and
fractal interconnects [74, 81–83]. Since the arc-shaped designs are very similar to
the wavy strategies in the previous section, here we mainly introduce the latter two.
Arc-shaped interconnects. Under stretching and compression, switch-on/off of out-
of-plane buckling of arc-shaped interconnects can accommodate the applied strain.
Initially, the arc-shaped interconnects were usually treated as clamped beams or
films, and the out-of-plane buckling displacement was assumed as sinusoidal profiles
[75]. After that, to break the limitation of the sinusoidal assumption for the large
displacements, a finite-deformation theoretical model was developed, which can
offer a more accurate prediction of the amplitude [84]. Wang et al. presented a
systematic analysis for the buckling behaviors and gave the critical criteria, by which
the buckling behaviors are determined by the relationship between pre-strain and
adhesion and the modes can be subdivided into global buckling, local buckling, and
no buckling [85].
Serpentine interconnects. The serpentine strategy was firstly applied in the mile-
stone work of epidermal electronics in 2011 [2], in which the concept of epidermal
electronics was defined. With the filamentary serpentine designs, the system provided
elastic, reversible responses to large-strain deformations with effective moduli
(<150 kPa), bending stiffnesses (<1 nN/m), and areal mass densities (<3.8 mg/cm2
).
Those merits are in orders of magnitude smaller than those possible with conventional
electronics or even with recently explored flexible devices. Thereafter, the filamen-
tary serpentine interconnects have been widely explored in flexible electronics. A
representative example is the soft microfluidic assemblies of sensors, circuits, and
radios for the skin, in which strain-isolated device components are connected by a
freestanding serpentine interconnecting network (Fig. 1.4b) [73].
A systematic and comprehensive investigation of mechanical behaviors in such
serpentine interconnects has been performed by researchers. In earlier research, the
serpentine interconnects were mostly freestanding and the stretchability relies on the
folding, tenuous, and thin designs. Zhang et al. developed analytical models to reveal
the buckling mechanisms in such stretchable serpentine microstructures, including
the buckling and post-buckling behaviors [80]. A scaling law was established to deter-
mine the critical buckling strain, by which symmetric and anti-symmetric buckling
modes were identified. Soon after, Zhang et al. further proposed a pre-strain approach
that can significantly improve the stretchability (more than two times) compared to](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-29-320.jpg)
![1.3 Structural Strategies 9
the situation without pre-strain [86]. The analytical model can not only effectively
predict the wavelength but also explain the influence of thickness, offering a rational
method to obtain the desired stretchability.
Moreover, in 2016, Su et al. introduced a different route to design the serpentine
interconnects, where tenuous and freestanding geometries were substituted by thick
and bonded layouts to enhance the mechanical and electrical performance [87]. The
in-plane and out-of-plane buckling modes were replaced by pure in-plane scissor-
like deformations. More specifically, with the increase of thickness, the deformation
mechanisms change from wrinkling (localized, multiwave, out-of-plane buckling) to
buckling (coupled out-of-plane buckling and twisting) and then to scissoring (pure
in-plane bending deformation). The scissor-like deformations significantly increased
stretchability from 20% for thin, buckling interconnects to ~ 100% for thick, scis-
soring interconnects. The findings provide a significant supplement and open a new
direction for the design of serpentine interconnects. In addition, Yang et al. proposed
a “cut-and-paste” process to manufacture the multiparametric epidermal sensing
systems based on the serpentine designs [88]. Pan et al. discussed the effect of
substrate thickness and concluded that the reduction of the substrate can improve the
stretchability [89].
Fractal interconnects. Aiming to reconcile the mutually exclusive requirements of
large stretchability and high-area coverage, fractal design concepts were introduced
for stretchable electronics. Before that, Zhang et al. showed the increase of surface
filling ratios by increasing the fractal order from 1 to 4 [81]. As with the other strate-
gies, fractal interconnects can also be divided into freestanding and bonding layouts.
For the freestanding situation, the deformations can be further divided into pure in-
plane stretching for large thickness-width ratio (>1) and spatial buckling for small
thickness-widthratio(<1/5).Fortheformercase,Zhangetal.gaverecursiveformulae
to describe the relationship between fractal orders and flexibility and stretchability
[81]. For the latter case, Xu et al. propounded ordered unraveling mechanisms to
interpret the deformations [82]. In addition to the abovementioned serpentine shape,
fractal designs can be applied to many other shapes [83, 90], e.g., zigzag, sinusoidal,
and horseshoe. Su et al. developed an analytic method to directly compute the elastic
energy and the tensile stiffness of fractal interconnects of arbitrary order n and in
arbitrary shape [83]. Furthermore, Dong et al. investigated the bonding configuration
of such interconnects with the elastomer substrate [91, 92].
By dint of the simultaneous large stretchability and high-area coverage, fractal
interconnects used as bridges between mechanically isolated islands have shown
unique advantages in stretchable lithium-ion batteries [82], epidermal electronics
(Fig. 1.4c) [74], and radio-frequency antennas [93]. Furthermore, fractal designs can
also enhance conformability on soft, curvilinear surfaces, which are meaningful for
applications in bio-systems. Xu et al. presented a fractal electrode array distributed
over a rabbit heart circumference to deliver cardiac electrical stimulation and sense
cardiac electrical activity [94]. Moreover, fractal electrodes are conformable to more
complex biological surfaces, in which the fractal electrode meshes can be directly and
chronically mounted on the complex surfaces of the auricle and the mastoid [95]. The](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-30-320.jpg)
![10 1 Structural Engineering of Flexible Electronics
case of the auricle proved another advantage of fractal interconnects, namely direc-
tional guided deformability to match the complex topology of the auricle. Specifi-
cally, all-vertical Peano curves were used to get selective high stretchability along
with the longitudinal coordinates.
1.3.3 Kirigami and Origami Strategy
Recently, the concept of Kirigami mechanical metamaterials (Kiri-MMs) has been
introduced into the design of flexible electronics [54]. The salient structural traits
including negative Poisson’s ratio [44, 96], ultra-stretchability [97–100], mechanical
programmability [101–105], and transformability from 2 to 3D [106–108] provide a
brand-newresearchstrategy.Specifically,theprogrammabilityenablesthepossibility
to fit arbitrary curvilinear surfaces [105], and the transformability from 2 to 3D is a
perfect solution to the contradiction between 2D planar processing technology and
3D conformal demands [106, 109]. Compared with other thin open-mesh serpentine
or island-bridge strategies introduced above, Kirigami structures have a high fill
factor, which are particularly desirable for high-density sensing or high-resolution
imaging [110]. In the early stage, graphene Kirigami was an impressive example to
demonstrate the salient traits of Kirigami [111], by which one-atom-thick graphene
sheets were transformed into resilient and morphing structures without sacrificing
electrical performance.
By now, Kirigami-inspired creative applications have become abundant, including
self-powered strain sensor (Fig. 1.5a) [112], soft crawler (Fig. 1.5b) [113], morphable
stent (Fig. 1.5c) [114], shoe grip (Fig. 1.5d) [115], adaptive imager (Fig. 1.5e) [110],
and flexible car shell [116]. The underlying mechanism of Kirigami structures relies
on the local buckling of hinges under low-energy loads to empower the whole struc-
tural reconfigurability. Rafsanjani and Bertoldi investigated the mechanisms of the
buckling-induced Kirigami and studied how the behavior evolves when the thickness
is progressively decreased [109]. However, all these creative buckling-driven traits
are poles apart from the fully conformal goals of flexible electronics that require
coating as smooth as possible. Current studies mostly directly make use of the
buckling-induced traits, but how to restrain buckling while keeping extraordinary
mechanical properties of Kiri-MMs remains open. As a note, despite non-stretchable
PI used in curvy, shape-adaptive imagers, the ultrathin thickness (~5 µm) makes the
devices sufficiently compliant [110]. To address this challenge, we show that using
soft material at the hinges of Kiri-MM can restrain this local buckling effect and
improve conformability [117], which will be presented in the following chapter.
As for the design methods, fractal Kirigami can further enhance the stretchability.
Programmable Kirigami mechanical metamaterials are becoming popular. Choi et al.
propounded a meaningful inverse Kirigami design method, in which the cutting
path was determined by objective and constrained functions derived from targeted
conformal shapes [105]. Most recently, based on the computer graphics, Jiang et al.
presented an affirmative solution to realize the targeted programmability of Kirigami](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-31-320.jpg)
![1.3 Structural Strategies 11
Fig. 1.5 Kirigami strategy: a self-powered strain sensor [112], b soft crawler [113], c morphable
stent [114], d shoe grip [115], and e adaptive imager [110]
sheet between arbitrary shapes, in particular between a 2D sheet and a 3D curved
surface [118]. This so-called inverse problem for Kirigami cut and fold patterns
is solved by drawing on a differential-geometric interpretation of the morph and
progress in geometric computing. Therefore, it is convinced that Kirigami will be a
powerful strategy to design the flexible electronics.
As the twin concept of Kirigami, Origami is another admired strategy to build
wonderful structures. Because of the ability to hold creases and bend at will, a 2D
sheet can be folded into arbitrary shapes. The mechanisms of Origami have been
widely studied for a very long time, including mathematics [119], geometry [120],
and compliant mechanisms [121]. Many Origami strategies have been proposed like
Miura-ori and its variants, Ron Resch’s tessellation, and square twist. Either way, the
kernel always roots in the crease patterns. Mechanically, the creases act as soft hinges
to connect the stiffer facets, and the low-energy deformation of the creases induces the
morphability of the whole Origami structures (Fig. 1.6a) [122]. Similar to previous
“island-bridge” strategies, the strain-free facets between the creases are uniquely
suited as the integration platform to mount active devices of flexible electronics.
Moreover, as Origami-based mechanisms often feature multiple discrete folding
motions, they enable the realization of programmable 3D multishapes. In a nutshell,
Origami has many ideal characteristics: monolithic preparation, scale-independent,
perfect self-assembly compatibility, and unlimited design space based on rich folding
patterns, making it very suitable for applications in flexible electronics.
Recently, Origami has been explored for applications in many engineering fields,
e.g., compactly deployable solar arrays for space applications [123], self-folding
crawling robots for machine manufacturing [124], and medical stents for biomedical](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-32-320.jpg)
![12 1 Structural Engineering of Flexible Electronics
Fig. 1.6 Origami strategy. a Transformable Origami with multiple degrees of freedom [122]. b
Ori-MMs-based silicon optoelectronics for hemispherical electronic eye systems. Reproduced with
permission [128]. c Ori-MMs-based conformal electronics made of non-stretchable materials [131].
d Origami-based electrothermal devices [132]. e Stretchable Origami robotic arm [133]. f Origami-
based wide-range flexible capacitive pressure sensors [43]
applications [125]. Its strut in flexible electronics may start from the Origami lithium-
ion batteries proposed in 2014 [126]. This Origami battery exhibited stable and
reliable performances under large cycles of mechanical deformations. By utilizing
printable ZnO nanowires and carbon electrodes, Lin et al. developed a stretchable
and deformable Origami photodetector array based on the Miura-ori strategy [127].
The Origami photodetector array can provide excellent capabilities of omnidirec-
tional photodetection. Furthermore, Zhang et al. developed Origami silicon opto-
electronics for dense, scalable, and compact hemispherical electronic eye systems
[128] (Fig. 1.6b), which were compatible with mature complementary metal–oxide–
semiconductor (CMOS) technologies that enable deployments in extremely high
density. In the same period, Origami was introduced to fabricate a flexible and fold-
able thermoelectric nanogenerator [129]. Most recently, Qi et al. displayed reconfig-
urable flexible electronics driven by Origami magnetic membranes [130]. In addition,
Origami can be adopted to realize full wrapping of conformal electronics made of
non-stretchable materials (Fig. 1.6c) [131], to structure electrothermal devices with
controllable multi-degrees-of-freedom shape morphing (Fig. 1.6d) [132], to enable
stretchable robotic arm with omnidirectional bending and twisting soft robotics
(Fig. 1.6e) [133], and to devise wide-range flexible capacitive pressure sensors
(Fig. 1.6f) [43]. Incidentally, a number of computer-aided tools to Origami such
as TreeMaker and Oripa have been developed. TreeMaker allows new Origami bases
to be designed for special purposes and Oripa tries to calculate the folded shape from
the crease pattern. All these achievements demonstrate the potential of the Origami
to develop spatial flexible electronics.](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-33-320.jpg)
![1.3 Structural Strategies 13
1.3.4 Buckling-Driven Assembly Strategy
Buckling-driven assembly strategy is another milestone of structural engineering of
flexible electronics (Fig. 1.7) [134]. It relies on the control of buckling to realize 2D-
to-3D transformation, with the release of prestrained elastomer substrate to provide
initial mechanical drive and vice versa. It can be reversibly stretched and buckled
between 2D and 3D configurations without degradation of performance even with
a large number of cyclic loadings [106, 135, 136]. The 3D stretchable multifunc-
tional photodetector is a convincing paradigm for the effectiveness of the buckling-
driven assembly strategy [137]. The interconnects of the device exploit a sandwich
configuration, with the graphene encased by two SU-8 layers, and then the SU-8
layers are buckled into a hemispherical structure, rendering a 3D arrangement of the
MoS2 patches that serve as photodetecting elements. The advantages include (i) the
concurrent tracking of the direction and intensity of the incident light, (ii) optically
transparent system allowing the detection of incident angles, and (iii) high geomet-
rical extensibility like an octagonal prism and an octagonal prismoid. Note that these
merits cannot be easily achieved by photodetector arrays in planar layouts.
Except for the compressive buckling, the 3D assembly can be also realized
by tensile buckling that can circumvent the pre-stretching limit [139]. When the
substrate is stretched uniaxially, the nonbonded regions of the 2D precursor are
delaminated from the substrate, resulting in a 3D transformation through coordinated
bending/twisting deformations and translational/rotational motions. The derivative
Fig. 1.7 Buckling-driven assembly strategy. a Schematic illustration of the assembly process
guided by controlled buckling [134]. b 3D Origami micro/nanostructures [138]. c 3D Kirigami
mesostructures [106]](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-34-320.jpg)
![14 1 Structural Engineering of Flexible Electronics
strain sensor shows a great increase in sensing range, from ≈9.8% to ≈50%. 3D elec-
trically small antennas (ESAs) provide another representative example [140]. ESAs
are antennas that are much smaller when compared to the operating wavelength,
which is of significance in miniaturized communication systems. As the planar ESAs
are usually limited in the bandwidths and the efficiencies because of the small volume
occupation of the Chu-sphere, the spatially integrated ESAs, such as hemispherical
ESA, can ideally occupy the Chu-sphere, thereby offering substantially improved
performance.
Furthermore, recent achievements of the buckling-driven assembly strategy show
that different releasing sequences and specially engineered precursor designs can
trigger multistable, transformable, and reconfigurable 3D electronics [136]. Fu et al.
illustrated morphable 3D mesostructures and microelectronic devices by multistable
buckling mechanics, in which mesostructures can be reshaped between different
geometries as well as those that can morph into three or more distinct states. The
sequential release of prestrain applied to the elastomer platform serves as the control
strategy. An adaptive radiofrequency circuit and a concealable electromagnetic
device provide examples of functionally reconfigurable microelectronic devices. By
the way, the initial 3D helical interconnects can also be an attractive design in the
flexible electronics due to high elastic stretchability and exceptionally low effective
modulus [141].
1.3.5 Structural Designs of Substrate
The above-discussed structural strategies focus on the design of functional devices,
rendering the conventional stiff, nonstretchable, and even brittle materials suffi-
ciently conformable to complex surfaces while maintaining high-performance elec-
trical properties. Structural engineering of substrates is another important branch
of flexible electronics. By now, there are many work concentrating on this field,
e.g., strain-isolation of the rigid devices from substrates [142–144], enhancement of
biocompatibility for biological applications [145–148], and functional supplement
of devices to support multifunctions [149, 150]. Here, three representative structural
designs of substrates, namely surface structural designs, cellular structural designs,
and embedded designs, are highlighted.
Surface strategies. In the extensively used “island-bridge” strategies as presented
above, the freestanding, tenuous, and complex interconnects have poor stability in
mechanical properties, as well as non-negligible electrical resistances and high levels
of power dissipation [87]. As for the bonded interconnects based on the thick bar
layouts, the challenge is low effective areal coverage that will reduce the integrated
density of devices and corresponding performance, especially for optoelectronic
devices like photovoltaics and photodetectors. The emerging Kirigami strategy is
effective to address such issues, but the local buckling at hinges will result in
conformal problems, which greatly increases the risk of delamination and even](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-35-320.jpg)
![1.3 Structural Strategies 15
detachment from the conformal surfaces, weakens the accuracy of measurement
and biocompatibility, and exacerbates the encapsulating difficulty.
The surface design of substrates is an alternative approach, which can free the
electronicsfromthedeformationsofthesubstratewhilekeepingahigharealcoverage
[142,151,152].Arepresentativelayoutisthatthedevicesmountonraisedislandsand
the interconnects between them buckle downward into separating trenches (Fig. 1.8a)
[142]. The trench regions absorb most of the externally applied strains, whereas the
top surfaces of the islands barely deform, resulting in minimal interfacial stresses
transmitted to devices mounted on top. Moreover, based on the optimizations that
replace the square islands with notched islands, Lee et al. demonstrated a high-
efficiency dual-junction GaInP/GaAs photovoltaics [152]. Afterward, Cantarella
et al. reported a mesa-shaped elastomeric substrate, supporting thin-film transistors
(TFTs) and logic circuits (inverters) [153]. During mechanical solicitations, the use
of such high-relief structures aims at localizing the strains on the substrate, around
the pillars and not on the pillars’ surface. In this way, devices can withstand different
modes of deformation with stable electrical performances, e.g., bending of up to
6 mm radius, stretching of 20% uniaxial strain, and 180° global twisting. Similarly,
a mogul-patterned elastomeric substrate was reported to improve the stretchability,
with bumps and valleys regularly positioned in hexagonal closed packed structures
[154]. This arrangement enables the layers and devices to be more stable, even under
multidirectional stretching conditions similar to that of skin.
The tripod-structured substrate is another available surface strategy, in which
the devices are suspended to the substrate [155–158]. As a result, the strain can
be significantly decreased by reducing physical contact. This configuration would
confer two advantages: (i) electrode materials of the microsupercapacitors could
Fig. 1.8 Engineering substrates. a Isolated microisland design [142]. b Tripod-structured substrate
[157]. c Toothed substrate design [159]. d Cellular substrate [160]. e Embedment of stiff platforms
in elastomeric substrate [143]. f Soft elastomers with ionic liquid-filled cavities as strain isolating
substrates [144]](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-36-320.jpg)
![16 1 Structural Engineering of Flexible Electronics
be stretched despite being intrinsically stiff, and (ii) the suspended wavy struc-
tures would reduce the strain concentration in the electrode fingers during the
stretching/relaxing processes. Such design is very useful to thin-film devices, e.g.,
graphene microribbons in microsupercapacitors (Fig. 1.8b) [157] and electrodes in
biointegrated electronics [158]. Except for the strain isolation of devices, surface
strategies can also be used for the strain reduction of interconnects. For example, a
toothed substrate has been proposed to release the mechanical constraints of serpen-
tine interconnects (Fig. 1.8c) [159]. The freestanding segments offer the desired
stretchability while the bonded segments enhance the mechanical stability. Indeed,
the position deviation of the serpentine interconnects to the toothed substrate may
weaken the stretchability, but the reduction is predictable and acceptable.
Cellularstrategies.Architectedcellulardesignisubiquitousinengineering,bywhich
light weight and high stiffness/strength can be achieved simultaneously [160, 161].
The architecture of the core is complex, with intricately shaped ligaments and gradi-
ents in density. Cellular substrates are widely used in flexible electronics, not only
because of the low modulus for the enhancement of stretchability and conforma-
bility, but also because of their unique ability to minimize disruptions to the natural
diffusive or convective flow of bio-fluids in advanced, bio-integrated implants [148,
162, 163]. Notably, the solid substrate/encapsulation in other structural strategies will
disrupt the natural diffusive or convective flows of bio-fluids through flexible elec-
tronics. Jang et al. introduced a cellular, composite, and deterministic soft network
that can be tailored precisely to match the non-linear properties of biological tissues
[163]. Lee et al. proposed a bio-inspired honeycomb cellular substrate to achieve
high permeability to facilitate solution exchange for use in biointegrated electronics
[148].
The overall elastic properties of such systems are often difficult to be determined
precisely, because they strongly depend on the detailed alignment and position of
the serpentine interconnects relative to the pores in the cellular substrate. A theo-
retical model of hierarchical lattices is developed to study the underlying relations
between the J-shaped stress–strain curves and the microstructure geometric param-
eters of hierarchical lattices [164]. Three common lattices (triangular, kagome, and
honeycomb) and their hierarchical variants were discussed. Chen et al. established
an analytic constitutive model by considering the cellular substrate as an equivalent
medium under finite stretching (Fig. 1.8d) [160], in which the lower bound of the
stretchability with respect to all alignments and positions of a representative intercon-
nect on the cellular substrate can be estimated. Furthermore, Zhao et al. developed an
analytical model for “zigzag” cellular substrates under finite deformation, achieving
higher compliance than the previously reported hexagonal cellular substrates [165].
The programmability of mechanical properties is another distinct advantage of
cellular substrates, which can adjust the Poisson’s ratio or stiffness to reinforce the
conformability on the targeted surfaces, especially for biomedical devices. Liu et al.
developed a soft cellular network [166], which can yield tailored isotropic Poisson’s
ratio from −1 to 1, with a tunable strain range from 0% to ≈90%. The theoretical
design methods define tailored network geometries to yield target Poisson ratios with](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-37-320.jpg)
![1.4 Structural Opportunities by Materials 17
desired strain ranges. Meanwhile, Zhang et al. introduced a class of soft mechanical
metamaterials that can achieve large effective negative swelling ratios, with desired
isotropic/anisotropic features [167].
Embedded strategies. The strain isolation can also be achieved by embedded designs
of substrates. Romeo et al. presented elastomeric substrates with embedded stiff
platforms to realize the strain isolation (Fig. 1.8e) [143], in which the devices were
deployed on the isolated surfaces overhead the embedded stiff platforms. When the
platform is significantly stiffer than the surrounding silicone matrix, the elastomer
volume above the platform is little strained when the matrix is macroscopically
stretched. Therefore, brittle materials deposited on the corresponding top surface are
not extensively stretched. Conversely, Ma et al. proposed soft elastomers with ionic
liquid-filled cavities as strain isolating substrates (Fig. 1.8f) [144]. Ionic liquids filled
a microfluidic space, or cavity, defined in a low modulus elastomeric substrate. The
contained liquid film mechanically isolates the underlying skin from the electronics
above, without any direct contact to either. Notably, ionic liquids were applied instead
of traditional liquids to eliminate any possibility for leakage or evaporation. The
liquids were positioned between the electronics and the skin, within an enclosed,
elastomeric microfluidic space, but not in direct contact with the active elements of
the system, to avoid any negative consequences on electronic performance.
1.4 Structural Opportunities by Materials
During the above discussion about the structural engineering of flexible electronics,
traditional active materials, like silicon, gold, silver, and carbon, have been repur-
posed through novel structural designs and integration processes. Structural engi-
neering that enables low-modulus mechanics of high-modulus materials has been
widely exploited to accommodate extreme mechanical deformations. The most repre-
sentative ‘island-bridge’ strategy consists of tenuous structurally stretchable (wavy,
serpentine, self-similar, coil, etc.) interconnects between rigid devices. Meanwhile,
material chemistry and engineering play key roles in the exploration and devel-
opment of flexible electronics, including new active materials, new organic-based
electronic components and elastomers, and new integration processes of both. New
supramolecules, polymers, hydrogels, and other biomaterials based on the latest
genetic and synthetic biology technologies have been proposed and used to get
high-performance flexible electronics. On the other hand, the material engineering
enables more novel structural realizations that are inaccessible before, for example,
advanced metamaterials (MMs) can provide an additional dimension of probability
and enable dynamically adjustable structural performance. As such, active metama-
terial is a hybrid structure-material strategy, which shows great potential to advance
flexible electronics. At present, there are six main types of materials for flexible
electronics: conductive metal oxides like ITO [168], metal nanowires [169], carbon
materials like carbon nanotubes [170] and graphene [171], conductive polymers](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-38-320.jpg)
![18 1 Structural Engineering of Flexible Electronics
[172], conductive and stretchable gels [173], and single-phase liquid metals and
liquid metal alloys [174]. Here, structural opportunities by these advanced materials
are briefly discussed.
About active mechanical metamaterials, several recent reviews gave very elab-
orate discussions from different perspectives [45, 175, 176]. Cai et al. proposed a
brand-new concept of ‘mechanomaterials’ to define the programming advanced func-
tional materials by leveraging the force-geometry-property relationships at multiple
scales [45]. Qi et al. made a deep and systematic summary of active mechanical
metamaterials [175], elucidating their underlying construction principles, classifica-
tions, and applications. Pishvar et al. surveyed the innovative multidisciplinary soft,
smart matter in the context of active mechanical metamaterials [176]. The applica-
tions also include topics focused here such as stealth cloak, electronic skin and soft
robot. Referring to active wave-based metamaterials, there is also a recent review
that provides a comprehensive introduction [177].
1.5 Summary
This chapter summarizes some of the most significant advances in structural engi-
neering of flexible electronics, including various strategies based on wavy, island-
bridge, Kirigami, Origami, and buckling-driven, with engineering substrates that
involve surface, cellular and embedded designs. These branches are growing very
fast, and it is time to review and analyze all these efforts, sum up the first stage, and
draw the next blueprint. That is the inspiration of this book. Under the guidance of
this principle, the following chapters introduce the themed efforts, focusing on the
underlying theory and method of structural designs for flexible electronics. Notably,
despite significant progress, challenges remain, especially in structural designs to
enable enhanced levels of comfort of wearable electronics, to achieve minimally
invasive bio-integration of implantable electronics, to offer a high degree of mechan-
ical deformability after solid encapsulation in soft robotics, and to withstand extreme
operating experiments in aircraft. Figure 1.9 gives the panorama of this book. We
do hope the readers enjoy the structural engineering in flexible electronics presented
in this book, and find it inspiring for their future research with blooming ideas and
progress to further usher the prosperity of this field. This research is absolutely multi-
disciplinary, and the related chemical, electrical, and biomedical investigations are
closely involved in each other.](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-39-320.jpg)
![Chapter 2
Buckling of Film-on-Substrate System
in Flexible Electronics
2.1 Introduction
The film-on-substrate system is a typical structure in flexible electronics, as shown in
Fig. 2.1. Reported studies on the film-on-substrate system have made large progress
in duality, crack, and bendability/stretchability [1–6]. Freestanding Si/metal thin film
ruptures when stretched beyond 1–2% [7]. However, when bonded onto an organic
substrate, it can sustain plastic deformation ten times larger than its fracture strain, yet
still remains electrically conductive [8, 9]. Park et al. [1] carried out theoretical and
experimental studies on bending in structure relevant to inorganic flexible electronics,
and the effects of edges and finite device sizes were also considered.
Buckling plays a key role in achieving the stretchability of flexible electronics,
which is a classic mechanical problem but is full of new vitality in recent years.
Most stretchable electronics are distributed, interconnected, and thin films mounted
on an elastomeric substrate. A commonly used strategy is the island-bridge system,
in which the electronic components reside at the islands and various interconnects
form the bridges, the latter provide the majority of stretchability. It is easy to buckle
for this type of structure during the fabrication processes and applications, and the
stretch deformation often leads to out-of-plane or lateral buckling, in-plane buck-
ling, or a combination of all. The critical buckling load, beyond which catastrophic
consequences usually occur, is one of the most important properties of engineering
structures.
A compressively strained elastic film bonded onto the compliant substrate will
form wrinkles, which is a simple approach for flexible integrated circuits to circum-
vent the limitations of stretchability and their low flexural rigidity. Khang et al. [3],
Kim et al. [4], Jiang et al. [10], and Sun et al. [11] studied the buckling behaviors
of film-on-substrate system and the controllability of the structure formation, and
developed stretchable and foldable silicon integrated circuits with high-performance.
Huang et al. [2], Huang et al. [12], and Lacour et al. [9] studied nonlinear buck-
ling behaviors and discussed the mechanisms of reversible stretchability of thin
metal films on elastomeric substrates. Circuits in wavy patterns offer fully reversible
© Science Press 2022
Y. Huang et al., Flexible Electronics,
https://doi.org/10.1007/978-981-19-6623-1_2
27](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-48-320.jpg)
![28 2 Buckling of Film-On-Substrate System in Flexible Electronics
Fig. 2.1 Schematic illustration of the process for building buckled thin film on elastomeric
substrates [3]
stretchability/compressibility without substantial strains in circuit materials them-
selves. The resulting mechanical advantages are critically important for achieving
stretchability. In these systems, strains at the circuit level can exceed the fracture
limits of almost all known electronic materials [3]. In this chapter, the model of the
film-on-substrate structure is established based on interface continuity.
The rest of this chapter is organized as follows. In Sect. 2.2, the formation
of film-on-substrate structure in flexible electronics is presented. In Sect. 2.3,
temperature-dependent global buckling analysis and structural design are discussed.
In Sect. 2.4, temperature-dependent local buckling analysis and critical condition are
demonstrated. Finally, in Sect. 2.5, a brief summary is drawn.
2.2 Formation of Film-on-Substrate Structure
The film-on-substrate structure in flexible electronics was first proposed by Khang
et al., in which wavy Si ribbon was obtained via prestrained PDMS [3]. Figure 2.1
presents the schematic illustration of the process for building stretchable single-
crystal Si devices on elastomeric substrates: (i) Fabricaiton of thin Si ribbon by
conventional photolithography, (ii) Covering with prestrained PDMS, and (iii)
Peeling back the PDMS and releasing the prestrain. After these procedures, self-
assembly, well-controlled, highly periodic, and stretchable wavy Si ribbon on PDMS
can be achieved.
The deflect is much larger than the thickness of film when the film buckles, which
can be modeled adequately by Von Karman plate theory [13]. Since the membrane
would be subject to lateral load during the conformal procedure, the additional strain
produced in the middle plane during bending should be taken into consideration. The
membrane strain εij is
εij = ε0
ij +
1
2
(
∂ui
∂xj
+
∂uj
∂xi
)
+
1
2
∂u3
∂xi
∂u3
∂xj
(2.1)
where i, j = 1, 2. ε0
ij is the initial strains, u1(x1, x2) and u2(x1, x2) are related to the in-
plane displacements and u3(x1, x2) is out-of-plane displacement. The displacement
of membrane is assumed as](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-49-320.jpg)
![2.2 Formation of Film-On-Substrate Structure 29
u3 = A cos kx1 (2.2)
Supposing the film is subjected to pressure in x1 direction
ε0
11 = −εprestrain, ε0
12 = ε0
22 = 0 (2.3)
Substitution of Eqs. (2.2) and (2.3) into (2.1) gives
ε11 = −ε
prestrain
11 +
∂u1
∂x1
+
1
2
[Ak sin(kx1)]2
,
ε22 =
∂u2
∂x2
,
ε12 =
1
2
(
∂u2
∂x1
+
∂u1
∂x2
)
(2.4)
Supposing that the strain is uniform in the film, the partial derivative of ε11 with
respect to x1 gets
0 =
∂2
u1
∂x2
1
+ A2
k3
sin(kx1) cos(kx1) (2.5)
Then
u1 =
1
8
A2
k sin(2kx1) (2.6)
By the same method, u2 = 0. The Eq. (2.4) becomes
ε11 =
1
4
A2
k2
− εprestrain, ε22 = 0, ε12 = 0 (2.7)
The internal forces of the film become
N11 = hfilmEfilm
(
1
4
A2
k2
− εprestrain
)
, N22 = 0, N12 = 0 (2.8)
where hfilm is the thickness of Si film. Thus the bending energy and membrane energy
can be obtained
Umembrane =
1
2
N11ε11=
1
2
hfilmEfilm
(
A2 π
λ2
2
− εprestrain
)2
Ubending =
k
2π
( 2π
k
0
Efilmh3
film
24
(
∂2
w
∂x2
1
)2
dx1 =
1
48
Efilmh3
filmA2
k4
(2.9)](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-50-320.jpg)
![30 2 Buckling of Film-On-Substrate System in Flexible Electronics
Supposing the interfacial shear stress is zero, the substrate energy can be written
as [12]
Usubstrate=
g
4
EsubstrateA2
k (2.10)
where g =
(3−4νsubstrate) cosh(2khsubstrate)+5−12νsubstrate+8ν2
substrate+2(khsubstrate)2
(6−8νsubstrate) sinh(2khsubstrate)−4khsubstrate
.
Therefore, the total energy is
Utotal = Umembrane+Ubending + Usubstrate
=
1
2
hfilmEfilm
(
A2 π
λ2
2
− εprestrain
)2
+
1
48
Efilmh3
filmA2
k4
+
g
4
EsubstrateA2
k (2.11)
According to the principle of minimum potential energy, namely a stable system
always stays at the minimum, minimizing the total energy with respect to the
amplitude and wave length can obtain the buckled configuration [3]
λbuckling=
2πhfilm
√
εcritical
, Abuckling = hfilm
/
εprestrain
εcritical
− 1 (2.12)
where εcritical=
[
3
8
Esubstrate
(
1−ν2
film
)
Efilm(1−ν2
substrate)
]2/ 3
is the critical strain for buckling, εprestrain is the
levelofprestrain,λbuckling isthewavelength,andAbuckling istheamplitude.ThePoisson
ratio is ν, the Young’s modulus is E, and the subscripts refer to properties of the film
or substrate. The thickness of the Si is hfilm. However, the “wavy” Si ribbons are
formed from spontaneous buckling with amplitudes and wavelengths determined by
material properties (e.g., moduli and thickness). Moreover, although the range of
acceptable strain is improved significantly (~20%) compared to that of silicon itself
(~1%), the stretchability is still too small for certain applications.
After that, to control the buckling geometries and enhance the stretchability, Sun
et al. [11] used lithographically defined surface adhesion sites together with elastic
deformations of a supporting substrate to achieve buckling configurations with deter-
ministic control over their geometries. In this case, the PDMS surface remains flat
after relaxation for both activated/inactivated regions. For vanishing displacement
and vanishing stress traction, the relaxed PDMS has vanishing energy. The buckling
amplitude A can be described as
Abuckling =
4
π
L1L2
/(
εprestrain − εcritical
)
, εprestrain > εcritical (2.13)
where εcritical = h2
filmπ2
/
12L2
1, which is identical to the Euler buckling strain for a
doubly clamped beam with length 2L1. The thin film does not buckle when εprestrain](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-51-320.jpg)
![2.2 Formation of Film-On-Substrate Structure 31
< εcritical. The maximum strain in the thin film is the bending strain that results from
the thin film curvature
εmax =
hfilmπ
L2
1
√
L1L2εprestrain (2.14)
Becauseofthesmalldeformationapproximationsandlinearstress–strainbehavior
used in derivation, the above results imply displacements that are tangential to the
local surface relief, yielding a displacement trajectory that has the shape of a wave
whose wavelength is fixed. However, they do not provide sufficient precision for prac-
tical applications with inevitable uncertainties, such as poorly defined film/substrate
interfaces and unknown mechanical properties in the films or substrates. To address
such limitations, Jiang et al. [10] proposed a buckling theory that accounts for
finite deformations and geometrical nonlinearities to yield a quantitatively accurate
description of the system. This buckling theory is different from previous models
in the following three important aspects: (i) The initial strain-free (or stress-free)
configurations for the substrate and film are different, (ii) The strain–displacement
relation in the substrate (as well as the film) is nonlinear, (iii) The stress–strain rela-
tion in the substrate is characterized by the nonlinear neo-Hookean constitutive law.
The wavelength and amplitude in initial buckling are
λbuckling=
2πhfilm
(
Efilm
3Esubstrate
)1/ 3
√
1 + εprestrain(1 + ξ)1/ 3
, Abuckling =
hfilm
/
εprestrain
εcritical
− 1
√
1 + εprestrain(1 + ξ)1/ 3
(2.15)
where ξ = 5εprestrain(1 + εprestrain)/32.
Figure 2.2 gives the experimentally measured and theoretically predicted wave-
length λ and amplitude A versus applied strain εapplied for a buckled Si thin-
film/PDMS substrate formed with a prestrain of 16.2%, and same other parame-
ters. The measured wavelength increases for tension and the measured amplitude
decreases, reaching zero once the tensile strain reaches the prestrain. The finite-
deformation buckling theory agrees well with experiments for both amplitude and
wavelength. The existing mechanical models also capture the amplitude trend but
deviate from the experimental results for large tensile strain (>10%).
Usually,thethicknesseffectofthesubstrateisnotconsideredbecausethesubstrate
is much thicker than the film. The film-on-substrate system is always considered as
a semi-infinite solid. In this case, it appears local buckling. However, when the
substrate becomes thinner, another buckling mode, namely, the global buckling will
be observed in experiments. Wang et al. investigated the underlying mechanisms of
local versus global buckling of thin films on elastomeric substrates, definitely giving
the critical condition separating the local and global buckling modes [14].
The critical condition of local buckling can be further rewritten as](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-52-320.jpg)
![32 2 Buckling of Film-On-Substrate System in Flexible Electronics
Fig. 2.2 Wavelength and amplitude of buckled structures of Si (100 nm thickness) on PDMS as a
function of the prestrain. The finite-deformation buckling theory yields wavelengths and amplitudes
that both agree well with experiments. Also shown are results from previous mechanical models
(i.e., small deformation limit) and the simple accordion model [10]
εlocal
critical =
(
3Esubstrate
8Efilm
)2/ 3
(2.16)
and that of global buckling is
ε
global
critical =
1
1+
1.2F0
critical
Gbeam(hsubstrate+hfilm)
F0
critical
EAbeam
(2.17)
where EAbeam=Esubstratehsubstrate+Efilmhfilm, Gbeam is the effective shear modulus of
the composite beam, which is approximately the shear modulus Gsubstrate of the
substrate since the film is very stiff Efilm ≫ Esubstrate and thin hfilm ≪ hsubstrate.
F0
critical=4π2
EIbeam
/
L2
is the critical buckling load neglecting the effect of shear,
EIbeam =
(
Efilmh2
film−Esubstrateh2
substrate
)
+4EsubstratehsubstrateEfilmhfilm(hsubstrate+hfilm)2
12(Esubstratehsubstrate+Efilmhfilm)
. When εlocal
critical <
ε
global
critical, local buckling occurs. When εlocal
critical > ε
global
critical, global buckling occurs.
Because ε
global
critical is related with the thickness of the substrate and the length of the
film-on-substrate structure, whether the film-on-substrate structure generates local
or global buckling is not determined only by the ratio of Young’s modulus of the
film and substrate, but also by the length of the film-on-substrate structure and the
thickness of the film and substrate.](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-53-320.jpg)
![2.3 Island-Bridge Structure of Stretchable Electronics 33
2.3 Island-Bridge Structure of Stretchable Electronics
In stretchable electronics, the island-bridge structure on a soft substrate plays an
important role in achieving large stretchability. A key issue in developing such a
system is to prevent the island-bridge structure from breaking during use because it
is composed of brittle semiconductor materials (e.g., silicon) which withstand very
small strains (~1%). Figure 2.3 schematically illustrates the fabrication of circuits
with noncoplanar mesh design on compliant substrates. The silicon (or other inor-
ganic material) islands, on which the active devices or circuits are fabricated, are
chemically bonded to a prestrained (e.g., 50%) elastometric substrate of a material
such as PDMS, while interconnects are loosely bonded. Releasing the prestrain leads
to compression, which causes the interconnects to buckle and move out of the plane
of the substrate to form arc-shaped structures. The poor adhesion of interconnects (to
PDMS) and their narrow geometries and low stiffness (compared to device islands)
cause the out-of-plane deformation to localize only to interconnects, and therefore
the strain in islands is very small.
2.3.1 Mechanical Model for the Bridge Structure
As shown in Fig. 2.4a, at a given prestrain εprestrain of the substrate, the bridge
with length Lbridge buckles to accommodate the release of prestrain, which yields the
distance Lbridge/(1 + εprestrain) between the two ends A and E. In view of the symmetry,
only part of the bridge with length Lbridge/4 is analyzed, as shown in Fig. 2.4b. This
is indeed a problem of large deflection of a buckled bar (the elastica). The governing
equation of the bridge is
EbridgeIbridge
dθ
ds
= P(wB − w) (2.18)
Fig. 2.3 Schematic illustration of fabrication process for stretchable electronics with the
noncoplanar mesh design on a compliant substrate; a mesh on a pre-stretched substrate; and b
buckled mesh after the release of pre-stretch in the substrate [15]](https://image.slidesharecdn.com/24452920-250512122920-f2cf1c2c/85/Flexible-Electronics-Theory-And-Method-Of-Structural-Design-Yongan-Huang-54-320.jpg)
Flexible Electronics Theory And Method Of Structural Design Yongan Huang
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- 5. Flexible Electronics YongAn Huang YeWang Su Shan Jiang Theory and Method of Structural Design
- 7. YongAn Huang · YeWang Su · Shan Jiang Flexible Electronics Theory and Method of Structural Design
- 8. YongAn Huang State Key Laboratory of Digital Manufacturing Equipment and Technology Huazhong University of Science and Technology Wuhan, China Shan Jiang Hangzhou Institute of Technology Xidian University Hangzhou, China YeWang Su Institute of Mechanics Chinese Academy of Sciences Beijing, China ISBN 978-981-19-6622-4 ISBN 978-981-19-6623-1 (eBook) https://doi.org/10.1007/978-981-19-6623-1 Jointly published with Science Press, Beijing, China. The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: Science Press. © Science Press 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore
- 9. Foreword Flexible electronics are electronics that can be stretched, bent, twisted, and deformed into arbitrary shapes. They break through the bottleneck and monopoly of traditional, rigid IC technologies and represent the next-generation electronics. Flexible elec- tronics were introduced about two decades ago and have attracted increasing interest since then, both because of their compelling physical properties and because of their potential applications. The topic of flexible electronics stands at the crossroad of physics (mechanics, photonics, electromechanical coupling, etc.) and engineering (material engineering, structure engineering, electrical engineering, etc.). Their representative applications include epidermal/implantable/wearable electronics for human health monitoring, artificial skin for robotics or human–robot interaction, and smart sensing skin for aircraft. The flexibility has become the trend of modern life. Should the current growth trend in flexible electronics research continue, it is not inconceivable to witness a near explosion in industrial interest, as what happened over half a century ago in the field of IC electronics and more recently in photonics. Structure engineering and material engineering are two main research branches of flexible electronics and mutually reinforcing. Compared to the intrinsically flex- ible and stretchable materials, structural engineering has proven its unique advan- tages, e.g., stretchable inorganic electronics. High-performance inorganic materials are ubiquitous in modern electronics, but their natural rigidity and brittle character (e.g., the fracture strain of silicon is only ~ 2%) limit the deformability of the resulting devices. Structural designs can render inorganic electronics into forms that provide large effective levels of deformability, while maintaining the high-performance elec- trical properties. After the first decade during which the topic remained mainly theoretical with a few proof-of-concept demonstrations appearing in the literature, the latest evolution has been toward applications. The mechanical explanations for various structural designs are now quite well understood, efficient numerical methods have been developed, and sufficient verifying experiments have been conducted. Thus, the topic of structure engineering of flexible electronics has become mature for books. This book is the first one about systematic introduction of structural designs of flexible electronics. It covers the state-of-the-art and comprehensive works on v
- 10. vi Foreword theoretical modeling, numerical simulations, and experiments of the authors and also provides a rather exhaustive perspective on the realm of flexible electronics. These achievements are very useful for further development of flexible electronics. I think this book is a timely and good summary for this rising field in which an increasing number of young researchers are putting their efforts. This is an excellent reference book for both academic research and industrial design of flexible electronics. Wuhan, China Han Ding
- 11. Preface Over the past decade, the area of flexible electronics has experienced great devel- opments since it opens up a series of unprecedented applications with broad inter- ests and potentials for impact. The most salient trait of flexible electronics is their deformability during the working process, making them potential candidates for applications in many fields like wearable electronics, epidermal electronics, human– machine interfaces, soft robotics, aircraft smart skin, other biomedical devices, and the related systems. The study of flexible electronics has thus become one of the most active and fast-growing interdisciplines in physics (e.g., solid mechanics, condensed matter physics) and engineering (e.g., electrical engineering, mechanical engineering, structural engineering). By now, one of the main challenges is to reduce the strains in the rigid inor- ganic electronic materials and metallic interconnects, while accommodating the large applied deformations. The various solutions may be summed up as two strate- gies: advanced functional materials and flexible micro-/nano- structural designs. Structural designs of devices can ensure the high-performance electrical proper- ties of inorganic electronics under complex deformation, such as stretchability, conformability, and stability. This book provides an overview of the underlying theory and method of structural designs for flexible/stretchable electronics. Based on the mechanical mechanism, this book discusses the main structural deforma- tion behaviors, including the buckling of film/fiber-on-substrate, self-similar design with/without substrate, conformal design on soft/rigid surfaces, stability design under stretching/compression, Kirigami-based conformal design, multiple neutral plane design. Moreover, the related advanced fabrication technologies, devices, and appli- cations are also presented. The review of the developments of flexible electronics is discussed in Chap. 1, focusing on the branch of structural designs. In Chap. 2, the buckling behaviors of typical film-on-substrate systems in flexible electronics are presented. Especially, an analytical mechanical model of the island– bridge structure is established and the accurate solution is obtained. A validated scaling law is found to reveal the dependence of the normalized maximum strain in the island on the prestrain of the substrate, which controls the mechanical failure of the island–bridge structure and provides a theoretical basis for fracture-safe design of vii
- 12. viii Preface stretchable electronics. Then, thermomechanical properties are discussed in detail, where the model of the film-on-substrate structure is established based on interface continuity and considered as a function of room, working, and deposit temperatures. In Chap. 3, the buckling behaviors of typical fiber-on-substrate systems in flex- ible electronics are demonstrated. A mechano-electrospinning (MES) technique is first proposed to fabricate large-area, high-performance stretchable piezoelectric nanowire devices without out-of-surface buckling or wrinkling, by which polyvinyli- dene difluoride nanofibers can be direct-written onto a pre-strained elastomeric substrate. Then, the inherent competing mechanism between out-of-/in-surface buck- ling of micro-/nanowires on elastomeric substrates is first uncovered. Theoretical analysis and numerical simulations are presented to discover the critical factors that govern the competition between the two buckling modes. In Chap. 4, fractal-inspired geometric designs in electrical interconnects are analyzed in theory to simultaneously achieve large stretchability and high aerial coverage for stretchable electronics. A universal mechanical theory based on the energy density is developed for the self-similar interconnects with a representative element of arbitrary shape to calculate the stiffness and to estimate the stretchability. The key parameters governing the tensile stiffness are identified. After that, the fractal-inspired space-filling structures of electronic materials (including monocrys- talline silicon) are demonstrated for electrophysiological sensors, precision monitors and actuators, and radio frequency antennas. In Chap. 5, the bonding configuration of self-similar serpentine interconnects on the elastomer substrate is considered. The stretchability of the order-2 self- similar interconnects bonded onto the PDMS substrate is studied through analyt- ical modeling, finite element simulations, and experiments. The scaling law is built to predict the stretchability of the structure. Then, the application of self-similar interconnects in surface electrodes is introduced to design surface electrodes with high mechanical adaptability (stretchability and conformability with skin) and high electrical sensitivity/stability which are usually a pair of paradoxes. In Chap. 6, the conformal behaviors of flexible electronics on rigid substrates are studied. A mechanical analysis framework based on the energy method is established for understanding the conformability of flexible electronics on target surfaces. On this basis, the specific derivation process for two representative practical cases of rigid substrates are displayed, including 1D conformability of membranes on rigid wavy substrates and 2D conformability of island–bridge structures on non-developable rigid surfaces. Effects of key factors containing geometric parameters of electrodes, areal coverage of electrodes, and external loads are disinterred. In Chap. 7, the conformal behaviors of flexible electronics on soft substrates are investigated. An interfacial mechanical model describing epidermal electronics and skin system is put forward. Similarly, the contact behaviors of three representative soft cases are reviewed, including (i) 1D conformability on soft substrate, (ii) 2D conformability on wavy soft substrate, and (iii) 2D conformability on complex soft substrate. Furthermore, local failure analysis of island during conformal process is given a special discussion for the 2D conformability on complex soft substrate. The
- 13. Preface ix conformability of epidermal electronics is validated by experiments with different substrate thickness, areal coverage, and external loadings. In Chap. 8, a significant in-plane design strategy for highly stretchable elec- tronics is propounded, in which thick bar geometries are used to replace conven- tional thin ribbon layouts to yield scissor-like deformations instead of in-plane or out-of-plane buckling modes. Systematic studies involving experimental work, finite element simulation, and analytical theory reveal the underlying mechanisms between three different deformation modes (wrinkling, buckling, and scissoring), for serpen- tine structures of hard materials on soft elastomeric substrates. Analytical studies of these designs identify key geometric parameters that govern the elastic stretchability and yield optimal values for metallic serpentine interconnects that reach levels of stretchability up to 350%. In Chap. 9, the prebuckling problems of thick serpentine interconnects are consid- ered. A systematic and straightforward theory (finite prebuckling deformation, FPD) is developed to analyze the FPD buckling behaviors of beams with the coupling of bending, twisting, and stretch/compression. As a comparison, various theoretical and numerical methods are applied to three classic problems, including lateral buckling of a three-point bending beam, lateral buckling of a pure bending beam, and Euler buck- ling. The proposed FPD buckling theory for beams can give a good prediction than the conventional buckling theories and numerical methods that always neglected the prebuckling deformation. Finally, an experiment is conducted to observe the actual effects. In Chap. 10, the buckling-driven self-assembly strategy for stretchable electronics is presented. A novel helix electrohydrodynamic printing technique is proposed, without photolithography and transfer printing processes. The buckling behaviors of the serpentine fibers are investigated by combining theoretical modeling, finite element analyses, and experiments. The critical geometric parameter governing the buckling behaviors from local buckling to global buckling is obtained. Finally, the application of the buckling-driven self-assembly strategy in a hyper-stretchable self- powered sensor is displayed to show their distinct advantages. In Chap. 11, a novel Kirigami assembly strategy that can address the conformal challenges of flexible electronics is shown, especially for those made of stiff, non- stretchable materials. Although aimed at forming a 3D curved electronic circuit, this strategy is fully compatible with the conventional 2D circuit designs and fabrication methods. It allows a 2D sheet to wrap a 3D surface conformally and completely, i.e., with a smooth coating surface and high areal coverage. The geometrical design algo- rithm for 2D-to-3D conformal mapping is elucidated. Finally, several 3D multifunc- tional sensing systems are fabricated to demonstrate the advantages of the proposed strategy. In Chap. 12, the mechanical behaviors of laminated structure-based flexible elec- tronics are first accurately described. An analytic mechanical model of the laminated structure is established to accurately predict the strain distribution of the structure and the locations of the neutral mechanical planes of the hard layers. A significant finding is revealed that shear deformation dominates in the soft adhesive layers of the
- 14. x Preface laminated structure of flexible electronics while the normal strain-induced deforma- tion is negligible. Moreover, the finite element method is used to prove the accuracy of the theoretical model. In addition, the effects of the membrane energy and bending energy of the soft layer are also investigated by incorporating or neglecting the shear energy. In Chap. 13, liquid metals are introduced into the design of stretchable electronics, focusing on microfluidic serpentine antennas for mechanically adaptive frequency modulation. Mechanical tuneability of resonance frequencies of the stretchable antennas is exhibited, including decreasing, stabilizing and increasing the reso- nance frequencies under stretching. The strain-isolated design and modular assembly of individual antenna sheets are presented to facilitate the practical applications. Furthermore, the serpentine design for liquid-metal antennas is introduced into the design of stretchable RFID tag, which can keep the working frequency stable under high stretching up to 50%. Finally, a ultrathin, flexible electromagnetic metasurface based on liquid metals is demonstrated. In Chap. 14, several representative applications of flexible electronics are displayed. First, the application of flexible electronics as E-tattoos for physiolog- ical sensing and therapeutics is given. Large-area, breathable, mechanically robust, and high-fidelity epidermal electrodes are reported. Then, the application of flexible electronics as implantable electronics for the cardiac membrane is demonstrated. Third, an intelligent flexible sensing (iFlexSense) skin, bio-inspired by the powerful sensing capacities of biological systems, is presented for airflow sensing and struc- tural health monitoring of the full-coverage, curved surface. Finally, the application of flexible electronics as robotic interface is shown. This book mainly includes the authors’ achievements in the topic of structural designs of flexible electronics, which is both its strength and its weakness. Many significant progresses by other researchers are covered but not discussed in detail. Though it probably contains many mistakes and misses certain developments and contributions, it can but only reflect the sincere knowledge of its authors. To all distinguished colleagues, collaborators and often friends, we wish to present our apologies for any omissions in our text. We are also deeply grateful for the many efforts and contributions from other contributors, including Dr. Wentao Dong, Dr. Xingquan Wang, Dr. Jianpeng Liu, Dr. Lin Xiao, PhD candidate Jiacheng Li, and master candidates Xuejun Liu and Yichen Liu. In addition, we want to express our gratitude to the support from the National Natural Science Foundation of China (Grant Nos. 51925503, 52105575,and12172359),NaturalScienceFoundationofHubeiProvince(GrantNo. 2020CFA028), China Postdoctoral Science Foundation (Grant Nos. 2020M672331 and 2022T150234), CAS Interdisciplinary Innovation Team (JCTD-2020-03), and the help of the publisher. Wuhan, China Beijing, China Hangzhou, China YongAn Huang YeWang Su Shan Jiang
- 15. Contents 1 Structural Engineering of Flexible Electronics . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Applications of Flexible Electronics . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Wearable Human Healthcare . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.2 Robotics and Haptic Interface . . . . . . . . . . . . . . . . . . . . . . 3 1.2.3 Smart Skin in Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Structural Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.1 Wavy Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.2 Island-Bridge Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.3 Kirigami and Origami Strategy . . . . . . . . . . . . . . . . . . . . . 10 1.3.4 Buckling-Driven Assembly Strategy . . . . . . . . . . . . . . . . . 13 1.3.5 Structural Designs of Substrate . . . . . . . . . . . . . . . . . . . . . 14 1.4 Structural Opportunities by Materials . . . . . . . . . . . . . . . . . . . . . . . 17 1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Buckling of Film-on-Substrate System in Flexible Electronics . . . . . 27 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Formation of Film-on-Substrate Structure . . . . . . . . . . . . . . . . . . . . 28 2.3 Island-Bridge Structure of Stretchable Electronics . . . . . . . . . . . . . 33 2.3.1 Mechanical Model for the Bridge Structure . . . . . . . . . . . 33 2.3.2 Mechanical Model for the Island Structure . . . . . . . . . . . 36 2.4 Temperature-Dependent Global Buckling Analysis and Structural Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.4.1 Geometrical Model and Governing Equations . . . . . . . . . 46 2.4.2 Structure Design Based on Temperature-Dependent Properties . . . . . . . . . . . . . . . 48 xi
- 16. xii Contents 2.4.3 Temperature-Dependent Local Buckling Analysis and Critical Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3 Buckling of Fiber-on-Substrate System in Flexible Electronics . . . . . 57 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Fabrication of Buckled Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2.1 Mechano-Electrospinning (MES) Technique for In-Surface Buckled Devices . . . . . . . . . . . . . . . . . . . . . 58 3.2.2 Direct-Writing of Fibers onto a Pre-strained PDMS . . . . 59 3.2.3 Materials and Experimental Set-Up . . . . . . . . . . . . . . . . . . 60 3.3 Buckling Behaviors of 1D Micro/Nanowires . . . . . . . . . . . . . . . . . 62 3.3.1 Out-of-Plane and In-Plane Buckling . . . . . . . . . . . . . . . . . 62 3.3.2 Mechanics of Out-of-/In-Surface Buckling . . . . . . . . . . . 64 3.3.3 Competition of Buckling Modes . . . . . . . . . . . . . . . . . . . . 68 3.4 In-Plane Buckled, Highly Stretchable Devices . . . . . . . . . . . . . . . . 78 3.5 Performance of the Fabricated Stretchable Piezoelectric Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4 Freestanding Fractal-Inspired Design for Stretchable Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2 Elasticity of Fractal Inspired Interconnects . . . . . . . . . . . . . . . . . . . 86 4.2.1 Elastic Analysis of Fractal Interconnects . . . . . . . . . . . . . 86 4.2.2 Experiments of the Fractal Structures . . . . . . . . . . . . . . . . 92 4.3 Fractal Designs in Stretchable Electronics . . . . . . . . . . . . . . . . . . . 94 4.3.1 Mechanics and Electronics with Peano-Based Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.3.2 Fractal-Based Epidermal Electronics . . . . . . . . . . . . . . . . 101 4.3.3 Radio-Frequency Devices with Fractal Layouts . . . . . . . 104 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5 Fractal-Inspired Design on Substrate for Stretchable Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2 Mechanical Modeling of SSIs on Soft Substrate . . . . . . . . . . . . . . 110 5.2.1 Maximum Strain of Order-2 SSI . . . . . . . . . . . . . . . . . . . . 110 5.2.2 The Scale Law Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.2.3 FEM Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.3 Self-Similar Design of Surface Electrodes . . . . . . . . . . . . . . . . . . . 115 5.3.1 Electromechanical Design of Self-Similar Surface Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
- 17. Contents xiii 5.3.2 Electromechanical Optimization Model . . . . . . . . . . . . . . 116 5.3.3 Feasible Range of the Geometric Parameters . . . . . . . . . . 118 5.3.4 Characterization of Mechanical Performance . . . . . . . . . 120 5.4 Self-Similar Design for Stretchable Wireless LC Strain Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.4.1 Self-Similar Design for Wireless LC Strain Sensor . . . . 124 5.4.2 Structural Stretchability of the Self-Similar Strain Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.4.3 Strain-Induced Tunable Inductance . . . . . . . . . . . . . . . . . . 127 5.4.4 Experimental Platform and Sensitivity Analysis . . . . . . . 130 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6 Conformal Design on Rigid Curved Substrate . . . . . . . . . . . . . . . . . . . . 137 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.2 Theoretical Analysis Based on Energy Method . . . . . . . . . . . . . . . 139 6.2.1 Energy Components of Thin Film . . . . . . . . . . . . . . . . . . . 139 6.2.2 Energy Components of Substrate . . . . . . . . . . . . . . . . . . . . 140 6.2.3 Total Energy of Flexible Electronics . . . . . . . . . . . . . . . . . 141 6.3 1D Conformability of Membranes on Rigid Wavy Substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.3.1 Analytical Interface Model by Work of Adhesion . . . . . . 142 6.3.2 Analytical Interface Model by Traction-Separation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.4 2D Conformability of Island-Bridge Structures on Non-Developable Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.4.1 Analytical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.4.2 Adhesion Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.4.3 Adhesion Experiment for Island on Rigid Surface . . . . . 157 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 7 Conformal Design on Soft Curved Substrate . . . . . . . . . . . . . . . . . . . . . 165 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.2 1D Conformability on Soft Substrates . . . . . . . . . . . . . . . . . . . . . . . 166 7.2.1 1D Conformability of Epidermal Electronics on Soft Skin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 7.2.2 1D Conformability of Epidermal Electronics on Soft Skin Under External Strain . . . . . . . . . . . . . . . . . . 169 7.3 2D Conformability on Wavy Soft Substrates . . . . . . . . . . . . . . . . . 171 7.3.1 2D Conformability of Epidermal Electronics on Soft Skin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7.3.2 The Effects of the Roughness and Elastic Modulus of the Skin on Conformability . . . . . . . . . . . . . . . . . . . . . . 175 7.3.3 The Effects of the Substrate Thickness on Conformability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
- 18. xiv Contents 7.3.4 The Effects of the Areal Coverage of Electrode on Conformability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7.3.5 The Effects of the External Load on Conformability . . . 179 7.4 2D Conformability on Complex Soft Substrates . . . . . . . . . . . . . . . 180 7.4.1 2D Conformability of Island on a Bicurvature Soft Substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.4.2 The Effects of Geometry Parameters on Stable Conformal Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 7.4.3 The Effects of Materials Parameters on Conformal Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 7.4.4 Contact Pressure in Conformal Contact . . . . . . . . . . . . . . 186 7.5 Local Failure Analysis of Island During Conformal Process . . . . 189 7.5.1 Conformal Strain in Island During Conformal Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.5.2 Wrinkling and Buckling Delamination During Conformal Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 7.5.3 Adhesion Experiment for Island on Soft Surface . . . . . . 193 7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 8 In-Plane Design of Serpentine Interconnect on Substrate . . . . . . . . . . 199 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 8.2 Thick Interconnects for Ultra-Large Stretchability . . . . . . . . . . . . . 200 8.3 Transition Between Wrinkling, Buckling and Scissoring . . . . . . . 203 8.3.1 Transition from Wrinkling to Buckling . . . . . . . . . . . . . . . 203 8.3.2 Transition from Buckling to Scissoring . . . . . . . . . . . . . . 206 8.4 Criteria for Three Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 8.4.1 Stretchability in the Wrinkling Mode . . . . . . . . . . . . . . . . 213 8.4.2 Stretchability in the Buckling Mode . . . . . . . . . . . . . . . . . 215 8.4.3 Stretchability in the Scissoring Mode . . . . . . . . . . . . . . . . 215 8.5 Some Applications of Thick Interconnects . . . . . . . . . . . . . . . . . . . 219 8.5.1 Interconnects for Stretchable Arrays of LEDs . . . . . . . . . 219 8.5.2 Interconnects for Stretchable Arrays of Solar Cells . . . . 221 8.5.3 Traces for Stretchable RF Antennas . . . . . . . . . . . . . . . . . 223 8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 9 In-Plane Design for Serpentine Interconnect Without Substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 9.2 Buckling of Stretchable Serpentine Interconnects . . . . . . . . . . . . . 231 9.3 FPD Buckling Theory of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 9.3.1 Geometric Relations for the Finite Deformation of 3D Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 9.3.2 Governing Equations for the FPD Buckling Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
- 19. Contents xv 9.4 Application of Three Specific Cases . . . . . . . . . . . . . . . . . . . . . . . . . 241 9.4.1 Lateral Buckling of a Three-Point-Bending Beam . . . . . 241 9.4.2 Lateral Buckling of a Pure Bending Beam . . . . . . . . . . . . 245 9.4.3 Euler Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 9.5 Sample Fabrication and Experimental Verification . . . . . . . . . . . . 251 9.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 10 Self-Assembly of Self-Similar Fibers for Stretchable Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 10.2 HE-Printing Technique for Fabrication of Self-Similar Nano/Microfibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 10.2.1 HE-Printing Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 10.2.2 Fabrication of Self-Similar Nano/Microfibers . . . . . . . . . 264 10.3 Buckling-Driven Self-Assembly of Self-Similar Fiber-Based Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 10.3.1 Buckling of Serpentine Fibers Under Uniaxial Prestrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 10.3.2 Buckling of Serpentine Fibers Under Biaxial Prestrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 10.3.3 Self-Assembly by Tuning In-/Out-of-Surface Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 10.4 Hyper-Stretchable Self-Powered Sensors Based on Self-Similar Piezoelectric Nano/Microfibers . . . . . . . . . . . . . . . 278 10.4.1 Hyper-Stretchable Self-Powered Sensors . . . . . . . . . . . . . 278 10.4.2 Architecture of an HSS and HE-Printing Technique . . . . 279 10.4.3 Characterizations of the HSS . . . . . . . . . . . . . . . . . . . . . . . 280 10.4.4 Applications of the HSS . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 10.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 11 Kirigami Strategy for Conformal Electronics . . . . . . . . . . . . . . . . . . . . 289 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 11.2 Self-Healing Kirigami Assembly Strategy . . . . . . . . . . . . . . . . . . . 290 11.2.1 Conformal Criterion for Kirigami Geometry . . . . . . . . . . 292 11.2.2 Geometrical Design Algorithm for 2D-to-3D Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 11.2.3 Preparation and Characterization of the Ag/PCL Self-Healing Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 11.2.4 Kirigami-Based Conformal Heater . . . . . . . . . . . . . . . . . . 302 11.2.5 Multifunctional Wind Sensing System . . . . . . . . . . . . . . . 304 11.3 Soft-Hinge Kirigami Metamaterials for Self-Adaptive Conformal Electronic Armor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 11.3.1 Deformation Mechanism of Soft-Hinge Kiri-MMs . . . . . 306
- 20. xvi Contents 11.3.2 Stretchability, Flexibility and Conformability of Soft-Hinge Kiri-MMs . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 11.3.3 Electrical Enhancements with Conductive Polymer Composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 11.3.4 Functional Soft-Hinge Kiri-MM E-armor Systems . . . . . 315 11.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 12 Neutral Layer Design for Flexible Electronics . . . . . . . . . . . . . . . . . . . . 321 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 12.2 Mechanics of Neutral Mechanical Plane . . . . . . . . . . . . . . . . . . . . . 322 12.3 The Effect of Length on Splitting of the Multiple Neutral Mechanical Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 12.4 The Effect of Boundary Conditions on Splitting of the Neutral Mechanical Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 12.4.1 Given Slopes Are Imposed at the Ends of the Hard Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 12.4.2 Given Slopes Are Imposed at the End Sections . . . . . . . . 341 12.5 Effects of the Membrane Energy and Bending Energy of the Middle Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 12.5.1 The Model Incorporating the Shear Energy, Membrane Energy and Bending Energy of the Middle Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 12.5.2 The Model Neglecting the Shear Energy of the Middle Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 12.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 13 Liquid Metal-Based Structure Design for Stretchable Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 13.2 Microfluidic Serpentine Antennas with Designed Mechanical Tunability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 13.2.1 Galium-Based Eutectic Alloys . . . . . . . . . . . . . . . . . . . . . . 353 13.2.2 The Design of Serpentine Microfluidic Antenna . . . . . . . 354 13.2.3 Fabrication of the Serpentine Microfluidic Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 13.2.4 Characterization of the Serpentine Microfluidic Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 13.3 Liquid–Metal Antennas with Stable Working Frequency for RFID Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 13.3.1 Serpentine Liquid–Metal Antennas for RFID . . . . . . . . . 363 13.3.2 Design of Stretchable RF Antennas . . . . . . . . . . . . . . . . . . 364 13.3.3 Relationship Between Stretchability and Resonant Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
- 21. Contents xvii 13.4 Liquid Metal Nanoparticles (LMNPs) for Ultrathin, Flexible Metasurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 13.4.1 Liquid Metal Metasurface . . . . . . . . . . . . . . . . . . . . . . . . . . 372 13.4.2 Sintering Process of LMNPs . . . . . . . . . . . . . . . . . . . . . . . 373 13.4.3 Electromagnetic Performance . . . . . . . . . . . . . . . . . . . . . . 376 13.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 14 Applications of Flexible Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 14.2 Application of Flexible Electronics as E-tattoos . . . . . . . . . . . . . . . 382 14.2.1 Low-Cost, µm-Thick, and Tape-Free E-tattoos . . . . . . . . 382 14.2.2 Characterization of Wearability and Motion Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 14.2.3 Applications of the Large-Area Epidermal Electrodes on Human Skin . . . . . . . . . . . . . . . . . . . . . . . . . 387 14.3 Application of Flexible Electronics as Implantable Cardiac Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 14.3.1 3D Multifunctional Integumentary Cardiac Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 14.3.2 Design of Conformability . . . . . . . . . . . . . . . . . . . . . . . . . . 390 14.3.3 Spatiotemporal Cardiac Measurements . . . . . . . . . . . . . . . 392 14.4 Application of Flexible Electronics as Aircraft Smart Skin . . . . . 396 14.4.1 Design of the Multifunctional, Flexible Sensing Skin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 14.4.2 External Airflow Multifunctional Perception . . . . . . . . . . 398 14.4.3 Internal Structural Health Monitoring . . . . . . . . . . . . . . . . 401 14.5 Application of Flexible Electronics as Robotic Interface . . . . . . . 403 14.5.1 Design and System Architecture of 3D-Shaped E-skin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 14.5.2 Fabrication of the 3D-Shaped E-skin . . . . . . . . . . . . . . . . 406 14.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
- 22. Chapter 1 Structural Engineering of Flexible Electronics 1.1 Introduction Over the last two decades, extensive efforts on advanced functional materials and flexible structural designs have been implemented to improve flexible electronics. Structural engineering of flexible electronics that contain conventional stiff, non- stretchable, and even brittle materials such as piezoelectrics and silicons has proven its unique advantages compared to the intrinsically flexible and stretchable mate- rials. Despite low yield strain (typically below 1%), such conventional materials can provide high performance and reliability. The structural engineering can bridge the gap between the large deformation and inorganic materials, by which low-modulus mechanics of high-modulus materials can be achieved to accommodate extreme mechanical deformations. While applied research in this area has been abundant in recent years, review from the standpoint of fundamental mechanics is needed to advance the field in new directions. For example, the new opportunities and chal- lenges of flexible electronics derived from the conceptual breakthroughs of emerging mechanical metamaterials. The rest of this chapter is organized as follows. In Sect. 1.2, several represen- tative application scenarios are demonstrated to show the significance of flexible electronics. In Sect. 1.3, an overview of historical developments of structural engi- neering of flexible electronics is presented with a focus on the pioneering achieve- ments, from initial wavy strategies to subsequent self-assembly strategies and then to flexible mechanical metamaterials. In Sect. 1.4, from the view of structural oppor- tunities, several common advanced functional materials are discussed. Finally, in Sect. 1.5, perspectives on the remaining challenges and open opportunities in the mechanically-guided structural engineering of flexible electronics are offered. © Science Press 2022 Y. Huang et al., Flexible Electronics, https://doi.org/10.1007/978-981-19-6623-1_1 1
- 23. 2 1 Structural Engineering of Flexible Electronics 1.2 Applications of Flexible Electronics Flexible electronics that can apply on complex curvilinear surfaces have aroused extensive interest and become one of the greatest concerned, cutting-edge interdis- ciplines. They have broad innovative application prospects like wearable electronics [1],epidermalelectronics[2],roboticskin[3],andaircraftsmartskin[4].Therequire- ments of flexible electronics differ from application to application, from bendability and rollability for easier handling of large area photovoltaics, to conformability onto irregular shapes, and to hybrid of stretchability, twistability, foldability, and deforma- bility for electronic skin. This section will present some representative application scenarios, offering insights into the significance of structural engineering of flexible electronics. 1.2.1 Wearable Human Healthcare Flexible electronics can be built into our clothes and accessories, attached to our skin, and even implanted in our bodies. The corresponding branches are wearable elec- tronics [5, 6, 7], epidermal electronics [2, 8, 9], and implantable electronics [10–13], respectively (Fig. 1.1). Such bioelectronics have been changing conventional medical diagnoses by endowing them with combined features of wearability, comfortability, remote operation, and timely feedback. Specifically, they can be utilized for contin- uous, noninvasive, real-time, and comfortable monitoring of vital biometric signs, which provide important clinically related information for disease diagnosis, preven- tive healthcare, and rehabilitation care [14, 15]. In a word, flexible electronics offer bright promise for next-generation healthcare and biomedical applications. Various physical and physiological signals can be measured by wearable elec- tronics, such as electrophysiological signals [16, 17], body motion [18], and muscle movement [19] (e.g., walking, jogging, and eye blinking). Such real-time monitoring of muscle activities is useful for clinical gait analysis and muscle fatigue evaluation, which can improve exercise performance and even prevent unexpected catastrophic situations.Withepidermalelectronics,vitalphysiologicalsignals,suchaspulse,heart and respiration rate, body temperature, and skin and breath moisture, can be contin- uously and long-timely tracked in a mechanically unfeelable manner [20, 21]. In this way, chronic diseases like diabetes and glaucoma can be effectively diagnosed and managed with noninvasive, transcutaneous therapeutic treatment. Moreover, with the advent of biocompatible and bioresorbable materials, implantable electronics have been developed for in vivo recording of internal conditions, such as intracranial pressure [22], biochemical constituents (e.g., metabolites and electrolytes) [23], and electrophysiological signals [24] (e.g., electrocardiogram, ECG; electromyogram, EMG; electroencephalogram, EEG; electrooculogram, EOG; and electroglottogram, EGG). Direct contact with the dynamic curvilinear tissues and organs can provide accurate neural and physiological signals.
- 24. 1.2 Applications of Flexible Electronics 3 Fig. 1.1 Flexible electronics can be used as wearable electronics [5–7], epidermal devices [2, 8, 9], and implantable electronics [12, 13] One of the most challenges of flexible electronics for human applications is how to maintain intimate and effective contact with the complex, time-dynamic surfaces of skin, tissues and organs during service [25]. For example, organs such as the heart, arteries, and alveoli undergo periodic areal strains of up to several tens of percent. Structural engineering and materials engineering are two main strategies. Structural engineering can circumvent the high threshold of intrinsically flexible/stretchable materials. Moreover, rational structural designs can not only enable the stretcha- bility of conventional stiff, non-stretchable, and even brittle materials, but realize the programmable deformability of such materials to better adapt the targeted surfaces. 1.2.2 Robotics and Haptic Interface The research of robotics has grown into a popular field in the past few years (Fig. 1.2a) [26]. Electronic skin (E-skin) is the most intuitive application of flexible electronics in robotics to enhance perceptibility and interactivity, which is emerging rapidly with
- 25. 4 1 Structural Engineering of Flexible Electronics the goal of matching or even surpassing the performance of human skin (Fig. 1.2b, c) [27, 28]. Moreover, soft robots can integrate E-skin and soft actuators to achieve integrated control of sensing and driving (Fig. 1.2d, e) [29–32], and programmability and multifunction [33]. With the increasing demand, flexible electronics have broad application prospects in the field of soft robots [34]. Mechanical compliance is a prerequisite for E-skin to accommodate complex dynamic environments while main- taining its multiple functionalities. In this regard, structural engineering of E-skin has been developed for compliant designs, enabling the integration of imperceptible sensing systems with great conformability to 3D surfaces. More importantly, analogy to human skin, E-skin should exhibit extraordinary sensing abilities for numerous tactile and thermal stimuli, object recognition, texture discrimination, slip detection, sensory-motor feedback, etc. The flexible sensors are Fig. 1.2 Flexible electronics in soft robotics. a Human–machine interaction [26]. b Electronic skin [28]. c Prosthetic skin [27]. d Self-powered soft robot [31]. e Soft, autonomous robots [32]. f Programmable, reprocessable multifunctional soft robots [33]
- 26. 1.2 Applications of Flexible Electronics 5 the key to realizing this goal. Taking the flexible pressure sensors as examples, microstructural engineering of the active layers is one of the most admirable strate- gies to enhance the performance of flexible sensors like sensitivity, linearity, and sensing range. Common microstructural designs include regular pyramids [29, 35], hemispheres [30], and pillars [36, 37]. Moreover, bioinspired hierarchical structural arrays [38], randomly distributed spinosum [39], and bioinspired cone arrays [40] have been proposed successively. Recently, Bai et al. reported an iontronic pressure sensor with ultra-broad-range high sensitivity based on graded intrafillable architec- ture [41]. Ji et al. synergistically optimized the sensitivity and linearity of flexible pressure sensors via double conductive layer and porous microdome array [42]. Not long ago, Liu et al. introduced Origami designs to broaden the sensing range of flex- iblesensors[43].Inthefuture,continuousbreakthroughsofstructuralengineeringfor the applications of flexible electronics in soft robotics will appear, such as the intro- duction of emerging mechanical metamaterials [44, 45], and the structural design of a driving-sensing integrated strategy suitable for soft robots. 1.2.3 Smart Skin in Aircraft Recently, the demands of flexible electronics applied in the field of aeronautics are increasing such as the flexible smart sensing skin [4, 46–49]. The flexible smart sensing skin can realize the in-situ measurement of aerodynamic parameters without changing the structure attributes and flow field environment, making it a promising candidate in wind tunnel test, unmanned driving, morphing aircraft, and other related fields. Compared with the traditional method of pressure taps [50], the pressure- sensitive paint [51], and the conventional smart skin [52], recent significant tech- nical advances in flexible electronics have overcome many historic drawbacks, e.g., avoidance of the inevitable structural damage and significant increase of weight, real-time and in-situ measurement, and high-density sensing network. However, unlike the applications in human and robots, the extreme operating environments of aircrafts impose strict limitations to the materials. The widely used intrinsi- cally flexible/stretchable materials, such as PDMS and Ecoflex, are incompetent and unsuitable for such situations due to insufficient strength, temperature sensitivity (low glass-transition temperature), high-frequency fitration, and so on. Therefore, the structural engineering of flexible electronics made of conventional stiff, stable, and high-performace materials is natural and sagacious choice. Furthermore, the intelligent flexible sensing (iFlexSense) skin is a key enabling technology for the wind tunnel test and the future “Fly-by-Feel” control of morphing aircraft (Fig. 1.3). It represents the next-generation skin of aircraft that can exhibit more powerful sensing functions. The smart skin can “feel”, “think”, and “react” in real time based on high-resolution state-sensing, awareness, and self-diagnostic capabilities, endowing the aircraft with the ability to change shape according to the fly states and structural health [4]. Undoubtedly, such advanced functions pose great challenges to the mechanical properties of the flexible smart sensing skin,
- 27. 6 1 Structural Engineering of Flexible Electronics Fig. 1.3 Intelligent flexible sensing (iFlexSense) skin for aircraft [54] especiallyforthedeformability.Itshouldholdextrafeatures:(i)capabilityofadaptive deformation with the infrastructural skin, (ii) low membrane stiffness to allow small actuation requirement, (iii) effectively sensing air loads while maintaining initial surface geometry, and (iv) high elasticity and recovery to allow multiple cycles of deformation [53]. Empirically and reasonably, the realization of these targets mainly relies on the structural engineering of flexible electronics. Predictably, future aircrafts with more controlled systems will lead to the increasing need for flexible electronics applied on the aircraft surface. 1.3 Structural Strategies 1.3.1 Wavy Strategy The wavy strategy was first proposed in the pioneering work of flexible electronics in 2006, in which flexible devices made of single-crystal silicon were buckled into microscale, periodic, and wavy shapes [55]. Thin silicon ribbons (20 ~ 320 nm) were firstly fabricated by lithographic processing and then transferred to the surface of a pre-strained elastomer substrate (e.g., PDMS). After releasing the pre-strain, the silicon ribbons were compressed to wavy structures. Wavy silicon can be reversibly stretched and compressed to large levels of strain without damage. The strains of the whole system during deformation are accommodated mainly through changes of the wavelengths and amplitudes of the wavy structures rather than the mechan- ical properties of materials. The wavy silicon ribbons of 100 nm thickness can be stretched up to 30% that overwhelms the 1% strain limit of silicon. Such strategy renders the conventional stiff, nonstretchable, and even brittle materials sufficiently deformable to act as flexible electronics. It not only allows large-scale stretchability
- 28. 1.3 Structural Strategies 7 without degradation in electronic performances, but also realizes good conforma- bility on complex surfaces. The wavy strategies pave the new way for the structural engineering of flexible electronics. Many mechanical models have been established to analyze the configuration of wavy structures. In the beginning, under the small-deformation assumption (<5%) and the principle of minimum potential energy, Khang et al. [55] and Huang et al. [56] developed an energetic method to describe the buckling behaviors, in which the out- of-plane displacement is assumed as a sinusoidal profile. Afterward, considering the practical large deformations (>5%), Jiang et al. [57] and Song et al. [58] developed a nonlinear finite-deformation model, by which the wavelength and amplitude of wavy profile can be predicted. Following this direction, the theoretical framework is gradually and continuously improved,andmoreandmoresituationsareconsidered,includingpost-buckling[59], finite-width effect [60], local versus global buckling [61], thermomechanical analysis [62, 63], adhesion-governed buckling [64], rigid/soft wavy surfaces [65, 66], 1D/2D wavy surfaces [67, 68], and moderately large-deflection theory [69]. Relying on the underlying mechanisms, the wavy designs are widely applied in flexible electronics. Initially, Khang et al. presented a stretchable single-crystal silicon p-n diode on a PDMS substrate at −11%(top), 0% (middle), and 11% (bottom) applied strains [55]. Furthermore, Kim et al. demonstrated stretchable and foldable silicon integrated circuits based on 2D wavy designs [70]. Ko et al. realized the wrapping of a silicon membrane circuit on a golf ball (Fig. 1.4a) [71]. Noteworthily, except for the full bonding method, the selective bonding method is also widely used to generate wavy structures on the elastomeric substrates. For example, Sun et al. constructed metal– semiconductor field-effect transistors based on buckled wavy GaAs ribbons [72]. Fig. 1.4 Various structure strategies in stretchable electronics: a Wavy [71], b serpentine [73], c self-similar [74] strategies
- 29. 8 1 Structural Engineering of Flexible Electronics 1.3.2 Island-Bridge Strategy Originating from and then beyond the wavy strategies, the island-bridge strategies were proposed to achieve higher levels of stretchability. The non-stretchable func- tional components reside at the low-strain islands and the stretchable interconnects form the bridges to accommodate the mechanical stretchability and electrical conduc- tivity. Due to the much lower stiffness of the bridges than the islands, the islands and adhered functional devices are mechanically isolated. In other words, the key point of island-bridge strategies lies in the design of bridges. By now, there are three representative successful cases, namely arc-shaped [75, 76], serpentine [77–80], and fractal interconnects [74, 81–83]. Since the arc-shaped designs are very similar to the wavy strategies in the previous section, here we mainly introduce the latter two. Arc-shaped interconnects. Under stretching and compression, switch-on/off of out- of-plane buckling of arc-shaped interconnects can accommodate the applied strain. Initially, the arc-shaped interconnects were usually treated as clamped beams or films, and the out-of-plane buckling displacement was assumed as sinusoidal profiles [75]. After that, to break the limitation of the sinusoidal assumption for the large displacements, a finite-deformation theoretical model was developed, which can offer a more accurate prediction of the amplitude [84]. Wang et al. presented a systematic analysis for the buckling behaviors and gave the critical criteria, by which the buckling behaviors are determined by the relationship between pre-strain and adhesion and the modes can be subdivided into global buckling, local buckling, and no buckling [85]. Serpentine interconnects. The serpentine strategy was firstly applied in the mile- stone work of epidermal electronics in 2011 [2], in which the concept of epidermal electronics was defined. With the filamentary serpentine designs, the system provided elastic, reversible responses to large-strain deformations with effective moduli (<150 kPa), bending stiffnesses (<1 nN/m), and areal mass densities (<3.8 mg/cm2 ). Those merits are in orders of magnitude smaller than those possible with conventional electronics or even with recently explored flexible devices. Thereafter, the filamen- tary serpentine interconnects have been widely explored in flexible electronics. A representative example is the soft microfluidic assemblies of sensors, circuits, and radios for the skin, in which strain-isolated device components are connected by a freestanding serpentine interconnecting network (Fig. 1.4b) [73]. A systematic and comprehensive investigation of mechanical behaviors in such serpentine interconnects has been performed by researchers. In earlier research, the serpentine interconnects were mostly freestanding and the stretchability relies on the folding, tenuous, and thin designs. Zhang et al. developed analytical models to reveal the buckling mechanisms in such stretchable serpentine microstructures, including the buckling and post-buckling behaviors [80]. A scaling law was established to deter- mine the critical buckling strain, by which symmetric and anti-symmetric buckling modes were identified. Soon after, Zhang et al. further proposed a pre-strain approach that can significantly improve the stretchability (more than two times) compared to
- 30. 1.3 Structural Strategies 9 the situation without pre-strain [86]. The analytical model can not only effectively predict the wavelength but also explain the influence of thickness, offering a rational method to obtain the desired stretchability. Moreover, in 2016, Su et al. introduced a different route to design the serpentine interconnects, where tenuous and freestanding geometries were substituted by thick and bonded layouts to enhance the mechanical and electrical performance [87]. The in-plane and out-of-plane buckling modes were replaced by pure in-plane scissor- like deformations. More specifically, with the increase of thickness, the deformation mechanisms change from wrinkling (localized, multiwave, out-of-plane buckling) to buckling (coupled out-of-plane buckling and twisting) and then to scissoring (pure in-plane bending deformation). The scissor-like deformations significantly increased stretchability from 20% for thin, buckling interconnects to ~ 100% for thick, scis- soring interconnects. The findings provide a significant supplement and open a new direction for the design of serpentine interconnects. In addition, Yang et al. proposed a “cut-and-paste” process to manufacture the multiparametric epidermal sensing systems based on the serpentine designs [88]. Pan et al. discussed the effect of substrate thickness and concluded that the reduction of the substrate can improve the stretchability [89]. Fractal interconnects. Aiming to reconcile the mutually exclusive requirements of large stretchability and high-area coverage, fractal design concepts were introduced for stretchable electronics. Before that, Zhang et al. showed the increase of surface filling ratios by increasing the fractal order from 1 to 4 [81]. As with the other strate- gies, fractal interconnects can also be divided into freestanding and bonding layouts. For the freestanding situation, the deformations can be further divided into pure in- plane stretching for large thickness-width ratio (>1) and spatial buckling for small thickness-widthratio(<1/5).Fortheformercase,Zhangetal.gaverecursiveformulae to describe the relationship between fractal orders and flexibility and stretchability [81]. For the latter case, Xu et al. propounded ordered unraveling mechanisms to interpret the deformations [82]. In addition to the abovementioned serpentine shape, fractal designs can be applied to many other shapes [83, 90], e.g., zigzag, sinusoidal, and horseshoe. Su et al. developed an analytic method to directly compute the elastic energy and the tensile stiffness of fractal interconnects of arbitrary order n and in arbitrary shape [83]. Furthermore, Dong et al. investigated the bonding configuration of such interconnects with the elastomer substrate [91, 92]. By dint of the simultaneous large stretchability and high-area coverage, fractal interconnects used as bridges between mechanically isolated islands have shown unique advantages in stretchable lithium-ion batteries [82], epidermal electronics (Fig. 1.4c) [74], and radio-frequency antennas [93]. Furthermore, fractal designs can also enhance conformability on soft, curvilinear surfaces, which are meaningful for applications in bio-systems. Xu et al. presented a fractal electrode array distributed over a rabbit heart circumference to deliver cardiac electrical stimulation and sense cardiac electrical activity [94]. Moreover, fractal electrodes are conformable to more complex biological surfaces, in which the fractal electrode meshes can be directly and chronically mounted on the complex surfaces of the auricle and the mastoid [95]. The
- 31. 10 1 Structural Engineering of Flexible Electronics case of the auricle proved another advantage of fractal interconnects, namely direc- tional guided deformability to match the complex topology of the auricle. Specifi- cally, all-vertical Peano curves were used to get selective high stretchability along with the longitudinal coordinates. 1.3.3 Kirigami and Origami Strategy Recently, the concept of Kirigami mechanical metamaterials (Kiri-MMs) has been introduced into the design of flexible electronics [54]. The salient structural traits including negative Poisson’s ratio [44, 96], ultra-stretchability [97–100], mechanical programmability [101–105], and transformability from 2 to 3D [106–108] provide a brand-newresearchstrategy.Specifically,theprogrammabilityenablesthepossibility to fit arbitrary curvilinear surfaces [105], and the transformability from 2 to 3D is a perfect solution to the contradiction between 2D planar processing technology and 3D conformal demands [106, 109]. Compared with other thin open-mesh serpentine or island-bridge strategies introduced above, Kirigami structures have a high fill factor, which are particularly desirable for high-density sensing or high-resolution imaging [110]. In the early stage, graphene Kirigami was an impressive example to demonstrate the salient traits of Kirigami [111], by which one-atom-thick graphene sheets were transformed into resilient and morphing structures without sacrificing electrical performance. By now, Kirigami-inspired creative applications have become abundant, including self-powered strain sensor (Fig. 1.5a) [112], soft crawler (Fig. 1.5b) [113], morphable stent (Fig. 1.5c) [114], shoe grip (Fig. 1.5d) [115], adaptive imager (Fig. 1.5e) [110], and flexible car shell [116]. The underlying mechanism of Kirigami structures relies on the local buckling of hinges under low-energy loads to empower the whole struc- tural reconfigurability. Rafsanjani and Bertoldi investigated the mechanisms of the buckling-induced Kirigami and studied how the behavior evolves when the thickness is progressively decreased [109]. However, all these creative buckling-driven traits are poles apart from the fully conformal goals of flexible electronics that require coating as smooth as possible. Current studies mostly directly make use of the buckling-induced traits, but how to restrain buckling while keeping extraordinary mechanical properties of Kiri-MMs remains open. As a note, despite non-stretchable PI used in curvy, shape-adaptive imagers, the ultrathin thickness (~5 µm) makes the devices sufficiently compliant [110]. To address this challenge, we show that using soft material at the hinges of Kiri-MM can restrain this local buckling effect and improve conformability [117], which will be presented in the following chapter. As for the design methods, fractal Kirigami can further enhance the stretchability. Programmable Kirigami mechanical metamaterials are becoming popular. Choi et al. propounded a meaningful inverse Kirigami design method, in which the cutting path was determined by objective and constrained functions derived from targeted conformal shapes [105]. Most recently, based on the computer graphics, Jiang et al. presented an affirmative solution to realize the targeted programmability of Kirigami
- 32. 1.3 Structural Strategies 11 Fig. 1.5 Kirigami strategy: a self-powered strain sensor [112], b soft crawler [113], c morphable stent [114], d shoe grip [115], and e adaptive imager [110] sheet between arbitrary shapes, in particular between a 2D sheet and a 3D curved surface [118]. This so-called inverse problem for Kirigami cut and fold patterns is solved by drawing on a differential-geometric interpretation of the morph and progress in geometric computing. Therefore, it is convinced that Kirigami will be a powerful strategy to design the flexible electronics. As the twin concept of Kirigami, Origami is another admired strategy to build wonderful structures. Because of the ability to hold creases and bend at will, a 2D sheet can be folded into arbitrary shapes. The mechanisms of Origami have been widely studied for a very long time, including mathematics [119], geometry [120], and compliant mechanisms [121]. Many Origami strategies have been proposed like Miura-ori and its variants, Ron Resch’s tessellation, and square twist. Either way, the kernel always roots in the crease patterns. Mechanically, the creases act as soft hinges to connect the stiffer facets, and the low-energy deformation of the creases induces the morphability of the whole Origami structures (Fig. 1.6a) [122]. Similar to previous “island-bridge” strategies, the strain-free facets between the creases are uniquely suited as the integration platform to mount active devices of flexible electronics. Moreover, as Origami-based mechanisms often feature multiple discrete folding motions, they enable the realization of programmable 3D multishapes. In a nutshell, Origami has many ideal characteristics: monolithic preparation, scale-independent, perfect self-assembly compatibility, and unlimited design space based on rich folding patterns, making it very suitable for applications in flexible electronics. Recently, Origami has been explored for applications in many engineering fields, e.g., compactly deployable solar arrays for space applications [123], self-folding crawling robots for machine manufacturing [124], and medical stents for biomedical
- 33. 12 1 Structural Engineering of Flexible Electronics Fig. 1.6 Origami strategy. a Transformable Origami with multiple degrees of freedom [122]. b Ori-MMs-based silicon optoelectronics for hemispherical electronic eye systems. Reproduced with permission [128]. c Ori-MMs-based conformal electronics made of non-stretchable materials [131]. d Origami-based electrothermal devices [132]. e Stretchable Origami robotic arm [133]. f Origami- based wide-range flexible capacitive pressure sensors [43] applications [125]. Its strut in flexible electronics may start from the Origami lithium- ion batteries proposed in 2014 [126]. This Origami battery exhibited stable and reliable performances under large cycles of mechanical deformations. By utilizing printable ZnO nanowires and carbon electrodes, Lin et al. developed a stretchable and deformable Origami photodetector array based on the Miura-ori strategy [127]. The Origami photodetector array can provide excellent capabilities of omnidirec- tional photodetection. Furthermore, Zhang et al. developed Origami silicon opto- electronics for dense, scalable, and compact hemispherical electronic eye systems [128] (Fig. 1.6b), which were compatible with mature complementary metal–oxide– semiconductor (CMOS) technologies that enable deployments in extremely high density. In the same period, Origami was introduced to fabricate a flexible and fold- able thermoelectric nanogenerator [129]. Most recently, Qi et al. displayed reconfig- urable flexible electronics driven by Origami magnetic membranes [130]. In addition, Origami can be adopted to realize full wrapping of conformal electronics made of non-stretchable materials (Fig. 1.6c) [131], to structure electrothermal devices with controllable multi-degrees-of-freedom shape morphing (Fig. 1.6d) [132], to enable stretchable robotic arm with omnidirectional bending and twisting soft robotics (Fig. 1.6e) [133], and to devise wide-range flexible capacitive pressure sensors (Fig. 1.6f) [43]. Incidentally, a number of computer-aided tools to Origami such as TreeMaker and Oripa have been developed. TreeMaker allows new Origami bases to be designed for special purposes and Oripa tries to calculate the folded shape from the crease pattern. All these achievements demonstrate the potential of the Origami to develop spatial flexible electronics.
- 34. 1.3 Structural Strategies 13 1.3.4 Buckling-Driven Assembly Strategy Buckling-driven assembly strategy is another milestone of structural engineering of flexible electronics (Fig. 1.7) [134]. It relies on the control of buckling to realize 2D- to-3D transformation, with the release of prestrained elastomer substrate to provide initial mechanical drive and vice versa. It can be reversibly stretched and buckled between 2D and 3D configurations without degradation of performance even with a large number of cyclic loadings [106, 135, 136]. The 3D stretchable multifunc- tional photodetector is a convincing paradigm for the effectiveness of the buckling- driven assembly strategy [137]. The interconnects of the device exploit a sandwich configuration, with the graphene encased by two SU-8 layers, and then the SU-8 layers are buckled into a hemispherical structure, rendering a 3D arrangement of the MoS2 patches that serve as photodetecting elements. The advantages include (i) the concurrent tracking of the direction and intensity of the incident light, (ii) optically transparent system allowing the detection of incident angles, and (iii) high geomet- rical extensibility like an octagonal prism and an octagonal prismoid. Note that these merits cannot be easily achieved by photodetector arrays in planar layouts. Except for the compressive buckling, the 3D assembly can be also realized by tensile buckling that can circumvent the pre-stretching limit [139]. When the substrate is stretched uniaxially, the nonbonded regions of the 2D precursor are delaminated from the substrate, resulting in a 3D transformation through coordinated bending/twisting deformations and translational/rotational motions. The derivative Fig. 1.7 Buckling-driven assembly strategy. a Schematic illustration of the assembly process guided by controlled buckling [134]. b 3D Origami micro/nanostructures [138]. c 3D Kirigami mesostructures [106]
- 35. 14 1 Structural Engineering of Flexible Electronics strain sensor shows a great increase in sensing range, from ≈9.8% to ≈50%. 3D elec- trically small antennas (ESAs) provide another representative example [140]. ESAs are antennas that are much smaller when compared to the operating wavelength, which is of significance in miniaturized communication systems. As the planar ESAs are usually limited in the bandwidths and the efficiencies because of the small volume occupation of the Chu-sphere, the spatially integrated ESAs, such as hemispherical ESA, can ideally occupy the Chu-sphere, thereby offering substantially improved performance. Furthermore, recent achievements of the buckling-driven assembly strategy show that different releasing sequences and specially engineered precursor designs can trigger multistable, transformable, and reconfigurable 3D electronics [136]. Fu et al. illustrated morphable 3D mesostructures and microelectronic devices by multistable buckling mechanics, in which mesostructures can be reshaped between different geometries as well as those that can morph into three or more distinct states. The sequential release of prestrain applied to the elastomer platform serves as the control strategy. An adaptive radiofrequency circuit and a concealable electromagnetic device provide examples of functionally reconfigurable microelectronic devices. By the way, the initial 3D helical interconnects can also be an attractive design in the flexible electronics due to high elastic stretchability and exceptionally low effective modulus [141]. 1.3.5 Structural Designs of Substrate The above-discussed structural strategies focus on the design of functional devices, rendering the conventional stiff, nonstretchable, and even brittle materials suffi- ciently conformable to complex surfaces while maintaining high-performance elec- trical properties. Structural engineering of substrates is another important branch of flexible electronics. By now, there are many work concentrating on this field, e.g., strain-isolation of the rigid devices from substrates [142–144], enhancement of biocompatibility for biological applications [145–148], and functional supplement of devices to support multifunctions [149, 150]. Here, three representative structural designs of substrates, namely surface structural designs, cellular structural designs, and embedded designs, are highlighted. Surface strategies. In the extensively used “island-bridge” strategies as presented above, the freestanding, tenuous, and complex interconnects have poor stability in mechanical properties, as well as non-negligible electrical resistances and high levels of power dissipation [87]. As for the bonded interconnects based on the thick bar layouts, the challenge is low effective areal coverage that will reduce the integrated density of devices and corresponding performance, especially for optoelectronic devices like photovoltaics and photodetectors. The emerging Kirigami strategy is effective to address such issues, but the local buckling at hinges will result in conformal problems, which greatly increases the risk of delamination and even
- 36. 1.3 Structural Strategies 15 detachment from the conformal surfaces, weakens the accuracy of measurement and biocompatibility, and exacerbates the encapsulating difficulty. The surface design of substrates is an alternative approach, which can free the electronicsfromthedeformationsofthesubstratewhilekeepingahigharealcoverage [142,151,152].Arepresentativelayoutisthatthedevicesmountonraisedislandsand the interconnects between them buckle downward into separating trenches (Fig. 1.8a) [142]. The trench regions absorb most of the externally applied strains, whereas the top surfaces of the islands barely deform, resulting in minimal interfacial stresses transmitted to devices mounted on top. Moreover, based on the optimizations that replace the square islands with notched islands, Lee et al. demonstrated a high- efficiency dual-junction GaInP/GaAs photovoltaics [152]. Afterward, Cantarella et al. reported a mesa-shaped elastomeric substrate, supporting thin-film transistors (TFTs) and logic circuits (inverters) [153]. During mechanical solicitations, the use of such high-relief structures aims at localizing the strains on the substrate, around the pillars and not on the pillars’ surface. In this way, devices can withstand different modes of deformation with stable electrical performances, e.g., bending of up to 6 mm radius, stretching of 20% uniaxial strain, and 180° global twisting. Similarly, a mogul-patterned elastomeric substrate was reported to improve the stretchability, with bumps and valleys regularly positioned in hexagonal closed packed structures [154]. This arrangement enables the layers and devices to be more stable, even under multidirectional stretching conditions similar to that of skin. The tripod-structured substrate is another available surface strategy, in which the devices are suspended to the substrate [155–158]. As a result, the strain can be significantly decreased by reducing physical contact. This configuration would confer two advantages: (i) electrode materials of the microsupercapacitors could Fig. 1.8 Engineering substrates. a Isolated microisland design [142]. b Tripod-structured substrate [157]. c Toothed substrate design [159]. d Cellular substrate [160]. e Embedment of stiff platforms in elastomeric substrate [143]. f Soft elastomers with ionic liquid-filled cavities as strain isolating substrates [144]
- 37. 16 1 Structural Engineering of Flexible Electronics be stretched despite being intrinsically stiff, and (ii) the suspended wavy struc- tures would reduce the strain concentration in the electrode fingers during the stretching/relaxing processes. Such design is very useful to thin-film devices, e.g., graphene microribbons in microsupercapacitors (Fig. 1.8b) [157] and electrodes in biointegrated electronics [158]. Except for the strain isolation of devices, surface strategies can also be used for the strain reduction of interconnects. For example, a toothed substrate has been proposed to release the mechanical constraints of serpen- tine interconnects (Fig. 1.8c) [159]. The freestanding segments offer the desired stretchability while the bonded segments enhance the mechanical stability. Indeed, the position deviation of the serpentine interconnects to the toothed substrate may weaken the stretchability, but the reduction is predictable and acceptable. Cellularstrategies.Architectedcellulardesignisubiquitousinengineering,bywhich light weight and high stiffness/strength can be achieved simultaneously [160, 161]. The architecture of the core is complex, with intricately shaped ligaments and gradi- ents in density. Cellular substrates are widely used in flexible electronics, not only because of the low modulus for the enhancement of stretchability and conforma- bility, but also because of their unique ability to minimize disruptions to the natural diffusive or convective flow of bio-fluids in advanced, bio-integrated implants [148, 162, 163]. Notably, the solid substrate/encapsulation in other structural strategies will disrupt the natural diffusive or convective flows of bio-fluids through flexible elec- tronics. Jang et al. introduced a cellular, composite, and deterministic soft network that can be tailored precisely to match the non-linear properties of biological tissues [163]. Lee et al. proposed a bio-inspired honeycomb cellular substrate to achieve high permeability to facilitate solution exchange for use in biointegrated electronics [148]. The overall elastic properties of such systems are often difficult to be determined precisely, because they strongly depend on the detailed alignment and position of the serpentine interconnects relative to the pores in the cellular substrate. A theo- retical model of hierarchical lattices is developed to study the underlying relations between the J-shaped stress–strain curves and the microstructure geometric param- eters of hierarchical lattices [164]. Three common lattices (triangular, kagome, and honeycomb) and their hierarchical variants were discussed. Chen et al. established an analytic constitutive model by considering the cellular substrate as an equivalent medium under finite stretching (Fig. 1.8d) [160], in which the lower bound of the stretchability with respect to all alignments and positions of a representative intercon- nect on the cellular substrate can be estimated. Furthermore, Zhao et al. developed an analytical model for “zigzag” cellular substrates under finite deformation, achieving higher compliance than the previously reported hexagonal cellular substrates [165]. The programmability of mechanical properties is another distinct advantage of cellular substrates, which can adjust the Poisson’s ratio or stiffness to reinforce the conformability on the targeted surfaces, especially for biomedical devices. Liu et al. developed a soft cellular network [166], which can yield tailored isotropic Poisson’s ratio from −1 to 1, with a tunable strain range from 0% to ≈90%. The theoretical design methods define tailored network geometries to yield target Poisson ratios with
- 38. 1.4 Structural Opportunities by Materials 17 desired strain ranges. Meanwhile, Zhang et al. introduced a class of soft mechanical metamaterials that can achieve large effective negative swelling ratios, with desired isotropic/anisotropic features [167]. Embedded strategies. The strain isolation can also be achieved by embedded designs of substrates. Romeo et al. presented elastomeric substrates with embedded stiff platforms to realize the strain isolation (Fig. 1.8e) [143], in which the devices were deployed on the isolated surfaces overhead the embedded stiff platforms. When the platform is significantly stiffer than the surrounding silicone matrix, the elastomer volume above the platform is little strained when the matrix is macroscopically stretched. Therefore, brittle materials deposited on the corresponding top surface are not extensively stretched. Conversely, Ma et al. proposed soft elastomers with ionic liquid-filled cavities as strain isolating substrates (Fig. 1.8f) [144]. Ionic liquids filled a microfluidic space, or cavity, defined in a low modulus elastomeric substrate. The contained liquid film mechanically isolates the underlying skin from the electronics above, without any direct contact to either. Notably, ionic liquids were applied instead of traditional liquids to eliminate any possibility for leakage or evaporation. The liquids were positioned between the electronics and the skin, within an enclosed, elastomeric microfluidic space, but not in direct contact with the active elements of the system, to avoid any negative consequences on electronic performance. 1.4 Structural Opportunities by Materials During the above discussion about the structural engineering of flexible electronics, traditional active materials, like silicon, gold, silver, and carbon, have been repur- posed through novel structural designs and integration processes. Structural engi- neering that enables low-modulus mechanics of high-modulus materials has been widely exploited to accommodate extreme mechanical deformations. The most repre- sentative ‘island-bridge’ strategy consists of tenuous structurally stretchable (wavy, serpentine, self-similar, coil, etc.) interconnects between rigid devices. Meanwhile, material chemistry and engineering play key roles in the exploration and devel- opment of flexible electronics, including new active materials, new organic-based electronic components and elastomers, and new integration processes of both. New supramolecules, polymers, hydrogels, and other biomaterials based on the latest genetic and synthetic biology technologies have been proposed and used to get high-performance flexible electronics. On the other hand, the material engineering enables more novel structural realizations that are inaccessible before, for example, advanced metamaterials (MMs) can provide an additional dimension of probability and enable dynamically adjustable structural performance. As such, active metama- terial is a hybrid structure-material strategy, which shows great potential to advance flexible electronics. At present, there are six main types of materials for flexible electronics: conductive metal oxides like ITO [168], metal nanowires [169], carbon materials like carbon nanotubes [170] and graphene [171], conductive polymers
- 39. 18 1 Structural Engineering of Flexible Electronics [172], conductive and stretchable gels [173], and single-phase liquid metals and liquid metal alloys [174]. Here, structural opportunities by these advanced materials are briefly discussed. About active mechanical metamaterials, several recent reviews gave very elab- orate discussions from different perspectives [45, 175, 176]. Cai et al. proposed a brand-new concept of ‘mechanomaterials’ to define the programming advanced func- tional materials by leveraging the force-geometry-property relationships at multiple scales [45]. Qi et al. made a deep and systematic summary of active mechanical metamaterials [175], elucidating their underlying construction principles, classifica- tions, and applications. Pishvar et al. surveyed the innovative multidisciplinary soft, smart matter in the context of active mechanical metamaterials [176]. The applica- tions also include topics focused here such as stealth cloak, electronic skin and soft robot. Referring to active wave-based metamaterials, there is also a recent review that provides a comprehensive introduction [177]. 1.5 Summary This chapter summarizes some of the most significant advances in structural engi- neering of flexible electronics, including various strategies based on wavy, island- bridge, Kirigami, Origami, and buckling-driven, with engineering substrates that involve surface, cellular and embedded designs. These branches are growing very fast, and it is time to review and analyze all these efforts, sum up the first stage, and draw the next blueprint. That is the inspiration of this book. Under the guidance of this principle, the following chapters introduce the themed efforts, focusing on the underlying theory and method of structural designs for flexible electronics. Notably, despite significant progress, challenges remain, especially in structural designs to enable enhanced levels of comfort of wearable electronics, to achieve minimally invasive bio-integration of implantable electronics, to offer a high degree of mechan- ical deformability after solid encapsulation in soft robotics, and to withstand extreme operating experiments in aircraft. Figure 1.9 gives the panorama of this book. We do hope the readers enjoy the structural engineering in flexible electronics presented in this book, and find it inspiring for their future research with blooming ideas and progress to further usher the prosperity of this field. This research is absolutely multi- disciplinary, and the related chemical, electrical, and biomedical investigations are closely involved in each other.
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- 48. Chapter 2 Buckling of Film-on-Substrate System in Flexible Electronics 2.1 Introduction The film-on-substrate system is a typical structure in flexible electronics, as shown in Fig. 2.1. Reported studies on the film-on-substrate system have made large progress in duality, crack, and bendability/stretchability [1–6]. Freestanding Si/metal thin film ruptures when stretched beyond 1–2% [7]. However, when bonded onto an organic substrate, it can sustain plastic deformation ten times larger than its fracture strain, yet still remains electrically conductive [8, 9]. Park et al. [1] carried out theoretical and experimental studies on bending in structure relevant to inorganic flexible electronics, and the effects of edges and finite device sizes were also considered. Buckling plays a key role in achieving the stretchability of flexible electronics, which is a classic mechanical problem but is full of new vitality in recent years. Most stretchable electronics are distributed, interconnected, and thin films mounted on an elastomeric substrate. A commonly used strategy is the island-bridge system, in which the electronic components reside at the islands and various interconnects form the bridges, the latter provide the majority of stretchability. It is easy to buckle for this type of structure during the fabrication processes and applications, and the stretch deformation often leads to out-of-plane or lateral buckling, in-plane buck- ling, or a combination of all. The critical buckling load, beyond which catastrophic consequences usually occur, is one of the most important properties of engineering structures. A compressively strained elastic film bonded onto the compliant substrate will form wrinkles, which is a simple approach for flexible integrated circuits to circum- vent the limitations of stretchability and their low flexural rigidity. Khang et al. [3], Kim et al. [4], Jiang et al. [10], and Sun et al. [11] studied the buckling behaviors of film-on-substrate system and the controllability of the structure formation, and developed stretchable and foldable silicon integrated circuits with high-performance. Huang et al. [2], Huang et al. [12], and Lacour et al. [9] studied nonlinear buck- ling behaviors and discussed the mechanisms of reversible stretchability of thin metal films on elastomeric substrates. Circuits in wavy patterns offer fully reversible © Science Press 2022 Y. Huang et al., Flexible Electronics, https://doi.org/10.1007/978-981-19-6623-1_2 27
- 49. 28 2 Buckling of Film-On-Substrate System in Flexible Electronics Fig. 2.1 Schematic illustration of the process for building buckled thin film on elastomeric substrates [3] stretchability/compressibility without substantial strains in circuit materials them- selves. The resulting mechanical advantages are critically important for achieving stretchability. In these systems, strains at the circuit level can exceed the fracture limits of almost all known electronic materials [3]. In this chapter, the model of the film-on-substrate structure is established based on interface continuity. The rest of this chapter is organized as follows. In Sect. 2.2, the formation of film-on-substrate structure in flexible electronics is presented. In Sect. 2.3, temperature-dependent global buckling analysis and structural design are discussed. In Sect. 2.4, temperature-dependent local buckling analysis and critical condition are demonstrated. Finally, in Sect. 2.5, a brief summary is drawn. 2.2 Formation of Film-on-Substrate Structure The film-on-substrate structure in flexible electronics was first proposed by Khang et al., in which wavy Si ribbon was obtained via prestrained PDMS [3]. Figure 2.1 presents the schematic illustration of the process for building stretchable single- crystal Si devices on elastomeric substrates: (i) Fabricaiton of thin Si ribbon by conventional photolithography, (ii) Covering with prestrained PDMS, and (iii) Peeling back the PDMS and releasing the prestrain. After these procedures, self- assembly, well-controlled, highly periodic, and stretchable wavy Si ribbon on PDMS can be achieved. The deflect is much larger than the thickness of film when the film buckles, which can be modeled adequately by Von Karman plate theory [13]. Since the membrane would be subject to lateral load during the conformal procedure, the additional strain produced in the middle plane during bending should be taken into consideration. The membrane strain εij is εij = ε0 ij + 1 2 ( ∂ui ∂xj + ∂uj ∂xi ) + 1 2 ∂u3 ∂xi ∂u3 ∂xj (2.1) where i, j = 1, 2. ε0 ij is the initial strains, u1(x1, x2) and u2(x1, x2) are related to the in- plane displacements and u3(x1, x2) is out-of-plane displacement. The displacement of membrane is assumed as
- 50. 2.2 Formation of Film-On-Substrate Structure 29 u3 = A cos kx1 (2.2) Supposing the film is subjected to pressure in x1 direction ε0 11 = −εprestrain, ε0 12 = ε0 22 = 0 (2.3) Substitution of Eqs. (2.2) and (2.3) into (2.1) gives ε11 = −ε prestrain 11 + ∂u1 ∂x1 + 1 2 [Ak sin(kx1)]2 , ε22 = ∂u2 ∂x2 , ε12 = 1 2 ( ∂u2 ∂x1 + ∂u1 ∂x2 ) (2.4) Supposing that the strain is uniform in the film, the partial derivative of ε11 with respect to x1 gets 0 = ∂2 u1 ∂x2 1 + A2 k3 sin(kx1) cos(kx1) (2.5) Then u1 = 1 8 A2 k sin(2kx1) (2.6) By the same method, u2 = 0. The Eq. (2.4) becomes ε11 = 1 4 A2 k2 − εprestrain, ε22 = 0, ε12 = 0 (2.7) The internal forces of the film become N11 = hfilmEfilm ( 1 4 A2 k2 − εprestrain ) , N22 = 0, N12 = 0 (2.8) where hfilm is the thickness of Si film. Thus the bending energy and membrane energy can be obtained Umembrane = 1 2 N11ε11= 1 2 hfilmEfilm ( A2 π λ2 2 − εprestrain )2 Ubending = k 2π ( 2π k 0 Efilmh3 film 24 ( ∂2 w ∂x2 1 )2 dx1 = 1 48 Efilmh3 filmA2 k4 (2.9)
- 51. 30 2 Buckling of Film-On-Substrate System in Flexible Electronics Supposing the interfacial shear stress is zero, the substrate energy can be written as [12] Usubstrate= g 4 EsubstrateA2 k (2.10) where g = (3−4νsubstrate) cosh(2khsubstrate)+5−12νsubstrate+8ν2 substrate+2(khsubstrate)2 (6−8νsubstrate) sinh(2khsubstrate)−4khsubstrate . Therefore, the total energy is Utotal = Umembrane+Ubending + Usubstrate = 1 2 hfilmEfilm ( A2 π λ2 2 − εprestrain )2 + 1 48 Efilmh3 filmA2 k4 + g 4 EsubstrateA2 k (2.11) According to the principle of minimum potential energy, namely a stable system always stays at the minimum, minimizing the total energy with respect to the amplitude and wave length can obtain the buckled configuration [3] λbuckling= 2πhfilm √ εcritical , Abuckling = hfilm / εprestrain εcritical − 1 (2.12) where εcritical= [ 3 8 Esubstrate ( 1−ν2 film ) Efilm(1−ν2 substrate) ]2/ 3 is the critical strain for buckling, εprestrain is the levelofprestrain,λbuckling isthewavelength,andAbuckling istheamplitude.ThePoisson ratio is ν, the Young’s modulus is E, and the subscripts refer to properties of the film or substrate. The thickness of the Si is hfilm. However, the “wavy” Si ribbons are formed from spontaneous buckling with amplitudes and wavelengths determined by material properties (e.g., moduli and thickness). Moreover, although the range of acceptable strain is improved significantly (~20%) compared to that of silicon itself (~1%), the stretchability is still too small for certain applications. After that, to control the buckling geometries and enhance the stretchability, Sun et al. [11] used lithographically defined surface adhesion sites together with elastic deformations of a supporting substrate to achieve buckling configurations with deter- ministic control over their geometries. In this case, the PDMS surface remains flat after relaxation for both activated/inactivated regions. For vanishing displacement and vanishing stress traction, the relaxed PDMS has vanishing energy. The buckling amplitude A can be described as Abuckling = 4 π L1L2 /( εprestrain − εcritical ) , εprestrain > εcritical (2.13) where εcritical = h2 filmπ2 / 12L2 1, which is identical to the Euler buckling strain for a doubly clamped beam with length 2L1. The thin film does not buckle when εprestrain
- 52. 2.2 Formation of Film-On-Substrate Structure 31 < εcritical. The maximum strain in the thin film is the bending strain that results from the thin film curvature εmax = hfilmπ L2 1 √ L1L2εprestrain (2.14) Becauseofthesmalldeformationapproximationsandlinearstress–strainbehavior used in derivation, the above results imply displacements that are tangential to the local surface relief, yielding a displacement trajectory that has the shape of a wave whose wavelength is fixed. However, they do not provide sufficient precision for prac- tical applications with inevitable uncertainties, such as poorly defined film/substrate interfaces and unknown mechanical properties in the films or substrates. To address such limitations, Jiang et al. [10] proposed a buckling theory that accounts for finite deformations and geometrical nonlinearities to yield a quantitatively accurate description of the system. This buckling theory is different from previous models in the following three important aspects: (i) The initial strain-free (or stress-free) configurations for the substrate and film are different, (ii) The strain–displacement relation in the substrate (as well as the film) is nonlinear, (iii) The stress–strain rela- tion in the substrate is characterized by the nonlinear neo-Hookean constitutive law. The wavelength and amplitude in initial buckling are λbuckling= 2πhfilm ( Efilm 3Esubstrate )1/ 3 √ 1 + εprestrain(1 + ξ)1/ 3 , Abuckling = hfilm / εprestrain εcritical − 1 √ 1 + εprestrain(1 + ξ)1/ 3 (2.15) where ξ = 5εprestrain(1 + εprestrain)/32. Figure 2.2 gives the experimentally measured and theoretically predicted wave- length λ and amplitude A versus applied strain εapplied for a buckled Si thin- film/PDMS substrate formed with a prestrain of 16.2%, and same other parame- ters. The measured wavelength increases for tension and the measured amplitude decreases, reaching zero once the tensile strain reaches the prestrain. The finite- deformation buckling theory agrees well with experiments for both amplitude and wavelength. The existing mechanical models also capture the amplitude trend but deviate from the experimental results for large tensile strain (>10%). Usually,thethicknesseffectofthesubstrateisnotconsideredbecausethesubstrate is much thicker than the film. The film-on-substrate system is always considered as a semi-infinite solid. In this case, it appears local buckling. However, when the substrate becomes thinner, another buckling mode, namely, the global buckling will be observed in experiments. Wang et al. investigated the underlying mechanisms of local versus global buckling of thin films on elastomeric substrates, definitely giving the critical condition separating the local and global buckling modes [14]. The critical condition of local buckling can be further rewritten as
- 53. 32 2 Buckling of Film-On-Substrate System in Flexible Electronics Fig. 2.2 Wavelength and amplitude of buckled structures of Si (100 nm thickness) on PDMS as a function of the prestrain. The finite-deformation buckling theory yields wavelengths and amplitudes that both agree well with experiments. Also shown are results from previous mechanical models (i.e., small deformation limit) and the simple accordion model [10] εlocal critical = ( 3Esubstrate 8Efilm )2/ 3 (2.16) and that of global buckling is ε global critical = 1 1+ 1.2F0 critical Gbeam(hsubstrate+hfilm) F0 critical EAbeam (2.17) where EAbeam=Esubstratehsubstrate+Efilmhfilm, Gbeam is the effective shear modulus of the composite beam, which is approximately the shear modulus Gsubstrate of the substrate since the film is very stiff Efilm ≫ Esubstrate and thin hfilm ≪ hsubstrate. F0 critical=4π2 EIbeam / L2 is the critical buckling load neglecting the effect of shear, EIbeam = ( Efilmh2 film−Esubstrateh2 substrate ) +4EsubstratehsubstrateEfilmhfilm(hsubstrate+hfilm)2 12(Esubstratehsubstrate+Efilmhfilm) . When εlocal critical < ε global critical, local buckling occurs. When εlocal critical > ε global critical, global buckling occurs. Because ε global critical is related with the thickness of the substrate and the length of the film-on-substrate structure, whether the film-on-substrate structure generates local or global buckling is not determined only by the ratio of Young’s modulus of the film and substrate, but also by the length of the film-on-substrate structure and the thickness of the film and substrate.
- 54. 2.3 Island-Bridge Structure of Stretchable Electronics 33 2.3 Island-Bridge Structure of Stretchable Electronics In stretchable electronics, the island-bridge structure on a soft substrate plays an important role in achieving large stretchability. A key issue in developing such a system is to prevent the island-bridge structure from breaking during use because it is composed of brittle semiconductor materials (e.g., silicon) which withstand very small strains (~1%). Figure 2.3 schematically illustrates the fabrication of circuits with noncoplanar mesh design on compliant substrates. The silicon (or other inor- ganic material) islands, on which the active devices or circuits are fabricated, are chemically bonded to a prestrained (e.g., 50%) elastometric substrate of a material such as PDMS, while interconnects are loosely bonded. Releasing the prestrain leads to compression, which causes the interconnects to buckle and move out of the plane of the substrate to form arc-shaped structures. The poor adhesion of interconnects (to PDMS) and their narrow geometries and low stiffness (compared to device islands) cause the out-of-plane deformation to localize only to interconnects, and therefore the strain in islands is very small. 2.3.1 Mechanical Model for the Bridge Structure As shown in Fig. 2.4a, at a given prestrain εprestrain of the substrate, the bridge with length Lbridge buckles to accommodate the release of prestrain, which yields the distance Lbridge/(1 + εprestrain) between the two ends A and E. In view of the symmetry, only part of the bridge with length Lbridge/4 is analyzed, as shown in Fig. 2.4b. This is indeed a problem of large deflection of a buckled bar (the elastica). The governing equation of the bridge is EbridgeIbridge dθ ds = P(wB − w) (2.18) Fig. 2.3 Schematic illustration of fabrication process for stretchable electronics with the noncoplanar mesh design on a compliant substrate; a mesh on a pre-stretched substrate; and b buckled mesh after the release of pre-stretch in the substrate [15]
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