Mechanics and thermal management of stretchable inorganic electronics

1Department of Engineering Mechanics and Soft Matter Research Center, Zhejiang University, Hangzhou 310027, China; 2Key Laboratory of Applied Mechanics, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China; 3Center for Mechanics and Materials, Tsinghua University, Beijing 100084, China and 4Department of Civil and Environmental Engineering, Department of Mechanical Engineering, Center for Engineering and Health, and Skin Disease Research Center, Northwestern University, Evanston, IL 60208, USA

Many new design concepts and enabling technologies have been developed in recent years to overcome the mismatch between the brittle nature of conventional electronics and soft/elastic requirements by stretchable electronics.In general, there are two complementary routes to develop stretchable electronics.In the first route, new organic semiconductor materials that are intrinsically compliant are used to replace the intrinsically brittle inorganic semiconductor materials [18][19][20][21][22][23][24].In the second route, conventional high-performance inorganic semiconductor materials, such as silicon and gallium arsenide, are designed in a novel structure and layout integrating with a compliant substrate to achieve stretchability.Mechanics and thermal management will help to develop robust methods for stretchable electronics by identifying the underlying mechanism of deformation and heat transfer, and establishing design guidelines and optimal strategies for experiment/fabrication.This paper will focus on the second route and provide a review on the recent advances in the mechanics and thermal management of stretchable inorganic electronics.

MECHANICS OF STRUCTURAL DESIGNS FOR STRETCHABLE INORGANIC ELECTRONICS
Several design strategies that utilize mechanics principles have been explored to make brittle inorganic semiconductor materials stretchable.Fig. 2 shows four representative structural designs for stretchable inorganic electronics: (1) wavy design [25]; (2) straight bridge-island design [13]; (3) serpentine bridge-island design [13]; (4) fractal bridgeisland design [7].Mechanics models associated with Figure 1.Stretchable electronic and optoelectronic systems.(a) CMOS circuits on PDMS substrate poked by a glass rod.Reproduced with permission from Ref. [10].Copyright 2010 AAAS.(b) Electronic eye camera with a hemispherical silicon photodiode imaging plane.Reproduced with permission from Ref. [11].Copyright 2008 Nature Publishing Group.(c) Hemispherical apposition compound eye camera inspired by arthropod eyes.Reproduced with permission from Ref. [12].Copyright 2013 Nature Publishing Group.(d) Encapsulated CMOS inverters deformed into a complex shape.Reproduced with permission from Ref. [13].Copyright 2008 the National Academy of Sciences of the USA.(e) Multifunctional epidermal electronic systems on a wrinkled skin.Reproduced with permission from Ref. [14].Copyright 2011 AAAS.(f) Stretchable ILEDs with serpentine bridges, tightly stretched onto the sharp tip of a pencil.Reproduced with permission from Ref. [15].Copyright 2010 Nature Publishing Group.(g) A lead zirconate titanate (PZT) mechanical energy harvester (MEH) on the bovine diaphragm.Reproduced with permission from Ref. [16].Copyright 2014 the National Academy of Sciences of the USA.(h) A multifunctional balloon catheter.Reproduced with permission from Ref. [17].Copyright 2011 Nature Publishing Group.
these designs have been established to identify the underlying deformation mechanism, and to establish design guidelines for experiments.The fundamental aspects of the mechanics associated with the above four representative designs are reviewed through discussions of analytic, finite element analysis (FEA) and experimental results.

Mechanics of wavy design
In this design, conventional high-performance inorganic semiconductor materials are designed in a wavy layout, which can be realized through the non-linear buckling process of thin films bonded to a prestrained compliant substrate.Fig. 3 schematically shows the fabrication process of wavy Si ribbons.The Si nanoribbons are first formed on a mother wafer by traditional lithographic processing.Etching the sacrificing layer between the thin film and the mother wafer eliminates their strong bonding such that Si ribbons are weakly bonded to the mother wafer.Contacting the prestrained compliant substrate to the thin films leads to their transfer to the compliant substrate.When the prestrain in the substrate is released, the substrate shrinks and induces a compression in the thin films to form the wavy structure, which could accommodate the external deformations through changes in wavelength and amplitude.Fig. 2a shows scanning electron micrographs of wavy, single-crystal Si ribbons on an elastomeric poly(dimethylsiloxane) (PDMS) substrate with a prestrain of ∼15%.In addition to Si thin film, this strategy has been demonstrated in other materials to achieve stretchability such as metal [26,27], carbon nanotubes [28,29], graphene [30] and ferroelectrics [31].
The formation of wavy ribbon structure can be well described by mechanics models.Under a relative small prestrain (ε pre < 5%), the mechanics model based on the small deformation theory, where the film is modeled as an elastic non-linear von Karman beam and the substrate is modeled as a semiinfinite elastic solid, gives the wavelength and amplitude as [32][33][34][35] where Ē = E /(1 − ν 2 ) is the plain strain modulus with E as Young's modulus and ν as Poisson's ratio, the subscripts f and s denote the film and substrate, respectively, h f is the film thickness, ε c = (3 Ē s / Ē f ) 2/3 /4 is the critical strain for buckling to occur, which is very small (e.g.0.034% for Si/PDMS system with E f = 130 GPa, ν f = 0.27, E s = 1.8 MPa and ν s = 0.48).The peak strain in the thin film can be approximated by the bending strain and is obtained by ε peak ≈ 2 √ ε pre ε c , which clearly indicates that ε peak is much smaller than ε pre due to the small value of ε c such that the film can survive when the system is under large deformation.For the buckled system subjected to the applied strain ε applied , the mechanics model based on small deformation theory predicts the wavelength and amplitude as and the peak strain in the ribbon is ε peak ≈ 2 (ε pre − ε applied )ε c .The wavelengths in Equations ( 1) and (2) are strain independent and agree well with experiments under a relative small prestrain [36].Under a relative large prestrain (ε pre > 5%), experiments show a strain-dependent wavelength due to the finite deformation in the compliant substrate [25,37].Jiang et al. [25] and Song et al. [38] developed an analytical model, where the difference of strain-free states for thin film and substrate, the non-linear strain-displacement relation in the substrate, and the non-linear constitutive model (neo-Hookean) in the substrate are accounted, to explain the strain-dependent wavelength.Cheng and Song [39] further simplified the analysis and showed that the difference of strain-free states for thin film and substrate has a dominant contribution to the straindependent wavelength while the non-linear straindisplacement relation and non-linear constitutive model in the substrate have negligible contributions.Their results are described below.The wavelength and amplitude are obtained as The peak strain in the buckled ribbon can be approximated by ε peak ≈ 2 ε pre ε c /(1 + ε pre ), which indicates that the peak strain in the ribbon is only 1.8% for 31% prestrain in the Si/PDMS system, i.e. the wavy Si ribbon can be stretched as large as 31%.For the buckled system subjected to the applied strain ε applied , the mechanics model in Cheng and Song [39] gives the wavelength and amplitude as The peak strain in the ribbon is approximated by ε peak ≈ 2 (ε pre − ε applied )ε c /(1 + ε pre ).The wavelengths in Equations ( 3) and ( 4) are strain dependent and agree well with finite element simulations and experiments as shown in Fig. 4.
The above buckling analysis describes, in a quantitative manner, the formation of wavy design and the peak strain in the ribbon, which are very helpful to predict the stretchability/compressibility of the system.Let ε fracture denotes the fracture strain of the film.The stretchability is the maximum applied tensile strain when the film fractures and it is well approximated by ε pre + ε fracture , where ε pre stretches the ribbon to be flat and the additional ε fracture fractures the film.The compressibility is the maximum applied compressive strain when the peak strain reaches ε fracture and it is well approximated by ε 2 fracture (1 + ε pre )/(4ε c ) − ε pre .
In addition to the above buckling analysis, several other mechanics issues such as free edge effect in Fig. 5a [40], finite ribbon width effect in Fig. 5b [41] and deformation mode in Fig. 5c [42] have also been investigated.Wavy ribbon could only provide stretchability along the ribbon direction and this could limit the applications where stretchability along all directions is needed.Following the same approach, Choi et al. [43] generated twodimensional wavy Si membrane as shown in Fig. 5d and the associated mechanics are carried out by many researchers [32,33,[44][45][46][47].

Mechanics of straight bridge-island design
The introduction of straight bridge-island design, where straight bridges are used to interconnect separately fabricated active devices (i.e.island), was demonstrated by Kim et al. [13] to further increase the system stretchability.Fig. 6 schematically illustrates the fabrication of the straight bridgeisland design on a compliant substrate.The mesh design is first formed on a mother wafer by traditional lithographic processing and then transfer printed onto a prestrained compliant substrate with the bridges loosely bonded to the substrate and the islands chemically bonded.Releasing the prestrain causes the bridge to buckle and move out of the plane of the substrate to form arcshaped structures.Under deformation, the buckled bridge, which has a much lower stiffness comparing with the island, deforms to accommodate external deformations to ensure small strains in the island.This strategy has been demonstrated in many systems such as stretchable circuits [13], electronic eye cameras [11], silicon curvilinear electronics [48] and stretchable solar cells [49].
Song et al. [50] developed an analytic model for straight bridge-island design to understand the underlying physics and to provide guidance for device design and optimization.The mechanics model is schematically illustrated in Fig. 7a.The bridge can be modeled as a beam with its two ends clamped by the islands.Let L 0 bridge and L bridge denote the distances between islands before and after the prestrain ε pre = (L 0 bridge − L bridge )/L bridge is released.Minimization of the bridge energy gives the amplitude A of the buckled bridge as where h bridge is the thickness of the bridge and the approximation holds for h bridge L 0 bridge .The peak strain in the bridge could be approximated by ε max bridge ≈ 2π h bridge ε pre /(1 + ε pre )/L 0 bridge , which is proportional to the bridge thickness to length ratio, h bridge /L 0 bridge .A thin and long bridge reduces the strain in the bridge.
The buckled bridge yields forces and bending moments at the ends, which are then applied to the islands.The deformation of the island is rather complex.The finite element method (FEM), combining with the dimensional analysis, is used to study the deformation of islands (L 0 island ×L 0 island ) on a compliant substrate, and gives the maximum strain in the island ε max island , where E island , ν island and h island are Young's modulus, Poisson's ratio and thickness of the island, respectively.Fig. 7b shows the distribution of the strain ε xx in islands (20 μm × 20 μm) when the bridge relaxes from 20 to 17.5 μm.It is shown that the maximum island strain occurs at the bridge/island boundary.A stiff and thick island reduces the strain in the island.
The stretchability and compressibility of the system can also be obtained from the model.The condition at which the buckled bridge returns to the flat state yields the stretchability . The condition at which the maximum strains in bridge and island reach the corresponding failure strain ε failure bridge and ε failure island , or the neighbor islands start to contact yields the compressibility where a = . Fig. 7c shows the stretchability and compressibility versus the prestrain for L 0 island = 20 μm, h island = 50 nm, L 0 bridge = 20 μm, h bridge = 50 nm, and ε failure bridge = ε failure island = 1%.The stretchability increases with the prestrain while the compressibility decreases.Fig. 7d shows the effect of bridge The above analytic model describes the deformations of straight bridge-island design subjected to the tension or compression along the bridge direction.The case of tension or compression not along the bridge direction is also studied by Su et al. [51] and Chen et al. [52].For other complex loadings such as twist, FEA is performed to obtain the strain distribution [13].In practice, the buckled bridge and devices islands need an encapsulation layer, which provides mechanical and environmental protection and minimizes any restrictions on free motion of the buckled bridge.Wu et al. [53] derived an analytic expression for the stretchability of encapsulated system and established design guidelines for the encapsulation layer.

Mechanics of serpentine bridge-island design
To expand the stretchability even further, serpentine bridge can be used.Comparing with straight bridge, the serpentine bridge can be stretched much further because large twist will be involved to accommodate larger deformations.Fig. 8a shows a schematic illustration of a serpentine bridge-island design, where periodically distributed posts are adopted to allow freely suspended serpentine bridges.Fig. 8b shows a representative serpentine bridge with m unit cells with one unit cell consisting of two half circles and two straight lines with the length l 2 and spacing l 1 .The cross-section of the bridge has the width w and thickness t.Zhang et al. [54] investigated the onset of buckling and postbuckling behaviors of serpentine interconnects and established scaling laws for the critical buckling strain and the limits of elastic behavior.For buckled serpentine bridge under stretching, the maximum strain in the bridge can be expressed as where the first term on the right-hand side of Equation (7) corresponds to the out-of-plane bending strain, the second term corresponds to the in-plane bending strain, g 1 and g 2 depend on the number (m) of unit cells and the length/spacing ratio α, and are determined by FEA.The condition at which the maximum strain reaches the yield strain ε yield of the bridge (e.g.0.3% for copper) gives the elastic stretchability where λ > 0 is the single positive solution of the following fourth-order algebraic equation Equation (9) indicates that the elasticstretchability depends on four non-dimensional parameters t/w, ε yield l 1 /w, m and α.The analytic predictions agree well with finite element simulations as shown in Fig. 9.It is shown that the elastic stretchability increases with the decrease of t/w, the increase of ε yield l 1 /w, and the increase of length/spacing ratio α.For m < 6, the elastic stretchability increases with m and is essentially independent of m for m ≥ 6.
Equation ( 9) defines the limit for elastic behavior in the serpentine interconnects.Once the applied strain exceeds the elastic limit, plastic deformation occurs.To obtain the total stretchability, Zhang et al. [54] adopted an elastic-ideally plastic constitutive model for the metal layer (e.g.copper) with the failure strain 10%.The predicted elastic stretchability and total stretchability for the serpentine bridge with m = 1, α = 2.4, t/w = 0.1 and l 1 /w = 10 are 43% and 81%, respectively, and they agree well with the experimental results (40% < ε elastic stretchability < 50% and 85% < ε total stretchability < 90%).
The above model describes the deformation of freely suspended serpentine bridges.Fully bonded bridges are also used to develop stretchable electronics to simplify the fabrication [55][56][57][58][59][60] by scarifying a certain level of stretchability/compressibility.The deformation mode of such fully bonded system is much more complicated than non-bonded case due to the constraints from substrate deformation.For this reason, the existing studies on fully bonded system mainly rely on finite element simulations.

Mechanics of fractal bridge-island design
The fractal bridge is introduced to achieve simultaneously even larger system-level stretchability and higher areal coverage of active devices due to its capability of covering a limited space by increasing the fractal order as illustrated by Xu et al. [7] in an ultra-stretchable battery.Fig. 10 shows an example of fractal bridge with a fractal order n up to 4. The first order pattern has a serpentine shape, and higher orders correspond to self-similar assemblies of the patterns in the previous orders.Let η denote the height/spacing aspect ratio at each order such that η = h (i ) /l (i ) (i = 1, . . ., n) with h (i ) as the height and l (i ) as the spacing.The height h (i ) is also related to the spacing l (i −1) and the number (m) of unit cells by h (i ) = 2ml (i −1) (i = 2, . . ., n).Then we have which indicates that a fractal bridge is characterized by one base length l (n) and three non-dimensional parameters: the fractal order n, the height/spacing ratio η and the number of unit cells m.The crosssection of the bridge has the width w and thickness t.The elastic stiffness of fractal interconnects could be determined by analytical models [61,62].Finite element simulations are first carried out for the first-order fractal bridge to identify the key pa- rameters on the elastic-stretchability.Fig. 11 shows the maximum principal strain versus the applied strain in the first-order fractal bridge.The maximum principal strain increases rapidly with the applied strain, and quickly reaches the yield strain 0.3% at which the bridge is still from complete unraveling with the elastic stretchability less than 20%.The width of bridge plays an important role on the elastic stretchability and a narrower interconnect could lead to a larger elastic stretchability.For example, the elastic stretchability increases from 2.3% to 17.6% when w/l (1) decrease from 0.8 to 0.2.FEA of a second-order fractal bridge as shown in Fig. 12a indicates that a mechanism of ordered unraveling occurs when the stretching is applied.The second-order structure unravels first bending and twisting while the first-order structure essentially does not deform.Only after the secondorder structure is fully extended corresponding to an applied strain of ∼150%, the first-order fractal bridge starts to unravel and continues until the elas-tic stretchability, corresponding to an applied strain of ∼300%, is reached.Zhang et al. [63] developed a hierarchical computational model (HCM) based on the mechanism of ordered unraveling for postbuckling analysis of fractal interconnects under stretching.The model, validated by experiments and conventional FEA, reduces the computational effects of traditional approaches by many orders of magnitude but with accurate predictions.For example, the computational times to finish the postbuckling analysis of the third-order fractal bridge on the same computer (eight CPUs) are 146 minutes for HCM and 2340 minutes for conventional FEA.
Fig. 12b illustrate that this HCM has good accuracy in predictions of the deformation and strains for the fractal bridges.Fig. 12c shows the elastic stretchability versus the order n of fractal bridge for m = 4, η = 8/ √ 11 and w/l (1) = 0.4.The elastic stretchability increases significantly from 10.7% to 192% when the order of bridge increases from 1 to 2. As n further increases (n ≥ 2), for each increase of n by 1, the elastic stretchability increases by ∼3 times and this scaling substantially improves the elastic limit.The elastic stretchability for fourth order is 2140%, which is about 200 times larger than that for the first order.It should be noted that the elastic stretchability is for the fractal bridge only not for the system.The system stretchability also depends on the length of island and the spacing between neighboring islands.For a system with an elastic stretchability 1000% of bridge and an island length/spacing ratio of 20, corresponding to an areal coverage of 90%, the system elastic stretchability can reach as high as ∼50%.

THERMAL MANAGEMENT OF STRETCHABLE INORGANIC ELECTRONICS
Integration of electronic/optoelectronic devices with the human body provides powerful diagnostic and therapeutic capabilities.Stretchable electronics make this possible due to its ability of bending, twisting and stretching, which could enable fully integration of devices with the soft tissue.The thermal properties of these devices in the tissue are critically important because excessive heating (even 1 • C-2 • C temperature increase) may cause tissue lesioning and induce adverse responses.Thermal management has been carried out to identify the underlying mechanism of heat transfer, and establish design guidelines to minimize the adverse thermal effects.We take microscale, inorganic light-emitting diodes (μ-ILED), which serve as the active device islands in the stretchable designs in previous section, as an example to overview the recent advances on heat management through discussions of analytic, FEA and experimental results.

Thermal analysis of μ-ILEDs in a constant operation
Conventional routes to develop LEDs involve epitaxial growth of active materials followed by wafer dicing and pick-and-up robotic manipulation into individually packaged components.These individual packaged components are then interconnected by bulk wire bonding and mounted on millimeter-scale heat sinks for thermal management.Although suitable for many uses, such designs do not allow practical realization of potentially valuable engineering options, such as stretchable LEDs, which requires large collections of small devices on unusual substrates.Kim et al. [64] used advanced method in epitaxial  The temperature increase from the ambient temperature in the μ-ILED is obtained analytically as where J 1 is the first-order Bessel function of the first kind and β(ξ ) is an analytic expression depending on material and geometry parameters.Equation ( 11) is further simplified by considering that the thickness of glass is much larger than other thicknesses (H g H m , H B , r 0 ), the thermal conductivity of metal is much larger than that of BCB (k m k B ), and the μ-ILED size is much larger than the metal and BCB layer thickness (r 0 H m , H B ) to a simple, analytic expression as which shows that the normalized temperature k m H m (T μ−ILED − T 0 )/Q in μ-ILED depends on only one non-dimensional parameter k g L/(k m H m ).
As shown in Fig. 13e, the approximate solution in Equation ( 12) agrees well with the accurate solution in Equation ( 11), 3D FEA and experimental measurements.This simple scaling law is very helpful for the thermal management of μ-ILEDs to minimize the adverse thermal effects.It suggests that thick metal layer or large metal thermal conductivity helps the heat dissipation in the system.
The above results for a single μ-ILED can also be used to find the temperature for μ-ILED arrays by the method of superposition, i.e.T array (r, z) = where T i (r, z) is the temperature distribution due to the ith μ-ILED.Figs 14a and b shows the surface temperature distribution for a conventional, macrosize LED (i.e. 1 mm × 1 mm) and an array of 100 μ-ILEDs (i.e. 100 μm × 100 μm) at a spacing of 2 mm with the same total input power density 400 W/cm 2 .The conventional LED would reach a temperature of over 1000 • C whereas the array of μ-ILEDs would operate at only ∼100 • C.This indicates that a small LED is very helpful to dissipate the heat.Fig. 14c shows the effect of spacing on the maximum temperature for an array of 100 μ-ILEDs (100 μm × 100 μm).The maximum temperature decreases with increasing spacing and reaches almost a constant when the spacing is larger than ∼500 μm, which indicates that the μ-ILED will have negligible effect on its neighboring μ-ILED.

Thermal analysis of μ-ILEDs in a pulsed operation
The thermal analysis discussed above is for μ-ILEDs in a continuous mode operation.Many other applications (e.g.backlight unit and optical communications) require a pulsed mode operation to take advantage of its high efficiency in thermal management.Moreover, pulsed mode operation can provide additional benefits, especially in applications of optogenetics, where the biological response can be suppressed by the continuous mode operation [66].Kim et al. [67] reported strategies to develop μ-ILEDs on unconventional substrates such as hydrogel to simulate biological tissue.Using state-of-the art techniques creates high-quality epitaxial material to define the individual μ-ILED followed by the technique of transfer printing to release the completed μ-ILED onto the hydrogel substrate with a polyimide (PI) layer on the top to ensure a good contact between μ-ILED and the substrate.Fig. 15a shows the layouts of a single μ-ILED on a PI layer attaching to a hydrogel substrate encapsulated by an SU8 layer, and the cross section is shown in Fig. 15b.The pulsed power applied to the μ-ILED is denoted by Q(t) = Q 0 U(t) with Q 0 as the peak power and U (t) as a unit pulsed power as shown in Fig. 15c.The duty cycle is defined as D = τ /t 0 , where τ is the pulse duration and t 0 is the period of the pulse.Fig. 15d shows the μ-ILED temperature versus time from the 3D FEA.It is shown that the μ-ILED temperature first increase in a fluctuation way, then reached saturation in a constant band after a few seconds.The maximum temperature after the saturation determines the performance of μ-ILEDs and is one of the most important thermal characteristics of the system.
Li et al. [68] developed an analytic model, validated by experiments and 3D FEA, to study the thermal properties of μ-ILED in a pulsed operation and derived a scaling law for the μ-ILED temperature after saturation.For simplicity, an axisymmetric model is adopted.The μ-ILED is modeled as a circular planar heat source at the SU8-PI interface with radius r 0 = L/ √ π with L as the size of μ-ILED.Due to the similar thermal conductivity and thermal diffusivity of PI and hydrogel, the PI layer and hydrogel substrate are modeled as a single hydrogel layer as shown in Fig. 15e.The periodic pulsed power can be expressed via its Fourier series by where ω = 2π/t 0 , a 0 = D = τ /t 0 , a n = sin(2nπ D)/(nπ ), and b n = [1 − cos(2nπ D)]/ (nπ ).The temperature increase from the ambient temperature, T = T − T ∞ , due to a pulsed power, can be obtained by the superposition of the solution due to a sinusoidal power Q 0 cos(ωt) [or Q 0 sin(ωt)], which can be written as the real (or imaginary) part of Q 0 e iωt .The temperature increase at saturation due to a sinusoidal power then takes the form θ (r, z; ω)e iωt , where θ (r, z; ω) = |θ (r, z; ω)|e iβ , and β is the phase angle depending on the real and imaginary part of θ (r, z; ω).The analytic model with negligible convection at the top surface of SU8 and ambient temperature at the bottom surface of hydrogel gives the μ-ILED temperature increase at saturation due to Q 0 e iωt as where f (ξ ) is an analytic expression depending on material and geometry parameters [66], J 1 is first-order Bessel function of the first kind, h is the thickness, and q = √ ω/α with α = k/(cρ) as the thermal diffusivity where k is the thermal conductivity, c is the specific heat capacity and ρ is the mass density.The subscript 'encap' denotes the SU8 encapsulation.
For a pulsed power, the temperature increase of μ-ILED is obtained by the method of superposition as where is the phase angle of θ LED (nω).The inset of Fig. 15d compares the μ-ILED temperature after saturation from Equation ( 14) and the accurate 3D FEA for the pulsed peak power Q 0 = 30 mW with the duty cycle D = 50% and the period t 0 = 1.0 ms.The good agreement of temperature between analytic and FEA validates the analytic model.The temperature increases during the pulse duration and then decreases between pulses, which defines a maximum and minimum temperature increase.Fig. 16a shows the maximum and minimum temperature in-creases of μ-ILED for various duty cycles.The good agreement among the analytic model, FEA and experiments further validates the analytic model.It is shown that T LED decreases with the decrease of the duty cycle D.For duty cycle D decreasing from 100% (i.e.constant power) to 1%, T LED decreases 99.8% from ∼202 • C to ∼0.5 • C, which suggests that μ-ILEDs can be operated in a pulsed mode at a high power density but with a low temperature increase.
The temperature increase of μ-ILED in a pulsed operation given in Equation ( 14) is rather complex.Equation ( 14) is further simplified by considering that ω/α encap is usually larger, or on the order of, a few μm −1 for polymer with period less than 1 ms as in applications of optogenetics, and h encap larger than, on the order of, a few μm to which shows that the normalized temperature increase of μ-ILED depends on three non-dimensional parameters: the normalized time t/t 0 , duty cycle D and πr 2 0 /(α sub t 0 ).The normalized maximum temperature increase small duty cycle D or large πr 2 0 /(α sub t 0 ) helps to reduce the adverse thermal effects.For large πr 2 0 /(α sub t 0 ) 1, the maximum temperature increase in μ-ILED is approximately linear with D, and is given by where πr 2 0 is the in-plane area of μ-ILED.This simple expression for the maximum temperature increase in terms of material, geometric and loading (heating) parameters may serve as guidelines for the thermal management design of μ-ILED.
Li et al. [69] further extended the above model to perform thermal management of μ-ILEDs in optogenetics.Figs 17a and b show injectable, cellularscale μ-ILEDs, which are inserted into the brain of mouse [70].The temperature predictions from the model (dot line) agree very well with experimental measurements (solid line) as in Fig. 17c.
is shown that the temperature increase is much lower than 1 • C, which is low enough to avoid tissue lesioning.

CONCLUSIONS
Fast developments and substantial achievements made on various aspects of stretchable electronics in the last decade have enabled many novel applications, which cannot be addressed by conventional design and critically rely on mechanicsguided heterogeneous integration of hard films (e.g.Si with the modulus of 130 GPa) and compliant substrate (e.g.PDMS with the modulus of 2 MPa or even smaller), with four representative strategies reviewed in this paper.As summarized in the paper, mechanics and thermal modeling have played crucial roles in these developments and achievements to guide the system and to ensure accurate operation.Recent developments in soft active materials (e.g.shape memory polymer and hydrogel) [71][72][73][74][75][76] could enable active shape change through the external stimuli such as light, heat and humidity, and thus realize new emerging applications (e.g.soft robotic) that are not possible in the current form of stretchable electronics on passive compliant substrates (e.g.PDMS).Moreover, the mechanics and thermal modeling reviewed here could play an important role in nanomaterial-enabled stretchable electronics and relatively little work has been done in this area [77,78].Many open challenges and opportunities exist for future research in these systems.Analytical and computational models are necessary for these new systems to avoid adverse strain and thermal effects with the constructing geometric parameters.

Figure 2 .REVIEWFigure 3 .
Figure 2. Four representative mechanics-guided design strategies to develop stretchable electronics.(a) Wavy design: scanning electron microscope (SEM) image of wavy, wrinkled, single-crystal Si ribbons on a PDMS substrate.Reproduced with permission from Ref. [25].Copyright 2007 The National Academy of Sciences of the USA.(b) Straight bridge-island design: SEM image of Si mesh with islands interconnected by straight bridges.Reproduced with permission from Ref. [13].Copyright 2008 The National Academy of Sciences of the USA.(c) Serpentine bridge-island design: SEM image of stretchable CMOS inverters with islands interconnected by serpentine bridges.Reproduced with permission from Ref. [13].Copyright 2008 The National Academy of Sciences of the USA.(d) Fractal bridge-island design: optical image of Al electrode pads interconnected by fractal bridges.Reproduced with permission from Ref. [7].Copyright 2013 Nature Publishing Group.

REVIEWFigure 5 .
Figure 5. Various mechanics issues associated with wavy design.(a) Atomic force microscope (AFM) image of buckled Si ribbons on a PDMS substrate showing a flat region near the edges due to the edge effect.Reproduced with permission from Ref. [40].Copyright 2007 American Institute of Physics.(b) Optical microscopy images of buckled Si ribbons showing the width-dependent buckling profile.Reproduced with permission from Ref. [41].Copyright 2008 Elsevier Ltd.(c) Critical strains separating the local and global buckling versus substrate thickness for Si/PDMS system.Reproduced with permission from Ref. [42].Copyright 2008 American Institute of Physics.(d) SEM image of 2D wavy Si nanomembrane on a PDMS substrate.Reproduced with permission from Ref. [43].Copyright 2007 American Chemical Society.

REVIEW Song et al. 133 Figure 6 .
Figure 6.Schematic illustration of the process for fabricating straight bridge-island design on a prestrained compliant substrate.Reproduced with permission from Ref. [50].Copyright 2009 American Institute of Physics.

Figure 7 .
Figure 7. (a) Schematic illustration of the mechanics model for straight bridge-island design.(b) Distribution of the strain ε x x in islands (20 μm × 20 μm) when the bridge relaxes from 20 to 17.5 μm.(c) Stretchability and compressibility versus the prestrain.(d) The effect of bridge length on the stretchability and compressibility.Reproduced with permission from Ref. [50].Copyright 2009 American Institute of Physics.

Figure 8 .
Figure 8.(a) Schematic diagram of serpentine bridge-island design.(b) Schematic illustration of geometric parameters for a representative serpentine bridge with m unit cells.Reproduced with permission from Ref. [54].Copyright 2013 the Royal Society of Chemistry.

Figure 9 .
Figure 9. (a) The elastic stretchability versus t/w for various ε yield l 1 /w.(b) The elastic stretchability versus α for various m.Reproduced with permission from Ref. [54].Copyright 2013 the Royal Society of Chemistry.

Figure 10 .
Figure 10.Geometric construction of a fractal bridge from order 1 to 4. Reproduced with permission from Ref. [61].Copyright 2014 Elsevier Ltd.

Figure 12 .Figure 13 .
Figure 12.(a) Optical images and corresponding predictions based on conventional FEA for a second-order fractal bridges under various stages of stretching.(b) Maximum principal strain versus the applied strain for second-order fractal bridge in (a) based on HCM and conventional FEA.(c) The elastic stretchability versus the order for fractal bridges from 1 to 4 with (m , η) = (4, 8/ √ 11), t/w = 0.03 and w/l (1) = 0.4.Reproduced with permission from Ref. [61].Copyright 2014 Elsevier Ltd.

Figure 14 .
Figure 14.(a) Temperature distribution for a macrosize LED (1 mm × 1 mm) at a constant heat flux of 400 W/cm 2 .(b) Temperature distribution for an array of 100 μ-ILEDs (100 μm × 100 μm) at a spacing of 2 mm and a constant heat flux of 400 W/cm 2 .(c) The μ-ILED temperature versus spacing for an array of 100 μ-ILEDs.Reproduced with permission from Ref. [64].Copyright 2011 the National Academy Sciences of the USA.
liftoff and deterministic assembly and successfully fabricated μ-ILEDs on different substrates.Figs 13a−c shows the schematic diagram of a μ-ILED structure with the μ-ILED, encapsulated by benzocyclobutene (BCB) and metal layers, on the top of a glass substrate.The thermal conductivity and thickness are denoted by k and H with subscripts m, B and g for metal, BCB and the glass substrate, respectively.Lu et al. [65] developed an analytic model to study the thermal properties of μ-ILEDs and derived simple expressions for calculating the device temperature increase in terms of the material and geometry parameters.For simplicity, the μ-ILED is modeled as a circular planar heat source with a radius of r 0 = L/ √ π and the input power of Q at the BCB-glass interface.Here, L is the in-plane size of μ-ILED.Fig. 13d compares the surface temperature distributions from analytic model to the experiments and three-dimensional finite element analysis (3D FEA) for the input power Q = 37.6 mW with the μ-ILED size L = 100 μm (r 0 = 56.4μm) and ambient temperature T 0 = 50 o C. The results clearly show that the analytical model has a good accuracy in predicting the temperature.

Figure 15 .
Figure 15.(a) Three-dimensional, and (b) cross-sectional illustrations of the μ-ILED on a hydrogel substrate.(c) Unit pulsed power with duration time τ and period t 0 .(d) Temperature obtained from FEA for the pulsed peak power 20 mW with 50% duty cycle and period 1 ms.The insert compares the temperature after saturation from the analytic model and FEA.(e) A schematic illustration of the analytical model.Reproduced with permission from Ref. [68].Copyright 2013 American Institute of Physics.

[ 1 +Figure 17 .
Figure 17.(a) SEM of an injectable array of μ-ILEDs.(b) Process of injection and release of the μ-ILEDs into the brain.(c) Change in brain temperature as a function of time from experiments (solid line) and the analytical model (dot line) at a peak power 8.65 mW in 3, 5, 10, 20 Hz pulses with 10 ms duration.Reproduced with permission from Ref. [70].Copyright 2013 AAAS.
Downloaded from https://academic.oup.com/nsr/article-abstract/3/1/128/2460279 by guest on 23 November 2018 REVIEW Figure 16.μ-ILED temperature increase versus duty cycle for the peak power 30 mW with period 1 ms.(b) The normalized maximum μ-ILED temperature increase at a pulsed power versus duty cycle.Reproduced with permission from Ref. [68].Copyright 2013 American Institute of Physics.