Stretchable Electronics E-Book

• Herausgeber: Wiley-VCH
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 718

Veröffentlichungsjahr: 2012

Beschreibung

On a daily basis, our requirements for technology become more innovative and creative and the field of electronics is helping to lead the
way to more advanced appliances. This book gathers and evaluates the materials, designs, models, and technologies that enable the fabrication of fully elastic electronic devices that can tolerate high strain. Written by some of the most outstanding scientists in the field, it lays down the undisputed knowledge on how to make electronics withstand stretching. This monograph provides a review of the specific applications that directly benefit from highly compliant electronics, including transistors, photonic devices, and sensors. In addition to stretchable devices, the topic of ultraflexible electronics is treated, highlighting its upcoming significance for the industrial-scale production of electronic goods for the consumer.

Divided into four parts covering:

* Theory
* Materials and Processes
* Circuit Boards
* Devices and Applications

An unprecedented overview of this thriving area of research that nobody in the field - or intending to enter it - can afford to miss.

Leseprobe

Table of Contents

Cover

Related Titles

Title page

Copyright page

Preface

List of Contributors

Part I: Theory

1 Theory for Stretchable Interconnects

1.1 Introduction

1.2 Mechanics of Stretchable Wavy Ribbons

1.3 Mechanics of Popup Structure

1.4 Mechanics of Interconnects in the Noncoplanar Mesh Design

1.5 Concluding Remarks

2 Mechanics of Twistable Electronics

2.1 Introduction

2.2 Postbuckling Theory

2.3 Postbuckling of Interconnect under Twist

2.4 Symmetric Buckling Mode

2.5 Antisymmetric Buckling Mode

2.6 Discussion and Concluding Remarks

Part II: Materials and Processes

3 Graphene for Stretchable Electronics

3.1 Introduction

3.2 Production of Graphene Films

3.3 Fabrication of Graphene Films on Substrates

3.4 Applications in Flexible and Stretchable Electronics

3.5 Concluding Remarks

4 Stretchable Thin-Film Electronics

4.1 Introduction

4.2 Silicone Rubber as a Substrate

4.3 Mechanical Architecture

4.4 Stretchable Metallization

4.5 Integrated Stretchable Thin-Film Devices

4.6 Outlook

5 Stretchable Piezoelectric Nanoribbons for Biocompatible Energy Harvesting

5.1 Energy Harvesting and Piezoelectric Materials

5.2 PZT Nanofabrication and Interfacing with Stretchable Substrates

5.3 Piezoelectric Characterization and Electrical Measurements

5.4 Summary

Part III: Circuit Boards

6 Modeling of Printed Circuit Board Inspired Stretchable Electronic Systems

6.1 Technology Development Considerations

6.2 Modeling and Simulation

7 Materials for Stretchable Electronics Compliant with Printed Circuit Board Fabrication

7.1 Introduction

8 Technologies and Processes Used in Printed Circuit Board Fabrication for the Realization of Stretchable Electronics

8.1 Lamination Technology

8.2 Molding Technology

9 Reliability and Application Scenarios of Stretchable Electronics Realized Using Printed Circuit Board Technologies

9.1 Application Considerations

9.2 Reliability

9.3 Application Scenarios

Part IV: Devices and Applications

10 Stretchable Electronic and Optoelectronic Devices Using Single-Crystal Inorganic Semiconductor Materials

10.1 Introduction

10.2 Stretchable Circuits

10.3 Application of Stretchable Designs to Microscale Inorganic Light Emitting Diodes (μ-ILEDs)

10.4 Biomedical Applications of Stretchable Electronics and Optoelectronics

10.5 Stretchable Digital Imagers and Solar Modules

10.6 Conclusions

11 Stretchable Organic Transistors

11.1 Introduction

11.2 Perforated Organic Transistor Active Matrix for Large-Area, Stretchable Sensors

11.3 Rubber-Like Stretchable Organic Transistor Active Matrix Using Elastic Conductors

11.4 Rubber-Like Organic Transistor Active Matrix Organic Light-Emitting Diode Display

11.5 Future Prospects

Acknowledgments

12 Power Supply, Generation, and Storage in Stretchable Electronics

12.1 Introduction

12.2 Radio Frequency Power Supplies

12.3 Power Generation

12.4 Power Storage

12.5 Summary

Acknowledgments

13 Soft Actuators

13.1 Introduction

13.2 Conducting Polymers

13.3 Ionic Polymer Metal Composites (IPMCs)

13.4 Nanocarbon Actuators

13.5 Applications

13.6 Conclusion

14 Elastomer-Based Pressure and Strain Sensors

14.1 Introduction

14.2 A Brief Elastomers Overview

14.3 Important Sensor Characteristics

14.4 Elastomeric Force Sensors

14.5 Active Pressure/Strain Sensors Systems

14.6 Applications

14.7 Outlook

15 Conformable Active Devices

15.1 Introduction

15.2 Printing Processes for Organic TFTs

15.3 Sensing and Memory Devices Based on Piezoelectric Polymer

15.4 Electronic Circuits

15.5 Curved Conformal Devices by a Cut-and-Bend Approach

15.6 Summary

Acknowledgments

16 Stretchable Neural Interfaces

16.1 Introduction

16.2 Overview of MEAs

16.3 Classes of SMEAs

16.4 Common Limitations for All SMEAs

16.5 Future Directions in Stretchable Neural Interfaces

16.6 Conclusion

17 Bio-based Materials as Templates for Electronic Devices

17.1 Introduction

17.2 Polysaccharide-Based Templates

17.3 Protein-Based Templates

17.4 DNA Templates

17.5 Virus Templates: Tobacco Mosaic Virus and M13 Bacteriophage

17.6 Summary

18 Organic Integrated Circuits for EMI Measurement

18.1 Introduction

18.2 Stretchable EMI Measurement Sheet

18.3 Silicon CMOS LSI for EMI Detection

18.4 Experimental Results and Discussion

18.5 Conclusion

Acknowledgments

Index

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The Editor

Prof. Takao Someya

University of Tokyo

Department of Electrical Engineering

7-3-1 Hongo, Bunkyo-ku

Tokyo 113-8656

Japan

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Preface

Today’s electronics are bulky and rigid since they are manufactured on rigid substrates such as glass and/or silicon; however, the next-generation electronics will be manufactured on polymeric foils, subsequently going to be flexible and even stretchable.

Objects that surround our everyday life have very complicated shapes. In fact, most ambient tools are composed of round/curvy surfaces rather than flat glass/silicon wafers. Imagine overlaying a machine such as a humanoid robot with a skin-like sensor sheet. For a joint part of a robot, the sensor sheet needs to be extremely stretchable to accommodate the twisting or bending motion of the joint. In order to mask the sensor sheet to a surface with complex, intricate curves, “stretchability” is the key. As a result, electronics that are as flexible as films, and as stretchy as rubber sheets, will soon replace the traditional solid electronics. Future electronics will be highly deformable and will adapt their shapes by stretching, shrinking, and wrinkling as desired. Expectations are unlimited and abundant. Emerging applications will be realized when new electronics adopt their stretchability.

This book, which has four parts, comprises 18 chapters. Part I introduces the theory of stretchable electronics and mechanics of twistable electronics. Here, deformation analysis and postbuckling theory are described. Part II puts together materials and processes. Graphene-based stretchable electrodes, polydimethyl­siloxane (PDMS) substrates, and stretchable piezoelectric nanoribbons are presented as candidate materials for stretchable electronics. In Part III, stretchable circuit boards and related technologies ranging from modeling, materials, processes, device reliability, and applications are reported. Part IV covers novel stretchable applications of stretchable electronics which are made of inorganic and organic semiconductors. In addition to stretchable transistor integrated circuits and other conformable active devices, stretchable actuators, sensors, and stretchable power sources are described. Bio-inspired and bio-medical applications are important and bio-based materials are introduced for electronics devices. Novel applications include stretchable systems to measure electromagnetic interference and signals from neurons.

I would like to express my sincere gratitude to all our colleagues and friends involved in the realization of this book. I greatly appreciate them for agreeing to devote their time and effort to submitting and reviewing chapters to ensure its success. I am also indebted to Wiley-VCH for publication of this book.

Takao SomeyaTokyo, October 2012

List of Contributors

Matthias AdlerFreudenberg Forschungsdienste SE & Co. KGHöhnerweg 2-469469 WeinheimGermany

Jong-Hyun AhnSungkyunkwan UniversitySchool of Advanced Materials Science and EngineeringSuwon, 440-746Korea

Ana Claudia AriasUniversity of CaliforniaEECS DepartmentBerkeley, CA 94720USA

Kinji AsakaNational Institute of Advanced Industrial Science and Technology (AIST)Health Research Institute1-8-31 MidorigaokaIkedaJapan

Zhenan BaoStanford UniversityDepartment of Chemical Engineering381 North South MallStanford, CA 94305USA

Siegfried BauerJohannes Kepler University LinzSoft Matter PhysicsAltenbergerstraße 694040 LinzAustria

Ruth BieringerFreudenberg Forschungsdienste SE & Co. KGHöhnerweg 2-469469 WeinheimGermany

Frederick BossuytUniversity Ghent9052 Gent-ZwijnaardeBelgium

Frederick BossuytCentre for Microsystems TechnologyGhent University and Interuniversity Microelectronics CentreTechnology Park, Building 914-A9052 Gent-ZwijnaardeBelgium

Wenzhe CaoPrinceton UniversityDepartment of Electrical Engineering and Princeton Institute for the Science and Technology of MaterialsF310 Engineering QuadOlden StreetPrinceton, NJ 08544USA

Zhichao FanTsinghua UniversityDepartment of Engineering MechanicsBeijing, 100084China

Mario GonzalezThe Interuniversity Microelectronics Center IMECKapeldreef 753001 LeuvenBelgium

Jürgen GüntherFreudenberg Forschungsdienste SE & Co. KGHöhnerweg 2-469469 WeinheimGermany

Yung-Yu HsuThe Interuniversity MicroElectronics Center IMECKapeldreef 753001 LeuvenBelgiumPresent address: MC10 Inc.36 Cameron Ave.MA 02140USA

Yonggang HuangNorthwestern UniversityDepartments of Civil and Environmental Engineering and Mechanical EngineeringEvanston, IL 60208USA

Keh-Chih HwangTsinghua UniversityDepartment of Engineering MechanicsBeijing, 100084China

Olle InganäsLinköping UniversityBiomolecular and Organic ElectronicsDepartment of Physics, Chemistry and Biology58183 LinköpingSweden

Koichi IshidaUniversity of TokyoTokyo 153-8505Japan

Houk JangSungkyunkwan UniversitySchool of Advanced Materials Science and EngineeringSuwon, 440-746Korea

Martin KaltenbrunnerJohannes Kepler University LinzSoft Matter PhysicsAltenbergerstraße 694040 LinzAustria

Woo Hyeun KangColumbia UniversityDepartment of Biomedical Engineering351 Engineering TerraceMC 89041210 Amsterdam AvenueNew York, NY 10027USA

Dae-Hyeong KimSeoul National UniversitySchool of Chemical and Biological EngineeringSeoul, 151-744Korea

Stéphanie P. LacourCenter for NeuroprostheticsEPFL | STI | IMT/IBI | LSBI Station 17CH-1015 Lausanne Switzerland

Seoung-Ki LeeSungkyunkwan UniversitySchool of Advanced Materials Science and EngineeringSuwon, 440-746Korea

Thomas LöherFraunhofer IZMGustav-Meyer-Allee 2513355 BerlinGermany

Nanshu LuDepartment of Aerospace Engineering and Engineering MechanicsUniversity of Texas at Austin405 N Mathews St.Austin, TX 78712USA

Stefan C.B. MannsfeldStanford UniversityDepartment of Chemical Engineering381 North South MallStanford, CA 94305USA

Michael C. McAlpinePrinceton UniversityDepartment of Mechanical and Aerospace EngineeringEngineering QuadOlden StreetPrinceton, NJ 08544USA

Barclay Morrison, IIIColumbia UniversityDepartment of Biomedical Engineering351 Engineering TerraceMC 89041210 Amsterdam AvenueNew York, NY 10027USA

Christian MüllerDepartment of Chemical and Biological Engineering/Polymer TechnologyChalmers University of Technology41296 GöteborgSweden

Thanh D. NguyenPrinceton UniversityDepartment of Mechanical and Aerospace EngineeringEngineering QuadOlden StreetPrinceton, NJ 08544USA

Prashant K. PurohitUniversity of Pennsylvania Department of Mechanical Engineering and Applied Mechanics220 South 33rd Street Philadelphia, PA 19104-6391USA

Yi QiPrinceton UniversityDepartment of Mechanical and Aerospace EngineeringEngineering QuadOlden StreetPrinceton, NJ 08544USA

John A. RogersUniversity of IllinoisDepartment of Materials Science and EngineeringUrbana, IL 61801USA

John A. RogersUniversity of Illinois at Urbana-ChampaignDepartment of Materials Science and EngineeringBeckman Institute for Advanced Science and Technology and Frederick Seitz Materials Research LaboratoryUrbana, IL 61801USA

Takayasu SakuraiUniversity of TokyoTokyo 153-8505Japan

Thomas SchauberFreudenberg Forschungsdienste SE & Co. KGHöhnerweg 2-469469 WeinheimGermany

Tsuyoshi SekitaniThe University of Tokyo Department of Electrical Engineering and Information Systems 7-3-1 Hongo Bunkyo-ku Tokyo 113-8656Japan

Takao SomeyaThe University of TokyoDepartment of Electrical Engineering and Information Systems 7-3-1 HongoBunkyo-kuTokyo 113-8656Japan

Jizhou SongUniversity of MiamiDepartment of Mechanical and Aerospace EngineeringCoral Gables, FL 33146USA

Robert A. StreetPalo Alto Research Center3333 Coyote Hill RoadPalo Alto, CA 94304USA

Yewang SuTsinghua UniversityDepartment of Engineering MechanicsBeijing, 100084China

Makoto TakamiyaUniversity of TokyoTokyo 153-8505Japan

Benjamin C.K. TeeStanford UniversityDepartment of Electrical EngineeringDavid Packard Building350 Serra MallStanford, CA 94305USA

Jan VanfleterenCentre for Microsystems TechnologyGhent University and Interuniversity Microelectronics CentreTechnology Park, Building 914-A9052 Gent-ZwijnaardeBelgium

Thomas VervustCentre for Microsystems TechnologyGhent University and Interuniversity Microelectronics CentreTechnology Park, Building 914-A9052 Gent-ZwijnaardeBelgium

Sigurd WagnerPrinceton UniversityDepartment of Electrical Engineering and Princeton Institute for the Science and Technology of MaterialsB422 Engineering QuadOlden StreetPrinceton, NJ 08544USA

Shuodao WangNorthwestern UniversityDepartment of Mechanical EngineeringEvanston, IL 60208USA

Jian WuTsinghua UniversityDepartment of Engineering MechanicsBeijing, 100084China

Chao YanSungkyunkwan UniversitySchool of Advanced Materials Science and EngineeringSuwon, 440-746Korea

Part ITheory

1

Theory for Stretchable Interconnects

Jizhou Song and Shuodao Wang

1.1 Introduction

A rapidly growing range of applications demand electronic systems that cannot be formed in the conventional manner on semiconductor wafers. The most prominent example is stretchable electronics, which has a performance equal to established technologies that use rigid semiconductor wafers, but in formats that can be stretched and compressed. It enables many application possibilities such as flexible displays [1], electronic eye camera [2–4], conformable skin sensors [5], smart surgical gloves [6], and structural health monitoring devices [7]. There are primarily two directions to make stretchable electronics. One is to use intrinsically stretchable materials such as organic materials [8–13]. However, the electrical per­formance of organic semiconductor materials is relatively poor comparing with the well-developed, high-performance inorganic electronic materials. The other direction to achieve stretchable electronics is to use conventional semiconductors, such as silicon, and make the system stretchable. The main challenge here is to make silicon-based structures stretchable since the brittleness of silicon makes it almost impossible to be stretched. Many researches bypassed this difficulty by using stretchable interconnects [14–22].

One of the most intuitive approaches to develop stretchable interconnects is to exploit out-of-plane deflection in thin layers to accommodate strains applied in the plane. Figure 1.1 illustrates some examples of this concept. In the first case (Figure 1.1a) [17, 24, 25] of stretchable wavy ribbons, the initially flat ribbons are bonded to a prestrained elastomeric substrate. The prestrain can be induced by mechanical (or thermal) stretch along the ribbon directions. Releasing the prestrain causes a compression in the ribbon, and this compression leads to a nonlinear buckling response and results in a wavy profile. When the wavy structure is subject to stretches, the amplitudes and periods of the waves change to accommodate the deformation. In the second case (Figure 1.1b) of popup structure [26], the ribbons can be designed to bond the prestretched elastomeric substrate only at certain locations. When the prestrain is released, the ribbon on the nonbonded regions delaminates from the substrate and forms popup profile. Compared to Figure 1.1a, this layout has the advantage that the wavelengths can be defined precisely with a level of engineering control to have higher stretchability.

Figure 1.1 SEM images of (a) stretchable wavy ribbons, (b) popup structure, (c) noncoplanar mesh design with straight interconnects, and (d) noncoplanar mesh design with serpentine interconnects.

(Reprinted with permission from Ref. [15] Copyright 2007 American Institute of Physics and Ref. [23] Copyright 2009 American Vacuum Society).

Combining the stretchable interconnects in Figure 1.1a (or Figure 1.1b) with rigid device islands, an interconnect-island structure [16, 19, 20, 22] can be developed to accommodate the deformations. Mechanical response to stretching or compression involves, primarily, deformations only in these interconnects, thereby avoiding unwanted strains in the regions of the active devices. Lacour et al. [16] and Kim et al. [19] developed a coplanar mesh design by using the wavelike interconnects, which are bonded with the substrate. Although such a coplanar mesh design can improve the stretchability to around 40%, the stretchability is still small for certain applications. Kim et al. [20] developed a nonco­planar mesh design (Figure 1.1c), consisting of device islands linked by popup interconnects for stretchable circuits, which can be stretched to rubber-like levels of strain (e.g., up to 100%). To further increase the stretchability, serpentine inter­connects [14, 15, 19–22] can be used. Compared to the straight interconnects, the serpentine ones can accommodate larger deformation because they are much longer and can involve large twist to reduce the strains in the interconnects. Figure 1.1d shows a SEM image of serpentine interconnects used in the noncoplanar mesh design.

For serpentine interconnects, there are no theoretical work, and many researchers have developed numerical models to study their deformations due to their complex geometries [14, 15, 19–22]. The related review is not the focus of this chapter. Here, we will review the theoretical aspects related to the designs in Figure 1.1a–c. Mechanics of stretchable wavy ribbons (Figure 1.1a) is described in Section 1.2. Analysis for small and large strains and width effect are discussed in this section. Section 1.3 describes the mechanics of popup structure (Figure 1.1b). Section 1.4 reviewed the mechanics of interconnects in the noncoplanar mesh design (Figure 1.1c). Interfacial adhesion and large deformation effect are also discussed in this section.

1.2 Mechanics of Stretchable Wavy Ribbons

The fabrication of stretchable wavy ribbons is illustrated in Figure 1.2. The flat ribbon is first chemically bonded to a prestrained compliant substrate. When the prestrain is released, the ribbon is compressed to generate the wavy layout through a nonlinear buckling response. These wavy layouts can accommodate external deformations through changes in wavelength and amplitude, which is also shown in Figure 1.2.

Figure 1.2 Schematic illustration of the process for fabricating buckled, or “wavy,” single crystal Si ribbons on a PDMS substrate.

(Reprinted with permission from Ref. [27] Copyright 2009 American Vacuum Society).

1.2.1 Small-Deformation Analysis

Several models [28, 29] have been developed to explain the mechanics of stretchable wavy ribbons under small deformations. For example, Huang et al. [29] developed an energy method to determine the buckling profile. The thin ribbon is modeled as an elastic nonlinear von Karman beam since its thickness is much smaller compared with other characteristic lengths (e.g., wavelength). The substrate is modeled as a semi-infinite solid because its thickness (∼mm) is much larger than that (∼µm) of film. The total energy consists of the bending energy Ub and membrane energy Um in the thin film and strain energy Us in the substrate.

For a stiff thin film (ribbon) with thickness hf, Young’s modulus Ef and Poisson’s ratio vf on a prestrained compliant substrate with prestrain εpre, Young’s modulus Es, and Poisson’s ratio vs, the wavy profile forms with the out-of-plane displacement:

(1.1) 

when the prestrain is released. Here, x1 is the coordinate along the ribbon direction, A is the amplitude, λ is the wavelength, and k = 2π/λ is the wave number. A and λ (or k) are to be determined by minimizing the total energy. The bending energy Ub can be obtained by

(1.2) 

where L0 and are the length and plane-strain modulus of the thin film, respectively.

The membrane strain ε11, which determines the membrane energy in the ribbon, is related to the in-plane displacement u1 and out-of-plane displacement w by εmembrane = du1/dx1 + (dw/dx1)2/2 − εpre. The membrane force Nmembrane is given by . The interfacial shear is negligible [29] and the force equilibrium equation becomes dN11/dx1 = 0, which gives a uniform membrane force and therefore a uniform membrane strain:

(1.3) 

The membrane energy Um in the film can then be obtained by

(1.4) 

The strain energy in the substrate is obtained by solving a semi-infinite solid subjected to the normal displacement in Eq. (1.1) and vanishing shear on its boundary, yielding

(1.5) 

where is the plane-strain modulus of the substrate.

Energy minimization of the total energy with respect to the amplitude A and wavelength λ, that is, ∂(Um + Ub + US)/∂A = ∂(Um + Ub + US)/∂λ = 0, gives

(1.6) 

where

(1.7) 

is the critical strain for buckling. When εpre < εc, no buckling occurs, and the ribbon remains flat. When εpre > εc, the ribbon buckles such that the membrane strain remains a constant εmembrane = −εc. The (maximum) bending strain is equal to the maximum curvature times the half thickness hf/2, that is, . The peak strain εpeak in the film is the summation of membrane and bending strains. In most cases of practical interest, the bending strain is much larger than the membrane strain. For example, the membrane strain is only 0.034% for the Si ribbon (Ef = 130 GPa, vf = 0.27) on PDMS substrate (Es = 1.8 MPa, vs = 0.48). Therefore, the peak strain can be approximated by

(1.8) 

Because the magnitude of critical strain is very small, the magnitude of the peak strain εpeak is much smaller than the prestrain εpre. For example, εpeak is only 1.8% for the Si/PDMS system when εpre = 23.8%. This provides an effective level of stretchability/compressibility of the system.

For the buckled system subjected to the applied strain εapplied, the above results can be obtained by simply replacing the prestrain εpre by εpre − εapplied. The wavelength and amplitude become

(1.9) 

and the peak strain in the ribbon is

(1.10) 

1.2.2 Finite-Deformation Analysis

The wavelengths in Eqs. (1.6) and (1.9) are constant and strain-independent, and have been widely used in high precision micro and nano-metrology methods [30, 31]. However, when the prestrain is large, the experiments [32–34] showed that the wavelength decreases with increasing prestrain. Figure 1.3 clearly shows this dependence for the Si/PDMS system. Jiang et al. [34] and Song et al. [35] pointed out that the strain-dependent wavelength is due to the finite deformation (i.e., large strain) in the compliant substrate and established a buckling theory that accounts for finite geometry change (i.e., different strain-free or stress-free states for the ribbon and substrate) as shown in Figure 1.4, nonlinear strain-displacement relation and nonlinear constitutive model for the substrate to explain this finite deformation effect.

Figure 1.3 Stacked plane-view AFM images of buckled Si ribbons (100 nm thick) on PDMS for different levels of prestrain.

(Reprinted with permission from Ref. [35] Copyright 2008 Elsevier Ltd).

Figure 1.4 Three sequential configurations for the thin film/substrate buckling process. The top figure shows the undeformed substrate with the original length L0, which represents the zero strain energy state. The middle figure shows the substrate deformed by the prestrain and the integrated film, which represents zero strain energy state for the thin film. The bottom figure shows the deformed (buckled) configuration.

(Reprinted with permission from Ref. [35] Copyright 2008 Elsevier Ltd).

The out-of-plane displacement of the buckled thin ribbon can be represented by

(1.11) 

in the strain-free configuration (middle figure, Figure 1.4) as well as in the relaxed configuration (bottom figure, Figure 1.4). The coordinate in the middle figure is related to x1 in the bottom figure by .

The thin ribbon is still modeled as a von Karman beam. Using similar approach in Section 1.2.1, the bending energy and membrane energy in the film can be obtained as

(1.12) 

and

(1.13) 

respectively, where (1 + εpre)L0 is the initial length of strain-free Si thin ribbon (middle figure, Figure 1.4).

The geometric and material nonlinearity are considered in the modeling of substrate. All the governing equations are in terms of the coordinates for the strain-free configuration of PDMS substrate (i.e., x1 and x3 in Figure 1.4). The Green strains EIJ in the substrate are related to the displacements u1(x1,x3) and u3(x1,x3) by

(1.14) 

where the subscripts I and J are 1 or 3. To account for the material nonlinearity, the Neo–Hookean constitutive law is used to represent the substrate

(1.15) 

where TIJ is the second Piola–Kirchhoff stress, and the strain energy density Ws takes the form . Here J is the volume change at a point and is the determinant of deformation gradient FiJ, is the trace of the left Cauchy–Green strain tensor BIJ = FIkFJk times J−2/3. The force equilibrium equation for finite deformation is

(1.16) 

The perturbation method is used to find the solutions for the substrate, and the strain energy is obtained by Song et al. [35] as

(1.17) 

where L0 is the original length of the substrate.

Minimization of the total energy gives the wavelength and amplitude

(1.18) 

where λ0 and A0 are, respectively, the wavelength and amplitude in Eq. (1.6) from small-deformation analysis, and ξ = 5εpre(1 + εpre)/32. Contrary to the small-deformation theory, the wavelength decreases with εpre, but the amplitude increases with εpre. Both wavelength and amplitude agree well with experimental data and finite element simulations without any parameter fitting as shown in Figure 1.5a.

Figure 1.5 (a) Wavelength and amplitude (b) membrane and peak strains of buckled Si ribbons (100 nm thick) on PDMS as functions of the prestrain.

(Reprinted with permission from Ref. [35] Copyright 2008 Elsevier Ltd).

The membrane and bending strain can be obtained as

(1.19) 

For large prestrain, the peak strain, which is summation of εmembrane and εbending, is given by

(1.20) 

Figure 1.5b shows εpeak and εmembrane as a function of εpre. Both the membrane and peak strains agree well with finite element analysis. Compared to the peak strain, the membrane strain is much smaller and negligible. Compared to the prestrain, the peak strain is much smaller, and therefore the system can provide an effective level of stretchability/compressibility. For example, for εfracture = 1.8%, the maximum allowable prestrain is obtained as ∼ 29% by εpeak = εfracture, which is almost 20 times larger than εfracture.

For the buckled system subjected to the applied strain εapplied, Song et al. [35] obtained the total energy of the system using the perturbation method and gave the wavelength and amplitude

(1.21) 

where ξ = 5(εpre − εapplied)(1 + εpre)/32. Figure 1.6a shows the wavelength and amplitude as a function of applied strain for a buckled Si/PDMS system formed at the prestrain 16.2%. Both amplitude and wavelength agree well with experimental data and finite element simulations. As the tensile strain increases, the wavelength increases, but the amplitude decreases. Once the tensile strain reaches the prestrain plus the critical strain, the amplitude becomes zero and further stretch of εfracture will fracture the film. Therefore, the stretchability is given by εpre + εfracture + εc. The membrane and peak strains in the ribbon are obtained as

(1.22) 

Figure 1.6 (a) Wavelength and amplitude (b) membrane and peak strains of buckled Si ribbons (100 nm thick) on PDMS formed with a prestrain of 16.2% as a function of the applied strain.

(Reprinted with permission from Ref. [35] Copyright 2008 Elsevier Ltd).

Figure 1.6b shows εpeak and εmembrane as functions of εapplied under the prestrain 16.2%. The analytical solutions agree well with finite element simulations.

The compressibility is the maximum applied compressive strain when the peak Si strain reaches εfracture, and it is well approximated by . Figure 1.7 shows the stretchability and compressibility versus the prestrain. The stretchability increases with increasing the prestrain, while the compressibility decreases. When the prestrain is 13.4%, the stetchability and compressibility is equal.

Figure 1.7 Strechability and compressibility of buckled Si ribbons (100 nm thickness) on PDMS.

(Reprinted with permission from Ref. [35] Copyright 2008 Elsevier Ltd).

1.2.3 Ribbon Width Effect

The analyses in the above sections assumed that the thin ribbon width is much larger than the wavelength such that the deformation is plane strain. However, this assumption may not hold for small-width ribbons. Figure 1.8 shows the strong effect of ribbon width effect for the Si/PDMS system. Figure 1.8a shows the plane-view (from top to bottom) AFM images of Si ribbons for different widths 2, 5, 20, 50, and 100 µm. It clearly shows that the wavelength increases with the increase of the ribbon width and approaches to a constant at a finite value. The linecut profiles from AFM measurements in Figure 1.8b for the 2 and 20 µm wide ribbons also shows this strong ribbon width effect

Figure 1.8 (a) Stacked plane-view AFM images of buckled Si ribbons for different widths of 2, 5, 20, 50, and 100 µm (from top to bottom). (b) AFM line-cut profiles along the buckled wavy Si ribbons for 2 and 20 µm wide ribbons.

(Reprinted with permission from Ref. [17] Copyright 2008 Elsevier Ltd).

Jiang et al. [27] studied the ribbon width effect on the buckling profile. The ribbon width is denoted by W as shown in Figure 1.9a. Similar to Section 1.2.1, the total energy of the system consists of membrane and bending energy in the film and strain energy in the substrate. The membrane energy and bending energy in Eqs. (1.2) and (1.4) still hold except that they need to be multiplied by the ribbon width W. The substrate is modeled as a three-dimensional, semi-infinite solid with traction-free surface except for the portion underneath the ribbon. The strain energy in the substrate can be obtained analytically as

(1.23) 

where

(1.24) 

is a nondimensional function, Yn (n = 0,1,2, …) is the Bessel function of the second kind, and Hn (n = 0,1,2, …) denotes the Struve function.

Figure 1.9 (a) Schematic illustration of the geometry and coordinate system for a buckled single thin film on PDMS substrate. W is the width of the thin film. (b) Wavelength and (c) amplitude of the buckling profile as functions of the width of silicon thin films. The theoretical analysis is shown in solid line, and the experimental data is shown in filled circles.

(Reprinted with permission from Ref. [17] Copyright 2008 Elsevier Ltd).

The energy minimization gives the following governing equation for the wave number k:

(1.25) 

From Eq. (1.25), we have

(1.26) 

where f is a nondimensional function be determined numerically by Eq. (1.25), which can be well approximated by the simple relation . Therefore, the wavelength λ = 2π/k is given by

(1.27) 

The energy minimization gives the amplitude as

(1.28) 

where . Figure 1.9b and c shows the buckling wavelength and amplitude versus the ribbon width for 100 nm thick Si ribbon under the prestrain 1.3%, respectively. The solid lines are the analytical solutions from Eqs. (1.27) and (1.28), and the experimental results are plotted by filled circles. Both wavelength and amplitude agree well with experiments. The width effect is negligible for wide ribbons (i.e., >50 µm). However, when the ribbon is narrow, the width effect is strong and cannot be ignored. For example, for 2 µm-wide ribbon, the buckling wavelength is 12.5 µm, and it will increase by 25% to 15.5 µm for 100 µm-wide ribbon.

1.3 Mechanics of Popup Structure

Figure 1.10 schematically illustrates the fabrication of popup structure on compliant substrates [23, 26], which combines lithographically patterned surface bonding chemistry and a buckling process. The ribbon is bonded to the prestrained substrate only at certain locations. Let Wact denote the width of activated regions, where chemical bonding occurs between the ribbon and the substrate, and Win denote the width of inactivated regions, where only weak van der Waals interactions occur at the interface as shown in Figure 1.10a. Thin ribbons are then attached to the prestrained and patterned PDMS substrate (Figure 1.10b) with the ribbon direction parallel to the prestreched direction. Releasing the prestrain leads to compression, which causes the ribbon on the inactivated regions to buckle and form the popup structure as shown in Figure 1.10c.

Figure 1.10 Processing steps for precisely controlled thin film buckling on elastomeric substrate. (a) Prestrained PDMS with periodic activated and inactivated patterns. L is the original length of PDMS, and ΔL is the extension. The widths of activated and inactivated sites are denoted as Wact and Win, respectively. (b) A thin film parallel to the prestrain direction is attached to the prestrained and patterned PDMS substrate. (c) The relaxation of the prestrain εpre in PDMS leads to buckles of thin film. The wavelength of the buckled film is 2L1, and the amplitude is A. 2L2 is the sum of activated and inactivated regions after relaxation.

(Reprinted with permission from Ref. [15] Copyright 2007 American Institute of Physics).

Jiang et al. [23] developed an analytical model to study the buckling behavior of such systems and to predict the maximum strain in the ribbons as a function of interfacial pattern. The buckling profile of the ribbon can be expressed as

(1.29) 

where A is the buckling amplitude to be determined, is the buckling wavelength, and is the sum of activated and inactivated regions after relaxation (Figure 1.10c). The bending and membrane energy in the thin film can be obtained as

(1.30) 

and

(1.31) 

respectively. It should be noticed that the substrate energy

(1.32) 

because the substrate has zero displacement at the interface where it remains intact and vanishing stress traction at the long and buckled portion.

Energy minimization of the total energy gives the amplitude as

(1.33) 

where is the critical strain for buckling, which is identical to the Euler buckling strain for a doubly clamped beam with length 2L1. The critical strain εc is usually a small number in most practical applications. For example, εc is on the order of 10−6 for a typical wavelength 2L1 ∼ 200 µm and ribbon thickness hf ∼ 0.1 µm. Therefore, the buckling amplitude A in Eq. (1.33) can be approxi­mately by

(1.34) 

which is completely determined by the interfacial patterns (Win and Wact) and the prestrain. The comparison of buckled profiles from analytical prediction (dot lines) and experiments is shown in Figure 1.11 for the case of Wact = 10 µm and Win = 190 µm. Both wavelength and amplitude agree well with experiments.

Figure 1.11 Buckled GaAs thin films on patterned PDMS substrate with Win = 10 µm and Win = 190 µm for different prestrain levels, 11.3%, 25.5%, 33.7%, and 56.0% (from top to bottom). The bold lines are the profiles of the buckled GaAs thin film predicted by the analytical solution.

(Reprinted with permission from Ref. [15] Copyright 2007 American Institute of Physics).

The maximum strain in the ribbon can be approximately by the bending strain since the membrane strain is negligible (∼10−6). The bending strain is equal to the maximum curvature times the half thickness and therefore, we have

(1.35) 

The maximum strain is much smaller than the prestrain. For example, for hf = 0.3 µm, Wact = 10 µm, Win = 400 µm, and εpre = 60%, εpeak is only 0.6%, which is two orders of magnitude smaller than the 60% prestrain. For much smaller active region (i.e., Wact Win), the maximum strain in Eq. (1.35) can be approximated by .

1.4 Mechanics of Interconnects in the Noncoplanar Mesh Design

Figure 1.12 schematically illustrates the fabrication of a noncoplanar mesh design consisting of stretchable interconnects and device islands [20]. The device islands are chemically bonded to a prestrained substrate, while the interconnects are loosely bonded. When the prestrain in the substrate is released, the interconnects buckle out of the surface and form arc-shaped structures. Due to the low adhesion, narrow geometries, and low stiffness of interconnects, the deformation localizes only to the interconnects and therefore the device islands experience small strains. Figure 1.13 shows the initial, strain-free configuration X of the interconnects and the buckled configuration x, respectively. The initial distance between the ends of interconnect is L0, and changes to L = L0/(1 + εpre) after the prestrain is released.

Figure 1.12 A schematic illustration of the process for fabricating noncoplanar mesh designs on a complaint substrate.

(Reprinted with permission from Ref. [23] Copyright 2009 American Institute of Physics).

Figure 1.13 Schematic diagram of mechanics model for the interconnect region of a noncoplanar mesh structure.

(Reprinted with permission from Ref. [23] Copyright 2009 American Institute of Physics).

1.4.1 Global Buckling of Interconnects

Song et al. [36] established a mechanics model to understand the buckling behavior of the interconnects. Compared with the model in Section 1.3, the geometry change is accounted here. The out-of-plane displacement, w, of the interconnect takes the form

(1.36) 

which satisfies vanishing displacement and slope at the two ends X = ±L0/2, and the amplitude A is to be determined by energy minimization. The bending energy and membrane energy can be obtained as

(1.37) 

and

(1.38) 

where ε = (L0 − L)/L0 is the compressive strain, E and h are the Young’s modulus and thickness of the interconnects, respectively. Minimization of total energy Uglobal = Umembrane + Ubending in the interconnect gives the amplitude

(1.39) 

where is the critical buckling strain for Euler buckling of a doubly clamped beam. The maximum (compressive) strain in the interconnect is the sum of bending (curvature * h/2) and membrane strains (−εc). Because the membrane strain is very small compared to the bending strain, the peak strain in the interconnects is given by

(1.40) 

In experiments, the initial interconnect length is L0 = 20 µm and after relaxation, the length becomes L = 17.5 µm, which corresponds to a compressive strain ε = 12.5%. The thickness of interconnects is h = 50 nm. The critical buckling strain is εc = 0.0021%. The predicted amplitude by Eq. (1.39) is 4.50 µm, which agrees well with the experimental value 4.76 µm. Equation (1.40) shows that thin and long interconnects give small maximum strain because it is proportional to the ratio of the interconnect thickness to the length, h/L0, which provides a design rule for the buckled interconnects.

1.4.2 Adhesion Effect on Buckling of Interconnects

It should be noticed that other buckling modes may occur if the prestrain is small. Figure 1.14 shows the top and cross-sectional views of a linear array of interconnected silicon islands on a PDMS substrate subjected to low, medium, and high levels of compressive strains. When the compressive strain is small, the interconnect remains flat, and there is no buckling. With the increase of the compressive strain, local buckling (i.e., small multiple waves) may occur. Further increasing the compressive strain up to 8.5%, local buckling transforms to global buckling as the small multiple waves merge together.

Figure 1.14 Cross-sectional and top views of a linear array of interconnected silicon islands on a PDMS substrate subjected to low, medium, and high levels of compressive strains.

(Reprinted with permission from Ref. [22] Copyright 2009 John Wiley and Sons).

Ko et al. [37] and Wang et al. [38] developed a mechanics model to explain the occurrence of different buckling modes by accounting for the adhesion between the interconnect and substrate. Prior to buckling, the interconnect remains flat and the total energy is

(1.41) 

where γ is the work of adhesion between the interconnect and substrate, the first term is the membrane energy Umembrane = EhL0ε2/2 and the second is the adhesion energy. For global buckling, the summation of Eqs. (1.37) and (1.38) gives the total energy as

(1.42) 

where . The critical strain for the transition from flat to global buckling can be obtained by Uflat = Uglobal as

(1.43) 

For local buckling, the interconnect buckles to form small multiple waves with amplitude a and wavelength l to be determined. The membrane energy is obtained as , the bending energy , and adhesion energy Uadhesion = −γ(L0 − l). Energy minimization gives the amplitude and the governing equation for l as

(1.44) 

Therefore, we have , where g is a nondimensional function to be determined numerically by Eq. (1.44). The total energy for local buckling is then obtained as

(1.45) 

The critical strain for the transition from flat to local buckling can be obtained by Uflat = Ulocal as

(1.46) 

Equations (1.43) and (1.46) give a simple criterion to predict buckling patterns. When the critical strain in Eq. (1.46) is larger than that in Eq. (1.43), that is, , global buckling occurs, and there is no local buckling. When the critical strain in Eq. (1.46) is smaller than that in Eq. (1.43), that is, , local buckling occurs first and as the compressive strain increases; global buckling occurs when Ulocal = Uglobal, which gives the critical strain for transition from local to global buckling as

(1.47) 

Figure 1.15 shows the normalized total energy for no buckling, local buckling, and global buckling versus the normalized compressive strain ε/εc. For the polyimide interconnect with E = 2.5 GPa, h = 1.4 µm, L = 150 µm, and the work adhesion γ = 0.16J m−2, which predicts local buckling first and then global buckling as the compressive strain increases. For the strain smaller than 0.78% (Eq. (1.46)), the total energy for no buckling is the lowest. Local buckling prevails until the compressive strain reaches 8.0% from Eq. (1.47), at which global buckling has the lowest energy. The two strains 0.78% and 8.0% are consistent with the ranges of strains for no, local, and global buckling modes observed in Figure 1.14.

Figure 1.15 Comparison of the energy curves for the global, local, and no buckling modes.

(Reprinted with permission from Ref. [39] Copyright 2010 The Royal Society of Chemistry).

1.4.3 Large Deformation Effect on Buckling of Interconnects

In Sections 1.4.1 and 1.4.2, the buckling profile of the ribbon is assumed to be a sinusoidal form, which satisfying vanishing displacement and slope at the two ends. Those results are referred as small deformation model. However, when the compressive strain is large, the buckling profile will deviate from sinusoidal form, and the ends may rotate since the substrate is very compliant. Chen et al. [40] developed a mechanics model to describe the deformation of the buckled thin film by discarding the assumptions of sinusoidal form for the buckling profile and zero rotation at the two ends. The nonvanishing rotation at the ends is accounted by a rotational spring with a spring constant k.

Figure 1.16a shows the initial, strain-free configuration of the interconnect with a length L0. The distance between two ends becomes L after buckling, and Figure 1.16b shows the deformed configuration and forces acting on the interconnect. The bending moment M0 at the ends is related to the rotation θ0 by M0 = kθ0. The doubly clamped and simply supported boundary corresponds to the two limit cases k → ∞ and k → 0, respectively. The intrinsic coordinate (s,θ) as shown in Figure 1.16b is used to describe the deformation of the interconnects. Here s is the arc length from the left end to a point on the deformed shape and θ is the slope angle at that point. The coordinate (x, y) is related to (s,θ) by dx/ds = cos θ and dy/ds = sin θ. The equilibrium equation of the beam is then given by

(1.48) 

where is bending rigidity, and P is the compressive load at the ends. The boundary conditions are

(1.49) 

Figure 1.16 Schematic diagram of mechanics model for the thin film with torsional springs at the two ends.

(Reprinted with permission from Ref. [3] Copyright 2011 The Chinese Society of Theoretical and Applied Mechanics).

Equations (1.1) and (1.2) can be written in nondimensional form as

(1.50) 

and

(1.51) 

where , , , , and .

Equations (1.50) and (1.51) give

(1.52) 

where C satisfies

(1.53) 

The plus and minus sign distinguish between buckling to the top and to the bottom. Here, the minus sign is considered. Equation (1.52) then becomes

(1.54) 

where sin(θ/2) = C sin φ. Integrating Eq. (1.54) from the end of the beam to the midlength gives

(1.55) 

where φ1 satisfies

(1.56) 

Equations (1.53), (1.55), and (1.56) give the solutions of C, , and φ1 for any given θ0. The shortening (i.e., compressive strain ε) and maximum deflection of the beam are then obtained as

(1.57) 

and

(1.58) 

Figure 1.17 shows normalized midspan deflection versus the compressive strain ε with different normalized torsional spring constant . The dotted line is from the previous small deformation model in Section 1.4.1 and the solid lines from Eqs. (1.57) and (1.58). The finite element results are also given for comparison. The current results (solid line) agree well with finite element simulations, while previous small deformation model overestimates the deflection as the compressive strain increases. It should be noted that is almost same for doubly clamped () and simply supported ends (), while becomes slightly larger for midvalue . For example, for is 3% larger than that for at ε = 50%.

Figure 1.17 The normalized midspan deflection versus the compressive strain ε with different normalized torsional spring constant .

(Reprinted with permission from Ref. [3] Copyright 2011 The Chinese Society of Theoretical and Applied Mechanics).

1.5 Concluding Remarks

We have reviewed the mechanics of the stretchable wavy ribbon, popup structure, and interconnects in the noncoplanar mesh design. Both the buckling geometry (wavelength and amplitude) and the maximum strains are obtained analytically. The solutions agree well with the experiments and finite element simulations and clearly show how wavy profile reduces the strain to achieve large stretchability.

Stretchable wavy ribbons: In this case, the ribbon is chemically bonded to the substrate and no delamination occurs. Both small-deformation and finite-deformation analysis are performed for this system. The finite-deformation model predicts a strain-dependent wavelength, while the small-deformation one gives a strain-independent wavelength. The finite width effects have been studied analytically. The experimental and analytical results show that both the buckling amplitude and wavelength increase with the film width.

Popup structure: In this case, the ribbon is only bonded to the substrate at certain locations. When the prestrain is released, the portion of the ribbon without bonding to the substrate delaminates from the substrate and forms the popup structure. The wavelength and amplitude only depend on the geometry and can be precisely controlled to lower the maximum strain to have larger stretchability.

Interconnects in the noncoplanar mesh designs: In this case, the popup interconnects, which is loosely bonded to the substrate, are used to link the device islands, which are chemically bonded to the substrate. The adhesion between the interconnects and substrate is accounted to explain different buckling patterns. The large deformation effect on the buckling of the interconnects is also considered.

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2

Mechanics of Twistable Electronics

Yewang Su, Jian Wu, Zhichao Fan, Keh-Chih Hwang, Yonggang Huang, and John A. Rogers

2.1 Introduction

A successful strategy to stretchable electronics uses postbuckling of stiff, inorganic films on compliant, polymeric substrates [1, 2]. Kim et al. [3] further improved by structuring the film into a mesh and bonding it to the substrate only at the nodes as shown in Figure 2.1a. Once buckled, the arc-shaped interconnects between the nodes can move freely out of the mesh plane to accommodate large deformation. For stretch and bend along the interconnects, this can be modeled by Euler-type postbuckling analysis [4]. For twist, the interconnects undergo rather complex buckling modes as shown in Figure 2.1b and are studied in the following to ensure that the maximum strain in interconnects are below their fracture limit. Simple, analytical expressions are obtained for the amplitude of and maximum strain in buckled interconnects, which are important to the design of stretchable electronics.

Figure 2.1 An island-bridge, mesh structure under twist. (a) A schematic diagram; (b) SEM image.

(Copyright 2008 National Academy of Sciences, USA).

2.2 Postbuckling Theory

Let (X, Y, Z) denote the Cartesian coordinates, Ei (i = 1, 2, 3) the corresponding unit vectors, and Z the central axis of the interconnect before deformation. A point X = (0, 0, Z) on the central axis moves to X + U = (U1, U2, U3 + Z) after deformation, where Ui(Z) (i = 1, 2, 3) are the displacements. The stretch along the central axis is

(2.1) 

such that dZ becomes λdZ after deformation, where . The unit vector along the deformed central axis is e3 = d(X + U)/(λdZ). The other two unit vectors e1 and e2 are related to the twist angle ϕ of each cross section by

(2.2) 

The twist curvature is given by [5]

(2.3) 

The curvatures κ1 and κ2 are related to the displacements Ui(Z) and twist angle ϕ by

(2.4) 

Let t and m denote, respectively, the force and bending moment (and torque) in the cross section Z after deformation. Equilibrium of forces and moment requires

(2.5) 

(2.6) 

The work conjugate of t3 and m are λ − 1 and , respectively, and they are related by the constitutive relations:

(2.7) 

where EA, EI1, and EI2, and C are the tensile, bending, and torsion stiffness, respectively.

Let and denote the critical forces, bending moments, and torque at the onset of buckling, at which the deformation is still small. The forces, bending moments, and torque during postbuckling can be written as and , where Δt and Δm are the changes beyond the onset of buckling. The stretch λ and curvatures are similarly expressed as the sum of critical values at and changes beyond the onset of buckling. All variables are then expanded in the Taylor power series of their changes during postbuckling. Koiter [6] showed that all terms that contribute up to the fourth power of the displacement (from the nonbuckled state) in the potential energy should be included in the postbuckling analysis, and the third and fourth power terms actually govern the postbuckling behavior.

2.3 Postbuckling of Interconnect under Twist

Figure 2.1a shows a mesh structure of device islands with interconnects on a substrate of thickness hs subjected to a twist θ (twist angle per unit length). The islands and interconnects are much thinner than the substrate such that the torsion stiffness of the system results mainly from the substrate. The maximum shear strain in the substrate is [7]

(2.8) 

For the interconnect length L and length of device island Lisland, this shear strain gives the shear displacement γs(L + Lisland) between the device islands, which in turn is imposed between the two ends of interconnect because the device islands are much thicker (wider) than the interconnect such that its deformation is negligible.

The nonzero forces, bending moments, and torque in the interconnect prior to the onset of buckling are

(2.9) 

where denote the critical twist at the onset of buckling, and EI1 is the bending stiffness of the interconnect in the plane of shear. Substitution of displacements and twist angle in Eqs. (2.1)–(2.4) into the equilibrium equations and constitutive relation (Eqs. (2.5)–(2.7)) gives the ordinary differential equations for the changes beyond the onset of buckling, which for simplicity are still denoted by U and ϕ in the following:

(2.10) 

(2.11) 

The boundary conditions at the two ends of the interconnect are

(2.12) 

(2.13) 

where Δθ is the change of twist beyond the onset of buckling.

Equations (2.10) and (2.12) constitute an eigenvalue problem for ϕ and U1. Elimination of U1 yields the equation for ϕ

(2.14)