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Acta Materialia 61 (2013) 7816–7827

Mechanics of ultra-stretchable self-similar serpentine interconnects

Yihui Zhang a,b,c,d,e,1, Haoran Fu f,a,b,c,d,1, Yewang Su a,b,c,d,e, Sheng Xu g,h,Huanyu Cheng a,b,c,d, Jonathan A. Fan g,h, Keh-Chih Hwang e, John A. Rogers g,h,⇑,

Yonggang Huang a,b,c,d,⇑

a Department of Civil and Environmental Engineering, Northwestern University, Evanston, IL 60208, USAb Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA

c Center for Engineering and Health, Northwestern University, Evanston, IL 60208, USAd Skin Disease Research Center, Northwestern University, Evanston, IL 60208, USA

e Center for Mechanics and Materials, Tsinghua University, Beijing 100084, Chinaf Department of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China

g Department of Materials Science and Engineering, University of Illinois at Urbana–Champaign, Urbana, IL 61801, USAh Frederick Seitz Materials Research Laboratory, University of Illinois at Urbana–Champaign, Urbana, IL 61801, USA

Received 21 July 2013; received in revised form 10 September 2013; accepted 10 September 2013Available online 9 October 2013

Abstract

Electrical interconnects that adopt self-similar, serpentine layouts offer exceptional levels of stretchability in systems that consist ofcollections of small, non-stretchable active devices in the so-called island–bridge design. This paper develops analytical models of flex-ibility and elastic stretchability for such structures, and establishes recursive formulae at different orders of self-similarity. The analyticsolutions agree well with finite element analysis, with both demonstrating that the elastic stretchability more than doubles when the orderof the self-similar structure increases by one. Design optimization yields 90% and 50% elastic stretchability for systems with surface fillingratios of 50% and 70% of active devices, respectively.� 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Stretchable electronics; Serpentine interconnect; Mechanical properties; Self-similarity; Analytical modeling

1. Introduction

Interest in the development of electronic and optoelec-tronic systems that offer elastic response to large strain(�1%) deformation has grown rapidly in recent years[1–10], due in part to a range of important applicationpossibilities that cannot be addressed with established

1359-6454/$36.00 � 2013 Acta Materialia Inc. Published by Elsevier Ltd. All

http://dx.doi.org/10.1016/j.actamat.2013.09.020

⇑ Corresponding authors. Address: Department of Materials Scienceand Engineering, University of Illinois at Urbana–Champaign, Urbana,IL 61801, USA (J.A. Rogers). Department of Civil and EnvironmentalEngineering, Northwestern University, Evanston, IL 60208, USA(Y. Huang).

E-mail addresses: jrogers@illinois.edu (J.A. Rogers), y-huang@northwestern.edu (Y. Huang).

1 These authors contributed equally to this work.

technologies, such as wearable photovoltaics [11], “epider-mal” health/wellness monitors [8], eyeball-like digital cam-eras [9,12] and sensitive robotic skins [13–15]. Many ofthese stretchable devices adopt the island–bridge design[8,12,16–18], where the active components are distributedin small, localized regions (i.e. islands) and are joined bynarrow, deformable electrical and/or mechanical intercon-nects (i.e. bridges). Under stretching conditions, the rela-tively stiff islands effectively isolate the active components(usually brittle materials) from strains that could causefracture (e.g. <1%); the bridge structures accommodatenearly all of the deformation [17–19]. For many practicaldevices, the island–bridge design must achieve simulta-neously two competing goals, i.e. high surface filling ratioof active devices, and high stretchability of the entire

rights reserved.

Y. Zhang et al. / Acta Materialia 61 (2013) 7816–7827 7817

system. Demonstrated design solutions involve either ser-pentine [1,8,17,20–27] or non-coplanar [12,18] intercon-nects. These technologies, however, typically give levels oftotal stretchability that are less than 50%, in systems thatdo not significantly sacrifice areal coverage. Recently, Xuet al. [19] illustrated an alternative type of interconnectdesign that exploits self-similar serpentine geometries(shown in Fig. 1a), a type of space-filling curve. Thisconcept enabled lithium-ion batteries with a biaxial stretch-ability of up to�300%, and areal coverages of active materi-als as high as �50%. Comprehensive experimental andnumerical investigations indicated that such self-similarserpentine interconnects possess improved levels of stretch-ability compared to traditional serpentine structures for agiven spacing between adjacent islands. The nature of the

a

b

n=1

n=2

n=3

n=4

( )2l

( )3h

( )2h

( )3l

( )4h

( )1h

( )1l

( )4l

Self-similar rectangular (m=4, η=2.41)

Fig. 1. (a) Optical images of the Al electrode pads and self-similar interconnprinting on a sheet of silicone (middle panel; oblique view, in a bent geometry)geometry), for a stretchable Li-ion battery; (b) schematic illustration on theschematic illustration on the geometric construction of the self-similar serpepermission from Xu et al. [19], � 2013, Nature Publishing Group.

space-filling geometry in these structures and the mecha-nisms for their ordered unraveling were found to play impor-tant roles.

This study aims at developing an analytic model tostudy the flexibility and elastic stretchability (referred tosimply as stretchability in the following) of self-similar ser-pentine interconnects, and to establish the design guidelinesfor optimizing the stretching limit. Here, we focus on thescenario that the interconnects are not bonded to the sup-porting substrate such that deformation can occur freelyand the interactions with the substrate can be neglected.Such freely suspended interconnects can be fabricated byeither of two methods: (i) molding surface relief structureson the elastomeric substrate [16,18,28] and bonding theislands onto the top of the raised relief; or (ii) designing

c Self-similar serpentine (m=4, η=2.41)

( )2l

( )3h

( )2h

( )3l

( )4h

( )1h

( )1l

( )4l

ects on Si wafer (left panel; top down view; �4 unit cells), after transfer, and with molded slurries of LiCoO2 (right panel; oblique view, in a bent

geometric construction of the self-similar rectangular interconnect; (c)ntine interconnect. The scale bars in (a) are 2 mm. (a) is reprinted with

7818 Y. Zhang et al. / Acta Materialia 61 (2013) 7816–7827

the mask of SiO2 deposition to enable selective bonding ofthe islands onto the substrate [29,30], while leaving theinterconnects with a minimum interaction with the sub-strate. The present study mainly focuses on relative thickinterconnects with the thickness comparable to the width,as required for applications that demand low electricalresistance, such as wireless inductive coils [19], and photo-voltaic modules [11]. In such cases, the deformation of theinterconnects is governed by in-plane bending, rather thanbuckling, when the system is under stretching. Here, thecritical buckling strain is large compared to the stretchabil-ity [31], such that buckling is not triggered within the rangeof acceptable deformations. This mechanics is qualitativelydifferent from that of the types of free-standing, thin ser-pentine interconnects that have been investigated previ-ously [17,31–33]. For free-standing, thick self-similarinterconnects, analytic models of the flexibility and stretch-ability are established in this study. The models are thenextended to arbitrary self-similar orders. The results estab-lish design guidelines for practical applications.

This paper is outlined as follows: Section 2 focuses onthe simplest geometric configuration, self-similar rectangu-lar interconnects, to illustrate the mechanics model for ana-lyzing the flexibility and stretchability. The analytic modelis extended to generalized self-similar rectangular and ser-pentine interconnects in Section 3. The stretchability ofself-similar interconnects is studied in Section 4. Section 5presents the optimal design of self-similar serpentine inter-connects for stretchable electronics to illustrate its advan-tage in achieving high system stretchability.

2. Self-similar rectangular interconnects

This section focuses on a geometrically simple self-sim-ilar interconnect in a rectangular configuration (as shownin Fig. 1b), to illustrate its structure, flexibility andstretchability. The rectangular interconnect is a variantof the traditional serpentine interconnect (top panel ofFig. 1c), and is convenient for constructing self-similarstructures because of its simple geometry. To determinethe flexibility of self-similar rectangular interconnects,the key is to establish the relation between the flexibilityof neighboring orders, i.e. the recursion formula. We firsttake the first order self-similar rectangular interconnect asan example to illustrate the model as in Section 2.2, andthen generalize the theoretical framework to the secondorder and arbitrary order in Sections 2.3 and 2.4,respectively.

2.1. Geometry

This subsection introduces the geometric construction ofself-similar rectangular interconnects. The first order (origi-nal) rectangular interconnect consists of two sets of straightwires that are perpendicular to each other and connected inseries, as shown in the black box of Fig. 1b. The secondorder rectangular interconnect, shown in the blue box of

Fig. 1b, is created by reducing the scale of the first orderinterconnect, rotating the structure by 90� and thenconnecting multiple copies of it in a fashion that repro-duces the layout of original geometry. The wide blue linein Fig. 1b represents the second order geometry that is sim-ilar to the first order rectangular geometry. By implement-ing the same algorithm, we can generate the third- andfourth order rectangular interconnects, as illustrated inthe red and purple boxes of Fig. 1b, where the red and pur-ple lines denote the third- and fourth order geometries,respectively.

For self-similar rectangular interconnects, let m denotethe number of unit cell and g the height/spacing aspectratio at each order. Therefore the lengths of horizontaland vertical lines of the ith order (i = 1, . . . ,n), l(i) and h(i)

(Fig. 1b), are related by

hðiÞ ¼ glðiÞ ð1ÞIn addition, the height of ith order geometry equals to thedistance between two ends of (i � 1)th order geometry, thatis

hðiÞ ¼ 2mlði�1Þ ði ¼ 2; . . . ; nÞ ð2ÞEqs. (1) and (2) give the length and height at any order interms of l(n), g and m, as

lðiÞ ¼ ðg=2mÞn�ilðnÞ; hðiÞ ¼ gðg=2mÞn�ilðnÞ; ði ¼ 1; . . . ; nÞð3Þ

This indicates that the geometry of an arbitrary self-similarrectangular interconnect is characterized by one baselength (l(n)) and three non-dimensional parameters, namelythe self-similar order (n), the height/spacing ratio (g) andnumber (m) of unit cell. It should be mentioned that, forn P 3, there is an additional constraint on the height/spac-ing ratio g because of the following relation, which can beobserved from the geometry of the third order rectangularinterconnect shown in Figs. 1 and 3 (to be discussed):

lðiÞ ¼ ð2mh þ 1Þlði�2Þ ði ¼ 3; . . . ; nÞ ð4Þwhere mh is the number of full unit cells in the structurerepresented by the horizontal part of the ith order geome-try (i = 3, . . . ,n). Eqs. (3) and (4) give the constraint on theheight/spacing ratio g for n P 3:

g ¼ 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mh þ 1p ði ¼ 3; . . . ; nÞ ð5Þ

i.e. the height/spacing ratio can only take some discretevalues for n P 3. Fig. 1b shows a set of self-similar rectan-gular interconnects, from n=1 to 4, with m = 4 andg ¼ 8=

ffiffiffiffiffi11p

.

2.2. Flexibility of first order rectangular interconnects

Fig. 2a shows a schematic illustration of the first orderself-similar rectangular interconnect with m unit cells andheight/spacing ratio g. As illustrated in Fig. 2b, a represen-

a

b

(1)

(1)

Fig. 2. (a) A freely suspended first order rectangular interconnect, clamped at the left end, and subject to an axial force N, shear force Q, and bendingmoment M, at the right end. (b) Exploded view and free body diagram of the kth unit cell of first order rectangular interconnect.

a Part I Part II Part III Part IV

Part V

2m

( )1l

( )1h

m

( )1l

( )1h12hm +

2m

( )2l

( )2h

Part I Part II Part III Part IV

Part V

b

m

Fig. 3. The exploded view of a representative unit cell for the (a) second order and (b) third order self-similar rectangular interconnect.

Y. Zhang et al. / Acta Materialia 61 (2013) 7816–7827 7819

7820 Y. Zhang et al. / Acta Materialia 61 (2013) 7816–7827

tative unit cell (e.g. the kth unit cell) of the first order struc-ture is composed of five straight wires (i.e. zeroth orderstructure) (Parts I–V). The vertical wires, Parts I and III,have a length of h(1)/2, and Part II has a length of h(1).The horizontal wires, Parts IV and V, have a length of l(1).

Consider the first order rectangular interconnectclamped at the left end, and subject to an axial force N

(along the direction between the two ends of the intercon-nect), a shear force Q (normal to N) and a bending momentM, at the right end, within the plane of interconnect, asshown in Fig. 2a. The width (w) and thickness (t) of the ser-pentine interconnect are usually much smaller than thelength such that the structure can be modeled as a curvedbeam. Let u and v denote the displacements at the rightend, along and normal to the axial direction of the inter-connect (parallel to N and to Q), respectively, and h isthe rotation angle (Fig. 2a). They are related to (N, Q,M) via the strain energy W(1) in the interconnect by

u

m

h

0B@

1CA ¼ @W ð1Þ=@N

@W ð1Þ=@Q

@W ð1Þ=@M

264

375 ¼

T ð1Þ11 T ð1Þ12 T ð1Þ13

T ð1Þ12 T ð1Þ22 T ð1Þ23

T ð1Þ13 T ð1Þ23 T ð1Þ33

2664

3775

N

Q

M

0B@

1CA

¼ Tð1ÞN

Q

M

0B@

1CA ð6Þ

where W(1) = (N, Q, M)T(1)(N, Q, M)T/2 is a quadraticfunction of N, Q, and M for linear elastic behavior of theinterconnect; and T(1) is the symmetric flexibility matrixof the first order interconnect and is to be determined.The strain energy also equals the sum of strain energyW(0) in all zeroth order interconnects (Parts I–V), i.e.

W ð1Þ ¼ W ð0Þ ¼Xm

k¼1

W Ik þ W II

k þ W IIIk þ W IV

k þ W Vk

� �ð7Þ

where W Ik to W V

k represent the strain energy of each compo-nent in the kth unit cell. For the zeroth order structure, i.e.a straight wire with length l and bending stiffness EI, thebeam theory gives the flexibility matrix as [34]

Tð0ÞðlÞ ¼ 1

6EI

0 0 0

0 2l3 3l2

0 3l2 6l

0B@

1CA ð8Þ

Here the membrane energy is neglected. The free body dia-gram of the kth unit cell of the first order interconnect(Fig. 2b) gives the axial force, shear force and bending mo-ment in each wire, and the strain energy of each zeroth or-der interconnect can then be obtained as

W Ik

W IIk

W IIIk

W IVk

W Vk

0BBBBBB@

1CCCCCCA¼ 1

2ðN ;Q;MÞ

DI Tð0Þ½hð1Þ=2�DT

I

DII Tð0Þ½hð1Þ�DT

II

DIII Tð0Þ½hð1Þ=2�DT

III

DIV Tð0Þ½lð1Þ�DTIV

DV Tð0Þ½lð1Þ�DTV

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;ðN ;Q;MÞT ð9Þ

where DI ¼0 1 �hð1Þ=2

1 0 �ð2m� 2k þ 2Þlð1Þ

0 0 �1

264

375;

DII ¼0 1 �hð1Þ=2

�1 0 ð2m� 2k þ 1Þlð1Þ

0 0 1

264

375;

DIII ¼0 1 0

1 0 �ð2m� 2kÞlð1Þ

0 0 �1

264

375;

DIV ¼1 0 hð1Þ=2

0 1 ð2m� 2k þ 1Þlð1Þ

0 0 1

2664

3775 and

DV ¼1 0 hð1Þ=2

0 �1 �ð2m� 2kÞlð1Þ

0 0 �1

264

375 ð10Þ

Substitution of Eq. (9) into Eq. (7) gives the recursiveformula between the flexibility matrices of first and zerothorder interconnects as

Tð1Þ ¼Xm

k¼1

DI Tð0Þ½hð1Þ=2�DT

I þDII Tð0Þ½hð1Þ�DT

II

þDIII Tð0Þ½hð1Þ=2�DT

III þDIV Tð0Þ½lð1Þ�DTIV þDV Tð0Þ½lð1Þ�DT

V

( )ð11Þ

Substitution of T(0) in Eq. (8) into the above equation givesa simple expression of the flexibility of first order intercon-nect in terms of the number of unit cells m, height/spacingratio g and l(1):

Tð1Þ m;g;lð1Þ� �

¼ 1

EI

m g3þ3g2

6½lð1Þ�3 m gðgþ2Þ

4½lð1Þ�3 0

m gðgþ2Þ4½lð1Þ�3 ð8m3þmÞgþ8m3

3½lð1Þ�3 2m2ðgþ1Þ½lð1Þ�2

0 2m2ðgþ1Þ½lð1Þ�2 2mðgþ1Þlð1Þ

8>>><>>>:

9>>>=>>>;ð12Þ

For the convenience of generalization to higher order (n)structure, the following non-dimensional form of flexibilitymatrix is adopted:

u=lðiÞ

m=lðiÞ

h

264

375 ¼ lðiÞ

EITðiÞ

NlðiÞ

QlðiÞ

M

264

375; i ¼ 1; . . . ; n ð13Þ

where TðiÞ is dimensionless, and Tð1Þ is given by

Tð1Þðm; gÞ ¼m g3þ3g2

6m gðgþ2Þ

40

m gðgþ2Þ4

ð8m3þmÞgþ8m3

32m2ðgþ 1Þ

0 2m2ðgþ 1Þ 2mðgþ 1Þ

2664

3775 ð14Þ

For the zeroth order structure, i.e. a straight wire oflength k, the non-dimensional flexibility matrix is definedas ðu=k; m=k; hÞT ¼ ðk=EIÞTð0ÞðNk;Qk;MÞT , where

Y. Zhang et al. / Acta Materialia 61 (2013) 7816–7827 7821

Tð0Þ ¼0 0 0

0 13

12

0 12

1

0B@

1CA ð15Þ

2.3. Flexibility of the second order rectangular interconnect

The recursive formula for the flexibility matrix of thesecond order interconnect is established in this section.A representative unit cell of the second order structureis composed of three first order structures (Parts I–III),and two straight wires (i.e. zeroth order structure) (PartsIV and V) with length of l(2), as illustrated in Fig. 3a.The first order structures, Parts I and III, consist of m/2(m is an even integer) unit cells, and Part II consists ofm unit cells.

The strain energy of the second order structure canbe expressed in terms of the dimensionless flexibilitymatrix as

W ð2Þ ¼ lð2Þ

2EINlð2Þ;Qlð2Þ;M� �

Tð2Þ Nlð2Þ;Qlð2Þ;M� �T

¼ lð2Þ

2EIðN ;Q;MÞ

lð2Þ 0 0

0 lð2Þ 0

0 0 1

264

375Tð2Þ

lð2Þ 0 0

0 lð2Þ 0

0 0 1

264

375ðN ;Q;MÞT

ð16Þ

where Tð2Þ is to be determined. The strain energy alsoequals the sum of strain energy in all first order (PartsI–III, Fig. 3a) and zeroth order (Parts IV and V, Fig. 3a)interconnects, i.e.

W ð2Þ ¼Xm

k¼1

W Ik þ W II

k þ W IIIk þ W IV

k þ W Vk

� �ð17Þ

where

W IIk ¼

lð1Þ

2EINlð1Þ;Qlð1Þ;M� �

DII Tð1Þðm; gÞDT

II Nlð1Þ;Qlð1Þ;M� �T

ð18Þis the strain energy in Part II (first order structure, m unit

cell) with DII ¼0 1 �m�1 0 ð4m� 4k þ 2Þmg�1

0 0 1

24

35 being the

normalized DII in Eq. (10) (with l(1) and h(1) replaced byl(2) and h(2), respectively);

W IVk ¼

lð2Þ

2EINlð2Þ;Qlð2Þ;M� �

DIV Tð0ÞDTIV Nlð2Þ;Qlð2Þ;M� �T

W Vk ¼

lð2Þ

2EINlð2Þ;Qlð2Þ;M� �

DV Tð0ÞDTV Nlð2Þ;Qlð2Þ;M� �T

ð19Þare the strain energy in Parts IV and V (zeroth order struc-

ture, length k ¼ lð2Þ) with DIV ¼1 0 g=20 1 2m� 2k þ 10 0 1

0@

1A

and DV ¼1 0 g=20 �1 �2mþ 2k0 0 �1

0@

1A being the normalized

DIV and DV in Eq. (10) (with l(1) and h(1) replaced by l(2)

and h(2), respectively);

W Ik ¼

lð1Þ

2EINlð1Þ;Qlð1Þ;M� �

DI Tð1Þ m

2; g

� �DT

I Nlð1Þ;Qlð1Þ;M� �T

W IIIk ¼

lð1Þ

2EINlð1Þ;Qlð1Þ;M� �

DIII Tð1Þ m

2; g

� �DT

III Nlð1Þ;Qlð1Þ;M� �T

ð20Þ

are the strain energy in Parts I and III (first order structure,

m/2 unit cell) with DI ¼0 1 �m1 0 �4ðm� k þ 1Þmg�1

0 0 �1

24

35 and

DIII ¼0 1 01 0 �4ðm� kÞmg�1

0 0 �1

24

35 being the normalized DI

and DIII in Eq. (10) [with l(1) and h(1) replaced by l(2) andh(2), respectively].

Substitution of Eqs. (18)–(20) into Eq. (17) gives therecursive formula for the flexibility matrix of second orderinterconnect as

Tð2Þ ¼ g2m

g2m 0 0

0 g2m 0

0 0 1

0BB@

1CCA

�Xm

k¼1

DI Tð1ÞKðmgÞþKTðmgÞT ð1Þ� �

DTI þDII Tð1ÞDT

II

þDIII Tð1ÞKðmgÞþKTðmgÞT ð1Þ� �

DTIII

8<:

9=;

�

g2m 0 0

0 g2m 0

0 0 1

0BB@

1CCAþX

m

k¼1

DIV Tð0ÞDTIV þDV Tð0ÞDT

V

� �ð21Þ

where

KðmgÞ ¼ 1

4

1 0 0

0 1 0

0 �mg 1

0B@

1CA ð22Þ

results from the identity

Tð1Þm2; g

� �¼ Tð1Þðm; gÞKðmgÞ þ KT ðmgÞTð1Þðm; gÞ ð23Þ

Substitution of Tð0Þ and Tð1Þ in Eqs. (15) and (14) into Eq.(21) gives Tð2Þ as

Tð2Þðm; gÞ ¼g2 g2þ2m2ðfþ2Þ

12mm4gðf þ 1Þ 0

m4gðf þ 1Þ g5ðgþ3Þþ8m2ðf�1Þþ64m4f

24m 2m2f

0 2m2f 2mf

2664

3775ð24Þ

where f = g2 + g + 1.

n=3

n=2

n=1

m (3)=1, (3)=3.1

m (2)=4, (2)=1.6

m (1)=4, (1)=3.8

Fig. 4. Schematic illustration on the geometric construction of a third order generalized self-similar serpentine interconnect.

7822 Y. Zhang et al. / Acta Materialia 61 (2013) 7816–7827

2.4. Flexibility of higher order rectangular interconnect

For the higher order (n P 3) rectangular interconnect, arepresentative unit cell is composed of three (n � 1) orderstructures (Parts I–III), and two (n � 2) order structures(Parts IV and V). The (n � 1) order structures, Parts I andIII, consist of m/2 (m is an even integer) unit cells, and PartII consists of m unit cells. The recursive formula (21) becomes2

TðnÞ ¼ g2m

g2m 0 0

0 g2m 0

0 0 1

0B@

1CA

�Xm

k¼1

DI T ðn�1ÞKðmgÞþKT ðmgÞT ðn�1Þ� �DT

I

þDII Tðn�1ÞDT

II

þDIII Tðn�1ÞKðmgÞþKT ðmgÞT ðn�1Þ� �DT

III

8><>:

9>=>;

�

g2m 0 0

0 g2m 0

0 0 1

0B@

1CAþ g2

4m2

Xm

k¼1

D�IV T ðn�2ÞD�TIV þD�V Tðn�2ÞD�TV

� �for n�3

ð25Þ

where D�IV ¼g2=ð4m2Þ 0 g=2

0 g2=ð4m2Þ 2m� 2k þ 1

0 0 1

264

375 and

D�V ¼g2=ð4m2Þ 0 g=2

0 �g2=ð4m2Þ �2mþ 2k

0 0 �1

264

375

3. Generalized self-similar interconnects

The analytic model for self-similar rectangular intercon-nects in Section 2 is extended to generalized self-similarrectangular and serpentine interconnects in this section.

3.1. Generalized self-similar rectangular interconnects

The generalized rectangular interconnect still exhibitsthe rectangular shape (shown in Fig. 4), but does notrequire the same height/spacing ratio across differentorders, nor the number of unit cell. Each order may have

2 The (n � 2)th order structures (e.g. Parts IV and V in Fig. 3b for thecase of n = 3) have (mh + 1/2) unit cells at the (n � 2)th order geometry.However, because the contribution of the (n � 2)th order structures to theoverall flexibility is much smaller than that of (n � 1)th order structures,the dimensionless flexibility of Parts IV and V can be approximated by theself-similar (n � 2)th order structures with m unit cells, which, as to beshown by finite element analysis, gives rather good accuracy.

its own height/spacing ratio g(i) and number of unit cellm(i) (i = 1, . . . ,n), where only m(n) can be an odd number,and m(1) to m(n) must be even numbers. Fig. 4 illustratesa generalized third order self-similar rectangular intercon-nect. For the nth order generalized self-similar rectangularinterconnect, the geometric relations (1)–(3) become

hðiÞ ¼ gðiÞlðiÞ ð26ÞhðiÞ ¼ 2mði�1Þlði�1Þ ði ¼ 2; . . . ; nÞ ð27Þ

lðiÞ ¼Yn�i

k¼1

gðn�kþ1Þ

2mðn�kÞ

" #lðnÞ; hðiÞ

¼ gðiÞYn�i

k¼1

gðn�kþ1Þ

2mðn�kÞ

" #lðnÞ; ði ¼ 1; . . . ; n� 1Þ ð28Þ

The flexibility matrix Tð0Þ in Eq. (15) remains the same,while m and g in Eq. (14) for Tð1Þ need to be replaced bym(1) and g(1), respectively. The recursive formulae for Tð2Þ

in Eq. (21) and TðnÞ (n P 3) in Eq. (25) now become

Tð2Þ ¼ gð2Þ

2mð1Þ

gð2Þ

2mð1Þ0 0

0 gð2Þ

2mð1Þ0

0 0 1

264

375

�Xmð2Þk¼1

Dð2ÞI

Tð1ÞK mð1Þgð1Þ� �

þKT mð1Þgð1Þ� �

Tð1Þ

( )Dð2ÞTI

þDð2ÞII Tð1ÞD

ð2ÞTII

þDð2ÞIII

Tð1ÞK mð1Þgð1Þ� �

þKT mð1Þgð1Þ� �

Tð1Þ

( )Dð2ÞTIII

* + gð2Þ

2mð1Þ0 0

0 gð2Þ

2mð1Þ0

0 0 1

264

375

þXmð2Þk¼1

Dð2ÞIV Tð0ÞD

ð2ÞTIV þD

ð2ÞV Tð0ÞD

ð2ÞTV

h ið29Þ

TðnÞ ¼ gðnÞ

2mðn�1Þ

gðnÞ

2mðn�1Þ 0 0

0 gðnÞ

2mðn�1Þ 0

0 0 1

26664

37775

�XmðnÞk¼1

DðnÞI

Tðn�1ÞK mðn�1Þgðn�1Þ� �þKT mðn�1Þgðn�1Þ� �

Tðn�1Þ

( )DðnÞTI

þDðnÞII Tðn�1ÞD

ðnÞTII

þDðnÞIII

Tðn�1ÞK mðn�1Þgðn�1Þ� �þKT ½mðn�1Þgðn�1Þ�Tðn�1Þ

( )DðnÞTIII

* +

gðnÞ

2mðn�1Þ 0 0

0 gðnÞ

2mðn�1Þ 0

0 0 1

26664

37775

þ gðnÞgðn�1Þ

4mðn�1Þmðn�2Þ

XmðnÞk¼1

DðnÞ�IV Tðn�2ÞD

ðnÞ�TIV þD

ðnÞ�V Tðn�2ÞD

ðnÞ�TV

h i; for n�3

ð30Þ

Y. Zhang et al. / Acta Materialia 61 (2013) 7816–7827 7823

where

DðnÞI ¼

0 1 �mðn�1Þ

1 0 �4½mðnÞ � k þ 1�mðn�1Þ½gðnÞ��1

0 0 �1

8><>:

9>=>;;

DðnÞII ¼

0 1 �mðn�1Þ

�1 0 ½4mðnÞ � 4k þ 2�mðn�1Þ½gðnÞ��1

0 0 1

8>><>>:

9>>=>>;

and DðnÞIII ¼

0 1 0

1 0 �4½mðnÞ � k�mðn�1Þ½gðnÞ��1

0 0 �1

264

375 for n P 2

Dð2ÞIV ¼

1 0 gð2Þ=2

0 1 2mð2Þ � 2k þ 1

0 0 1

264

375;

Dð2ÞV ¼

1 0 gð2Þ=2

0 �1 �2mð2Þ þ 2k

0 0 �1

264

375 and

D�ðnÞIV ¼

gðn�1ÞgðnÞ=½4mðn�2Þmðn�1Þ� 0 gðnÞ=2

0 gðn�1ÞgðnÞ=½4mðn�2Þmðn�1Þ� 2mðnÞ �2kþ1

0 0 1

264

375

and D�ðnÞV ¼

gðn�1ÞgðnÞ= 4mðn�2Þmðn�1Þ� �0 gðnÞ=2

0 �gðn�1ÞgðnÞ= 4mðn�2Þmðn�1Þ� ��2mðnÞ þ2k

0 0 �1

264

375

for nP3

3.2. Generalized self-similar serpentine interconnects

Fig. 1b and c shows the generalized self-similar serpen-tine interconnect, which replaces the sharp corners in therectangular configuration by half circles, as in Xu et al.’sexperiments [19]. The first order serpentine interconnectconsists of straight wires (length h(1)–l(1)) connected by halfcircles (diameter l(1)), as shown in the black box of Fig. 1c.A representative unit cell of the second order serpentineinterconnect, as shown in the blue box of Fig. 1c, is com-posed of two (horizontal) straight wires of length l(2) and

a b

1 2 3 40

100

200

300

400

500

600 Analytic FEA

Self-similar order n

( )nijT Serpentine

Rectangular

( )33nT

( )11nT

( )12nT

T

Fig. 5. The effect of self-similar order on the flexibility: the dimensionless flexself-similar order. In the FEA, the width is fixed as w = 0.4l(1) for the structur

three (vertical) first order serpentine interconnects (twowith lengths h(2)/2 and one with length of h(2)). The flexibil-ity matrix Tð0Þ for straight wires is still given in Eq. (15),and the flexibility matrix Tð1Þ for the first order serpentineinterconnect is obtained as [31]

T ð1Þ ¼mð1Þ

24

4g3þ6pg2þ24gþ3p 6ðg2þpgþ2Þ 0

6ðg2þpgþ2Þ 32½mð1Þ�2ð2gþpÞþ8gþp 48mð1ÞðgþpÞ0 48mð1ÞðgþpÞ 24ð2gþpÞ

8><>:

9>=>;ð31Þ

where g = g(1) � 1.The second to fourth (and higher) order geometries all

exhibit a rectangular geometry (shown in Fig. 1c), whichindicates that, strictly speaking, the self-similarity onlystarts at the second order interconnects. Comparison ofthe self-similar serpentine structure (Fig. 1c) to the rectan-gular one (Fig. 1b) suggests that only their first ordergeometries are different. Therefore, the recursive formulaein Eqs. (29) and (30) still hold for the self-similar serpentinestructure.

Substitution of Tð0Þ in Eq. (15) and Tð1Þ in Eq. (14) intoEq. (31) gives Tð2Þ as

T ð2Þðm;gÞ¼mð2Þ

24

6 gð2Þ� �2ð4�pÞþ6 gð2Þ

mð1Þ

h i3

T ð1Þ22 3gð2Þðpþ2Þ 0

3gð2Þðpþ2Þ 32 mð2Þ� �2

pþ4p�8þ6 gð2Þ

mð1Þ

h i3

T ð1Þ11 24mð2Þp

0 24mð2Þp 24p

266664

377775

ð32Þ

where p = g(2)[2g(1) + p � 2] + 2.Fig. 5a and b shows the components of non-dimensional

flexibility vs. the order (n) for self-similar rectangular andserpentine interconnects for the height/spacing ratiog ¼ 8=

ffiffiffiffiffi11p

and number of unit cell m = 4. The rectangularinterconnect is slightly softer than the serpentine one. Theanalytic results are validated by FEA, which is also shownin Fig. 5a and b, for copper interconnect with the elasticmodulus ECu = 119 GPa and Poisson’s ratio mCu = 0.34.The component T13 is always zero, and is therefore notshown. The other five flexibility components all increasewith n, and are more than doubled for each n increasingby 1. For n from 1 to 4, these components increase by morethan 17 times, indicating that the higher order interconnectbecomes much softer than the lower order one.

1 2 3 40

2000

4000

6000

8000

10000

12000 Analytic FEA

( )nij Serpentine

Rectangular

( )22nT

( )23nT

Self-similar order n

ibility components T ðnÞ11 ; TðnÞ12 and T ðnÞ33

� �(a) and T ðnÞ22 and T ðnÞ23

� �(b) vs. the

es of different orders.

7824 Y. Zhang et al. / Acta Materialia 61 (2013) 7816–7827

4. Stretchability

The interconnect usually spans the space between tworigid device islands (e.g. in Fig. 1a), corresponding toclamped boundary conditions at the two ends. For stretch-ing u0 of the self-similar interconnect (with n orders), theboundary conditions are u = u0, v = 0 and h = 0, and Eq.(13) then gives the reaction forces, N and Q, and bendingmoment M as

N

Q

M

8>><>>:

9>>=>>; ¼

EI

lðnÞ� �3

u0

T ðnÞ11 T ðnÞ22 T ðnÞ33 � T ðnÞ11 T ðnÞ23

h i2

� T ðnÞ33 T ðnÞ12

h i2

�

T ðnÞ22 T ðnÞ33 � T ðnÞ23

h i2

�T ðnÞ12 T ðnÞ33

lðnÞT ðnÞ12 T ðnÞ23

8>>>><>>>>:

9>>>>=>>>>;

ð33Þ

since T ðnÞ13 ¼ 0. The maximum strains for the rectangularand serpentine configurations are analyzed separately inSections 4.1 and 4.2. Since no experiment result is availableregarding the stretchability of relative thick self-similarrectangular or serpentine interconnects, we only comparethe analytic results to the finite element analysis (FEA) re-sults for validation. The experimental measurement of thestretchability and comparison to analytic results will beconsidered in our future work.

4.1. Generalized self-similar rectangular interconnects

For the first order rectangular interconnect, it can beshown that the maximum strain occurs at the third nearestcorners from the loading points, as illustrated in Fig. S.1a(Electronic Supplementary material), which is well sup-ported by FEA results. The maximum strain in the inter-connect can be then obtained accurately as

emax ¼w 2M þ Nhð1Þ þ 2Qlð1Þ� �

4EIð34aÞ

For higher order structures with n P 2, the maximumstrain can be well approximated by

emax w 2M þ NhðnÞ þ 2QlðnÞ � Qhðn�1Þ� �

4EIðfor n P 2Þ

ð34bÞ

estretchability ¼ 2eyield lð1Þ

mð1Þw

T ð1Þ11

T ð1Þ22

T ð1Þ33�T ð1Þ

11T ð1Þ

23½ �2�T ð1Þ

33T ð1Þ

12½ �2

2T ð1Þ12

T ð1Þ23�3T ð1Þ

12T ð1Þ

33þ T ð1Þ

22T ð1Þ

33� T ð1Þ

23½ �2

� ðgð1Þ�1Þþ

ffiffiffiffiffiffiffiffiffiffiffiffiT ð1Þ

22T ð1

33

�q

Based on the yield criterion emax = eyield, where eyield is theyield strain of the interconnect material (e.g. 0.3% forcopper [35]), the stretchability of the generalized self-simi-lar rectangular interconnect is obtained as

eð1Þstretchability ¼eyield lð1Þ

wgð1Þ

12

�16 mð1Þ� �2

gð1Þ þ 1� �

gð1Þ þ 3� �

� gð1Þ þ 6� �2

4 mð1Þ½ �2 gð1Þ þ 1½ � þ 3mð1Þ gð1Þ þ 2½ � � gð1Þ � 6

ð35aÞ

eðnÞstretchability ¼eyield lðnÞ

w2

mðnÞ

T ðnÞ11 T ðnÞ23

h i2

þT ðnÞ33 T ðnÞ12

h i2

�T ðnÞ11 T ðnÞ22 T ðnÞ33

2T ðnÞ12 T ðnÞ33 �T ðnÞ23

h iþ T ðnÞ23

h i2

�T ðnÞ22 T ðnÞ33

gðnÞ � gðn�1ÞgðnÞ

2mðn�1Þ T ðnÞ12 T ðnÞ33

��������

��������ðfor n P 2Þ ð35bÞ

When the applied strain is smaller than the stretchability,the interconnect undergoes linear, reversible deformations,and no plastic deformation would accumulate, such thatthe interconnect would not suffer from plastic fatigue undercyclic loadings. Eqs. (35a) and (35b) show clearly that thestretchability is linearly proportional to eyieldl(n)/w. There-fore, in order to enhance the stretchability, it is better toadopt a metallic material with high yield strength and rel-ative low elastic modulus to give a high yield strain, suchas the nano-grained-size copper, or transforming metalnanocomposites [36].

4.2. Generalized self-similar serpentine interconnects

For first order serpentine interconnect, as shown inFig. S.1b (Electronic Supplementary material), the maxi-mum strain always occurs at the nearest or second nearesthalf circle from the two ends. Let uð0 6 u 6 pÞ representthe location of this half circle. The bending strain on thecircle can be given by

eðuÞ¼w 2MþN ½hð1Þ�lð1Þ�þ3Qlð1Þþlð1ÞðN sinu�QcosuÞ�

4EIð36Þ

It reaches the maximum at u = tan�1(�N/Q), and themaximum strain is given by

emax ¼w 2M þ N ½hð1Þ � lð1Þ� þ 3Qlð1Þ þ lð1Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN 2 þ Q2

pn o4EI

ð37ÞThe stretchability of the first order serpentine intercon-

nect is then obtained as (via Eq. (33)):

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÞ� T ð1Þ

23½ �2 2

þ T ð1Þ12½ �

2T ð1Þ

33½ �2: ð38Þ

1 2 3 40

10

20

30

40

50

60 Analytic FEA

( )stretchability

nyield

w

l

εε

ε stretchability

Serpentine

Rectangular

Self-similar order n

Fig. 8. The dimensionless stretchability as a function of the self-similarorder. In the FEA, the width is fixed as w = 0.4l(1) for the structures ofdifferent orders.

1 2 3 4 5 6 7 80

5

10

15

20

25

Analytic FEA

ε stretchability

m(1)=10m(1)=4m(1)=2

Height/spacing ratio (η (1))

…

m (1) unit cells, height/spacing ratio η (1)=h(1)/l(1)

1l

1h

m(1)→∞

Fig. 6. The dimensionless stretchability vs. the height/spacing ratio (g(1)) fordifferent number (m(1)) of unit cells for the first order serpentine interconnect.

Y. Zhang et al. / Acta Materialia 61 (2013) 7816–7827 7825

The normalized stretchability estretchabilityw/[eyieldl(1)]depends only on the height/spacing ratio g(1) and numberof unit cell m(1). It increases with both g(1) and m(1), asshown in Fig. 6, and saturates to

estretchability ¼eyield lð1Þ

w4½g

ð1Þ�3þ6ðp�2Þ½gð1Þ�2�12ðp�3Þgð1Þ þ9p�28

12gð1Þ

ð39Þ

for m(1)!1 (also shown in Fig. 6).

1 2 3 4 5 6 7 80

10

20

30

40

50

60 Analytic FEA

ε stretchability

m(2)=10m(2)=4m(2)=2m(2)=1

Height/spacing ratio (η(2))

…

( ) ( )1 18, 2m η= =

Fig. 7. The dimensionless stretchability of the second order serpentineinterconnect vs. the height/spacing ratio (g(2)) for different number (m(2))of unit cells, with (m(1), g(1)) = (8, 2).

For higher order (n P 2) serpentine interconnects,Eq. (35b), together with the corresponding flexibilitymatrix T ð2Þ in Eq. (32) and T ðnÞ in Eq. (30) for serpentineinterconnects, give an excellent approximation to thestretchability as compared to the FEA shown in Fig. 7.

Fig. 8 shows the normalized stretchability, estretchabilityw/[eyieldl(n)], vs. the order n for self-similar rectangular andserpentine interconnects, where the height/spacing ratiog ¼ 8=

ffiffiffiffiffi11p

and number of unit cell m = 4 at differentorders. The stretchability is more than doubled for each n

increasing by 1, indicating the elastic limit of the intercon-nect can be well improved by adopting higher order self-similar design. Fig. 8 also shows that the analytic modelagrees very well with the FEA results.

The analytic models and FEA results above are all forinfinitesimal deformation. Table S.1 (Electronic Supple-mentary material) shows that the effect of finite deforma-tion on stretchability (determined by FEA) is negligiblefor both first and second order serpentine interconnects,with various combinations of geometric parameters. There-fore, the analytic models above give good estimations ofthe stretchability. In real fabrications, the microscale self-similar serpentine interconnect may have imperfectionsdue to lithography defects, especially along the sidewallsof the lines, and such geometric imperfections will increasefor decreased pattern size (i.e. metal width and roundingradius) that may occur when increasing the self similarorder. These geometric imperfections are not accountedfor in the present study.

5. Optimal design of self-similar serpentine interconnects forstretchable electronics

Two competing goals of stretchable electronics [19,37]are (1) high surface filling ratio of active devices, whichrequires small spacing between the device islands(Fig. 9a); and (2) large stretchability of the system, whichdemands large spacing between the device islands. Priorapproaches based on buckling of straight or conventionalserpentine interconnects achieve �100% stretchability

1 2 3 40

20

40

60

80

100

Analytic FEA

a

b

ε stretchability

(%)

Number (m(2)) of unit cell

m(2)=4, m(1)=16

H

H/2 H/2L

d

2rrounding

w

Fig. 9. Design optimization of the second order serpentine interconnects for island–bridge structure. (a) Schematic of the island–bridge structure with a8 � 8 array, and illustration on the geometric parameters; (b) the maximum stretchability vs. the number (m(2)) of unit cells (left panel), and the optimizedconfiguration (right panel).

7826 Y. Zhang et al. / Acta Materialia 61 (2013) 7816–7827

[17,18,28,30]. The stretchability (esystemstretchability) of the system is

related to that (einterconnnectstretchability ) of the interconnect by

esystemstretchability ¼ einterconnnect

stretchability 1�ffiffiffif

p� �ð40Þ

where f denotes the surface filling ratio. For �50% surfacefilling ratio of active devices, the �100% stretchability ofthe interconnect translates to �30% stretchability of thesystem, which is low for some biomedical applications ofstretchable electronics (to skin, heart or elbow). The ana-lytic models in Sections 3 and 4 can guide the design of gen-eralized self-similar interconnects to simultaneouslyachieve the two competing goals above.

The second order serpentine interconnects are studied toillustrate the design optimization in a square-shaped deviceisland with a representative size H = 1 mm and the surfacefilling ratio of 50% (Fig. 9a). The photolithography tech-nology [38,39] for fabricating the metal interconnect posessome constraints, such as the width w P 10 lm, roundingradius rrounding P 10 lm and the distance between neighbor-ing arcs d P 5 lm (Fig. 9a). Other geometric parametersare optimized to achieve large stretchability. Fig. 9b showsthat the stretchability increases with the number of unitcells m(2). The right panel of Fig. 9b shows the optimal

design, which gives �308% stretchability of the intercon-nect, and corresponds to �90% stretchability of the system,outperforming the previous designs using buckled intercon-nects [18,28]. Even for a much larger surface filling ratio70%, Eq. (40) still gives �50% stretchability of the system.

6. Conclusions

This paper develops the analytic models of flexibilityand stretchability for the self-similar interconnects. Afterthe straightforward design optimization, the analytic mod-els, validated by FEA, show that the higher order self-sim-ilar interconnect gives very large stretchability of thesystem, such as �90% for 50% surface filling ratio of activedevices, or >50% stretchability for 70% surface filling ratio.The analytic models are useful for the development ofstretchable electronics that simultaneously demand largeareal coverage of active devices, such as stretchable photo-voltaics [11] and electronic eyeball cameras [12]. The con-cept of self-similar serpentine configuration could befurther combined with other strategies of stretchabilityenhancement, e.g. the control of wrinkling patterns, to givean enhanced level of stretchability for interconnectsbonded to the substrate.

Y. Zhang et al. / Acta Materialia 61 (2013) 7816–7827 7827

Acknowledgements

Y.H. and J.A.R. acknowledge the support from NSF(DMR-1242240). K.C.H. and Y.H. acknowledge the sup-port from NSFC.

Appendix A. Supplementary material

Supplementary data associated with this article can befound, in the online version, at http://dx.doi.org/10.1016/j.actamat.2013.09.020.

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