ARTICLE

Received 13 Dec 2015 | Accepted 1 Dec 2016 | Published 20 Jan 2017

The shape of telephone cord blistersYong Ni1, Senjiang Yu2, Hongyuan Jiang1 & Linghui He1

Formation of telephone cord blisters as a result of buckling delamination is widely observed in

many compressed film-substrate systems. Here we report a universal morphological feature

of such blisters characterized by their sequential sectional profiles exhibiting a butterfly shape

using atomic force microscopy. Two kinds of buckle morphologies, light and heavy telephone

cord blisters, are observed and differentiated by measurable geometrical parameters. Based

on the Foppl-von Karman plate theory, the observed three-dimensional features of the

telephone cord blister are predicted by the proposed approximate analytical model and

simulation. The latter further replicates growth and coalescence of the telephone cord into

complex buckling delamination patterns observed in the experiment.

DOI: 10.1038/ncomms14138 OPEN

1 CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics, University of Science and Technology of China,Hefei, Anhui 230026, China. 2 Department of Physics, China Jiliang University, Hangzhou 310018, China. Correspondence and requests for materials shouldbe addressed to H.J. (email: jianghy@ustc.edu.cn).

NATURE COMMUNICATIONS | 8:14138 | DOI: 10.1038/ncomms14138 | www.nature.com/naturecommunications 1

Detailed origins of postbuckling phenomena have beena subject of discussions for several decades1. The bucklesusually involve multiple instabilities and exhibit

intriguing morphologies2,3. When a film deposited on a subs-trate is subject to a large residual compression, it bucklesand delaminates away from the substrate, and the blisterswith straight-sided, circular, telephone cord (TC) or network-like patterns are formed4,5. Predicting the shape of such blisters isof scientific interest for stress-driven pattern formation andimportant for avoiding structural failure in various film-substratesystems4. Among various manifestations of this phenomenon,the TC blister is the most frequently observed in many differentfilm-substrate systems5. This motivates a search to identifyuniversal features of TC blisters. Recently, TC formation hasbeen considered as a development from a secondary instabilityof straight-sided or circular blisters. The instability conditionhas been analytically addressed by the Foppl-von Karman(FvK) plate model6–9. Further, it has been found that theTC blister can also grow with the appearance of sags by anoscillatory process and its edge unnecessarily starts straight10,11.So far, a convincing solution for the three-dimensional(3D) TC blister morphology remains a challenge because oflimited methods for handling nonlinear elasticity of thin plate,although a series of numerical models and experiments haveprovided valuable insights into the morphological features ofTC blisters10–25.

The elegant instability analysis of a straight-sided blisterpredicted a constant undulation-period-to-width ratio of theTC buckle under the clamping boundary conditions close tothe experimental observation8. In addition, the ratio may benot constant and change with the interfacial adhesion andthe film-to-substrate modulus ratio11–16. The zigzag undulationfeature of the TC buckle has been replicated by nume-rical investigations5,15–19. A comprehensive study for the3D TC morphology suggested that it could be approximatelyviewed as a sequence of connected segments of a circularbuckle pinned at its centre12. However, such approxima-tion predicted the ridge of the TC blister to be discontinuous,different from our following refined experimental observationswherein the ridge is continuous and its height is periodicallychanging.

Here we aim to probe the 3D features of the TC morphologyusing a combination of analytical model, numerical simulationand atomic force microscopy (AFM) study. As shown in Fig. 1a,b,two typical TC blister shapes, light and heavy TC blisters, canbe observed in compressed SiAlNx films on glass substrates.Our experimental observation based on the optical imagesin Fig. 1a,b indicates that the projected area of the delaminationzone of the TC blister is similar to the area swept by a segment ofwidth 2b perpendicularly to a sinusoidal centreline. We thenassume that the shape of the TC blister can thus be modelledas the postbuckling morphology of the thin plate clampedalong such delamination boundary under equal biaxial residualcompression (Fig. 1c). An approximate analytical solution forthe shape of the TC blister is obtained in the following part.We argue that the shape of the TC blister can be characterizedby several sectional profiles perpendicular to the centrelineof the delamination zone (Fig. 1c). In contrast to the fact thatthe sectional profile of the straight-sided blister is alwayssymmetric, the corresponding profile of the TC blister becomesasymmetric, dependent on the wavy amplitude of the centreline.The sequential sectional profiles of the TC blister exhibita butterfly shape. Such geometric feature becomes moresignificant in the case of larger waviness amplitude of thecentreline, corresponding to a transition from the light TC blisterto the heavy TC blister.

ResultsAnalytical solution to the 3D profile of TC blisters. We firstlydetermine the 3D shape of the TC blister by solving theFvK equations for a plate of thickness h clamped alongthe delamination boundary in a curvilinear coordinate (Fig. 1c).The centreline of the delamination zone is described byr0 ¼ x;A sin 2px=lð Þ; 0½ �, where A and l are the amplitude andthe wavelength of the wavy centreline, respectively. The shape ofthe TC buckles reflected by the profile of the mid-plane of thethin plate after deformation can be expressed as

r ¼ x� xþ uð Þ sin y;A sin 2px=lð Þþ xþ uð Þ cos y;w½ � ð1Þwhere y (x) is defined as the angle between the tangential ofthe centreline and x axis, (s, x) is a curvilinear coordinate, u(s, x)is the displacement along x direction and w(s, x) is the deflectionof the plate. Here we have assumed that the displacementchange along the S direction is negligible compared to u(s, x)and w(s, x). The edges of the TC buckles are clamped at x¼±b.We also assume that the in-plane displacement is small, thatis, u � x, u dy

ds � @u@x, and the out-of-plane deflection is relatively

large. Based on the observed morphology of the TC buckles,we make another important assumption that u and w changeslowly in the S direction, that is, @u

@s � @u@x and @w

@s � @w@x.

We derive the FvK equations with the boundary condition inthe curvilinear coordinate by minimizing the total elastic energyin the plate (see the details in the Supplementary Note 1 forthe analytical solution to the TC buckle). The strain energy of theTC buckle within a period can be written as

F ¼Z s0

0

Z b

� bfffiffiffigp

d xds ð2Þ

where g is the determinant of the metric tensor gab and thereforeffiffiffigp

dxds is the element of the area, f is the elastic strain energy

b

b

Centreline

Centreline

Substrate

s�

w

h b1 b2

z xy

B-line

C-line D-line

�m

a b

c

Figure 1 | Schematic model for the shape of telephone cord blisters.

Fitting the projected area of (a) a light TC blister and (b) a heavy TC blister

observed in SiAlNx films on glass substrates. (c) Sketch of the FvK model in

a curvilinear coordinate under the clamping boundary condition along the

boundary of the projected area. Scale bars in (a) and (b) are 30 and 50mm,

respectively.

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14138

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density per unit area having the form

f ¼ h2

Dabrmgabgrmþh3

24DabrmDkabDkrm ð3Þ

where gab, Dkab and Dabrm are defined as the mid-plane Lagrangestrain tensor, the curvature change tensor due to deformationand the tensor of the elastic constants in the curvilinear coordi-nate, respectively. They can be explicitly expressed as a functionof the variables s, x, u and w. Here the prime denotesdifferentiation with respect to x. If we only keep the leading orderterm of u0 þ 1

2 w0ð Þ2, the element of areaffiffiffigp � 1� xy;s with

y;s ¼ �Aq2 sin qxð Þ1þA2q2 cos2 qxð Þ½ �3=2. The arc length of the centreline s(x) is

given by ds ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þA2q2 cos2 qxð Þ

pdx, with q¼ 2p/l. We also

assume that s¼ 0 at x¼ 0 and s¼ s0 at x¼ 2p/q. FromdF=du ¼ 0, we have the in-plane equilibrium equation of theplate

t0a ¼ 0 ð4Þwhere ta ¼ �Ef h

ffiffiffigp

1þ nð Þemþ u0 þ 12 w0ð Þ2

� �is the membrane

stress along the direction of x in the film. �Ef ¼ E1� n2 is plane strain

modulus with E, v the Young’s modulus and the Poisson’s ratio ofthe plate, respectively. em is the uniform equibiaxial residualstrain in the plate. Equation (4) indicates the membrane stress ta

should be a constant. Similarly, from dF=dw ¼ 0 we obtain theout-of-plane equilibrium equation of the plate

1� xy;s� �

w00� �00 � lw0 þ

y2;s

1� xy;sw0

" #0¼ 0 ð5Þ

where l ¼ 12ta�Ef h3. By solving equations (4) and (5) as an eigenvalue

problem, we get an approximate solution to the deflection of theplate up to the first order

wðx; xÞ � w012

1� 14y;sx

� �1þ cos

pbx

h iþ p

4by;s b2� x2� �

sinpbx

� �

ð6Þwhere w0 is an unknown constant. Equation (6) is reduced towðx; xÞ ¼ 1

2 w0 1þ cos pb x� �� �

for the buckling of a straight stripwith uniform width (y,s¼ 0) (ref. 6). Therefore, w0 representsthe maximum deflection of the plate when y,s¼ 0. Note thatta ¼ �

�Ef h3

12pb

� �2at y,s¼ 0. Integrate this equation from � b to b,

and notice that u¼ 0 at x¼±b, one can get

w20 ¼ �

640 12b2em 1þ nð Þþ h2p2½ �480p2þ � 465� 110p2þ 16p4ð Þb2y2

;s

ð7Þ

The characteristic shape of TC blisters. Equation (6) togetherwith equation (7) provides for the first time an approximateanalytical solution to the 3D profile of the TC buckle after thespecific delamination area is available. To test whether thesolution can predict the 3D feature of the TC blister shape ornot, Fig. 2 plots several sectional profiles of a buckled tantalum(Ta) film on glass substrates perpendicular to the centrelineshown in the inserted modified AFM image with the sizeof 40 mm� 40 mm. The result in Fig. 2 shows that the sequenceof the sectional profiles of the TC blister perpendicular to thecentreline exhibits a butterfly shape reminiscent of the butterflycurve of strain versus applied load in shape memory alloys.The asymmetry is characterized by two different half-separationsdenoted by b1 and b2 based on the ridge line of the TC blister(see Fig. 1c). The asymmetry has a maximum in the profile alongB-line or D-line, while the profile along the C-line is symmetric.In addition, the maximum deflection in each sectional profilecorresponding to the point at the ridge line is not constant. Ithas the smallest value in the profile along the C-line and the

largest value in the profile along the B-line or D-line. Thisgeometric feature of the TC blister is different from the previousreport12 wherein the ridge line is viewed as a discontinuouscontour line. Figure 2 demonstrates that the theoretical resultobtained by equation (6) predicts the asymmetric 3D featureof the TC blister, and the deflection amplitude variation of theTC ridge line. This is consistent with the AFM measurementsafter the measured parameters of the buckled Ta film is givenas h¼ 225 nm, b¼ 10mm, l/b¼ 2, A/l¼ 0.08, and the residualstrain in the Ta film is assumed to be em¼ 0.004. The deflectionamplitude of the ridge line given by equation (6) is under-estimated because of the assumption of rigid substrate andclamping boundary condition. In fact, the substrate compliancecould increase the maximum deflection26. We further numeri-cally track the postbuckling morphology of the compressed plateon the substrate by setting the same predelamination zone, thesame residual stress and the same other material parameters asthose in the case of the analytical solution. The film-to-substratemodulus ratio is set to be 2.5 (about 70 GPa for the glass substrateand 175 GPa for Ta film) based on the continuum model15. Thenumerical result in Fig. 2 matches the experimentally observedmorphological feature of the TC blister even more closely.

The approximate solution in equation (6) also predicts theasymmetry denoted by b1/b2 and the variation of the maximumdeflection at the ridge of the TC blister denoted by wmax

C�line=wmaxB�line

as a function of the waviness of the centreline denoted by A/l. Inthe limit of A/l¼ 0, b1/b2¼ 1 and wmax

C�line=wmaxB�line ¼ 1, it is the

case of a straight-sided blister. The stability analysis for thestraight-sided blister only predicted the onset stage duringformation of the TC blister, namely, the amplitude of theundulation is infinitely small corresponding to A/l-0 (ref. 8).The current solution can apply to the case with finite value of A/l.Figure 3 shows that different undulatory shapes of the TC blisterscan be described by two measurable geometrical parameters b1/b2

and wmaxC�line=wmax

B�line as a function of A/l with the comparisonbetween theory, simulation and experimental observations invarious films including Ta, SiAlNx, Fe and Ni on glass substrates.With the increase of A/l, the values of b1/b2 and wmax

C�line=wmaxB�line

more significantly deviate from one, corresponding to the largerasymmetry. However, we must note that the approximateanalytical solution is reliable only when the value of A/l is very

1,200

1,000

800

600

400

z (n

m)

200

0

4 8 12 16� (µm)

B-line

B

D

C

C-line

D-line

Open symbol: ExperimentSolid symbol: Simulation

Line: Theory for rigid substrate

20 24

Figure 2 | Characterization of the TC blister shape. Several sectional

profiles perpendicular to the centreline are shown in the inserted modified

AFM image 40mm�40mm, which is updated by manually performing

planefit offline of the original AFM image obtained by line-by-line

(256 lines) scanning modes. The results from theory, simulation and

AFM measurements are compared in a Ta film on glass substrates.

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small since it proceeds by an asymptotic expansion in y,s (see thedetails in the Supplementary Note 1 for analytical solution tothe TC buckle). From equation (5) it indicates that the solutionmay be questionable since there is a singularity as 1� xy;s¼ 0.The singularity disappears when A=lo1= 2p2ð Þ ¼ 0:05 at l¼ 2b.For the larger value of A/l, the numerical solution15 is more closeto the experimental data, as shown in Fig. 3. It is found that thevariations of b1/b2 and wmax

C�line=wmaxB�line tend to be saturated,

respectively, after A/l40.08, in contrast to the monotonousdeviation from one predicted by the approximate analyticalsolution. The result in Fig. 3 thus demonstrates that b1/b2 andwmax

C�line=wmaxB�line are another two measurable geometric parameters

to characterize the undulatory morphology of the TC blisterbeside of l/b and A/l. The dependence of b1/b2 andwmax

C�line=wmaxB�line on A/l is attributed to the fact that the wavy

centreline of the delamination zone breaks the symmetry

with respect to the straight centreline and introduces aposition-dependent curvature as indicated in SupplementaryEquation (10) in the buckled film wherein the buckling across thecentreline may be not symmetric anymore.

Light and heavy TC blisters. Figure 4 demonstrates that there aretwo kinds of buckle morphologies, light and heavy TC blisters,differentiated by the value of A/l. The upper row in Fig. 4 isthe experimental observation in Ta and SiAlNx films onglass substrates, respectively. The lower row in Fig. 4 is thesimulated postbuckling morphology given b/h¼ 49, l/b¼ 2 andem ¼ 0:005 at different values of A/l. It is found that with theincrease of A/l, the buckle really turns out to be ubiquitouslyundulated from light to heavy TC blisters. The obtained resultis consistent with the experimental observation shownin Supplementary Movie 1, where an initially straight blisterprogressively grows into the familiar TC structure. We believethat the transition point between the light and heavy TC blistersdepends on the value of A/l. In the light TC blister, the value ofA/l is very small, the width of the delamination area is close to2b, the delamination boundary is smooth and the values of b1/b2

and wmaxC�line=wmax

B�line monotonously deviate from one. In contrast,in the heavy TC blister, the value of A/l is much larger, thedelamination zone significantly widens, the values of b1/b2

and wmaxC�line=wmax

B�line are saturated, and the delamination boundaryshows a cusp indicating the existence of a singularity(see Supplementary Fig. 3).

In our analytical approach, l/b and A/l extracted fromthe projected area of the TC blister are assumed to be the inputparameters instead of the prediction. Based on the assumption,the equilibrium values of b1/b2 and wmax

C�line=wmaxB�line that quantify

the typical 3D features of the TC blister are determined bythe postbuckling solution given delamination area. We canobtain the values of l/b and A/l from the experimentalobservation. The measurement of l/b is straightforward12.While the measurement of A/l could be obtained by trackingthe undulation of the ridge line denoted by A0=l, which valuecould be roughly viewed as that of A/l (Fig. 1a,b). The results

1.2 Theory 1.2

1.0

0.8

0.6

ExperimentSimulation

1.0

0.8

b 1/b

2

0.6

0.0 0.1 0.2

A/�

0.3

wm

ax /wm

ax

C

-lineB

-line

Figure 3 | Measurable geometrical parameters. b1/b2 and wmaxC�line=wmax

B�line

as a function of A/l characterize different undulatory shapes of the

TC blisters: comparison between theory, simulation and experimental

observations in various films including Ta, SiAlNx, Fe and Ni on glass

substrates.

Light TC blister

5

4

3

2

1

0A/�= 0.04 A/�= 0.08 A/�= 0.4

Heavy TC blister

a

d e f

b c

Figure 4 | Morphologies of light and heavy TC blisters. (a–c) AFM images. Scale bars are 10mm. (d–f) Numerical simulation results with parameters

b/h¼49, l/b¼ 2, em¼0.005, and A/l¼0.04, 0.08 and 0.4.

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shown in Supplementary Fig. 2 indicate that both valuesof l/b and A0=l in various observed TC blisters tend to besaturated and fall into a narrow range. The value of the former isbetween 1.6 and 2.8, consistent with the report in the literature16.The value of the latter is between 0.1 and 0.3. Since the TC blistercan be developed from a secondary buckling instability ofa straight-sided blister, where the instability mode predictsl/bE2 given the parameters b and A¼ 0 (ref. 8), we believe thatthe waviness period of the TC blister may be inherited from theinstability wavelength. The deviation from l/bE2 is attributedthat l/b also depends on the adhesion with a non-trivialrelationship14,16. Usually the increase of A/l releases more theelastic energy in the biaxially compressed film deposited on thesubstrate, while it has to increase the mixed-mode-dependentadhesion energy. The competition between them sets theequilibrium value of A/l. In addition, large value of A/l in theTC blisters is hard to report. It is possible because the case with alarge value of A/l definitely increases the stress concentrationwhich may lead to ridge crack20 or further buckling bifurcation,and the buckle is not TC blister any more15.

Up to now, our modelling to the shape of the TC blisteris obtained under the particular assumption on the shape ofthe delamination region. In fact, the delamination area isunnecessary to be sinusoidal shaped, especially during theoscillatory growth process of the TC blister11. Usually,determination of the buckle–delamination morphology can beseparated into two steps. The first step is to derive a postbucklingsolution given the delamination zone. The second step is todetermine the delamination zone at equilibrium. If the edge ofthe delamination zone is described as the interfacial crack front,the delamination zone at equilibrium is determined asthe buckling-mediated energy release rate is equal to theinterface toughness. For the straight-sided blister, both steps aredone analytically6. However, the analytical solution to the secondstep for the TC blister is not available.

In fact, both the equilibrium delamination area and postbuck-ling morphology in thin films deposited on substrates can alsobe theoretically determined in principle. If we further takeinto account the contribution of the adhesion energy and theelastic strain energy in the substrate, all the equilibrium valuesof l/b, A/l, b1/b2 and wmax

C�line=wmaxB�line that quantify the TC blister

shape with less restriction can be numerically determinedby minimizing the total free energy in the film-substrate system.Our current numerical simulations rely on the recently developedcontinuum modelling to track the general morphologicalevolution of the buckle delamination without any restrictionof the delamination zone15. In this approach, the concurrentbuckling and delaminating processes are formulated usingthe time-dependent Ginzburg–Landau kinetic equations, drivenby minimizing the film-substrate total free energy, includingthe elastic energies in both the film and the substrate, andthe mixed-mode interfacial adhesion between them11,15.The effect of substrate elasticity and interfacial adhesion on theshape of the TC blister can be taken into account, which isneglected in the analytical approach. The modelling andsimulation approach is outlined in Supplementary Note 3 andsome results are shown in Supplementary Figs 4–6.The coupling behaviour between buckling and delaminationin a film deposited on substrates with higher compressive stressesbecomes more complex20,22, and the TC blister may exhibitbeyond the sinusoidal configuration. Our numerical simula-tions not only recover the growth process of the TC blisterfrom an initially circular blister7 but also capture a richcoalescence behaviour accompanied with the increase of thebuckling width during further propagation of the TC buckle.The TC buckle becomes larger and larger with the appea-

rance of several spikes and/or daughter TC buckle at theouter undulated edge, consistent with our experimentalobservation (see Supplementary Fig. 6 and SupplementaryMovies 2 and 3).

DiscussionIn summary, the refined 3D morphological features of theTC buckle are elucidated by using AFM characterization,approximate analytical model and numerical simulations.We confirm that the shape of the TC blister can be modelledas the postbuckling morphology of the compressed plate clampedalong the delamination front given the parameters l/b andA/l. Two measurable geometrical parameters b1/b2 andwmax

C�line=wmaxB�line are proposed to characterize a so-called ‘butterfly

shape’ of the sequential sectional profiles, which are universalin both the light and heavy TC blisters. The above featuresare successfully reproduced by our approximate analyticalmodel and numerical simulations. Furthermore, how the fullynonlinear buckle-driven delamination process leads to themorphological evolution of the TC blister is captured bynumerical simulations and experimental observation. The presentwork provides insight into the 3D shape of TC buckles.

MethodsFormation of the TC buckle as a result of the buckle-driven delaminationprocess was simulated using the phase field method where the film buckles intoan equilibrium buckle–delamination configuration driven by minimizing the totalfree energy. The total free energy of the film-substrate system is established byincorporating Green function method for the substrate elasticity, FvK plate theoryfor nonlinear film deformation and cohesive zone model for mixed-modeinterfacial adhesion. The total free energy of the film-substrate system including thefilm, the substrate and the interface can be expressed as a functional of the out-of-plane displacement of the film z(x, t) and the displacement jump vector across theinterface L x; tð Þ as U tot ¼ U film þU sub þU int; where Ufilm is the elastic strainenergy including the bending and stretching energies in the film, Usub is the elasticenergy of the substrate and Uint is the adhesion energy between the film and thesubstrate. Their detailed expressions are found in ref. 15. Following ref. 11, we mayalso adopt alternative cohesive zone model with bilinear traction versus separationlaw for better description of mixed-mode dependence of interface adhesion. Thedynamic equations for z;Li denoted as the time-dependent Ginzburg–Landaukinetic equations to describe the minimization process of the total free energy aregiven by: @z=@t ¼ �GzdU tot=dz and @Li=@t ¼ �GLidU tot=dLi , where t denotestime and Gz , GLi are the kinetic coefficients that characterize the relaxation rates ofthe buckling and delamination processes in the overdamped dynamics. Thisgradient flow form of the dynamics guarantees a monotonous minimizationprocess of Utot and ensures a convergent solution of the dynamic equations,whose steady-state solutions provide the configuration at the equilibrium. We solvethe reduced forms in a computational cell with periodic boundary conditionsusing input parameters: e0

ab, mf =ms, vf ¼ 0:3, vs ¼ 0:5, G�Li¼ G�Li

=Gz ¼ 0:1,dn ¼ dt ¼ 0:2h, g�n ¼ gn=ednms and g�t ¼ gt=ednms . The grid spacing isDx ¼ Dy ¼ h and the time step is Dt ¼ 0:1t=t, with t ¼ h=Gzms. A small randomfluctuation mimicking the thermal fluctuation is used to facilitate nucleation ofbuckling delamination from the pre-existing interfacial delamination region, inwhich zero interface toughness is assumed.

Data availability. The data that support the findings of this study are availablefrom the corresponding author on request.

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AcknowledgementsThis work was supported by the National Natural Science Foundation of China(Grant Nos 11222219, 11472262, 11132009, 11472271, 11622222 and 11204283), theCollaborative Innovation Center of Suzhou Nano Science and Technology, theFundamental Research Funds for the Central Universities (Grant Nos WK2090050027and WK2480000001) and the Strategic Priority Research Program of the ChineseAcademy of Sciences (Grant No. XDB22040502 and XDB22040403).

Author contributionsY. Ni, S. Yu, H. Jiang and L. He initiated and supervised the project. H. Jiang conceivedthe theory. Y. Ni performed phase field simulation. S. Yu performed AFM measurement.All authors analysed the data and wrote the manuscript.

Additional informationSupplementary Information accompanies this paper at http://www.nature.com/naturecommunications

Competing financial interests: The authors declare no competing financialinterests.

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How to cite this article: Ni, Y. et al. The shape of telephone cord blisters. Nat. Commun.8, 14138 doi: 10.1038/ncomms14138 (2017).

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r The Author(s) 2017

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14138

6 NATURE COMMUNICATIONS | 8:14138 | DOI: 10.1038/ncomms14138 | www.nature.com/naturecommunications

1

Supplementary Figure 1. sketch of the FvK model in a curvilinear coordinate under

the clamping boundary condition along the boundary of the projected area.

substrate

m

h

B-line

C-line D-line

1b 2b

z

s

centerline

wxy

2

(a)

(b)

Supplementary Figure 2. Plot of /A and / 2b as a function of in

various TC blister buckles

3

Supplementary Figure 3. Plot of the side undulation curve under different

values of /A

0 50 100 150 20050

100

150

200

A/A/

A/

A/

x/h

b/h

2b

4

Supplementary Figure 4. The simulated contour map of the post-buckling

morphology under different value of /A from 0 to 0.38 with the increase of 0.02

0 0.02 0.04 0.06 0.08

0.1 0.12 0.14 0.16 0.18

0.2 0.22 0.24 0.26 0.28

0.3 0.32 0.34 0.36 0.38

5

Supplementary Figure 5. Effect of the modulus ratio between film and

substrate on the shape of TC blister

2f

s

5

f

s

20

f

s

6

Supplementary Figure 6. The growth of a TC blister obtained by numerical

simulation

7

Supplementary Note 1: Analytical solution to the telephone cord buckle

As shown Supplementary Figure 1, the centerline of the telephone cord can be given

by 0 , sin 2 / ,0x A x r where A and are the amplitude and wavelength

of the centerline, respectively. The tangent vector of the centerline is

0 1, cos ,0d

Aq qxdx

r

with 2 /q . Therefore, the distance between the two

nearby points is

2 2 2 2 20 0 1 cosd d

ds A q qx dxdx dx

r r

(S.1)

Define the angle ( )x between the tangential of the centerline and x axis so that

2 2 2

2 2 2

1cos

1 cos

cossin

1 cos

dx

ds A q qx

Aq qxdy

ds A q qx

(S.2)

Therefore, the shape of the telephone cord buckles can be described as

sin , sin cos ,x u A qx u w r (S.3)

where ,s is a curvilinear coordinate and ,u s is the displacement along

direction and ,w s is the deflection of the plate. Here, we have assumed that the

displacement along the s direction is negligible compared to ,u s and ,w s .

The edge of the telephone cord buckles are clamped at b . By considering Eq.

(S.2), the local tangential vectors can be written as

1

2

1 cos sin , 1 sin cos ,

1 sin , 1 cos ,

d u d u wu u

s ds s ds s s

u u w

rm

rm

(S.4)

In this work, we assume the in-plane displacement is small, i.e., u and

8

d uu

ds

, and the out-of-plane deflection is relatively large. From the morphology

of telephone cord buckles, we make another important assumption that u and w

change slowly in the s direction, i.e., u u

s

and

w w

s

. Therefore, the two

tangential vectors can be simplified to

1 , ,

2

1 cos , 1 sin ,0

1 sin , 1 cos ,

s s

u u w

m

m (S.5)

where ,s

d

ds

,

uu

and

ww

. Here the prime denotes differentiation with

respect to . The metric tensor in the deformed film is given by

2

,

2 2

1 0

0 1

sg

u w

m m (S.6)

Similarly, we can calculate the tangential vectors and therefore obtain the metric

tensor in the undeformed film as

2

,1 0

0 1

sg

(S.7)

The mid-plane Lagrange strain tensor defined as

2

,

2

1 01

12 02

m s

m

m

g g gu w

(S.8)

where m is the uniform equi-biaxial mismatch stress. Here we have used the

assumption that 1u . The corresponding mixed component of the mid-plane

Lagrange strain tensor is

2

0

10

2

m

m

gu w

(S.9)

9

where g is the inverse of the metric tensor in the undeformed film, i.e.,

g g

. The curvature tensors in the deformed and undeformed film are

, ,1 0

and 00

s sw

w

(S.10)

Therefore, the curvature change tensor is

, ,1 0

0

s sw

w

(S.11)

So the Lagrange strain tensor is defined as

3E x (S.12)

with two nonzero components as

2

11 , 3 , ,

2

22 3

1 1

1

2

m s s s

m

E x w

E u w x w

(S.13)

where 3x is the distance from the mid-plane of the film. The tensor the elastic

constants can be defined as

21 2

2(1 )

ED g g g g g g

(S.14)

where E and are the Young's modulus and Poisson's ratio, respectively. The

three nonzero components of D are

211

1111

42 2

,

222

2222

2 2

11 221122

22 2

,

1

1 1 1

1 1

1

1 1 1

s

s

E g ED

E g ED

Eg g ED

(S.15)

The stress-resultant tensor and the internal moment tensor are

10

3

12

T hD

hM D

(S.16)

Therefore, the elastic strain energy density per unit area is

3

22 22

2

23

2, ,

22

,,

2 24

1 12

2 1 2 2

2

24 1 11

m m m m

s s

ss

h hD D

h Eu w u w

w w wh Ew

(S.17)

The stain energy of the telephone cord buckles within a period is

0

0

s b

bgd ds

(S.18)

where g is the determinant of the metric tensor g and therefore gd ds is the

element of the area. If we only keep the leading order term of 21

2u w , the

element of area ,1 sg . The arc-length of the centerline s x is given by

2 2 21 cosds A q qx dx . We also assume that 0s at 0x and 0s s at

2 /x q . From the principle of minimum potential energy, we can obtain the

equilibrium equations. From / 0u , we have

0at (S.19)

where 21

12

a f mt E h g u w

is the in-plane normal force in the film.

21f

EE

is plane strain modulus. Eq. (S.19) indicates the in-plane normal force

at should be a constant. Similarly, from / 0w we obtain another differential

equation

2

,

,

,

1 01

s

s

s

w w w

(S.20)

11

where 3

12 a

f

t

E h . Apparently, this is an eigenvalue problem. Here, we seek a

perturbative solution to Eq. (S.20) of the form

(0) (1)

,

(0) (1)

,

n n s n

n n s nw w w

(S.21)

where n and nw are the n-th eigenvalue and eigenfunction of Eq. (S.20).

Substituting Eq. (S.21) into Eq. (S.20) and comparing the power of ,s gives the

following sequence of equations:

4 (0) 2 (0)

(0)

4 20n n

n

w w

(S.22)

4 (1) 2 (1) 2 (0) 2 (0)2(0) (1)

4 2 2 2 2

n n n nn n

w w w w

(S.23)

with boundary conditions

( )

( ) 0 and 0 ati

i nn

ww b

(S.24)

Eq. (S.22) is the zeroth order approximation of the problem. In this case, the

centerline is a straight line (, 0s ). The eigenvlaue and eigenfunciton of Eq. (S.22)

are

2

(0)

n

n

b

(S.25)

1(0) 1

1 cos3

n

n

nw

bb

(S.26)

(0)

nw form an orthonormal basis, i.e.,

(0) (0)

1 ( )

0 ( )

b

m nb

m nw w d

m n

(S.27)

By multiplying Eq. (S.23) by (0)

mw and integrating it from b to b , one can find

12

2 (0)2(0)

2 2

(1)

2 (0)(0)

2

0

bn

nb

nb

nn

b

ww d

ww d

(S.28)

when m n . By substituting (1) 0n into Eq. (S.23) and solving the equation

directly, one can get

(1) 2 2 (0)

0

11 cos sin

4 3

n

n n

n n nw b C w

b b bb

(S.29)

where 0C is an unknown constant. By multiplying Eq. (S.23) by (0)

mw ( m n ) and

integrating it from b to b , one can find

2 (0)

(0) (0) (1)

20

bm

n m nb

ww d

(S.30)

Notice that (0) (0)

n m when m n . Therefore, we get 2 (0)

(1)

20

bm

nb

ww d

, which

indicates 0 0C . Therefore, when 1n , we get an approximation of the deflection

of the plate as

2 2

0 , ,

1 1( , ) 1 1 cos sin

2 4 4s sw x w b

b b b

(S.31)

where 0w is an unknown constant. Eq. (S.31) is reduced to

0

1( , ) 1 cos

2w x w

b

for the buckling of a straight strip with uniform width

(, 0s ). Therefore, 0w represents the maximum deflection of the plate when

, 0s .

Notice that 23 3

2 111

2 12 12

f f

a f m

E h E ht E h g u w

b

. Integrate

this equation from b to b , and notice that 0u at b , one can get

13

2 2 2

2

0 2 2 4 2 2

,

640 12 1

480 465 110 16

m

s

b hw

b

(S.32)

Eq.(S.31) together with Eq.(S.32) provides an analytical solution to the 3D profile of

the TC buckle after its projected area is available.

Supplementary Note 2: Measurable /A and / b in various TC blisters

shown in Supplementary Figure 2

We have measured the values of /A and / b in various TC blisters

according to the optical or AFM images reported in the literatures [1-27]. The values

of /A and / b for each kind of materials are estimated as the average by

several measurements.

Supplementary Note 3: Method to simulate evolution of the telephone cord

buckle

Supplementary Figure 3 shows the side undulation curve under different values

of /A at b/h=49, 2/ b to set the pre-delamination area in our numerical

simulation.

We perform the numerical simulation for the effect of /A on the

post-buckling morphology of the compressed plate on the substrate given the

parameters0

m , 0.005m / 2.5f s , 0.3fv , 0.5sv ,

* / 0.01i i , b/h=49, 2/ b , as shown in Supplementary Figure 4. The

simulated result in Figure 3 about the dependence of 1 2/b b and max max/C line B linew w on

/A is extracted from the data in Supplementary Figure 4.

Supplementary Figure 5 shows the effect of modulus ratio between film and

14

substrate on the post-buckling morphology of the compressed plate on the substrate

given the parameters 0.005m , 0.3fv , 0.5sv , , b/h=49, 2/ b , / =0.08A .

It is found that the larger modulus ratio leads to larger energy releasing rate and

maximum deflection of the TC blister. Interestingly, the asymmetry characterized by

max max/C line B linew w becomes weak as the substrate is compliant, i.e. / 20f s . This

may indicate that it does not always happens. More detailed study is needed in future.

The simulated result in Supplementary Figure 6 is obtained by numerically

solving the above outlined continuum model in a computational cell with periodic

boundary conditions using input parameters: the cell size 2048h×2048h with an initial

circular delamination nuclei of radius r=5h , 0

m , 0.022m / 5f s ,

0.3fv , 0.5sv , * / 0.01

i i , 0.2n h , t n ,

/ 0.02n n n se and / 0.1t t n se .

15

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