Cao Z. (Ed.) Thin Film Growth: Physics, materials science and applications

328 Thin ﬁ lm growth

This curve is in good agreement with the experimental pro les, a sinusoid

with a vertical shift of height H between its edges (Foucher et al., 2006).

The buckle pro le is asymmetric in such a way that the maximal de ection

is not located in the middle of the structure. In the  nal state, the width of

the buckle is given by:

2

bb

H

bb* = 2bb

–

ta

n

q

n

q

n

[13.23]

After buckling, the width that can be experimentally measured is 2b*. in

order to quantify this asymmetry, it is useful to de ne a new axis labelled

(0*y*), y* varying between – b* and b* (see Fig. 13.11). The maximum of

z

H

0

y

–b b

– H

tanq

13.10 Theoretical proﬁ le of a straight-sided buckle localized above a

step of height H given by (y + v

2

(y), w

2

(y)).

z z

H

y

–b

–b*

b*

b

y*

– H

tanq

w

max

Dv

2

(y

max

)

y

max

0

0*l*

13.11 Theoretical proﬁ le of a straight-sided buckle localized above a

step of height H given by (y + v

2

(y), w

2

(y)). A new axis is labelled by

(0*y*), taking into account the in-plane displacement on the right-

hand side. The asymmetry of the structure is given by the parameter

l* corresponding to the abscissa of the maximal deﬂ ection in the

ﬁ nal state.

ThinFilm-Zexian-13.indd 328 7/1/11 9:44:48 AM



329Understanding substrate plasticity and buckling of thin ﬁ lms

the de ection, given by w

max

= w

2

(y

max

), is thus localized in y

max

+ v

2

(y

max

).

the initial abscissa y

max

of the point associated with the maximal de ection

is given by:

y

b

wH

ma

x

=

a

r

cco

s

wH – wH

wH +wH

m

wH

m

wH

2

wH

2

wH

m

wH

m

wH

2

wH

2

wH

p

wH

p

wH

p

wH

p

wH

22

wH

22

wH

22

wH

22

wH

wH +wH

22

wH +wH

Ê

Ë

Á

Ê

Á

Ê

Ë

Á

Ë

ˆ

¯

˜

ˆ

˜

ˆ

¯

˜

¯

[13.24]

The maximal de exion of the buckle is then given by:

ww

H

y

b

ma

ww

ma

ww

xm

ww

xm

ww

ma

x

ww =ww

ww

xm

ww =ww

xm

ww

xm

ww

xm

ww ww

xm

ww

+

1 +

2

Ê

Ë

Ê

Ë

Ê

Á

Ê

Ë

Á

Ë

Ê

Ë

Ê

Á

Ê

Ë

Ê

ˆ

¯

ˆ

¯

ˆ

˜

ˆ

¯

˜

¯

ˆ

¯

ˆ

˜

ˆ

¯

ˆ

[13.25]

the abscissa of w

max

in the  nal benchmark (y*O*z) is labelled l*:

l* = b – b* + y

max

+ v

2

(y

max

) [13.26]

Experimentally, H << b in such a way that the parameter of asymmetry l*

is very small (Foucher et al., 2006).

in conclusion, the fundamental solution w

1

(y) is only valid for H ≤ H

c

and the buckled solution w

2

(y) is the only solution for H ≥ H

c

. Whereas the

fundamental and the buckle solutions coexist in the classical case of buckling

on plane substrates, this is not the case for crystalline substrates.

13.4 Discussion

13.4.1 Localization of buckling structures

In order to explain the localization of the buckling structures on the areas

of dislocation emergence observed in Figs 13.5 and 13.12, the criteria for

buckling on step areas has been compared with those for plane areas. The

critical step height H

c

is associated to a critical initial stress value, labelled

s

c

step

and given by:

Step structure

Plane area

10 µm

13.12 A 20 nm thick nickel ﬁ lm deposited on a single LiF crystal

strained at –2%. AFM image showing the localization of the buckles

above the steps formed during the dislocations in the substrate.

ThinFilm-Zexian-13.indd 329 7/1/11 9:44:48 AM



330 Thin ﬁ lm growth

ss

p

q

c

ss

c

ss

step

ss

step

ss

c

()

ss

()

ss

=

ss

1 –

3

–

()H()

ss

()

ss

H

ss

()

ss

b

H

b

h

2

23

H23H

4

ta

n

q

n

q

Ê

Ë

Á

Ê

Á

Ê

Ë

Á

Ë

ˆ

¯

˜

ˆ

˜

ˆ

¯

˜

¯

Ê

Ë

ÁÁÁ

Ê

Á

Ê

Á

Ê

Á

Ê

Ë

Á

Ë

Á

Ë

Á

Ë

ˆ

¯

˜

ˆ

˜

ˆ

¯

˜

¯

È

Î

Í

È

Í

È

Í

Î

Í

Î

Í

˘

˚

˙

˘

˙

˘

˙

˚

˙

˚

˙

2

[13.27]

if s

c

step

is lower than s

c

, the buckling preferentially occurs on the steps. This

condition is veri ed when the angle of emergence q is lower than a critical

angle q

c

given by (Foucher et al., 2006):

q

c

=

8

3

ar

ctan

b

H

[13.28]

as previously stated, the shape ratio H/b is small. For instance, for H/b =

1/10 the critical angle is equal to 87.85°. For the  lm and the substrate used

in the experiments shown in Fig. 13.5 (nickel thin  lm on LiF single crystal),

this crystallographic angle is equal to 45°, explaining thus the localization

of the buckles above the steps. Moreover, for a parallelepiped sample, the

calculation of the Schmidt’s factor leads to an angle q between 30° and

60°. The relation s

c

step

< s

c

is thus always veri ed whatever the crystalline

substrate. However, it is possible to consider particular con gurations where

the angle of emergence is equal to 90°, as shown in Fig. 13.13. Such a

con guration could then be used in order to prevent buckling induced by

substrate plasticity.

13.4.2 Slip systems

Several slip systems can be activated and form steps of various orientations

during crystalline plasticity. In the case where slip systems are activated

beneath the same buckle, the boundary conditions need to be modi ed as:

q = 90°

45°

13.13 Example of the effects of uniaxial strain on a thin ﬁ lm

deposited on a crystalline substrate where the angle between the

active slip planes and the surface is 90°. In this case, q > q

c

in such a

way that the ﬁ lm does not buckle on the step.

ThinFilm-Zexian-13.indd 330 7/1/11 9:44:49 AM



331Understanding substrate plasticity and buckling of thin ﬁ lms

vb

H

()

vb()vb

= –

i

r

i

j

d

j

SS

H

SS

H

– SS –

i

SS

i

ta

nt

qq

i

qq

i

j

qq

j

SS

qq

SS

– SS –

qq

– SS –

nt

qq

nt

i

nt

i

qq

i

nt

i

j

nt

j

qq

j

nt

j

an

qq

an

[13.29]

wb

HH

()

wb()wb

= –

i

j

d

SS

HHSSHH

HH – HHSSHH – HH

HH

i

HHSSHH

i

HH

r

SS

r

HH

r

HHSSHH

r

HH

[13.30]

with H

i

r

the height of the rising step i and H

j

d

the height of the descending

step j as shown in Fig. 13.14. The shape equation of the buckle can be

determined as outlined in Section 13.3.2.

13.4.3 Tensile tests

Complementary to the previous sections, two other solutions can be found,

related to the case where the  lm is unstressed on the step (s

yy

= 0) and the

case where the  lm is in tension (s

yy

> 0). These solutions are obtained for

thin  lms undergoing tensile internal stresses after deposition or when the

step height H is negative. Experimentally, H < 0 corresponds to a tensile test;

the vertical displacement induced by the step in y = + b being thus oriented

to the bottom, as shown in Fig. 13.15. The parameter H is thus equal to

N ¥ h

e

in compression and to – N ¥ h

e

in tension. Note that the angle q is

set as 0 < q ≤ 90°.

The solution for the unstressed  lm (s

yy

= 0) is given by:

wy

H

y

b

y

b

wy

nul

wy

3

()

wy()wy

=

–

3

– –

+

+ 1

2

–

2

–

2

3

2

3

Ê

Ë

–

Ë

–

Á

–

Á

–

Ê

Á

Ê

Ë

Á

Ë

–

Ë

–

Á

–

Ë

–

ˆ

¯

˜

ˆ

˜

ˆ

¯

˜

¯

[13.31]

z

H

1

H

2

0

y

13.14 Sketch showing a straight-sided buckle formed during the

activation of two symmetric slip systems.

ThinFilm-Zexian-13.indd 331 7/1/11 9:44:50 AM



332 Thin ﬁ lm growth

vy

H

b

y

b

y

b

vy

nul

vy

()

vy()vy

= –

3

160

3

– 10

+

2

53

Ê

Ë

Ê

Ë

Ê

Á

Ê

Ë

Á

Ë

Ê

Ë

Ê

Á

Ê

Ë

Ê

ˆ

¯

ˆ

¯

ˆ

˜

ˆ

¯

˜

¯

ˆ

¯

ˆ

˜

ˆ

¯

ˆ

Ê

53

Ê

53

Ë

Ê

Ë

Ê

Á

Ê

Ë

Á

Ë

Ê

Ë

Ê

Á

Ê

Ë

Ê

ˆ

53

ˆ

53

¯

ˆ

¯

ˆ

˜

ˆ

¯

˜

¯

ˆ

¯

ˆ

˜

ˆ

¯

ˆ

777

–

2tan

+ 1

y

b

H

y

b

È

Î

Í

È

Í

È

Í

Î

Í

Î

Í

˘

˚

˙

˘

˙

˘

˙

˚

˙

˚

˙

Ê

Ë

Ê

Ë

Ê

Á

Ê

Ë

Á

Ë

Ê

Ë

Ê

Á

Ê

Ë

Ê

ˆ

¯

ˆ

¯

ˆ

˜

ˆ

¯

˜

¯

ˆ

¯

ˆ

˜

ˆ

¯

ˆ

q

[13.32]

the solution (y + v

null

(y), w

null

(y)) is plotted in Fig. 13.16. It corresponds

to a  lm following the step structure. It is associated with a step height H

0

given by:

H

b

E

b

0

f

22

b

22

b

=

–

20

(

–

)

b) b

22

)

22

b

22

b) b

22

b

+

5

33

–

33

–

1

5

3

)

0

)

ta

33ta33

nt

EntE

f

nt

f

E

f

EntE

f

E

f

nt

f

(

nt

(

)

nt

)

33nt33

3nt3

0

nt

0

)

0

)

nt

)

0

)

an

q

33

q

33

nt

q

nt

33nt33

q

33nt33

ns

f

ns

f

22

ns

22

)

ns

)

22

)

22

ns

22

)

22

q

an

q

an

Ê

Ë

nt

Ë

nt

Á

Ê

Á

Ê

Ë

Á

Ë

nt

Ë

nt

Á

nt

Ë

nt

ˆ

¯¯¯

˜

ˆ

˜

ˆ

¯¯¯

˜

¯¯¯

2

[13.33]

the function of H = H

0

can also be written as a condition on the initial

internal stress s

0

, as s

0

= S

0

with:

S

0

2

()

=

1 –

–

1

2tan

+

3

20

f

2

()H()

E

f

E

f

H

b

H

b

n

q

Ê

Ë

Á

Ê

Á

Ê

Ë

Á

Ë

ˆ

¯

˜

ˆ

˜

ˆ

¯

˜

¯

È

Î

Í

È

Í

È

Í

Î

Í

Î

Í

˘

˚˚˚

˙

˘

˙

˘

˙

˚˚˚

˙

˚˚˚

˙

[13.34]

–b

z

b

y

0

e

0

e

1

> e

0

H

tanq

–

H = –N · h

e

q

13.15 Evolution of a thin ﬁ lm deposited on a crystalline substrate

during a tensile test. The emergence of dislocations leads to the

formation of a negative step of height H = –N·h

e

associated with an

elongation of –H/tanq > 0 (0 < q < 90°).

–b

y

b

H

0

H

2

z

– H

tanq

13.16 Theoretical proﬁ le of an unstressed thin ﬁ lm (s

yy

= 0) on a

step formed during the emergence of dislocations from a crystalline

substrate during tensile strain (y + v

null

(y), w

null

(y)).

ThinFilm-Zexian-13.indd 332 7/1/11 9:44:51 AM



333Understanding substrate plasticity and buckling of thin ﬁ lms

The solution associated with a thin  lm in tension is given by:

wy

H

bb

e

ee

b

by

wy

te

wy

n

wy

2

(–

by(–by

)(

+)

by+)by

()

wy()wy

=

1

1+

bb+–bb

ee– ee

+

2

gg

bb

gg

bb

bb+–bb

gg

bb+–bb

g

b

g

b

gg

ee

gg

ee

by

gg

by

ee

by

ee

gg

ee

by

ee

(–

gg

(–

ee

(–

ee

gg

ee

(–

ee

by(–by

gg

by(–by

ee

by

ee

(–

ee

by

ee

gg

ee

by

ee

(–

ee

by

ee

)(

gg

)(

ee

)(

ee

gg

ee

)(

ee

ee– ee

gg

ee– ee

()

bb()bb

+–()+–

bb+–bb()bb+–bb

1()1

gg

()

gg

bb

gg

bb()bb

gg

bb

bb+–bb

gg

bb+–bb()bb+–bb

gg

bb+–bb

(

ggg

g

(1

+)

g

+)

g

)+

ey

+)ey+)

g

+)

g

ey

g

+)

g

b

g

b

g

+)

b

+)

+)ey+)

b

+)ey+)

2

+)

2

+)

+)ey+)

2

+)ey+)

1

È

Î

Í

È

Í

È

Î

Í

Î

˘

˚

˙

˘

˙

˘

˚

˙

˚

[13.35]

vy

AH

yb

vy

te

vy

n

vy

22

AH

22

AH

()

vy()vy

= –

8

[–

[– [–

z

gg

yb

gg

yb

gg

yb

gg

yb

1

(

ee( ee

yb

(

yb

(

yb

––

(

––

(

ee ee( ee ee

ee+ ee( ee+ ee

– ( –

ee – ee( ee – ee

)( )

gg

(

gg

ee

gg

ee( ee

gg

ee

yb

gg

yb

(

yb

gg

yb

ee

yb

ee

gg

ee

yb

ee( ee

yb

ee

gg

ee

yb

ee

ee+ ee

gg

ee+ ee( ee+ ee

gg

ee+ ee

yb

+

yb

gg

yb

+

yb

(

yb

+

yb

gg

yb

+

yb

ee

yb

ee+ ee

yb

ee

gg

ee

yb

ee+ ee

yb

ee( ee

yb

ee+ ee

yb

ee

gg

ee

yb

ee+ ee

yb

ee

gg

(

gg

ee

gg

ee( ee

gg

ee

yb

gg

yb

(

yb

gg

yb

ee

yb

ee

gg

ee

yb

ee( ee

yb

ee

gg

ee

yb

ee

––

gg

––

(

––

gg

––

yb––yb

gg

yb––yb

(

yb––yb

gg

yb––yb

ee – ee

gg

ee – ee( ee – ee

gg

ee – ee

ee

yb

ee – ee

yb

ee

gg

ee

yb

ee – ee

yb

ee( ee

yb

ee – ee

yb

ee

gg

ee

yb

ee – ee

yb

ee

+

(

22

–2

3

zz

(

zz

(

–

zz

–

)]

zz

)]

+

zz

+

22

zz

22

–2

zz

–2

zz

gg

zz

22

zz

22

gg

22

zz

22

zz

gg

zz

2

zz

(

zz

(

2

(

zz

(

zz

ee

zz

+

zz

+ ee +

zz

+

22

zz

22

ee

22

zz

22

zz

gg

zz

ee

zz

gg

zz

+

zz

+

gg

+

zz

+ ee +

zz

+

gg

+

zz

+

22

zz

22

gg

22

zz

22

ee

22

zz

22

gg

22

zz

22

zz

ee

zz

–

zz

– ee –

zz

–

–2

zz

–2

ee

–2

zz

–2

zz

gg

zz

ee

zz

gg

zz

–

zz

–

gg

–

zz

– ee –

zz

–

gg

–

zz

–

y

( y(

yb

22yb22

zz

yb

zz

22

zz

22yb22

zz

22

zz

gg

zz

yb

zz

gg

zz

22

zz

22

gg

22

zz

22yb22

zz

22

gg

22

zz

22

+

22

+

zz

+

22

+

gg

+

22

+

zz

+

22

+

yb

+

22

+

zz

+

22

+

gg

+

22

+

zz

+

22

+

zz

gg

zz

ee

zz

gg

zz

yb

zz

gg

zz

ee

zz

gg

zz

+

zz

+

gg

+

zz

+ ee +

zz

+

gg

+

zz

+

yb

+

zz

+

gg

+

zz

+ ee +

zz

+

gg

+

zz

+

22

zz

22

gg

22

zz

22

ee

22

zz

22

gg

22

zz

22yb22

zz

22

gg

22

zz

22

ee

22

zz

22

gg

22

zz

22

+

22

+

zz

+

22

+

gg

+

22

+

zz

+

22

+ ee +

22

+

zz

+

22

+

gg

+

22

+

zz

+

22

+

yb

+

22

+

zz

+

22

+

gg

+

22

+

zz

+

22

+ ee +

22

+

zz

+

22

+

gg

+

22

+

zz

+

22

+

yb

zz

yb

zz

gg

zz

yb

zz

gg

zz

2yb2

zz

2

zz

yb

zz

2

zz

gg

zz

2

zz

gg

zz

yb

zz

gg

zz

2

zz

gg

zz

gg

zz

ee

zz

gg

zz

yb

zz

gg

zz

ee

zz

gg

zz

–

zz

–

gg

–

zz

– ee –

zz

–

gg

–

zz

–

yb

–

zz

–

gg

–

zz

– ee –

zz

–

gg

–

zz

–

zz

gg

zz

yb

zz

gg

zz

-

zz

gg

zz

yb

zz

gg

zz

+

++ +

)

b

[13.36]

Where

g

ss

2

yy

ss

yy

ss

=

(

ss

–

ss

)

h

D

0

A

bb

e

=

1 +

bb+ (bb

– 1)

b

1

2

gg

bb

gg

bb

gg

bb bb

gg

bb bb

+ (

gg

+ (

bb+ (bb

gg

bb+ (bb

g

b

g

b

B = g(1 + e

2gb

)

z

1

= 2Be

gb

z

g

2

=

2

e

b

g

b

g

z

n

gs

g

3

f

2

f

2

gs

2

gs

0

22

=

1 –

+

gs

+

gs

–

8

(

22

(

22

2

22

E

f

E

f

D

h

AH

22

AH

22

Be

g

Be

g

22

Be

22

g

22

g

Be

g

22

g

( Be(

22

(

22

Be

22

(

22

+ Be+

22

+

22

Be

22

+

22

2Be2

22

2

22

Be

22

2

22

Ê

Ë

Á

Ê

Á

Ê

Ë

Á

Ë

ˆ

¯

˜

ˆ

˜

ˆ

¯

˜

¯

22

2222

2

)

22

)

22

–

11

22

11

22

12

g

22

g

22

g

b

22b22

22

g

22b22

g

22

h

22

h

22

the solution (y+v

ten

(y), w

ten

(y)) is plotted in Fig. 13.17. It corresponds to

a  lm stretched between its edges and is associated with a step height H <

H

0

(s

0

< S

0

).

Note that these solutions are based on the FvK theory of thin plates which

does not take into account the effect of the substrate and the adhesion.

However, Foucher and Coupeau (2007) have shown that the emergence of

dislocations is modi ed by the presence of the  lm in such a way that the

step pro le appears to follow the equilibrium shape of the unbuckled  lm.

The interaction between thin  lms and substrate dislocations was discussed

previously by Junqua and Grilhé (1997).

13.4.4 Experiment versus modelling: characterization of

the mechanical properties of thin ﬁ lms

The main difference between the modelling and the compression test experiments

comes from the initial interfacial delamination hypothesis that does not exist

in the experiment. This delamination is obtained experimentally when external

ThinFilm-Zexian-13.indd 333 7/1/11 9:44:52 AM



334 Thin ﬁ lm growth

compression leads to an increase in the elastic energy stored in the  lm. When

the stored energy is high enough to counterbalance the adhesion energy, the

 lm buckles. The experimentally buckling critical stress

s

c

ex

p

is consequently

found higher than the theoretical one. This variation in the onset of buckling

is shown by a jump from unbuckled to buckled state in the curve

ds

(|

ds

(|

ds

|)

yy

0

,

as shown in Fig. 13.18. Once the interface is delaminated, the unstuck part

of the  lm can be modelled as a thin plate and the curve

ds

(|

ds

(|

ds

|)

yy

0

follows

the theoretical curve. it is noted that if the initial internal stress s

0

is smaller

than the critical stress s

c

, the  lm can go down onto the substrate during

the release of the external applied stress.

this discrepancy has been used in the past to estimate the adhesion

energy G (coupeau et al., 1999a) between the  lm and the substrate using

the following equation:

G

=

(

–

)(

–

)

f

2

cc

–

cc

–

h

(

h

(

E

(

E

(

f

E

f

2

(

2

(

1

ns

)(

ns

)(

f

ns

f

2

ns

2

s

cc

s

cc

ex

p

[13.37]

The model has also been used to estimate the Young’s modulus or the initial

internal stress in the  lm (Matuda et al., 1981). Indeed, the internal stress

in the  lm is related to the width and the de ection of the buckle by the

equation:

s

p

n

d

0

f

22

=

12

1 –

3

4

22

4

22

+ 1

22

d

22

d

3

22

3

22

f

22

f

2

E

f

E

f

22

E

22

f

22

f

E

f

22

f

h

22

h

22

h

b

Ê

22

Ê

22

Ë

22

Ë

22

Ê

Ë

Ê

Á

Ê

Ë

Á

Ë

22

Ë

22

Á

22

Ë

22

Ê

Ë

Ê

Á

Ê

Ë

Ê

ˆ

¯

ˆ

¯

ˆ

˜

ˆ

¯

˜

¯

ˆ

¯

ˆ

˜

ˆ

¯

ˆ

Ê

Ë

Á

Ê

Á

Ê

Ë

Á

Ë

ˆ

¯

˜

ˆ

˜

ˆ

¯

˜

¯

[13.38]

If the internal stress has been determined using another technique, Eq.

13.38 can be used to estimate the Young’s modulus of the  lm. Finally, the

–b

y

b

H

0

H

2

z

– H

tanq

s

0

= 0

s

0

= 0.5 Â

0

13.17 Theoretical proﬁ les of a thin ﬁ lm strained in tension (s

yy

>

0) on a step formed during the emergence of dislocations from a

crystalline substrate under tensile strain (y + v

ten

(y), w

ten

(y)), for

various initial stress intensities s

0

(s

0

= 0.5 ∑

0

and s

0

= 0).

ThinFilm-Zexian-13.indd 334 7/1/11 9:44:53 AM



335Understanding substrate plasticity and buckling of thin ﬁ lms

measurement of the geometric parameters of the buckles appears to be an

interesting way to estimate the energy of adhesion, the internal stress or the

Young’s modulus, parameters that are relatively dif cult to determine by

other techniques. However, the compliance of the substrate, considered to be

in nitely rigid in the model, has been shown to be a limiting process. Indeed,

experiments carried out with thin metallic  lms deposited on polymeric

substrates show that the maximal de ection of the buckles was experimentally

higher than predicted and that the  lm sinks into the substrate at the base

of the buckles (Parry et al., 2005). It has even been shown that this effect

could induce cracking of the  lm and plastic strain in the substrate (Cordill

et al., 2005). The characterization of the mechanical properties of the  lm

appears thus dif cult in the case of polymeric substrates.

The critical stress for buckling in the region of steps can be used to

estimate the adhesion energy using the formula:

G

=

2

(

1 –

)(

–

)

f

2

c

ex

p

c

step

h

E

f

E

f

ns

)(

ns

)(

f

ns

f

2

ns

2

s

[13.39]

Theory

First loading

Unloading and next loadings

d

s

0

s

c

s

c

exp

aG

s

0

yy

13.18 Schematic diagram showing the effect of adhesion on the

evolution of buckle height d versus the applied stress on its edges.

During the ﬁ rst loading, the ﬁ lm adheres to the substrate and the

stress must be greater than the theoretical critical stress s

c

for the

ﬁ lm to buckle. After delamination, the ﬁ lm buckles suddenly for a

critical stress s

c

exp

. There is a jump from unbuckled to buckled state

(arrow) and the deﬂ exion follows the theoretical curve for thin

plates. If the initial internal stress s

0

is lower than the critical stress

s

c

, the ﬁ lm touches the substrate after relaxation of the applied

stress and the deﬂ exion follows the theoretical curve for the next

loadings.

ThinFilm-Zexian-13.indd 335 7/1/11 9:44:54 AM



336 Thin ﬁ lm growth

By neglecting the asymmetry of the structure, it can be set that w

max

~ w

m

+ H/2. The initial internal stress can then be determined using the following

formula:

s

p

np

p

0

f

np

f

np

2

np

2

np

ma

x

=

12

1 –

3

4

np

4

np

–

+–

8

2

np

2

np

2

+–

2

+–

2

np

2

np

3

+–

3

+–

E

f

E

f

h

np

h

np

w

H

HH

+–

HH

+–

8HH8

2

HH

2

Ê

Ë

np

Ë

np

Á

Ê

Á

Ê

Ë

Á

Ë

np

Ë

np

Á

np

Ë

np

ˆ

¯

np

¯

np

˜

ˆ

˜

ˆ

¯

˜

¯

np

¯

np

˜

np

¯

np

222

ta

n

*+

+1

2t

an

q

b

H

h

2th2t

2t

an

q

an

q

Ê

Ë

Á

Ê

Á

Ê

Ë

Á

Ë

ˆ

¯

˜

ˆ

˜

ˆ

¯

˜

¯

Ê

Ë

np

Ë

np

Á

Ê

Á

Ê

Ë

Á

Ë

np

Ë

np

Á

np

Ë

np

ˆ

¯

˜

ˆ

˜

ˆ

¯

˜

¯

È

Î

np

Î

np

Í

È

Í

È

Í

np

Í

np

Î

Í

Î

np

Î

np

Í

np

Î

np

Í

˘

˚

˙

˘

˙

˘

˙

˚

˙

˚

˙

¥

2*

22*2

bH

n bHn

+ bH+

bH

2*bH2*

tabHta

n bHn

q

bH

q

bH

n bHn

q

n bHn

Ê

Ë

Á

Ê

Á

Ê

Ë

Á

Ë

ˆ

¯

˜

ˆ

˜

ˆ

¯

˜

¯

2

[13.40]

Crystalline materials being less compliant than polymers, the mechanical

properties of the  lm can be characterized from the geometric parameters of

the buckles with a better precision. Buckling induced by substrate crystalline

plasticity is thus a good way to estimate the adhesion energy and the initial

internal stress.

13.5 Conclusions

In this chapter dedicated to thin  lm buckling phenomena, it has been shown

that buckling can occur spontaneously or under the action of external forces.

Compressive tests carried out to induce buckling show that the nature of the

substrate plays a key role. For polymeric substrates, the buckling leads to

straight-sided buckles perpendicular to the applied stress. These structures can

be modelled analytically using the FvK theory of thin plates. However, it is

shown that the experimental and analytical onsets of buckling are different,

due to the adhesion energy neglected in the model, and that the experimental

shape is slightly modi ed by the compliance of the substrate.

Experiments carried out on crystalline substrates show the formation of

pseudo straight-sided buckles, shifted vertically on their edges, localized on

the steps formed by substrate plasticity. This shift modi es the analytical

model. It is shown that the critical stress for buckling is decreased on the area

of dislocation emergence explaining the localization of the buckles observed

experimentally. The model has been extended for all kinds of external and

initial internal stresses in the  lm. All the solutions are displayed in Table

13.1.

Finally, it is shown that compressive tests can be used to estimate the

adhesion energy and the internal stress from the difference between the

experimental and theoretical onsets of buckling and from the shape of the

buckles. Since it is dif cult to experimentally determine the mechanical

properties of thin  lms, buckling induced by compression tests appears to

be a good way to characterize.

ThinFilm-Zexian-13.indd 336 7/1/11 9:44:54 AM



Table 13.1 Vertical displacement w of a thin ﬁlm in the equilibrium state on a crystalline substrate, whatever the initial internal

stress in the ﬁlm and the substrate deformation

Tensile test Spontaneous Compression test

H < H

0

,

s

0

< S

0

H = H

0

,

s

0

= S

0

H

c

> H > H

0

,

s

c

step

>s

0

> S

0

H > H

c

,

s

0

> s

c

step

H = 0,

s

0

< s

c

H = 0,

s

0

> s

c

H < H

0

,

s

0

< S

0

H = H

0

,

s

0

= S

0

H

c

> H > H

0

,

s

c

step

> s

0

> S

0

H > H

c

,

s

0

> s

c

step

s

0

< 0 w

ten

X X X w = 0 X w

ten

w

null

w

1

w

2

s

0

= 0 w

ten

w = 0 X X w = 0 X X w = 0 w

1

w

2

s

0

> 0 w

ten

w

null

w

1

w

2

w = 0 w

b

X X w

1

w

2

X = no solution.

ThinFilm-Zexian-13.indd 337 7/1/11 9:44:54 AM



Cao Z. (Ed.) Thin Film Growth: Physics, materials science and applications

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