Full text of DOI:10.1051/epjap:2000113

Eur. Phys. J. AP 10, 3–7 (2000)
THE EUROPEAN
PHYSICAL JOURNAL
APPLIED PHYSICS
ꢀ
c EDP Sciences 2000
Interactive study of straight-sided buckling patterns in thin films
under compressive stress
a
F. Cleymand , C. Coupeau, J. Colin, and J. Grilhe
LMP UMR 6630, SP2MI, bd Curie, T ´e l ´e port 2, BP 179, 86960 Futuroscope Cedex, France
Received: 17 January 2000 / Accepted: 9 March 2000
Abstract. In situ atomic force microscopy observations have been carried out of thin films under external
compressive stress. Straight-sided buckling patterns arise perpendicular to the compression axis which tend
to attract one another during propagation a few hundred nanometers apart. The mechanisms whereby these
debonding patterns interact have been investigated taking into account the elastic energy of both the film
and the substrate. The equilibrium distance between two straight-sided wrinkles has been determined;
good agreement has been obtained between the experimental results and the mechanics involved.
PACS. 61.16.Ch Scanning probe microscopy: scanning tunneling, atomic force, scanning optical, magnetic
force, etc. – 68.35.Gy Mechanical and acoustical properties; adhesion – 68.55.-a Thin film structure and
morphology
1
Introduction
Thin films and coatings produced by sputtering improve
the mechanical properties of materials even though they
often introduce residual stresses of magnitude greater than
the elastic limit of the bulk material [1]. The sign and
magnitude of these residual stresses, as well as film thick-
ness, adhesion energy and flaw distributions at the inter-
face may cause the nucleation and the growth of debond-
ing patterns. During the past few years, various shapes
of spontaneous buckling just after deposition have been
observed such as blisters, straight-sided or “worm-like”
wrinkles, for many different film/substrate systems [2–6].
Fig. 1. Specimen configuration.
Recent optical observations [7,8] have been performed on 2 Experimental procedure
thin thermally grown alumina layers formed on a com-
3
04 L S.S. thin films were deposited on polycarbonate
mercial bond coat and have shown the need to charac-
terise the buckle propagation by a buckling map and three
non-dimensional indices. Although the buckling growth
and propagation of an individual blister from an initially
debonded patch is now well-established, few experimental
results of the mechanical behaviour of thin films have been
reported, generally based on optical microscopy observa-
tions [7–10].
substrates at room temperature using an argon ion beam
sputtering technique operating in a vacuum chamber. The
sample configuration is presented in Figure 1. The choice
of substrate was made depending on its elastic behaviour
over a large range of strain and its dimensions are nomi-
3
nally 2.5 × 2.5 × 5 mm . Before deposition, the polycar-
bonate substrates were cleaned in a neutral solution for
◦
4
8 hours followed by annealing at 120 C under an Ar
In this paper, the surface evolution of stressed 304 L
Stainless Steel (S.S.) thin films on polycarbonate sub-
strates has been investigated and characterised by atomic
force microscope (AFM) [11] using an experimental ap-
paratus allowing the in situ observation of the specimen
during compression. Elastic energy calculations have been
carried out to model the interaction of straight-sided wrin-
kles observed during the compressive tests.
atmosphere to decrease the humidity content. The faces
destined to be compressed were then mechanically pol-
ished by standard metallographic procedure.
The sputtering system has been described in detail
in reference [12]. The deposition conditions are as fol-
−
8
lows: the starting pressure is 2 × 10
torr and during
−4
deposition, the pressure is maintained at about 10 torr.
Both deposition rate (0.05 nm/s) and film thickness h are
controlled by a quartz oscillator and the actual thickness
a
e-mail: Cleymand@lmp.univ-poitiers.fr

4
The European Physical Journal Applied Physics
Fig. 2. Optical image of bunching mechanism (α, β, γ) in
04 L S.S. thin film ∆σ correspond to the external applied
stress.
3
of the as-deposited layer is determined using a reflecting
X-ray method [13] or by a profilometer from step heights.
For curvature and for X-ray diffraction measurements, the
films are also deposited on a silicon wafer, 200 µm thick.
In situ AFM observations have been carried out us-
ing an experimental apparatus described by Coupeau
et al. [14–17]. The system operates at room temperature
and pressure and consists of an AFM operated in conjunc-
tion with a compression device. The specimen surface is
scanned continuously during deformation. Strains and ap-
plied stresses are determined to a high degree of accuracy
and resolution, permitting the correlation of AFM image
to the compression curve.
Fig. 3. Evolution of the 304 L S.S. thin film during deforma-
tion. The scan size is 16.6 µm×16.6 µm and the external stress
Unless otherwise specified, the AFM images presented
in this paper have been taken in error signal mode; these
∆
σ generated in the film during the experiment is indicated
images are thus not calibrated in the z-axis but present a below.
great visual interest because of fine detail enhancement.
V-shaped cantilevers with theoretical normal spring con-
size of each image is about 16.6 µm × 16.6 µm and the
stant of about 0.1 N/m and sharpened pyramidal silicon
nitride tips were used in this study.
stresses ∆σ generated during the experiment are indi-
cated. It is to be emphasised that the total stress σ in
the film is the sum of ∆σ and of the internal stress σi,
estimated by the curvature method to be of the order of
3
Experimental results
1
–2 GPa.
The experiments were conducted especially on well-
adhering thin films where no debonding patterns were ob-
served on the as-deposited films.
σ = σ + ∆σ.
(1)
i
This value has however not been confirmed by X-ray
An optical image is presented in Figure 2 of a diffraction due to a column texture in the film induced by
3
04 L S.S. thin film after the application of complete com- the deposition process. The specific structures labelled A
pression cycle. Optical observations show that the whole and B can be used to relate the position of the images
specimen surface exhibits buckling features. Instead of ir- (Figs. 3a–3d).
regular buckled regions found near the substrate edges,
The buckling process arises above a measured criti-
∗
regular straight-sided wrinkles are observed and have a di- cal stress of ∆σc = 0.33 GPa, with the appearance of
rection of progress perpendicular to the compression axis. the first straight wrinkle (Fig. 3b). As described previ-
No telephone cord structures are observed. Moreover the ously, this buckling pattern propagates perpendicular to
“
bunching” of straight wrinkles due to the attraction of the compression axis and appear to be locked at point A
two wrinkles propagating in parallel directions is clearly probably due to a crystallographic defect in the film. As
observed (α, β, γ) and seems to be the usual phenomenon the applied stress increases, a second straight wrinkle ap-
at the specimen surface. This process has been studied us- pears parallel to the first one but in the opposite direction
ing the experimental apparatus described in Section 2 al- and seems to be both attracted by the first, as observed in
lowing the in situ observations of specimen surfaces during Figures 3c–3f, and slowed down considering its propaga-
deformation.
tion perpendicular to the compression axis. It is noted that
In Figure 3 is presented the evolution of a 304 L S.S. a remaining well-adhering part of the thin film of distance
film, 58 nm thick, under uniaxial compression. The scan d between the wrinkles is almost always observed during

F. Cleymand et al.: Interactive study of buckling patterns
5
a point force [24] whose intensity is a function of the film
thickness h and of the stress σ (Fig. 6):
2
h
Fx = ꢀ σhδ(x − xi)sgn (y)
(2)
where δ(x − xi) is the Dirac function applied at point
|
y|
(xi, 0), and sgn (y) is the function defined by sgn(y) =
.
y
The classical theory of elasticity can now be used to
calculate the stress concentration due to the two blisters
in the substrate. The total stress in the substrate due to
the two couples of surface steps is then the sum of two
terms:
relax, sub
ij
relax, A
relax, B
σ
= σij
+ σij
(3)
Fig. 4. Experimental variation as a function of the external
stress ∆σ generated in the film of the equilibrium distance d
between two wrinkles.
relax, A
where σij
relax, B
and σij
are the stresses in the substrate
due to the punctual forces associated with blisters A and B
respectively. The stress in the x-direction in the substrate,
relax, sub
xx
the bunching process. Distance d dependence on stress
σ
, can be expressed as a function of b the half-
is shown in Figure 4; d decreases with increase of stress width of the blisters and d the distance between the two
and an asymptotic value of the order of 800 nm is deter- blisters by:
mined. A three-dimensional AFM image of two straight-
sided wrinkles is shown in Figure 5; their cross-sections
obtained by averaging consecutive profiles perpendicular
to the wrinkles are included showing a well-adhering part
is present between the buckling patterns. The half-width
b of the straight-sided wrinkle is determined to be approx-
imately 1.3 µm.
2σh
π
relax, sub
xx
σ
=
ꢀ
2
2
(
2x − d − 2b)
(2x − d)
×
−
2
2 2
2
2 2
(
(2x − d − 2b) + y )
((2x − d) + y )
ꢁ
2
2
(
2x + d)
(2x + d + 2b)
+
−
.
(4)
2
2 2
2
2 2
(
(2x + d) + y )
((2x + d + 2b) + y )
This stress due to the two blisters decreases as a function
of 1/d (see Fig. 7) and relaxes the residual stress in the
substrate at a distance of the order of 2b around each
blister.
4
Study of the interaction between two
wrinkles
The interaction between two wrinkles is modelled and the
To determine the critical distance between two wrin-
equilibrium distance between two wrinkles determined, kles, the forces acting on the blisters have been deter-
considering the elastic contribution of the substrate. In mined.
previous papers [2,7,10,18–22], it has been demonstrated
First, the interaction force between two straight-sided
that a critical stress exists to buckle the film and for the blisters has been deduced from the classical expression of
blisters to propagate, by calculating the energy release the elastic energy stored in the substrate:
rate of the thin film delamination. Nevertheless, these now
Z
1
2
well-established models consider only the effect of the sub-
strate through the adhesion energy and cannot really ex-
plain the mechanical behaviour observed; that is to say the
relax, sub relax, sub
ε
we =
σ
dV
(5)
is
ij
ij
v
relax, sub
existence of an equilibrium distance between two wrinkles. where V is the volume of the substrate, and εij
In the present approach, it is considered that in ad- the total strain in the substrate due to the punctual forces
dition to the elastic energy stored in the film, the elastic associated with the blisters A and B. Using the Ostrograd-
effect of the substrate is of fundamental importance due sky’s theorem, this volume integral of energy has been
to the stress concentration in the substrate. The elastic transformed into a surface integral:
energy of film/substrate system has been calculated for
Z
1
2
different relative positions of the two wrinkles of constant
shape, i.e. for b and δ constants. This implies in our calcu-
lation that the energy of the well-bonded (adherent) parts
of the film is constant and does not introduce any term in
the expression of the force equilibrium conditions deter-
mined in the following. The elastic stresses in the substrate
have been determined, approximating each blister to two
surface steps [23]. In this case, each of the two steps rep-
resenting a blister on a stressed film can be modelled by
A B
i
we =
njσ u dS
(6)
ij
S
where nj is the j-surface normal.
The elastic energy has been calculated and the inter-
action force has been deduced to be:
2
16 (1 − ν )
S
2 2 2
∂
We
σ h b
d(d + 2b)(d + 4b)
FA = −
= −
(7)
∂
d
π
ES

6
The European Physical Journal Applied Physics
Fig. 5. Three-dimensional AFM image of the two straight-sided wrinkles width their two cross-sections.
Fig. 7. Variation of the reduced σxx stress component in the
substrate as a function of x, for y = −b and for d = 10b.
Fig. 6. Two-dimensional model for the study of the interaction
between two straight-sided blisters.
Since the stress in the film around the fixed wrinkle
E
f
relax, film relax, substrat
is also in average relaxed, σij
= E
σ
, a
ij
S
d
repulsive force can be determined for y = 0 and x = 2 +b.
In fact, the relaxation of the energy in the film is more
where ES, νS are Young modulus and Poisson’s ratio re- important when the moving blister propagates in a region
spectively of the substrate.
of high residual stress, free of wrinkles, than in the vicinity

F. Cleymand et al.: Interactive study of buckling patterns
7
5
Conclusion
The mechanical behaviour of thin films on substrates and
in particular the interaction between two blisters during
film deformation has been observed in situ, taking advan-
tage of the high resolution offered by the AFM. Blister
interactions have been modelled by consideration of the
elastic effects of the film and the substrate. Attractive and
repulsive forces due to the substrates and the films respec-
tively have been established and an equilibrium distance
has been determined in a good agreement with experimen-
tal observations.
The author would like to thank L. Cartz for reading the
manuscript and making useful suggestions.
Fig. 8. Theoretical variation of the equilibrium distance d be-
tween two wrinkles as a function of the stress σ, for h = 58 nm,
E
f
= 160 GPa, ν
f
S S
= 0.28, E = 5.5 GPa, ν = 0.45, b = 1.3 µm
and σ = 0.28 GPa.
c
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