数理解析研究所講究録 1231 巻 2001 年 56-63

56
Spatio-Temporal Patterns in an Adhesive Tape at Peeling
山崎義弘 (Yoshihiro Yamazaki) 1 , 戸田昭彦 (Akihiko Toda)*

Dept. of Physics, Chuo Univ.
*

Fac. of Integrated Arts and Sciences, Hiroshima Univ.

Abstract
We investigated the dynamical behavior of an adhesive tape at peeling. And adynamical-morphological phase diagram as functions of peel speed and spring constant
was obtained. The spati0-temporal patterns turn out to be classified roughly into fourtypes; low-speed pattern, high-speed pattern, oscillatory pattern, and spati0-temporal intermittent pattern.
keywords :adhesion, dynamical phase diagram, stick-slip motion, fingering instability,
spati0-temporal intermittency

Introduction
Adhesion is acommon phenomenon in our daily life and is characterized as amechanicalbehavior when we separate two materials sticking to each other. It is, however, considered tobe difficult to understand adhesive behaviors quantitatively since we must take into accountnot only amicroscopic change of surface properties but also amacroscopic deformation of anadhesive itself [1]. In order to characterize the mechanical properties of an adhesive, speed-loadcurve is measured by peeling an adhesive tape. This measurement has been carried out as anindustrial test in general. Furthermore this measurement is also important from the viewpointof soft matter physics.
Recently, the dynamical properties of an adhesive tape has been studied by observing themorphology of apeeling front of adhesive [2]. In the case of adhesive failure which meansthat the adhesive does not remain on the surface of the substrate after peeling, two sortsof morphologies are observed depending on the peel speed and load. In one state, adhesive
undergoes alarge deformation and forms atunnel-like morphology. The large deformation

causes the energy dissipation and peel speed becomes low. In the other state, adhesive failure
occurs before the large deformation of adhesive. Therefore the peel speed is larger than thatof the former state, and no morphology is observed. The interesting point is that there is an
1 yamazaki@seagull.phys.chu0-u.ac.jp

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oscillatory region between these two states with respect to peel load. If the experiment withaconstant speed within the intermediate region is carried out, then these two states emerge

alternately in time. We expect that this oscillatory motion in an adhesive tape is also able tounderstand within the common physical framework to other stick-silp phenomena [3] such as
dry friction of granular matter [4], paper-paper slip [5], earthquake [6], and melt fracture inpolymeric liquid [7].
In this Letter, we report the results of our experiments for the further investigation and try
to understand the peeling behavior by constructing adynamical phase diagram.

Experimental Setup
The experimental setup is shown schematically in Fig. 1. AtwO-ply adhesive tape was stickedon asolid plate and the top tape was connected to an elastic spring whose spring constant isdenoted by$k$in this figure. The elastic spring was pulled upward at one end with aconstant

speed $V$ . Therefore the peeling angle in our experiment is 90’. It is noted that we consider thepeeling of an adhesive tape not from the surface of asolid plate but from the back surface ofthe other adhesive tape. We used an adhesive tape with cross-linked adhesive (Denko Corporation). The width of the tape is$\mu \mathrm{m}$thickness. The temperature was 25$25\mathrm{m}\mathrm{m}$$\pm 1\circ \mathrm{c}$$\mathrm{N}\mathrm{o}.31\mathrm{D}$, Nittoand the substrate is aPET film with 25

and humidity was not controlled. It is known

that moisture has little effect on the property of adhesive we used [8]. As control parameters

we changed

$k$

and

$V$

, and we measured the peel load $F$ and observed the patterns left on thesurface of apeeled tape.

Results
Speed-load curves in three different cases, namely, $k=2.9\cross 10^{2},2.9\cross 10^{3}$ , and 2.4$\mathrm{N}/\mathrm{m}$$\cross 10^{4}$

are shown in Fig.2. From Fig.2(a), it is found that two stationary regions represented byaset of white circles (region$\mathrm{A}:V<0.8\mathrm{m}\mathrm{m}/\min$, region$\mathrm{B}:V>2.0\mathrm{m}\mathrm{m}/\min$) exist. In eachregion, peel load increases as peel speed becomes high. The strength of the peel load in theseregions seems to be independent of the spring constant. In the intermediate region between

regions Aand$\mathrm{B}$, oscillatory motion occurs as shown in Fig.2(a). The speed-load behavior in

Fig.2(a) is consistent with the experimental results by Urahama [2]. In addtion we found that58
the intermediate oscillatory region becomes narrower and finally vanishes as$k$increases (seeFig.2(b) and (c) . We also confirmed that the fluctuations in peel load from the average are$)$gradually diminished as$k$increases at afixed peel speed. Therefore we expect that there exists

acritical spring constant as afunction of peel speed at which aHopf bifurcation occurs.
The difference between region Aand region$\mathrm{B}$emerges not only in aspeed-load curve but

also in amorphology of adhesive during peeling. Figure 3shows the typical failure front of anadhesive tape in peeling process. Figs.
and$\mathrm{B}$$3(\mathrm{a})$and (b) are failure fronts in the case of regions A, respectively. The direction of peeling is down. In the case of region$\mathrm{A}$, it is found thatoval holes aligning regularly are formed in front of the failure front. These holes represent crosssections of tunnels made of adhesive as pointed out by Urahama. On the other hand, in thecase of region , no hole is formed and the failure front becomes rough. The difference in the$\mathrm{B}$failure front between these two regions is considered to be caused by the visc0-elastic behaviorof adhesive. In region$\mathrm{A}$, peel speed is low and adhesive is able to deform largely. In region$\mathrm{B}$,however, peel speed is so high that the failure occurs without large deformation in most partof adhesive. Instead of atunnel formation as in the case of region$\mathrm{A}$, fibrils are formed by thelarge deformation only in the limited local part of adhesive. These deformations in both speed

regions occur in the vicinity of the failure front and remain even after peeling. Therefore, we

can identify apeeling process by observing an adhesive tape after peeling.
Figure$4(\mathrm{a})$and (b) show the typical patterns in apeeled adhesive tape in regions Aand$\mathrm{B}$,respectively. In Fig.4(a), astriped pattern is seen. White lines represent tunnels of adhesive,and some defects exist where awhite line is terminated or is splitting into two or three lines.
The splitting of awhite line occurs due to the tip splitting of an oval air hole in Fig.3(a)induced by fingering instability [9]. In contrast, no pattern is observed in Fig.4(b) because
large deformation occurs only in alimited local region of adhesive.

In the intermediate speed region between two regions Aand$\mathrm{B}$, we found two characteristic

pattens depending on the spring constant as shown in Fig.5. In these figures, peeling progresses

from top to bottom. Morphologies observed in the case of regions Aandblack regions in Fig.5, respectively. Figurein the case of region Aand$\mathrm{B}$$5(\mathrm{a})$$\mathrm{B}$are seen as white andis astriped pattern alternating morphologies, and this striped pattern is obtained in the case where the springconstant is small. It is found that this pattern is spatially uniform and periodic emergence of59
two morphologies causes the oscillation of peel load. As the spring constant becomes bigger,

spatial uniformity in the pattern is lost and black and white regions vary not only temporally

but also spatially as shown in Fig.5(b). In the case of ahard spring, the deviation of peel load
must be reduced and asteady state in peel speed and load is realized. The crucial point fromFig.5(b) is that asteady state is realized not by emergence of anew morphology formed bypeeled adhesive but by coexistence of two morphologies obtained in the case of region Aand
B. Therefore, if we change apeel speed, anew steady state is achieved by changing the ratioof white region to black region as shown in Fig.6.
Then, It turns out to be that the spati0-temporal patterns can be classified into four types;

low-speed pattern, high-speed pattern, oscillatory pattern, and spati0-temporal intermittent
pattern. Actually the dynamical-morphological phase diagram in V-k space was obtained asshown in Fig.7. Low-speed and high-speed patterns correspond to morphologies in the case ofregions Aand$\mathrm{B}$, respectively. And oscillatory pattern is characterized as aspatially uniform

striped pattern.

Summary and Discussion
We investigated the dynamical behavior in the peeling process of an adhesive tape from amorphological viewpoint. The features are summerized as follows.
(i) By constructing adynamical-morphological phase diagram in V-k space the spati0-temporal

patterns of apeeled adhesive tape can be classified into four types; low-speed patternhigh-speed pattern$(V>V_{\mathrm{H}})$

temporal intermittent pattern
low-speed pattern, and$V_{\mathrm{H}}$$(V<V_{\mathrm{L}})$,, oscillatory pattern $(V_{\mathrm{L}}<V<V_{\mathrm{H}}, k<k_{\mathrm{c}}(V))$ , and spatial$(V_{\mathrm{L}}<V<V_{\mathrm{H}}, k>k_{\mathrm{c}}(V))$.

$V_{\mathrm{L}}$is the upper limit speed for theis the lower limit speed for the high-speed pattern.

$k_{\mathrm{c}}$is the upperlimit spring constant for aspatially uniform striped pattern.
(ii) From aspeed-load measurement, atransition from asteady state to an oscillatory state

occurs at acritical spring constantrelation between$k_{\mathrm{c}}$and$k^{*}$$k^{*}$.

$k^{*}$

is expected to be aHopf bifurcation point. The

is not clear.

(iii) Asteady state in aspeed-load curve in the case of$V_{\mathrm{L}}<V<V_{\mathrm{H}}$and $k>k^{*}$ is realized

by controlling the rate of aspatial occupation of morphologies in low-speed and high-speed
patterns at atime. And this ratio depends on the peel speed

60The failure front in aspati0-temporal intermittent pattern is shown in Fig.8. The peelingprocess progresses downward. It is found from this figure that the failure in high-speed region

occurs in the middle region of the front line and two kinds of failure fronts as shown in Fig.3
coexist. The important point is that the positions of the failure fronts in two states are different
from each other. Therefore this difference is considered to have an influence on the morphologies
in the vicinity of the interface between two states. In constructing amodel for the dynamicalbehavior of an adhesive, we must take into account not only the effect of spring as aglobal

interaction but also the spatial variation of the failure front as alocal interaction.

Acknowledgments
The authors thank Dr. T. Mizuguchi and Mr. A. Nishimoto for their fruitful discussions.
The authors are grateful to Prof. T. Ohta and Prof. M. Matsushita for their enlightening
discussions.

References[1] C. Gay and L. Leibler. Physics Today, 52(11):48-52, 1999.
[2] Y. Urahama. J. Adhesion, 31:47-58, 1989.

[3] B. N. J. Persson. Sliding Fkiction. Springer-Verlag, Berlin, 1998.
[4] S. Nasuno, A. Kudrolli, A. Bak, and J. P. Gollub. Phys. Rev. E, 58:2161-2171, 1998.

[5] F. Heslot, T. Baumberger, B. Perrin, B. Caroli, and C. Caroli. Phys. Rev. E, 49:4973-4987,1994.
[6] J. M. Carlson and J. S. Langer. Phys. Rev. Lett, 62:2632-2635,1989.
[7] K. Sato and A. Toda. J. Phys. Soc. Jpn., 68:77-85, 1999.
[8] Y. Urahama. private communication. 2001.
[9] M. Rabaud, S. Michalland, and Y. Couder. Phys. Rev. Lett, 64:184-187, 1990

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Figure 1: Schematic representation of the experimental setup.

ooo

15$\overline{\frac{\mathrm{z}}{\triangleleft_{\phi}[mathring]_{A}}}$$\mathrm{o}$$\mathrm{o}\circ$$\circ$$\circ 0\circ$$\mathrm{o}$$\overline{"\omega v}$$\mathrm{o}_{0\nwarrow_{b\mathrm{o}^{\circ}}0^{\circ^{\mathrm{O}}}}$

1$10^{0}$Peel Speed(a)

Peel Speed$[ \mathrm{m}\mathrm{m}/\min]$$k=2.9\cross 10^{2}\mathrm{N}/\mathrm{m}$(b)Peel SPeed$[ \mathrm{m}\mathrm{m}/\min]$(c)$k=2.9\cross 10^{3}\mathrm{N}/\mathrm{m}$$10^{1}$$[ \mathrm{m}\mathrm{m}/\min]$$k=2.4\cross 10^{4}\mathrm{N}/\mathrm{m}$Figure 2: Peel load as afunction of peel speed in three cases with adifferent spring constant.
, (c)
(a)
, (b)
. The bar represents theamplitude of the oscillatory motion in peel load at each fixed peel speed.
$V$$k=2.9\cross 10^{3}\mathrm{N}/\mathrm{m}$$k=2.9\cross 10^{2}\mathrm{N}/\mathrm{m}$(a)$k=2.4\cross 10^{4}\mathrm{N}/\mathrm{m}$(b)Figure 3: Failure fronts at peeling. Peeling progresses from top to bottom, (a) the morphologyof adhesive in the case of region A. (b) the morphology of adhesive in the case of region B. The.
actual size of these figures is 440$\mu \mathrm{m}\cross 290\mu \mathrm{m}$62(b)(a)

Figure 4: Snapshots of typical patterns in apeeled adhesive tape, (a) the case of region A. (b).
the case of region B. The actual size of these figures is 5.0 mm$\cross 5.0\mathrm{m}\mathrm{m}$

(b)

(a)

Figure 5: Snapshots of typical patterns in the intermediate regions of peel speed between region,. (b) spati0-temporal
Aand B. (a) oscillatory pattern;,.
The
actualsize of these figures isintermittent pattern;.
25 mm$k=2.9\cross 10^{3}\mathrm{N}/\mathrm{m}$$k=2.7\cross 10^{3}\mathrm{N}/\mathrm{m}$$V=0.8\mathrm{m}\mathrm{m}/\mathrm{s}\mathrm{e}\mathrm{c}$$V=1.6\mathrm{m}\mathrm{m}/\mathrm{s}\mathrm{e}\mathrm{c}$$\cross 19\mathrm{m}\mathrm{m}$Figure 6: Another example of aspati0-temporal intermittent pattern with adifferent condition.
from Fig.5(b). The actual size of this figure is 15 mm$\cross 11\mathrm{m}\mathrm{m}$63a$\grave{\Xi}$$\overline{\epsilon 6}$$*\mathrm{r}\dot{n}$$\overline{\circ 0}$$\underline{\mathrm{o}}\mathfrak{v}$$.\infty\approx \mathrm{E}$Peel Speed$[ \mathrm{m}\mathrm{m}/\min]$Figure 7: Dynamical-morphological phase diagram in V-k space.

Figure 8: Afailure front in the case of aspati0-temporal intermittent pattern. Two kindsof failure morphologies, namely, morphologies in low-speed pattern and high-speed pattern.
coexist. The actual size of this figure is 1.9 mm$\cross 0.75\mathrm{m}\mathrm{m}$