The enigma of the two interleaved phonebooks

Hector Alarcon,1, 2 Thomas Salez,2 Christophe Poulard,1 Jean-Francis

Bloch,3 Elie Raphael,2 Kari Dalnoki-Veress,2, 4 and Frederic Restagno1, ∗

1Laboratoire de Physique des Solides, CNRS & Universite Paris-Sud, 91405 Orsay Cedex.2Laboratoire de Physico-Chimie Theorique, UMR CNRS 7083 Gulliver,

ESPCI ParisTech, PSL Research University, Paris, France.3Universite de Grenoble Alpes, LGP2, CS 10065, 38402 Saint-Martin-d’Heres, France.

4Department of Physics & Astronomy, McMaster University, Hamilton, Canada.

Many of us are familiar with a classical demon-stration of the strength of friction: take twophonebooks, interleave their sheets and try toseparate them by pulling on their spines. Thisdemonstration has been carried out spectacularlyby attempting to pull the books apart with peo-ple, trucks, lifting a car [1], and even two militarytanks [2], only to fail and suggest that the innerfriction between these sheets prevails. The sim-ple explanation often given is that gravity pro-vides the normal force that generates the tangen-tial friction, but this hypothesis is easily provento be wrong as there is no discernible differencebetween such an experiment carried out in thevertical or horizontal direction. In this Letter,we study the force needed to separate two booksas a function of the number of sheets, the thick-ness of the sheets, and the interleaving distance.In particular, we show that the force required toseparate the books increases more than exponen-tially with the number of sheets. The strengthof the system is to be due to the operator: theperson, car, truck, or tank, amplifies any smallfriction arising from the normal force acting onthe boundaries of the stack. We present a sim-ple model that captures all the data into a self-similar master curve. The model depends onone single dimensionless amplification parameter,and thus gives insight into the mechanisms atplay in this deceivingly complex system. In ad-dition to solving a long-standing familiar enigmarelated to the classical problem of friction, thismodel system provides a framework within whichone can accurately measure friction forces andcoefficients at low loads, and opens the way tothe technologically-relevant engineering of fric-tion in complex assemblies from the macro to thenanoscale.

The first-known systematic studies of friction were car-ried out five centuries ago by da Vinci [3, 4] who discov-ered basic rules which were later confirmed by Amon-tons [5]. In particular, these laws establish that the fric-tion force is independent of the contact area and pro-portional to the applied load during sliding, the propor-tionality constant being the coefficient of kinetic friction.

(a) (b)

(c)

hn

hn-1

A

B

θn

FIG. 1. The enigma of the two interleaved phonebooks,experimental setup, and theoretical model. a) Liftinga car with two interleaved phonebooks [1]. b) Schematic ofthe experimental setup. c) Schematic of the interleaved booksintroducing the local traction force Tn exerted by the operatoron the nth paper sheet. The two vertical lines represent theclamps, while the other lines represent the central lines of eachsheet.

Coulomb rediscovered these laws and further determinedthat, during sliding, friction is independent of the rel-ative speed between the surfaces [6]. This simple setof rules, collectively known as Amontons-Coulomb (AC)laws, has been well studied in macroscopic experimentsover the centuries. During the last decades, efforts onthe micro and nanoscale have resulted in a resurgence ofactivity in tribology. For example, tools like the surfaceforce apparatus and the atomic force microscope havefuelled experimental efforts [7–9], as well as advancedtheoretical treatments that go well beyond phenomenol-ogy [10–12]. Interest in the development of micro-electro-

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mechanical-systems and mechanical devices that oper-ate on small length scales has driven much of this re-search. At the extreme limit, down to the nanoscale, itwas found that the energy dissipation in friction dependson both electronic and phononic contributions, and thatdifferences in the electron-phonon coupling between sin-gle and bilayer sheets of graphene result in variationsin friction [13]. Furthermore, friction was probed in ex-periments on multi-walled carbon nanotubes and boronnitride nanotubes [14–16]. In such investigations, theinner tubes could be slid out of the outer tube reveal-ing vanishingly-small molecular friction for carbon, anda much stronger, area-dependent, molecular friction forboron nitride [16]. Clearly, such works reveal drastic de-partures from the simple AC laws, as one approaches thenanoscale. Moreover, friction can be further complicatedby a stick-slip response, as seen in many cases [17, 18],including layers of paper [19, 20]. Remarkably, althoughthese complex non-linear phenomena may be observed atmicrometric scales, it is often the case on larger lengthscales that the simple AC laws with which we are so fa-miliar are valid (see reviews [8, 10, 18, 21, 22]).

In this Letter, we focus on the following stubborn pop-ular enigma. As shown in Fig. 1(a), an entire car can belifted using a coupling element prepared from two inter-leaved phonebooks [1]. In order to investigate this strik-ing observation, we developed a well-controlled system oftwo identical books, both made up of 2M � 1 identicalsheets of paper with width w = 12 cm, length L = 25 cm,and thickness ε = 0.10 mm. The two books are perfectlyinterleaved sheet-by-sheet and are then separated in avertical orientation, while the total traction force T ismeasured (see Fig. 1(b)). The length of overlap betweenthe two books is denoted by L−d, where d is the separa-tion distance from the clamp of each book to the contactzone (see Fig. 1(c)). Consistent with the actual experi-mental parameters, we make the simplifying assumptionthat d is large compared to the total thickness 2Mε ofone book, so that the angle θn made by the nth sheet asit traverses from the clamp to the contact zone is small.Therefore, L−d is nearly identical for all sheets. Finally,the books are separated at constant velocity (typically1 mm/min) and we have found, in accordance with theAC laws, that the velocity does not significantly affectthe results.

As the books are pulled apart, an initial maximumtraction force is first reached before they start to slidewith respect to each other. In Fig. 2, we show the rawtotal traction force T , measured during constant-velocitysliding, as a function of the distance d, for seven experi-ments with M ranging from 12 to 100. As observed, thesmaller d or the larger M , the larger the traction, andthose dependences are highly nonlinear. Furthermore,the amplification of friction is far from being a small ef-fect: a single experiment spans over three decades in thetraction force. Additionally, a tenfold increase in the

50 150 25010−1

101

103

d [mm]

T[N

]

0 50 100 15010−1

101

103

49.8 50.2 50.6

500

800

M

M50

FIG. 2. Measured total traction force as a function ofseparation distance for various numbers of sheets. Thenumber of sheets for each of the two identical books is 2M ,where M = 12, 16, 23, 27, 50, 75, and 100, with M increasingas indicated by the arrow. The left inset shows a zoomed re-gion for M = 50 highlighting stick-slip friction. The rightinset shows three experiments with M = 50, and varying ini-tial separation distances d0 = 11, 14.5, and 22 mm (verticalarrows).

number of sheets (e.g. M = 12 and M = 100) induces afour orders of magnitude increase in the traction force. Inthe left inset of Fig. 2, we can see clear evidence of stick-slip [19, 20] for an experiment with M = 50. However,the difference between the local maxima and minima,which results from the difference between the coefficientsof static and kinetic friction, is negligible in comparisonto the global amplitude in T . We can thus neglect thiseffect, as well as the difference in the coefficients of staticand kinetic friction. Furthermore, in the right inset ofFig. 2, we see that the experiments are reproducible andindependent of the initial separation distance d0.

These results can be explained using simple geometri-cal and mechanical arguments, for which Fig. 1(c) pro-vides detailed notations. Each sheet of a given book isindexed by n ranging from n = 1 in the middle of thebook to n = M at one extremity. The problem is sym-metric with respect to the central line of the book. Wedefine Hn ≡ hn/d = nε/d, where hn is the shift in posi-tion of the nth sheet in the contact zone with respect toits position of clamping. As a consequence, the tilt angleθn of the nth sheet satisfies sin θn = Hn/

√1 +H2

n and

cos θn = 1/√

1 +H2n. Essentially, the tilting of each in-

dividual sheet n in the intermediate region between theclamping and contact zones converts part of the localtraction Tn exerted on it by the operator (at point A) intoa supplementary local normal force Tn tan θn = HnTnexerted on the stack below it (at point B). Therefore,according to AC laws, this leads to a self-induced addi-tional inner friction force that resists the traction: themore the operator pulls, the higher the frictional resis-

3

tance. At onset of sliding, the change in traction with nreads:

Tn − Tn+1 = 4µHn Tn , (1)

where µ is the coefficient of kinetic friction, and the twofactors of 2 come from the identical contributions of thetwo books and the two pages of one sheet. Finally, theboundary condition is given by TM = T ∗, where T ∗ isthe unknown traction exerted on the outer sheet, andthe total traction force is defined by T = 2

∑Mk=1 Tk.

We now introduce the variable z = n/M , as well as thedimensionless amplification parameter α = 2µεM2/d,and use a continuous description by replacing Tn by T (z),since M � 1. We thus obtain the ODE:

T ′(z) + 2αz T (z) = 0 . (2)

Together with the boundary condition T (1) = T ∗,we get by integration the local traction force T (z) =T ∗ exp[α(1 − z2)]. Finally, we obtain the total traction

force T = 2M∫ 1

0dz T (z) in the self-similar form:

T2MT ∗

=

√π

4αexp(α) erf

(√α). (3)

This expression is remarkable fort two reasons. First, ittends to 1 as α→ 0, which means that it represents thegeometrical amplification gain of friction with respect toa simple linear collection of 2M independent flat sheetswith local friction T ∗. Secondly, it depends solely andalmost exponentially on the amplification parameter α,which thus appears as the central dimensionless param-eter of this study.

We find that the data is well described by Eq. (3), us-ing a single value T ∗ = 0.01 N of the microscopic tractionforce, and with a single coefficient of kinetic friction fitfreely to each traction experiment. A typical example isshown in the inset of Fig. 3, for which an excellent fitis obtained over more than two decades in the tractionforce, with a constant value µ = 0.73 ± 0.02. The rangeover which our simple model is applicable, with a con-stant µ, is clear from the fit, and deviations are observedat small overlap between the books. Note that the tinyforce T ∗ acting on the outmost sheets is a crucial bound-ary condition. It is this finite force, corresponding to nomore than the weight of a butterfly, that is self-amplifiedby the operator – either in a well-controlled experimentas we have carried out here, or when lifting a car withphonebooks [1]. In our experiment, the boundary forceT ∗ originates from the elasticity of the paper: the outersheets have a tendency to be flat and resist slightly thebowing induced in the contact region, thus creating asmall normal force resulting in a small friction force.

The coefficient of kinetic friction is a phenomenologicalquantity whose value depends on several parameters [23].For metal-on-metal [24] and paper-on-paper [25], µ is

10−2 10−1 100 1010

0.5

1

1.5

2

2.5

3

[N]

µ 50 100 150100

101

102

103

d [ mm ]

T[Ν

]

FIG. 3. Coefficient of kinetic friction as a function ofthe average effective load per page. (circles) Values ofµ obtained by fitting each experiment performed on the inter-leaved books to Eq. (3), as a function of the average effectiveload 〈N〉 defined in text. Since µ is load-dependent, we indi-cate the traction range over which Eq. (3) was fit to the databy the grey horizontal lines. (squares) Values of µ obtainedindependently using a tribometer for two sheets of the samepaper. In this case, 〈N〉 is simply the externally-applied load,and the error bars correspond to the standard deviation of themeasurements that have been repeated at least five times. Theinset shows a fit to Eq. (3) (dashed line) of a traction experi-ment (plain line), for M = 50.

known to be dependent on the normal load. Specifically,µ increases with decreasing load and can even be higherthan 1 for small loads, due to adhesion forces [26]. Thisfact is confirmed in Fig. 3, where we plot the coefficientof kinetic friction obtained from a standard tribology ex-periment between two sheets of paper, as a function ofthe applied normal load (squares). Let us now comparethis result with the interleaved-book case. As explainedabove, our model is based on the AC laws and considersa single average µ for each experiment, that is constantirrespective of the load. This hypothesis is equivalent totaking a coarse-grained approach, even though the loadincreases towards the centre of the books and at smalld. The previous assumption is sufficient to describe thedata, as evidenced by the fit in the inset of Fig. 3. Then,we define through AC laws an effective load per page,N = T /(4Mµ), which would correspond to the load ap-plied on each page if the 2M sheets were parallel (i.e.all θn = 0). Note that, by defining the microscopic loadN ∗ ≡ T ∗/(2µ), Eq. (3) also provides the amplificationgain N/N ∗ in normal load. The average traction force〈T 〉 of a given experiment is calculated over the rangeof applicability of the theory, and used to obtain the av-erage effective load per page 〈N〉. The result µ(〈N〉) isshown in Fig. 3 (circles), with the ranges of applicabil-ity indicated by the grey horizontal lines. In short, asingle value of µ could be used to adequately describeeach T (d) experiment with the theory, even though theload varied over the indicated range. The consistency

4

0 2 4 6 810 -1

1

10 3

α

T (2

MT

∗ )−1

FIG. 4. Self-similar master curve of the rescaled totaltraction force as a function of the amplification param-eter. To produce this figure, each of the data sets shown inFig. 2 (same colour code) was fitted to Eq. (3) (dashed line),using T ∗ = 0.01 N, and µ as a free parameter (see Fig. 3).

between the simple tribology data and that of the in-terleaved books provides confidence in the model. Fur-thermore, it is noteworthy that the classic phonebookdemonstration probes a broad range of loads, that typi-cally consists of two decades, and a nearly sixfold changein the coefficient of kinetic friction.

The self-similar Eq. (3) suggests that with T ∗ =0.01 N, and the best fit values of µ for each experiment(see Fig. 3), all data should collapse when normalised ap-propriately. Indeed, in Fig. 4, we plot the rescaled totaltraction force as a function of the amplification parame-ter, and observe a single master curve for all experiments,with varying d and N .

The tremendous geometrical amplification of frictionthrough the interleaving of the 2M sheets of each bookbears a resemblance to two everyday examples. First, asailor can moor a large ship simply by wrapping a ropearound a cylinder known as a capstan [27, 28]. Note thatits principle might be relevant to the interaction betweenDNA and a bacteriophage capsid [29]. The second ex-ample is a simple toy known as the Chinese finger trap,where a woven helical braid is loosely wrapped around afinger. The braid tightens and the finger is trapped as itis pulled. The trapping mechanism results from a simpleconversion of the traction force to an orthogonal compo-nent, which enhances the load and thus the friction. Thistype of braid is applicable to sutures in surgery [30], andis also thought to play a role in adhesive proteins [31].

Crucial to the geometrical amplification of friction withinterleaved books are the angles θn that the sheets makeas they approach the contact region. It is through theseangles that the traction forces result in loads perpen-dicular to the paper-paper interfaces, and thus in largeself-created friction forces. This fact is easily verified:one can realise an interleaved-book system with θn = 0,

by removing alternating sheets in two notepads. In sucha case, the books can be easily pulled apart, consistentwith the theory presented.

Finally, it is clear from the normalised data in Fig. 4that the experiments deviate from the model at verysmall loads. This is the result of assuming a coefficientof kinetic friction that is independent of the load. Whileit is certainly possible to include numerically an ad-hocload-dependent coefficient of kinetic friction, this wouldbe at the cost of simplicity and additional free param-eters. In this Letter, we have instead opted to capturethe essential physics needed to elucidate the interleaved-phonebook enigma, and we reveal the key dimensionlessparameter α = 2µεM2/d of the problem.

Experimental Methods

The experiments were carried out on paper sheets man-ufactured by InacopiaTM, with thickness ε = 0.10 mm.Books were prepared by stacking sheets and holding theassembly at one end with a rigid aluminium clamp. Theinterleaved books were mounted by their clamps verti-cally in a traction instrument (Adamel Lhomargy DY32).The displacement was measured (with an accuracy of10 µm), along with the total traction force T . Sincethe experiments explored different ranges of force, threesensors were used with maximum forces of 10 N, 100 N,and 1000 N. All the measurements were carried out atambient humidity, which varied from 30 to 60%. Thoughhumidity affects the friction of paper [26], this effect wasnegligible as indicated by the reproducibility of the ex-periments (see inset of Fig. 2).

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Acknowledgments The authors thank the televisionchannel France 5 and the 2P2L producing company, forthe scientific program “On n’est pas que des cobayes” [1]which stimulated this study through the participationof F.R. and C.P., as well as the French ANR (WAFPIproject: ANR-11-BS04-0030) and NSERC of Canada forfunding.

Author Contributions H.A. performed theinterleaved-book experiments; J.F.B. performed theclassical paper friction experiments; T.S., E.R. andK.D.-V. developed the theoretical model; C.P. and F.R.designed the study; H.A., T.S., E.R., K.D.-V. and F.R.wrote the manuscript; all the authors discussed theresults and the content of the article.