Energy distributions and effective temperatures in the packing of elastic sheets
aa r X i v : . [ phy s i c s . c l a ss - ph ] D ec Energy distributions and effective temperatures in the packing ofelastic sheets
S. Deboeuf, M. Adda-Bedia and
A. Boudaoud
Laboratoire de Physique Statistique de l’Ecole Normale Sup´erieure, CNRS UMR 8550 - 24 rue Lhomond, 75005 Paris,France
PACS – Continuum mechanics of solids / Static buckling and instability
PACS – Continuum mechanics of solids / Random phenomena and media
PACS – Thin film structure and morphology
PACS – Thermodynamics and statistical mechanics
Abstract. - The packing of elastic sheets is investigated in a quasi two-dimensional experimentalsetup: a sheet is pulled through a rigid hole acting as a container, so that its configuration ismostly prescribed by the cross-section of the sheet in the plane of the hole. The characterisationof the packed configuration is made possible by using refined image analysis. The geometricalproperties and energies of the branches forming the cross-section are broadly distributed. We finddistributions of energy with exponential tails. This setup naturally divides the system into twosub-systems: in contact with the container and within the bulk. While the geometrical propertiesof the sub-systems differ, their energy distributions are identical, indicating ’thermal’ homogeneityand allowing the definition of effective temperatures from the characteristic scales of the energydistributions.
Introduction. –
The challenges raised by out-of-equilibrium systems are exemplified by granular materi-als [1] and glasses [2, 3], featuring complex energy land-scapes and aging. Energy flow, thermal equilibration, andthe statistical properties of energy in such systems can becharacterised by various effective temperatures [4–6]; how-ever, previous experimental studies [7–12] only measureda temperature based on the ratio between fluctuations andresponse of the system. Here we present experiments on amacroscopic out-of-equilibrium system, namely the pack-ing of elastic sheets into quasi two-dimensional contain-ers [13] and focus on the statistical properties of the con-figurations. We measure the distributions of geometricaland energetic properties and show thermal homogeneitywithin the system although its geometrical properties arenot uniform, enabling the definition of effective tempera-tures from the distributions of energy. Thus we obtaina macroscopic experimental system that could be usedto test out-of-equilibrium statistical physics. Our resultsbear on the packing of flexible structures such as elas-tic rods [14–17], crumpled paper [18–21], folded leaves inbuds [22], chromatin in cell nuclei [23] or DNA in viralcapsids [15, 24].At equilibrium, systems with a large number of degrees of freedom are characterised by a single temperature T .On the one hand, the energy of one degree of freedom fol-lows Boltzmann’s distribution, the mean energy being pro-portional to T . On the other hand, T might be measuredusing the fluctuation-dissipation theorem (FDT), relatingfluctuations of an observable to its response to an externalfield. By analogy, two main effective temperatures wereintroduced for systems out of equilibrium. The approachof Edwards [4] amounts to the replacement of T by an ef-fective temperature in the distribution of energies; it canbe extended to intensive thermodynamic parameters as-sociated with global conserved quantities [6, 25–27]. Thegeneralisation of the FDT [5] gives another effective tem-perature, which can be measured [7–12,28]. In many mod-els, Edwards’ and FDT temperatures are equal [29–32] orproportional [33]. An experimental measurement of Ed-wards’ temperature seems to be lacking as it is difficultto obtain energy distributions. Here we measure energydistributions in the packing of elastic sheets. Experimental set-up. –
Fig. 1 a represents the ex-perimental set-up, as introduced in [13], which was in-spired by the study of single d-cones [34]. We use circu-lar polyester (polyethylene terephtalate) sheets of Young’smodulus measured as E = 5 GPa, density 1 . , var-p-1. Deboeuf et al. a)b)c)d) Fig. 1: The experiment. a) Sketch of the set-up showingthe radius of the sheet r , its thickness h , the radius of thehole R and the control parameter Z ; the force F is mea-sured with a dynamometer. b) Thresholded picture of a hor-izontal cross-section of a configuration from set i of experi-ments. c) Analysed cross-section, showing the existence ofmulti-branches stacks delimited by two junction points. Thenumber of branches is indicated near each stack. d)
3D re-construction of the same configuration, assuming exact self-similarity of shape. ious radii r ∼
30 cm and thicknesses h ∼ µ m (seeTable 1). At each realisation, a sheet is pulled from itscentre through a circular rigid hole of radius R ∼
20 mm.The values of the parameters ( r, h, R ) for each set of ex-periments are given in Table 1. The hole is machinedthrough a Plexiglas plate and its edges are rounded to forma toroidal convex shape, to avoid damaging the sheet. Thecenter of the sheet is pierced and fixed to a dynamometerby means of a threaded mount of radius 0 . . Z ,between the pulling point and the plane of the hole is ourmain control parameter. The measurement of the pullingforce F during the compaction directly yields the work in-jected in the system W = R Z F d z . This injected energyserves to pack the sheet and is dissipated through fric-tion. The coefficients of friction for polyester/Plexiglassand polyester/polyester were measured as 0 .
37 and 0 . πZ , that grows within a disk of radius R as Z is in-creased. The experiment allows isotropic confinement topacking ratios P as high as 0 .
11, where P = 2 Zh/R isthe ratio of cross-sectional area of the sheet 2 πZh to thearea of the hole πR .In principle, configurations can be visualised from be-low. However this turns out to be inconvenient as parts ofthe sheet assemble into thick bundles and the edge of thesheet does not lie in a single plane. Therefore we resortto a hot wire cutting tool to obtain cross-sections for onevalue of the control parameter Z m (given in Table 1). Withgreat care, one obtains neat cuts without perturbing theconfiguration. The cross-section is digitised with a scan-ner at a resolution of 50 pixels per mm. A thresholdingresults in a binary image, in which empty spaces of sur-face area larger than (10 h ) are kept, which removes lightnoise from the raw image, as shown in Fig. 1 b . The binaryimage is skeletonized (reduced to a one pixel thick skele-ton); junction points are then defined as pixels with atleast three neighbours. Two neighbouring junction pointsdelimit a stack of branches in close contact. The nextstep is to determine the number of branches in each of the M stacks. The conservation of the number of branchesat each junction point yields 2 M/ M/ h =50 µ m) correspondsto 5 pixels. We keep the M/ h ( µ m) r (cm) B (J) κ c (mm − ) R (mm) Z m (cm) P R P N br i
50 33 7 10 − ii
50 33 7 10 − iii
125 22 1 10 − Table 1: Material parameters for the sheets used in experiments: thickness h , radius r , bending stiffness B and plastic thresholdcurvature κ c ; Control parameters: hole radius R , maximal pulling distance Z m , and packing ratio P = 2 hZ m /R ; Total numbersof realisations R and of branches P N br , on which statistical analyses are based. E ( J ) Fig. 2: Total elastic energy E (measured from the geometryaccording to Eq. 1) and injected work W (measured from thepulling force F ) for the three sets of experiments: i ( ◦ ), ii ( (cid:3) )and iii ( ♦ ). The straight lines are linear fits to each set. solution of the linear M × M system yields the number ofbranches in each stack. We reopened a few configurations(5 per set of experiments) and checked by counting thenumber of branches in each stack: we found no error forsets ii and iii , and an error of ± i . These errors are smallthanks to the fact that the number of branches in a stackis an integer. Thus, we obtain both the geometry and thetopology of the sheet (Fig.1 c , d ).When repeating the experiment with the same experi-mental parameters, a whole variety of shapes is generated,which calls for a statistical approach and an ensemble anal-ysis. We systematically performed and analysed three setsof experiments with a number of realisations R ∼
Total energy. –
We first consider the global ener-getic quantities, injected work W and elastic energy E ,and their correlations. Assuming the shape of the foldedsheet to be exactly self-similar, a cross-section prescribesthe energy of the whole sheet as follows [35]. Using thepolar coordinates ( ρ, θ ) on the initially plane sheet, thebranches, located in the plane ρ ≃ Z m , have a local cur-vature κ ( θ ), and correspond to an angular sector on thesheet, where the curvature is c ( ρ, θ ) = Z m κ ( θ ) /ρ , assum-ing the hole to be small ( R ≪ Z m ). The bending energy E of the whole sheet is B Z rR c Z π c ρ d θ d ρ = BZ m (cid:18) rR c (cid:19)Z πZ m κ ( t ) d t (1)where we introduced t = Z m θ , the curvilinear coordinatein the hole cross-section. The logarithmic prefactor knownfor d-cones [35] contains as cutoffs the radii of the core ofthe cone R c and of the sheet r . In actual experiments, theself-similarity is not exact as some generators end belowthe mount. This affects the logarithmic prefactor throughthe effective value of R c , which might lead to an errorin the overall multiplicative factor of order 1 in the es-timation of the total elastic energy. Here, we chose toestimate R c as the radius of the mount (0 . κ dependence ofthe energy (Eq. 1) was replaced by a linear dependance κ c (2 κ − κ c ) for curvatures greater than the plastic thresh-old κ c (Table 1), measured as in [19].Fig. 2 shows that the bending energy E and the injectedwork W are correlated. Indeed, for each set of experi-ments, E is roughly proportional to W , showing that thestored elastic energy E can be controlled with an externalforce. The unphysical values (mostly in set iii ) such that E > W can be mainly ascribed to the choice of R c as theradius of the mount; choosing, instead, R c of the order ofthe hole radius would shift all data below the line E = W .Another possible source of bias comes from the estimationof the energy of the very few branches with local high cur-vatures ( κ ≫ κ c ) that contribute significantly to the totalenergy. We stress however that these two sources of errordo not affect the statistics of energy as discussed below.Fig. 2 also shows that the global quantities E and W fluctuate over the realisations of a given set as the sys-tem explores its configurational space. Energy dissipationoccurs by friction between layers and with the container,and through discontinuous bifurcations [16] correspondingto reorganisations of the configurations when the confine-ment is increased. The evolution of the overall slope of E ( W ) suggests that the dissipated fraction of energy in-creases with confinement; indeed the more compact set i has the smallest slope. Furthermore, the injected work W is history-dependent as it fluctuates for a given value ofthe elastic energy E . This illustrates the multistability ofthe system, suggesting a complex energy landscape.p-3. Deboeuf et al. −4 −3 −2 −1 l (mm) ρ ( l ) a) −3 −2 −1 −4 −3 −2 −1 κ m (mm −1 ) ρ ( κ m ) −2 −1 −1 b) Fig. 3: Statistics of the geometrical properties for the three setsof experiments: i ( ◦ / • ), ii ( (cid:3) / (cid:4) ) and iii ( ♦ / (cid:7) ) respectivelyin periphery/bulk. a) Experimental pdfs ρ ( ℓ ) of the length ℓ of branches and exponential distributions f E ( ℓ ), Eq. (2) ofthe same mean as the experimental data. For the periphery ρ ( ℓ ) is multiplied by 10 for clarity. Means h ℓ i = 3 .
3, 4 . i ; 4 .
6, 6 . ii ; and 9 .
6, 16 mm for set iii ,respectively for the bulk and the periphery. The inset showsthe two sub-systems: branches in the bulk (green) and at theperiphery (dark). b) Experimental pdfs ρ ( κ m ) of the meancurvature of branches κ m , with same symbols as in a . For thebulk (main panel), the Gamma distribution f G ( κ m ), Eq. (3)with the same mean and variance as the experimental datais plotted; its exponent is α = 0 .
43, 0 .
51 and 0 .
62 and itsmean is h κ m i = 0 .
16, 0 .
12 and 0 .
08 mm − , for sets i , ii and iii , respectively. For the periphery (inset), the distributionsare peaked at the curvature of the containers 1 /R , shown byvertical lines. Statistics of the geometry. –
In the following wedetail the main statistical properties measured over all re-alisations of a given set to insure convergence of the statis-tics. Note that the statistics over one configuration arecompatible with ensemble statistics. Within elastic the-ory of rods, the equilibrium state of a confined rod resultsfrom the torque balance of each branch, whereas the in-teraction between neighbouring branches is mediated bytheir extremities where contact forces and friction comeinto play [16, 36]. This fact allows unambiguously to con-sider branches as the elementary particles of the systemcomparatively to other topologic or geometric decomposi- tion such as contact points or loops which have been usedpreviously [14, 17].As the container constrains the curvature of branches inthe periphery, it is natural to split the system into two sub-systems (inset in Fig. 3 a ): branches with/without contactwith the container, which we will refer to as periphery and bulk , respectively. The two sub-systems roughly containthe same number of branches (60% in periphery). In thefollowing, we measure probability distribution functions ρ ( x ) in each sub-system; we compare these distributionsto analytical pdfs f ( x ) with the same average and varianceas experimental data, instead of direct curve fitting. Theerror bars δx and δρ of the experimental pdfs ρ ( x ), shownin Figs. 3 and 4, are given by the bin width δx and the es-timated standard deviation δρ = ρ/ √ n of the correspond-ing histograms n ( x ). The total number of branches (inperiphery and bulk) for a given set of experiments variesbetween ∼ and ∼ (see Table 1), which allowsfor accurate statistics.Fig. 3 a shows that the lengths ℓ of branches are well-described by exponential distributions f E ( ℓ ) = 1 /µ exp ( − ℓ/µ ) , (2)both in periphery and in bulk. It appears that the valueof the averaged length h ℓ i = µ is significantly larger forbranches at the periphery than in the bulk.Next, we consider the absolute value of the average cur-vature κ m of each branch (Fig. 3 b ). For the bulk (mainpanel of Fig. 3 b ), the distribution ρ ( κ m ) is characterisedby an exponential tail and a weak power law for smallcurvatures, which is well described by a Gamma law withdensity f α,χ G ( κ m ) = ( κ m /χ ) α Γ( α ) κ m exp (cid:18) − κ m χ (cid:19) , (3)where Γ stands for Euler’s Gamma function. In contrast,for the periphery (inset of Fig. 3 b ), the distribution ρ ( κ m )is peaked around the value 1 /R given by the container.Thus, the geometrical properties of the periphery and thebulk are significantly different. Statistics of the energy. –
Each configuration is atmechanical equilibrium, so that any branch can be char-acterised by its elastic energy. The energy e of the branchcorresponds to that of an angular sector on the sheetand is calculated using Eq. (1), with limits of integration t ∈ (0 , ℓ ), where ℓ is the branch length. Surprisingly, theprobability distribution functions ρ ( e ) in periphery andin bulk coincide, as shown in Fig. 4 a , b and c for thethree sets of experiments i , ii and iii respectively. De-spite the heterogeneous geometry of the branches, whichcontributes to their energy through length and curvature,the energy is homogeneous inside the whole system. Thesedistributions are characterised by a power-law divergenceat small values and by exponential tails, as shown by thelog-log and log-lin scales in inset and main panels of Fig. 4.p-4nergy distributions and effective temperatures in the packing of elastic sheets −4 −3 −2 −1 e (J) ρ ( e ) −5 −4 −3 −2 −1 −2 a) −3 −2 −1 e (J) ρ ( e ) −5 −4 −3 −2 −1 −2 b) −3 −2 −1 e (J) ρ ( e ) −5 −4 −3 −2 −1 −2 c) Fig. 4: Statistics of the energy. Experimental pdfs ρ ( e ) ofthe energy e of the branches for the three sets of experiments: i ( a ), ii ( b ) and iii ( c ) in log-lin (main panels) and log-logscales (insets). The distributions are given separately for thetwo sub-systems: bulk ( • , (cid:4) and (cid:7) ) and periphery ( ◦ , (cid:3) and ♦ ). The lines are Gamma distributions Eq. (3) with the samemean and variance as the experimental data. The parametersof the distributions are reported in Table 2. Thus, it is natural to compare them to Gamma distribu-tions. Indeed, they are well-described by Gamma laws f α e ,χ e G ( e ) as given by Eq. (3), with the same average andvariance as the experimental data (see Fig. 4). However,we found exponents α e < α e > r/R c ) of Eq. (1) nor to the plastic threshold κ c . The former amounts to a normalisation of the averageenergy of a given set, while the value of the latter does notchange the statistics since it affects only a few branches. Discussion. –
We investigated the close packing ofelastic sheets in a quasi two-dimensional experimentalsetup allowing the statistical study of the geometry andthe energy of the resulting configurations. These quanti-ties are broadly distributed, suggesting a complex energylandscape. We identified branches as natural elementaryparticles : the shape of a branch is completely prescribedby its length and boundary conditions. The interactionbetween branches is mediated by the contact forces attheir extremities, which is reminiscent of granular pack-ing. The presence of the rigid container led us to splitthe system into two sub-systems: periphery and bulk. Itturns out that the energy of branches is the only quantitywhich is identically distributed in the two sub-systems,even though the geometrical properties differ. This homo-geneity of the distributions of energy is our central result.This property might be an indication of thermal equilibra-tion. Future work should address this important question.Moreover, the energy distributions of the different setsof experiments are characterised by an exponential tailthat is reminiscent of Boltzmann distributions. Conse-quently, the distributions of energy allow to define effec-tive temperatures for each set of experiments: the meanenergy per branch h e i and the characteristic energy givenby the exponential tails χ e . The effective temperatures areordered as h e i < χ e for each set of experiments (Table 2);the sets of temperatures are close for the two sets of ex-periments with the same thickness h and bending stiffness B ( i and ii ), whereas these correspond to very differentpacking ratios ( P = 0 .
11 and 0 . B might be relevant for the value ofthe effective temperatures. However, more work is neededwith this respect because of the inaccurate estimation ofthe overall logarithmic multiplicative factor in Eq. (1).As stated above and shown in Table 2, the exponents α e < f BE ( e ) = g ( e )exp ( βe ) − g ( e ) is thedensity of states. In the case of noninteracting bosons g ( e ) ∼ e ( d − / where d is the space dimension, which canlead to a divergent behaviour of the distribution at smallenergies. Thus the distributions measured here could beinterpreted as obeying a Bose-Einstein statistics with ap-5. Deboeuf et al. h e i (mJ) χ e (mJ) α e Table 2: Effective temperatures: h e i is the mean energy per elementary particle , i.e. per branch; χ e and α e are given bythe tail and the exponent of the Gamma distribution of energyin Fig. 4. power-law g ( e ). A rationale would be as follows: manybranches may be in the same state (when belonging tothe same stack) as bosons; the number of branches is un-prescribed so that the ‘chemical potential’ is zero.Thermal homogeneity suggests a description of the sys-tem in terms of statistical physics. However, our systemis obviously not ergodic, as it must be driven by injectingwork in order to explore the phase space. This drivinghas some similarities with the slow shearing of colloidalglasses [28] or granular materials [10, 30]; however, it isnot stationary and restricts the accessible phase space ateach reconfiguration of the sheet. As in other glassy sys-tems, two different time scales characterise the dynamics:a very slow one associated with the driving and a quickone corresponding to the reconfiguration to local mechan-ical equilibrium. Finally, further experimental and the-oretical work is needed to explain our observations andto confirm our interpretations. Can one predict the dis-tributions from first principles? How universal are thesedistributions? What controls the effective temperaturesmeasured here? ∗ ∗ ∗ We are grateful to G. Angot and J. Da Silva-Quintas fortheir experimental and technical help. This study was sup-ported by the EU through the NEST MechPlant project.LPS is associated with the universities of Paris VI andVII.
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