Pentagon deposits unpack under gentle tapping
aa r X i v : . [ c ond - m a t . s o f t ] O c t epl draft Pentagon deposits unpack under gentle tapping
Ana M. Vidales , , Luis A. Pugnaloni and Irene Ippolito Laboratorio de Ciencia de Superficies y Medios Porosos, Departamento de F´ısica, Universidad Nacional de San Luisy CONICET Groupe Mati`ere Condens´ee et Mat´eriaux, UMR CNRS 6626, Universit´e de Rennes I, F-35042 Rennes Cedex, France Instituto de F´ısica de L´ıquidos y Sistemas Biol´ogicos (UNLP-CONICET), cc. 565, 1900 La Plata, Argentina Grupo de Medios Porosos, Facultad de Ingenier´ıa, Universidad de Buenos Aires, Paseo Col´on 850, 1063 BuenosAires, Argentina and CONICET (Argentina).
PACS – Porous materials; granular materials
PACS – Granular flow: mixing, segregation and stratification
PACS – Computer simulation
Abstract. - We present results from simulations of regular pentagons arranged in a rectangulardie. The particles are subjected to vertical tapping. We study the behavior of the packing fraction,number of contacts and arch distributions as a function of the tapping amplitude. Pentagons showpeculiar features as compared with disks. As a general rule, pentagons tend to form less archesthan disks. Nevertheless, as the tapping amplitude is decreased, the typical size of the pentagonarches grows significantly. As a consequence, a pentagon packing reduces its packing fractionwhen tapped gently in contrast with the behavior found in rounded particle deposits.
Introduction. –
The study of compaction of gran-ular matter under vertical tapping is a subject of muchdebate and consideration. Setting apart all issues relatedto the slow relaxation shown by these systems, and consid-ering only the steady state regime whenever achieved, theinvestigated granular deposits attain rather high packingfractions when tapped very gently. Moreover, these sys-tems display a rapid reduction in the packing fraction astapping intensity is increased followed by a smooth in-crease at large intensities. On the one hand, The initialdecrease in packing fraction has been observed in simula-tion of spheres [1], in 3D experiments with glass beads [2,3]and in simulation of disks [4, 5]. On the other hand, thesmooth increase at large intensities has been pointed out insimulations of disks [4, 5] and in experiments on 2D pack-ings [6]. Also, a hint of this smooth increase can be appre-ciated in early 3D experiments (see figure 2 in Ref. [2]). Insome cases, the steady state may be obtained by extendedconstant intensity tapping; however, other experimentalconfigurations may require a suitable annealing in orderto achieve the so called ’reversible branch’ [3].Up to now, few studies of this type have been carried outon pointed objects. A first experimental investigation usesassemblies of spheres to build up more complex objectswhich however retain the smooth edges of the constituents [7].Here, we show that packings of pentagons simulatedthrough a pseudo molecular dynamics method (PMDM)present a response to vertical tapping which is significantlydifferent from that observed in packings of rounded grainslike disks or spheres. We base our assertions on the com-parisons with disk packings obtained with an analogousmethod.It is worth mentioning that studies on pentagon assem-blies do exist [8–10]. These make special emphasis onthe crystallization of these systems. However, these ex-periments and simulations consider systems which relaxcontinuously under the effect of a background vibration(either thermal or mechanical agitation).
The model. –
The main satages of our simulationsconsist in: (a) the generation of an irregular base, (b) se-quential deposition of pentagons to create an initial pack-ing, (c) vertical tapping obtained through vertical expan-sion followed by small random rearrangements, and (d)non-sequential (simultaneous) deposition of the pentagonsusing a PMDM.We sample 1000 regular pentagons from a uniform sizedistribution (5% dispersion). A number of them are placedat the bottom of a rectangular die in a disorder way in or-der to create an irregular base. Arranged in this manner,p-1na M. Vidales et al. A=3.0A=2.0
A=1.7
A=1.6A=1.5 A=1.4 A=1.3A=1.2 A=1.1 number of taps pa ck i ng f r a c t i on Fig. 1: Packing fraction of pentagons as a function of the num-ber of taps. The different curves correspond to different am-plitudes A , as indicated on the plot. the N base particles fix the wall-to-wall width of the diewhich is about 40 particle diameters. These pentagonsremain still over the course of the tapping protocol. Theremaining pentagons are poured one at a time from thetop of the die and from random horizontal positions andrandom orientations. Each grain falls following a steepestdescendent algorithm. When a pentagon touches an al-ready deposited particle, it is allowed to rotate about thecontact point until a new contact is made or until the con-tact point no longer constrains the downward motion ofthe particle which is deemed to fall freely again. If a parti-cle has reached two contacts such that the x -coordinate ofits center of mass lies between them, the pentagon is con-sidered stable. Otherwise, the pentagon will be allowed torotate around the contact point with lower y-coordinate.Side walls are considered without friction.Once the initial configuration is obtained, a tapping pro-cess is carried out by using an algorithm that mimics theeffect of a vertical tap of amplitude A . The system is ex-panded by vertically scaling all the y -coordinates of theparticle centers by a factor A >
1. Base particles are notsubjected to this expansion. We then introduce a hori-zontal random noise for those particles touching any ofthe walls of the die. This is done by attempting to dis-place each of these particles a random distance in the range[0 , A −
1] towards the center of the die in the x -direction.Only if the new position of a pentagon does not originatean overlap with neighbor pentagons the move is accepted.Each of these particles has only one chance to move. Thisprocess mimics in some way the shaking that grains suf-fer in a real experiment because of the collisions with thewalls. Note that the amplitude of the random moves isproportional to the amplitude of the expansion.After expansion and random rearrangements, the par- ticles are allowed to deposit non-sequentially (i.e. simul-taneously rather than one at a time) following an algo-rithm similar to that designed by Manna and Khakhar fordisks [11, 12]. In brief, this is a pseudo dynamic methodthat consists in small falls and rolls of the grains untilthey come to rest by contacting other particles or the sys-tem boundaries. Particles are moved one at a time butthey perform only small moves that do not perturb to asignificant extent the ulterior motion of the other parti-cles in the system. For very small particle displacementsthis method yields a realistic simultaneous deposition ofthe grains. Details on the convergence of the results fordecreasing values of the size of the particle displacementswill be presented elsewere.Once all pentagons come to rest, the system is verti-cally expanded again and a new cycle begins. After alarge number of taps, the packing attains a steady statewhose characteristic parameters fluctuate around equilib-rium values.We study the packing fraction φ , coordination number < z > , and the arch size distribution n ( s ) of the de-posits. To identify arches one needs first to identify thetwo supporting particles of each pentagon in the packing.Then, arches can be identified in the usual way [5]: wefirst find all mutually stable particles —which we defineas directly connected— and then we find the arches aschains of connected particles. Two pentagons A and Bare mutually stable if A supports B and B supports A.Unlike disk deposits generated through PMDM, pentagonpackings present capriciously shaped arches. Results and discussion. –
The packing fraction ofthe pentagon deposits is plotted against the number oftaps for various tapping amplitudes in Fig. 1. It is clearlyseen that, as the amplitude is increased, compaction is en-hanced. This trend is similar to the one found by Knightet al. [13], where they observed an increase in the packingfraction with the tapping intensity. However, our systemreaches a clear plateau after a moderate number of tapsirrespective of the tapping amplitude while in Ref. [13] thesteady state was hardly achieved for high tapping ampli-tudes and definitely not reached for low A . Experiments in2D packings [14] of disks show a much faster equilibrationthan the 3D packings of Ref. [13].We show snapshots of part of two packings in Fig. 2.Part ( a ) shows a picture of part of the whole assemblyof a deposit of 1000 pentagons after being shaken 5 × times at A = 1 .
2. Part ( b ) shows the same situation but for A = 1 .
7. Arches formed among particles are indicated bysegments and will be discussed below. It can be seen thatthe final equilibrium positions of the particles in each caseare quite different. At low A , the creation of long archesdue to blocked rollings of the particles gives as a resulta lower φ in comparison with that shown for a packingtapped at higher amplitudes. Moving the particles fartherappart during expansion allows them to rearrange betterand to increase side-to-side contacts.p-2entagon deposits (b) (a) Fig. 2: Examples of two packing tapped during 5 x times.We only show part of the 1,000 particles assemblies. Archesare indicated by segments. (a) A = 1 . A = 1 . In Fig. 3 we plot the final values of φ , obtained whenthe system attains the steady state regime (averaging overthe last 1000 taps) as a function of the tapping amplitude.We compare results with the same experiment carried outon disks [5] (using the same size dispersion, number ofparticles and die size) and with a pentagon limiting caseobtained as follows. We rise all pentagons up to a largeheight and let them fall again, one at a time and in order ofheight (the lower particle first). This process leads to thehighest compaction. The tapped deposits approach thisvalue of φ when A is increased, as seen in Fig. 3. Sincethe deposition is sequential, pentagons do not form archesat all in the limiting case.There are two clear distinctions between the behaviorshown by disks and that displayed by pentagons. Firstly,disks attain larger packing fractions at all tapping am-plitudes. This is to be expected since pentagons, if notcarefully arranged, tend to leave large interstitial spaces.Secondly, while disks present a nonmonotonic dependenceof φ versus A , pentagons show a monotonic increase inthe packing fraction. At high values of A both systemsincrease φ with increasing tapping amplitudes and even-tually reach a maximum plateau value. For low A we findthat disks tend to order and so increase φ as A is de-creased [5]. A minimun in the packing fraction of disks isthen located at intermediate values of A . However, thisfeature is not present in pentagon packings. Pentagonsseem not to order at low A , and φ does not present aminimum as in disk packings.Realistic molecular dynamic simulations of the tappingof pentagon packings yield higher densities overall due tothe particular thermal like vibrations this type of simula- sequential depositionlimit for pentagons(0.7285) f i na l pa ck i ng f r a c t i on A Fig. 3: Final values of the packing fraction obtained by av-eraging over the last 1 ,
000 taps as a function of the tappingamplitude for disks (circles) and pentagons (pentagons). Thehorizontal dotted line corresponds to the sequential depositionlimiting case for pentagons (see text for details). tion suffer until equilibrium is achieved [15]. This featureresembles the high densification of packings obtained inRef. [8]. The same effect is observed when realistic molec-ular dynamics of disks [4] are compared with correspond-ing PMDM [5]. However, PMDM has been shown to yieldthe same general trends observed in realistic moleculardynamics (compare Ref. [4] with Ref. [5]).In pentagon packings, we find that the number of archespresents a monotonic decrease with increasing A in con-trast with the behavior of disks that present a maximum atthe same tapping amplitude where the minimum packingfraction is achieved. It is particularly interesting that at A < . A . We confirmhere that for low A pentagons have a larger tendency toform large arches (up to 20 particles), whereas disks formarches of less than 10 particles. A detailed study of theparticle-particle contacts and the formation of arches willbe presented elsewhere.In order to assess whether the tapping protocol appliedto the packings is significant in the results discused above,we have carried out an annealed tapping on our packingsto compare with the constant tapping discused up to thispoint. We start from a sequentially deposited packing andp-3na M. Vidales et al. -5 -4 -3 -2 -1 s n ( s ) Fig. 4: Distribution of arch sizes for disks (circles) and pen-tagons (pentagons) at A = 1 . A = 3 . then tap the system at variable amplitude. The amplitudewas increased from A = 1 . A = 1 . . ,
000 taps where applied at each amplitude value.Then, the same protocol was followed but for decreasingamplitudes. No evidence of hysteresis nor irreversibilityis found in our results. We also found that the annealingcurves coincide with the constant tapping results of Fig.3. Both, disks and pentagons, attain a unique packingfraction value for given tapping amplitude no matters thehistory of the tapping protocol.Previous simulations on disks [5] and experiments onglass beads [2] do show an irreversible branch in this typeof experiment. It is important to note that in the caseof previous simulations [5] the annealing was conductedin a different manner since the tapping amplitude was in-creased in a quasi-continuum fashion and a single tap wasapplied at each value of A . This prevented the disk pack-ing from reaching the steady state at each value of A . Inthe present work we give sufficient time for the system toreach the steady state at each amplitude. On the otherhand, the annealing experiments by Nowak et al. [2] wereconducted in much the same way as our simulations, how-ever, their system presented a very slow relaxation thateffectively prevented the packing from ’equilibration’ ateach tapping amplitude.To get a closer insight into the ’peculiar’ behavior ofpentagons (i.e. the reduction of packing fraction as A di-minishes) we show in Fig. 5 the evolution of a pentagondeposit after a sudden reduction in tapping amplitude.After 5000 taps applied to the system with A = 1 . A = 1 . A induces a rapid reduction in packing fraction associ-ated with an increase in the size of the arches formed (see A=1.50 tap=5000
Number of taps P a ck i ng f r a c t i on A=1.10 tap=5001
Fig. 5: Packing fraction of pentagons as a function of the num-ber of taps before and after a sudden reduction in tappingamplitude. From t = 1 to t = 5000 the packing is tapped with A = 1 .
5, from t = 5000 on the amplitude is set to A = 1 . insets in Fig 5), in contrast with the behavior generallyobserved in deposits of disks. This seemingly paradoxi-cal effect is in fact simple to explain. Arches —which arethe main void-forming structures— are more easily createdwhen particles start deposition from an initial high densityexpanded configuration. At low A , the expanded config-uration leaves particles very close to each other and thismake particles to meet each other more often during depo-sition, enhancing the probability of arch formation. Thishas been discused recently by Roussel et al. [16] and it hasbeen observed by Blumenfeld et al. [6] in experiments ofcompaction in two dimensional granular systems. Conclusions. –
We have shown that, for pentagons,either through constant tapping or annealing, the steadystate of the packing presents a monotonically increasingpacking fraction with tapping intensity. However, disksand spheres display a clear initial reduction in the packingfraction as tapping intensity is increased followed by asmooth increase at large amplitudes [1–6]. Such findingreveals that the complexity of pentagon deposition leads toan unexpectedly simpler behavior of the packing fractionas compared with simpler systems.To our understanding, the behavior of rounded par-ticles —which increase density on reduction of tappingintensity— are indeed puzzling; while pentagons seem tobehave as expected. If grains fall from a highly compactexpanded configuration they should form more arches, andhence reduce packing fraction. Rounded particles do notfollow this pattern as has been observed in experimentsand simulations of various kinds. Although the behaviorof rounded particles seems to be considered as reasonablefor most workers, no thorough discusion on this has beenp-4entagon deposits A pa ck i ng f r a c t i on Fig. 6: Packing fraction for disks as a function of A for twovalues of size dispersion: 5% (circles), and 50% (squares). given in the literature. Most authors explain the effect onthe basis that large taps create voids but do not explainhow these voids are created from a mechanical point ofview. According to the detailed discussion presented byRoussel et al. [16] large taps should destroy arches (andthen voids). We believe that the ’reasonable’ behavior isthat large taps eliminate arches and voids; however, at lowtapping amplitudes, we presume that this phenomenoncompetes with the crystal-like ordering that reduces archformation in disk packings in our simulations.We have tested the hypothesis that partial orderingleads the nonmonotonic behavior of disks and spheres.However, some trial simulations carried out with ratherpolydisperse disks that are known to show frustration oforder still present the same nonmonotonic features, al-though less marked than in monosized disks (see Fig. 6).A sensible explanation for the formation of large archesat low tapping amplitude should in principle shed light onthis issue. At present we can only suggest that pentagons(and any other pointed particles) have a larger tendencyto multiparticle collisions. Multiparticle collisions arenecessary (although not sufficient) to form many-particlearches. These multiparticle collisions are enhanced bytwo factors: (a) the fact that pentagons may approacheach other closer than disks (recall that a side-to-side con-tact leaves pentagon centers separated by ≈ . ∗ ∗ ∗ AMV thanks to the Groupe Mati`ere Condens´ee et Mat´eriaux and to the Universit´e de Rennes I, France,for their hospitality and support during this investiga-tion. Special thanks to Luc Oger, Patrick Richard andRodolfo U˜nac for valuable suggestions and discussions.LAP acknowledges financial support from CONICET (Ar-gentina). AMV and II acknowledge financial support fromCONICET (Argentina) under project PIP 5496.
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