Bridges in three-dimensional granular packings: experiments and simulations
Yixin Cao, Bandan Chakrabortty, G. C. Barker, Anita Mehta, Yujie Wang
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Bridges in three-dimensional granular packings: experiments andsimulations
Y. X. Cao , B. Chakrabortty , G. C. Barker , A. Mehta and Y. J. Wang
Physics Department, Shanghai Jiao Tong University - 800 Dongchuan Rd. Shanghai 200240, China Theory Department, S N Bose National Centre - Block JD Sector III, Salt Lake, Calcutta 700098, India Institute of Food Research, Norwich Research Park - Norwich NR4 7UA, UK
PACS – Static sandpiles; granular compaction
PACS – Structures and organization in complex systems
PACS – Porous materials; granular materials
Abstract – In this Letter, we present the first experimental study of bridge structures in three-dimensional dry granular packings. When bridges are small, they are predominantly ‘linear’, andhave an exponential size distribution. Larger, predominantly ‘complex’ bridges, are confirmed tofollow a power-law size distribution. Our experiments, which use X-ray tomography, are in goodagreement with the simulations presented here, for the distribution of sizes, end-to-end lengths,base extensions and orientations of predominantly linear bridges. Quantitative differences betweenthe present experiment and earlier simulations suggest that packing fraction is an importantdeterminant of bridge structure.
The study of random granular packings remains an ac-tive research field [1]. For packings containing frictionalgrains, it is now well established that cooperative struc-tures such as bridges, are ubiquitous: these are defined ascollective structures where neighboring grains rely on eachother for mutual stability [2]. In other words, bridges arestructures within a random close packed deposit that areinconsistent with the results of sequential deposition. Theword bridge is descriptive (arch would be equally valid)and is used widely within powder and bulk processing.Bridges are a measureable property of complex three di-mensional granular structures and so their investigationcan help build a more complete picture of the relationshipbetween deposition processes and deposit structures.Apart from their intrinsic interest, bridges are inti-mately related to the phenomenon of compaction [3]; it hasbeen shown that much of the compaction to high densitiesin shaken packings occurs via the cooperative dynamics ofbridge collapse [4]. Bridges can be further classified [5] aslinear or complex (see Fig. 1), depending on the topologyof their backbones or contact networks: complex bridgeshave backbones with loops and/or branches, while linearbridges do not. Their dynamical implications are equally (a)
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E-mail: [email protected] (c)
E-mail: [email protected]
Fig. 1: (a) A typical linear bridge. (b) The backbone of thelinear bridge shown in (a). (c) A typical complex bridge. (d)The backbone of the complex bridge shown in (c). interesting: it has been suggested that grain motion closeto the jamming transition [6] is via the motion of ‘dy-namic linear chains’ [7,8], which are akin to linear bridges[5]. Force networks in anisotropically sheared static pack-ings [9, 10] show fractal dimensions similar to those of lin-p-1 a r X i v : . [ c ond - m a t . s o f t ] M a y . X. Cao et al. ear bridges formed under gravity, thus reinforcing connec-tions that have already been made [3, 11, 12] between lin-ear bridges and force chains. It should be emphasised thatthese connections are still somewhat qualitative, and thatthe precise characterisation of the relationship betweenforce chains and bridges is an important and emergentresearch area. That there must be such a connection isevident: bridges are structures within a granular packing,whereas force chains involve combinations of complex in-terparticle forces constrained by the structure. While thecommon element is the underlying contact network, thisalone does not define force chains — in that many forcenetworks are consistent with a mechanically stable struc-ture containing bridges. From this point of view, studiesof force chains and bridges are necessarily complementaryin nature, and the study of bridges forms a valuable con-straint on possible structures involving force chains.Recent experimental advances have allowed for the non-invasive imaging of structure in dry granular media, usingfor example magnetic resonance imaging or positron emis-sion tomography [13, 14]. Although interesting attemptshave been made to use these tools to characterise forcechain distributions, they rely on indirect geometric mea-sures of, say force chain lengths [15], rather than directmeasurements of force. Bridge structures have also re-cently been probed experimentally in colloidal packings[16] and compared with the results of computer simula-tions of shaken (dry) granular media [5]; while the exper-imental results are of interest in and of themselves, it hasto be remembered that colloids are governed by thermalenergy, while temperature does not govern the dynamicsof dry granular media. Instead, a perturbation such asshaking or tapping has to be applied [17] in order to gen-erate particulate motion in dry granular packings; suchathermal perturbation could conceivably generate its ownparticularities in bridge structure, absent in the colloidalcase. It is therefore important to characterise the statis-tics of bridges in the steady state of shaken dry granularpackings in 3 d and to compare them with the results ofcomputer simulations modelling exactly the same physicalsituation; this is the main purposes of this Letter, wherewe use direct and non-invasive tomographic measurementsto characterise bridge structures in dry granular packings.In the experiments, two packings (one monodisperse(Duke Scientific, USA) and one polydisperse) of glassbeads were used, with packing fractions of 0 .
623 and 0 . ± µ m and 300 ± µ m respectively: there were ∼ ∼ ,
000 cycles using a commercial shaker to en-sure that a steady state had been reached. The tappingprotocol involved a single 30-Hz sine wave at a rate of1 Hz with an effective tapping amplitude of 2 .
82 g. AnX-ray microtomography machine (MicroXCT-200, XradiaInc., USA) was used to make structural measurements,with 1200 projection images taken on the samples. The effective spatial resolution of the detector was 6 . × . µm after optical magnification (2 × ). The tomography-reconstructed 3 d images were analyzed by a marker-basedwatershed imaging segmentation technique [18, 19] to en-sure an accurate determination of contacting neighbourswhich is crucial for the identification of bridges. The wa-tershed algorithm represents a grey-scale image as a topo-logical surface, where the grey-scale value of each pixel isinterpreted as its ‘altitude’. The initial image is separatedinto a binary image which comprises a ‘solid’ area and abackground; the key step is to transform each solid areainto a single ‘catchment basin’. Before this process is car-ried out, the standard ‘erosion’ and ‘dilation’ steps werecarried out to remove the noise. Afterwards, a ‘recon-struction’ step is invoked before the distance transform(‘bwdist’) is performed to compute the nearest distancebetween two pixels corresponding to the background, thusidentifying the limits of the solid phase, or ‘catchmentbasin’. The edges of each solid phase, i.e. each grain, areidentified as a watershed ridge line (see white lines in lastcolour panel of Fig. 2). Fig. 2: (color online) Image processing steps from the raw re-constructed image to the segmented image using a marker-based watershed algorithm.
The identification of bridges in experiment and simula-tion followed the overall algorithm used in [5, 11, 20, 21].Sphere coordinates at the end of the stabilization phaseare transformed into a list of at least three contacts foreach particle. Each particle in a contact list is next identi-fied with a unique set of three other particles that provideits supporting base, using a criterion favouring the stabi-lizing triplet with the lowest centroid. Once these stabi-lizing triplets are identified, one looks for sets of particlesthat appear in each other’s stabilizing triplets: these areclearly particles which are mutually stabilizing, and thusidentified with bridge particles. Finally, clusters of mutualstabilizations can be identified, using a linked list structureas in other aggregation applications, to reveal a unique setof bridges in a static close packing. The statistics of theseclusters can be used to get the kind of information onbridge structure such as sizes, orientations and topologies,referred to below.Our experimental findings suggest that linear bridgesp-2ridges in 3 d granular packings Fig. 3: (color online) (a) Log-linear plot of experimental sizedistributions P ( n ) lin of linear bridges: the full and dashedlines represent exponential-law fits for mono- and polydispersebeads respectively. The corresponding exponent values are α = 0 . ± .
02 (monodisperse) and α = 0 . ± .
04 (poly-disperse). (b) Log-log plot of experimental size distributions P ( n ) comp for all bridges. The power-law fits to the largelycomplex bridges for n ≥ τ = 2 . ± . τ = 1 . ± . P ( n ) lin for linear bridges in packings at differ-ent densities φ from computer simulations of shaken monodis-perse spheres. The black line is an exponential-law fit as inpanel (a) to the results for φ = 0 .
62, yielding α = 0 . ± . α = 0 . ± .
02. (d) Log-log plot of size dis tributions P ( n ) comp for all bridges in packings at different densities φ ,from computer simulations of shaken monodisperse spheres.The black line represents a power-law fit for n ≥ φ = 0 .
62, giving τ = 2 . ± .
1. For φ = 0 . , . , . τ = 1 . ± . , . ± . , . ± . τ appears to vary strongly with packing fraction. predominate for sizes of up to n ≈
10, which are char-acterised by a simple exponential distribution P ( n ) lin ∼ e − αn (Fig. 3(a)). For larger sizes n , complex bridges pre-dominate, which are in turn characterised by a power-lawdistribution P ( n ) comp ∼ n − τ (Fig. 3(b)). These resultsare robust to the presence of polydispersity: remarkably,even the associated exponents agree — within error bars—in the two cases, with α = 0 . ± .
02 and τ = 2 . ± . α = 0 . ± .
04 and τ = 1 . ± . φ considered in the ear-lier simulations and in the current experiment (0 .
56 and0 .
62 respectively); since packing fraction is a fundamental structural descriptor of granular media, it could reason-ably lead to quantitative, if not qualitative differences inthe size distribution of granular bridges. The simulationsdescribed below investigate this issue, and confirm thisdependence.Accordingly, we generated configurations correspondingto φ = 0 .
58, 0 .
59, 0 .
60 and 0 .
62 using a well-establishedhybrid Monte Carlo sphere-shaking algorithm [22]. Thesimulations were performed on 1630 spheres in a rectan-gular cell with lateral periodic boundaries and a hard dis-ordered base, using 100 different random initial configura-tions per shaking amplitude. Size checks were performed,and qualitatively similar configurations were obtained fora couple of different system sizes. Care was taken to usestable configurations in the steady state, saving about 200stable configurations (picked out every 500 cycles to avoidcorrelation effects) for bridge identification and analysis.Our findings are shown in Figs. 3(c) and (d) where we findthat there continues to be excellent qualitative agreementbetween experiment and simulation. Quantitatively, weobserve an interesting trend: the agreement between ex-periment and simulation gets better as the correspondingpacking fractions converge. In fact for φ = 0 .
62, our simu-lations yield values of α = 0 . ± .
01 and τ = 2 . ± . α values between experimentsand simulations conducted at the same packing fractionsuggests that this is the most important parameter con-trolling the behaviour of linear bridges. We also noticethat for the three lower packing fractions, the exponent τ show a consistently increasing trend as a function of φ .However, at φ = 0 .
62, the exponent τ from experiment ismore compatible with the trend than that from simula-tion. This is possibly due to the onset of ordering in thesimulations [1].All these trends persist in measurements of the otherbridge structure descriptors considered below; in the fol-lowing we focus on linear bridges, leaving the detailed ex-amination of complex bridges to future work. The firstof our descriptors is the base extension, whose definitionwe review. If all possible connected triplets of base par-ticles for a particular bridge are considered, the vectorsum of their normals is defined to be the direction of the main axis of the bridge, typically inclined at some angleΘ (the orientation angle) to the z -axis. The base exten-sion b (see Fig. 4) is defined as the radius of gyration ofthe base-particles about the z -axis , and is a measure of thespanning or jamming potential of a bridge [5].In Fig. 5, we present experimental and simulation re-sults obtained for the base extension distributions of lin-ear bridges. Figs. 5 (a) and (c) correspond respectively toexperimental and simulation plots of p ( b | n ) (normalisedprobability distributions of b conditional on bridge size n )vs. b for different bridge sizes n . They are qualitativelysimilar, showing sharply peaked distributions which flat-ten out with increasing bridge sizes, as seen in earlier sim-ulations [5]. Figs. 5 (b) and (d) correspond respectivelyp-3. X. Cao et al. Fig. 4: (color online) The base extension of a bridge. to experimental and simulation plots of log p ( b ) (cumu-lative probability distribution of b ) vs. the normalisedvariable b/ < b > , with < b > the mean extension ofbridge bases. Both show the exponential tail in the dis-tribution function noted in the results of earlier simula-tions [5], suggesting that bridges with small base exten-sions are not favored. This result appears to be robustboth with respect to polydispersity in the experimentalresults and packing fractions in the simulations. It alsoreinforces earlier suggestions [3] of deep connections be-tween force chains and linear bridges: the cumulative dis-tributions of force chains in anisotropically sheared gran-ular systems [10, 23] as well as MD simulations of anal-ogous particle packings [24, 25] show very similar expo-nential tails. Recent simulations have directly confirmedthe connection between force chains and linear bridges,by suggesting that forces are principally transmitted byparticles in bridges [12].Another important quantity related to a linear bridgeof size n is obviously its ‘span’, i.e. its rms end-to-endlength R n . Our results for this are presented in Fig. 6(a)(experiment) and Fig. 6(b)(simulations). As expected, wefind the scaling law R n ∼ n ν in both experiment andsimulation. We obtain exponent values ν = 0 . ± . ν = 0 . ± .
01 from simulations.Again, the agreement between experiment and simulationsis remarkable; additionally, we do not observe a strongdependence on packing fraction. Given that ν ∼ .
59 for a3 d self-avoiding random walk, this suggests that the linearbridges that we have examined here look – within errorbars – exactly like self-avoiding walks in three dimensions.Finally, we examine via experiment and simulation, thenormalised distribution for the mean angle Θ made by alinear bridge with the z -axis, as well as that of its variance[5] in Fig. 7. In Figs. 7(a) and (c), we plot respectivelyexperimental and numerical results for p (Θ | n ), the ori-entational distribution conditional on n . The cumulativedistributions p (Θ), also plotted in these figures, closely re-semble each other as well as confirming earlier results [5]. Fig. 5: (color online) Probability distributions, for monodis-perse beads, of base extensions b for linear bridges. (a) Exper-imental plot of p ( b | n ) vs. b for different n . (b) Experimentalplot of the log of the cumulative distribution, log ( p ( b )) vs. b/ < b > . (c) Plot of p ( b | n ) vs. b for different n , generatedfrom computer simulations of shaken monodisperse spheres.(d) Plot of log ( p ( b )) vs. b/ < b > from computer simulationsof monodisperse sphere packings at different densities. Notable features are a peak around 20 o as well as the de-crease of Θ with increasing bridge size, which [5] suggeststhat larger linear bridges form domes. In Figs. 7(b) and(d) we plot the variance of the mean orientational angle, (cid:10) Θ (cid:11) , against size n . According to [5], (cid:10) Θ (cid:11) obeys (cid:10) Θ (cid:11) ( s ) = 2 σ eq as − e − as a s + (cid:0) σ − σ eq (cid:1) (1 − e − as ) a s , (1) where σ eq is the equilibrium value of the variance of thelink angle. This is plotted as the full line in the figures,where the symbols represent results for experiment andsimulation respectively. These look similar to each other,and also to the results of earlier simulations [5].Before concluding, we mention that while we hope ourinvestigations of bridge structures might shed some lighton possible relationships between structural signaturesand force networks in the future, we do not for the presentinclude any explicit force information directly. Our in-vestigations of structure are in addition to ongoing forcechain investigations [26], since stable structures are a con-straint on force chains, and the two taken together mightlead some day to the emergence of a holistic picture ofheterogeneities in granular systems.The experiments reported in this Letter show satisfy-ing qualitative agreement with the results of present andearlier [5] simulations, in the sense that the forms of dis-tributions of quantities ranging from sizes to orientationsand base extensions are robustly the same in every case.This is already quite good for such a complex field, andsuggests that there is truth in the idea of bridges beingp-4ridges in 3 d granular packings Fig. 6: (color online) Plot of the (log of the) end-to-end length,ln R n vs. ln( n −
1) for linear bridges in monodisperse spherepackings. The full lines represent power-law fits. For the ex-perimental plot (a) ν = 0 . ± .
02, while (b) simulations forall the densities give ν = 0 . ± . good characterisers of spatial heterogeneities in granularmedia. Quantitative differences, where these exist, be-tween the current experiment and earlier simulations [5]have been successfully ascribed to the fact that the datawere taken at rather different packing fractions φ : thesimulations reported in this paper have been conductedat a range of packing fractions and manifest this depen-dence, such that there is excellent agreement between sim-ulation and experiment at matching φ . This reinforcesthe intuitive idea that packing fractions should influencethe details of bridge structure (while leaving the qualita-tive features unchanged); higher packing fractions, for in-stance, probably constrain linear bridges to be fully three-dimensional rather than leaving them the choice to beplanar. We might expect that for complex bridges, thefactors at play (apart from the robust power-law size dis-tribution) may indeed be more complex: since these arebranched structures, we might expect that the coordina-tion number, might have an important role to play in pre-cise quantitative measurements of structure, as indicatedby our measurements of the exponent τ . We hope to carryout a detailed study of complex bridges in future work. ∗ ∗ ∗ Some of the initial work has been carried out at theBL13W1 beamline of Shanghai Synchrotron Radiation Fa-cility (SSRF), and the work is supported by the Chi-nese National Science Foundation No. 11175121, Shang-hai Pujiang Program (10PJ1405600), program for NewCentury Excellent Talents (NCET) in University, Na-tional Basic Research Program of China (973 Program;2010CB834300).
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