A granular-matter model is exactly solved, where disks of two sizes and weights in alternating sequence are confined to a narrow channel. The axis of the channel is horizontal and its plane vertical. Disk sizes and channel width are such that under jamming no disks remain loose and all disks touch one wall. Jammed microstates are characterized via statistically interacting particles constructed out of two-disk tiles. Jammed macrostates depend on measures of expansion work, gravitational potential energy, and intensity of random agitations before jamming. The dependence of configurational entropy on excess volume exhibits a critical point

D Liu

D-V Anghel

FDM Haldane

P Lu

RK Bowles

SS Ashwin

Y-S Wu

English

arXiv

Liu, Dan

Müller, Gerhard

DigitalCommons@URI

University of Rhode Island DigitalCommons@URI Physics Faculty Publications Physics 2020 Jammed Disks of Two Sizes in a Narrow Channel Dan Liu Gerhard Müller University of Rhode Island, gmuller@uri.edu This Conference Proceeding is brought to you for free and open access by the Physics at DigitalCommons@URI. It has been accepted for inclusion in Physics Faculty Publications by an authorized administrator of DigitalCommons@URI. For more information, please contact digitalcommons@etal.uri.edu. Citation/Publisher Attribution Liu D., Müller G. (2020) Jammed Disks of Two Sizes in a Narrow Channel. In: Zuriguel I., Garcimartin A., Cruz R. (eds) Traffic and Granular Flow 2019. Springer Proceedings in Physics, vol 252. Springer, Cham. https://doi.org/10.1007/978-3-030-55973-1_48 Available at: https://doi.org/10.1007/978-3-030-55973-1_48 Follow this and additional works at: https://digitalcommons.uri.edu/phys_facpubs The University of Rhode Island Faculty have made this article openly available. Please let us know how Open Access to this research benefits you. This is a pre-publication author manuscript of the final, published article. Terms of Use This article is made available under the terms and conditions applicable towards Open Access Policy Articles, as set forth in our Terms of Use. Jammed disks of two sizes in a narrow channelDan Liu and Gerhard MüllerAbstract A granular-matter model is exactly solved, where disks of two sizes andweights in alternating sequence are confined to a narrow channel. The axis of thechannel is horizontal and its plane vertical. Disk sizes and channel width are suchthat under jamming no disks remain loose and all disks touch one wall. Jammedmicrostates are characterized via statistically interacting particles constructed outof two-disk tiles. Jammed macrostates depend on measures of expansion work,gravitational potential energy, and intensity of random agitations before jamming.The dependence of configurational entropy on excess volume exhibits a critical point.Introduction The research reported here builds on two previous studies [1, 2] em-ploying the same methodology of exact analysis in the framework of configurationalstatistics. They, in turn, were inspired by work on jammed disks in a narrow channelbased on different methods of analysis [3, 4, 5] that yielded intriguing results. Thefocus on disks of two sizes and weights in alternating sequence, for which new exactresults are being presented here, is intended to be a first step toward a scenariowhere the channel contains such disks in a random sequence – a realistic goal in theframework of the same methodology.Geometry Disks of two sizes with diameters σL ≥ σS in alternating sequence arebeing jammed in a channel of width H. The following two conditions guarantee that(i) the disk sequence remains invariant before jamming, (ii) all jammed disks havewall contact, and (iii) jamming leaves no disks loose:Dan Liu, Department of Physics, University of Hartford, West Hartford, CT 06117, USAe-mail: dliu@hartford.eduGerhard Müller, Department of Physics, University of Rhode Island, Kingston RI 02881, USAe-mail: gmuller@uri.edu12 Dan Liu and Gerhard Müller14<σSσL≤ 1, 1 <HσL<121 +σSσL+√(1 +σSσL)2− 1 , (1)All jammed microstates can be assembled from 8 tiles composed of two disks withone disk overlapping (Table 1). Adding a tile to an already existing stringmust satisfythe successor rule to maintain mechanical stability under jamming forces.Table 1 Distinct tiles that constitute jammedmicrostates of disk sequences subject to the conditions(1). Mechanical stability rule: v must be followed by w or 2 etc. Motifs pertain toσL = 2,σS = 1.4,H = 2.5. Volume of tiles: Vc = 12 (σL + σS) +√H(σL + σS − H), Vf = 12 (σL + σS) +√σLσS(assuming unit cross section of channel).motif ID rule vol. motif ID rule vol. motif ID rule vol. motif ID rule vol.v w, 2 Vc 1 3, 4 Vf 3 v Vf 5 3, 4 Vcw v, 1 Vc 2 5 Vf 4 5, 6 Vc 6 w, 2 VfUnder mild assumptions the microstate of minimum volume with N (large/small)disk pairs is composed of an alternating sequence, vwvw· · · v, of just two tiles.We declare it to be the reference state for statistically interacting particles in thisapplication. All other (jammed) microstates can be generated by the activation ofquasi-particles from this reference state. We have identified M = 5 species ofparticles that serve this purpose (Table 2). Adopting the taxonomy of Ref. [6] wedistinguish between the categories of hosts and tags.Table 2 Five species of quasi-particles. The hostsm = 1, . . . , 4 modify the reference state whereasthe tagm = 5modifies any one of the hosts. The ID lists the overlapping tiles involved. The activationenergy εm is relevant before jamming and the excess volume ∆Vm after jamming.ID m cat. ∆Vm εm ID m cat. ∆Vm εm13 1 host 2Vt 2pVt − γS 2546 4 host 2Vt 2pVt − γS + 2γL146 2 host 2Vt 2pVt − γS + γL 45 5 tag 0 γL − γS253 3 host 2Vt 2pVt − γS + γLParticles from species m = 1, . . . , 4 can be placed directly into the reference state(pseudo-vacuum), meaning that it is possible to add a tile v or w to the left or tothe right as follows: w13v, w146w, v253v, v2546w. Particles from species m = 5are parasitic in the sense that they can only be placed inside a particle from speciesm = 1, . . . , 4, at exactly one position and with one disk overlapping as follow: 1|45|3,Jammed disks of two sizes in a narrow channel 31|45|46, 25|45|3, 25|45|46. The number of tag particles that can be added insidethe same host is only limited by the size of the number N of disk pairs in the system.For example, two tags inside the first host reads 1|4545|3. The minimum number oftiles v or w between two host can be two as in 13vw146, one as in 13v13, or zero asin 146253.Energetics The activation of every host particle extends the total volume afterjamming by the amount 2Vt, where Vt  Vf − Vc (see Tables 1 and 2). Placing atag does not change the volume. The activation energy εm assigned to a particlefrom species m pertains to the state of random agitation before jamming. It consistsof work against the ambient pressure exerted by pistons and gravitational potentialenergy, all relative to the reference state.We are free to choose the mass density of small and large disks. Therefore,the gravitational potential energies γL and γS are independent parameters as is theexpansion work 2pVt. A fourth parameter is the intensity Tk of random agitations.The jamming protocol is explained in [1]. All results coming out of configurationalstatistics will only depend on three (dimensionless) ratios of four energy parameters:β 2pVtTk, ΓL γL2pVt, ΓS γS2pVt. (2)Combinatorics The quasi-particles identified in Table 2 are statistically interactingin the sense that activating one particle affects the number dm of open slots for theactivation of further particles from each species. This interaction can be accountedfor by a multiplicity expression for jammedmicrostates involving a generalized Pauliprinciple [7] in the form, [6, 8, 9],W({Nm}) =M∏m=1(dm + Nm − 1Nm), dm = Am −M∑m′=1gmm′(Nm′ − δmm′), (3)with capacity constants Am and statistical interaction coefficients gmm′ as listed inTable 3, and where Nm is the number of activated particles from species m.The initial capacity for hosts grows linearly with the number of disks. It is zerofor tags, which can only be activated inside hosts. Activating a host (m′ = 1, . . . , 4)Table 3 Capacity constants and statistical interaction coefficients for each particle species.m Am1 N − 22 N − 33 N − 24 N − 35 0gmm′ 1 2 3 4 51 2 2 1 2 12 1 2 1 1 13 2 2 2 2 14 1 2 1 2 15 −1 −1 −1 −1 04 Dan Liu and Gerhard Müllerremoves one or two slots for activating a further host (m = 1, . . . , 4) but adds oneslot for activating a tag (m = 5). Activating a tag (m′ = 5) removes one slot foractivating hosts (m = 1, . . . , 4) but leaves the number of slots for activating a furthertag (m = 5) invariant.Statistical mechanics Wecan express the excess volume and the entropy as functionsof the average particle content {〈Nm〉} of a jammed macrostate as follows [9]:V − Vref =M∑m=1〈Nm〉∆Vm, Ym  Am −M∑m′=1gmm′ 〈Nm′〉. (4a)S = kBM∑m=1[ (〈Nm〉 + Ym)ln(〈Nm〉 + Ym)− 〈Nm〉 ln〈Nm〉 − Ym lnYm]. (4b)The average particle numbers are the solutions of the linear equations [8, 9],wm〈Nm〉 +M∑m′=1gmm′ 〈Nm′〉 = Am. (5)The wm are non-negative solutions of the algebraic equations [6, 8, 9],eεm/Tk = (1 + wm)M∏m′=1(1 + w−1m′)−gm′m . (6)The analytic solution of Eqs. (6), too unwieldy for display, gives us explicit ex-pressions for the scaled excess volume, V̄  (V − Vref)/NVt, the scaled entropy,S̄  S/NkB, and the particle densities, N̄m  〈Nm〉/N , as functions of β, ΓL, ΓS.Results There is space to present one case, ΓL = ΓS, at the border between tworegimes, large and small disks have equal gravitational potential energy when theytouch the same wall. Only two of the parameters (2) are independent. Increasing βmeans reducing the intensity of random agitations before jamming and increasingΓL means increasing the effects of gravity (e.g. by tilting the the plane of the channelfrom horizontal toward vertical).In the high-intensity limit, β→ 0, we have the most disordered macrostate:N̄1 = · · · = N̄4 =112, N̄5 =16, V̄ =23, S̄ = ln 3 (β = 0). (7)It is instructive to plot the the population densities versus volume [Fig. 1(a)-(c)].For N̄1, . . . , N̄4, β = 0 is realized at the points where multiple curves merge. For N̄5that point is at the top of the dotted line. Increasing β toward infinity means movingalong any path toward V̄ = 0 if ΓL < 1 or toward V̄ = 1 if ΓL > 1. The path stopsat V̄ = 12 if ΓL = 1, which signals criticality. The following conservation law onlyholds for the case considered here:Jammed disks of two sizes in a narrow channel 50.0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.40.5VN1,N5N1N5(a)0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.01.2VSln3ln21/2 2/3(d)0.0 0.2 0.4 0.6 0.8 1.00.000.020.040.060.080.10VN2,N3(b)0.0 0.2 0.4 0.6 0.8 1.00.000.020.040.060.080.10VN4(c)Fig. 1 Population densities N̄m ,m = 1, . . . , 5 versus excess volume V̄ for fixed values 0, 0.5, 0.75,1.0, 1.25, 1.5 of parameter ΓL = ΓS from bottom up in panel (a) and from left to right in panels (b)and (c)]. The dotted line connect the points pertaining to β = 0 of N̄1 and N̄5. The dashed diagonalrepresents the sum N̄1 + N̄2 + N̄3 + N̄4. (d) Entropy S̄ versus excess volume V̄ for fixed values 0,0.5, 0.75, 0.9, 1.0, 1.1, 1.25, 1.5 of parameter ΓL = ΓS (from left to right).N̄tot 5∑m=1N̄m =12(ΓL = ΓS). (8)Its validity is illustrated by the two diagonal lines in Fig. 1(a), where the dashedline represents the population density of all hosts combined. The tags m = 5 donot contribute to excess volume. Their numbers depend on V̄ nevertheless, via theconservation law (8). Any increase in volume caused by the activation of one or theother host necessarily crowds out one tag.The results show that hosts m = 2, 3, 4 only contribute significantly at highintensity. The three distinct macrostates associated with β = ∞ are located at thecorners or the center of Fig. 1(a). All have N̄2 = N̄3 = N̄4 = 0.• The state with N̄1 = 0 and N̄5 = 12 is realized for ΓL < 1 and has V̄ = 0,S̄ = 0. It is a doublet consisting of the reference state vwvw · · · vwv and the state14545 · · · 4546. The former contains no particles, N̄5 = 0, and the latter one host2 and a macroscopic number of tags inside, amounting to N̄5 = 1.• The state with N̄1 = 12 and N̄5 = 0 is realized for ΓL > 1 and has V̄ = 1, S̄ = 0.It is a singlet packed with hosts 1: 13vw13vw · · · 13v. Hosts 1 proliferate becausethey have negative activation energies.• The state with N̄1 = N̄5 = 14 is realized for ΓL = 1 and has V̄ =12 , S̄ = ln 2. Itis highly degenerate. The hosts 1 are randomly distributed between vacuum tiles6 Dan Liu and Gerhard Müllerwith a random number of tags inside. Hosts 1 and tags 5 have zero activationenergies whereas the hosts, 2, 3, 4 have positive activation energies.With all this information in place we are ready to interpret a graphical represen-tation of entropy versus excess energy [Fig. 1(d)]. The parameter β runs from zeroto infinity along each path from the top down. All paths start at coordinates V̄ = 23 ,S̄ = ln 3. Paths for ΓL < 1 end at V̄ = 0, S̄ = 0, and paths for ΓL > 1 at V̄ = 1, S̄ = 0.At critical gravity, ΓL = 1, both volume and entropy decrease monotonically but endat the critical values V̄ = 12 , S̄ = ln 2.Compact analytic expressions for the curves in Fig. 1 pertaining to zero gravityand critical gravity are available. For ΓL = 0 we haveN̄1 = · · · = N̄4 =18V̄, N̄5 =12(1 − V̄), S̄ =(V̄ − 1)ln(2V̄− 2)+ ln(2V̄), (9)where the range of volume is 0 ≤ V̄ ≤ 23 . For ΓL = 1 we haveN̄1 =(V̄ − 1)22V̄, N̄2 = N̄3 =32− V̄ −12V̄, N̄4 = 2V̄ +12V̄− 2, (10a)N̄5 =12(1 − V̄), S̄ = 2(V̄ − 1) ln(1 − V̄2V̄ − 1)− ln(2V̄ − 1), (10b)across the more restricted range 12 ≤ V̄ ≤23 of volume.Outlook Qualitatively different jamming patterns pertain to the regimes ΓL < ΓS oflight large disks and ΓL > ΓS of heavy large disks. Yet different jamming patternsare expected when the analysis is generalized to periodic and aperiodic sequencesincluding random sequences of large and small disks (with modified jamming pro-tocols).References1. N. Gundlach, M. Karbach, D. Liu, and G. Müller, J. Stat. Mech. P04018 (2013).2. C. Moore, D. Liu, B. Ballnus, M. Karbach, G. Müller, J. Stat. Mech. P04008 (2014).3. R. K. Bowles and I. Saika-Voivod, Phys. Rev. E 73, 011503 (2006).4. S. S. Ashwin and R. K. Bowles, Phys. Rev. Lett. 102, 235701 (2009).5. R. K. Bowles and S. S. Ashwin, Phys. Rev. E 83, 031302 (2011).6. D. Liu, P. Lu, G. Müller, and M. Karbach, Phys. Rev. E 84, 021136 (2011).7. F. D. M. Haldane, Phys. Rev. Lett. 67, 937 (1991).8. Y.-S. Wu, Phys. Rev. Lett. 73, 922 (1994).9. S. B. Isakov, Phys. Rev. Lett. 73, 2150 (1994); Mod. Phys. Lett. B 8, 319 (1994).10. D.-V. Anghel, J. Phys. A 40, F1013 (2007); Europhys. Lett. 87, 60009 (2009).11. P. Lu, J. Vanasse, C. Piecuch, M. Karbach, and G. Müller, J. Phys. A 41, 265003 (2008).12. P. Lu, D. Liu, G. Müller, and M. Karbach Condens. Matter Phys. 15, 13001 (2012).13. D. Liu, J. Vanasse, G. Müller, and M. Karbach, Phys. Rev. E 85, 011144 (2012).

Jammed Disks of Two Sizes in a Narrow Channel

Crossref

Jammed Disks of Two Sizes in a Narrow Channel

Abstract

Similar works

Full text

Available Versions

DigitalCommons@URI

Crossref