Correlated rigidity percolation in fractal lattices
CCorrelated rigidity percolation in fractal lattices
Shae Machlus, Shang Zhang, and Xiaoming Mao Department of Physics, University of Chicago, Chicago, Illinois, 60637, USA Department of Physics, University of Michigan, Ann Arbor, Michigan, 48109, USA (Dated: August 31, 2020)Rigidity percolation (RP) is the emergence of mechanical stability in networks. Motivated by the experi-mentally observed fractal nature of materials like colloidal gels and disordered fiber networks, we study RP ina fractal network. Specifically, we calculate the critical packing fractions of site-diluted lattices of Sierpi´nskigaskets (SG’s) with varying degrees of fractal iteration. Our results suggest that although the correlation lengthexponent and fractal dimension of the RP of these lattices are identical to that of the regular triangular lattice,the critical volume fraction is dramatically lower due to the fractal nature of the network. Furthermore, wedevelop a simplified model for an SG lattice based on the fragility analysis of a single SG. This simplifiedmodel provides an upper bound for the critical packing fractions of the full fractal lattice, and this upper boundis strictly obeyed by the disorder averaged RP threshold of the fractal lattices. Our results characterize rigidityin ultra-low-density fractal networks.
I. INTRODUCTION
Soft disordered solids are ubiquitous; they exist in manyforms such as colloidal gels, fiber networks, colloidal glasses,emulsions, aerogels, polymer melts, and foams. These classesof materials make up biological tissues, food products, cos-metic products, and materials like paper and nonwoven fab-ric. Some of these soft materials need only a very low densityof solid particles to become rigid. In particular, colloidal gelscan exhibit nonzero shear rigidity at a wide range of volumefractions [1–8], which can be below 1% in the case of bloodclots [6].Classical RP transitions are associated with much highervalues of critical volume fractions φ c for a material to berigid [9–13], so how can these ultra-low-density materialsexhibit rigidity? Previous work suggested that the answerto this question lies in how the particles are spatially corre-lated to each other–the Warren truss, for example, transmitsstress very efficiently and can achieve rigidity at φ c = φ c . While thetype of correlation used in [13] was not enough for describ-ing rigidity in ultra-low-density solids, it suggested that theremay be another sort of spatial correlation that is both physi-cally realistic and allows the system to achieve an arbitrarilylow value of φ c . We conjecture that a recursive correlation(which generates a fractal network) would be a promising can-didate for describing rigidity at ultra-low-densities because (i)fractals are low density while still being connected, and theycan be rigid, and (ii) experimental evidence suggests that lowdensity disordered solids (coagulated blood, for example) canindeed be fractal as a result of the non-equilibrium process inwhich the material is assembled [1, 8, 14–18].In this paper, we show that a model fractal network, theSierpi´nski gasket lattice (SGL), does indeed achieve rigidityat arbitrarily low volume fractions. This result is supportedanalytically by simple calculation on the undiluted SGL andnumerically on the randomly diluted SGL by using the pebblegame algorithm. We also calculate the correlation length and fractal dimension critical exponents for RP in this lattice andfind that the universality class of the rigidity phase transitionin the lattice is the same as that for the regular triangular lat-tice. We further propose a simple non-fractal model, the RPof which yields a strict upper bound to the disorder-averagedcritical volume fraction of the SGL. II. MODEL
We use a lattice that achieves an arbitrarily low volumefraction while still exhibiting rigidity at full site occupancy.Motivated by the experimentally observed fractal structure offiber networks and colloidal gels [8, 15, 19], we consider alattice of Sierpi´nski gaskets (SG’s), as shown in Fig. 1. Vibra-tional modes and spin phase transitions have been studied onthis lattice [20–26]. This is a rich lattice to study since thereare three length scales: (i) the size of the smallest triangle inan SG which we always set as 1, (ii) the length of the edge ofan SG 2 n , and (iii) the length of the lattice L = s n . s is thenumber of SG’s on one side of the lattice, and n is the num-ber of times the SG pattern repeats on itself, what we call thefractal iteration number. We emphasize that L is measured inunits of the smallest triangle of an SG since the length of thesmallest triangle is always 1, independent of n . Also note that n = φ SGL undiluted = π √ n + − n . (1)This result is derived in Appendix A, and it is obtained byassuming that each site is occupied by a disk whose diameterequals the bond length between neighboring sites, pictured inFig. 1(c). It follows thatlim n → ∞ φ SGL undiluted ( n ) = . (2)An arbitrarily large n corresponds to an arbitrarily small φ SGL undiluted , so the SGL is indeed a suitable model to studythe emergence of rigidity in ultra-low-density networks. Asingle SG, of any n , is isostatic–it has 3 trivial zero modes a r X i v : . [ c ond - m a t . s o f t ] A ug IG. 1. (a,b) Sierpi´nski gasket (SG) of fractal iteration n = ,
5. (c,d) Lattices of SG’s are models forultra-low-density networks at n = ,
5. In (c)semi-transparent purple disks represent the physical particleswe are modeling. The diameter of each particle is equal tothe bond length, which we set to 1.and no states of self stress [27]. The coordination numberof the undiluted lattice under periodic boundary conditions (cid:104) z (cid:105) undiluted can be calculated as a function of n . (cid:104) z (cid:105) undiluted = + ( x − ) x , (3)where x = ( n + − ) / n -level SG where n ≥
1. At n =
0, the lattice is a regulartriangular lattice, so (cid:104) z (cid:105) undiluted =
6. The coordination numberdecreases from 6 to 4 as n goes from 0 to ∞ .We dilute the SGL by removing randomly chosen sites. Ifa site is removed, all of the bonds attached to that site arealso removed. The occupancy fraction p SGL is the ratio of thenumber of occupied sites to the number of sites present in acompletely filled SGL. As shown in Appendix A, the volumefraction of the diluted SGL is then φ SGL = p SGL φ SGL undiluted . (4)We emphasize that while the occupancy fraction p SGL isthe ratio of the number of occupied sites to total number ofsites (unoccupied and occupied), the volume fraction φ SGL isthe ratio of the occupied space to the total space covered bythe lattice. Because the volume fraction of the undiluted SGL φ SGL undiluted vanishes in the n → ∞ limit, φ SGL can approach 0even when p SGL is of O ( ) . III. METHOD & RESULTS
In order to study the RP in the diluted SGL, we execute thepebble game algorithm [28, 29] on SGL’s at n = , , , , n , weconsider 4 different system sizes L which were chosen so thatthe lattices have approximately 250, 1000, 4000, and 16,000particles (sites) (although at n = n = n , we reference L = n + (cid:114) N n + − , (5)which is immediate from Eqs. (A4) and (A5) (Appendix A),to choose an integer valued side length L for each target sys-tem size (in terms of the total number of sites) and fractaliteration n .For each n and L , we generate 200 samples of SGL’s. Eachone represents a realization of disordered dilution. For eachsample, we use the pebble game algorithm to determine thecritical occupancy fraction p c , SGL , which, by increasing p SGL ,is achieved when a spanning rigid cluster first appears. Wealso record the mass of the spanning rigid cluster M c , SGL whenit first occurrs in each sample. We then average over the 200samples to obtain the averaged quantities, (cid:104) M c , SGL ( n , L ) (cid:105) and (cid:104) p c , SGL ( n , L ) (cid:105) , for each n and L . We also measure the fluctua-tion of the transition point ∆ p c , SGL = (cid:113) (cid:104) p c , SGL ( n , L ) (cid:105) − (cid:104) p c , SGL ( n , L ) (cid:105) . (6)Our previous study of correlated RP on the triangular lat-tice [13] showed that the short-range spatial correlation onlyshifts the transition point and does not change the universalityclass of RP in the triangular lattice. Following this result, wemake the assumption that RP in the SGL is also a continuoustransition, with the mass of the infinite rigid cluster being theorder parameter. This assumption is verified by our scalingresults below.We invoke finite-size scaling relations [13, 30] to calculatethe critical exponents associated with the rigidity phase tran-sition. The correlation length exponent ν SGL and the fractaldimension d f , SGL are calculated as the slopes of linear fits oflog-log plots of (cid:104) M c , SGL (cid:105) and ∆ p c , SGL versus L , according tothe finite size scaling relations (cid:104) M c , SGL ( n , L ) (cid:105) ∝ L d f , SGL , (7) ∆ p c , SGL ∝ L − / ν SGL (8)(Appendix B). Note that these relations give a calculation of d f , SGL and ν SGL for each n .We find ν SGL and d f , SGL for the SGL rigidity phase transi-tion are the same as for the rigidity phase transition in the reg-ular triangular lattice [9] as shown in Fig. 2. This observationis consistent with results on RP in lattices with spatial corre-lations [13], where the critical exponents remain the same asin classical RP, and the short-ranged spatial correlation can beviewed as an irrelevant perturbation. Here, the fractals in eachunit cell can also be viewed as a short range feature, which doIG. 2. (a) The correlation length exponent ν SGL at n = , , , ,
5. (b) The spanning rigid cluster fractaldimension d f , SGL for the five values of n . The red lines showthese exponents for classical RP in the regular triangularlattice ( ν = . ± .
06 and d f = . ± .
02) [9]. The errorbars are 95% confidence intervals.not change the divergent length scale at the transition. We as-sert that the large-scale fractal structure of the spanning rigidcluster in the infinite system size limit overwhelms the localfractal structure of the SG’s, so d f , SGL is the same as in theregular triangular lattice case instead of being the fractal di-mension of the SG. We also verify that our assumption (thephase transition is continuous) is well justified since the phasetransition belongs to the same universality class as [13].We extract the critical occupancy fraction at the infinite sys-tem size limit p c , SGL ( n , L = ∞ ) by linearly extrapolating thefinite critical occupancy fractions p c , SGL ( n , L ) for each n asa function of L − / ν . The p c , SGL ( n , L = ∞ ) are simply the y-intercepts of these linear fits which are displayed in Fig. 3.Further information about this process can be found in Ap-pendix C of [13].We find that the critical occupancy fraction p c , SGL ( n , L = ∞ ) approaches 1 as n increases while the critical volume frac-tion φ c , SGL ( n , L = ∞ ) approaches 0 [following the relation in FIG. 3. Extracting p c , SGL ( n , L = ∞ ) from the linearextrapolation of the finite-size critical occupancy fractions p c , SGL ( n , L ) as a function of L − / ν SGL where ν SGL = . p c , SGL ( n , L = ∞ ) . The error bars are 95% confidenceintervals. n p c , SGL ( n , L = ∞ ) φ c , SGL ( n , L = ∞ ) . ± .
002 0 . ± . . ± .
004 0 . ± . . ± .
004 0 . ± . . ± .
004 0 . ± . . ± .
005 0 . ± . Table I. The critical occupancy and volume fractions for theSGL’s for n = , , , , p c , SGL ( n , L = ∞ ) and φ c , SGL ( n , L = ∞ ) . As n increases, p c , SGL ( n , L = ∞ ) → φ c , SGL ( n , L = ∞ ) →
0. The errorvalues are 95% confidence intervals.Eq. (4)], indicating that these disordered fractal structures ex-hibit rigidity at vanishing volume fractions. These results areshown in Table I.
IV. INTERPRETATION
The fact that the p c , SGL ( n , L = ∞ ) ’s approach 1 as n in-creases is a reflection of both the fragility of a single SG–forany value of n , removing any non-corner site of an SG segre-gates the three corners of the SG into three separate rigid clus-ters (Appendix C), and the result [Eq. 3] that (cid:104) z (cid:105) approachesthe critical value of 4 as n increases. The latter point revealsthat the SGL is asymptotically a Maxwell lattice (i.e., latticesthat satisfy (cid:104) z (cid:105) = d and are thus at the verge of mechanicalinstability [27, 31]) as n → ∞ .These observations motivate a simplified model of theSGL–the triangle plate lattice (TPL). The TPL is a regular tri-angular lattice consisting of upwards-pointing rigid trianglesIG. 4. The triangular plate model (TPL) is a regulartriangular lattice which has been diluted in units of upwardspointing equilateral triangles (black).hinged at their tips. In other words, if we view it as a regularbond-dilution RP in a triangular lattice, the items which arebeing diluted are groups of three bonds which together forman upwards pointing triangle. Figure 4 is an example of whata diluted TPL can look like.There is one main feature that separates the TPL from theSGL: in the SGL an SG with a site removed may still be an es-sential part of the spanning rigid cluster. In the TPL, a vacanttriangle cannot transmit rigidity. Because of this differencethe critical packing fraction of the TPL is used to calculate astrict upper bound on that of the SGL.All p ’s that follow in this section should be taken to be inthe infinite system size limit. The relationship between p c , SGL and the critical packing fraction for the TPL p c , TPL is as fol-lows: consider an SGL and a TPL, where the SG’s in the SGLand the triangle plates in the TPL are the same size. Let thetwo lattices also be of equal size. A removed upwards pointingtriangle from the TPL corresponds to at least one removed sitefrom the SGL. Letting the number of triangles/SG’s present ineither lattice be N ∆ and the number of sites present in a singleSG be x = ( n + − ) /
2, the critical occupancy fractions forthe two lattices are related by xN ∆ ( − p c , SGL ) ≥ N ∆ ( − p c , TPL ) . (9)The number of removed sites at the critical point in the SGLis at least the number of removed triangles at the critical pointin the TPL. The “=” sign is only satisfied if removing eachsite from the SGL corresponds to removing a distinct triangleplate from the TPL. This is not always the case because (i)multiple removed sites in the SGL can belong to the sameSG, and, as we discussed above, (ii) a “broken” SG can stillcontribute to the rigidity of the lattice. As a result, the TPLprovides an upper bound of the critical occupancy in the SGL, P c , SGL . Explicitly, p c , SGL ≤ − − p c , TPL x ≡ P c , SGL . (10)We perform the pebble game routine on the TPL and ex-ecute the same finite scaling procedures that we did for the FIG. 5. The difference between the upper bound on p c , SGL given by the TPL, P c , SGL , and the measured p c , SGL becomessmaller as n increases. The error bars are 95% confidenceintervals.SGL. We find that p c , TPL = . ± .
005 and ν TPL = . ± .
1. The errors given are 95% confidence intervals. p c , TPL and ν TPL both lie within error bars of the corresponding variablesfor the regular triangular lattice in the case of bond dilution[9]. The upper bounds on the p c , SGL ’s predicted by the TPLare obeyed for all tested values of n and tightly obeyed forlarger values of n (Fig. 5). It is worth pointing out that this isa strict upper bound in the sense of disorder averaged criticaloccupancy. It does not necessarily hold for individual sam-ples. V. CONCLUSIONS AND DISCUSSIONS
In this paper we show that by introducing fractal local struc-tures, rigidity can exist at an arbitrarily low volume fractionof solid particles. Using a periodic lattice model consistingof Sierpi´nski gaskets, we find that as the fractal iteration in-creases, the critical site occupancy fraction for rigidity in-creases, while the critical volume fraction decreases, allow-ing rigidity at progressively lower volume fractions. We alsoshow that the RP transition in this fractal lattice remains inthe same universality class as the classical RP transition whenlength is measured in units of the sides of the smallest trian-gles. We interpret this result by mapping the RP on this fractallattice into the RP of a simple triangle plate model, based onthe fragility of a single SG. This mapping gives a strict upperbound of the critical volume fraction of the fractal lattice.Our results may shed light on the origin of rigidity in ultra-low volume fraction soft solids, such as hydrogels and aero-gels. A simple way to understand this phenomena is to realizethat, even in a dense disordered solid such as granular mat-ter or colloidal glass, stress is often carried by a very smallfraction of the solid content, i.e., force chains [32–34], whileother components do not significantly contribute to the elas-icity. Thus, by introducing appropriate spatial correlation be-tween the solid particles, a material can be constructed with-out filling the space which is not needed for rigidity. Inter-estingly, interactions and non-equilibrium processes (such ashydrodynamics of the solvent) occuring during the formationof these ultra-low volume fraction solids appear to naturallyachieve this goal of arranging particles in very efficient waysof transmitting stress. It is of our interest to understand howthis occurs in these experimental systems in the future.The model we discuss here is a two-dimensional lattice.A curious question that immediately arises is what happensin three dimensions. The SG has a direct three-dimensionalgeneralization: the Sierpi´nski tetrahedron (ST), which is con-structed by iteratively hinging tips of four tetrahedra togetherto form a bigger tetrahedra (which has an octahedron of emptyspace in the middle). Each face of an ST is an SG. Interest-ingly, there is a mechanical analogy between the SG and theST: each internal node in the ST has six bonds, satisfying theMaxwell condition (cid:104) z (cid:105) = d , while the four tip nodes eachhave three bonds ( z = z =
12, taking the whole structure to (cid:104) z (cid:105) > ACKNOWLEDGEMENTS
We thank the National Science Foundation for their supportthrough Grant No. DMR-1609051 (S. Z. and X. M.) and NSFPHY 1852239 “Summer Undergraduate Research in Physicsand Astrophysics at the University of Michigan” (S. M.)
A. CALCULATING φ SGL
The volume fraction, an area fraction for d =
2, is the ratioof space taken up by the occupied sites to the space enclosedwithin the unit cell. φ SGL is the volume fraction of the lattice, N occ is the number of occupied sites in the lattice, A v is thearea covered by a single site, and A is the total area coveredby the lattice. φ SGL ≡ N occ A v A . (A1) FIG. 6. (a) The correlation length exponent ν SGL and (b) thefractal dimension d f , SGL for the SGL are both obtained fromthe slopes of the linear fits for each n according to Eqs. (7)and (8). The error bars are 95% confidence intervals.The lattice is a rhombus with side length L , so A = √ L . (A2)Additionally, we define the occupancy fraction p SGL as p SGL ≡ N occ N total , (A3)where N total is the total number of sites (occupied and unoc-cupied) in the lattice. For an SGL with periodic boundaryconditions, N total is given by N total = s (cid:18) n + − (cid:19) , (A4)where n is the number of fractal iterations, and s is the lengthof the lattice in units of SG’s. We set the distance betweenneighboring sites on the lattice to be 1. Due to the fractalstructure of an SG, L = s n . (A5)Since the length between sites is 1, we also know that A v = π (cid:18) (cid:19) . (A6)IG. 7. (a) An n = n = φ SGL = π √ n + − n p SGL . (A7) B. CALCULATING CRITICAL EXPONENTS
Given the finite size scaling relations Eqs. (7) and (8), wecan calculate the correlation length exponent ν SGL and the fractal dimension d f , SGL for the SGL, as shown in Fig. 6.
C. FRAGILITY OF AN SG
We use induction to prove that removing any non-cornersite in an SG will segregate the 3 corner sites into differentrigid clusters. Consider an n = n -level SG. Consider now an SG of fractaliteration n +
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