Scaling Theory for Mechanical Critical Behavior in Fiber Networks

Scaling theory for mechanical critical behavior in ﬁber networks

Jordan Shivers,1, 2 Sadjad Arzash,1, 2 Abhinav Sharma,3 and Fred C. MacKintosh1, 2, 4 1Department of Chemical and Biomolecular Engineering, Rice University, Houston, TX 77005, USA 2Center for Theoretical Biological Physics, Rice University, Houston, TX 77030, USA 3Leibniz Institute of Polymer Research Dresden, Dresden, Germany 4Departments of Chemistry and Physics & Astronomy, Rice University, Houston, TX 77005, USA As a function of connectivity, spring networks exhibit a critical transition between floppy and rigid phases at an isostatic threshold. For connectivity below this threshold, fiber networks were recently shown theoretically to exhibit a rigidity transition with corresponding critical signatures as a function of strain. Experimental collagen networks were also shown to be consistent with these predictions. We develop a scaling theory for this strain-controlled transition. Using a real-space renormalization approach, we determine relations between the critical exponents governing the transition, which we verify for the strain-controlled transition using numerical simulations of both triangular lattice-based and packing-derived fiber networks.

It has long been recognized that varying connectivity can that vary as a power of |z − zc| for z > zc [6, 7, 13]. In the lead to a rigidity transition in networks such as those formed floppy or sub-isostatic regime with z < zc, such systems can by springlike, central force (CF) connections between nodes. be stabilized by introducing additional interactions [7, 8] or Maxwell introduced a counting argument for the onset of by imposing stress or strain [14, 15]. It was recently shown rigidity for such systems in d dimensions with N nodes, in that sub-isostatic networks undergo a transition from floppy which the number of degrees of freedom dN is balanced by to rigid as a function of shear strain γ [11, 16]. Moreover, this the number of constraints Nz/2, where z is the average co- fundamentally nonlinear transition was identified as a line of ordination number of the network [1]. The transition at this critical points characterized by a z−dependent threshold γc(z), isostatic point of marginal stability has been shown to ex- as sketched in Fig. 1a. Above this strain threshold, the differ- hibit signatures of criticality. Such a balance of constraints ential or tangent shear modulus K = dσ/dγ scales as a power f and degrees of freedom is important for understanding rigid- law in strain, with K ∼ |γ − γc| . Introducing bending inter- ity percolation and jamming [2–6]. Even in networks with actions with rigidity κ between nearest neighbor bonds stabi- additional interactions that lead to stability below the CF iso- lizes sub-isostatic networks below the critical strain, leading static point, the mechanical response can still exhibit strong to K ∼ κ for γ < γc. Both of these regimes are captured by the signatures of criticality in the vicinity of the CF isostatic point scaling form [17] [7–10]. More recently, criticality has been shown in fiber net- f φ works as a function of strain for systems well below the iso- K ≈ |γ − γc| G± κ/|γ − γc| (1) static point [11]. in which the branches of the scaling function G± account for While both jammed particle packings and fiber networks the strain regimes above and below γc. This scaling form exhibit athermal (T = 0) mechanical phase transitions and su- was also shown to successfully capture the nonlinear strain perficially similar critical behavior, these systems have strong stiffening behavior observed in experiments on reconstituted qualitative differences. There is growing evidence that the jamming transition is mean-field [6, 12], while fiber networks to date have shown non-mean-field behavior [8–11]. Goodrich et al. recently proposed a scaling theory and performed numerical simulations of jamming to both demonstrate mean- field exponents and support the conclusion that the upper critical dimension du = 2 for the jamming transition [12]. Al- though many aspects of the critical behavior of fiber networks, including various critical exponents, have been quantified, a arXiv:1807.01205v2 [cond-mat.soft] 4 Jul 2018 theory has been lacking to identify critical exponents or even scaling relations among exponents. Here, we develop a scaling theory for both the sub-isostatic, strain controlled transition, as well as the transition in z near the isostatic point for athermal fiber networks. We derive scaling relations among FIG. 1. (a) Schematic phase diagram depicting the state of mechanical rigidity of a central force network as a function of coordination the various exponents and demonstrate good agreement with number z and applied shear strain γ. The arrow A depicts the strain- numerical simulations. Interestingly, our results imply that controlled transition and B depicts the transition at the isostatic point. the upper critical dimension for fiber networks is du > 2, in With the addition of bending interactions, the floppy region becomes contrast with jamming packings. instead bending-dominated, but the critical curve γc(z) vs. z remains Near the isostatic point with average coordination number the same. (b) Portion of a a triangular network and (c) a 2D packing- derived network, both diluted to z = 3.3 < z . z = zc = 2d, spring networks exhibit shear linear moduli G c 2 networks of collagen, a filamentous protein that provides me- Near the critical strain, networks exhibit large, nonaffine chanical rigidity to tissues as the primary structural compo- internal rearrangements in response to small changes in ap- nent of the extracellular matrix [16]. Collagen constitutes an plied strain [11, 16]. In the absence of thermal fluctuations excellent experimental model system on which to study this in such athermal networks, these nonaffine strain fluctuations transition, as it forms elastic networks that are deeply sub- are analogous to divergent fluctuations in other critical sys- isostatic (z ≈ 3.4 [18], whereas zc = 6 in 3D) in which in- tems. In response to an incremental strain strain step δγ, the dividual fibrils have sufficiently high bending moduli to be nonaffine displacement of the nodes is expected to be propor- treated as athermal elastic rods. tional to δγ. Thus, the nonaffine fluctuations can be captured 2 Scaling theory — For the strain-controlled transition at a by δΓ ∼ h δu − δuA i/δγ2, where δu − δuA represents the de- fixed z < zc (arrow A in Fig. 1a), we define a reduced vari- viation relative to a purely affine displacement δuA. For large able t = γ − γc that vanishes at the transition and let h(t, κ) systems with small κ, δΓ diverges as t → 0 [16]. This diver- denote the Hamiltonian or elastic energy per unit cell. This gence can also be derived from the energy h(t, κ) in the limit energy depends on the bending stiffness κ that also vanishes of small bending stiffness κ, as follows. For small bending at the transition. Assuming the system becomes critical as energy, the nonaffine displacements δu2 are determined by the t, κ → 0, we consider the real-space renormalization of the minimization of h, which should then be due to purely bend- system when scaled by a factor L to form a block or effective ing: h ∼ κδu2 ∼ κδγ2δΓ. Thus, the nonaffine fluctuations are unit cell composed of Ld original cells, where d is the dimen- predicted to diverge as sionality of the system [19]. Under this transformation, the energy per block becomes h(t0, κ0) = Ldh(t, κ), where t0 and κ0 ∂ ∂2 δΓ ∼ h(t, κ) ∼ |t|−λ, (6) are renormalized values of the respective parameters. We as- ∂κ ∂t2 sume that the parameters evolve according to t → t0 = tLx and κ → κ0 = κLy, where the exponents x, y can be assumed to be with the same exponent λ = φ − f as in Eq. (5). positive, since the system must evolve away from criticality. Computational model — In order to test the scaling rela- Combining these, we find the elastic energy tions derived above, we study two complementary models of fiber networks: triangular lattice-based networks and jammed −d x y h(t, κ) = L h(tL , κL ). (2) packing-derived networks. Our triangular networks consist of fibers of length W arranged on a periodic triangular lattice The stress is simply given by the derivative with respect to with lattice spacing l0 = 1, with freely-hinging crosslinks at- strain of the elastic energy per volume, which is proportional taching overlapping fibers. To avoid singular mechanical be- to h(t, κ). Thus, σ ∼ ∂ h ∼ ∂ h(t, κ) ∼ L−d+xh (tLx, κLy) and ∂γ ∂t 1,0 havior deriving from infinitely long fibers, we initially cut a the stiffness single randomly chosen bond on each fiber, yielding an ini- ∂ ∂2 tial network coordination number z approaching 6 from be- K = σ ∼ h(t, κ) ∼ L−d+2xh (tLx, κLy), (3) ∂γ ∂t2 2,0 low with increasing system size [8, 20]. We prepare packing- derived networks by populating a periodic square unit cell 2 where hn,m refers to the combined n-th partial with respect to of side length W with N = W randomly placed, bidisperse t and m-th partial with respect to κ of h. Being derivatives of disks with soft repulsive interactions and with a ratio of radii the energy with respect to the control variable γ, the stress and of 1: 1.4. The disks are uniformly expanded until precisely stiffness are analogous to the entropy and heat capacity for a the point at which the system exhibits a finite bulk modulus, thermal system with phase transition controlled by tempera- after which a contact network excluding rattlers is generated ture. If we let L = |t|−1/x, then the correlation length scales [6, 21, 22]. Sufficiently large networks prepared using this −ν according to ξ ∼ L ∼ |t| , from which we can identify the protocol have an initial connectivity z ≈ zc [23]. correlation length exponent ν = 1/x. Thus, the stiffness can For both network structures, we reduce z to a value below be expressed as in Eq. (1), where G± (s) ∼ h2,0(±1, s) and the isostatic threshold by bond dilution and removal of dan- gling ends and clusters that do not contribute to the network’s f = dν − 2 and φ = yν. (4) bulk mechanical response [24]. We use a purely random dilution process, in contrast with special cutting protocols that The first of these is a hyperscaling relation analogous to that have been used previously to suppress variation in local con- for the heat capacity exponent for thermal critical phenomena, nectivity and promote mean-field behavior [7, 25]. We per- but in which f > 0 corresponds to nonlinear stiffness K that form simulations on ensembles of at least 50 network realiza- is continuous. For γ > γ , we expect that h (1, s) is approx- c 2,0 tions each for triangular networks and 30 networks each for imately constant for s 1, so that K ∼ |γ − γ | f , while for c packing-derived networks, both with z = 3.3. Unless other- γ < γ we expect that h (−1, s) ∼ s for s 1, so that c 2,0 wise stated, we use triangular networks of size W = 140 and −λ packing-derived networks of size W = 120. Examples of the K ∼ κ|γ − γc| , (5) final disordered network structures are shown in Fig. 1b-c. consistent with a bend-dominated elastic regime. Moreover, We treat each bond as a Hookean spring with 1D modulus the susceptibility-like exponent is expected to be λ = φ − f . µ, such that the total contribution of stretching to the network 3

(a) 0.5 = 0 -2 (b) > energy is c = 10 -1 c

) -2.5 10

2 c = 10 10 f = 10-3 2 P( -3.5 1 − = 10 -2 µ X li j li j,0 0 10 10 0 -4 4 < HS = (7) 0.2 0.25 = 10 10 c 2 l 1 K i j,0 c K hi ji -3 -2 10 10 = 10-4.5 102 K/ = 10-5 in which li j and li j,0 are the length and rest length, respectively, -4 = 10-5.5 -4 10 10 100 = 10-6 of the bond connecting nodes i and j. Bending interactions -2 -1 = 0 10 10 -6 c | | are included between pairs of nearest-neighbor bonds, which 10 10-5 are treated as angular springs with bending modulus κ. For 10-1 100 10-2 100 triangular networks, bending interactions are only considered between pairs of bonds along each fiber, which are initially FIG. 2. (a) Differential shear modulus K vs. shear strain for tri- collinear, whereas for packing-derived networks we account angular networks of size W = 140 and connectivity z = 3.3, with for bending interactions between all nearest-neighbor bonds, varying reduced bending stiffness κ. The dashed line indicates the observed critical strain γ for the ensemble. The inset shows the as in typical bond-bending networks. Thus, the total contribu- c probability distribution for the measured γ values for 50 individual tion of bending to the network energy is c network samples with κ = 0. (b) For γ > γc and with decreasing κ, K converges to the form K ∼ |γ − γ | f , with f = 0.73 ± 0.04. These 2 c κ X θi jk − θi jk,0 data are for the same networks as in (a). Inset: In the low-κ limit H = (8) and below γ , K/κ converges to a power law in |∆γ| with exponent B 2 l c hi jki i jk,0 f − φ ≈ −1.5. in which θi jk and θi jk,0 are the angle and rest angle, respec- then estimate φ by averaging values computed from two sep- tively, between bonds i j and jk, and li jk,0 = (li j,0 + l jk,0)/2. As we are interested only in the relative contributions of bend- arate scaling predictions, as follows. For γ < γc, we show the ing and stretching, we set µ = 1 and vary the dimensionless results for Eq. (5) in the inset to Fig. 2b. We also note that bending stiffness κ [26]. continuity of K as a function of strain near γc requires that f /φ f /φ We apply incremental simple shear strain steps using Lees- G±(s) ∼ h2,0(±1, s) ∼ s for large s. Thus, K(γc) ∼ κ , as Edwards periodic boundary conditions [27], minimizing the shown in the insets of Fig. 3a-b. Averaging the φ values computed from these corresponding fits, together with our previ- total network energy H = HS + HB at each step using the FIRE algorithm [28]. We compute the stress tensor as a func- ously determined value for f , we estimate φ = 2.26 ± 0.09 for tion of strain as triangular networks and φ = 2.05 ± 0.08 for packing-derived networks. These values of f and φ are used in Figs. 3a-b, 1 X σ = f r (9) which demonstrate the collapse according to Eq. (1). αβ 2A i j,α i j,β hi ji (a) Triangular (b) Packing-derived 100 f/ 100 f/ in which ri j is the vector connecting nodes i and j, fi j is the K~ K~ j i A force exerted by node on node [29], and is the area of -2 -2 -f -f 10 10 | | -1 the periodic√ box containing the network. For the triangular 10 ) 2 -2 ) c 10 c lattice, A = ( 3 /2)W , and for packing-derived networks, A = -4 -4 -2 K| 10 K| 10 10 K( 2 K( W . The differential shear modulus K is computed as K = f/ f/ -3 10 1 -3 1 -6 -6 10 ∂σxy/∂γ. To symmetrize K, we shear each network in both 10 f = 0.73 10 f = 0.68 -5 -3 -5 -3 the γ > 0 and γ < 0 directions. Figure 2a shows K(γ) for = 2.26 10 10 = 2.05 10 10 -5 0 5 -5 0 5 triangular networks with varying bending rigidity. 10 10 - 10 10 10 - 10 Results — First, we consider the scaling of K as a func- | | | | tion of strain near γc. We determine γc for individual samples FIG. 3. Plotting the K vs. |∆γ| data for both (a) triangular networks as the strain corresponding to the onset of finite K in the CF and (b) packing-derived networks according to the Widom-like scal- (κ = 0) limit, and utilize the mean of the resulting distribution, ing predicted by Eq. 1, and using the values of f and φ determined hγci, for our scaling analysis. The γc distribution for triangu- previously, yields a successful collapse for both systems. The dashed lar networks of size W = 140 is shown in Figure 2a. We lines in each panel have slope 0 and the dotted lines have slope 1. In- f /φ observe that with increasing system size, the width of the γc sets: At the critical strain, K ∼ κ . distribution decreases [24]. f We then determine f from K ∼ |γ − γc| in the low-κ limit. We compute the nonaffine fluctuations δΓ as We obtain a distribution of estimated f values using sample- 1 X δΓ = kδuNAk2 (10) specific K curves and γc values for networks with κ = 0, Nl2δγ2 i yielding an estimate of f = 0.73 ± 0.04 for triangular net- c i works, as shown in Fig. 2b with decreasing κ. Similarly, for in which N is the number of nodes, lc is the average bond NA A packing-derived networks we find f = 0.68 ± 0.04 [24]. We length, and δui = δui − δui is the nonaffine component of 4 the displacement of node i due to the imposed incremental same values of the exponents x and y as determined for the strain δγ. Plotting δΓ vs. γ − γc in Fig. 4a, we observe agree- strain-controlled transition. We can again determine the stress ment with the scaling predicted from Eq. (6) using the f and σ and stiffness K as in Eq. (3). By letting L = |∆|−1/w, we φ values determined above. Importantly, as predicted, the cor- again identify the correlation length exponent ν0 = 1/w and responding critical exponent λ = φ − f is the same as for Eq. find

(5), with λ ' 1.5 [30]. Further, we observe that near γc, the 0 0 f /φ−1 K ∼ |∆| f h (0, κ/|t|φ , ±1), (12) expected scaling δΓ(γc) ∼ κ appears to be satisfied [24]. 2,0,0 It is apparent from Fig. 4a that the divergence of the fluc- where tuations is suppressed by finite-size effects. This is consistent −ν with a diverging correlation length ξ ∼ |t| . Critical effects f 0 = (d − 2x)ν0, φ0 = yν0. (13) such as the divergence of δΓ will be limited as the correlation length becomes comparable to the system size W, correspond- Moreover, following similar arguments as above, it can be −1/ν −λ0 0 0 0 ing to a value of t ∼ tW = W . Thus, the maximum value shown that δΓ ∼ |∆| , where λ = φ − f [24, 31], consistent of δΓ increases as δΓ ∼ W(φ− f )/ν (Fig. 4a inset). From least- with the values f 0 ' 1.4 ± 0.1, φ0 ' 3.0 ± 0.2, ν0 ' 1.4 ± 0.2, squares fits to this scaling for both triangular and packing- and λ0 ' 2.2 ± 0.4 reported in Ref. [8]. While our approach derived networks with κ = 0 and κ = 10−7, combined with uses the elastic energy, it is interesting to note that prior work our estimates for φ and f , we determine that ν = 1.3 ± 0.2 on rigidity percolation has suggested the use of the number of for both systems. We then verify that this leads to a scaling floppy modes as a free energy [32]. collapse in a plot of δΓW( f −φ)/ν vs tW1/ν for both systems with Conclusion — The scaling theory and relations derived κ = 0, as shown in Fig. 4b, and with finite κ [24]. This finite- here for the strain- and connectivity-controlled rigidity tran- size scaling is consistent with the (hyperscaling-like) relation sitions in athermal fiber networks are consistent with our nu- f = 2ν − 2 in 2D from Eq. (4). merical results, as well as those reported previously observations near the isostatic point [8–10]. Interestingly, for the (a) 1 (b) 1 Triangular Packing-derived subisostatic, strain-controlled transition, we observe that sim- 10 1 10 ulations of both triangular networks and packing-derived net- 2 = 1.32 10 100 works exhibit consistent non-mean-field exponents. This, to- )/ 100 10-1 gether with agreement with the hyperscaling relation in Eq. 104 1 (f- ) ( -2 d 0 10 10 W -6 -4 -2 0 2 4 (4) suggest that the upper critical dimension for fiber net-

2 W = 50 d works is du > 2, in contrast with jammed networks at the iso- 10 = 0 10-1 max(d W = 70 -7 = 10 W = 100 static point [12]. Our observations, combined with prior scal- -2 W = 140 = 1.32 10 1 2 ing with similar exponents for alternate subisostatic network 10 W 10 W = 200 10-2 structures, including 2D and 3D phantom networks, branched 10-2 10-1 100 -5 0 5 10 15 20 W1/ (honeycomb) networks, and Mikado networks [11, 33], suggest that non-mean-field behavior might be ubiquitous in FIG. 4. (a) Near the critical strain, the nonaffinity scales as subisostatic networks, irrespective of the local network struc- δΓ ∼ |∆γ| f −φ. These data correspond to triangular networks with ture. Interestingly, the hyperscaling relation in Eq. (4), to- − κ = 10 7 and z = 3.3, with varying system size. Inset: Nonaffine gether with the observation that f > 0, suggests that fiber net- fluctuations are limited by the system size. For small or zero κ, the works satisfy the Harris criterion [34], which would imply that maximum of δΓ scales as max(δΓ) ∼ W(φ− f )/ν, with ν = 1.3 ± 0.2. (b) (φ− f )/ν 1/ν such networks should be insensitive to disorder. Further work Plots of δΓ/W vs. (γ − γc)W for triangular networks and (inset) packing-derived networks for with κ = 0 demonstrate successful will be needed to test this hypothesis, as well as the scaling re- scaling collapse using the f and φ values determined earlier, with ν lations derived here in 3D. This will likely require finite-size values determined from the scaling relation. scaling to identify the correlation length exponent, which has not been possible to date in 3D. Near the isostatic point — For networks near the isostatic This work was supported in part by the National Science transition at z = z , we can define a dimensionless distance c Foundation Center for Theoretical Biological Physics (Grant ∆ = z − z from the isostatic point and let h(γ, κ, ∆) be the c PHY-1427654). The authors acknowledge helpful discussions Hamiltonian or elastic energy per unit cell. At the isostatic with Andrea Liu, Tom Lubensky and Michael Rubinstein, as point, since γ = 0, t above reduces to the strain γ. Assuming c well as discussions with Edan Lerner and Robbie Rens on cen- the system becomes critical as γ, κ, ∆ → 0, we can follow a tral force packing-derived networks. similar real-space renormalization procedure as above, resulting in

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