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Mechanics of Disordered Fiber Networks Xiaoming Mao* Department of Physics, University of Michigan, 450 Church Street, Ann Arbor, Michigan 48109, United States *E-mail:
[email protected].
Disordered network models of fibers tied together by crosslinks capture the essential structure of many materials in nature, such as the cytoskeleton and the extracellular matrix as networks of biopolymers, and manmade materials such as paper and felt. In this chapter, we discuss fundamentals of mechanical properties of disordered fiber networks. A basic concept we introduce is the central-force isostatic point, where in the limit of ignoring fiber bending stiffness, the network is on the verge of mechanical instability. Networks with connectivity lower than this point are bending dominated when they are deformed, whereas dense networks with connectivity above this point are stretching dominated. Using this concept, we explain phase diagrams of linear elasticity of fiber networks, strain stiffening in the nonlinear elasticity regime, as well as unusual fracturing processes of these fiber networks.
Introduction Fiber networks are disordered networks composed of long fibers and crosslinks that tie the fibers together (Figure 1a). This simple conceptual model of fiber networks captures the essential structure of a wide collection of materials in nature, including various biopolymer network gels in the cytoskeleton and the extracellular matrix (1–4) (Figure 1b), and manmade materials, such as felt, paper, and buckypaper which shares a similar structure with regular paper but consist of carbon nanotubes instead of cellulose fibers (5). Many different types of gels, such as colloidal gels consisting three dimensional dilute networks of colloidal particles, hydrogels which are polymer networks swelled by water, and © 2018 American Chemical Society
aerogels composed of porous solid nanostructures with numerous air pockets, all share fibrous nature in their microscopic structure, and could be characterized as fiber networks.
Figure 1. Mechanics of fiber networks. (a) An illustration of the fiber networks model consisting long fibers tied by crosslinks. (b) Confocal reflectance image of collagen network (green) as an example of fiber network material. Three cancer cells (red) pull on the collagen network, driving it into nonlinear elasticity regime. Scale bar 20μm Adapted with permission from ref. (6). Copyright 2017 Springer Nature. (c) Storage (G′) and loss (G″) shear modulus of various fiber network gels as functions of shear strain γ showing strain stiffening. Adapted with permission from ref. (1). Copyright 2005 Springer Nature. (d) Storage modulus as a function of strain for collagen gels at different collagen concentration. Adapted with permission from ref. (2). Copyright 2009 PLOS. (see color insert)
Fiber networks display intriguing mechanical properties. A rather universal phenomenon among many fiber networks is “strain stiffening”, which means under shear strain, the shear modulus of the network greatly increases, often by more than an order of magnitude, before the network fails (Figure 1c). Even in the linear elasticity regime, shear modulus of fiber networks can also change very sensitively to parameters such as fiber density (Figure 1d). Such huge change of shear modulus as control parameters change indicates that there is a mechanical critical point underlying this phenomenon. In this chapter we will discuss this mechanical critical point, named “central-force isostatic point” (CFIP), and its profound implications to the elasticity of various fiber networks. 200
In Sec. 2 we introduce foundations of mechanics of fiber networks, the CFIP, and discuss its implications in the linear elasticity regime. In Sec. 3 we discuss implications of the CFIP for nonlinear elasticity, and In Sec. 4 we discuss interesting behaviors of these fiber networks when they fail under pressure, where the CFIP plays an important role again.
Maxwell Counting and Linear Elasticity of Fiber Networks To elucidate mechanics of fiber networks, let us start by considering a minimal model, where details of the actual fibers and crosslinks are ignored, and we focus on the essential physics of the mechanical properties of the network. In this minimal model, we have Nfiber fibers which are modelled as straight slender elastic rods. Each fiber i has ni crosslinks on it, which are modelled as free hinges connecting fiber i with other fibers. These crosslinks make sure the fibers cross but post no constraints on their rotation. The Hamiltonian of this minimal fiber network model can then be written as
where the first term is the stretching energy and the second term is the bending energy of the fibers. In the first term, is the position of crosslink m on fiber i, li,m and μi,m are the rest length and the stretching stiffness of segment m (defined as the segment between crosslinks m and m + 1 on fiber i. In the second term, the sum rums over all inner crosslinks on each fiber. θi,m is the bending angle of fiber i at crosslink m. Here we have discretized the bending energy of the fibers such that the fiber segments are straight between neighboring crosslinks and bend at finite angles at each crosslink. In Mao et al (7) we have carefully discussed this approximation from the full continuous bending energy of the fibers as elastic rods. Because fibers are long slender rods, it is straightforward to see that it’s much easier to bend than to stretch them, so (here we refer to general segments so we dropped the subscripts i, m). It is thus very interesting to consider the limit of where the system becomes a central-force (CF) network. “Central force” means that the Hamiltonian only depends on the distances between the crosslinks [keeping only the first term in Eq.(1)]. An example of taking this limit is shown in Figure 2a. 201
Mechanics of CF networks has been extensively studied since J. C. Maxwell, who wrote down the equation for the number of zero modes (normal modes of zero energy) of a CF network
where N is the number of sites (corresponding to crosslinks in the fiber network), d is the spatial dimension which gives the number of degrees of freedom per site as a point particle in d dimensions, and Nb is the number of bonds (corresponding to fiber segments in the fiber network model). This equation relates the number of zero modes of a CF network to the “unconstrained” degrees of freedom, and this follows directly from normal mode analysis of the equations of motion of the network (8). Maxwell’s counting rule [Eq.(2)] leads to a simple equation for the verge of mechanical instability in CF networks
where ázñ is the mean coordination number of the sites (mean number of arms from a site). This follows from Eq.(2) by considering degrees of freedom and averaged number of constraints per site. In general, when ázñ < 2d the CF network has more constraints than degrees of freedom and is stable, whereas when ázñ < 2d there must be zero modes. The special point where ázñ = 2d marks a mechanical critical point of a CF network separating stable and unstable phases, and is thus called the central-force isostatic point (CFIP) (9–12). Networks in the CF limit with ázñ < 2d are in general floppy and have vanishing elastic moduli. Interestingly, when we turn bending stiffness of the fibers, κ, back on, network elastic moduli are restored, because the zero modes bend the fibers and will now cost bending elastic energy. This regime is thus called “bending dominated.” Of course, the resulting elastic moduli in this regime are proportional to the fiber bending stiffness, G~κ and much smaller than elastic moduli of networks with ázñ > 2d (the “stretching dominated” regime), which is proportional to stretch stiffness of the fibers, G~μ. Mechanical phase transitions at the CFIP has been studied from the 1980s in the literature of glass transitions (10, 14–16). More recently this concept has been applied to understand fiber networks with bending stiffness (7, 9, 13, 17). Fiber network models based on diluted periodic lattices have been used to analytically study this mechanical critical point via effective medium theory. With good agreement between the effective medium theory and numerical simulations, phase diagrams were obtained for these fiber networks, as shown in Figure 2 b-e.
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Figure 2. The central force isostatic point and phase diagrams for fiber networks linear elasticity. (a) Mapping a fiber network to a CF network with the same geometry by turning off the bending stiffness. (b) An example of a diluted triangular lattice under shear deformation. Color shows deviation from affine (i.e. uniform) deformation: blue for small deviation and red for large nonaffine (non-uniform) deformations. (c) Linear elasticity phase diagram for lattice in (b). (d) An example of a diluted kagome lattice, the phase diagram of which is shown in (e). Note axes in (c) and (e) are related to one another by a 90° rotation. (b-c) Adapted with permission from ref. (9). Copyright 2011 Springer Nature. (d-e) Adapted with permission from ref. (13). Copyright 2013 American Physical Society. Labels in the phase diagram have been modified to follow the notation of this chapter. (see color insert) In particular, two types of two-dimensional lattices, the triangular (18) and the kagome lattices have been used. The main difference between these two lattices is that the maximum coordination number (the coordination number of the undiluted lattice), zm is zm = 6 > 2d for the triangular lattice (Figure 2 b-c) and zm = 4 = 2d for the kagome lattice (Figure 2 d-e). Bonds in the lattices are randomly removed with probability 1 − p so the mean coordination number is ázñ = pzm for a network. 203
This difference in zm makes a big difference in the resulting phase diagrams of the two types of lattices. For the triangular lattice the CFIP is a continuous transition in the middle of the ázñ axis, giving rise to a critical regime where bending and stretching effects are coupled in shear deformations (7, 9). On the other hand, for the kagome lattice, the CFIP is the maximum of ázñ (corresponding to p = 1) and features a discontinuous transition (13). For both lattices, there is also a rigidity transition at lower coordination zb = pbzm where fiber networks with κ > 0 loses rigidity. These two lattices show very similar behaviors in their bending dominated regimes, and can both be used to describe diluted fiber networks. The triangular lattice model has the extra regime where ázñ > 2d which can be applied to networks where crosslinks can tie more than two fibers together. In Broedersz et al (9), a three-dimensional diluted face-center cubic lattice was also studied, which displays a phase diagram similar to the triangular lattice. Besides these lattices, off-lattice disordered fiber networks models, such as the Mikado model, has also been extensively used in the literature to study mechanics of fiber networks (19, 20). Now let’s consider implications of the results from these diluted lattice models to real fiber networks. First, if the crosslinks in the network ties only two fibers together (which is the case for most fiber networks), we have zm = 4 which = 2d in two dimensions and < 2d in three dimensions. This puts these networks almost always in the bending dominated regime, where G~κ and the network is very soft in the linear elasticity regime. Second, in many experiments the control parameter is the fiber concentration (e.g., in Figure 1d). As analyzed in Sharma et al (21), the fiber concentration controls both the mesh size of the network and the bending stiffness of the fibers. For a fiber network in the bending dominated regime close to the CFIP, the shear modulus is very sensitive to the change of these parameters. This explains why the linear shear modulus of fiber networks displays big spreads at different fiber concentrations (Figure 1d).
Nonlinear Elasticity of Fiber Networks As we mentioned in the introduction, strain stiffening, i.e., shear modulus increases as strain increases in the nonlinear elasticity regime, is a ubiquitous phenomenon in most fiber networks (1). There are two main causes for strain stiffening. First, as discussed in Storm et al and Mackintosh et al (1, 22), fiber networks are often in a thermal environment and fiber segments experience thermal undulations. In other words, the equilibrium state we consider should be a state where fiber segments have excess length for thermal fluctuations. When the network is under strain, these fibers are being stretched. In the first stage, the thermal undulations are being stretched, and the shear modulus is controlled by entropic elasticity, G~kBT which is small (similar to rubber elasticity). As the strain progress, thermal undulations are stretched out and the fiber segments become straight. Any further strain will stretch the fibers directly, so the shear modulus is controlled by the stretching modulus G~μ which is much larger. 204
Figure 3. Strain stiffening of fiber networks. (a) An illustration of soft bending modes being exhausted by shear deformation and any further deformation will stretch fibers. Red dashed lines show fiber segments becoming aligned with the strain. Adapted with permission from ref. (25). Copyright 2015 American Physical Society. (b) Three-dimensional phase diagram for the diluted triangular lattice model in the space of probability of each bond to be present, p (which relates to network connectivity through ázñ zmp), bending stiffness κ and shear strain γ. The p − κ plane is the same as the linear elasticity phase diagram as shown in Figure 2, and the γ axis represents nonlinear strain. The yellow dot denotes the CFIP. As strain increases, bending dominated networks enters the nonlinear regime and eventually become stretching dominated (26). In the limit of κ = 0, the bending-dominated to nonlinear crossover becomes a continuous transition with a line of critical points (red solid curve starting from the CFIP), as pointed out in Sharma et al. (21) Point Ω represents κ = γ = 0 and p = pb (defined in the previous section). (c) Similar phase diagram for the kagome lattice, where the CFIP is at p = 1 and there is no critical regime. (b-c). Adapted with permission from ref. (26). Copyright 2016 Royal Society of Chemistry. (see color insert)
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The second cause, as first discussed in Onck et al and Wyart et al (23, 24), is an athermal scenario based on the geometry of the network. As the strain increases, bending modes become “exhausted” and fibers are becoming stretched so the shear modulus changes from G~κ to G~μ and greatly increases (see illustration of this effect in Figure 3a). The second scenario is easy to understand from the linear elasticity phase diagrams we discussed in the previous section: if a network start in the linear regime as bending dominated (because they have ázñ < 2d and small bending stiffness), they can crossover to stretching dominated in the nonlinear regime. Which of the two scenarios applies for a real fiber network depends on the network itself. One simple quantity to consider is the persistence length of the fibers in the network, lp which characterizes the length scale under which the fiber stays straight under thermal fluctuations. If the persistence length is much smaller than the mesh size, strong thermal undulations are present on fiber segments. In the linear elasticity regime, the network is soft due to both thermal undulations and bending modes, and in the nonlinear regime, both of these two effects need to be exhausted before the network enters the stretching dominated fiber segments regime and shear modulus increases. On the other hand, if are essentially straight between crosslinks, and strain stiffening is mainly caused by bending to stretching crossover. It is instructive to carefully investigate this strain-induced bending-tostretching crossover using the diluted lattice models again, which are convenient for theory and simulation (21, 25–27). Nonlinear elasticity of these lattice models (which captures essential physics of fiber networks) are characterized by the phase diagrams in Figure 3. In particular, the triangular lattice where the CFIP is a continuous transition displays a nonlinear regime between bending and stretching dominated regimes, whereas the kagome lattice does not. Moreover, in the limit of κ = 0, the lattices are completely floppy below the CFIP, and as strain increases, stiffness emerge as a continuous phase transition, as pointed out in Sharma et al. (21) Strain stiffening in fiber networks studied here shares a lot of similarities with nonlinear elasticity of colloidal gels (28). Another feature related to strain stiffening in fiber networks is the alignment of fibers in the strain direction (Figure 3a) as the bending modes are being exhausted. This alignment can be described be a nematic like order parameter as discussed in Vader et al and Feng et al (2, 25) and plays an important role in guiding cell motion in the extracellular matrix, a phenomena called “contact guidance” (2, 29–31).
Fracturing of Fiber Networks The interesting nonlinear elasticity of fiber networks governed by the crossover from bending to stretching dominated regimes, as discussed in the previous section, implies unusual phenomena when these networks fail under pressure.
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Figure 4. Fracturing of fiber networks. (a) Stress concentration near crack tips in dense fiber networks above the CFIP (using the diluted triangular lattice model), with the resulting Griffith crack shown in (b). (c) and (d) show force chains and distributed damage in dilute fiber networks below the CFIP. Bonds are colored according to force; broken bonds are also marked. Bonds with a larger force are thicker. Adapted with permission from ref. (32). Copyright 2017 American Physical Society. (see color insert)
Failure of conventional brittle materials is characterized by a long, straight crack, a phenomena called “crack nucleation”, due to stress focusing effect at crack tips, as explained by A. A. Griffith in 1921 (33). Surprisingly, it is found that fiber networks below the CFIP displays very different phenomena, where cracks do not nucleate and damages are distributed over a divergent length scale, even though microscopic breaking events of fiber segments are set to be brittle (32). The origin of this interesting phenomenon is the mechanism of how rigidity emerge as strain increases in these fiber networks. Let us first consider the CF network (κ = 0)), which has no rigidity (G = 0) below the CFIP. In a numerical 207
study in Zhange et al (32), a diluted triangular lattice model was used, and it is observed that as strain increases beyond a critical value γc (red line in Figure 3 b-c) the network starts to have rigidity as force chains emerge which bear stress. which means it will break If each fiber segment has an extension threshold when it is extended beyond (1 + λ) of its original length, fibers in the force chain will break first. Interestingly, after the first force chains break, new force chains will emerge, and their locations are not correlated with the previous ones. Thus, the network enters a “steady state” where new force chains replaces broken ones, and the damage is distributed over the whole sample with no crack nucleation, as shown in Figure 4. This phenomenon occurs for all fiber networks below the CFIP. On the other hand, for networks above the CFIP, crack nucleates at a length scale, which only diverge as the network approaches the CFIP from above (34). When a small bending stiffness κ is added, this phenomenon of distributed damage in fiber networks remains true, because the origin of this phenomenon, the steady state where force chains emerge-break-emerge, is robust against the addition of bending stiffness. It is only when the bending stiffness becomes comparable to the stretching stiffness, the network will enter the linear stretching dominated regime, and the Griffith scenario of stress focusing and crack nucleation is recovered (32).
Conclusion and Discussion In this chapter, we reviewed studies of fundamental mechanical properties of disordered fiber networks, which models a wide variety of materials, such as biopolymer gels and manmade porous media. A number of interesting mechanical properties, such as strain-stiffening, sensitive change of elastic moduli as a function of fiber concentration, distributed damage during fracturing process, can emerge from an underlying mechanical critical point, the CFIP, where the network is at the verge of mechanical stability in the central-force limit. Predictions based on the CFIP has been verified in experiments on biopolymer gels (21, 27). The comparison between theoretical predictions based on the CFIP and measured mechanical properties in various types of real gels can help elucidate the structures and mechanics at microscopic length scales of these gels. This also opens the door to a number of intriguing questions for future study, such as the cytoskeleton and the extracellular matrix as active fiber network under driving, as well as the design of mechanical metamaterials with well controlled nonlinear mechanical response and fracturing process (35).
Acknowledgments The author was supported by the National Science Foundation under grant DMR-1609051. 208
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