LOS COMPONENTES DEL ENTRENAMIENTO DEPORTIVO

The picture of a floppy matrix embedding stiffer inclusions breaks down when the stiff filaments become independently rigidity percolated: the point where defor- mation of the stiffelements becomes inevitable. We may estimate the percolation threshold by a counting argument [61]. Equating the number of degrees of freedom of the sti_fffilaments to the number of constraints due to crosslinks between sti_fffil- aments gives (forL/`c=6) a thresholdcs=0.56. This marks the transition from the low to the highcsregime and coincides roughly with the rise in the linear modulusK0 (figure 3.3a). Two separate obervations confirm the onset of stiffdominance: Firstly,

cs=0.56 is the point at which the non-affinity, which we attribute to the floppy ma- trix attempting to work around the stifffraction, begins to plateau at the level of the bending dominated response of a purely stiffnetwork. Secondly, the critical strainγc levels offaround this same value ofcs. In the range of stiffness ratios (Rp) accessi- ble to the simulations the percolation is rather “soft” and represents a smooth cross- over phenomenon. The approach towards the singular percolation limit,Rp= ∞, has for example been studied in simulations of mixed random resistor networks [62]. To address the analogous problem we compare our results with theoretical considera- tions, in which the parameterRpcan be tuned to arbitrarily large values. The “floppy- mode” theory [36] has recently been shown to capture quite well the elasticity in one- component isotropic [24, 63, 64] as well as anisotropic networks [65]. Within this the- oretical framework the calculation of the network elastic modulus is reduced to the description of a “test” filament in an array of pinning sites. The coupling strength to these sites,k, represents the elastic modulus of the network and has to be calcu- lated self-consistently. To generalize this model to the case of composite networks we use two different test chains with coupling parameterskf andks, representing floppy and stifffilaments, respectively, see [66] for details of the calculation. The use of two different coupling strengths quite naturally takes into account the load parti- tioning encountered in the simulations. The network modulus,k=csks+(1−cs)kf, is obtained by solving the two equations

kf/s' * min y ( W_bf/s[y(s)]₊1 2 n ∑ i=1 k_αi(y(si)−y¯i )2)+ , (3.1)

wherek_αi =ks,kf with probabilitycs and 1−cs, respectively. The two energy con-

tributions on the rhs of equation (3.1) reflect the competition between the bending energy of the (floppy or stiff) filament,W_bf/s, and the energy due to deformation of the surrounding medium by displacing the pinning sites (located at arclength po- sitionsi along the filament). The nonlinear entropic stretching elasticity is not in- cluded in these equations. The minimization is to be performed over the contour of

3.4 Conclusions 63

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Figure 3.5: The scaled initial stiffness as a function ofcs, obtained by the floppy-mode model forRp=16, 64, 1000,∞(from top to bottom). For comparison, the data from simulations are given by the symbols.

the filament,y(s), the angular brackets specify the disorder average over the network structure.

Figure 3.5 displays the results from this calculation for various stiffness ratiosRp, showing a sharp percolation transition in the limitRp → ∞. The model compares well with the simulation data, even though the entropic stretching elasticity is not ac- counted for. This indicates that bending is likely the dominant factor in determining the rise of the linear elastic modulus, in agreement with the proposed mechanism of load-partitioning and the observed increase of the non-affinity.

3.4 Conclusions

In conclusion, our results demonstrate that the mechanical behavior of filamentous composites is considerably richer than the simple proportional mixing of properties. The fact that the floppy and stiffnetworks are physically linked causes a strongly non- linear coupling between the strain fields which deeply affects composite mechan- ics. This may explain the ubiquity of composites in structural biological applications: slight variations in composition cause large changes in mechanical behavior. This high susceptibility makes the composite architecture an attractive motif for biologi- cal regulation. Likewise, the “best of both worlds“ aspect may be exploited by Nature: composites combine the initial softness of their most compliant components with the ultimate toughness of the stiffest elements. This greatly enhances the stiffness range of nonlinearly elastic materials. Moreover, composites do so in a manner that could never be attained in monodisperse materials, since linear and nonlinear prop-

erties of composites are determined by two physically different materials and there- fore may be independently varied. This possibility of independently tuning the linear and nonlinear behavior also has considerable potential for the design of biomimetic or bio-inspired synthetic materials and deserves further exploration.

While exploring the physics of composite networks, we are faced with the fact that internal stresses, percolation and floppy modes are important for the character- ization of our networks. This raises questions on the presence and impact of internal stresses in networks, the ridigly percolation transition and the occurence and char- acteristics of floppy modes. In the next chapter we will therefore return to single- component networks and adress these issues.

C h a p t e r

4

Internal stresses, normal modes and

non-affinity of networks

We numerically investigate deformations and normal modes of three-dimensional networks of semiflexible biopolymers as a function of average crosslink coordina- tion numberz and relative strength of bending and stretching energies. Our net- works consist of filaments that in equilibrium are in a state of internal stress, and they exhibit shear rigidity below the Maxwell isostatic point. In contrast to two- dimensional networks, ours exhibit nonaffine bending-dominated response in all rigid states, including those approaching the maximum ofz=4 as long as bending energies are smaller than stretching ones.

4.1 Introduction

As discussed in introductory chapter, networks of semiflexible biopolymers [4, 7, 16, 67] are important for determining and controlling the mechanical properties of eu- karyotic cells. Understanding their properties, in particular the relation between me- chanical response and network architecture, has been a major goal of biophysics re- search. Networks of semiflexible biopolymers consist of long filaments of average lengthLlinked two at a time by crosslinks so that each is connected to at most four others [15, 27, 32, 35, 68]. The crosslinks we consider, which we will refer to as nodes, allow free rotations of filaments relative to each other. They divide the filaments into a series of segments, of average length`c, that give rise to a central force between nodes determined by the force-extension curve of a semiflexible polymer. In addi- tion, bending forces favor parallel alignment of contiguous segments on the same fil- ament meeting at a common node. As shown in chapter 2, the mechanical properties of networks of semiflexible biopolymers depend on their connectivity, parameterized by the average coordination numberzof their nodes or by the ratioL/`c, on their in- teraction parameters, and on their architecture.

Networks of semiflexible polymers have much in common with those that occur in network glasses [69–71]. They are both continuous random networks [72]; they both have nodes with maximum coordination number 4; and they are both stabilized below the central-force rigidity threshold by bending forces, between all bonds in the latter but only between segments in the same filament in the former. Careful mode counting and study of the mode structure of network glasses [69–71] and other ran- dom systems such as hard spheres near the jamming transition [73–75] have provided fundamental insight into the physics of these systems. They have also been used in the study of two dimensional networks of semiflexible biopolymers [23, 36]. In this chapter, we undertake a similar study of simulated three-dimensional networks of semiflexible biopolymers [54] as a function of their connectivity and interaction pa- rameters, and we analyze the zero-frequency shear modulus and the mode structure as a function of these parameters.