OTROS PAÍSES O TERRITORIOS

Networks formed by semiexible laments undergo strain stiening (Figure13) and the dierential shear modulus dσ/dγ in dependence of the prestress σ can be de-

scribed by a power-lawdσ/dγ∼σ3/2[86]; which has its origins in the nonlinear FE

of the single laments -similar to the argument of nite extensibility above, caused by non-gaussian statistics of the deformed chains. Therefore the stiening can be tracked down to the straightening of entropic uctuations of individual chains [89]. Assuming homogeneous, isotropic networks, which are strained uniformly, Storm et al. demonstrated that theoretical predictions based on this entropic model are able to provide good agreement to the nonlinear rheological data of a wide spectrum of biopolymers [74]. This model is in this paragraph further referred to as the ane network model (AN).

Figure 13: Shear moduli vs. strain measurements [74]; observing strain-stiening for a series of cross-linked biopolymer networks.

Such networks typically also exhibit negative normal stresses: when sheared be- tween two plates, they tend to pull the plate together [90]. This property is directly related to the nonlinear strain-stiening behaviour of biopolymer gels - a natural consequence of the nonlinear force-extension of semiexible laments (Figure 14). This behaviour is observed inter alia for F-actin, collagen, neurolaments, matrigel [90] and brin [91,92] and is in stark contrast to a wide range of elastic solids where the torsion of a wire results in an axial elongation (Poynting eect [93]).

Figure 14: Schematic diagram: deformation leads to negative normal stress in an isotropic network of semiexible laments [90]; laments are elongated (red) and equally many are compressed (yellow), therefore an overall negative normal force arises due to the nonlinear force-extension relation of the single laments.

One explanation for strain stiening, as already mentioned above, is the non- linearity in the force extension arising from the lament's thermal uctuations [89, 86, 74, 94]. The extension of a semiexible chain in the high force regime is described by Eq. 1.28, where a divergence of the forcef∼ |δl−lC|−2is observed

as the end-to-end distance approaches the contour length. As already argued in Section 1.2.7, the extension of a chain segment under ane deformations is propor- tional to the strain γ and therefore df /dγ ∼f3/2 scales with a power law of 3/2.

Estimating again, that this force gives rise to the shear stress σ∼f /l2_M, leads to

dσ dγ ∼

f3/2 l2_M ∼lMσ

3/2 _(1.37)

And nally, for a network of sti chains l_M can be linked to the polymer concen-

tration c_P via l_M ∼ c−1_P /2 [95], the dierential shear modulus shows the following

scaling properties

dσ dγ ∼c

−1/2

P σ3/2 (1.38)

which could be experimentally conrmed e.g. for actin [89]. Storm et al. [74] observe the strain stiening in biomaterials such as actin, collagen, brin, neurolaments and vimentin and speculate this could thereby help in living organisms to prevent large deformations that could threaten tissue integrity (Figure13). Further experi- ments on the anity of deformations in brin networks [91] obtain results that are consistent with the entropic model for non-linear elasticity of semiexible polymer networks and show that strain-stiening does not require non-ane deformations.

Intermediate lament networks are ionically cross-linked; their nonlinear elastic- ity has purely entropic origins for neurolaments, and an additional enthalpic con- tribution due to backbone stretching for vimentin [96,97]. The backbone stretching

causes the dierential modulus to fall below the dσ/dγ ∼ σ3/2 _{behaviour at high}

prestresses. Strain stiening with a similar deviation at high stresses is also found by Blundell and Terentjev [98] in their elaborations on semiexible polymers with a mean inextensibility. Their results show three dierent scaling regimes: a lin- ear entropic regime at low forces, a nonlinear entropic regime at intermediate forces (causing the slope of3/2) and a linear mechanical regime at high forces (Figure15).

Figure 15: The normalised stiness as a function of the applied normalised force for a semiexible lament [98].

On the other hand a stiening in networks formed by much thicker and stier - bres, where the entropic uctuations can be neglected, can be modelled by non-ane network rearrangements, which control the transition from a bending-dominated re- sponse at small strains to a stretching-dominated response at large strains. This model is referred to as discrete network model (DN) [99]. A more elaborate three- dimensional analysis of this mechanism nds that the strain-stiening depends ad- ditionally on the network architecture through the local topology around cross-links [100]. Another way to describe this purely geometrical eect is to address the highly non-ane response of soft transverse oppy modes [79]. State-of-the-art simula- tion studies of three-dimensional cross linked elastic bres [101] also show stiening without entropic contributions.

Kang et al. [92] nd that brin gels are suitable systems to test the two dierent models of strain stiening, because the brin monomers assemble under dierent conditions to form either thermally uctuating protobrils with persistence length on the order of the network mesh size, or thicker rigid bres. The AN model does not allow the normal stress magnitude to exceed the shear stress, whereas the athermal"

model does. The ratio of normal stress to shear stress, which is strain dependent,

was found to be a parameter to experimentally determine which of the two models applies. Computational studies of van Dillen et al. [102] compare the two opposing models. At low and intermediate lament densities, the DN model deviates from the AN model as a result of non ane motions and bending deformations of the laments in the network, resulting in a DN-stiness that is signicantly smaller than the calculated AN-stiness. In contrast the stiness resulting from the DN calcula- tions does approach the ane value at high network densities. These observations suggest that, besides entropic stiening in segments between cross-links, changes in the network architecture can play a crucial role in its mechanical response, especially at low and intermediate densities.