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Many factors lead to non-affine deformation in hydrogels. Inhomogeneities can play a major role in the degree of non-affinity in polymer gels. Didonna and Lubensky demonstrated theoret- ically [23] that variations in local elasticity lead to spatially correlated non-affine deformation in random, elastic media. The magnitude of this non-affinity was predicted to be proportional to the variance in local elastic moduli [23]. Experiments [8] indicate that network inhomogeneities formed during sample preparation are a major source of non-affinity in flexible polymer gels such as PA gels. Fig. 2.3(a) is a schematic of some common network imperfections that can lead

to inhomogeneities in flexible polymer gels: (a) closed loops of polymer chain, wherein a cross- linking unit is attached to the ends of the same polymer chain instead of connecting two chains together, (b) dangling polymer chain ends, (c) cross-links reacting among themselves instead of with polymer chains, and (d) polymer chain entanglements that tend to slip under external loading [22]. Inhomogeneity in cross-link and polymer concentrations may also occur during polymerization. The size of the inhomogeneities can range anywhere from tens of nanometers to a micrometer [48] which determines the non-affinity length-scale, i.e., the length-scale above which the gels deform affinely.

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Figure 2.3: Inhomogeneities in cross-linked polymer gels.

Inhomogeneities are not restricted to flexible polymer gels only. Semiflexible polymer net-

works, be theyin vivocollagen scaffolds [49], orin vitrofibrin gels [74], may have an additional

sources of non-affinity,viz., spatial inhomogeneity in the gel. Microscopic inhomogeneities in

under shear. Such inhomogeneities may be inherent, or may depend on deformation proto- cols [36, 90]. Deformations larger than the average size of these inhomogeneities are seen to be essentially affine [49].

Experimentally, observed deformations in a polymer gel under external load can be affine or non-affine depending on the length-scale examined [125]. Different polymer gel classes have

different “important” length-scales,viz., persistence length of the constituent polymers, end-to-

end length of filaments, mesh-size, etc.

For an isotropic, cross-linked polymer network [47, 45], the macroscopic elasticity parame- ters like the shear and the Young’s moduli can be seen to depend on the bending and stretching

moduli, i.e.,κ andµ, respectively, of the constituent polymer filaments. For a filament of arc

length, s, the total length, δl(s), the Hamiltonian per unit length, δsin the simplified linear

regime can be written as

δ_H δs = µ 2 δl δs 2 +κ 2 δθ δs 2 , (2.5)

whereθ(s)is the angle the filament makes atswith thexˆaxis. In the limitκ _→ 0, the system

becomes a network of flexible polymers where all network deformations occur through stretch-

ing of individual polymer filaments. At the other extreme, when κ _{→ ∞}, the energetic cost

of filament bending is prohibitive and network deformations are again stretching-dominated. In the intermediate regime, however, there is a transition from bending-dominated to stretching-

dominated deformation as the ratio ofκ/µdecreases. For such isotropic, cross-linked polymer

networks, there is an intrinsic non-affine length-scale, λ = lc(lc/lb)1/3 [47, 45], depending

on gel morphology. Here, lc is the average length of polymer chain between cross-links, and

lb= p

filament length andλ. At largel/λ, i.e., when the network is highly crosslinked, network defor-

mation is affine. Conversely, for a loosely crosslinked network,l/λis small and deformation is

non-affine.

Biopolymer filaments can bundle together under certain conditions, e.g., pH [128] and shear

[66]. Formation of bundles changes the value oflb, and hence, the mode of deformation in the lo-

cality of the bundles. Of course, a polymer network with filament bundles randomly interspersed must be inhomogeneous on the length-scale of the filament bundles. The non-affinity measure is also affected by the applied strain: under extensional forces, non-affinity has been measured to increase with increasing strain [36], and under shear, non-affinity decreases as strain increases [155].

Simulations of 2D athermal networks of rigid rods [115] mimic gels consisting of stiff poly- mer filaments. Under shear, such a system exhibit filament-bending at low strains, and network rearrangements under high strains, both causing non-affine deformations. Such shear-induced network rearrangements are depicted in Fig. 2.3(c). Network rearrangements were observed in collagenous tissue, albeit under uniaxial extension [36], especially when loaded perpendicular

to the natural occurring alignment of collagen fibers; here, the stiff fibers reorienteden masse

to align with the direction of extension. Such non-affine bending and rearrangement in (non- covalently bonded) stiff collagen filaments that form the underlying substrate have been shown to have profound effects on the shape, proliferation and motility of mammalian cells [146].

There are yet other sources of non-affine deformation. The effect of network connectivity

on elasticity and non-affinity has been investigated by Broedersz, et al. [11] using a lattice-

conditions [90], and gelation kinetics [8], also have influence on gel morphology and hence non-affinity measures.