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Modeling (Visco)elasticity of Macromolecular and Biomacromolecular Networks

Fanlong Meng1,2,3

1Chinese Academy of Sciences, CAS Key Laboratory for Theoretical Physics, Institute of Theoretical Physics, Beijing 100190, China

2University of Chinese Academy of Sciences, School of Physical Sciences, Beijing 100049, China

3University of Chinese Academy of Sciences, Wenzhou Institute, Wenzhou 325000, China

When polymers are permanently crosslinked into a network, e.g., by covalent bonds which possess a very high energy barrier to break, then the network is elastic and can sustain mechanical loadings, resulting from the conformational entropy of the polymers if not considering other specific intramolecular/intermolecular interactions. By replacing the permanent crosslinks with the transient ones (low energy barriers to break and/or re-form), which can be either physical crosslinks formed by physical interactions (hydrogen bonds, guest-host interactions, hydrophobicity, etc.) or chemical ones whose breakage energy barrier is low (e.g., by adding catalysts), then the crosslinked network can dynamically re-organize its topological structure by the crosslink breakage/re-formation; the network can be treated as complex fluids, exhibiting interesting viscoelastic properties. Due to their characteristic rheological properties, the products made of the macromolecular networks (both permanently and transiently crosslinked ones) are widely utilised in various applications such as rubber bands, tires, self-healing materials, etc. As a special class of the macromolecular networks, the biomacromolecular networks are ubiquitous in nature such as cytoskeletons and extracellular matrices, which are relevant with various bio-functions including shape maintenance of cells, cell division, and movements. In the following, we will introduce the well-known theories of permanent macromolecular networks consisting of flexible polymers in Section 1.1 and those of permanent biomacromolecular networks consisting of semiflexible polymers in Section 1.2, discuss the viscoelastic responses of transient macromolecular networks in Section 1.3, and then finish this chapter with a brief discussion about some possible developments in the future.

1.1 Permanent Macromolecular Networks

As polymers are the main entity of the macromolecular network, the physical properties of a single polymer are very important in determining the overall responses of the network.

1.1.1 Mechanic Properties of a Single Polymer Chain

For a single polymer, one can define a persistence length [1, 2] to quantify the bending rigidity of the polymer (, with as the bending rigidity, as the Boltzmann constant, and as the temperature), over which the correlations in the tangent direction along the polymer are lost. A relevant physical quantity is Kuhn length [1], which is , over which the polymer can be treated as freely joint. By comparing the contour length and persistence length (as shown in Figure 1.1), polymers can be categorized into flexible (), semiflexible () and rigid () ones. Usually, synthetic polymers such as polyethylene can be taken as flexible and bio-polymers of interested lengths such as microtubules are semiflexible. There exist many famous models for describing the mechanic properties of a single polymer [1, 3-6], and three of them are listed here, two for flexible polymers and one for semiflexible polymers.

• Gaussian chain. By assuming a flexible polymer with a bead-spring structure (shown in Figure 1.1a) where the lengths of the springs can fluctuate obeying a Gaussian distribution and the orientations of the springs are independent of the others, one can get the probability of finding the polymer which consists of monomers (or springs in the bead-spring description) with the end-to-end vector, , as shown in Figure 1.1 [3], as: , where is the monomer size. Then the free energy of the polymer as a function of the end-to-end vector, , is: , which describes the Gaussian chain as a Hookean spring with the elastic constant: . Straightforwardly, one can obtain the force-extension relation as: (1.1)

  As noted, the Gaussian chain can be stretched infinitely with if the tensile force is large enough, which is not correct for realistic polymers. Actually, this model can only describe the cases where the polymer undergoes small deformations rather than finite ones, i.e., .

  Figure 1.1 Illustration of a (a) flexible polymer () (the inset shows the bead-spring structure), (b) semiflexible polymer (), and (c) rigid polymer ().

• Langevin chain. For a flexible polymer chain which is stretched finitely, , one can adopt the freely joint chain model, where the length of the springs in Figure 1.1a is a constant (infinitely rigid and not deformable) and the orientation of each spring is independent of others. Then the force-extension relation can be obtained as [6]: (1.2) where is the Langevin function. One can easily show that Equation (1.2) reduces to Equation (1.1) if .
• Semiflexible chain. For semiflexible polymer chains, one needs to consider the bending rigidity (the orientations of the springs are dependent on those of their neighbors if still taking the bead-spring picture for an intuitive understanding). A semiflexible polymer of contour length can be coordinated as with the arc length coordinate along the polymer, and the bending energy of the semiflexible polymer can be written as: where is the bending modulus, and is the local curvature at . There are different models reviewed in Ref. [7, 8] for describing the elasticity of a single semiflexible polymer, such as Marko-Siggia model [9], Ha-Thirumalai model [10], MacKintosh-Käs-Janmey model [11], and Blundell-Terentjev model (shown below as an example) [12]. By taking the mean inextensibility assumption, one can obtain the free energy of a semiflexible polymer as a function of the end-to-end factor of the polymer , approximated as (details in Ref. [12]): , where describes the bending rigidity of the polymer. From this, one can obtain the force-extension relation as: (1.3) Obviously, the force diverges at large when the polymer is stretched to its length limit.

With the mechanical models of a single polymer as exemplified above, one can try to construct theoretical models for macromolecular networks by taking account of the crosslinked structure of the network. The theories for a permanently crosslinked network have been developed for a relatively long time, and there are many successful models as reviewed in Ref. [13, 14]. The theoretical models of permanently crosslinked macromolecular networks can be roughly categorised into two types: statistical ones based on assumed network structures (Section 1.1.2) and phenomenological ones with fitting parameters well matching experimental observations (Section 1.1.3).

1.1.2 Statistical Models

With the mechanic models of a single polymer as shown above, one can try to construct the constitutive models of a macromolecular network with proper assumptions of the network structure. Here, we introduce four models with different network architectures, as shown in Figure 1.2.

Figure 1.2 Four different architectures of macromolecular networks: (a) 1-chain model (full network model), (b) 3-chain model, (c) 4-chain model, (d) 8-chain model.

• 1-chain model (also called as full network model). 1-chain model may be the most straightforward assumption of the network architecture, where the orientation of the polymer connecting two neighboring crosslinks (one located at the center of the sphere and the other at the sphere surface) is randomly distributed [15]. Then, the free energy density of the macromolecular network can be obtained by averaging the energy contributions of all polymers. Suppose the network is uniformly deformed, and the material point located at is displaced to a new location where is the deformation gradient tensor. Note that can be written as with as an orthogonal tensor and as a diagonal tensor; in other words, the deformation can be decomposed into a stretch denoted by and a rotation denoted by . The diagonal components () of , satisfy the relation: where is the first invariant which will be used later. Upon deformations, the sphere of radius (denoting the mesh size of the network) will be deformed into an ellipsoid, where the lengths of the three semi-axes become , , and , respectively. The energy density of the network can be calculated as: (1.4) where is the density of the subchains (polymers connecting two neighboring crosslinks), is the deformation ratio at orientation , and is the energy density of a single polymer, which can be that of Gaussian chain model, Langevin chain model, semiflexible chain model, etc. Then, by averaging the contributions of all polymers, one can obtain the free energy...