Study of Disordered Biopolymer Gels: From Statistical Mechanics to Network Models
Creators: Moosavian, Seyedhashem
Publisher: name: University of Alberta Library
Year: 2024

<p>Disordered biopolymer gels, such as those synthesized from polysaccharide and gelatin, play a crucial role in biomedical applications, particularly in tissue engineering. During the gelation process of these gels, polymer chains associate in the presence of gelling agents, forming physical cross-links known as the junction zones. In contrast to rubber-like networks, the resulting network comprises two main regions: the ordered region due to the junction zones and the amorphous region due to the unassociated chains. Under thermal fluctuations and/or external loading, the number and locations of junction zones can change leading to <code>zipping&#39;&#39; (lengthening, i.e., expansion of the junction zones) and</code>unzipping&#39;&#39; (shortening, i.e., shrinkage of the junction zones). This gives rise to intriguing features in biopolymer gels such as healing and damage-like energy dissipation. Despite the recognition of zipping and unzipping in such gels, the development of mathematical models that incorporate the microscopic mechanisms into the material&#39;s macroscopic mechanical properties is still in its early stages.<br>
The current study is devoted to providing a systematic framework that describes the overall behaviour of the biopolymer gels with zipping/unzipping under mechanical loading.<br><br>
The entire polymer network can be envisaged as a collection of coil-rod structures serving as the building blocks, where the coil and rod are representative of the disordered and ordered zones, respectively. <br>
 The coil-rod structure is modelled by a rod attached to a freely jointed chain, where the length of the rod and the number of segments in the freely jointed chain are variable. During the zipping/unzipping process, segments can be exchanged between the coil and the rod, and the extent of exchange is governed by the binding energy of segments to form the junction zone.</p>

<p>Prior to identifying coil-rod structures, an in-depth study of the statistical mechanics of the freely-jointed chain model is conducted. The analysis reveals that the conventional approach to deriving the canonical ensemble of this structure relies on the condition of <code>fixed displacement&#39;&#39; rather than the desirable</code>fixed end-to-end distance&#39;&#39;. This nuanced distinction becomes negligible as the number of constituent segments increases. <br>
 However, for coil-rod structures, this distinction between ensembles must be carefully taken into account, as the predictions from each ensemble result in markedly different force-extension relationships. The precise treatment of coil-rod structures establishes the foundation for the subsequent construction of a grand-canonical ensemble with a fixed end-to-end distance but a variable number of segments.<br>
The latter ensemble can appropriately formulate the zipping/unzipping when the individual chain, such as DNA is subjected to mechanical force. The proposed force-extension relationship for a single coil-rod can be integrated to generate stress-stretch relations for the collection of coil-rod structures forming a network.<br>
However, the presence of solvent and its interaction with the coil-rod structures lead to phenomena such as network swelling, thereby complicating the overall response. Another critical aspect pertains to the identification of a state devoid of residual stress, commonly known as the stress-free state. These concepts will be rigorously examined from the perspective of micro-scale network models.<br>
Experimental evidence demonstrates that the disordered biopolymer gel exhibits dissipation under cyclic loading. Furthermore, upon complete removal of the load, the sample experiences residual strain, commonly referred to as permanent set. These phenomena can be captured by the zipping/unzipping  within the coil-rod structures in the proposed network model. Despite this model&#39;s ability to predict material response, it lacks explicit interactions between coils and rods.<br>
For this reason, the canonical ensemble of the more explicit network, where two coils share the same rod as a junction zone, is also elaborated upon. It is demonstrated that if the constituent freely-jointed chains follow a Gaussian distribution, this system can be equivalently described as a collection of coil-rod structures.</p>

<p>The proposed multiscale formulation not only advances understanding of disordered biopolymer gels and underlying mechanism but also lays the groundwork for modeling hybrid gels that include coil-rod structures as a component.</p>