Linear and Non-Linear Rheology of Single and Double Cross-Linked Biopolymer Networks under Viscous Shear Flow

Technical Summary: Linear and Non-Linear Rheology of Single- and Double-Cross-Linked Biopolymer Networks under Viscous Shear Flow

Problem Statement
Biopolymer networks, such as actin, collagen, and fibrin hydrogels, are critical for applications in tissue engineering and wound healing. Their macroscopic mechanical properties are governed by complex microstructures. While previous studies have modeled single networks or specific double-network systems (e.g., Alginate/Polyacrylamide), gaps remain in understanding the interplay between intra-fibre and inter-fibre interactions in intertwined double networks under oscillatory shear. Specifically, prior numerical approaches often fixed intraspecies interactions while varying interspecies interactions, potentially overlooking the significance of intraspecies dynamics. Furthermore, the mechanisms behind "two-step yielding" (double peaks in stress-strain curves) in double networks have been attributed primarily to hydrogen bonds or transient crosslinkers; however, the role of permanent crosslinkers and initial network morphology in generating these features requires further investigation.

Methodology
The authors developed a modified numerical tool based on Slender Body Theory (SBT) for Stokes flow to simulate the mechanical behavior of single- and double-cross-linked biopolymer networks. Key methodological components include:

• Fibre Discretization: Fibres are approximated using Chebyshev polynomial interpolation to define continuous profiles and tangent vectors. A resampling technique generates uniform binding points along the fibres to facilitate crosslinking.
• Hydrodynamics and Forces: The model incorporates Rotne-Prager-Yamakawa (RPY) kernels to evaluate hydrodynamic interactions. Forces acting on the fibres include bending forces (proportional to the fourth derivative of the profile), steric repulsion (modeled via a potential density function to prevent non-physical fibre crossing), and crosslinker forces (modeled as springs between uniform points).
• Network Construction: Percolated networks were generated using a dynamic cross-linking model driven by Brownian motion with specific binding/unbinding rates. Once the structure reached a steady state, these rates were set to zero to create permanent links, and the networks were subjected to oscillatory shear flow.
• Simulation Conditions: The study simulated three types of single networks and two types of double networks (with varying persistence lengths for subnetworks) under oscillatory shear. Simulations were run at two strain amplitudes: 5% (linear regime) and 100% (nonlinear regime). The study also varied the shear flow direction (x, y, or z) and the initial "seed number" (morphology) to test isotropy and structural dependence.
• Analysis: Stress-strain relationships were analyzed, and Fourier decomposition was performed on the stress response to evaluate shear moduli and nonlinearity effects (higher-order modes).

Key Results

• Linear vs. Nonlinear Superposition: In the linear regime (5% strain), the stress-strain response of a double network cannot be predicted by simply superimposing the results of the corresponding single networks. However, in the nonlinear regime (100% strain), the double network behavior approaches the superposition of the single network results.
• Two-Step Yielding and Double Peaks: Contrary to previous literature attributing double peaks in double networks solely to hydrogen bonds or transient crosslinkers, this study found that double peaks (two-step yielding) in double networks with permanent crosslinkers are not guaranteed. The presence of double peaks was found to be highly dependent on the initial network morphology (seed number) and fibre alignment. Notably, double peaks were also observed in single networks under nonlinear conditions, suggesting that morphology plays a more significant role than previously thought.
• Isotropy: The constructed networks were found to be largely isotropic; changing the direction of the shear flow (x, y, or z) did not significantly alter the stress-strain response or the presence of double peaks.
• Fourier Analysis:
  • Linear Regime (5%): For single networks, the response is dominated by the first Fourier mode. For double networks, the zero mode must also be considered.
  • Nonlinear Regime (100%): Significant nonlinearity is observed, requiring Fourier coefficients from the zero to the fifth modes to accurately capture the stress waveform. The amplitude of these higher-order modes is considerable and varies with the seed number, indicating that nonlinearity is strongly linked to the initial structural morphology rather than just the flow conditions.

Significance and Claims
The paper claims that the mechanical behavior of biopolymer networks, particularly double networks, is not solely dictated by the type of crosslinking (e.g., hydrogen bonds vs. permanent links) but is significantly influenced by the initial structural morphology (seed number) and fibre alignment. The study demonstrates that "two-step yielding" can occur in single networks and is not an exclusive signature of specific double-network chemistries.

By providing a numerical framework that accounts for both intra-fibre and inter-fibre interactions under various flow conditions, the authors suggest these results can assist scientists in better designing fibrous materials for specific applications, such as wound healing and tissue engineering. The work fills a gap in the literature by systematically studying intertwined networks with both intra- and inter-species interactions, challenging the assumption that double peaks are exclusively caused by transient crosslinkers.