Energy Barriers for Reversible Chain Scission and Healing under Tension with Displacement Control

Technical Summary: Energy Barriers for Reversible Chain Scission and Healing under Tension with Displacement Control

Problem Statement
Polymer chain scission is a fundamental mechanism governing the fracture of soft materials like elastomers and hydrogels. While single-molecule force spectroscopy experiments have established that chain scission is probabilistic and dependent on loading rates and environmental conditions, traditional theoretical descriptions often assume force-controlled conditions. Under a prescribed force, the free energy landscape of a polymer chain typically possesses only one local minimum (the intact state) and one transition state, rendering chain scission irreversible and precluding the possibility of healing. However, experimental observations of self-healing in soft materials suggest that chain scission can be reversible. This paper addresses the theoretical gap by investigating whether reversible chain scission and healing are possible under displacement-controlled conditions, where the end-to-end distance of the polymer chain is prescribed rather than the force.

Methodology
The authors develop a "breakable freely-jointed chain" (FJC) model to analyze the thermodynamics and kinetics of a single polymer chain under tension.

• Model Formulation: The chain consists of $n+1$ identical links (Kuhn segments). The model assumes that scission occurs when a single arbitrary link dissociates while the remaining $n$ links behave as rigid segments. The breakable link is assigned a Lennard-Jones (LJ) potential to account for bond stretching and dissociation energy.
• Thermodynamic Framework: The system is treated as a thermal reservoir at temperature $T$. The free energy $A(x; y)$ is derived as a function of the internal variable $x$ (the length of the breakable link) and the control parameter $y$ (the prescribed end-to-end distance). The derivation integrates the potential energy of the breakable link with the entropic contributions of the rigid chain fragments.
• Statistical Approximations: Two statistical treatments for the rigid fragments are employed:
  1. Gaussian Approximation: Valid for small stretches, yielding a closed-form free energy function.
  2. Non-Gaussian (Kuhn-Grun) Approximation: Utilizes the inverse Langevin function to account for finite extensibility at large stretches, providing a more accurate description near the contour length.
• Kinetic Analysis: Using the derived free energy landscape, the authors calculate energy barriers for scission ($E_s$) and healing ($E_h$). They employ a kinetic approach based on first-order rate equations (Arrhenius law) to predict the survival probability of the chain over time. Additionally, Kinetic Monte Carlo (KMC) simulations are conducted to visualize the stochastic nature of scission and healing events and to validate the analytical rate equations.

Key Contributions and Results

1. Reversibility under Displacement Control: The study demonstrates that under displacement control, the free energy function exhibits two local minima (intact and dissociated states) separated by a local maximum (transition state) for a range of end-to-end distances. This dual-minimum landscape allows for the definition of distinct energy barriers for both chain scission ($E_s$) and chain healing ($E_h$), a feature absent in force-controlled models where healing is energetically impossible.
2. Nonlinear Energy Barriers:
   • Scission Barrier ($E_s$): Starts near the bond energy at zero displacement and decreases approximately quadratically as the displacement increases, vanishing at a critical displacement ($y_c \approx n \cdot l$).
   • Healing Barrier ($E_h$): Starts at a low level (comparable to $k_B T$) and increases approximately quadratically with displacement. Crucially, $E_h$ is highly dependent on chain length (entropic nature) but largely independent of the bond energy, whereas $E_s$ is dominated by the link's bond energy.
3. Rate-Dependent Strength: The model predicts that the rupture force of a polymer chain is stochastic and rate-dependent.
   • At extremely low pulling rates, the system approaches equilibrium, and the average rupture force is determined by the equilibrium probability distribution. This lower bound is found to be 2–3 orders of magnitude lower than the theoretical strength of the covalent bonds.
   • At high pulling rates, the chain behaves more like a rigid rod, and the rupture force increases, but it remains significantly below the theoretical upper bound (the force required to break the bond instantaneously with zero energy barrier) even at rates typical of molecular dynamics simulations.
4. Critical Displacement: The model identifies a critical end-to-end distance beyond which the intact state minimum disappears, leading to instantaneous chain scission. This critical distance is insensitive to bond energy but scales with chain length.

Significance and Claims
The paper claims that the assumption of force-controlled conditions in previous models has obscured the possibility of reversible chain scission and healing. By shifting to displacement control, the authors provide a thermodynamic basis for self-healing in polymer networks, showing that healing is energetically favorable when the displacement is below a critical threshold.

The significance of these findings lies in their potential to explain the rate-dependent fracture and self-healing behaviors observed in soft materials. The authors suggest that incorporating these statistical and kinetic mechanisms into macroscopic polymer network models could improve the understanding of crack resistance and damage evolution, particularly under cyclic loading where reversible scission and healing play a role. The work establishes that the macroscopic strength of polymer networks is not merely a function of bond strength but is governed by the statistical kinetics of chain scission and the entropic barriers to healing, which are strongly influenced by loading rates and chain architecture.