Non-Commutative Morphology Trajectories in Polymer Durability

Polymer durability under cyclic mechanical and environmental loading is governed not by static material state, but by the explicit path taken through a minimal, observable morphology state vector.

1. Problem Statement

Conventional polymer durability models implicitly assume that mechanical and environmental loading effects commute: that total exposure governs outcome regardless of order. In real systems, polymers routinely violate this assumption. Identical materials, exposed to the same total stress and environment but in different sequences, exhibit divergent durability, failure modes, and service lifetimes.

This work formalizes that divergence as a non-commutative trajectory through morphology space. The order of loading matters because polymer morphology evolves on service-relevant time scales, and that evolution feeds back into subsequent mechanical and environmental response.

2. Scope and Regime Boundaries

Applies to

  • Glassy amorphous polymers (e.g., polycarbonate, polystyrene)
  • Semi-crystalline thermoplastics (e.g., PE, PP)
  • Physically crosslinked elastomers (TPU, ionomers)
  • Supramolecular and hydrogen-bonded polymer networks

Does Not Apply to

  • Inorganics and metallic glasses
  • Fully crystalline polymers without dynamic morphology
  • Systems dominated by rapid chemical degradation

Degrades or Becomes Conditional

  • Filler-dominated fracture regimes
  • Ultra-high strain-rate deformation
  • Monotonic loading in inert environments

3. Minimal Morphology State Vector

Polymer morphology is represented as a minimal, observable state vector:

M = (φ, d_cl, ρ_d, σ_int)
  • φ: ordered or crystalline phase fraction
  • d_cl: characteristic domain or cluster size
  • ρ_d: accumulated defect or microvoid density
  • σ_int: effective interfacial cohesion

These variables evolve under load and environment and are jointly responsible for stress transfer, energy dissipation, and crack initiation or arrest.

4. Non-Commutativity Claim

In trajectory-sensitive polymer systems, environmental cycling (E) and mechanical cycling (M) do not commute:

E → M ≠ M → E

Even when total cycle counts and exposure magnitudes are equal, the resulting morphology states and durability outcomes differ because intermediate morphology evolution alters subsequent response.

5. Decisive Experimental Test

Prepare identical specimens from a trajectory-sensitive polymer system.

  • Group A: Environmental cycling followed by mechanical fatigue
  • Group B: Mechanical fatigue followed by environmental cycling

Measure final morphology state vector components and durability metrics. Divergent outcomes under matched total exposure falsify commutative assumptions and validate trajectory governance.

6. Falsification Conditions

  1. If durability and morphology are invariant to load order across multiple polymer classes, the framework fails.
  2. If irreversible damage accumulates independently of trajectory, morphology path is not governing.

7. What This Framework Does Not Claim

This framework does not provide universal lifetime prediction, nor does it replace chemistry-limited degradation models. It applies only where morphology evolution competes with service timescales.

8. Edge of Knowledge Judgment

CONDITIONAL GO.The framework introduces explicit falsifiability, a minimal state vector, and a decisive experimental test. It elevates morphology trajectory from implicit background assumption to load-bearing explanatory variable while remaining bounded and non-universal.

This document is part of the Edge of Knowledge research series. Revisions occur only through explicit versioning to preserve epistemic continuity.