Unconventionally fast transport through sliding dynamics of rodlike particles in macromolecular networks

The transport of guest particles in confinement environments of macromolecular networks holds significant importance across biological science, materials science, and soft matter physics. Selective permeation of bacteria or antibiotics across mucus, nanoparticle organization within polymer-network-based nanocomposites, and efficient drug delivery through extracellular matrix all depend on transport dynamics in confined spaces. Physically, dynamics are governed by an effective free energy landscape arising from particle–network strand interactions and elastic deformation of strands. For spherical particles, the size ratio between network mesh and particle controls regimes including Brownian, hopping, and trapped diffusion. For rodlike particles, anisotropy introduces additional length scales that complicate dynamics; many biological entities (bacteria, viruses, nucleic acids, proteins, polypeptides) are rodlike. Most prior studies examined thin rods with diameters much smaller than mesh size, minimizing elastic penalties, but the opposite limit—rod diameter comparable to or larger than mesh size—is common (e.g., rodlike bacteria in mucus show d/ax ≈ 0.8–2.0). In this thick-rod limit, entropic free energy barriers due to conformational penalties of deformed strands can dominate, potentially leading to new regimes. The authors combine experiment, simulation, and theory to systematically explore dynamics of rods whose diameters are comparable to mesh size in cross-linked networks, aiming to clarify mechanisms governing transport in these essential systems.

• Prior understanding for spherical particles shows diffusion regimes (Brownian, hopping, trapped) controlled by particle-to-mesh size ratio, with the free energy landscape determined by interactions and strand elasticity.
• For rodlike particles, previous work largely focused on thin rods (d ≪ ax), where elastic deformation contributes little; in such cases, diffusivity typically decreases monotonically with rod length.
• In biological contexts, many rodlike species experience d/ax near unity or larger (≈0.8–2.0) in mucus, suggesting strong entropic penalties from deforming network strands.
• Existing reports on thin nanorods in polymer melts show monotonic length dependence of diffusion; the behavior of thick rods in cross-linked networks with comparable diameters remained unclear.
• Concepts from protein sliding along DNA and anomalous yet Brownian diffusion in crowded media provide theoretical analogs for possible regimes between hopping and Brownian motion.

Experiments:

• Network: Polyethylene glycol diacrylate (PEGDA) hydrogel synthesized under UV irradiation (365 nm, 30 W, 10 min), 4 wt% PEGDA (20 kDa) with LAP photoinitiator. Post-cure swelling in water for 48 h. Average mesh size ax ≈ 21.0 ± 1.8 nm (from literature correlations).
• Particles: PEG-capped gold nanorods (Au-NRs). Lengths L ≈ 30.6 ± 3.4 to 61.4 ± 7.7 nm; diameters including grafted layer ≈ 19.1 nm (TEM measured).
• Imaging: Dark-field microscopy (Olympus BX51). Water: 20 Hz for 20 s (400 frames). Network: 10 Hz. Trajectory detection via Crocker–Grier algorithm; linking via nearest-neighbor tracker. Longitudinal axis identified via eigendecomposition. Rotational dynamics neglected (rotation waiting time much longer than observation window). Computed MSD along rod axis, diffusion coefficients (D), and displacement probability distribution functions Gz(z,t).

Simulations (DPD):

• Cross-linked hexafunctional network in a cubic box 42.76 r per side; number density ρ=3 r^-3. Time integration via modified velocity-Verlet; kBT=1, γ=4.5, Δt=0.02 τ, τ=(mr^2/kBT)^1/2.
• Mesh size ax ≈ 3.35 r. Rods built as rigid bodies of beads. Interactions set initially equal between rod–strand and like species to emphasize entropic effects; varied arp to test enthalpic contributions (arp=20 attractive, 30 repulsive; like-like aii≈25).
• Parameter ranges: d/ax ≈ 1.3–1.9; L/ax ≈ 1.5–4.4 (to suppress rotations). Also thin-rod control with d/ax=0.18. Mesh polydispersity explored via coefficient of variation CV=σax/ax of 0.3, 0.7, and 0.8 by varying strand length distributions.
• Outputs: MSD, diffusion coefficients, trajectories along z (rod axis), G2(z,t) displacement distributions, and non-Gaussian parameter α2(t)=(1/3)⟨Δz^4⟩/⟨Δz^2⟩^2 − 1.

Theory:

• Coupled rod–network partition function based on Gaussian chain network elasticity with hard-core monomer–rod exclusion. Mesh size ax=b N^1/2 taken as unit length. Free energy F computed; free energy change ΔF(z)=F(z)−Fmin along z (rod center position within a unit cell).
• Defined barrier Ub=ΔFmax−ΔFmin; mapped Ub across (d/ax, L/ax). Interpreted regimes via Kramers’ theory: Ub>kBT → hopping/trapped; 0<Ub<kBT → fast sliding; Ub=0 → Brownian.
• Microscopic dynamics described by generalized Fokker–Planck and continuous-time random walk (Montroll–Weiss). Derived DPDF Gz(z,t) forms for hopping (exponential tails with discrete-lattice peaks), sliding (Bessel-function kernel producing shallow irregular peaks), and Brownian (Gaussian).

• Experiments (PEGDA network, d/ax≈1.0±0.1): Translational diffusivity along the rod axis depends nonmonotonically on L/ax. Rods with lengths near integer multiples of mesh size (commensurate), e.g., L/ax≈2.02±0.18 and 2.97±0.35, exhibit significantly faster diffusion than noncommensurate cases (L/ax≈1.46±0.16, 2.51±0.27, 2.63±0.31).
• D/D0 vs L/ax displays maxima at commensurate lengths, revealing an unconventionally fast regime absent in thin rods.
• Displacement distributions: Noncommensurate rods show G(z,t) with regular peaks spaced by ~1.0 ax (hopping between cells); commensurate rods show shallow, irregular peaks (fast non-Gaussian sliding). MSD remains Fickian (∝t) in fast regime while DPDF is non-Gaussian (anomalous yet Brownian).
• Simulations reproduce experiments: For d/ax=1.4, L/ax=1.5, 2.6, 3.5 show hopping (MSD plateau then diffusion); L/ax=2.1, 3.1, 4.1 show direct ballistic-to-Fickian crossover (fast). Peak spacing in hopping G2(z,t) ≈ ax; fast regime lacks sharp hops and shows weak non-Gaussianity with α2(t) intermediate between hopping and Brownian.
• Dynamic diagram across (d/ax, L/ax): Within each interval L/ax∈[n,n+1], noncommensurate lengths yield hopping (or trapped when d/ax≳1.6), while near integer L/ax the fast sliding regime appears, confirming nonmonotonicity and optimal L for thick-rod transport.
• Thin-rod control (d/ax=0.18): Diffusion coefficient decreases monotonically with increasing L/ax, consistent with prior reports, emphasizing that nonmonotonicity is a thick-rod, entropically driven effect.
• Entropic dominance and robustness: Varying rod–strand interaction strength (arp=20 attractive vs 30 repulsive) preserves the emergence of fast dynamics at commensurate L/ax. Mesh polydispersity (CV=0.3, 0.7) still yields fast dynamics; even at CV=0.8 fast dynamics emerges for slightly thicker rods (d/ax=1.5).
• Theory maps the free energy barrier Ub across parameters: Noncommensurate rods face Ub>kBT (hopping/trapped), while commensurate rods fall into a valley 0<Ub<kBT (fast sliding). Theoretical boundary Ub=kBT aligns with simulation regime transitions. Analytical DPDFs derived for hopping, sliding, and Brownian agree with observed distributions.
• Practical implication: Length commensuration provides a principle to optimize rodlike particle transport efficiency in macromolecular networks (e.g., informing design of nanorod-based drug delivery).

The study resolves how rodlike particle geometry relative to network mesh size governs transport regimes in cross-linked macromolecular networks. When thick rods (d≈ax) are noncommensurate with mesh size, deformation of surrounding strands imposes an entropic free energy barrier Ub exceeding kBT, producing constrained motion with intermittent hops between cells (non-Gaussian DPDF with regular peaks). At commensurate lengths (L≈n ax), the number and arrangement of deformed loops minimize the entropic penalty, lowering Ub into 0<Ub<kBT. Thermal fluctuations then drive a continuous sliding motion along the network structure: MSD is Fickian while the DPDF remains weakly non-Gaussian, an anomalous-yet-Brownian regime lying between hopping and classical Brownian motion. The theoretical Ub landscape and analytical DPDFs quantitatively rationalize the simulation and experimental observations and delineate regime boundaries. These findings address the central question by attributing the nonmonotonic diffusivity–length relation to a rod-length-dependent entropic barrier, establishing length commensuration as the key design lever. The results are broadly relevant to transport in biological matrices (e.g., mucus) and synthetic gels and suggest strategies to maximize penetration and mobility by tuning rod dimensions to network mesh. The framework may extend to active rods experiencing similar landscapes, informing understanding of nonequilibrium transport and dissipative self-assembly in heterogeneous media.

By integrating single-particle experiments, DPD simulations, and theory, the work establishes that thick rodlike particles (d≈ax) in macromolecular networks exhibit a nonmonotonic diffusion dependence on length. When rod length is commensurate with the network mesh (L≈n ax), transport enters a fast sliding regime with Fickian MSD but non-Gaussian displacement statistics—an anomalous yet Brownian behavior—arising from an entropic free energy barrier reduced to below kBT. Noncommensurate lengths encounter higher barriers causing hopping or trapping. A theoretical free energy landscape quantifies these regimes, and analytical forms of the displacement distributions connect sliding to hopping and Brownian dynamics. This identifies length commensuration as a principle for optimizing rod transport in biological and synthetic networks, with potential utility in nanocomposite design and drug delivery. Future work could test generality across diverse network chemistries and mechanics, explore broader polydispersity and dynamic networks, incorporate rotational dynamics over longer timescales, and extend to active rod propulsion in confined media.

• Rotational dynamics were deemed negligible within the experimental observation window due to long rotation waiting times; thus, analysis focused on longitudinal translation, which may omit coupling effects at longer timescales.
• Experimental validation used PEGDA networks and PEG-capped Au nanorods; while mesh polydispersity and interaction strengths were varied in simulations, real biological networks may involve additional chemistries and dynamic remodeling not explicitly modeled.
• Simulations emphasized entropic contributions (initially equating rod–strand and like-species interactions), with limited exploration of strong enthalpic interactions beyond two representative arp values.
• Theoretical analysis relies on Gaussian-chain elasticity and idealized cell geometry to compute free energy barriers; deviations from ideal network statistics could shift quantitative boundaries.
• Sliding and hopping were characterized along the rod axis; transverse and rotational components and possible coupling under different confinement conditions were not systematically explored.