The transport of particles within macromolecular networks is a fundamental problem with broad implications across biological and materials science. In biological systems, this governs processes like bacterial or antibiotic permeation through mucus (Button et al., 2012; Orell et al., 2013; Lieleg & Ribbeck, 2011), while in materials science, it dictates the organization of nanoparticles in polymer-network-based nanocomposites (Gardel et al., 2004; Bailey & Winey, 2020), influencing material properties. Efficient drug delivery systems targeting tissues with extracellular matrices also depend critically on understanding and manipulating this particle transport (Theocharis et al., 2016; Thorne et al., 2008; Fröhlich & Roblegg, 2009). The dynamics are governed by the interplay between particle-network interactions and network elastic deformation (Cai et al., 2015; Xu et al., 2021; Wong et al., 2004; Dell & Schweizer, 2014; Dai et al., 2022). For spherical particles, diffusion regimes vary based on size ratio with the network mesh (Brownian, hopping, trapped dynamics), but this simple picture doesn't hold for rodlike particles due to anisotropic shape introducing multiple length scales (Chiappini et al., 2020; Han et al., 2006; Bhattacharjee & Datta, 2019; Alvarez et al., 2017). Many biological particles, including bacteria, viruses, and proteins, exhibit rodlike geometry. Most previous studies on rodlike particle transport focused on rods with diameters much smaller than the network mesh size (Choi et al., 2015; Wang et al., 2021; Brochard Wyart & de Gennes, 2000; Karatrantos et al., 2019; Yu et al., 2016), simplifying the dynamics by minimizing the role of elastic energy. However, the opposite limit—rods with diameters comparable to or larger than the mesh size—is common, as evidenced by the analysis of bacteria in various types of mucus (Durack et al., 2017; Vijay et al., 2018; Duncan et al., 2016; Yamada et al., 2014; Fahy & Dickey, 2010; Matsui et al., 2005; Smith et al., 2021; Yildiz et al., 2015; Nhu et al., 2021; Pelaseyed et al., 2014; Constantino et al., 2016). In this case, entropic free energy barriers due to strand deformation might dominate, leading to unknown dynamical regimes. This study addresses this critical gap by investigating the transport of thick rodlike particles in macromolecular networks using a combined experimental, simulation, and theoretical approach.

Previous research on the transport of rod-like particles in macromolecular networks primarily focused on the regime where the rod diameter is significantly smaller than the network mesh size. These studies highlighted the effects of rod length and network architecture on diffusion, generally observing a monotonic decrease in diffusivity with increasing rod length (Choi et al., 2015; Wang et al., 2021). The influence of the network's viscoelastic properties on the diffusion process has also been investigated (Brochard Wyart & de Gennes, 2000; Karatrantos et al., 2019), often observing anomalous diffusion behaviors. Some studies looked into the role of rod rotation in the overall transport, finding it significant in certain regimes (Yu et al., 2016). However, a systematic investigation of the dynamics of thick rods, whose diameters are comparable or larger than the mesh size, has been lacking. The current study aims to address this gap, exploring the interplay between rod dimensions, network structure, and the resultant diffusion mechanisms.

This study employed a multi-pronged approach combining experiments, molecular simulations, and theoretical modeling. **Experiments:** Single-particle tracking (SPT) was performed on gold nanorods (Au-NRs) with varying lengths and diameters in a synthesized polyethylene glycol diacrylate (PEGDA) network. The PEGDA network, chosen for its biocompatibility, was synthesized via UV irradiation, and its mesh size was characterized. Dark-field microscopy was used to track the Au-NRs, and the Crocker-Grier algorithm was employed for particle detection and trajectory analysis. The mesh size of the PEGDA network was estimated to be 21.0 ± 1.8 nm. The Au-NRs had lengths ranging from 30.6 ± 3.4 nm to 61.4 ± 7.7 nm and diameters (including grafted-layer thickness) of around 19.1 nm. Mean squared displacement (MSD) and displacement probability distribution function (DPDF) were calculated to characterize the diffusion behavior. **Simulations:** Dissipative particle dynamics (DPD) simulations were conducted to explore the microscopic dynamics of rodlike particles in a cross-linked network. A hexafunctional network was modeled, and the rod-strand interaction was set to capture the entropic nature of the interplay. Various rod lengths and diameters were simulated, maintaining diameter comparable to the mesh size. MSD, DPDF, and non-Gaussian parameter (a1) were calculated to analyze the diffusion regimes. The simulation parameters included a bead representing a cluster of molecules, with the time evolution governed by Newton's equations of motion. Forces were comprised of conservative, dissipative, and random components. The parameters were carefully selected to reproduce the hydrodynamic behavior observed in the experiment. **Theoretical Modeling:** A theoretical model was developed to analyze the free energy landscape and dynamic regimes based on the length scales of the rod and network. This involved considering a canonical ensemble of a rod in a macromolecular network, coupling particle effects into the theory of network elasticity (Deam & Edwards, 1976; Schmid, 2013). The hard-core monomer-rod interaction was explicitly included. The free energy change was calculated as a function of rod position to determine the energy barriers involved in different dynamic regimes. A generalized Fokker-Planck equation and the Montroll-Weiss equation (Metzler et al., 1999; Montroll & Weiss, 1965) were used to analyze the microscopic dynamics and obtain the time-displacement distribution for hopping, sliding, and Brownian regimes. The Kramers’ escape theory (Hänggi et al., 1990) was used to explain the transition between regimes based on the height of the free energy barrier.

The key findings of this research demonstrate a novel and unexpected behavior in the diffusion of rod-like particles within macromolecular networks. Specifically: 1. **Non-monotonic Diffusion Dependence on Rod Length:** Unlike the previously observed monotonic decrease in diffusivity with increasing rod length for thin rods, this study revealed a non-monotonic relationship for thick rods (diameter comparable to or larger than the network mesh size). The diffusivity exhibited a peak when the rod length was an integer multiple of the network mesh size. This implies that size commensuration plays a crucial role in determining the transport dynamics of thick rods in the macromolecular network. 2. **Sliding Dynamics:** The unexpectedly fast diffusion observed for rods with lengths commensurate with the mesh size was attributed to a previously uncharacterized sliding mechanism. This implies that the rod moves through the network not by hopping between network cells but rather by a continuous sliding motion along the network strands. 3. **Anomalous Yet Brownian Diffusion:** The sliding diffusion was identified as anomalous yet Brownian. While the mean squared displacement (MSD) exhibited a linear time dependence characteristic of Brownian motion, the displacement probability distribution function (DPDF) departed from Gaussian statistics. This suggests that while the average behavior is similar to Brownian motion, individual trajectories show strong deviations and heterogeneous behavior, likely influenced by the underlying structural features of the network. 4. **Entropic Origin of Free Energy Barrier:** The theoretical analysis revealed that the transitions between the different dynamic regimes (hopping, sliding, Brownian) are dictated by the height of the free energy barrier, which is strongly influenced by the entropic contribution stemming from the deformation of network strands induced by the rod. For commensurate rods, this barrier is significantly lower than for non-commensurate rods, thus favoring the fast sliding mechanism. Simulations examining different particle-strand interaction parameters further corroborated the predominantly entropic nature of these effects. The free energy barrier is strongly dependent on rod length and diameter, providing a mechanistic basis for the observed length commensuration effect. The simulations and experimental results were in good agreement with the theoretical predictions, indicating the robustness and accuracy of the model. 5. **Generality of the Findings:** The effects observed were shown to extend beyond networks with perfectly uniform mesh sizes. Simulations using networks with a realistic distribution of mesh sizes found that the key effects of commensuration persisted, suggesting the importance of the size effect even in less idealized contexts.

This study significantly advances our understanding of particle transport in macromolecular networks, particularly for the previously understudied case of thick rodlike particles. The discovery of a non-monotonic diffusion dependence on rod length challenges existing models based solely on monotonic behavior and highlights the critical role of length commensuration with the network mesh size. The identification of sliding dynamics as a dominant mechanism opens new avenues for investigating anomalous yet Brownian diffusion. The entropic origin of the free energy barriers provides a mechanistic understanding of the observed behavior. The findings have implications for designing efficient drug delivery systems, optimizing nanocomposite materials, and understanding biological transport processes. Future studies could explore the influence of other factors, such as network flexibility, particle surface chemistry, and active transport, on these dynamics. Furthermore, investigating this effect in biological systems would provide strong validation for the theory and extend our understanding of size effects in complex environments.

This research demonstrates an unconventional fast transport mechanism—sliding dynamics—for thick rodlike particles in macromolecular networks. The fast transport is observed when the rod length is an integer multiple of the network mesh size, highlighting the importance of size commensuration. The underlying mechanism is rooted in an entropic free energy barrier, confirming the significant impact of the entropic contribution from the deformation of network strands by the thick rod. This study provides a comprehensive understanding of the dynamics, incorporating experimental validation, detailed simulations, and theoretical modeling, with implications for optimizing particle transport in various applications ranging from materials science to drug delivery.

The current study focused on a specific type of macromolecular network (PEGDA) and a specific type of rodlike particle (Au-NRs). While the PEGDA network provides a good model system due to its biocompatibility, extending the findings to other network types (e.g., different cross-linking densities, flexibility, and chemical compositions) is necessary to assess the generality of the conclusions. Similarly, the use of Au-NRs as model particles might influence the particle-network interactions. Further investigation using different rodlike materials is recommended. The theoretical model, while providing a robust framework, relies on simplifying assumptions such as ideal Gaussian chains and specific interaction potentials. Relaxing these assumptions and incorporating more realistic network models could improve the quantitative accuracy of the predictions.