Enhanced Heterogeneous Diffusion of Nanoparticles in Semiflexible Networks

Mechano-diffusion of particles in stretchable hydrogels

Precise control over particle diffusion is promising for diverse modern technologies. Traditionally, particle diffusion is governed by the inherent properties of a liquid medium, limiting versatility and controllability. Here, we report a mechano-diffusion mechanism that harnesses mechanical deformation to control particle diffusion in stretchable hydrogels with a significantly enlarged tuning ratio and a highly expanded tuning freedom. The working principle is to leverage the mechanical deformation of stretchable hydrogels for modulating the polymer network's geometric transformation and the polymer chain's energy modulation, which synergistically tunes the energy barrier for particle diffusion. Using a model particle-hydrogel material system and a customized mechano-diffusion characterization platform, we demonstrate that tension loads can enhance the diffusivity of gold nanoparticles up to 22 times, far exceeding that in traditional liquid medium and by external fields. Additionally, we show particle diffusion in hydrogels can be manipulated spatiotemporally by controlling the hydrogels' stress state and loading rate. To further push the limit of the mechano-diffusion, we use experiment, theory, and simulation to explore particle diffusion in biaxially stretched hydrogels, simultaneously expanding the mesh size and reducing the energy barrier. The enlarged tuning ratio and expanded tuning freedom enable a model-guided drug delivery system for pressure-controlled release of drug molecules. Understanding this spatiotemporal mechano-diffusion mechanism will provide insights pertinent to a broad range of biological and synthetic soft materials.

A single-particle energy-conserving dissipative particle dynamics approach for simulating thermophoresis of nanoparticles in polymer networks

The transport of nanoparticles in polymer networks has critical implications in biology and medicine, especially through thermophoresis in response to temperature gradients. This study presents a single-particle energy-conserving dissipative particle dynamics (seDPD) method by integrating a single-particle model into the energy-conserving DPD model to simulate the mesoscopic thermophoretic behavior of nanoparticles in polymer matrices. We first validate the newly developed seDPD model through comparisons with analytical solutions for nanoparticle viscosity, thermal diffusivity, and hydrodynamic drag and then demonstrate the effectiveness of the seDPD model in capturing thermophoretic forces induced by temperature gradients. The results show that nanoparticles driven by the Soret forces exhibit unique transport characteristics, such as drift velocity and diffusivity, leading to a significant acceleration of nanoparticle diffusion in the polymer network, which has been known as the giant acceleration of diffusion. Quantifying how nanoparticles move in flexible polymer networks sheds light on the interaction dynamics of nanoparticles within polymer networks, providing insight into nanoparticle behavior in complex environments that could be leveraged in various applications from drug delivery to material design.

Stick, Slide, or Bounce: Charge Density Controls Nanoparticle Diffusion

The diffusion and interaction dynamics of charged nanoparticles (NPs) within charged polymer networks are crucial for understanding various biological and biomedical applications. Using a combination of coarse-grained molecular dynamics simulations and experimental diffusion studies, we investigate the effects of the NP size, relative surface charge density (ζ), and concentration on the NP permeation length and time. We propose a scaling law for the relative diffusion of NPs with respect to concentration and ζ, highlighting how these factors influence the NP movement within the network. The analyses reveal that concentration and ζ significantly affect NP permeation length and time, with ζ being critical, as critical as concentration. This finding is corroborated by controlled release experiments. Further, we categorize NP dynamics into sticking, sliding, and bouncing regimes, demonstrating how variations in ζ, concentration, and NP size control these behaviors. Through normalized attachment time (NAT) analyses, we elucidate the roles of electrostatic interactions, steric hindrance, and hydrodynamic forces in governing NP dynamics. These insights provide guidance for optimizing NP design in targeted drug delivery and advanced material applications, enhancing our understanding of NP behavior in complex environments.

Nanorod Diffusion near the Solid-Liquid Interface with Varied Wall Nonuniformity

The complex diffusion behaviors of rod-shaped nanoparticles near the solid-liquid interface are closely related to various biological processes and technological applications. Despite recent advancements in understanding the diffusion dynamics of nanoparticles near some specific solid-liquid interfaces, systematical studies to tune the interfacial interaction or fabricating nonuniform wall to see their effects on the nanorod (NR) diffusion are still lacking. This work utilized molecular dynamics simulations to investigate the rotational and translational diffusion dynamics of a single NR near the solid-liquid interface. We constructed a patterned wall featuring adjustable nonuniformity, which was accomplished by modifying the interaction between NR and the wall, noting that the resulting nonuniformity limits both the translational and rotational diffusion of NR, evident from decreases in diffusion coefficients and exponents. By trajectory analysis, we categorized the diffusion modes of NRs near the patterned wall with varied nonuniformities into three types: Fickian diffusion, desorption-mediated flight, and in-plane diffusion. Furthermore, energy analysis based on the adsorption-desorption mechanism has demonstrated that the three diffusion states are driven by interactions between the NR and the wall, which are primarily influenced by rotational diffusion. These results could significantly deepen the understanding of anisotropic nanoparticle interfacial diffusion and would provide new insights into the transport mechanisms of nanoparticles within confined environments.

Rational Design of DNA Hydrogels Based on Molecular Dynamics of Polymers

In recent years, DNA has emerged as a fascinating building material to engineer hydrogel due to its excellent programmability, which has gained considerable attention in biomedical applications. Understanding the structure-property relationship and underlying molecular determinants of DNA hydrogel is essential to precisely tailor its macroscopic properties at molecular level. In this review, the rational design principles of DNA molecular networks based on molecular dynamics of polymers on the temporal scale, which can be engineered via the backbone rigidity and crosslinking kinetics, are highlighted. By elucidating the underlying molecular mechanisms and theories, it is aimed to provide a comprehensive overview of how the tunable DNA backbone rigidity and the crosslinking kinetics lead to desirable macroscopic properties of DNA hydrogels, including mechanical properties, diffusive permeability, swelling behaviors, and dynamic features. Furthermore, it is also discussed how the tunable macroscopic properties make DNA hydrogels promising candidates for biomedical applications, such as cell culture, tissue engineering, bio-sensing, and drug delivery.

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

Transport of rodlike particles in confinement environments of macromolecular networks plays crucial roles in many important biological processes and technological applications. The relevant understanding has been limited to thin rods with diameter much smaller than network mesh size, although the opposite case, of which the dynamical behaviors and underlying physical mechanisms remain unclear, is ubiquitous. Here, we solve this issue by combining experiments, simulations and theory. We find a nonmonotonic dependence of translational diffusion on rod length, characterized by length commensuration-governed unconventionally fast dynamics which is in striking contrast to the monotonic dependence for thin rods. Our results clarify that such a fast diffusion of thick rods with length of integral multiple of mesh size follows sliding dynamics and demonstrate it to be anomalous yet Brownian. Moreover, good agreement between theoretical analysis and simulations corroborates that the sliding dynamics is an intermediate regime between hopping and Brownian dynamics, and provides a mechanistic interpretation based on the rod-length dependent entropic free energy barrier. The findings yield a principle, that is, length commensuration, for optimal design of rodlike particles with highly efficient transport in confined environments of macromolecular networks, and might enrich the physics of the diffusion dynamics in heterogeneous media.

Penetrant shape effects on activated dynamics and selectivity in polymer melts and networks based on self-consistent cooperative hopping theory

We generalize and apply the microscopic self-consistent cooperative hopping theory for activated penetrant dynamics in polymer melts and crosslinked networks to address the role of highly variable non-spherical molecular shape. The focus is on vastly different shaped penetrants that have identical space filling volume in order to isolate how non-spherical shape explicitly modifies dynamics over a wide range of temperature down to the kinetic glass transition temperature. The theory relates intramolecular and intermolecular structure and kinetic constraints, and reveals how different solvation packing of polymer monomers around variable shaped penetrants impact penetrant hopping. A highly shape-dependent penetrant activated relaxation, including alpha time decoupling and trajectory level cooperativity of the hopping process, is predicted in the deeply supercooled regime for relatively larger penetrants which is sensitive to whether the polymer matrix is a melt or heavily crosslinked network. In contrast, for smaller size penetrants or at high/medium temperatures the effect of isochoric penetrant shape is relatively weak. We propose an aspect ratio variable that organizes how penetrant shape influences the activated relaxation times, leading to a (near) collapse or master curve. The relative absolute values of the penetrant relaxation time (inverse hopping rate) in polymer melts versus in crosslinked networks are found to be opposite when compared at a common absolute temperature versus when they are compared at a fixed value of distance from the glass transition based on the variable Tg/T with Tg the glass transition temperature. Quantitative comparison with recent diffusion experiments on chemically complex molecular penetrants of variable shape but fixed volume in crosslinked networks reveals good agreement, and testable new predictions are made. Extension of the theoretical approach to more complex systems of high experimental interest are discussed, including applications to realize selective transport in membrane separation applications.

Simulation Study on the Mechanism of Intermediate Subdiffusion of Diffusive Particles in Crowded Systems

The intermediate subdiffusion of diffusive particles in crowded systems is studied for two model systems: the continuous time random walk (CTRW) model and the obstruction-binding model. For the CTRW model with an arbitrarily given longest waiting time τmax, we find that the diffusive particle exhibits subdiffusion below τmax and recovers normal diffusion above τmax. For the obstruction-binding model with randomly distributed attractive obstacles, the diffusion of the diffusive particle is dependent on the binding energy and the density of obstacles. Interestingly, diffusion curves for different binding strengths can be overlapped by rescaling the simulation time, indicating that the diffusive particle in the obstruction-binding model can change from the intermediate subdiffusion to the normal diffusion at a long-term simulation scale. The results of the two model systems show that the diffusive particles only exhibit intermediate subdiffusion below the longest waiting time. Therefore, long timescale subdiffusion would only be observed in the CTRW model with an infinitely long waiting time and in the obstruction-binding model with an infinitely large binding strength.

Non-normalizable quasiequilibrium states under fractional dynamics

Particles anomalously diffusing in contact with a thermal bath are initially released from an asymptotically flat potential well. For temperatures that are sufficiently low compared to the potential depth, the dynamical and thermodynamical observables of the system remain almost constant for long times. We show how these stagnated states are characterized as non-normalizable quasiequilibrium (NNQE) states. We use the fractional-time Fokker-Planck equation (FTFPE) and continuous-time random walk approaches to calculate ensemble averages. We obtain analytical estimates of the durations of NNQE states, depending on the fractional order, from approximate theoretical solutions of the FTFPE. We study and compare two types of observables, the mean square displacement typically used to characterize diffusion, and the thermodynamic energy. We show that the typical timescales for transient stagnation depend exponentially on the value of the depth of the potential well, in units of temperature, multiplied by a function of the fractional exponent.

Exploring nanofibrous networks with x-ray photon correlation spectroscopy through a digital twin

We demonstrate a framework of interpreting data from x-ray photon correlation spectroscopy experiments with the aid of numerical simulations to describe nanoscale dynamics in soft matter. This is exemplified with the transport of passive tracer gold nanoparticles in networks of charge-stabilized cellulose nanofibers. The main structure of dynamic modes in reciprocal space could be replicated with a simulated system of confined Brownian motion, a digital twin, allowing for a direct measurement of important effective material properties describing the local environment of the tracers.

Converting Nanotoxicity Data to Information Using Artificial Intelligence and Simulation

Decades of nanotoxicology research have generated extensive and diverse data sets. However, data is not equal to information. The question is how to extract critical information buried in vast data streams. Here we show that artificial intelligence (AI) and molecular simulation play key roles in transforming nanotoxicity data into critical information, i.e., constructing the quantitative nanostructure (physicochemical properties)-toxicity relationships, and elucidating the toxicity-related molecular mechanisms. For AI and molecular simulation to realize their full impacts in this mission, several obstacles must be overcome. These include the paucity of high-quality nanomaterials (NMs) and standardized nanotoxicity data, the lack of model-friendly databases, the scarcity of specific and universal nanodescriptors, and the inability to simulate NMs at realistic spatial and temporal scales. This review provides a comprehensive and representative, but not exhaustive, summary of the current capability gaps and tools required to fill these formidable gaps. Specifically, we discuss the applications of AI and molecular simulation, which can address the large-scale data challenge for nanotoxicology research. The need for model-friendly nanotoxicity databases, powerful nanodescriptors, new modeling approaches, molecular mechanism analysis, and design of the next-generation NMs are also critically discussed. Finally, we provide a perspective on future trends and challenges.

How Segmental Dynamics and Mesh Confinement Determine the Selective Diffusivity of Molecules in Cross-Linked Dense Polymer Networks

The diffusion of molecules ("penetrants") of variable size, shape, and chemistry through dense cross-linked polymer networks is a fundamental scientific problem broadly relevant in materials, polymer, physical, and biological chemistry. Relevant applications include separation membranes, barrier materials, drug delivery, and nanofiltration. A major open question is the relationship between transport, thermodynamic state, and penetrant and polymer chemical structure. Here we combine experiment, simulation, and theory to unravel these competing effects on penetrant transport in rubbery and supercooled polymer permanent networks over a wide range of cross-link densities, size ratios, and temperatures. The crucial importance of the coupling of local penetrant hopping to polymer structural relaxation and the secondary importance of mesh confinement effects are established. Network cross-links strongly slow down nm-scale polymer relaxation, which greatly retards the activated penetrant diffusion. The demonstrated good agreement between experiment, simulation, and theory provides strong support for the size ratio (penetrant diameter to the polymer Kuhn length) as a key variable and the usefulness of coarse-grained simulation and theoretical models that average over Angstrom scale structure. The developed theory provides an understanding of the physical processes underlying the behaviors observed in experiment and simulation and suggests new strategies for enhancing selective polymer membrane design.

Rheological Role of Stiff Nanorings on Concurrently Strengthening and Toughening Polymer Nanocomposites

Nanorings, which are increasingly uncovered in natural systems and synthesized in man-made materials, exhibit dynamics distinct from those known for linear chains. We show in this study that, when immersed in a polymer melt matrix, segments of a stiff nanoring (SNR) have more facilitated subdiffusion, i.e., with a larger scaling exponent in the mean squared displacement, than those belonging to one flexible counterpart, while the whole SNR is more suppressed by its surroundings. It is revealed that adding SNRs contributes to achieving the long-anticipated rheological objective of sol- and gel-like characteristics at high and low shearing frequencies, respectively. This study suggests the promising prospect of exploiting SNRs to concurrently strengthen and toughen target polymer nanocomposites.

Configurational Entropy-Enabled Thermostability of Cell Membranes in Extremophiles: From Molecular Mechanism to Bioinspired Design

Understanding physicochemical interactions and mechanisms related to the cell membranes of lives under extreme conditions is of essential importance but remains scarcely explored. Here, using a combination of computer simulations and experiments, we demonstrate that the structural integrity and controllable permeability of cell membranes at high temperatures are predominantly directed by configurational entropy emerging from distorted intermolecular organization of bipolar tethered lipids peculiar to the extremophiles. Detailed simulations across multiple scales─from an all-atom exploration of molecular mechanism to a mesoscale examination of its universal nature─suggest that this configurational entropy effect can be generalized to diverse systems, such as block copolymers. This offers biomimetic inspiration for designing heat-tolerant materials based on entropy, as validated by our experiments of synthetic polymers. The findings provide new insight into the basic nature of the mechanism underlying the adaptation of organisms to extreme conditions and might open paths for designed materials inspired by entropic effects in biological systems.

Hopping Behavior Mediates the Anomalous Confined Diffusion of Nanoparticles in Porous Hydrogels

Diffusion is an essential means of mass transport in porous materials such as hydrogels, which are appealing in various biomedical applications. Herein, we investigate the diffusive motion of nanoparticles (NPs) in porous hydrogels to provide a microscopic view of confined diffusion. Based on the mean square displacement from particle tracking experiments, we elucidate the anomalous diffusion dynamics of the embedded NPs and reveal the heterogeneous pore structures in hydrogels. The results demonstrate that diffusive NPs can intermittently escape from single pores through void connective pathways and exhibit non-Gaussian displacement probability distribution. We simulate this scenario using the Monte Carlo method and clarify the existence of hopping events in porous diffusion. The resultant anomalous diffusion can be fully depicted by combining the hopping mechanism and the hydrodynamic effect. Our results highlight the hopping behavior through the connective pathways and establish a hybrid model to predict NP transport in porous environments.

"Shutter" Effects Enhance Protein Diffusion in Dynamic and Rigid Molecular Networks

Hydrogels have been widely applied to understand the fundamental functions and mechanism of a natural extracellular matrix (ECM). However, revealing the high permeability of ECM through synthetic hydrogels is still challenged by constructing analogue networks with rigid and dynamic properties. Here, in this study, taking advantage of the rigidity and dynamic binding of DNA building blocks, we have designed a model hydrogel system with structural similarity to ECM, leading to enhanced diffusion for proteins compared with a synthetic polyacrylamide (PAAm) hydrogel. The molecular diffusion behaviors in such a rigid and dynamic network have been investigated both in experiments and simulations, and the dependence of diffusion coefficients with respect to molecular size exhibits a unique transition from a power law to an exponential function. A "shutter" model based on the rigid and dynamic molecular network has been proposed, which has successfully revealed how the rigidity and dynamic bond exchange determine the diffusion mechanism, potentially providing a novel perspective to understand the possible mechanism of enhanced diffusion behaviors in ECM.

A prediction model for nanoparticle diffusion behavior in fibrous materials considering steric and hydrodynamic resistances

Precise prediction of the hindered diffusion behavior of electroneutral particles in fibrous media plays a critical role in the development of drugs, polymer membranes, and porous electrodes. However, the random microstructure and unknown coupling relationship of multiple resistance mechanisms lead to the lack of a universal prediction model. In this work, a dual-resistance model is proposed by reconstructed pore-scale simulations, which presents the coexistence of steric and hydrodynamic resistances in the multiplication form. The simulation results show that the relationship between steric resistance and structural parameters (porosity, fiber radius, and particle radius) is exponential, while that for hydrodynamic resistance is polynomial. The dominant diffusion resistance is found to change from hydrodynamic to steric resistance with a decrease in porosity. The fluorescent polystyrene microsphere diffusivity in a series of SiO₂ fibrous media is determined by single-particle tracking experiments, quantitatively confirming the dual-resistance model. The present model can be used for rapid diffusivity prediction and fibrous membrane and drug design.

Understanding Interfacial Nanoparticle Organization through Simulation and Theory: A Review

Understanding the behaviors of nanoparticles at interfaces is crucial not only for the design of novel nanostructured materials with superior properties but also for a better understanding of many biological systems where nanoscale objects such as drug molecules, viruses, and proteins can interact with various interfaces. Theoretical studies and tailored computer simulations offer unique approaches to investigating the evolution and formation of structures as well as to determining structure-property relationships regarding the interfacial nanostructures. In this feature article, we summarize our efforts to exploit computational approaches as well as theoretical modeling in understanding the organization of nanoscale objects at the interfaces of various systems. First, we present the latest research advances and state-of-the-art computational techniques for the simulation of nanoparticles at interfaces. Then we introduce the applications of multiscale modeling and simulation methods as well as theoretical analysis to explore the basic science and the fundamental principles in the interfacial nanoparticle organization, covering the interfaces of polymer, nanoscience, biomacromolecules, and biomembranes. Finally, we discuss future directions to signify the framework in tailoring the interfacial organization of nanoparticles based on the computational design. This feature article could promote further efforts toward fundamental research and the wide applications of theoretical approaches in designing interfacial assemblies for new types of functional nanomaterials and beyond.

Molecular Simulation of Hopping Mechanisms of Nanoparticles in Regular Cross-Linked Polymer Networks

We use coarse-grained simulations to explore the diffusion mechanism of nanoparticles with different sizes at various nanoparticle-polymer interactions in regular cross-linked polymer networks. The long time diffusivities of nanoparticles show a non-monotonic tendency at various nanoparticle-polymer interactions, due to the intermittent hopping of nanoparticles through network cells. The preferred locations of small nanoparticles switch from the cell centers to the corner of cells as they interact with network more strongly, which results in the hopping energy barrier between different cells switching from cell center localization to adsorption on networks. Steric hindrance seriously hampers large nanoparticles from hopping to neighboring network cells, the interactions between nanoparticle and network enhance the network deformability and also affect the hopping of nanoparticles. The multiple constraint mechanisms result in the non-monotonic diffusivities of nanoparticles with different interactions and non-Brownian motions at different time scales. Our work illustrates the hopping mechanisms of nanoparticles in polymer networks from thermodynamic and dynamic points of view.

Topology mediates transport of nanoparticles in macromolecular networks

Diffusion transport of nanoparticles in confined environments of macromolecular networks is common in diverse physical systems and regulates many biological responses. Macromolecular networks possess various topologies, featured by different numbers of degrees and genera. Although the network topologies can be manipulated from a molecular level, how the topology impacts the transport of nanoparticles in macromolecular networks remains unexplored. Here, we develop theoretical approaches combined with simulations to study nanoparticle transport in a model system consisting of network cells with defined topologies. We find that the topology of network cells has a profound effect on the free energy landscape experienced by a nanoparticle in the network cells, exhibiting various scaling laws dictated by the topology. Furthermore, the examination of the impact of cell topology on the detailed behavior of nanoparticle dynamics leads to different dynamical regimes that go beyond the particulars regarding the local network loop. The results might alter the conventional picture of the physical origin of transport in networks.

Macromolecular networks relevant for biological processes and technological applications, are often characterized by complex architectures. The authors uncover the impact of topology on the properties of nanoparticle transport in macromolecular networks.

Diffusion of polymer-grafted nanoparticles with dynamical fluctuations in unentangled polymer melts

The dynamics of polymer-grafted nanoparticles (PGNPs) in melts of unentangled linear chains were investigated by means of coarse-grained molecular dynamics simulations. The results demonstrated that the graft monomers closer to the particle surface relax more slowly than those farther away due to the constraint of the grafted surface and the confinement of the neighboring chains. Such heterogeneous relaxations of the surrounding environment would perturb the particle motion, making them fluctuating around their centers before they can diffuse through the melt. During such intermediate-time stage, the dynamics is subdiffusive while the distribution of particle displacements is Gaussian, which can be described by the popular fractional Brownian motion model. For the long-time Fickian diffusion, we found that the diffusivity D decreases with increasing grafting density Σ_g, grafted chain length N_g, and matrix chain length N_m. This is due to the fact that the diffusivity is controlled by the viscous drag of an effective core, consisting of the NP and the non-draining layer of graft segments, and that of the free-draining graft layer outside the "core". With increasing Σ_g, the PGNPs become harder with greater effective size and thinner free draining layer, resulting in a reduction in D. At extremely high Σ_g, the diffusivity can even be estimated by the diameter-renormalized Stokes-Einstein (SE) relation. With increasing N_g, both the effective core size and the thickness of the free-draining layer increase, leading to a reduction in diffusivity by D ∼ N-γg with 0.5 < γ < 1. Increasing N_m would lead to the enlargement of the effective core size but meanwhile result in the reduction of the free-draining layer thickness due to autophobic dewetting. The counteraction between these two opposite effects leads to only a slight reduction in the diffusivity, significantly different from the typical SE behavior where D ∼ N_m^-1. These findings bear significance in unraveling the fundamental physics of the anomalous dynamics of PGNPs in various polymers, including biological and synthetic.

Molecular simulation of the diffusion mechanism of nanorods in cross-linked networks

We study the diffusion of rod-shaped nanocarriers with different rigidities and aspect ratios in cross-linked networks using coarse-grained molecular dynamics (CGMD) simulations. The diffusivity of the nanorods increases with a reduction in the rigidities of the nanorods and network, as well as with an increasing aspect ratio with respect to the same volume fraction of the nanorods. The nanorods show an anisotropic diffusion pathway through translocating along their major axes at short time scales, and the anisotropy of diffusion decreases at long time scales. Meanwhile, the diffusion of the nanorods shows a sub-diffusion regime that deviates from Brownian motion in most cases due to the trapping of the nanorods in a cage composed of the network. The nanorod could hop when it escapes from the cage and the hopping behavior depends on the rigidities of both the nanorod and network, as well as the local network density. The rotational motion of the trapped nanorod also enhances the probability of hopping. Our results may help in the understanding of the microscopic mechanism for the diffusion of rod-shaped and other relevant nanocarriers, in a cross-linked network environment.