Brownian yet non-Gaussian diffusion of a light particle in heavy gas: Lorentz-gas-based analysis.

Non-Gaussian diffusion was recently observed in a gas mixture with mass and fraction contrast [F. Nakai et al., Phys. Rev. E 107, 014605 (2023)2470-004510.1103/PhysRevE.107.014605]. The mean-square displacement of a minor gas particle with a small mass is linear in time, while the displacement distribution deviates from the Gaussian distribution, which is called the Brownian yet non-Gaussian diffusion. In this work, we theoretically analyze this case where the mass contrast is sufficiently large. Major heavy particles can be interpreted as immobile obstacles, and a minor light particle behaves like a Lorentz gas particle within an intermediate timescale. Despite the similarity between the gas mixture and the conventional Lorentz gas system, the Lorentz gas description cannot fully describe the Brownian yet non-Gaussian diffusion. A successful description can be achieved through a canonical ensemble average of the statistical quantities of the Lorentz gas over the initial speed. Furhter, we show that the van Hove correlation function has a nonexponential tail, which is contrary to the exponential tail observed in various systems.

Percolation-induced gel-gel phase separation in a dilute polymer network.

Cosmic large-scale structures, animal flocks and living tissues can be considered non-equilibrium organized systems created by dissipative processes. Replicating such properties in artificial systems is still difficult. Herein we report a dissipative network formation process in a dilute polymer-water mixture that leads to percolation-induced gel-gel phase separation. The dilute system, which forms a monophase structure at the percolation threshold, spontaneously separates into two co-continuous gel phases with a submillimetre scale (a dilute-percolated gel) during the deswelling process after the completion of the gelation reaction. The dilute-percolated gel, which contains 99% water, exhibits unexpected hydrophobicity and induces the development of adipose-like tissues in subcutaneous tissues. These findings support the development of dissipative structures with advanced functionalities for distinct applications, ranging from physical chemistry to tissue engineering.

Increase in rod diffusivity emerges even in Markovian nature.

Rod-shaped particles embedded in certain matrices have been reported to exhibit an increase in their center of mass diffusivity upon increasing the matrix density. This increase has been considered to be caused by a kinetic constraint in analogy with tube models. We investigate a mobile rodlike particle in a sea of immobile point obstacles using a kinetic Monte Carlo scheme equipped with a Markovian process, that generates gaslike collision statistics, so that such kinetic constraints do essentially not exist. Even in such a system, provided the particle's aspect ratio exceeds a threshold value of about 24, the unusual increase in the rod diffusivity emerges. This result implies that the kinetic constraint is not a necessary condition for the increase in the diffusivity.

Fluctuating diffusivity emerges even in binary gas mixtures.

Diffusivity in some soft matter and biological systems changes with time, called the fluctuating diffusivity. In this work, we propose a novel origin for fluctuating diffusivity based on stochastic simulations of binary gas mixtures. In this system, the fraction of one component is significantly small, and the mass of the minor component molecule is different from that of the major component. The minor component exhibits fluctuating diffusivity when its mass is sufficiently smaller than that of the major component. We elucidate that this fluctuating diffusivity is caused by the time scale separation between the relaxation of the velocity direction and the speed of the minor component molecule.

Brownian simulations for tetra-gel-type phantom networks composed of prepolymers with bidisperse arm length.

We studied the effect of arm length contrast of prepolymers on the mechanical properties of tetra-branched networks via Brownian dynamics simulations. We employed a bead-spring model without the excluded volume interactions, and we did not consider the solvent explicitly. Each examined 4-arm star branch prepolymer has uneven arm lengths to attain two-against-two (2a2) or one-against-three (1a3) configurations. The arm length contrast was varied from 38-2 to 20-20 for 2a2, and from 5-25 to 65-5 for 1a3, with the fixed total bead number of 81, including the single bead located at the branch point for prepolymers. We distributed 400 molecules in the simulation box with periodic boundary conditions, and the bead number density was fixed at 4. We created polymer networks by cross-end-coupling of equilibrated tetra-branched prepolymers. To mimic the experiments of tetra gels, we discriminated the molecules into two types and allowed the reaction only between different types of molecules at their end beads. The final conversion ratio was more than 99%, at which unreacted dangling ends are negligible. We found that the fraction of double linkage, in which two of the four arms connect a pair of branch points, increases from 3% to 15% by increasing the arm length contrast. We stretched the resultant tetra-type networks to obtain the ratio of mechanically effective strands. We found that the ratio is 96% for the monodisperse system, decreasing to 90% for high arm length contrast. We introduced bond scission according to the bond stretching to observe the network fracture under sufficiently slow elongation. The fracture behavior was not correlated with the fraction of double linkage because the scission occurs at single linkages.

Application of projection operator method to coarse-grained dynamics with transient potential.

We show that the coarse-grained dynamics model with the time-dependent and fluctuating potential (transient potential) can be derived from the microscopic Hamiltonian dynamics. The concept of the transient potential was first introduced rather phenomenologically, and its relation to the underlying microscopic dynamics has not been clarified yet. This is in contrast to the generalized Langevin equation, the relation of which to the microscopic dynamics is well-established. In this work, we show that the dynamic equations with the transient potential can be derived for the coupled oscillator model, without any approximations. It is known that the dynamics of the coupled oscillator model can be exactly described by the generalized Langevin-type equations. This fact implies that the dynamic equations with the transient potential can be utilized as a coarse-grained dynamics model in a similar way to the generalized Langevin equation. Then we show that the dynamic equations for the transient potential can also be formally derived for the microscopic Hamiltonian dynamics, without any approximations. We use the projection operator method for the coarse-grained variables and transient potential. The dynamic equations for the coarse-grained positions and momenta are similar to those in the Hamiltonian dynamics, but the interaction potential is replaced by the transient potential. The dynamic equation for the transient potential is the generalized Langevin equation with the memory effect. Our result justifies the use of the transient potential to describe the coarse-grained dynamics. We propose several approximations to obtain the simplified dynamics model. We show that, under several approximations, the dynamic equation for the transient potential reduces to the relatively simple Markovian dynamic equation for the potential parameters. We also show that with several additional approximations, the approximate dynamics model further reduces to the Markovian Langevin-type equations with the transient potential.

Effects of Slip-Spring Parameters and Rouse Bead Density on Polymer Dynamics in Multichain Slip-Spring Simulations.

The multichain slip-spring (MCSS) model is one of the coarse-grained models of polymers developed in the niche between bead-spring models and tube type descriptions. In this model, polymers are represented by Rouse chains connected by virtual springs that temporally connect the chains, hop along the chain, and are constructed and annihilated at the chain ends. Earlier studies have shown that MCSS simulations can nicely reproduce entangled and unentangled polymer dynamics. However, the model parameters have been chosen arbitrarily, and their effects have not been reported. In this study, for the first time, we systematically investigated the effects of model parameters: fugacity of virtual springs, its intensity, and the Rouse bead density. We validated the employed simulation code by confirming that the statistics of the system follow the theoretical setup. Namely, the virtual spring density is correctly controlled, and polymer chains exhibit ideal chain statistics irrespective of the chosen parameter values. For diffusion and linear viscoelasticity, simulation results obtained for different parameters can be superposed with each other by conversion factors for the bead number per chain and units of length, time, and modulus. These conversion factors follow scaling laws concerning the number of Rouse segments between two consecutive anchoring points of virtual springs along the polymer chain. Besides, diffusion and viscoelasticity excellently agree with literature data for the standard bead-spring simulation. These results imply that the coarse-graining level for the MCSS model can be arbitrarily chosen and controlled by model parameters.

Analysis of Elongational Viscosity of Entangled Poly (Propylene Carbonate) Melts by Primitive Chain Network Simulations.

It has been established that the elongational rheology of polymers depends on their chemistry. However, the analysis of experimental data has been reported for only a few polymers. In this study, we analyzed the elongational viscosity of poly (propylene carbonate) (PPC) melts in terms of monomeric friction via primitive chain network simulations. By incorporating a small polydispersity of materials, the linear viscoelastic response was semi-quantitatively reproduced. Owing to this agreement, we determined units of time and modulus to carry out elongational simulations. The simulation with constant monomeric friction overestimated elongational viscosity, whereas it nicely captured the experimental data if friction decreased with increasing segment orientation. To see the effect of chemistry, we also conducted the simulation for a polystyrene (PS) melt, which has a similar entanglement number per chain and a polydispersity index. The results imply that PPC and PS behave similarly in terms of the reduction of friction under fast deformations.

Short-time dynamics of a tracer in an ideal gas.

A small tagged particle immersed in a fluid exhibits Brownian motion and diffuses on a long timescale. Meanwhile, on a short timescale, the dynamics of the tagged particle cannot be simply described by the usual generalized Langevin equation with Gaussian noise, since the number of collisions between the tagged particle and fluid particles is rather small. On such a timescale, we should explicitly consider individual collision events between the tagged particle and the surrounding fluid particles. In this study we analyze the short-time dynamics of a tagged particle in an ideal gas, where we do not have static or hydrodynamic correlations between fluid particles. We perform event-driven hard-sphere simulations and show that the short-time dynamics of the tagged particle is correlated even under such an idealized situation. Namely, the velocity autocorrelation function becomes negative when the tagged particle is relatively light and the fluid density is relatively high. This result can be attributed to the dynamical correlation between collision events. To investigate the physical mechanism which causes the dynamical correlation, we analyze the correlation between successive collision events. We find that the tagged particle can collide with the same ideal-gas particle several times and such collisions cause a strong dynamical correlation for the velocity.

Primitive chain network simulations for the interrupted shear response of entangled polymeric liquids.

The non-linear viscoelastic response under interrupted shear flows is one of the interesting characteristics of entangled polymers. In particular, the stress overshoot in the resumed shear has been discussed concerning the recovery of the entanglement network in some studies. In this study, we performed multichain slip-link simulations to observe the molecular structure of an entangled polymer melt. After confirming the reasonable reproducibility of our simulation with the literature data, we analyzed the molecular characteristics following the decoupling approximation. We reasonably found that the segment orientation dominates the stress overshoot even under the resumed shear with minor contributions from the segment stretch and entanglement density. We defined the mitigation function for the recovery of the stress overshoot as a function of the rest time and compared it with the relaxation of the molecular quantities after the initial shear. As a result, we have found that the mitigation of the stress overshoot coincides with the relaxation of entanglement density.

Coarse-graining of microscopic dynamics into a mesoscopic transient potential model.

We show that a mesoscopic coarse-grained dynamics model which incorporates the transient potential can be formally derived from an underlying microscopic dynamics model. As a microscopic dynamics model, we employ the overdamped Langevin equation. By utilizing the path probability and the Onsager-Machlup type action, we calculate the path probability for the coarse-grained mesoscopic degrees of freedom. The action for the mesoscopic degrees of freedom can be simplified by incorporating the transient potential. Then the dynamic equation for the mesoscopic degrees of freedom can be simply described by the Langevin equation with the transient potential (LETP). As a simple and analytically tractable approximation, we introduce additional degrees of freedom which express the state of the transient potential. Then we approximately express the dynamics of the system as the the combination of the LETP and the dynamics model for the transient potential. The resulting dynamics model has the same dynamical structure as the responsive particle dynamics type models [W. J. Briels, Soft Matter 5, 4401 (2009)1744-683X10.1039/b911310j] and the multichain slip-spring type models [T. Uneyama and Y. Masubuchi, J. Chem. Phys. 137, 154902 (2012)JCPSA60021-960610.1063/1.4758320]. As a demonstration, we apply our coarse-graining method with the LETP to a single particle dynamics in a supercooled liquid, and compare the results of the LETP with the molecular dynamics simulations and other coarse-graining models.

Brownian motion with alternately fluctuating diffusivity: Stretched-exponential and power-law relaxation.

We investigate Brownian motion with diffusivity alternately fluctuating between fast and slow states. We assume that sojourn-time distributions of these two states are given by exponential or power-law distributions. We develop a theory of alternating renewal processes to study a relaxation function which is expressed with an integral of the diffusivity over time. This relaxation function can be related to a position correlation function if the particle is in a harmonic potential and to the self-intermediate scattering function if the potential force is absent. It is theoretically shown that, at short times, the exponential relaxation or the stretched-exponential relaxation are observed depending on the power-law index of the sojourn-time distributions. In contrast, at long times, a power-law decay with an exponential cutoff is observed. The dependencies on the initial ensembles (i.e., equilibrium or nonequilibrium initial ensembles) are also elucidated. These theoretical results are consistent with numerical simulations.

Unveiling diffusive states from center-of-mass trajectories in glassy dynamics.

We propose a method to detect alternating diffusive states undergoing a free diffusive state and a trapped state described by the Ornstein-Uhlenbeck process. Using a stochastic model with alternating diffusive states, a phenomenological model of glassy dynamics, we show that control parameters in the method may be determined by the mean square displacement and the non-Gaussianity parameter. Our method works when diffusivities for the two states are clearly distinct and all the states last longer than a specified relaxation time. Applying our method to molecular dynamics simulation data of supercooled liquids, we show that trapped states last for a long time and the sojourn-time distribution for trapped states becomes a power-law form as the temperature approaches the glass temperature.

Retardation of the reaction kinetics of polymers due to entanglement in the post-gel stage in multi-chain slip-spring simulations.

Although the reaction kinetics of network formation of polymers has been extensively investigated, the role of entanglement between polymers has not been fully elucidated yet. In this study, we discuss the effect of entanglement via multi-chain slip-spring simulations, in which Rouse chains are dispersed in space and connected by slip-springs that mimic the entanglement. For stoichiometric conditions for the systems containing pre-polymers and cross-linkers, the simulations without slip-springs exhibited reaction kinetics that is consistent with the earlier mean-field theory. Meanwhile, the inclusion of slip-springs in the system retards the reaction in the post-gel stage after the percolation of the system. According to the analysis of the network structure, the reaction in the post-gel stage is dominated by the tethered chains. The entanglement indirectly retards the reaction kinetics through the suppression of tethered chain dynamics.

Relaxation functions of the Ornstein-Uhlenbeck process with fluctuating diffusivity.

We study the relaxation behavior of the Ornstein-Uhlenbeck (OU) process with time-dependent and fluctuating diffusivity. In this process, the dynamics of the position vector is modeled by the Langevin equation with a linear restoring force and a fluctuating diffusivity (FD). This process can be interpreted as a simple model of relaxational dynamics with internal degrees of freedom or in a heterogeneous environment. By utilizing the functional integral expression and the transfer matrix method, we show that the relaxation function can be expressed in terms of the eigenvalues and eigenfunctions of the transfer matrix for general FD processes. We apply our general theory to two simple FD processes where the FD is described by the Markovian two-state model or an OU-type process. We show analytic expressions of the relaxation functions in these models and their asymptotic forms. We also show that the relaxation behavior of the OU process with an FD is qualitatively different from those obtained from conventional models such as the generalized Langevin equation.

A transient bond model for dynamic constraints in meso-scale coarse-grained systems.

The dynamical properties of entangled polymers originate from the dynamic constraints due to the uncrossability between polymer chains. We propose a highly coarse-grained simulation model with transient bonds for such dynamically constrained systems. Based on the ideas of the responsive particle dynamics (RaPiD) model [P. Kindt and W. J. Briels, J. Chem. Phys. 127, 134901 (2007)] and the multi-chain slip-spring model [T. Uneyama and Y. Masubuchi, J. Chem. Phys. 137, 154902 (2012)], we construct the RaPiD type transient bond model as a coarse-grained slip-spring model. In our model, a polymer chain is expressed as a single particle, and particles are connected by transient bonds. The transient bonds modulate the dynamics of particles, but they do not affect static properties in equilibrium. We show the relation between parameters for the entangled polymer systems and those for the transient bond model. By performing simulations based on the transient bond model, we show how model parameters affect the linear viscoelastic behavior and the diffusion behavior. We also show that the viscoelastic behavior of entangled polymer systems can be well reproduced by the transient bond model.

Comparison among multi-chain models for entangled polymer dynamics.

Although lots of coarse-grained models have been proposed to trace the long-term behaviors of entangled polymers, compatibility among the different models has not been frequently discussed. In this study, some dynamical and static quantities, such as diffusion, relaxation modulus, chain dimension, and entanglement density, were examined for the multi-chain slip-link model (primitive chain network model) and the multi-chain slip-spring model, and the results were compared with those reported for the standard bead-spring model. For the diffusion, three models are compatible with scale-conversion parameters for units of length, time and bead (segment) number (or the molecular weight). The relaxation modulus is also compatible given that the model dependence can be accommodated by the entanglement density and the additional scale-conversion for the unit of modulus. The chain dimension is reasonably coincident with small deviations due to the weak non-Gaussianity of the models. Apart from these plausible compatibilities, significant discrepancies have been found for the inter-chain cross-correlations in the relaxation modulus.

Orientational cross correlations between entangled branch polymers in primitive chain network simulations.

Although it has not been frequently discussed, contributions of the orientational cross-correlation (OCC) between entangled polymers are not negligible in the relaxation modulus. In the present study, OCC contributions were investigated for 4- and 6-arm star-branched and H-branched polymers by means of multi-chain slip-link simulations. Owing to the molecular-level description of the simulation, the segment orientation was traced separately for each molecule as well as each subchain composing the molecules. Then, the OCC was calculated between different molecules and different subchains. The results revealed that the amount of OCC between different molecules is virtually identical to that of linear polymers regardless of the branching structure. The OCC between constituent subchains of the same molecule is significantly smaller than the OCC between different molecules, although its intensity and time-dependent behavior depend on the branching structure as well as the molecular weight. These results lend support to the single-chain models given that the OCC effects are embedded into the stress-optical coefficient, which is independent of the branching structure.

Fluctuation analysis of time-averaged mean-square displacement for the Langevin equation with time-dependent and fluctuating diffusivity.

The mean-square displacement (MSD) is widely utilized to study the dynamical properties of stochastic processes. The time-averaged MSD (TAMSD) provides some information on the dynamics which cannot be extracted from the ensemble-averaged MSD. In particular, the relative standard deviation (RSD) of the TAMSD can be utilized to study the long-time relaxation behavior. In this work, we consider a class of Langevin equations which are multiplicatively coupled to time-dependent and fluctuating diffusivities. Various interesting dynamics models such as entangled polymers and supercooled liquids can be interpreted as the Langevin equations with time-dependent and fluctuating diffusivities. We derive a general formula for the RSD of the TAMSD for the Langevin equation with the time-dependent and fluctuating diffusivity. We show that the RSD can be expressed in terms of the correlation function of the diffusivity. The RSD exhibits the crossover at the long time region. The crossover time is related to a weighted average relaxation time for the diffusivity. Thus the crossover time gives some information on the relaxation time of fluctuating diffusivity which cannot be extracted from the ensemble-averaged MSD. We discuss the universality and possible applications of the formula via some simple examples.

Self-consistent field model simulations for statistics of amorphous polymer chains in crystalline lamellar structures.

We calculate statistical properties of amorphous polymer chains between crystalline lamellae by self-consistent field model simulations. In our model, an amorphous subchain is modelled as a polymer chain of which ends are grafted onto the crystal-amorphous interfaces. The crystal-amorphous interfaces are expressed as impenetrable surfaces. We incorporate the interaction between segments to satisfy the incompressible condition for the segment density field. The simulation results show that amorphous polymer chains feel thin potential layers, which are mainly repulsive, near the crystal-amorphous interfaces. The impenetrable and incompressible conditions affect the statistics of polymer chains and the chain statistics becomes qualitatively different from the ideal Gaussian chain statistics without any constraints. We show the effects of the system size and the graft density to statistical quantities. We also show that the tie subchain statistics obey rather simple statistics.