Stokes-Einstein Research Articles

Stokes-Einstein relation for binary mixtures

The applicability of the Stokes-Einstein (SE) relation in two-dimensional finite binary mixtures is tested using the Langevin simulation method. Calculations of viscosity and self-diffusion coefficients are compared between monodisperse and binary systems. It is shown that adapted definitions of coupling strength Γ and screening parameter κ allow to describe the transport properties for both monodisperse and binary systems consistently. Further, it is discussed how the mixing ratio and charge ratio of the binary mixture affects transport properties.

Relaxation time scales of interfacial water upon fluid to ripple to gel phase transitions of bilayers.

The slow relaxation of interface water (IW) across three primary phases of membranes is relevant to understand the influence of IW on membrane functions at supercooled conditions. To this objective, a total of ∼16.26μs all-atom molecular dynamics simulations of 1,2-dimyristoyl-sn-glycerol-3-phosphocholine lipid membranes are carried out. A supercooling-driven drastic slow-down in heterogeneity time scales of the IW is found at the fluid to the ripple to the gel phase transitions of the membranes. At both fluid-to-ripple-to-gel phase transitions, the IW undergoes two dynamic crossovers in Arrhenius behavior with the highest activation energy at the gel phase due to the highest number of hydrogen bonds. Interestingly, the Stokes-Einstein (SE) relation is conserved for the IW near all three phases of the membranes for the time scales derived from the diffusion exponents and the non-Gaussian parameters. However, the SErelation breaks for the time scale obtained from the self-intermediate scattering functions. The behavioral difference in different time scales is universal and found to be an intrinsic property of glass. The first dynamical transition in the α relaxation time of the IW is associated with an increase in the Gibbs energy of activation of hydrogen bond breaking with locally distorted tetrahedral structures, unlike the bulk water. Thus, our analyses unveil the nature of the relaxation time scales of the IW across membrane phase transitions in comparison with the bulk water. The results will be useful to understand the activities and survival of complex biomembranes under supercooled conditions in the future.

Structural and dynamical properties of thermoplastic polyurethane/fullerene nanocomposites: a molecular dynamics simulations study.

In this work, the structural and dynamical properties of thermoplastic polyurethane (TPU)/fullerene (C60) nanocomposites are investigated using atomistic molecular dynamics simulations, focusing on the glass transition, thermal expansion, polymer mobility, polymer-C60 interactions, and diffusion behavior of C60. The results show a slight increase in the glass transition temperature (Tg) with increasing C60 weight fraction (wt%), attributed to hindered polymer dynamics, and a remarkable reduction in the coefficient of thermal expansion above Tg. Results of the mean squared displacement and the time decay of bond-reorientation autocorrelation indicate that the mobility of TPU hard segments is more restricted than that of soft segments, owing to the electrostatic attractions and the π-π stacking between isocyanate groups and C60 molecules. Analysis of TPU-C60 interaction energy reveals that the electrostatic interactions are weakened with an increase in the C60 wt%, while the van der Waals contributions become more significant due to the TPU-C60 interfacial characteristics. Further analysis shows that the translational and rotational diffusion of C60 are both increasingly suppressed with the increase of C60 wt%, indicating a violation of Stokes-Einstein (SE) and Stokes-Einstein-Debye (SED) relations, presumably due to the polymer chain-mediated hydrodynamic interactions arising from chain bridges between neighboring C60 particles. This is highlighted by a stronger decoupling of translational-rotational diffusion and a lower ratio of translational-rotational diffusion coefficient (DT/DR) with increasing C60 wt%. This work elucidates an atomistic understanding of the structure and properties of polymer/C60 nanocomposites.

Viscosity decoupling does not guarantee dynamic heterogeneity: A way out

Generally, dynamic heterogeneity in a media is identified by recognizing the viscosity decoupling of the dynamics (i.e., p ≠ 1 in the equation logτx=plogηT+logC). While dynamically heterogeneous media will show a breakdown from the Stokes-Einstein (SE) relationship (p ≠ 1), the vice-versa is not automatically true. One should be cautious in relating viscosity decoupling to dynamic heterogeneity. Herein, we developed a novel analysis for identifying dynamic heterogeneity in a better way. The analysis is based on selective probing of different diffussion time-regimes in a heterogeneous system from a single measurement and observing their temperature evolution. To prove our concept, we studied temperature dependent translational diffusion of a fluorophore using fluorescence correlation spectroscopy. We used two molecular liquids and two complex deep eutectic solvents in our study. All four solvents show viscosity decoupling, albeit to a different extent. Following the previous knowledge, this could be explained by dynamic heterogeneity in all these solvents. However, our analysis suggests that two molecular liquids, DMF and glycerol, are dynamically homogenous and the two DESs, acetamide/urea and lauric-acid/menthol, are dynamically heterogeneous. We conclude that difference in the extent of viscosity decoupling for different diffusion time-regimes might be a better criteria to identify dynamic heterogeneity, compared to the traditional way. We believe that the newly developed approach will be a step-forward in determining the dynamic heterogeneity from viscosity decoupling.

Dispersion and Diffusion Mechanism of Nanofillers with Different Geometries in Bottlebrush Polymers: Insights from Molecular Dynamics Simulation.

The dispersion and diffusion mechanism of nanofillers in polymer nanocomposites (PNCs) are crucial for understanding the properties of PNCs, which is of great significance for the design of novel materials. Herein, we investigate the dispersion and diffusion behavior of two geometries of nanofillers, namely, spherical nanoparticles (SNPs) and nanorods (NRs), in bottlebrush polymers by utilizing coarse-grained molecular dynamics simulations. With the increase of the interaction strength between the nanofiller and polymer (εnp), both the SNPs and NRs experience a typical "aggregated phase-dispersed phase-bridged phase" state transition in the bottlebrush polymer matrix. We evaluate the validity of the Stokes-Einstein (SE) equation for predicting the diffusion coefficient of nanofillers in bottlebrush polymers. The results demonstrate that the SE predictions are slightly larger than the simulated values for small SNP sizes because the local viscosity that is felt by small SNPs in the densely grafted bottlebrush polymer does not differ much from the macroscopic viscosity. The relative size of the length of the NRs (L) and the radius of gyration (Rg) of the bottlebrush polymer play a key role in the diffusion of NRs. In addition, we characterize the anisotropic diffusion of NRs to analyze their translational and rotational diffusion. The motion of NRs in the direction perpendicular to the end-to-end vector is more hindered, indicating that there is a strong coupling between the rotation of NRs and the motion of the polymer. The NR motion shows stronger anisotropic diffusion at short time scales because of the steric effects generated by side chains of the bottlebrush polymer. In general, our results provide a fundamental understanding of the dispersion of nanofillers and the microscopic mechanism of nanofiller diffusion in bottlebrush polymers.

The effective hydrodynamic radius in the Stokes–Einstein relation is not a constant

Variants based on the assumption of effective hydrodynamic radius being a constant are usually adopted to test the Stokes–Einstein (SE) relation. The rationality of the assumption is examined by performing molecular dynamics simulations with the truncated Lennard-Jones-like (TLJ) model, Kob–Andersen model and ortho-terphenyl (OTP) model. The results indicate the assumption is generally not established except for special case. The effective hydrodynamic radius is observed to increase with decreasing temperature for TLJ model but is decreased for Kob–Andersen and OTP model; and which is almost a constant for TLJ particle with enough rigidity. The variant of SE relation is invalid for the three models except for the TLJ particle with enough rigidity. We propose similar inconsistency may be also existed in other liquids and the assumption should be critically evaluated when adopted to test the SE relation.

Study on the Translational Diffusion of Transient Species in 1-Alkyl-3-methylimidazolium-Based Ionic Liquids by Transient Grating Spectroscopy

Abstract Translational diffusion coefficients of two transient species in 1-alkyl-3-methylimidazolium-based ionic liquids (Cnmim+) with different alkyl chain lengths (n = 4, 8, 12) and the bis(trifluoromethanesulfonyl)imide anion were measured by transient grating spectroscopy. The systems of interest are the benzophenone ketyl radical (BPK) produced by the hydrogen-abstraction reaction of benzophenone (BP) and the zwitterionic merocyanine dye (MC) produced by the photoisomerization of 1,3,3-trimethylindolino-6′-nitrobenzopyrylospiran (SP). The diffusion coefficients of BPK and MC closely obey the Stokes-Einstein (SE) relation in the ionic liquids regardless of the alkyl chain lengths of the cations, whereas the diffusion coefficients of BP are larger than those predicted by the SE equation. The temperature dependences of the translational diffusion coefficients of BP, BPK, and MC reveal that the diffusional activation energies of BPK and MC are larger than those of BP in all solvents. The origin of this difference is discussed in terms of intermolecular interactions within the ionic-liquid solvents.

Single molecule demonstration of Debye–Stokes–Einstein breakdown in polystyrene near the glass transition temperature

Rotational-translational decoupling, in which translational motion is apparently enhanced over rotational motion in violation of Stokes-Einstein (SE) and Debye-Stokes-Einstein (DSE) predictions, has been observed in materials near their glass transition temperatures (Tg). This has been posited to result from ensemble averaging in the context of dynamic heterogeneity. In this work, ensemble and single molecule experiments are performed in parallel on a fluorescent probe in high molecular weight polystyrene near its Tg. Ensemble results show decoupling onset at approximately 1.15Tg, increasing to over three orders of magnitude at Tg. Single molecule measurements also show a high degree of decoupling, with typical molecules at Tg showing translational diffusion coefficients nearly 400 times higher than expected from SE/DSE predictions. At the single molecule level, higher degree of breakdown is associated with particularly mobile molecules and anisotropic trajectories, providing support for anomalous diffusion as a critical driver of rotational-translational decoupling and SE/DSE breakdown.

Atomic transport properties of silicon melt at high temperature

Atomic transport properties, in particular, the self-diffusion and shear viscosity of silicon melt at high temperatures near the melting point are studied employing classical molecular dynamics simulations based on multiple empirical interatomic potentials. We find that the diffusion and viscosity coefficients of silicon melt predicted by the empirical potentials follow the Stokes-Einstein (SE) relation although both of which deviate from the experimental values. The discrepancy between the calculated and experimental diffusion coefficients are analyzed using the SE relation. We demonstrate that the discrepancy is partially attributed to the underestimation of the self-diffusion coefficients by the empirical potentials. Further, we analyze the microstructure of the melt based on several topological techniques and confirm that the (over) underestimation of the (viscosity) diffusion coefficients is caused by the overemphasis of the covalent bonding in the silicon melt.

Coupling between Translational Diffusion of a Solute and Dynamics of the Heterogeneous Structure: Higher Alcohols and Ionic Liquids.

Translational diffusion of nonpolar monoatomic solutes in a room-temperature ionic liquid and 1-octanol was studied by molecular dynamics simulation. The diffusion coefficient was evaluated in two different ways: (1) from the mean-square displacement of a freely diffusing solute and (2) from the time correlation function of force acting on a fixed solute. The diffusion of the free solute is much greater than the prediction of the Stokes-Einstein (SE) relation when the size of the solute is small, as has been reported by many experimental works. In contrast, the friction on fixed small solutes follows the SE relation. The mechanism of the solute diffusion in both solvents was then analyzed based on the coupling between the translational motion of the solute and the collective dynamics of the heterogeneous intermediate-range structure characteristic to these solvents. Analysis revealed that the coupling is present in all systems, but the relaxation is fast in the cases of free and small solutes. This suggests that the coupling can relax through the motion of the solute when the solute is free and small, while the relaxation of the heterogeneous structure itself is required for large or fixed solutes. The difference in the relaxation dynamics of the friction on the solute and the shear viscosity is explained as the coupling with different dynamic modes of the solvent. Therefore, the validity of the SE relation may not be a good criterion to judge whether the mechanisms of the diffusion and the viscosity are the same or not.

Spinodal limits of supercooled liquid Al deduced from configuration heredity of crystal clusters

Whether one-component supercooled liquid encounters a spinodal limit at a sufficiently low temperature has been a controversial issue. To explore the homogeneous nucleation limit (HNL) in liquid Al, a series of molecular dynamics simulations have been performed, and a unique technique able to distinguish the critical nucleus from various embryos has been adopted to measure the crystal nucleation time τN of supercooled liquids and evaluate the nucleation energy barrier ΔEcri∗ in isothermal crystallization. The estimated kinetic spinodal (i.e., HNL) temperature Tks and thermodynamic spinodal temperature Tts are 0.51 Tm and 0.45 Tm, respectively. At the HNL, the nucleation energy barrier ΔEcri∗ is discovered to be very small, and a critical slowdown behavior is also observed near Tks. A careful analysis for the Stokes-Einstein (SE) relation reveals the temperature TSE of SE relation breakdown is only 0.02 Tm higher than Tks, and the occurrence of HNL is not affected by whether the SE relation breaks down or not. These results are consistent with previous reports. To some extent, the small gap between TSE and Tks can be attributed to a weak glass formation ability of liquid Al.

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 Σg, grafted chain length Ng, and matrix chain length Nm. 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 Σg, the PGNPs become harder with greater effective size and thinner free draining layer, resulting in a reduction in D. At extremely high Σg, the diffusivity can even be estimated by the diameter-renormalized Stokes-Einstein (SE) relation. With increasing Ng, 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 Nm 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 ∼ Nm-1. These findings bear significance in unraveling the fundamental physics of the anomalous dynamics of PGNPs in various polymers, including biological and synthetic.

Molecular size and shape effects: Tracer diffusion and the Stokes-Einstein relation

Molecular size and shape effects are evaluated on the Stokes–Einstein (SE) relation from the viewpoint of molecular differences between the solute and solvent. To this end, dilute solutions of both an n-alkane in Ar and Ar in an n-alkane were simulated using the molecular dynamics method. The molecular-based SE relation derived for the Lennard-Jones liquid mixtures was found to be applicable to both systems, which shows that the effects of the differences in the size and interaction energy are predominant but that those in the shape are negligible in these systems.

Influence of atomic-scale structure on the transport, ordering, and thermodynamics of partially ordered Cu-In alloys

We demonstrate the atomic-level structural properties of liquid Cu-In alloy as a function of In concentration using Square-Well (SW) model potential function under random phase approximation. It was found that the concentration-dependent structure factor S(k) and its Fourier transform, radial distribution function g(r) are in good agreement with the results obtained by the X-ray diffraction technique. Transport coefficients like diffusion coefficient and shear viscosity of liquid Cu-In alloys were estimated through computed structural functions and SW potential function. Interdiffusion coefficients in the melts were computed by solving the well-known Einstein's equation for Brownian particles with SW pair potential under linear trajectory principal. The composition-dependent shear viscosity is calculated using computed diffusion data in modified Stokes-Einstein (SE) relation. Specifically, we demonstrate a breakdown of SE relation at low In concentration in liquid Cu-In alloy. Using model calculations, we investigate the thermodynamic properties of alloy such as enthalpy of mixing, Gibbs free energy of mixing, and entropy of mixing. The concentration-concentration fluctuation at the zero momentum vector SCC(0)shows that there is a strong chemical bond between hetero-atoms in the Cu-rich region of liquid Cu-In alloys. It is worth mentioning here that the computed values of SCC(0), without using any adjusting or experimental parameter are in excellent agreement with experimental values measured by activity data.

Excess entropy and Stokes-Einstein relation in simple fluids.

The Stokes-Einstein (SE) relation between the self-diffusion and shear viscosity coefficients operates in sufficiently dense liquids not too far from the liquid-solid phase transition. By considering four simple model systems with very different pairwise interaction potentials (Lennard-Jones, Coulomb, Debye-Hückel or screened Coulomb, and the hard sphere limit) we identify where exactly on the respective phase diagrams the SE relation holds. It appears that the reduced excess entropy s_{ex} can be used as a suitable indicator of the validity of the SE relation. In all cases considered the onset of SE relation validity occurs at approximately s_{ex}≲-2. In addition, we demonstrate that the line separating gaslike and liquidlike fluid behaviours on the phase diagram is roughly characterized by s_{ex}≃-1.

Decoupling between Solvent Viscosity and Diffusion of a Small Solute Induced by Self-Motion

The self-diffusion of a monatomic solute in liquid 1-octanol and n-tetradecane was investigated by means of a molecular dynamics simulation. The diffusion coefficient of a solute as small as argon is much greater than that obtained from the hydrodynamic-based Stokes-Einstein (SE) relation, as was reported experimentally. A relaxation of the memory function of a freely diffusing solute is much faster than that of the autocorrelation function of a shear stress. However, the SE behavior is recovered when the solute is spatially fixed, and the diffusion coefficient is calculated from the force-force autocorrelation function. A relaxation of the autocorrelation function of the force also follows that of shear stress. The fast diffusion of a small solute is thus ascribed to the decoupling between the structural relaxation of solvent and the solute diffusion induced by the self-motion of the solute.

Verification of the Stokes-Einstein relation in liquid noble metals over a wide range of temperatures

The validity of Stokes-Einstein (SE) relations in liquid noble metals namely Cu, Ag and Au over a wide temperature range has been studied using Dzugutov's scaling scheme and Faber's hard-sphere (HS) theory of transport coefficients. Basic ingredients of those theories are the temperature-dependent effective HS diameter and excess entropy. To determine them, we have applied variational modified hypernetted chain theory in conjunction with effective interionic interaction derived from embedded atomic method (EAM) with both temperature-dependent and independent adjustable parameters α and b. Obtained ingredients using α(T) and b(T) are close to the available experimental data than those obtained using fixed α and b with an empirical relation. Calculated transport coefficients are in good agreement with experimental data when scaling scheme with α(T) and b(T) has been considered. Simplified SE relations from transport coefficients hold when scaling scheme has been applied with α(T) and b(T) and lies around the slip boundary line.

Identification of time scales of the violation of the Stokes–Einstein relation in Yukawa liquids

We investigate the origin of the violation of the Stokes–Einstein (SE) relation in two-dimensional Yukawa liquids. Using comprehensive molecular dynamics simulations, we identify the time scales supporting the violation of the SE relation D∝(η/T)−1, where D is the self-diffusion coefficient and η is the shear viscosity. We first compute the self-intermediate scattering function Fs(k,t), the non-Gaussian parameter α2, and the autocorrelation function of the shear stress Cη(t). The time scales obtained from these functions include the structural relaxation time τα, the peak time of the non-Gaussian parameter τα2, and the shear stress relaxation time τη. We find that τη is coupled with D for all temperatures indicating the SE preservation; however, τα and τα2 are decoupled with D at low temperatures indicating the SE violation. Surprisingly, we find that the origins of this violation are related to the non-exponential behavior of the autocorrelation function of the shear stress and non-Gaussian behavior of the distribution function of particle displacements. These results confirm dynamic heterogeneity that occurs in two-dimensional Yukawa liquids that reflect the presence of regions in which dust particles move faster than the rest when the liquid cools to below the phase transition temperature.

Unveiling Carbon Dioxide and Ethanol Diffusion in Carbonated Water-Ethanol Mixtures by Molecular Dynamics Simulations.

The diffusion of carbon dioxide (CO) and ethanol (EtOH) is a fundamental transport process behind the formation and growth of CO bubbles in sparkling beverages and the release of organoleptic compounds at the liquid free surface. In the present study, CO and EtOH diffusion coefficients are computed from molecular dynamics (MD) simulations and compared with experimental values derived from the Stokes-Einstein (SE) relation on the basis of viscometry experiments and hydrodynamic radii deduced from former nuclear magnetic resonance (NMR) measurements. These diffusion coefficients steadily increase with temperature and decrease as the concentration of ethanol rises. The agreement between theory and experiment is suitable for CO. Theoretical EtOH diffusion coefficients tend to overestimate slightly experimental values, although the agreement can be improved by changing the hydrodynamic radius used to evaluate experimental diffusion coefficients. This apparent disagreement should not rely on limitations of the MD simulations nor on the approximations made to evaluate theoretical diffusion coefficients. Improvement of the molecular models, as well as additional NMR measurements on sparkling beverages at several temperatures and ethanol concentrations, would help solve this issue.

Modified Stokes–Einstein Equation for Molecular Self-Diffusion Based on Entropy Scaling

The Stokes–Einstein (SE) equation is often applied as an approximation of self-diffusion coefficients D of molecular species based on the shear viscosity of fluids. The SE relation gives rough esti...