Stokes-Einstein Relation Research Articles

Stokes–Einstein Relation in Different Models of Water

The purpose of this paper is to discuss to which extent a microscopic version of the Stokes–Einstein (SE) relation without the hydrodynamic radius applies to liquid water. We demonstrate that the self-diffusion and shear viscosity data for five popular water models, recently reported by Ando [J. Chem. Phys. 159, 101102 (2023)], are in excellent agreement with the SE relation. The agreement with experimental results is also quite impressive. The limitations on the applicability of the SE relation are briefly discussed.

Diffusion and Viscosity in Mixed Protein Solutions.

The viscosity and diffusion properties of crowded protein systems were investigated with molecular dynamics simulations of SH3 mixtures with different crowders, and results were compared with experimental data. The simulations accurately reproduced experimental trends across a wide range of protein concentrations, including highly crowded environments up to 300 g/L. Notably, viscosity increased with crowding but varied little between different crowder types, while diffusion rates were significantly reduced depending on protein-protein interaction strength. Analysis using the Stokes-Einstein relation indicated that the reduction in diffusion exceeded what was expected from viscosity changes alone, with the additional slow-down attributable to transient cluster formation driven by weakly attractive interactions. Contact kinetics analysis further revealed that longer-lived interactions contributed more significantly to reduced diffusion rates than short-lived interactions. This study also highlights the accuracy of current computational methodologies for capturing the dynamics of proteins in highly concentrated solutions and provides insights into the molecular mechanisms affecting protein mobility in crowded environments.

The Dynamic Diversity and Invariance of Ab Initio Water.

Comprehending water dynamics is crucial in various fields, such as water desalination, ion separation, electrocatalysis, and biochemical processes. While ab initio molecular dynamics (AIMD) accurately portray water's structure, computing its dynamic properties over nanosecond time scales proves cost-prohibitive. This study employs machine learning potentials (MLPs) to accurately determine the dynamic properties of liquid water with ab initio accuracy. Our findings reveal diversity in the calculated diffusion coefficient (D) and viscosity of water (η) across different methodologies. Specifically, while the GGA, meta-GGA, and hybrid functional methods struggle to predict dynamic properties under ambient conditions, methods on the higher level of Jacob's ladder of DFT approximation perform significantly better. Intriguingly, we discovered that both D and η adhere to the established Stokes-Einstein (SE) relation for all of the ab initio water. The diversity observed across different methods can be attributed to distinct structural entropy, affirming the applicability of excess entropy scaling relations across all functionals. The correlation between D and η provides valuable insights for identifying the ideal temperature to accurately replicate the dynamic properties of liquid water. Furthermore, our findings can validate the rationale behind employing artificially high temperatures in the simulation of water via AIMD. These outcomes not only pave the path to designing better functionals for water but also underscore the significance of water's many-body characteristics.

Calibration-free disposable electrochemical sensor with co-facing electrodes for viscosity monitoring of plasma samples

Plasma viscosity measurement is crucial in clinical diagnostics, providing insights into blood rheology and health status. Traditional methods, such as capillary and rotational viscometers, require large sample volumes and complex calibration. This study presents a novel disposable electrochemical sensor with co-facing electrodes for viscosity monitoring of plasma samples. The sensor independently determines the diffusion coefficient (D) of the electroactive test molecule ferrocyanide, eliminating the need for calibration curves. This enables the subsequent calculation of the solution's viscosity via the Stokes-Einstein relation. The sensor's performance was validated against a standard quartz-crystal microbalance method, demonstrating high accuracy and reliability. It maintained consistent measurements despite the presence of common electroactive plasma interferents such as ascorbic acid (≤ 160 μM), dopamine (< 9.0 nM), uric acid (< 1.0 mM), and urea (< 15.6 mM). Although these interferents impacted D at concentrations exceeding twice the maximum levels typically present in plasma, the sensor exhibited robust performance under normal physiological conditions and standard interferent concentrations. The sensor offers rapid response times of less than 20 seconds, requires only 7 μL of sample, and is cost-effective, making it suitable for point-of-care applications. Bland-Altman plot analysis confirmed its precision, showing a mean difference of 0.00877 and narrow limits of agreement compared to the standard method used.

Anomalous behaviour of transport properties in a supercooled water–glycerol mixture

Glycerol acts as a natural cryoprotectant by depressing the temperature of ice nucleation and slowing down the dynamics of water mixtures. In this work, we characterise dynamics – diffusion, viscosity and hydrogen bond dynamics – as well as density anomaly and structure of water mixtures with 1%–50% (w/w) glycerol at low temperatures via molecular dynamics simulations using all-atom and coarse-grained models. Simulations reveal distinct violations of the Stokes–Einstein relation in the low-temperature regime for water and glycerol. Deviations are positive for water at all concentrations, and positive for glycerol in very dilute solutions but turning negative in concentrated ones. The all-atom and coarse-grained models reveal an unexpected crossover in the dynamics of the 1% and 10 % (w/w) glycerol at the lowest simulated temperatures. This crossover manifests in the diffusion coefficients of water and glycerol, as well as in the viscosity and lifetime of hydrogen bonds in water. We interpret that the crossover originates on the opposing dependence with glycerol concentration of the two factors controlling the solutions' slow down: the increase in tetrahedrally coordinated water and the dynamics and clustering of the glycerol molecules. We anticipate that this dynamic crossover will also occur for solution of water with other polyols.

Experimental and statistical methods for microrheological characterization of heterogeneity in human respiratory mucus mimics of health and disease progression

Human respiratory mucus (HRM) is extremely soft, compelling passive microrheology for linear viscoelastic characterization. We focus this study on the use of passive microrheology to characterize HRM heterogeneity, a phenomenon in normal HRM that becomes extreme during cystic fibrosis (CF) disease. Specifically, a fraction of the mucin polymers comprising HRM phase-separate into insoluble structures, called flakes, dispersed in mucin-depleted solution. We first reconstitute HRM samples to the MUC5B:MUC5AC mucin ratios consistent with normal and CF clinical samples, which we show recapitulate progressive flake formation and heterogeneity. We then employ passive particle tracking with 200 nm and 1 μm diameter beads in each reconstituted sample. To robustly analyze the tracking data, we introduce statistical denoising methods for low signal-to-noise tracking data within flakes, tested and verified using model-generated synthetic data. These statistical methods provide a fractional Brownian motion classifier of all successfully denoised, tracked beads in flakes and the dilute solution. From the ensemble of classifier data, per bead diameter and mucus sample, we then employ clustering methods to learn and infer multiple levels of heterogeneity: (i) tracked bead data within vs. outside flakes and (ii) within-flake data buried within or distinguishable from the experimental noise floor. Simulated data consistent with experimental data (within and outside flakes) are used to explore form(s) of the generalized Stokes–Einstein relation (GSER) that recover the dynamic moduli of homogeneous and heterogeneous truth sets of purely flakelike, dilute solution, and mixture samples. The appropriate form of GSER is applied to experimental data to show (i) flakes are heterogeneous with gel and sol domains; (ii) dilute solutions are heterogeneous with only sol domains; and (iii) flake and dilute solution properties vary with probe diameter.

Diffusion and Viscosity in Mixed Protein Solutions.

The viscosity and diffusion properties of crowded protein systems were investigated with molecular dynamics simulations of SH3 mixtures with different crowders, and results were compared with experimental data. The simulations accurately reproduced experimental trends across a wide range of protein concentrations, including highly crowded environments up to 300 g/L. Notably, viscosity increased with crowding but varied little between different crowder types, while diffusion rates were significantly reduced depending on protein-protein interaction strength. Analysis using the Stokes-Einstein relation indicated that the reduction in diffusion exceeded what was expected from viscosity changes alone, with the additional slow-down attributable to transient cluster formation driven by weakly attractive interactions. Contact kinetics analysis further revealed that longer-lived interactions contributed more significantly to reduced diffusion rates than short-lived interactions. This study also highlights the accuracy of current computational methodologies for capturing the dynamics of proteins in highly concentrated solutions and provides insights into the molecular mechanisms affecting protein mobility in crowded environments.

Composition-dependent diffusion and viscosity behavior in liquid Ti–Al–Ni ternary alloys

We conducted ab initio molecular dynamics simulations to investigate the dynamic properties of the liquid Ti–Al–Ni ternary alloy system at 2033 K. Through simulations across the entire composition range, we revealed the complex compositional dependence of self-diffusion coefficients and viscosity. The self-diffusion coefficients of Ti, Al, and Ni exhibit nonlinear distribution characteristics, while viscosity shows non-monotonic compositional dependence, with lower viscosity in Al-rich regions and higher viscosity in Ni-rich regions. To understand the structural origins of these behaviors, we analyzed chemical short-range order and local fivefold symmetry structures using Warren-Cowley parameters and fivefold symmetry parameters, uncovering a close correlation between atomic-scale structural features and dynamic properties. Evaluation of the Stokes-Einstein relation revealed varying degrees of deviation from its predicted viscosity values across composition regions. We observed a negative correlation between free volume fraction and viscosity, with varying strengths in different regions. These findings provide new perspectives for understanding the complex dynamic properties of Ti–Al–Ni liquid alloys.

Diffusion Coefficients of Coated Plasmonic Nanoparticles in Viscous Environment.

The Stokes-Einstein relationship (SER) is not valid anymore in polymeric solutions for nanoparticles. It is thus important to characterize theirdiffusion properties to get a finer understanding of their behavior and to better tune their attributes for biomedical applications. The diffusion of gold and silver nanoparticles with citrate, hyaluronic acid, methyl-polyethylene glycol, and antibody-polyethylene glycol coatings is studied in hyaluronic-based viscous solutions. The diffusion coefficient D is estimated from the Brownian motion thanks to a cost-effective side-illumination device. It is determined that the nanoparticles (hydrodynamic radius rh: 30-135nm) diffuse up to 4-5 times faster than expected using the SER with a macroscopic viscosity from 1 to 30 mPa·s. It is shown that the adapted Huggins equation is a good model to describe the diffusion behavior of nanoparticles using an effective viscosity ηeff given by where where E is the polymer correlation length, Rh the polymer hydrodynamic radius and ηs the solvent viscosity. The values of k and a are given and allow to obtain D with an error of 10-20%. The impact of chemical interactions on the model parameter values are also highlighted, especially due to electrostatic interactions between the polymer and the nanoparticles.

Dynamic spectroscopic and optical characterization and modeling of bovine serum albumin corona during interaction with N-hydroxysulfo-succinimide-covalently functionalized gold nanourchins.

In this study, bovine serum albumin (BSA) is used as a globular protein model to examine the conformational changes that occur during the interaction of BSA with N-hydroxysulfo-succinimide (sodium salt)-functionalized gold nanourchins (GNUs), for which dynamic spectroscopic techniques are employed. The results showed that the absorbance of phosphate-buffered saline-BSA at 278 nm decreased when a GNU was added to the solution due to adsorption, and it decreased further when the GNU was increased. The intensity and width of the peak of local surface plasmon resonance increased, indicating the effect of corona formation. Dynamic UV-vis spectroscopy and scattering revealed a nonlinear behavior of BSA-GNU interaction. The bioplasmonic solution resulted in higher transmission and scattering than the BSA solution. Fourier transform-near-infrared spectra exhibited several bands due to overtones and combinations of the amide group and different proportions of α-helix and β-sheet components in BSA before and after the addition of the GNU. Time-resolved fluorescence spectroscopy demonstrated an initial increase in blueshifted emission, followed by a redshifted quenching of two major peaks of Tyr and tryptophan (Trp). The binding and dissociation constants were determined as Kb = 2.17 × 1010 M-1 and Kd = 4.6 × 10-11, respectively, using the Stern-Volmer relation. Both the dynamic CMOS-based imaging and the cadmium sulfide sensors demonstrated a nonlinear response of bioplasmonic solution. By increasing the GNU, the resistance of the solution decreased in the order of A > S1 > S3, where S3 exhibited the highest initial transmission with a longer desorption time. MATLAB modeling showed 80% surface coverage by the protein in 15 s at 0.05M, equivalent to a thickness of 1.7 nm, which was in agreement with the value determined by using the Stokes-Einstein relation.

Disorder-Induced Anomalous Mobility Enhancement in Confined Geometries.

Strong, scale-free disorder disrupts typical transport properties like the Stokes-Einstein relation and linear response, leading to anomalous diffusion observed in amorphous materials, glasses, living cells, and other systems. Our study reveals that the combination of scale-free quenched disorder and geometrical constraints induces unconventional single-particle mobility behavior. Specifically, in a two-dimensional channel with width w, under external drive, tighter geometrical constraints (smaller w) enhance mobility. We derive an explicit form of the response to an external force by utilizing the double-subordination approach for the quenched trap model. The observed mobility enhancement occurs in the low-temperature regime where the distribution of localization times is scale-free.

Diffusion, viscosity, and linear rheology of valence-limited disordered fluids.

We numerically investigate the dynamics and linear rheology of disordered systems made of patchy particles, focusing on the role of valence, temperature, and bonding mechanism. We demonstrate that the dynamics is enslaved to bonding, giving rise to an activated behavior at low temperatures. By independently computing the diffusion constant and the viscosity from the simulations, we also confirm the validity of the Stokes-Einstein relation in valence-limited systems, with two caveats: (i) the diffusion constant requires a finite-size correction, at least at the intermediate density we investigate, and (ii) there is the onset of a breakdown that appears at the lowest temperatures considered. Finally, our results show that the storage and loss moduli of mixtures of divalent and M-valent particles exhibit an apparent power-law dependence on frequency, hinting at the possibility of using the composition to finely tune the rheological response of these materials. Our results compare well with literature experimental data on valence-limited DNA nanostars. In addition, the wealth of data we present and analyze here will help develop and test theoretical frameworks aimed at describing the dynamics of flexible limited-valence particles that self-assemble into disordered networks.

The role of memory-dependent friction and solvent viscosity in isomerization kinetics in viscogenic media

Molecular isomerization kinetics in liquid solvent depends on a complex interplay between the solvent friction acting on the molecule, internal dissipation effects (also known as internal friction), the viscosity of the solvent, and the dihedral free energy profile. Due to the absence of accurate techniques to directly evaluate isomerization friction, it has not been possible to explore these relationships in full. By combining extensive molecular dynamics simulations with friction memory-kernel extraction techniques we consider a variety of small, isomerising molecules under a range of different viscogenic conditions and directly evaluate the viscosity dependence of the friction acting on a rotating dihedral. We reveal that the influence of different viscogenic media on isomerization kinetics can be dramatically different, even when measured at the same viscosity. This is due to the dynamic solute-solvent coupling, mediated by time-dependent friction memory kernels. We also show that deviations from the linear dependence of isomerization rates on solvent viscosity, which are often simply attributed to internal friction effects, are due to the simultaneous violation of two fundamental relationships: the Stokes-Einstein relation and the overdamped Kramers prediction for the barrier-crossing rate, both of which require explicit knowledge of friction.

Exploration of Stokes hydrodynamic law at molecular length scales.

The celebrated generalized Stokes law predicts that the velocity of a particle pulled through a liquid by an external force, Fex, is directly proportional to the force and inversely proportional to the friction ζ acted by the medium on the particle. We investigate the range of validity of the generalized Stokes law at molecular length scales by employing computer simulations to calculate friction by pulling a tagged particle with a constant force. We thus calculate friction for two model interaction potentials, Lennard-Jones and soft sphere, for several particle sizes, ranging from radius (a) smaller than the solvent particles to three times larger. We next obtain friction from diffusion (D) by using Einstein's relation between diffusion and friction ζ in an unperturbed liquid. We find a quantitative agreement between the two at a small-to-intermediate pulling force regime for all the sizes studied. The law does break down at a large pulling force beyond a threshold value. Importantly, the range of validity of Stokes' scheme to obtain friction increases substantially if we turn off the attractive part of the interaction potential. Additionally, we calculate the viscosity (η) of the unperturbed liquid and find a good agreement with the Stokes-Einstein relation ζ = Cηa for the viscosity dependence with a value of C close to 5 π, which is intermediate between the slip and stick boundary condition.

Nanoscopic Interfacial Hydrogel Viscoelasticity Revealed from Comparison of Macroscopic and Microscopic Rheology.

Deviations between macrorheological and particle-based microrheological measurements are often considered to be a nuisance and neglected. We study aqueous poly(ethylene oxide) (PEO) hydrogels for varying PEO concentrations and chain lengths that contain microscopic tracer particles and show that these deviations reveal the nanoscopic viscoelastic properties of the particle-hydrogel interface. Based on the transient Stokes equation, we first demonstrate that the deviations are not due to finite particle radius, compressibility, or surface-slip effects. Small-angle neutron scattering rules out hydrogel heterogeneities. Instead, we show that a generalized Stokes-Einstein relation, accounting for an interfacial shell around tracers with viscoelastic properties that deviate from bulk, consistently explains our macrorheological and microrheological measurements. The extracted shell diameter is comparable to the PEO end-to-end distance, indicating the importance of dangling chain ends. Our methodology reveals the nanoscopic interfacial rheology of hydrogels and is applicable to different kinds of viscoelastic fluids and particles.

Freezing density scaling of transport coefficients in the Weeks-Chandler-Andersen fluid.

It is shown that the transport coefficients (self-diffusion, shear viscosity, and thermal conductivity) of the Weeks-Chandler-Andersen (WCA) fluid along isotherms exhibit a freezing density scaling (FDS). The functional form of this FDS is essentially the same or closely related to those in the Lennard-Jones fluid, hard-sphere fluid, and some liquefied noble gases. This proves that this FDS represents a quasi-universal corresponding state principle for simple classical fluids with steep interactions. Some related aspects, such as a Stokes-Einstein relation without a hydrodynamic diameter and gas-to-liquid dynamical crossover, are briefly discussed. Simple fitting formulas for the transport coefficients of the dense WCA fluid are suggested.

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.

Calculating High-Pressure PAO4 Viscosity with Equilibrium Molecular Dynamics Simulations

The development of optimized lubricants is hindered by missing knowledge of fluid properties, in particular the viscosity, in the range of extreme pressures and temperatures relevant for application. Molecular dynamics simulations can be used to calculate viscosity, but the necessary computational effort imposes practical limits for high viscosities. In this study, the viscosity of PAO4 oil was extracted from equilibrium molecular dynamics simulations as a function of pressure and temperature reaching viscosities up to 20 Pas. Three calculation methods based on different microscopic expressions for the viscosity were used. The methods exhibit considerably different performance with respect to preciseness and computational efficiency. The highest viscosities were found to be calculated most efficiently via the Stokes–Einstein relation, by computing the diffusion coefficient from the velocity correlation function. This offers a new, more effective route to push viscosity calculations in equilibrium molecular dynamics simulations to higher pressure systems.Graphical

Relaxation and diffusion of an ionic plasticizer in amorphous poly(vinylpyrrolidone).

The present work focuses on the dynamics of the ionic constituents of 1-propyl-3-methyl-imidazolium-bis-(trifluormethylsulfonyl)-imide (PT), a paradigmatic ionic liquid, as an additive in poly(vinylpyrrolidone) (PVP). Hence, the resulting product can be regarded as a polymer electrolyte as well as an amorphous dispersion. Leveraging dielectric spectroscopy and oscillatory shear rheology, complemented by differential scanning calorimetry, the spectral shapes and the relaxation maps of the supercooled PVP-PT mixtures are accessed in their full compositional range. The study also presents dielectric and shear responses of neat PVP with a molecular weight of 2500 g mol-1. We discuss the plasticizing role of the PT additive and the decoupling between ionic dynamics and segmental relaxation in these mixtures. The extracted relaxation times, steady-state viscosities, and conductivities are employed to estimate the translational diffusivities of the ionic penetrants by means of the Stokes-Einstein, Nernst-Einstein, and Almond-West relations. While some of the estimated diffusivities agree with each other, some do not, pointing to the importance of the chosen hydrodynamic approximations and the type of response considered for the analysis. The present extensive dielectric, rheological, and calorimetric study enables a deeper understanding of relaxation and transport of ionic ingredients in polymers, particularly in the slow-dynamics regime which is difficult to access experimentally by direct-diffusivity probes.

Transport and Energetics of Carbon Dioxide in Ionic Liquids at Aqueous Interfaces.

A major hurdle in utilizing carbon dioxide (CO2) lies in separating it from industrial flue gas mixtures and finding suitable storage methods that enable its application in various industries. To address this issue, we utilized a combination of molecular dynamics simulations and experiments to investigate the behavior of CO2 in common room-temperature ionic liquids (RTIL) when in contact with aqueous interfaces. Our investigation of RTILs, [EMIM][TFSI] and [OMIM][TFSI], and their interaction with a pure water layer mimics the environment of a previously developed ultrathin enzymatic liquid membrane for CO2 separation. We analyzed diffusion constants and viscosity, which reveals that CO2 molecules exhibit faster mobility within the selected ILs compared to what would be predicted solely based on the viscosity of the liquids using the standard Einstein-Stokes relation. Moreover, we calculated the free energy of translocation for various species across the aqueous-IL interface, including CO2 and HCO3-. Free energy profiles demonstrate that CO2 exhibits a more favorable partitioning behavior in the RTILs compared to that in pure water, while a significant barrier hinders the movement of HCO3- from the aqueous layer. Experimental measurement of the CO2 transport in the RTILs corroborates the model. These findings strongly suggest that hydrophobic RTILs could serve as a promising option for selectively transporting CO2 from aqueous media and concentrating it as a preliminary step toward storage.