Stokes-Einstein Research Articles

Dynamics of methanol in ionic liquids: validity of the Stokes-Einstein and Stokes-Einstein-Debye relations.

The validity of Stokes-Einstein (SE) and Stokes-Einstein-Debye (SED) relations for methanol in the physical environment of the ionic liquid (IL) 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide is studied by means of nuclear magnetic resonance (NMR) relaxation time experiments, viscosity measurements and molecular dynamics (MD) simulations. The reorientational correlation times of the hydroxyl groups of pure methanol and of methanol in the IL/methanol mixtures were determined. For that purpose an approach for estimating NMR deuteron quadrupole coupling constants, presented by Wendt and Farrar (Mol. Phys. 1998, 95, 1077-1081), was confirmed. The self-diffusion coefficients of methanol were taken from the MD simulations. The viscosities of all systems were then measured and the SE and SED relations validated. For pure methanol both relations are valid, whereas they become increasingly invalid with increasing IL concentration, as indicated by effective volumes and radii that are too low. The deviation from the SE and SED relations could be related to dynamical heterogeneities described by the non-Gaussian parameter α(t) obtained from MD simulations. For pure methanol, α(t) is close to zero in accord with the validity of both relations. With increasing IL concentration the dynamical heterogeneities of methanol increase strongly. The times t* at the maximum of α(t) increase linearly with the relative number of methanol monomers in the mixtures. Thus, the dynamical heterogeneities are largest for single methanol molecules fully embedded in the IL environment. In their own environment methanol molecules are highly mobile, whereas in the IL-rich region the mobility is strongly reduced leading to the non-validity of SE and SED relations.

Effects of the solvation structure on diffusion of a large particle in a binary mixture studied by perturbation theory

We study the effects of the solvation structure on the diffusion of a large particle in a binary mixture. Using our recently developed perturbation theory, we calculate the diffusion coefficient of a large hard-sphere solute particle immersed in a binary solvent mixture of hard spheres with two different sizes. The calculation results show that the Stokes–Einstein (SE) relation breaks down in the hard-sphere system. When the size ratio of binary solvent spheres is three or more, the deviation from the SE relation increases with the packing fraction of larger solvent spheres. In contrast, at the size ratio of two, the diffusion coefficient approaches the value predicted by the SE relation as larger solvent spheres are added. We show that the large deviation from the SE relation is caused by the high density of larger solvent spheres around the solute sphere.

Perturbation Theory of Large-Particle Diffusion in a Binary Solvent Mixture

We study the diffusion of a large spherical particle immersed in a binary compressive liquid mixture using a perturbation theory. We focus on the breakdown of the Stokes–Einstein (SE) relation caused by the microscopic solvation structure of binary solvent particles around a solute particle. In order to consider the solvation structure, we solve multicomponent generalized Langevin equations by singular perturbation expansion. Then, we assume that solvent particles are much smaller than the solute particle. Solving the equations, we express the diffusion coefficient analytically using the radial distribution functions of a binary mixture. The expression shows the breakdown of the SE relation if the density distribution of a binary solvent is inhomogeneous around a solute particle. Actually, we show that the SE relation breaks down when a large hard sphere diffuses in a binary hard-sphere mixture. We observe the large deviation from the SE relation, which is a result specific to the binary solvent.

Distribution of diffusion constants and Stokes-Einstein violation in supercooled liquids.

It is widely believed that the breakdown of the Stokes-Einstein (SE) relation between the translational diffusivity and the shear viscosity in supercooled liquids is due to the development of dynamic heterogeneity, i.e., the presence of both slow and fast moving particles in the system. In this study we directly calculate the distribution of the diffusivity for a model system for different temperatures in the supercooled regime. We find that with decreasing temperature, the distribution evolves from Gaussian to bimodal indicating that on the time scale of the typical relaxation time, mobile (fluid like) and less mobile (solid like) particles in the system can be unambiguously identified. We also show that less mobile particles obey the Stokes-Einstein relation even in the supercooled regime and it is the mobile particles which show strong violation of the Stokes-Einstein relation in agreement with the previous studies on different model glass forming systems. Motivated by some of the recent studies where an ideal glass transition is proposed by randomly pinning some fraction of particles, we then studied the SE breakdown as a function of random pinning concentration in our model system. We showed that degree of SE breakdown increases quite dramatically with increasing pinning concentration, thereby providing a new way to unravel the puzzles of SE violation in supercooled liquids in greater details.

Transport and Electrochemical Properties of Three Quaternary Ammonium Ionic Liquids and Lithium Salts Doping Effects Studied by NMR Spectroscopy

Ionic liquids (IL) composed of a quaternary ammonium cation having an ether chain, N,N-diethyl-N-methyl-N-(2-methoxyethyl)ammonium (DEME) are electrochemically stable and used in electric double layer capacitors and also one of important candidates to use lithium secondary batteries. In this study, three DEME-based ILs with anions BF4, CF3BF3, and N(SO2CF3)2 were studied by measuring temperature dependences of ionic conductivity σ, viscosity η and density ρ. Also ion diffusion coefficients Danion and Dcation were obtained by NMR spectroscopy in the wide temperature range. Using the classical Stokes–Einstein (SE) and Nernst–Einstein (NE) equations, the relationships between η and D, and σ and D were evaluated. The lithium salt doping effects were studied by 7Li NMR spectroscopy. The lithium ion diffusion was slower than other ion diffusion at every temperature. Arrhenius-type plots of 7Li T1 showed minima in the three doped samples. Then one-jump distance of lithium ion was estimated in the temperature range between 273 and 373 K.

Electrical, thermal, and species transport properties of liquid eutectic Ga-In and Ga-In-Sn from first principles

Using ab initio molecular dynamics, the atomic structure and transport properties of eutectic Ga-In and Ga-In-Sn are investigated. The Kubo-Greenwood (K-G) and the Ziman-Faber (Z-F) formulations and the Wiedemann-Franz (W-F) law are used for the electrical and electronic thermal conductivity. The species diffusivity and the viscosity are also predicted using the mean square displacement and the Stokes-Einstein (S-E) relation. Alloying Ga causes more disordered structure, i.e., broadening the atomic distance near the In and Sn atoms, which reduces the transport properties and the melting temperature. The K-G treatment shows excellent agreement with the experimental results while Z-F treatment formula slightly overestimates the electrical conductivity. The predicted thermal conductivity also shows good agreement with the experiments. The species diffusivity and the viscosity are slightly reduced by the alloying of Ga with In and Sn atoms. Good agreements are found with available experimental results and new predicted transport-property results are provided.

Single molecule confocal fluorescence lifetime correlation spectroscopy for accurate nanoparticle size determination.

We report on an experimental procedure in confocal single molecule fluorescence lifetime correlation spectroscopy (FLCS) to determine the range of excitation power and molecular or particulate concentration in solution under which the application of an unmodified model autocorrelation function is justified. This procedure enables fitting of the autocorrelation to an accurate model to measure diffusion length (r) and diffusion time (τD) of single molecules in solution. We also report on the pinhole size dependency of r and τD in a confocal FLCS platform. This procedure determines a set of experimental parameters with which the Stokes-Einstein (S-E) equation accurately measures the hydrodynamic radii of spherical nanoparticles, enabling the determination of the particle size range for which the hydrodynamic radius by the S-E equation measures the real particle radius.

A relationship between diffusivity and cooperativity of supercooled liquids in the proximity of glass transition

In structurally disordered glass-forming liquids, the Stokes–Einstein (SE) law for translational diffusivity is violated, in particular, in the proximity of the glass transition temperature. Usually, the violation of the SE law is explained as due to the decoupling of the diffusivity from the viscous flow. Meanwhile, one of the prominent features observed in the supercooled liquids is the drastic increase of the viscosity with the decrease of temperature. The origin of the increase of the viscosity is attributed to the increase of the molecular motion cooperativity. In the present study, it is shown that the translational diffusivity of glass-forming liquids can be correlated with the cooperativity involved in the thermally activated viscous flow. In the present analysis, the cooperativity is defined in the framework of the bond strength–coordination number fluctuation (BSCNF) model of the viscosity. The result of the analysis suggests that the translational diffusivity is driven by the bond-breaking and bond-switching between the structural units that compose the network structure.

Structure, equation of state, diffusion and viscosity of warm dense Fe under the conditions of a giant planet core

Fe exists abundantly in the universe. In particular, the dynamic structures and transport properties of warm dense Fe are crucial for understanding the evolution and structures of giant planets. In this paper, we present the ionic structures, equation of states, diffusion and viscosity of Fe at two typical densities of 33.385 and 45 g cm−3 in the temperature range of 1–10 eV, giving the data by the first principles calculations using quantum Langevin molecular dynamics. Furthermore, the validation of the Stokes–Einstein (SE) relation in this regime is discussed, showing the importance of choosing the effective atomic diameter. The results remind us of the careful usage of the SE relation under extreme conditions.

Breakdown of the Stokes-Einstein relation in two, three, and four dimensions

The breakdown of the Stokes-Einstein (SE) relation between diffusivity and viscosity at low temperatures is considered to be one of the hallmarks of glassy dynamics in liquids. Theoretical analyses relate this breakdown with the presence of heterogeneous dynamics, and by extension, with the fragility of glass formers. We perform an investigation of the breakdown of the SE relation in 2, 3, and 4 dimensions in order to understand these interrelations. Results from simulations of model glass formers show that the degree of the breakdown of the SE relation decreases with increasing spatial dimensionality. The breakdown itself can be rationalized via the difference between the activation free energies for diffusivity and viscosity (or relaxation times) in the Adam-Gibbs relation in three and four dimensions. The behavior in two dimensions also can be understood in terms of a generalized Adam-Gibbs relation that is observed in previous work. We calculate various measures of heterogeneity of dynamics and find that the degree of the SE breakdown and measures of heterogeneity of dynamics are generally well correlated but with some exceptions. The two-dimensional systems we study show deviations from the pattern of behavior of the three- and four-dimensional systems both at high and low temperatures. The fragility of the studied liquids is found to increase with spatial dimensionality, contrary to the expectation based on the association of fragility with heterogeneous dynamics.

Test of Stokes-Einstein Formula for a Tracer in a Mesoscopic Solvent by Molecular Dynamics Simulation

particles interact through multi-particle collision events which take place at discrete time intervals. Between such collision events the particles undergo free streaming motion. The dynamics conserves mass, momentum and energy and yields the exact hydrodynamic equations of motion for the conserved fields on long distance and time scales. One may consider the dynamics of solute molecules in this mesoscopic solvent and because the solvent is described at an effective particle level the solute and solvent molecules interact through intermolecular forces rather than through boundary conditions. This leads to a hybrid description of the dynamics where solute molecules evolve by Newton’s equations of motion but the solvent evolves through the multi-particle mesoscale dynamics. 2,3 Calculations of the diffusion and friction coefficients by molecular dynamics (MD) simulations are well-studied problems that have been addressed many times. Nevertheless, the computation of the friction coefficient from MD simulations involves a number of subtle issues for finite-size systems as discussed in several recent papers. 4-6 The problems center around the definition of the friction coefficient in terms of the projected dynamics and its relation to the fixed-particle friction coefficient for a massive Brownian particle. The estimates of the friction coefficients have been shown to depend on the order in which the mass of the tracer and the solvent particle number N are taken to infinity. For finite-size systems one must investigate how large N must be to obtain a reliable estimate of the friction; this typically requires very large MD simulations. Similarly, the estimate of the diffusion coefficient from the velocity autocorrelation function requires large scale simulations, especially for large tracers, due to the importance of hydrodynamic contributions. In recent papers, 7-9 the friction and diffusion coefficients of a tracer in a Lennard- Jones (LJ) solvent were evaluated by equilibrium molecular dynamics simulations in a microcanonical ensemble. The solvent molecules of diameter σ1 interact with each other through a repulsive LJ force and the tracer of diameter σ2 interacts with the solvent molecules through the same repulsive LJ force except a different LJ size or diameter parameter. These works were motivated by determination of the diffusion (D) and friction (ζ) coefficients of the tracer in the thermodynamic limit (N→∞) and by test of the Stokes-Einstein (SE) formula: D =

Probe diffusion of single-walled carbon nanotubes in semidilute solutions of polyacrylonitrile homo- and copolymers: Effects of topological constraints and polymer/Nanorod interactions

With varying the molecular weight, copolymer chemical structure and concentration, the probe diffusion behavior of single-walled carbon nanotubes (SWCNTs) in N, N-dimethylacetamide (DMAc) solutions of polyacrylonitrile (PAN) was studied by dynamic light scattering (DLS) technique. In combination with the systematic viscosity property investigation for PAN/DMAc system, the probe diffusion results provided insight to understand the interactions between PAN and SWCNTs. It was found that, due to depletion interaction, the SWCNT diffusion positively deviates from the Stokes–Einstein (SE) relation in PAN homopolymer solutions. However, in PAN copolymer solution, the SWCNT diffusion mostly obeys the SE relation, which was speculated as a result of hydrogen-bonding formation between the carboxylic acid functional groups in PAN copolymers and on purified SWCNTs. Moreover, due to the unique features of SWCNT probes – very long, extremely rigid, and adjustable bundle diameter, the SWCNT probe diffusion unambiguously revealed the topological constraints of polymer network responsible for the retardation of thin rod diffusion. Quantitative analysis indicated that the correlation length rather than the “tube” diameter of polymer liquid is the critical length scale to dictate the diffusion retardation of SWCNTs in polymer solutions.

Multinuclear NMR Studies on Translational and Rotational Motion for Two Ionic Liquids Composed of BF4 Anion

Two ionic liquids (ILs) based on the BF(4)(-) anion are studied by (1)H, (11)B, and (19)F NMR spectroscopy by measuring self-diffusion coefficients (D) and spin-lattice relaxation times (T(1)). The cations are 1-ethyl-3-methylimidazolium (EMIm) and 1-butyl-3-methylimidazolium (BMIm). Since two NMR nuclei ((11)B and (19)F) of BF(4)(-) exhibit narrow lines and high sensitivity, the (11)B and (19)F NMR measurements of D(BF4) and T(1)(BF(4)) were performed in a wide temperature range. The temperature-dependent behaviors of T(1)((19)F) and T(1)((11)B) were remarkably different, although the values of D(BF4)((19)F) and D(BF4)((11)B) almost agreed. Since the Arrhenius plots of T(1)'s for (1)H, (19)F, and (11)B exhibited T(1) minima, the correlation times τ(c)((1)H), τ(c)((19)F), and τ(c)((11)B) were evaluated. The D(cation) and D(BF(4)) were plotted against 1/τ(c)((1)H) and 1/τ(c)((19)F), respectively, and the relationships between translational and rotational motion are discussed. The translational diffusion of the cations is related to molecular librational motion and that of BF(4) is coupled with reorientational motion. The τ(c)((11)B) derived from (11)B T(1) can be attributed to a local jump. From the plots of the classical Stokes-Einstein (SE) equation, the empirical c values, which were originally derived by theoretical boundary conditions, were estimated for each ion. The empirical c(BF(4)) was about 4.4(5), while the c values of the cations were smaller than 4.

Does polar interaction influence medium viscosity? A computer simulation investigation using model liquids

Molecular dynamics simulations of model liquids interacting via Lennard–Jones (L–J) and Stockmayer (SM) interactions have been carried out to explore the effects of the longer-ranged dipole–dipole interaction on solvent viscosity and diffusion. Switching on of the dipolar interaction at a fixed density and temperature has been found to increase the viscosity over that of the LJ liquid, the extent of increase being a few percent to as large as ∼60% depending on the magnitude of the solvent dipole moment used in the SM potential. The simulated translational and rotational diffusion coefficients show strong dipole moment and temperature dependences, eventhough effects of these parameters on solvent–solvent radial distribution function are moderate. Interestingly, a partial solute–solvent decoupling is observed when the simulated translational and rotational diffusion coefficients are connected to the simulated viscosity coefficients via the Stokes–Einstein (SE) and Stokes–Einstein–Debye (SED) relations. In the limit of large dipole moment, simulated self-part of the van Hove correlation function at intermediate times reveals a departure from the Gaussian distribution with particle displacement. This suggests that dynamic heterogeneity is one of the reasons for the departure of centre-of-mass diffusion from the SE relation in these model systems.

Communication: Fast and local predictors of the violation of the Stokes-Einstein law in polymers and supercooled liquids

The violation of the Stokes-Einstein (SE) law is investigated in a melt of linear chains by extensive molecular-dynamics simulations. It is found that the SE breakdown is signaled (with 5% uncertainty) by the monomer mean-square displacement <u(2)> on the picosecond time scale. On this time scale the displacements of the next-next-nearest neighbors are uncorrelated. It is shown that: (i) the SE breakdown occurs when <u(2)> is smaller than the breadth of the distribution of the square displacements to escape from the first-neighbors cage, (ii) the dynamical heterogeneity affects the form of the master curve of the universal scaling between the structural relaxation and <u(2)>.

Local Structure Evolution and its Connection to Thermodynamic and Transport Properties of 1-Butyl-3-methylimidazolium Tetrafluoroborate and Water Mixtures by Molecular Dynamics Simulations

Our recently developed improved united atom force field shows a good quality to reproduce both the static and transport properties of neat ionic liquids (ILs). Combined with the TIP4P-Ew water model, the force field is used to simulate the mixture of 1-butyl-3-methylimidazolium tetrafluoroborate ([C(4)mim][BF(4)]) and water without further optimization to adjust any cross parameters. Liquid densities of the mixture are well predicted over the entire concentration range at temperatures from 298.15 to 353.15 K. Simulations also reproduce the positive values of excess volumes and excess enthalpies, as well as their increase with temperature. The simulated viscosities are in good agreement with experimental values, especially in the water-rich region. We found three distinct regions by analyzing the concentration dependent self-diffusion coefficients via Stokes-Einstein (SE) relation, indicating the mixture experiences significant microheterogeneity with the adding of water. This observation is well connected to the structure features obtained in simulations, such as radial distribution functions (RDFs), spatial distribution functions (SDFs) and water clustering analysis. At the water mole fraction (x(2)) less than 0.2, most of the water molecules are isolated in the polar cation-anion network in ionic liquids. With the increase of x(2) from 0.2 to 0.8, large water cluster forms and eventually percolates the whole system. When x(2) > 0.8, ionic liquids show a moderate degree of aggregation (with maximum around 0.9 to 0.95) before the cations and anions are fully dissolved in water.

Theory of nanoparticle diffusion in unentangled and entangled polymer melts

We propose a statistical dynamical theory for the violation of the hydrodynamic Stokes-Einstein (SE) diffusion law for a spherical nanoparticle in entangled and unentangled polymer melts based on a combination of mode coupling, Brownian motion, and polymer physics ideas. The non-hydrodynamic friction coefficient is related to microscopic equilibrium structure and the length-scale-dependent polymer melt collective density fluctuation relaxation time. When local packing correlations are neglected, analytic scaling laws (with numerical prefactors) in various regimes are derived for the non-hydrodynamic diffusivity as a function of particle size, polymer radius-of-gyration, tube diameter, degree of entanglement, melt density, and temperature. Entanglement effects are the origin of large SE violations (orders of magnitude mobility enhancement) which smoothly increase as the ratio of particle radius to tube diameter decreases. Various crossover conditions for the recovery of the SE law are derived, which are qualitatively distinct for unentangled and entangled melts. The dynamical influence of packing correlations due to both repulsive and interfacial attractive forces is investigated. A central finding is that melt packing fraction, temperature, and interfacial attraction strength all influence the SE violation in qualitatively different directions depending on whether the polymers are entangled or not. Entangled systems exhibit seemingly anomalous trends as a function of these variables as a consequence of the non-diffusive nature of collective density fluctuation relaxation and the different response of polymer-particle structural correlations to adsorption on the mesoscopic entanglement length scale. The theory is in surprisingly good agreement with recent melt experiments, and new parametric studies are suggested.

Diffusion of the 35S isotope in soda–lime–silica and sodium trisilicate glass melts

The diffusivity of sulphur in nominal 10 Na2O–16 CaO–74 SiO2 (NCS) and 26 Na2O–74 SiO2 (NS3) melts was investigated in the temperature range 1273–1473K using the 35S radioactive isotope in a sandwich setup. Samples were sealed in platinum capsules and run with vertical alignment at 100MPa confining pressure in an internally heated gas pressure vessel. Using the lowest diffusion coefficient D (m2s−1) for each temperature the Arrhenian relations logD=−(4.6±0.3)−(216±7)kJ mol−1/RT for NCS and logD=−(6.3±0.6)−(167±17)kJ mol−1/RT for NS3 were determined. Viscosity of the melts was used to test the applicability of the Stokes–Einstein (SE) and the Eyring (EY) equations to sulphur diffusion. The SE equation yields unrealistically low radii of the diffusing particles, a consequent of the non-molecular structure of the silicate melts. On the other hand, the Eyring relation yields reasonable jump distances of 550pm (NCS) and 750pm (NS3) by fitting the diffusion data to the EY equation. These large values imply that sulphate ions (ionic diameter=290pm) migrate as large entities through the silicate network.

Temperature and Hydration-Dependent Rotational and Translational Dynamics of a Polyether Oligomer

Temperature-dependent rotational diffusion of tetraethylene glycol dimethyl ether (TEGDE) is measured by optical heterodyne-detected optical Kerr effect (OHD-OKE) spectroscopy and compared to previous measurements of rotational diffusion as a function of water content. Both types of data, temperature-dependent and hydration-dependent, follow the Debye-Stokes-Einstein (DSE) equation and agree quantitatively with hydrodynamic calculations. Of particular importance is the result that both types of data show nearly identical dependence on the viscosity divided by the temperature (η/T). We also compare the translational diffusion constants as previously measured by pulsed field gradient spin-echo (PFG-SE) NMR as a function of both temperature and water content. The temperature-dependent data follow the Stokes-Einstein (SE) equation. Similar to the rotation, the low water content mixtures obey the SE equation and show the same proportionality to η/T as the temperature-dependent data. At higher water fractions, the data do not obey the SE equation. The principal results are that the influence of temperature on dry TEGDE orientational relaxation is the same as the influence of water content at fixed temperature, and that the influence of temperature on translational diffusion of dry TEGDE is the same as the influence of water content over a range of relatively low water concentrations. The results demonstrate that there are no large TEGDE structural changes or specific, long-lived water-polyether interactions in the solutions over the entire concentration range.

Fractional Exponent of the Modified Stokes-Einstein Relation in the Metallic Glass-Forming Melt Pd&lt;sub&gt;43&lt;/sub&gt;Cu&lt;sub&gt;27&lt;/sub&gt;Ni&lt;sub&gt;10&lt;/sub&gt;P&lt;sub&gt;20&lt;/sub&gt;

The diffusion coefficient in the metallic glass-forming systems such as Pd-Cu-Ni-P exhibits a marked deviation from the Stokes-Einstein (SE) relation in the proximity of the glass transition temperature. Such a deviation is characterized by the fractional exponent p of the modified SE expression. For the material Pd43Cu27Ni10P20, it has been reported that it takes the value p = 0.75. In this work, it is shown that the value of p is highly correlated with the ratio ED / ENB, where ED and ENB are the activation energies for diffusion coefficient D and cooperativity NB defined by the Bond Strength-Coordination Number Fluctuation (BSCNF) model. The present paper reports that for the metallic glass-forming melt Pd43Cu27Ni10P20, the fractional exponent p can be calculated accurately within the framework of the BSCNF model.