Stokes-Einstein Equation Research Articles

Breakdown of the Stokes-Einstein Equation for Solutions of Water in Oil Reverse Micelles.

An experimental study is presented for the reverse micellar system of 15% by mass polydisperse hexaethylene glycol monodecylether (C10E6) in cyclohexane with varying amounts of added water up to 4% by mass. Measurements of viscosity and self-diffusion coefficients were taken as a function of temperature between 10 and 45 °C at varying sample water loads but fixed C10E6/cyclohexane composition. The results were used to inspect the validity of the Stokes-Einstein equation for this system. Unreasonably small reverse average micelle radii and aggregation numbers were obtained with the Stokes-Einstein equation, but reasonable values for these quantities were obtained using the ratio of surfactant-to-cyclohexane self-diffusion coefficients. While bulk viscosity increased with increasing water load, a concurrent expected decrease of self-diffusion coefficient was only observed for the surfactant and water but not for cyclohexane, which showed independence of water load. Moreover, a spread of self-diffusion coefficients was observed for the protons associated with the ethylene oxide repeat unit in samples with polydisperse C10E6 but not in a sample with monodisperse C10E6. These findings were interpreted by the presence of reverse micelle to reverse micelle hopping motions that with higher water load become increasingly selective toward C10E6 molecules with short ethylene oxide repeat units, while those with long ethylene oxide repeat units remain trapped within the reverse micelle because of the increased hydrogen bonding interactions with the water inside the growing core of the reverse micelle. Despite the observed breakdown of the Stokes-Einstein equation, the temperature dependence of the viscosities and self-diffusion coefficients was found to follow Arrhenius behavior over the investigated range of temperatures.

Hydration Thermodynamics and Hydrodynamic (Stokes) Radius of the Lanthanide Ions

Many biochemical and physiological properties depend on the size of ions and the thermodynamic quantities of ion hydration. The diffusion coefficient (D) of lanthanide (III) ions (Ln+3) in solution assumed (1.558-1.618 ×10−9 m2 s−1) by Einstein–Smoluchowski relation. The association constant (KA) of Ln+3 ions was calculated (210.3-215.3 dm3 mole-1) using the Shedlovsky method, and the hydrodynamic radius calculated (1.515-1.569 ×10−10 m) by the Stokes-Einstein equation. The thermodynamic parameters (ΔGo, ΔSo) also calculated by used suitable relations, while ΔHo, values are obtained from the literature. ΔGo, for ion hydration, has negative values in the range (13.25-13.30 KJ/mole), and a negative ΔSo, results have been shown in the limit (11.016-12.506 KJ/ K. mole).

Investigating the Process and Mechanism of Molecular Transport within a Representative Solvent-Filled Metal-Organic Framework.

Effective permeation into, and diffusive mass transport within, solvent-filled metal-organic frameworks (MOFs) is critical in applications such as MOF-based chemical catalysis of condensed-phase reactions. In this work, we studied the entry from solution of a luminescent probe molecule, 1,3,5,7-tetramethyl-4,4-difluoroboradiazaindacene (BODIPY), into the 1D channel-type, zirconium-based MOF NU-1008 and subsequent transport of the probe through the MOF. Measurements were accomplished via in situ confocal fluorescence microscopy of individual crystallites, where the evolution of the fluorescence response from the crystallite was followed as functions of both time and location within the crystallite. From the confocal data, intracrystalline transport of BODIPY is well-described by one-dimensional diffusion along the channel direction. Varying the chemical identity of the solvent revealed an inverse dependence of probe-molecule diffusivity on bulk-solvent viscosity, qualitatively consistent with expectations from the Stokes-Einstein equation for molecular diffusion. At a more quantitative level, however, measured diffusion coefficients are about 100-fold smaller than expected from Stokes-Einstein, pointing to substantial channel-confinement effects. Evaluation of the confocal data also reveals a non-negligible mass transport resistance, i.e., surface barrier, associated with the probe molecule leaving the solution and permeating the exterior surface of the MOF. Permeation by the probe entails displacement of solvent from the MOF channels. The magnitude of the resistance increases with the size of the solvent molecule. This work draws attention to the importance of MOF structure, external-surface barriers, and solvent molecule identity to the overall transport process in MOFs, which should assist in understanding the performance of MOFs in applications such as condensed-phase heterogeneous catalysis.

Chloroantimonate electrochemistry in dichloromethane

Antimony is a technologically relevant element, which is present in many semiconductor materials. Electrodeposition of such materials offers a potential route for cheaper and less wasteful manufacturing, and is especially suited for micro- and nano-feature sizes with complex geometries. Previous work has shown the applicability of electrodeposition of p-block metals and metalloids from the weakly coordinating solvent dichloromethane, using halometallate precursors. Here we more thoroughly investigate the behaviour of the tetrabutylammonium chloroantimonate precursor, [TBA][SbCl4]. We use voltammetry at a stationary macroelectrode, rotating disc electrode and electrochemical quartz crystal microbalance, as well as microelectrodes and highlight the advantages of microelectrodes in this context. Using [TBA]Cl as a background electrolyte it is found that the diffusion coefficients calculated from the rotating disc electrode and microelectrodes are similar. Due to the possibility of mixed speciation with an excess of Cl−, tetrabutylammonium tetrafluoroborate, [TBA][TFB], is also used as a background electrolyte and it is found that the diffusion coefficient does not change. Using a modified form of the Stokes-Einstein equation, that takes into account the shape of the solute and the relative sizes of the solute and solvent, the diffusion coefficients of the antimony precursor and decamethylferrocene are consistent with their relative sizes. Electrodeposition onto large surface area platinum and titanium nitride substrates using either background electrolyte results in amorphous deposits of elemental antimony with similar morphology.

Estimation of the Diffusion Coefficient and Hydrodynamic Radius (Stokes Radius) for Inorganic Ions in Solution Depending on Molar Conductivity as Electro-Analytical Technique-A Review

Diffusion coefficient (D) and radiuses of inorganic charge ions (R) in different environments is of importance in scientific fields, explaining certain features of ions. D values were obtained using the Einstein-Smoluchowski equation depending on limiting molar conductivity (λo) of inorganic charge ions. Inorganic ions have D values at the range of 1−3×10−9 m2.s-1,except hydroxyl and hydrogen ion (5.2×10−9 and 9.1×10−9 m2.s-1, respectively). The hydrodynamic radius R for diffusing ions was calculated using the Stokes–Einstein equation based on D values. Inorganic ionshaveR, values at the range of 1−3×10−10 m, except hydroxyl and hydrogen ion (R=0.47×10−10 and 0.27×10−10 m, respectively).

Microbubbles Size Distribution at Different Palm Oil Mill Effluent (POME) Temperatures for Oil Recovery Study

Palm Oil Mill Effluent (POME) is the largest contributor of biomass from the palm oil milling industry. Conventional method of POME treatment using ponding system should be improved because of huge land resource requirement. In this study, microbubbles technology was applied to understand the recovery rate of residual oil from POME at different operating temperatures. Temperature for POME was set at 27 oC, 30 oC and 50 oC to determine the microbubble size distributions and characteristics at different POME temperature. At each temperature, the size of microbubbles was measured based on six size range; <10 μm, (11-20) μm, (21-30) μm, (31-40) μm, (41-50) μm and >50 μm. The results showed that at different temperatures, the microbubbles size distribution varies and the smallest group of microbubbles (<10μm) was generated at 50oC. According to Stoke-Einstein equation, at higher temperature, smaller size of microbubbles is generated because of the gas diffusion factor into liquid.

Fundamental investigation of the gas permeation mechanism of facilitated transport membranes with Co(salen)-containing ionic liquid as O2 carriers

Metal-containing ionic liquids (MCILs) composed of a cobalt(II)–salen complex and ionic liquid-based axial ligands (ligand ILs) can chemically absorb O2 and act as O2 carriers in facilitated transport membranes. O2 and N2 permeation tests using supported MCIL membranes with various MCILs revealed that the effective diffusion coefficients of N2 and O2–MCIL complexes did not follow to the trend expected by power law theories against viscosity (e.g. the Stokes–Einstein equation), because the MCILs are not in normal liquid state due to the strong intermolecular interaction, but instead showed a tendency predicted by the free volume theory. This means the MCILs behave as a solid rather than a liquid, and N2, O2, and the O2–MCIL complexes would diffuse through the free volume of the MCILs. The solubility selectivity and the diffusion selectivity of the supported MCIL membrane decreased and increased, respectively, with the increase on MCIL molecular wight (free volume). Thus, there was trade-off relationship between the solution selectivity and the diffusion selectivity. It was suggested that the most important criteria for designing MCILs for highly selective O2 permeation in supported MCIL membranes is the use of an O2-absorbable metal complex without the strong intermolecular interaction.

(Invited) Control of Diffusion Behavior of Metal Ions Close to Electrode By Using of Local Structure of Ionic Liquid Electrolyte

Ionic liquids (ILs) have attracted keen attention due to its applicability to various kinds of electrochemical devices. However, we still lack enough knowledge on the interface structure and spatial distribution of chemical species during the electrochemical processes. In our previous study, we succeeded in the observation of the spatial distribution of XPS spectra close to the electrode under the applied potential using scanning electrochemical photoelectron spectroscopy (Scanning EC-PES) we originally developed. Interestingly, the shape of the diffusion layer (The spatial distribution of the concentration of the metal ions) was largely different from the concave function usually supposed in the case of aqueous solution. By a numerical simulation, we revealed that the characteristic shape of the diffusion layer of IL electrolyte was formed due to the fact that the metal ions diffused by the fast hopping-like mechanism, in which the apparent diffusion coefficient increased with increasing of the concentration of 'holes' (vacancies formed between IL molecules) acting as hopping sites. Based on these results, we proposed the new diffusion model in which hopping diffusion and conventional diffusion (which follows Stokes-Einstein equation) coexist. The most prominent feature of our diffusion model is that the local structure of ionic liquid (hole density, arrangement of the holes and/or IL molecules) strongly affect the diffusion behavior of metal ions.On the other hand, it can be anticipated that the local structure of ILs can be systematically controlled by the IL species (e.g. the alkyl chain length of imidazole cation, shape and size of anion), additive ions and/or temperature of solution. By using of these techniques for controlling the local structure of ILs, we can strictly control the diffusion behavior of metal ions close to the electrode. In other words, we will be able to propose the new strategy for controlling the diffusion behavior of solutes in ionic liquid solutions close to the electrode. In this study, we investigated the factors for controlling the diffusion behavior of metal ion in IL electrolyte based on our proposing diffusion model, and revealed that the behavior of metal ions can be controlled by some factors mentioned above.

Kinetics of static immersed leaching of low-grade sea-type evaporites based on theoretical and experimental investigation of unsteady-state mass transfer

In the immersed leaching process of low-grade sea-type evaporate wastes, the relative proportions of the solute contents in the mining brine continuously change over time, which makes the composition of the brine from leaching unpredictable. This strange phenomenon has been lacking a reasonable explanation for many years. To investigate this issue, the immersed dissolution and diffusion experiments of four typical sea-type evaporites NaCl, Na2SO4·10H2O, MgCl2·6H2O, and MgSO4·7H2O were performed in laboratory and pilot scale, respectively. Based on Fick’s second law, Stokes–Einstein equation, Thomas equation, Jones–Dole equation etc., a mathematical model was established to describe the mass transfer process. The results show that the unsteady-state mass transfer of each salt component in static water has three stages: linear fast mass transfer before the first 720 h, nonlinear unsteady-state mass transfer from 720 h to 7000 h, and pseudo-linear slow mass transfer after 7000 h. In every stage, the diffusion of Na+ and Cl− was faster than that of Mg2+ and SO42−, respectively, but their local diffusion rates were changing at different magnitudes, and their local viscosity and local mass transfer coefficients also varied to a great extent and underwent dynamic changes. At a distance of 60 mm from the evaporite-brine interface, the saturation degree reached by NaCl is approximately 1.23, 1.50, and 1.58 times that of Na2SO4, MgCl2, and MgSO4 when the diffusion time is 720 h, respectively. The local mass transfer coefficients of NaCl can reach approximately 1.48, 3.01, and 2.85 times that of Na2SO4, MgCl2, and MgSO4, respectively. The aforementioned results reveal that the unconstant change of the ratios of solute contents of the leaching brine resulted from the dynamic change in the local mass transfer rate of each component. The difference in concentration between Na+ and Mg2+ ions becomes more apparent when the diffusion distance is below 90 mm within the first 120 h of the diffusion time. Under the comprehensive influence of these factors, when NaCl or MgSO4·7H2O is present in the mixed evaporites, as the leaching time increases, the Na+/Mg2+ ratio changes continuously in the leaching brine owing to variations in the dissolution rate and magnitude of changes of components such as NaCl and MgSO4. Therefore, brines with different solute content ratios can be obtained as the leaching process proceeds.

Fluorescence microscopy tracking of dyes, nanoparticles and quantum dots during growth of polymer spherulites

The distribution of additives and nanoparticles in a semicrystalline polymer is largely dependent on the complex hierarchical polymer morphology and the kinetics of its development. Here we show by in-situ fluorescence microscopy (FM), coupled with polarized optical and scanning electron microscopy, that a substantial fraction of the additive (Nile Red, NR) and even NR-labelled silica nanoparticles (NP) and quantum dots (QD) was pushed ahead of the growing spherulites during melt-crystallization. For the 35 and 200 nm NPs this was unexpected because their diffusion coefficient based on Stokes-Einstein equation should have been 2–3 orders of magnitude slower than the rate at which spherulites were growing. Another surprising finding was that much of the initially rejected NR and some QDs subsequently re-entered the spherulites through back-diffusion, posing the question why they were then rejected in the first place. The excessive initial rejection of NR and QDs and the unexpectedly rapid migration of NPs are both explained by the additive preferentially filling and being carried along by the polymer depletion zone in the melt ahead of the growing spherulite, and the high negative pressure in these zones. The effectiveness of FM in detecting minute cracks and cavities is also demonstrated. The results also show that the most severe clustering of additives occurs where spherulites did not nucleate, a problem preventable by the addition of nucleating agent.

Measurement of the hydrodynamic radii of PEE-G dendrons by diffusion spectroscopy on a benchtop NMR spectrometer.

Benchtop nuclear magnetic resonance (NMR) spectroscopy is a useful tool for the rapid determination of the self-diffusion coefficient and the hydrodynamic radius of dendrons. The self-diffusion coefficients of the first four generations of poly ethoxy ethyl glycinamide (PEE-G) dendrons are measured by diffusion-ordered spectroscopy (DOSY) on a benchtop NMR equipped with diffusion gradient coils. The hydrodynamic radii of the dendrons are calculated via the Stokes-Einstein equation. The effects of solvent and pH are determined with the hydrodynamic radius increasing with generation and decreasing upon neutralization of an acidic solution. These measurements provide valuable information for biological and pharmaceutical applications of dendrons.

A model for droplet heating and evaporation of water-in-oil emulsified fuel

Water-in-oil (W/O) emulsified fuel is a promising alternative fuel by inducing flash boiling of water at high temperature that can improve the atomization of fuel spray. The heating and evaporation process of emulsified fuel droplet is affected by the diffusion and coalescence of small dispersed water droplets in oil during the heating. In this study, a model is developed for the droplet heating and evaporation of W/O emulsified fuel with these key physics considered. The diffusivity of dispersed water droplets in oil is calculated using the Stokes-Einstein equation. The deactivation temperature of surfactant is selected as a criterion for the water coalescence since the water coalescence is due to the deactivation of surfactant. The process of water coalescence is simplified that the dispersed water droplets coalesce into one single water sub-droplet at the center of the oil droplet instantly as the droplet temperature reaches the deactivation temperature of surfactant. The model is validated against experimental data of single droplet under different heating temperatures, surfactant concentrations, and sizes of dispersed water droplets. Based on the proposed model, the effects of fuel properties and heating conditions on the droplet heating and evaporation are analyzed.

Transient cavity dynamics and divergence from the Stokes-Einstein equation in organic aerosol.

The diffusion of small molecules through viscous matrices formed by large organic molecules is important across a range of domains, including pharmaceutical science, materials chemistry, and atmospheric science, impacting on, for example, the formation of amorphous and crystalline phases. Here we report significant breakdowns in the Stokes–Einstein (SE) equation from measurements of the diffusion of water (spanning 5 decades) and viscosity (spanning 12 decades) in saccharide aerosol droplets. Molecular dynamics simulations show water diffusion is not continuous, but proceeds by discrete hops between transient cavities that arise and dissipate as a result of dynamical fluctuations within the saccharide lattice. The ratio of transient cavity volume to solvent volume increases with size of molecules making up the lattice, increasing divergence from SE predictions. This improved mechanistic understanding of diffusion in viscous matrices explains, for example, why organic compounds equilibrate according to SE predictions and water equilibrates more rapidly in aerosols.

Nanoparticle Tracking Analysis of β-Casein Nanocarriers.

Nanoparticle tracking analysis is an excellent tool for the characterization of mono- and polydisperse nanoparticle systems within the 10-2000nm size range. The suitability of this technique relies on its ability to track all particles in solution simultaneously based on their Brownian motion giving an accurate size distribution. The tracked rate of particle movement is related to the particle's hydrodynamic radius using the Stokes-Einstein equation for determining the size distribution. Here we describe the characterization β-casein nanocarriers encapsulating a model hydrophobic compound, 8-anilino-1-naphthalenesulfonic acid, and the natural bioactive curcumin using the Malvern NanoSight NS300. Utilizing both normal light scattering and fluorescent modes of the NS300 enabled the differentiation of particles that had encapsulated the two fluorescent molecules and provided an accurate size distribution of the nanocarriers.

Comparison of the abnormal diffusion characteristics of tumor cells

The random dynamics of two different tumor cell types on a substrate were observed under a microscope and analyzed based on three parameters—mean square displacement (MSD), turning angle distribution (TAD), and velocity autocorrelation function (VACF)—under well-controlled experimental conditions wherein only thermal energy existed. A two-dimensional analysis on the substrate showed linear MSD but a lowered diffusion coefficient compared to the prediction by Stokes–Einstein equation due to particle substrate interaction. Moreover, anomalous characteristics based on the MSD and TAD of the two different cells, A549 (adenocarcinomic human alveolar basal epithelial cells) and MCF-7 (breast cancer cell line; estrogen, progesterone receptors +, HER2 −), were observed, and the nonlinear MSD at a short time scale was detected. This result is attributed to the localization error from the bright-field image of a different cell taken, while the cell moved randomly on the substrate. Parameters related to random process and localization errors were employed to distinguish the two cell types. This study could be useful for the characterization of cellular properties and cell types.

How many ferrocene units of multi-ferrocenyl complexes can react with the electrode?

Ferrocenylcarboxylic acid Fe, Co, and Ni complexes were synthesized as model complexes to investigate how many ferrocene units of multi-ferrocenyl system can react with the electrode. The molecular structures of model complexes were characterized by X-ray single-crystal diffraction, the diffusion coefficient of the complexes and ferrocene was determined by the diffusion-ordered spectroscopy and Einstein–Stokes equation, the electrode reaction numbers of ferrocene units were determined by cyclic voltammetry and the Randles–Sevcik equation. A model of electrode reaction of multi-ferrocenyl system was proposed in this paper, which was used to explore how many ferrocene units of multi-ferrocenyl complexes can react with the electrode.

Digital holography interferometry for measuring the mass diffusion coefficients of N2 in RP-3 and RP-5 jet fuels

Purpose The purpose of this study is to measure the mass diffusion coefficient of nitrogen in jet fuel using digital holography interferometry for cost-effective designing and modeling of the aircraft tank inerting system. Design/methodology/approach The mass diffusion coefficients of N2 in RP-3 and RP-5 jet fuels were measured by digital holography interferometry at temperatures ranging from 278.15 to 343.15 K. The Arrhenius equation is used to adequately describe the relationship between mass diffusion coefficients and temperature. The viscosities of RP-3 and RP-5 jet fuels were also measured to examine the accuracy of the Stokes–Einstein model in calculating mass diffusion coefficients. Findings As temperature increases from 278.15 to 343.15 K, the mass diffusion coefficients increase 4.23-fold for N2 in RP-3 jet fuel and 5.13-fold for N2 in RP-5 jet fuel. The value of Dµ/T is not constant as the Stokes–Einstein equation expressed, but is a weak linear function of temperature. Practical implications A more accurate diffusion model is proposed by fitting the measured Dµ/T with the temperature and calculating the mass diffusion coefficients of N2 in RP-3 and RP-5 jet fuels within 10 per cent relative deviation. Originality/value A measurement system for mass diffusion coefficients of N2 in RP-3 and RP-5 jet fuels was constructed based on the digital holography interferometry. The mass diffusion coefficient can be expressed by a uniform polynomial function of temperature and viscosity.

Characterization of the non-Arrhenius behavior of supercooled liquids by modeling nonadditive stochastic systems.

The characterization of the formation mechanisms of amorphous solids is a large avenue for research, since understanding its non-Arrhenius behavior is challenging to overcome. In this context, we present one path toward modeling the diffusive processes in supercooled liquids near glass transition through a class of nonhomogeneous continuity equations, providing a consistent theoretical basis for the physical interpretation of its non-Arrhenius behavior. More precisely, we obtain the generalized drag and diffusion coefficients that allow us to model a wide range of non-Arrhenius processes. This provides a reliable measurement of the degree of fragility of the system and an estimation of the fragile-to-strong transition in glass-forming liquids, as well as a generalized Stokes-Einstein equation, leading to a better understanding of the classical and quantum effects on the dynamics of nonadditive stochastic systems.

Decoupling of Translational Diffusion from the Viscosity of Supercooled Water: Role of Translational Jump Diffusion.

Some experiments have witnessed gradual decoupling of viscosity from the translational self-diffusion of supercooled water with decreasing temperature. This indicates the breakdown of the Stokes-Einstein equation in supercooled water. While some theoretical and computer simulation studies indicated the jump translation of the molecules as a probable origin of the above decoupling, direct quantitative evidence is still lacking. Through a molecular dynamics (MD) simulation study, along with careful consideration of translational jump motion, we have found the most definite proof of increasing relevance of translational jump diffusion in the above decoupling phenomena. By separating the jump-only diffusion contribution from the overall diffusion of the water, we obtain the residual diffusion coefficient, which remains strongly coupled to the viscosity of the medium at the supercooled regime. These new findings can help to elucidate many experimental studies featuring molecular transport properties, where strong diffusion-viscosity decoupling is present.

Extraction of Reliable Molecular Information from Diffusion NMR Spectroscopy: Hydrodynamic Volume or Molecular Mass?

Measuring accurate translational self-diffusion coefficients (Dt ) by NMR techniques with modern spectrometers has become rather routine. In contrast, the derivation of reliable molecular information therefrom still remains a nontrivial task. In this paper, two established approaches to estimating molecular size in terms of hydrodynamic volume (VH ) or molecular weight (M) are compared. Ad hoc designed experiments allowed the critical aspects of their application to be explored by translating relatively complex theoretical principles into practical take-home messages. For instance, comparing the Dt values of three isosteric Cp2 MCl2 complexes (Cp=cyclopentadienyl, M=Ti, Zr, Hf), having significantly different molecular mass, provided an empirical demonstration that VH is the critical molecular property affecting Dt . This central concept served to clarify the assumptions behind the derivation of Dt =ƒ(M) power laws from the Stokes-Einstein equation. Some pitfalls in establishing log (Dt ) versus log (M) linear correlations for a set of species have been highlighted by further investigations of selected examples. The effectiveness of the Stokes-Einstein equation itself in describing the aggregation or polymerization of differently shaped species has been explored by comparing, for example, a ball-shaped silsesquioxane cage with its cigar-like dimeric form, or styrene with polystyrene macromolecules.