Modified Stokes-Einstein Equation Research Articles

Diffusion in weakly coordinating solvents

Weakly coordinating solvents are of interest for the electrodeposition of p-block semiconductors for application in electronic devices. p-block complexes typically have weakly coordinated ligands that are easily displaced, making them incompatible with strongly Lewis basic solvents. In this work we use electrochemical measurements at microelectrodes to study diffusion in weakly coordinating solvents. Diffusion coefficients of the metallocenes decamethylferrocene, decamethylferrocenium hexafluorophosphate, cobaltocenium hexafluorophosphate, and the electrodeposition precursors tetrachloroantimonate(III) and tetrachlorobismuthate(III) were measured. The values are analyzed using the modified Stokes-Einstein equation and compared with the theoretical upper, Dmax, and lower, Dmin, bounds of the diffusion coefficients. This approach allows the interpretation of D values, whilst avoiding dealing with some of the uncertainties associated with molecular size in the Stokes-Einstein equation. The neutral decamethylferrocene was found to obey the Stokes-Einstein equation whereas the charged metallocene species had values which were less than the theoretical minimum, which was attributed to a larger than expected particle size caused by ion pairing. The importance of considering the modifications of the Stokes-Einstein equation is also highlighted.

Diffusion of Squalene in Nonaqueous Solvents.

Capillary flow techniques have been used to determine the translational diffusion constant, D, of squalene in seven alkanes and five cyclohexanes. The alkanes are n-hexane, n-octane, n-decane, n-dodecane, n-tetradecane, 2,2,4,4,6,8,8-heptamethylnonane (isocetane), and 2,6,10,14-tetramethylpentadecane (pristane). The cyclohexanes are cyclohexane, n-butylcyclohexane, n-hexylcyclohexane, n-octylcyclohexane, and n-dodecylcyclohexane. When combined with published data in CD2Cl2, ethyl acetate, n-hexadecane, squalane, n-octane–squalane mixtures, and supercritical CO2, the 35 diffusion constants and viscosities, η, vary by factors of ∼230 and ∼500, respectively. A fit to the modified Stokes–Einstein equation (MSE, D/T = ASE/ηp) gives an average absolute percentage difference (AAPD) of 7.72% between the experimental and calculated D values where p and ASE are constants, T is the absolute temperature, and the AAPD is the average value of (102) (|Dcalcd – Dexptl|/Dexptl). Two other MSE fits using subsets of the 35 diffusion constants may be useful for (a) estimating the viscosity of the hydrophobic core of lipid droplets, where squalene is a naturally occurring component, and (b) providing estimates of the D values needed to design extraction processes by which squalene is obtained from plant oils. The Wilke–Chang equation also was considered and found to give larger AAPDs than the corresponding MSE fits.

Density, viscosity, physical CO2 diffusivity, and CO2 absorption capacity of novel blended N-methyl-4-piperidinol and piperazine solvent

In the present work, novel blended piperazine (PZ) and N-methyl-4-piperidinol (MPDL) solvent was investigated in terms of density, viscosity, physical diffusivity of CO2, and CO2 absorption capacity. Density and viscosity were measured over PZ/MPDL concentration ratios of 5/25, 10/20, and 15/15 %wt. and temperatures of 313, 323, and 333 K. Physical diffusivity of CO2 was calculated based on viscosity data by the modified Stokes–Einstein equation. Lastly, CO2 absorption capacity (mol CO2/mol amine) was experimentally determined in a temperature-controlled absorption reactor at 313 K and 10% v/v CO2. The results showed that density and viscosity of novel blended PZ-MPDL solvent increased as PZ concentration ratio increased and decreased as temperature increased. On another hand, physical diffusivity of CO2 decreased as PZ concentration ratio increased and increased as temperature increased. Based on the physical properties data (i.e., density, viscosity, and physical diffusivity of CO2), it can be summarized that the studied physical properties of PZ-MPDL were in the same ranges with those of conventional amine solvents. Additionally, it was found that CO2 absorption capacity of PZ-MPDL can be improved by increasing PZ concentration ratio. As a result, 15/15 %wt. PZ-MPDL showed the highest CO2 absorption capacity (0.741 mol CO2/mol amine) among the three studied concentrations. It also possessed 49% higher CO2 absorption capacity than the conventional benchmarking 30 %wt. monoethanolamine (MEA).

Experimental data and modeling for density and viscosity of carbon dioxide (CO2)-loaded and -unloaded aqueous blend of 2-(ethylamino)ethanol (EAE) and aminoethylethanolamine (AEEA) for post-combustion CO2 capture

Amine based chemical absorption is the most developed technique for post-combustion CO2 capture from flue gases of low CO2 partial pressure. Density and viscosity data of CO2 – loaded and –unloaded absorbent are important in kinetics study and design the absorption column. Density and viscosity of CO2 loaded and –unloaded aqueous blend of 2-(ethylamino)ethanol (EAE) + aminoethylethanolamine (AEEA) were obtained experimentally in the temperature range of 293.15 to 323.15 K with 5 K temperature interval at atmospheric pressure. Concentration of aqueous EAE + AEEA blend was 10 wt%, 20 wt%, and 30 wt% with 7/3 weight ratio of EAE/AEEA. Excess volume was calculated by using experimental density data and correlated with Redlich-Kister type equation. Correlations were developed to calculate density and viscosity. For CO2-loaded and –unloaded aqueous EAE + AEEA blend, correlations predicted data were with average absolute deviation percentage (AAD %) of 0.1286, 0.0208, respectively for density while with AAD % 4.744, 4.433, respectively for viscosity. Wieland model was also correlated to CO2-loaded viscosity data and AAD % was 2.852 for this model. Moreover, diffusivity of CO2 into the aqueous EAE + AEEA blend was calculated using modified Stokes-Einstein equation.

Diffusion coefficients of eucalyptol at infinite dilution in compressed liquid ethanol and in supercritical CO2/ethanol mixtures

Tracer diffusivities, D12, of eucalyptol in liquid ethanol and in supercritical carbon dioxide (SC-CO2) mixed with ethanol (8.0wt.%) were measured in the temperature range 303.15–333.15K, using the chromatographic peak broadening technique. The obtained diffusivities ranged from 0.912×10−5 to 1.578×10−5cm2s−1 for the binary system (pressure range 1–100bar) and from 0.547×10−4 to 1.042×10−4cm2s−1 for the ternary system (pressure range 150–275bar). The dependence of D12 on temperature, pressure, solvent density, and Strokes-Einstein coordinates were deeply examined. Such trends were interpreted and discussed in terms of free volume theories and hydrodynamic fundamentals. Various prediction/correlation models were assessed to calculate D12, with the best results (deviations of 1.20–7.00%) obtained with Dymond–Hildebrand–Batschinski equation, the Tracer Liu-Silva-Macedo (TLSMd) model, four correlations of Magalhães et al., and a modified Stokes-Einstein equation. Lai-Tan and Wilke-Chang correlations attained reliable results (errors of 8.36 and 8.00%) in the case of eucalyptol/SC-CO2/ethanol system.

Transport and Surface Properties of Liquid Metals using their Diffusion Data

The atomic diffusion in liquid Na, K, Cs, Mg, Al, In and Pb have been evaluated from the well-known Einstein’s formula of self-diffusion coefficient, D  D  kBT ⎞ under square well (SW) interaction. The friction coefficient, ( of liquid    ⎝ ⎠ metals has been computed on the basis of Helfand-Rice-Nachtrieb approach under Helfand’s linear trajectory (LT) approximation. Shear viscosity of liquid metals were determined using modified Stokes-Einstein equation for SW potential. Dzugutov’s scaling was modified for SW interaction and has been employed to compute excess entropy of the considered liquids. The isothermal compressibility of liquid metals was determined using equation of state of SW potential. Surface entropy of these liquids has been determined through temperature derivative of surface tension. Dzugutov’s scaling law has also been tested in these liquid metals. It is found that the various transport and surface properties of these liquid metals extracted from the diffusion coefficients are in a good agreement with the experimental data.

Development of low viscous ionic liquids: the dependence of the viscosity on the mass of the ions

Straightforward syntheses of selected non-deuterated, partially deuterated, and fully deuterated room temperature ionic liquids (ILs) with the anions tris(pentafluoroethyl)trifluorophosphate, tetracyanoborate, and methyl sulfate were developed. The viscosity and the density of these ionic liquids were measured at various temperatures. The dissociation rate of the ionic liquid, the sizes of the cation and the anion, and the nature of the inter-ionic interaction are not influenced by deuteration, but viscosity is. All deuterated ionic liquids possess a higher viscosity and density in comparison to non-deuterated ones. The explanation of this phenomenon is given and discussed based on the modified Stokes-Einstein equation with a parameter which reflects the mass of the diffused particles. This new knowledge supports the recent development of novel low viscous inert ionic liquids which are of particular interest for the application in energy conversion devices. Deuterated ionic liquids are valuable media for the investigation of chemical reactions, natural products, and bio-materials by means of the NMR spectroscopy.

The Diffusion of Tc-99 in Beishan Granite-Temperature Effect

In the safety assessment of a potential site for high-level radioactive wastes (HLW) disposal, the investigation on the geochemical behaviors of key radionuclides with the possibility for releasing from the potential repository is an important aspect. Due to the high mobility of technetium under most repository conditions, lots of research works were performed to investigate the diffusion of technetium in different potential rocks. In spite of these studies, there remains a lack of data addressing temperature effects. In this paper, the diffusion of 99Tc in Beishan granite at temperatures from 25℃ to 55℃ was studied with laboratory small scale diffusion devices. The experimental data were fitted with a finite difference scheme to get the effective diffusion coefficient (De) of . The results indicated that the relationship of De with temperatures could be described as the modified Stokes-Einstein equation, and the formation factor of Beishan granite was constant in the temperature range of 25℃ - 55℃ with the value of (3.91 ± 1.77) × 10-4.

Measurements and Correlations of Diffusivities of Nitrous Oxide and Carbon Dioxide in Monoethanolamine + Water by Laminar Liquid Jet

The molecular diffusivities of nitrous oxide (N2O) with aqueous monoethanolamine (MEA) solutions up to 12 M were studied over a temperature range from 298.15 to 333.15 K under atmospheric pressure using a laminar liquid jet absorber. The diffusivities of CO2 in aqueous MEA solutions were calculated by the “N2O analogy” method. A simple and effective thermal control technique was used to control the temperatures of gas and liquid in the laminar liquid jet absorber. The rates of absorption were determined by measuring the flow of gas needed to replace the gas absorbed. The results showed that the diffusivities of both N2O and CO2 into aqueous MEA solution decrease with the increase of the concentration of MEA, and increase with an increase of the temperature of the solution. The relationship between the diffusivity and the viscosity of the solution roughly agrees with the modified Stokes–Einstein equation, but an exponent mathematical model was employed to simulate the diffusivity data and shows a better ag...

A modified Stokes-Einstein equation for Aβ aggregation

BackgroundIn all amyloid diseases, protein aggregates have been implicated fully or partly, in the etiology of the disease. Due to their significance in human pathologies, there have been unprecedented efforts towards physiochemical understanding of aggregation and amyloid formation over the last two decades. An important relation from which hydrodynamic radii of the aggregate is routinely measured is the classic Stokes-Einstein equation. Here, we report a modification in the classical Stokes-Einstein equation using a mixture theory approach, in order to accommodate the changes in viscosity of the solvent due to the changes in solute size and shape, to implement a more realistic model for Aβ aggregation involved in Alzheimer’s disease. Specifically, we have focused on validating this model in protofibrill lateral association reactions along the aggregation pathway, which has been experimentally well characterized.ResultsThe modified Stokes-Einstein equation incorporates an effective viscosity for the mixture consisting of the macromolecules and solvent where the lateral association reaction occurs. This effective viscosity is modeled as a function of the volume fractions of the different species of molecules. The novelty of our model is that in addition to the volume fractions, it incorporates previously published reports on the dimensions of the protofibrils and their aggregates to formulate a more appropriate shape rather than mere spheres. The net result is that the diffusion coefficient which is inversely proportional to the viscosity of the system is now dependent on the concentration of the different molecules as well as their proper shapes. Comparison with experiments for variations in diffusion coefficients over time reveals very similar trends.ConclusionsWe argue that the standard Stokes-Einstein’s equation is insufficient to understand the temporal variations in diffusion when trying to understand the aggregation behavior of Aβ42 proteins. Our modifications also involve inclusion of improved shape factors of molecules and more appropriate viscosities. The modification we are reporting is not only useful in Aβ aggregation but also will be important for accurate measurements in all protein aggregation systems.

Diffusion of Organic Solutes in Squalane

The translational diffusion constants, D, of 26 hydrocarbons have been determined in squalane (2,6,10,15,19,23-hexamethyltetracosane) at room temperature using capillary flow techniques. These new data and previously published room-temperature D values for the same solutes in some (or all) of the n-alkanes n-C(6)-n-C(16) constitute a study of solute diffusion in media spanning a 100-fold change in viscosity; at 23 °C, η = 0.31 cP for n-C(6), 3.2 cP for n-C(16), and 30 cP for squalane. The D values in the n-alkanes and squalane show deviations from the Stokes-Einstein relation, D = k(B)T/(6πηr); the values of r, a solute's hydrodynamic radius, decrease as the viscosity increases. The deviations increase as the solute size decreases and are analyzed by fitting the diffusion constants to the modified Stokes-Einstein equation, D/T = A(SE)/η(p). Fits involving the n-alkane-only and combined n-alkane-squalane D values give comparable results with values of p < 1 that increase as the solute size increases; p = 1 for the Stokes-Einstein limit. The deviations from Stokes-Einstein behavior also are discussed in terms of the relative sizes of the solutes, the n-alkanes, and squalane.

Diffusion NMR studies on neutral and cationic square planar palladium(II) complexes

Diffusion NMR investigations were carried out in CD 2Cl 2 for a series of neutral ( 1– 7) and cationic ( 8– 10) square planar palladium complexes. Diffusion data were elaborated through a modified Stokes–Einstein equation that takes into account the size and shape of molecules. The hydrodynamic volume at infinite dilution of all complexes was found to be similar to the crystallographic volume and always much larger than the van der Waals volume. The self-aggregation tendency of [Pd(N,C −)( N, N)][PF 6] ionic complexes [(N,C −) = (C 6H 4-(Ph)C(O)-C N-Et); 8, ( N, N) = 2,2′-bipirydine; 9, ( N, N) = (2,6-( iPr) 2-C 6H 3)N C(Me)-C(Me) N(2,6-( iPr) 2-C 6H 3); 10, ( N, N) = (2,6-( iPr) 2-C 6H 3)N C(R′)-C(R′) N(2,6-( iPr) 2-C 6H 3), R′ 2 = naphthalene-1,8-diyl] was investigated by performing 1H and 19F diffusion experiments as a function of the concentration. Clear evidence for the formation of ion triples containing two cationic units was obtained for 8, most likely due to the establishment of a weak Pd⋯O interaction. The tendency to form ion triples was much reduced in 9 and 10, having an increased steric hindrance in the apical positions. While 9 showed the usual tendency to afford a mixture of free ions and ion pairs, solvated ions were the predominant species in the case of 10 even at high concentration values (approaching 100 mM).

N-Methyl-N-Allylpyrrolidinium Based Ionic Liquids for Solvent-Free Dye-Sensitized Solar Cells

We prepared four new ionic liquids consisting of N-methyl-N-allylpyrrolidinium cation in conjunction with anions including iodide, nitrate, thiocyanate, and dicyanamide, respectively, and measured their physical properties of density, viscosity, and conductivity. Owing to the relatively lower melting point of electroactive N-methyl-N-allylpyrrolidinium iodide, in combination with three other nonelectroactive ionic liquids, we could construct solvent-free electrolytes possessing high iodide concentrations for dye-sensitized solar cells. We correlated temperature-dependent electrolyte viscosity with molar conductivity and triiodide mobility through applying an empirical Walden's rule and a modified Stokes-Einstein equation, respectively. We have further found that these anions (nitrate, thiocyanate, and dicyanamide) have different influences on surface states and electron transport in the mesoporous titania film, resulting in different photovoltages and photocurrents of dye-sensitized solar cells.

Solubility and diffusivity of 1,1,1,2-tetrafluoroethane in room-temperature ionic liquids

The solubility and diffusivity of 1,1,1,2-tetrafluoroethane (R-134a) in seven room-temperature ionic liquids (RTILs) are presented. Among them, five of the RTILs were prepared for the first time with three new fluorocarbon sulfonate anions, and two were commercially available (1-butyl-3-methylimidazolium hexafluorophosphate was previously studied with R-134a). The gas absorption measurements were made using a gravimetric microbalance. Four isotherms (283.15, 298.15, 323.15, and 348.15 K) were measured at pressures from 0.01 to 0.35 MPa. Two of the newly synthesized ionic liquids, tetradecyl(trihexyl)phosphonium 1,1,2-trifluoro-2-(perfluoroethoxy)ethanesulfonate and tributyl(tetradecyl)phosphonium 1,1,2,3,3,3-hexafluoropropanesulfonate had the strongest interaction (negative deviations from Raoult's Law) with R-134a. Experimental gas solubility data were successfully correlated with the nonrandom two-liquid (NRTL) solution model. The time-dependent absorption data was used to calculate diffusivities that were analyzed using a model based on a modified Stokes–Einstein equation. The derived molecular size for R-134a is 2–3 times larger than the known size. Magnitudes in the observed diffusion coefficients are 10 −10 to 10 −11 m 2 s −1, which are about 10–100 times lower than typical values, found in various organic liquids.

Kinetics of removal of carbon dioxide by aqueous 2-amino-2-methyl-1,3-propanediol

Kinetics of CO 2 with a sterically hindered amine, 2-amino-2-methyl-1,3-propanediol (AMPD), in aqueous solutions were studied at temperatures of 303, 313, and 318 K and over the concentration ranging from 5 to 25 wt% AMPD using a wetted-wall column absorber. The N 2O analogy was applied to estimate the physicochemical properties such as Henry's law constant and diffusivity of CO 2 in amine solutions. The measured diffusion coefficients of N 2O in aqueous AMPD solutions were correlated using a modified Stokes–Einstein equation. On the basis of the zwitterion mechanism, the zwitterion deprotonation and second-order rate constants were calculated and presented at each experimental condition. The second-order rate constants, k 2, were found to be 382, 665, and 977 m 3 kmol −1 s −1 at 303, 313, and 318 K , respectively. The activation energy for the reaction of CO 2 with AMPD was calculated to be 38.3 kJ mol −1 with an absolute error of 3%.

Enhancement of tracer diffusion in polypropylene glycol

Tracer diffusion coefficients of camphorquinone in polypropylene glycol (PPG) with molecular weight 400, 1000, 2000, and 4000 have been measured as a function of temperature by using the forced Rayleigh scattering technique. At a fixed temperature, the diffusion coefficient decreases with increasing molecular weight of PPG. A master curve is obtained when the diffusion data are normalized with respect to the viscosity of the PPG liquid. A modified Stokes–Einstein equation for tracer diffusion is used to calculate the effective hydrodynamic radius R*. Below room temperature, R* is found to decrease with decreasing temperature. The decrease of R* is discussed in terms of dynamic heterogeneity and rotation–translation coupling mechanisms. The rotation–translation coupling mechanism has recently been proposed to account for the enhancement of the tracer diffusion coefficient in the molecular liquid in the supercooled state. Comparing R* calculated for CQ in PPG with that for the same tracer in salol in the supercooled state over the same temperature range, one finds that R* in PPG decreases consistently less rapidly than in salol. The result is consistent with the fact that the rotation–translation coupling in PPG is considerably weaker than in liquid salol.

Local density effects on the hyperfine splitting constant and line width of TEMPO radical in gaseous and supercritical carbon dioxide

The density dependences of the nitrogen hyperfine coupling constant ( a N) and line width (Δ H pp) of the 2,2,6,6-tetramethylpiperidine-1-oxyl (TEMPO) free radical have been investigated at near-infinite dilution in gaseous and supercritical carbon dioxide using steady-state EPR spectroscopy. The local density augmentation around the radical is quantified based on the density dependences of the a N. The viscosity dependences of the Δ H pp are reproduced by the modified Stokes–Einstein equation, suggesting the local solute-fluid density augmentation.

Kinetics of removal of carbonyl sulfide by aqueous monoethanolamine.

The kinetics of the reaction between carbonyl sulfide and aqueous monoethanolamine (MEA) were studied over a range in temperature (298-348 K) and amine concentrations (5-20 wt%) using a wetted-sphere absorber. The experimental data were interpreted using a zwitterion mechanism. The key physicochemical properties needed to interpret the data are the solubility and diffusivity of COS in the aqueous amine solution. These properties were estimated using the N2O analogy method. Experimental values of N2O solubility were correlated using an extended scaled-particle model, and the measured N2O diffusion coefficients were correlated using a modified Stokes-Einstein equation. Solution densities and viscosities were also measured and correlated in this work. On the basis of the zwitterion mechanism whose rate-limiting step was the deprotonation of a zwitterion, the Arrhenius relationship between the third-order rate constant and the temperature was well correlated with an absolute mean deviation of 0.3%. It could be thus concluded that the overall reaction rate was first-order in the COS concentration and second-order in the MEA concentration.

A MATHEMATICAL MODEL TO PREDICT EXTRACTION BEHAVIOR OF METAL IONS BETWEEN A POLYMER GEL AND AN AQUEOUS PHASE

ABSTRACT A mathematical model to predict the extraction behavior of metal ion between a polymer gel and an aqueous solution was proposed. It consists of the Flory-Huggins formula for evaluating thermodynamically the physico-chemical properties of polymer gel, the modified Stokes-Einstein equation to evaluate the mass transfer rate of metal ion into polymer gel and the equation to evaluate the extraction equilibrium. The extraction of lanthanide elements, Nd(III), Sm(III) and Gd(III), from an aqueous solution containing nitrate ion was carried out by the use of SDB (styrene-divinylbenzene copolymer) gel swollen with a bidentate organophosphorus compound, CMP (dihexyl-N,N-diethylcarbamoylmethylphosphonate). The binary extraction and the effect of the crosslinking degree of SDB gel on the extraction rate were examined. These experimental results were in agreement with the predictions calculated by the proposed model. It was confirmed that the extraction behavior of lanthanide ions into- the SDB gel was predicted accurately, when the physico-chemical properties of SDB gel, such as the affinity between SDB and CMP ( ϰ ) and the crosslinking degree ( ν ), and a coefficient defined in the modified Stokes-Einstein equation (K0) were known. This model is available as a tool to design an extraction chromatographic process using polymer gel.

On the relative motion of a particle in a swarm of different particles

The classification of dissimilar particles in a liquid suspension is exploited in many industrial unit operations. To understand and predict fluidization phenomena in general and particle classification in particular, adequate expressions for each of the governing forces—gravity, buoyancy and drag forces—are required. A possible formulation of the effective driving force and friction in multicomponent mixtures is discussed in this paper. A new expression is derived, which is based on a steady-state force balance for the classifying particle. The expression requires only the usual fluidization data of a single particle species such as the terminal velocity and the Richardson and Zaki index. The new expression is compared with some existing classification models with respect to their predictive value for a wide range of classification data. The new expression proved to be accurate over the entire range of known experimental data and can be extended easily to multicomponent fluidized suspensions. We have found strong evidence that the exact formulation of the buoyancy acting on the classifying particle originates from the bulk density of the fluidized suspension, regardless of the ratio of particle sizes. To clarify the relation between the “overall” drag force and the ratio of particle sizes, the “overall” friction on the classifying particle is decomposed into a solid-liquid and a solid-solid component. The particle-particle interaction force is proportional to the product of the hold-up of the fluidized particles and the slip velocity of dense and fluidized particles. In agreement with other authors, we found an increased friction with a decreasing ratio of classifying and fluidized particles. The particle-particle interaction coefficient was correlated with a modified Stokes-Einstein equation.