The Kramers-Moyal expansion for electrochemical stochastic diffusion

It is proved that there is a general stochastic equation, according to which any random process in the transient mode can be presented by spatially homogeneous Kramers-Moyal expansion. In the electrochemical stochastic diffusion, an integral of the fluctuation component of electrode potential over the time plays the role of spatial coordinate. Based on these two facts, we derived a spatially homogeneous Kramers-Moyal expansion for the propagator of electrochemical stochastic diffusion. By using the limiting transition to long observation times, we obtained a time and spatially homogeneous asymptotic Kramers-Moyal expansion for the propagator of asymmetric non-Gaussian electrochemical stochastic diffusion. Under the conditions of Gaussian electrochemical noise, the asymptotic Kramers-Moyal expansion turns into the Einstein stochastic diffusion equation. The method of determining time and spatially homogeneous asymptotic Kramers-Moyal expansion for the propagator of asymmetric non-Gaussian electrochemical stochastic diffusion may be useful in the stochastic theory of slow electrochemical discharge and in the electrochemical noise diagnostics.     

Kinetics of the grating formation in holographic polymer-dispersed liquid crystals: NMR measurement of diffusion coefficients

Polymer films with embedded liquid crystal inclusions (polymer-dispersed liquid crystals) are superb composites for addressable windows, flexible displays and optical storage. Their scattering behavior and electro-optic properties depend essentially on the shape and size of the liquid crystal inclusions, which are typically formed by phase separation from a multicomponent homogeneous mixture. Here, pulsed field gradient NMR is used to measure the self-diffusion coefficients of the liquid crystal and a photo-reactive monomer, which compose such a precursor mixture. The kinetics of holographic grating formation in this mixture can be predicted by inserting the NMR diffusion coefficient of the monomer and the polymerization rate in a reaction diffusion model. The ratio of diffusion rate over reaction rate is found to be in the limiting regime in which the kinetics of the grating formation is not sensitive to this parameter.     

Diffusion and Relaxation Dynamics of Supercooled Polymer Melts

The dynamic properties of polymer melts are investigated in the range of normal liquid regime to the supercooled liquid regime. The polymer is modeled as a coarse-grained bead-spring model with chain length ranging from 5 to 160. The mean squared displacement and non-Gaussian parameter are used to describe the self diffusion of polymer beads. We find slow dynamics with decreasing temperature and increasing chain length. The time evolution of non-Gaussian parameters shows two peaks(or one peak one shoulder) in the α-relaxation time, τα, regime and sub-diffusion time regime, respectively, where the first primary peak indicates the dynamic heterogeneity stemmed from the motion of beads, and the secondary peak is the result of correlated motion along a polymer chain. Moreover, the relaxation of polymer beads shows clear two-step decay in supercooled melts and the dynamics shows growing heterogeneity with decreasing temperature. As chain length is increased, a peak of the dynamic susceptibility occurs, and the peak height,χ*4, increases and then reaches a plateau. The curves of the height of the first peak of α_2, α _2~*, versus τ and the curves of χ_4~*α versus τα follow two master curves for different chain lengths. Our results indicate the similarity of dynamic heterogeneity dominated by the motion of single bead even the chain length is different. It is interesting to find that the Stokes-Einstein(SE) relation between τα and diffusion coefficient D, D~τ-1 q, is highly length-scale dependent. The SE relation breaks down in both normal melts regime and supercooled regime at large magnitude of wave vectors, attributed to the non-Brownian motion arising from the chain connectivity and growing heterogeneity due to supercooling. However, the SE relation is reconstructed when the probing length scale is large(at small magnitude of wave vectors). Our results show a hierarchical physical picture of the supercooled polymeric dynamics.     

Refinements in the characterization of the heterogeneous dynamics of Li ions in lithium metasilicate

We have performed the molecular dynamics simulations of ionically conducting lithium metasilicate, Li(2)SiO(3), to get a more in depth understanding of the heterogeneous ion dynamics by separating out the partial contributions from localized and diffusive ions to the mean square displacement (MSD) , the non-Gaussian parameter alpha(2)(t), and the van Hove function G(s)(r,t). Several different cage sizes l(c) have been used for the definition of localized ions. Behaviors of fast ions are obtained by the subtraction of the localized component from the r(2)(t) of all ions, and accelerated dynamics is found in the resultant subensemble. The fractional power law of MSD is explained by the geometrical correlation between successive jumps. The waiting time distribution of jumps also plays a role in determining but does not affect the exponent of its fractional power law time dependence. Partial non-Gaussian parameters are found to be instructive to learn how long length-scale motions contribute to various quantities. As a function of time, the partial non-Gaussian parameter for the localized ions exhibits a maximum at around t(x2), the onset time of the fractional power law regime of . The position of the maximum is slightly dependent on the choice of l(c). The power law increases in the non-Gaussian parameter before the maximum are attributed to the Levy distribution of length scales of successive (long) jumps. The decreases with time, after the maximum has been reached, are due to large back correlation of motions of different length scales. The dynamics of fast ions with superlinear dependence in their MSD also start at time around the maximum. Also investigated are the changes of the characteristic times demarcating different regimes of on increasing temperatures from the glassy state to the liquid state. Relation between the activation energies for short time and long time regimes of is in accord with interpretation of ion dynamics by the coupling model.     

Relaxation processes in semidilute solutions of polymers in liquid crystal solvents

We investigate the relaxation phenomena in a polymer (polystyrene)/liquid crystal (4-cyano-4'-n-octyl-biphenyl) system, in its homogeneous isotropic phase near the isotropic-isotropic, isotropic-nematic, and isotropic-smectic coexistence curve, using both polarized and depolarized photon correlation spectroscopy (PCS). We study this system for different polystyrene molecular weights (4750, 12 500, and 65 000 g/mol), different compositions (50, 40, 30, and 10% polystyrene (PS) by weight), and different temperatures close to phase boundaries. First of all, we determine the phase diagrams of this system for the different molecular weights. The shape of the phase diagrams strongly depends on the molecular weight. However, in all cases, at low temperatures, these systems separate into an almost pure liquid crystalline (LC) phase and polystyrene-rich phase. PCS measurements show that the relaxation processes in the homogeneous phase are not affected by the proximity of the nematic, or smectic, boundaries (even at a temperature of 0.1 degrees C above the phase separation in two phases). In polarized PCS experiments, we always see three relaxation processes well separated in time: one, very fast, with a relaxation time of the order of 10(-5) s; a second one with a relaxation time within the range 10(-2)-10(-3) s; and a last one, very slow, with a relaxation time of the order of 1 s. Both the fast and slow modes are independent of the wave vector magnitude, while the intermediate relaxation process is diffusive. In depolarized PCS experiments, the intermediate mode disappears and only the fast and slow relaxation processes remain, and they are independent of the magnitude of the wave vector. The diffusive mode is the classical diffusive mode, which is associated with the diffusion of polymer chains in all polymer solutions. The fast mode is due to the rotational diffusion of 4-cyano-4'-n-octyl-biphenyl (8CB) molecules close to polystyrene chains (transient network). Finally, we assign the slowest mode to reorientational processes of small aggregates of PS chains that are not dissolved in 8CB.     

Excitation transfer induced spectral diffusion and the influence of structural spectral diffusion

The theory of vibrational excitation transfer, which causes spectral diffusion and is also influenced by structural spectral diffusion, is developed and applied to systems consisting of vibrational chromophores. Excitation transfer induced spectral diffusion is the time-dependent change in vibrational frequency induced by an excitation on an initially excited molecule jumping to other molecules that have different vibrational frequencies within the inhomogeneously broadened vibrational absorption line. The excitation transfer process is modeled as Fo?rster resonant transfer, which depends on the overlap of the homogeneous spectra of the donating and accepting vibrational chromophores. Because the absorption line is inhomogeneously broadened, two molecules in close proximity can have overlaps of their homogeneous lines that range from substantial to very little. In the absence of structural dynamics, the overlap of the homogeneous lines of the donating and accepting vibrational chromophores would be fixed. However, dynamics of the medium that contains the vibrational chromophores, e.g., a liquid solvent or a surrounding protein, produce spectral diffusion. Spectral diffusion causes the position of a molecule's homogeneous line within the inhomogeneous spectrum to change with time. Therefore, the overlap of donating and accepting molecules' homogeneous lines is time dependent, which must be taken into account in the excitation transfer theory. The excitation transfer problem is solved for inhomogeneous lines with fluctuating homogeneous line frequencies. The method allows the simultaneous treatment of both excitation transfer induced spectral diffusion and structural fluctuation induced spectral diffusion. It is found that the excitation transfer process is enhanced by the stochastic fluctuations in frequencies. It is shown how a measurement of spectral diffusion can be separated into the two types of spectral diffusion, which permits the structural spectral diffusion to be determined in the presence of excitation transfer spectral diffusion. Various approximations and computational methodologies are explored.     

The concentration dependence of the diffusion and permeability in a homogeneous membrane

For the sorption and diffusion coefficient dependence on the concentration of the penetrant the transport properties of a homogeneous medium are calculated. The diffusion current is assumed to be proportional to the negative gradient of the chemical potential. This is in contrast with the first Fick's law that assumes this current to be proportional to the negative gradient of the concentration of the penetrant. The difference between the two cases depends on the concentration dependence of the sorption coefficient. In a homogeneous membrane the chemical potential formulation leads to an equation which is very similar to the Fickian expression. The apparent diffusion coefficient, however, depends not onlly on the transport resistance but also on the deviation of the sorption coefficient from constancy.     

Distribution functions for filaments under tension

We develop a biased Monte Carlo simulation technique to measure the distribution functions of the extension and the end-to-end distance of fluctuating filaments stretched by external force. The method is applicable for arbitrary ratio of the persistence length to the contour length and for arbitrary forces, and also for the case of steric constraints, such as an external wall. The fundamental idea underlying the algorithm is to account explicitly for the length-scale dependence of the effective elastic moduli. We find that orientational fluctuations and wall effects produce non-Gaussian distributions for nearly rigid filaments in the small to intermediate force regime. The simulation results are tested against analytic expressions for the force-extension curves, both in the semiflexible and nearly stiff limits.     

An NMR determination of CO diffusion on platinum electrocatalysts

We report the first direct measurement of CO diffusion on nanoparticle Pt electrocatalysts at the solid/liquid interface, carried out using 13C nuclear magnetic resonance (NMR) with a spin-labeling pulse sequence. Diffusion parameters were measured in the temperature range of 253-293 K for CO adsorbed on commercial Pt-black under saturation coverage. 2H NMR of the same system indicates that the electrolyte remains in the liquid state at temperatures where the CO diffusion experiments were performed. The CO diffusion parameters follow typical Arrhenius behavior with an activation energy of 6.0 +/- 0.4 kcal/mol and a pre-exponential factor of (1.1 +/- 0.6) x 10-8 cm2/s. Exchange between different CO populations, driven by a chemical potential gradient, is suggested to be the main mechanism for CO diffusion. The presence of the electrolyte medium considerably slows down the diffusion of CO as compared to that seen on surfaces of bulk metals under UHV conditions. This work opens up a new approach to the study of surface diffusion of adsorbed molecules on nanoparticle electrode catalysts, including the possibility of correlating diffusion parameters to catalytic activity in real world applications of broad general interest.     

Determination of Bulk and Surface Diffusion Coefficients from Experimental Data for Thin Liquid Film Drainage

This note presents a method for the determination of the surface diffusion coefficient and surface diffusion flux. The theoretical considerations are based on the Onsager linear theory for the definition of the surface diffusion flux and on the Einstein theorem for the definition of the surface diffusion parameter. In this interpretation the surface diffusion coefficient differs from the one commonly defined in the literature. It does not depend on the surfactant concentration and it is a function only of the type of surfactant and the liquid/liquid interface. The theoretical calculations indicate that the effect of the surface diffusion on the film drainage is stronger than that predicted by previous theoretical studies. The experimental data for thin liquid film drainage in the case of low surfactant concentration in the continuous phase could be used for the calculation of the bulk and surface diffusion coefficients. In the present study we utilized the experimental data for the drainage of nitrobenzene films stabilized by different concentrations of dodecanol. Copyright 2000 Academic Press.     

The effect of riboflavin-photoinduced degradation of alginate matrices on the diffusion of poly(oxyethylene) probes in the polymer network

The effect of irradiation, in the wavelength range of 310-800 nm, on the tracer diffusion of poly(oxyethylene) (POE) of different molecular weights embedded in various alginate matrices with the photosensitizer riboflavin (RF) present, has been investigated with the aid of pulsed field gradient spin-echo NMR (PGSE-NMR). Both alginate solutions of different polymer concentration were studied as well as corresponding acid gels of alginate produced by introduction of different amount of glucono-δ-lactone (GDL). In 2 wt.% alginate solutions, the values of the tracer diffusion coefficient suggest a strong obstruction effect as the probe molecular weight increases. Faster probe diffusion was observed for the irradiated samples, which indicates a photochemical scission of the polymer chains and the formation of a fragmented polymer network that facilitates the migration of the tracer chains. A semidilute alginate/RF solution was transformed into a gel by adding sufficient amount of GDL. GDL lowers the pH of the solution under the pK_a of alginate, favouring intermolecular associations and the evolution of a less homogeneous network with more open structure. Therefore, the POE chains were shown to diffuse faster in the acid gel matrix than in the corresponding more homogeneous alginate solutions. The photochemically induced cleavage again promoted faster migration of the probe chains in the irradiated samples. The probe diffusion of the eight-arm star-shaped POE sample revealed an augmented obstruction effect with increasing alginate concentration and higher values of the diffusion coefficient were found in gels. The evolution of a tighter network inhibits the diffusion of the probe molecules. At lower alginate concentrations the values of the tracer diffusion coefficients are higher for the irradiated samples than for the non-exposed systems.     

Voids and necks in liquid ammonia and their roles in diffusion of ions of varying size

Voids in a medium are defined as the regions that are located outside an appropriately defined occupied space associated with molecules. Dynamical properties like diffusion can be related to the structure and distribution of voids present in the medium. This work deals with an analysis of voids and diffusion in liquid ammonia. The analysis of voids is done by the construction of Voronoi polyhedra and Delaunay tessellation. We have performed a series of molecular dynamics simulations of monovalent cations and anions of varying size in liquid ammonia at two different temperatures of 210 and 240 K to investigate the effects of ion size on the diffusion of ions and roles of voids in determining the observed diffusion behavior. It is found that with the increase of ion size, the diffusion coefficients first increase and then pass through a maximum similar to the behavior observed earlier for diffusion in water. The observed results are explained in terms of passage through voids and necks that are present in liquid ammonia. © 2012 Wiley Periodicals, Inc.     

Static and pulsed field gradient nuclear magnetic resonance studies of water diffusion in protein matrices

Static field gradient and pulsed field gradient NMR are used to study the temperature dependence of water diffusion in myoglobin and lysozyme matrices for low hydration levels of about 0.3?g/g. We show that in order to determine reliable self-diffusion coefficients D in a broad temperature range, it is very important to consider an exchange of magnetization between water and protein protons, often denoted as cross relaxation. Specifically, upon cooling, the observed stimulated-echo decays, which reflect water diffusion near ambient temperature, become more and more governed by cross relaxation. We demonstrate that comparison of experimental results for inhomogeneous and homogeneous magnetic fields enables successful separation of diffusion and relaxation contributions to the stimulated-echo decays. Making use of this possibility, we find that in the temperature range 230-300?K, the temperature-dependent diffusivities D exhibit a Vogel-Fulcher-Tammann behavior, where water diffusion in the studied protein matrices is substantially slower than in the bulk. By comparing present and previous data, we discuss relations between translational and rotational motions and between short-range and long-range water dynamics in protein matrices. In addition, we critically examine the significance of results from previous applications of NMR diffusometry to the temperature-dependent water diffusion in protein matrices.     

Application of Surface Diffusion Model to the Adsorption of Dyes on Bagasse Pith

A homogeneous solid phase diffusion model (HSDM) has been developed using a computer to predict the performance of a batch adsorber. The computer program utilises a semi-analytical solution for a two resistance model based on external mass transfer and homogeneous solid phase diffusion. The model has been successfully applied to four adsorption systems, namely, the adsorption of AB25, AR114, BB69 and BR22 onto pith. The method produces excellent correlations between experimental and theoretical concentration decay curves in batch adsorbers. The model developed presents a solution using a single solid diffusion coefficient and a single external mass transfer coefficient which are sufficient to characterise the system within a range of initial dye concentration, 25–300 mg · dm3 and solid/liquid ratios (w/v) 0.25–2.     

Reaction and Morphology Development Influenced by Diffusion in a Thermoplastic/Thermoset Blend

Diglycidyl ether of bisphenol A (DGEBA) and 4,4′-methylenebis [2,6-diethylaniline] (MDEA) are miscible in polystyrene at 177 °C. We have studied how their diffusion rate in a molten polystyrene matrix influences their polymerization rate and the morphology of the thermoset particles formed at the end of the reaction. The global composition of the blend was 60 wt% of PS and 40 wt% of epoxy-amine. The diffusional control of the reaction was evidenced by comparing the time of reaction of an initially homogeneous mixture with that of different bi-layer samples. The reaction was controled by the diffusion for relatively thick layers (>0.3. mm). A gradient of morphology was obtained due to the diffusionnal control of the reaction. The asymetricity of this gradient may be explained by three factors: differences in diffusion coefficients, in thermodynamic interactions and in viscosity.     

Diffusion properties of liquid crystal-based microemulsions

Microscopic diffusion processes in thermotropic 5CB liquid crystals (LC) with imbedded surfactant-stabilized water microemulsions are studied using pulsed field gradient nuclear magnetic resonance (PFG NMR). The experiments are performed in a temperature range around the isotropic-nematic transition temperature of the LC. The temperature dependence of the diffusivities of the liquid crystal and surfactant molecules remains almost unchanged in the whole temperature range studied. With varying water content, the diffusivities of the surfactant molecules are found to be almost invariable, indicating that the surfactant diffusivities remain essentially unaffected by whether a microemulsion is formed or the surfactant molecules diffuse as individual species. At the same time, the formation of the microemulsion is found to be crucial for the macroscopic separation of the mixture into LC- and surfactant-rich phases.     

Relation between hydraulic permeability and diffusion in homogeneous swollen membranes

The question of viscous flow versus molecular diffusion mechanisms for pressure-induced liquid transport through membranes is critically examined for the specific case of homogeneous swollen membranes. It is shown that previous attempts to compute diffusion coefficients from hydraulic permeabilities for such systems have used an equation which is grossly in error. It estimates diffusion coefficients which are orders of magnitude too high and often exceed self-diffusion coefficients. This has frequently led to the conclusion that viscous flow predominates. The origins of the errors in this equation are indicated, and a substitute equation is developed which gives diffusion coefficients well below that for self-diffusion when applied to literature data. As a result it is concluded that molecular diffusion is the dominant mechanism in homogeneous systems.     

Dynamic Gradient Directed Molecular Transport and Concentration in Hydrogel Films

Materials which selectively transport molecules along defined paths offer new opportunities for concentrating, processing and sensing chemical and biological agents. Here, we present the use of traveling ionic waves to drive molecular transport and concentration of hydrophilic molecules entrained within a hydrogel. The traveling ionic wave is triggered by the spatially localized introduction of ions, which through a dissipative ion exchange process, converts quaternary ammonium groups in the hydrogel from hydrophilic to hydrophobic. Through a reaction–diffusion process, the hydrophobic region expands with a sharp transition at the leading edge; it is this sharp gradient in hydrophilicity that drives the transport of hydrophilic molecules dispersed within the film. The traveling wave moved up to 450 μm within 30 min, while the gradient length remained 20 μm over this time. As an example of the potential of molecular concentration using this approach, a 70‐fold concentration of a hydrophilic dye was demonstrated.     

Diffusive and hydraulic permeabilities of water in water-swollen polymer membranes

The diffusive permeability of water P, which relates to diffusive flux of water under a concentration gradient of water (measured by diffusion of tritiated water), and the hydraulic permeability of water K, which relates to the water flux under a hydraulic pressure gradient are defined. For the case of diffusive transport one has P = KRT/ν₁, where ν₁ is the molar volume of water. The relationship between P and K was investigated as a function of hydration H, i.e., the volume fraction of water in swollen polymer membranes. The following characteristic features of water permeability are revealed. (a) In the lowhydration region (H < 0.2), water permeates by diffusion even under an applied hydraulic pressure gradient and KRT/ν₁ = P. (b) In the higher hydration region KRT/ν₁ is greater than P, and the ratio ω = KRT/ν₁P increases nearly exponentially with decrease of (1-H)/H. Water in this region moves partly by bulk flow under an applied hydraulic pressure gradient but moves only by diffusion in the absence of a pressure gradient. (c) The dependence of log P on (1-H)/H is nearly linear in regions of both high and low hydration but the slopes are different. The transition occurs in about the same H range where the discrepancy between P and KRT/ν₁ becomes significant. Excellent agreement was found between the experimental data for P as a function of H and the theoretical prediction based on the free-volume concept of diffusive transport in hydrated homogeneous membranes.     

Relaxation processes in mixtures of liquid crystals and polymers near phase boundaries and during phase separation

We present experimental studies of the relaxation of concentration fluctuations in a semidilute solution of polystyrene (PS) (30% by weight) in 4-cyano-4'-n-octyl-biphenyl (8CB) (70% by weight) using the photon correlation spectroscopy (PCS). In the homogeneous phase there are two modes of relaxation. The slow one (typical time scale is taus = 0.001 s) is due to the diffusion of polymer chains (of molecular mass 65,000) in the LC matrix (of molecular mass 290), while the fast one has the time scale of the order of tauf approximately 0.00001 s. The amplitude of the fast mode is much weaker than the one for the slow mode. Moreover it does not depend on the scattering wave vector, q. The value of the diffusion coefficient, Dc = 1/(tausq2) for the slow mode decreases with temperature according to the Arhenius law until we reach the coexistence curve. Its value close to the coexistence is Dc = 4 x 10(5) nm2/s and the activation energy in the homogeneous mixture is Ec=127 kJ/mol. If we gradually undercool the mixture below the coexistence into the metastable two-phase region without inducing the phase separation we find unexpectedly that Dc does not change with temperature even 4 degrees below the coexistence curve. The characteristic time of the fast mode does not depend on the scattering wave vector indicating that it is related to the transient gel structure. We have shown that it is possible to measure the short time relaxation of concentration fluctuations during the phase separation in the mixture. At low temperature close to the isotropic-nematic phase transition we have observed that the relaxation is well separated in time from the typical time of the domain growth. This relaxation mode is characterized by the large diffusion coefficient D = 2 x 10(8) nm2/s. The mode probably comes from the coupling between the orientational dynamics of liquid crystals and the transient gel structure of polymers.