Long-time Diffusion Research Articles

One-dimensional diffusion of colloids in polymer solutions

For colloid-polymer systems confined in one-dimensional channels, the diffusion of the colloidal particles is obtained by tracking individual particles using enhanced video microscopy and digital image analysis. For short times, the diffusion is normal, of the Fickian type, with mean-squared displacement varying linearly with time. For long times, however, the mean-squared displacement is found to increase more slowly with time, being proportional to the square root of time, in agreement with the theoretical prediction for diffusion of hard rods in one dimension in which mutual crossing of the particles cannot take place. The crossover from short-time to long-time diffusion is observed and is found to depend on the colloid and polymer concentrations. Unexpectedly, for small polymer-to-colloid size ratios, it is the polymer rather than the colloid concentration which has a leading effect on the colloid diffusion.

Computer simulations of diffusion and dynamics of short-chain polyelectrolytes

Brownian dynamics simulations are conducted to investigate the diffusional and dynamic properties of polyelectrolytes in dilute salt-free solutions. The polyelectrolyte molecule is represented by a bead-spring chain in a primitive model. The long-range hydrodynamic and Coulomb interactions are both taken into consideration through the Ewald summations for the first time. The major finding of our simulations is that the dependence of the long-time chain diffusivity on the Coulomb interaction strength is very different from that of the Kirkwood short-time diffusivity, which simply shows a trend nearly opposite to the chain size. When ignoring the hydrodynamic interaction (HI), the coupling effect between the chain and its counterions gives rise to a noticeable increase in the long-time diffusivity at intermediate electrostatic interaction strengths. However, the incorporation of HI suppresses this effect to a degree that one can no longer discern it. Moreover, the rotational relaxation is found to show a dependence opposite to that of the gyration radius relaxation.

Application of an Obstruction-Scaling Model To Diffusion of Vitamin B12and Proteins in Semidilute Alginate Solutions

Pulsed field gradient nuclear magnetic resonance (PFG-NMR) was used to measure tracer diffusion coefficients of vitamin B12 and globular proteins (myoglobin, β-lactoglobulin, ovalbumin, and bovine serum albumin) in semidilute alginate solutions under physiological conditions (pH 7.4, ionic strength 0.22 M). The long-time diffusion behavior can be predicted by an obstruction-scaling model, which assumes that the semidilute polymer solutions can be treated as a static network during the diffusion time scale, and the contributions of hydrodynamic interactions and electrostatic interactions to diffusion can be neglected. Arguments are given in support of these assumptions, which are expected to be valid only when the size of the probe is much larger than the polymer chain. The model was also applied to literature data of protein diffusion in flexible and inflexible polymer solutions, and it was found to be predictive. Moreover, it is demonstrated that, at equal polymer solution mesh sizes, probe diffusion is not influenced by the polymer chain flexibility.

Tracer diffusion and cluster formation in mixtures of spheres and rotating rods

On the basis of Brownian dynamics computer simulation studies, we investigate thestructure and dynamics of two-dimensional mixtures of rods driven by an external torqueand spheres. First, we show that the tracer long-time diffusion of spheres in a system ofrotating rods can be efficiently tuned via the rod density. It behaves non-monotonically forincreasing rod density. In particular, the sphere diffusion becomes strongly enhanced acrossthe jamming transition of the rods. Second, we show that rotating rods in a densesuspension of spheres form aggregates of rod clusters which are rotating jointly. Thecluster size is in reasonable agreement with the predictions of a simple theory.

CYCLCROP indirect 13C mapping for long-time and chemical-shift selective diffusion measurements in complex hydrocarbon systems

Cyclic cross-polarization from a proton magnetization to 13C and from there back to proton coherences permits the indirect, 13C chemical shift selective detection of hydrocarbon compounds in the proton NMR channel. This excitation technique can be combined with elements of one-, two- or three-dimensional magnetic resonance imaging permitting the measurement of time-resolved spatial distributions of hydrocarbon components. Beginning this sort of CYCLCROP mapping experiment with a non-equilibrium distribution of the constituents in the system allows one to study the time evolution of the concentrations of all components that can be identified by characteristic 13C resonance lines. As applications, studies of ingress, mixing, gel formation, transport and metabolism in living plants, long-time inter- and self-diffusion in complex hydrocarbon systems are suggested. As a test experiment, the diffusion of methanol in swollen polymethylmethacrylate was examined.

Simulations of concentrated suspensions of rigid fibers: Relationship between short-time diffusivities and the long-time rotational diffusion

Brownian dynamics simulations of the behavior of suspensions of fibers demonstrate that the scaling of the rotational diffusivity with respect to the number density (nL3) is a sensitive function of the thickness and the parameter L2D(R0)/D(T0), where D(R0) is the rotational diffusivity at infinite dilution, D(T0) is the average center-of-mass diffusivity at infinite dilution, and L is the fiber length. Existing theories for the long-time rotational diffusivities of rigid fibers in the semidilute and concentrated regimes fail to accurately account for the relationship with the dilute values of the rotational and translational diffusivities of the various physical models used to simulate the fibers. The concentration regime studied in this work ranges from a number density of nL3 approximately 0-150, which is below the transition from an isotropic to nematic state. The effect of the fiber thickness was studied by performing simulations of rods with aspect ratios (fiber length over diameter) of 25, 50, and 500, as well as performing projections for infinitely thin fibers. The excluded volume of the rods was enforced through the use of short-range potentials. For a rod with an aspect ratio of 50 with a parameter of L2D(R0)/D(T0)=9, which corresponds to a slender-body model of the individual fibers, the rotational diffusivity (D(R)) scales as D(R)/D(R0) approximately (nL3)(-1.9) in the concentration regime of 70 < or = nL3 < or = 150. Similarly with a parameter of L2D(R0)/D(T0)=4, corresponding to a rigid-dumbbell model, the rotational diffusivity scales as D(R)/D(R0) approximately (nL3)(-1.1) over the same range of concentrations. For rods with aspect ratios of 25, it is observed that a difference in the scaling is seen for L2D(R0)/D(T0) approximately < 8, with higher values of this ratio exhibiting essentially the same scaling. Additional values of the ratio L2D(R0)/D(T0) were investigated to determine the overall behavior of the suspension dynamics with respect to this parameter. These findings resolve discrepancies between simulation results for rotational diffusivities reported by previous investigators and provide new insights for the development of an accurate theory for the diffusivity of rigid rods suspended in solution.

Gas Diffusion in Polycrystalline Silicalite Membranes Investigated by 1H Pulse Field-Gradient NMR

1H pulse field-gradient (PFG) spin-echo NMR was performed to measure the diffusivity of methane in a polycrystalline MFI-type silicalite membrane. Measured diffusivities decreased with an increase in the diffusion distance and converged to the constant value. This result suggests the presence of a transport barrier in the membrane. The long-time diffusivity in the membrane was 3.7 x 10(-9) m2/s, which was a factor of 3 smaller than reported values in a single crystal. The distance between the transport barriers was estimated to be much larger than 6 mum from the relationship of diffusivity with displacement. It should be noted that the estimated distances were larger than the smallest dimension of the crystals appearing in the membrane surface. Gas permeation and pervaporation tests were carried out on the same sample for which NMR measurements were taken. The estimated methane flux using measured long-time diffusivity by the permeation theory overestimated the experimental value, although it is closer to the experimental value than the value estimated using the short-time diffusivity. These results mean that the methane diffusivity in a silicalite membrane is much smaller than that in a single crystal.

Breather-induced anomalous charge diffusion

We present results on the diffusive motion of a charge interacting with the nonlinear dynamics of a thermalized underlying lattice. Signatures of anomalous diffusive properties are found at relatively high temperatures, where highly nonlinear excitations are present. A sublinear diffusion and a plateau appear before the standard long-time diffusion during the evolution of the mean-squared displacement and a significant degree of heterogeneity is exhibited among individual trajectories. Both properties are connected with the existence of vibrational hot spots (breather or multibreather excitations). Transport parameters of the charge are strongly affected in this case, as can be exemplified by the significant suppression of the diffusion coefficient D. The variation of D with temperature follows a stretched exponential law. The results are contrasted with those of the linearized case, in the absence of breathers. Such anomalous diffusion of a charge coupled to a thermalized lattice may be relevant in low-dimensional soft materials with strong anharmonicities, such as biomolecules, conducting polymers, etc.

Experimental study of short- and long-time diffusion regimes of spherical particles in carboxymethylcellulose solutions

The behavior of latex particles in carboxymethylcellulose (CMC) solutions with different probe radii and solution concentrations is investigated. The diffusion processes of probe particles are studied by means of dynamic light scattering experiments. The existence of two diffusion regimes (a short- time regime and a long- time regime) is established for concentrations higher than the threshold value which changes for each probe dimension. Information is also provided about the relative weight of the probe population undergoing the two diffusion processes. This experimental evidence is discussed in connection with the existence of a defined volume (cage) in which short-time diffusion is the characteristic motion, while the long-time diffusion process dominates the motion over all the cages. Furthermore, there was found to be some deviation from Stokes–Einstein behavior.

Brownian motion of finite-inertia particles in a simple shear flow

Simultaneous diffusive and inertial motion of Brownian particles in laminar Couette flow is investigated via Lagrangian and Eulerian descriptions to determine the effect of particle inertia on diffusive transport in the long-time limit. The classical fluctuation dissipation theorem is used to calculate the amplitude of random-force correlations, thereby neglecting corrections of the order of the molecular relaxation time to the inverse shear rate. In the diffusive limit (time much greater than the particle relaxation time) the fluctuating particle-velocity autocorrelations functions are found to be stationary in time, the correlation in the streamwise direction being an exponential multiplied by an algebraic function and the cross correlation nonsymmetric in the time difference. The analytic, nonperturbative, evaluation of the particle-phase total pressure, which is calculated to be second order in the Stokes number (a dimensionless measure of particle inertia), shows that the particle phase behaves as a non-Newtonian fluid. The generalized Smoluchowski convective-diffusion equation, determined analytically from a combination of the particle-phase pressure tensor and the inertial acceleration term, contains a shear-dependent cross derivative term and an additional term along the streamwise direction, quadratic in the particle Stokes number. The long-time diffusion coefficients associated with the particle flux relative to the carrier flow are found to depend on particle inertia such that the streamwise diffusion coefficient becomes negative with increasing Stokes number, whereas one of the cross coefficients is always negative. The total diffusion coefficients measuring the rate of change of particle mean-square displacement are always positive as expected from general stability arguments.

A Brownian dynamics study on the self-diffusion of charged tracers in dilute polyelectrolyte solutions

Brownian dynamics simulations with hydrodynamic interactions are conducted to investigate the self-diffusion of charged tracer particles in a dilute solution of charged polymers, which are modeled by bead-spring chains. The Debye-Hückel approximation is used for the electrostatic interactions. The hydrodynamic interactions are implemented by the Ewald summation of the Rotne-Prager tensor. Our simulations find that the difference in short- and long-time diffusivities is very slight in uncharged short-chain solutions. For charged systems, to the contrary, the difference becomes considerable. The short-time diffusivity is found to increase with increasing chain length, while an opposite behavior is obtained for the long-time diffusivity. The former is attributed to the hydrodynamic screening among beads in a same chain due to the bead connectivity. The latter is explained by the memory effect arising from the electrostatic repulsion and chain length. The incorporation of hydrodynamic interactions improves the agreement between the simulation prediction and the experimental result.

Diffusion of spheres in crowded suspensions of rods

Translational tracer diffusion of spherical macromolecules in crowded suspensions of rodlike colloids is investigated. Experiments are done using several kinds of spherical tracers in fd-virus suspensions. A wide range of size ratios L/2a of the length L of the rods and the diameter 2a of the tracer sphere is covered by combining several experimental methods: fluorescence correlation spectroscopy for small tracer spheres, dynamic light scattering for intermediate sized spheres, and video microscopy for large spheres. Fluorescence correlation spectroscopy is shown to measure long-time diffusion only for relatively small tracer spheres. Scaling of diffusion coefficients with a/xi, predicted for static networks, is not found for our dynamical network of rods (with xi the mesh size of the network). Self-diffusion of tracer spheres in the dynamical network of freely suspended rods is thus fundamentally different as compared to cross-linked networks. A theory is developed for the rod-concentration dependence of the translational diffusion coefficient at low rod concentrations for freely suspended rods. The proposed theory is based on a variational solution of the appropriate Smoluchowski equation without hydrodynamic interactions. The theory can, in principle, be further developed to describe diffusion through dynamical networks at higher rod concentrations with the inclusion of hydrodynamic interactions. Quantitative agreement with the experiments is found for large tracer spheres, and qualitative agreement for smaller spheres. This is probably due to the increasing importance of hydrodynamic interactions as compared to direct interactions as the size of the tracer sphere decreases.

Structures and dynamics of thermosensitive microgel suspensions studied with three-dimensional cross-correlated light scattering

The structure factors, short- and long-time diffusion coefficients, and hydrodynamic interactions of concentrated poly(N-isopropylacryamide) microgel suspensions were measured with simultaneous static and dynamic three-dimensional cross-correlated light scattering. The data are interpreted through comparison to hard sphere theory. The structure factors are known to be described well by the hard sphere approximation. When the structure factor is fit to an effective hard sphere volume fraction and radius, the diffusion and hydrodynamic interactions are also well described by the hard sphere model. We demonstrate that one single hard sphere volume fraction is sufficient to describe the microgel structures, hydrodynamic interactions, and long- and short-time collective diffusion coefficients. This result is surprising because the particle form of the microgels at these temperatures is not rigid, but rather "fuzzy" spheres with dangling polymer chains.

Memory effects in collective dynamics of Brownian suspensions.

We obtain macroscopic equations for average suspension velocity and particle current in a Brownian suspension valid on long time scales for which the memory effects are important. The coefficients in these equations depend solely on local properties of the medium. This formalism allows one to obtain well-defined theoretical expressions for transport coefficients, free of the integrals diverging with the size of the system. As an example, the expression for long-time collective diffusion coefficient is derived and the memory contribution to this coefficient is estimated.

Dynamics of caged microemulsion droplets: a neutron spin echo and dynamic light scattering study

We report a neutron spin echo (NSE) and dynamic light scattering (DLS) study of droplet dynamics in a microemulsion at high droplet packing fraction ( Φ droplet∼62%). NSE measurements in range qr⪢1 allow to investigate the short-range Brownian motion of droplets trapped within the cages formed by their nearest neighbors. For a particular scattering contrast, NSE also reveals, for q values around qr∼ π, the shape fluctuations of the caged droplets. In range qr⪡1, DLS correlation functions reveal two independent diffusive modes with well separated decay times. Comparing DLS results with NSE and pulsed-field gradient NMR measurements allows to associate both DLS modes with collective dynamics of ‘transient-trapped’ droplets, revealing strong similarity to the long-time and short-time diffusion modes observed by light scattering in concentrated colloidal suspensions below the glass transition.

Brownian dynamics: divergence of mobility tensor

The mobility tensor for many spheres suspended in a viscous fluid is considered. An analytical formula for the divergence of this tensor is derived. It is then applied in calculations of long-time collective diffusion coefficient of hard-sphere suspension by means of Brownian dynamics method.

Paradoxes of steady-state and pulse operational mode characteristics of silicon detectors irradiated by ultra-high doses of γ-rays

Detectors processed from standard and oxygenated Si are compared with respect to radiation hardness to ultra-high dose of γ-rays up to 1.76Grad. The detectors are processed by different manufactures: Silicon Detector Development and Processing Lab (SDDPL) of BNL (USA), CIS (Germany) and Micron Semiconductor (UK). Oxygenation was performed either by using High Temperature, Long Time (HTLT) oxidation in oxygen-based ambient, or by short-time high-temperature oxidation in O2 followed by a long-time drive-in diffusion in N2. It has been shown that radiation hardness improvement in oxygenated detectors irradiated by γ-rays actually extends now up to ultra-high dose of 1.76Grad. Effects of space charge sign inversion (SCSI) and linear build-up of negative space charge with γ-ray dose have been observed in standard Si detectors, similar to the case of neutron/proton irradiation. In contrast to standard Si in detectors, it has been revealed that in oxygenated Si detectors, positive space charge is accumulated with increasing dose up to 1.76Grad with no SCSI (“positive space charge detectors”), which is unique for γ-irradiation. The advantage for using oxygenated Si detectors, as compared to standard Si detectors in practical applications in the ultra-high dose range of 1–1.76Grad, has been demonstrated in terms of the profit in the reduction of full depletion voltage by a factor of 3–4, and in the reduction of leakage current by a factor of 3.2–5. In the pulse operational mode, however, polarization effect has been observed for oxygenated detectors irradiated to this ultra-high dose range of 1–1.76Grad at room temperature, indicating the upper dose limit for a “damageless” oxygenated Si detector is about 1Grad. The fact that this polarization occurs at RT is a paradox, since for other types of radiations it occurs only at cryogenic temperatures.

Bulk mediated surface diffusion: finite bulk case

Within the framework of a Master Equation scheme, we address the dynamics of adsorbed molecules (a fundamental issue in surface physics) and study the diffusion of particles in a finite cubic lattice whose boundaries are at the z = 1 and the z = L planes where L = 2; 3; 4; ..., while the x and y directions are unbounded. As we are interested in the effective diffusion process at the interface z = 1, we calculate analytically the conditional probability for finding the particle on the z = 1 plane as well as the surface dispersion as a function of time and compare these results with Monte Carlo simulations finding an excellent agreement. These results show that: there exists an optimal number of layers that maximizes 〈r 2(t)〉 on the interface; for a small number the layers the long-time effective diffusivity on the interface is normal, crossing over abruptly towards a subdiffusive behavior as the number of layers increases.

Cross-section model for cold neutron scattering in solid and liquid methane

Incoherent neutron scattering cross-sections for solid CH 4 in the temperature range of 20.4– 90.7 K and liquid CH 4 at temperatures between 90.7 and 111.7 K are evaluated. A space–time correlation approach is used to describe a double-differential scattering cross-section which is basically expressed by a generalized frequency distribution. The cross-section model includes molecular translations and rotations as well as intramolecular vibrations. The former are concerned with very short-time free-gas like translation, short-lived vibration and long-time diffusion (only in liquid state). The latter consists of short-time free rotation and long-time isotropic rotational diffusion. Numerical calculations on double-differential and total cross-sections are carried out for incident neutron energies covered 0.1 μeV to 10 eV . Good agreement with experimental results at many different temperatures is found.

Structure and dynamics of lithium silicate melts: molecular dynamics simulations

We present molecular dynamics simulations of lithium silicate melts (Li2O)x(SiO2)1−x with Li2O concentrations x = 0.01, x = 0.1 and x = 0.5 in equilibrium conditions for temperatures down to 1500 K. The partial pair correlation functions are determined and compared with recent experimental data. The dynamics of the individual species is characterized ia the mean square displacement. From the temperature dependence of the long-time diffusion constant we extract information about its activation energy. Interestingly, for x = 0.5 the extrapolation of the diffusion constant to temperatures below 700 K shows good agreement with experimental tracer diffusion data. In agreement with experiment, the length scale beyond which the dynamics becomes diffusive systematically varies with lithium concentration. Analysis of the van Hove self correlation function reveals how the occurrence of hopping processes as well as the presence of dynamic heterogeneities at low temperatures is correlated with the Li2O concentration.