Lattice density functional theory of molecular diffusion

A density functional theory of diffusion is developed for lattice fluids with molecular flux as a functional of the density distribution. The formalism coincides exactly with the generalized Ono-Kondo density functional theory when there is no gradient of chemical potential, i.e., at equilibrium. Away from equilibrium, it gives Fick's first law in the absence of a potential energy gradient, and it departs from Fickian behavior consistently with the Maxwell-Stefan formulation. The theory is applied to model a nanopore, predicting nonequilibrium phase transitions and the role of surface diffusion in the transport of capillary condensate.     

Stationary flow, first passage times, and macroscopic Fick's first diffusion law: application to flow enhancement by particle trapping

A generalized macroscopic Fick's first diffusion law is derived which describes steady-state particle flow between two baths explicitly as a function of the concentration gradient, acting as the thermodynamic driving force, times a functional of the first passage time. The latter is shown to be the ratio of the number of particles trapped between the baths and the first passage time. Particle trapping is shown to be a powerful mechanism by which flow can be enhanced. This is analyzed for two examples: a potential and an entropy trap.     

Modeling diffusion in miscible polymer blend films

Recent experiments designed to probe polymer transport in the bulk and in the vicinity of surfaces have examined the interdiffusion of multilayer sandwiches of isotopically labeled polymers. The measured time dependent concentration profiles normal to the surface are typically fit to Fick's law, with a single fitting parameter, the mutual binary diffusion coefficient (MBDC). The resulting MBDCs are found to vary over a broad range of film thicknesses and time, with the time dependence being viewed as a unique signature of the reptation mechanism of long chain motion, and the thickness dependence being attributed to the slowing down of chain dynamics near surfaces. Since the experiments are conducted at finite concentration, the MBDC, which is a product of the bare mobility and the concentration derivative of the chemical potential, could be dominated by the time and thickness dependence of this second term (which is ignored in Fick's law). To quantify this conjecture we consider the more rigorous Cahn formulation of the diffusion problem in terms of chemical potential gradients. We use square gradient theory to evaluate chemical potentials, and fit the resulting time dependent concentration profiles to the analytical solution of Fick's law. By thus mimicking the experimental analysis we find that the apparent MBDCs vary with time as t(-1/2) at short times, in good agreement with existing experiments. We show that this time dependence reflects the system's desire to minimize concentration gradients, a fact ignored in Fick's law. Since these arguments make no reference to the mechanism of chain motion, we argue that the time dependence of MBDC derived from interdiffusion experiments does not provide unequivocal support for the reptation mechanism of long chain transport. The MBDC values, which also vary with the degree of confinement, are predicted to increase with decreasing thickness for model parameters corresponding to experimental systems. In contrast, since the experimental fits yield an opposite trend, we suggest that the bare mobility of the chains decreases strongly with decreasing thickness. These findings strongly support the idea that the chains are "pinned" irreversibly to the surfaces, in good agreement with other, independent experiments.     

Transport of matter and energy in a mesoscopic thermo-hydrodynamic approach

We derive a thermo-hydrodynamic theory for particles and energy flow, based on a nonequilibrium grand-canonical ensemble formalism. The time-dependent kinetic coefficients are explicitly given in terms of microscopic mechanical quantities. The time evolution equations describing the coupled flow of energy and particles are derived. The second-rank tensorial fluxes of current of energy and particles present in the nonequilibrium ensemble are nondiagonal. We obtain a generalized Fick's Law, which presents the effect of the energy flow on the particle diffusion equation.     

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.     

Fluxes of non-interacting and strongly repelling particles through a single conical channel: Analytical results and their numerical tests

Using a diffusion model of particle dynamics in the channel, we study entropic effects in channel-facilitated transport. We derive general expressions for the fluxes of non-interacting particles and particles that strongly repel each other through the channel of varying cross section area, assuming that the transport is driven by the difference in particle concentrations on the two sides of the membrane. For a special case of a right truncated cone expanding in the left-to-right direction, we show how the fluxes depend on the geometric parameters of the channel and on the particle concentrations. For non-interacting particles the flux is direction-independent in the sense that inversion of the concentration difference leads to the inversion of the direction of the flux without changing its magnitude. This symmetry is broken for repelling particles: The flux in the left-to-right direction exceeds its right-to-left counterpart. Our theoretical predictions are supported by three-dimensional Brownian dynamics simulations.     

Effect of particle shape on crossflow filtration flux

In an effort to further increase the understanding of crossflow filtration, experiments were performed on the influence of particle shape on permeation flux. Five particles of similar density and size distribution but of different shapes were used to test the influence of particle shape, while varying experimental parameters such as crossflow velocity, filtration pressure, solids concentration, membrane morphology and pore size. Particle shape was found to influence the equilibrium flux by the structure of the cake layer formed. Irregularly shaped particles such as branched carbon particles provided higher fluxes due to the high voidage cakes. More regularly shaped particles such as glass spheres resulted in lower fluxes. Platelet aluminium particles had relatively high filtration rates due to the gaps between the plates. The effects of the other experimental parameters typically showed results consistent with previous publications. Using the measured cake mass, a theoretical model based on D'Arcy and Kozeny gave reliable filtration flux compared to the experimental results.     

Driving force for hydraulic and pervaporative transport in homogeneous membranes

Experiments were designed to demonstrate that the chemical potential gradient required for liquid transport through swollen network polymer membranes manifests itself as a concentration gradient and that the rate of transport is independent of how this gradient is established. The fluxes of various liquids through a crosslinked rubber membrane were measured in hydraulic and pervaporation modes of permeation. The pressure applied downstream in the latter act simply to fix the activity of the liquid in the downstream membrane surface. The experiments show the flux is a unique function of this activity, and it does not matter how it is established. Sorption data were used to convert these results into a plot of flux versus concentration differential across the membrane which was analyzed by Fick's law using a model for the concentration dependence of the diffusion coefficient. Measured ceiling fluxes for pervaporation for a number of liquids were found to be the same as those estimated from hydraulic permeation data. A simple mathematical representation for an ideal system is used as a pedagogical device to demonstrate the conclusions.     

Analysis of constant permeate flow filtration using dead-end hollow fiber membranes

Dead-end filtration of colloids using hollow fibers has been analysed theoretically and experimentally. A mathematical model for constant flux filtration using dead-end hollow fiber membranes has been developed by combining the Hagen–Poiseuille equation, the (standard) filtration equation, and cake filtration theory of Petsev et al. [D.N. Petsev, V.M. Starov, I.B. Ivanov, Concentrated dispersions of charged colloidal particles: sedimentation, ultrafiltration and diffusion, Colloid Surf. A: Physicochem. Eng. Aspects, 81 (1993) 65–81.] to describe the time dependence of the filtration behavior of hollow fiber membranes experiencing particle deposition on their surface. Instead of using traditional constitutive equations, the resistance of the cake layer formed by the deposited colloids has been directly correlated to the cake structure. This structure is determined by application of a force balance on a particle in the cake layer combined with the assumption that an electrostatically stable cake layer of mono-sized particles would be ordered in a regular packing geometry of minimum energy. The developed model has been used to identify the relationship between the filtration behavior of the hollow fiber membrane and the particle properties, fiber size, and imposed average flux. Filtration experiments using polystyrene latex particles of relatively narrow size distribution with a single dead-end hollow fiber membrane demonstrate good consistency between experimental results and model prediction. The developed model has been used to simulate the distribution of the cake resistance, transmembrane pressure, and flux along the hollow fiber membrane and used to assess the effect of fiber size, particle size, zeta potential, and the average imposed flux on the suction pressure-time profiles, flux, and cake resistance distributions. These results provide new insights into the filtration behavior of the hollow fiber membrane under constant flux conditions.     

PARTICLE SIZE DISTRIBUTION IN CONTINUOUS EMULSION POLYMERIZATION WITH FREE RADICAL DESORPTION

A mathematical model of particle size distribution in continuous emulsion polymerization which accounts for the free radical desorption from polymer particles is presented. The desorption rate is based on the diffusion theories which suggest the rate coefficient should be inversely proportional to the surface area of the polymer particles. The number density and total particle number are estimated by our model.

The average number of radicals per particle approaches Smith-Ewart case II In the range of large particle sizes. A means for predicting the nature of average desorption rate is proposed, and it seems to be influenced by concentrations of emulsifier and initiator, and residence times as well     

The influence of the membrane zeta potential on the critical flux for crossflow microfiltration of particle suspensions

It was investigated how the value of zeta potential of microfiltration membranes influenced the critical flux while filtering a suspension of silica particles. Ceramic membranes of three different materials were used and measurements were performed at two different values of pH for each material corresponding to two different values of zeta potential for each material.It was found that neither the zeta potential of the membrane nor the zeta potential of the particles influenced the observed critical flux. The critical flux increased linearly with the wall shear stress and decreased with increasing particle concentration.Expressions were obtained for the critical flux based on two different particle transport mechanisms: particle rolling (torque-balance model) and shear-induced diffusion. It was found that neither of the expressions could explain experimental critical fluxes adequately.     

Measuring the mutual diffusion coefficient for dodecyl acrylate in low molecular weight poly(dodecyl acrylate) with laser line deflection (Wiener's Method) and the fluorescence of pyrene

Diffusion of small molecules into glassy polymers is quite complicated and almost always non-Fickian. Little work has been done with the diffusion of low molecular weight polymers that are liquids at room temperature (such as poly(dodecyl acrylate)) into their miscible monomers. We have studied three molecular weights under 20 000 to determine if poly(dodecyl acrylate) diffusion into dodecyl acrylate could be treated with Fick's law and if so to determine the values of the diffusion coefficients. We compare two methods for measuring the diffusion of dodecyl acrylate into poly(dodecyl acrylate): We used laser line deflection (Wiener's method) and improved upon the method from published reports. We also used the dependence of pyrene's fluorescence on the viscosity to measure the concentration distribution, and thus to extract the diffusion coefficient. After an initial relaxation period, diffusion in all cases followed Fick's law with a single concentration-independent diffusion coefficient. Comparison of the diffusion coefficients obtained by both methods yielded the same order of magnitude for the diffusion coefficients (10(-7) cm2/s) and showed the same trend in the dependence on the average molecular weight of the polymer (a decrease in the diffusion coefficient with an increase in the molecular weight).     

Effect of soluble polymer binder on particle distribution in a drying particulate coating

Soluble polymer is frequently added to inorganic particle suspensions to provide mechanical strength and adhesiveness to particulate coatings. To engineer coating microstructure, it is essential to understand how drying conditions and dispersion composition influence particle and polymer distribution in a drying coating. Here, a 1D model revealing the transient concentration profiles of particles and soluble polymer in a drying suspension is proposed. Sedimentation, evaporation and diffusion govern particle movement with the presence of soluble polymer influencing the evaporation rate and solution viscosity. Results are summarized in drying regime maps that predict particle accumulation at the free surface or near the substrate as conditions vary. Calculations and experiments based on a model system of poly(vinyl alcohol) (PVA), silica particles and water reveal that the addition of PVA slows the sedimentation and diffusion of the particles during drying such that accumulation of particles at the free surface is more likely.     

Particle bridging between oil and water interfaces

Particle bridging between a water drop and a flat oil-water interface has been observed when the drop is brought into contact with the interface, leading to the formation of a dense particle monolayer of disc shape (namely, particle disc) that prevents the drop from coalescing into the bulk water phase. Unlike previous observations where particles from opposite interfaces appear to register with each other before bridging, the present experiment demonstrates that the particle registry is not a necessity for bridging. In many cases, the particles from one of the interfaces were repelled away from the contact region, leaving behind the particles from the other interface to bridge the two interfaces. This is confirmed by particle bridging experiments between two interfaces covered with different sized particles, and between a particle-covered interface and a clean interface. The dynamics associated with the growth of the particle disc due to particle bridging follows a power law relationship between the radius of the disc and time: r proportional, variant t0.32+/-0.03. A scaling analysis assuming capillary attraction as the driving force and a hydrodynamic resistance leads to the power law r proportional, variant t1/3, in good agreement with the experiment. In addition, we found that binary mixtures of two different sized particles can undergo phase segregation driven by the particle bridging process.     

On the behavior of the electrostatic colloidal interaction in the membrane filtration of latex suspensions

In order to investigate effects of the colloidal interaction in the membrane filtrations, the dead-end ultrafiltration of latex colloids was conducted with fully retentive membranes. Experimental results concerning the permeate flux during the filtration indicate that the void fraction of cake layer increased with the decrease of the ionic strength, due to the expanded Debye double layer thickness around the particles. The concentration dependence of the gradient diffusion coefficient of colloidal particles has been examined as a function of solution ionic strength. The NVT Monte Carlo simulation was applied on the bulk suspension so as to determine the thermodynamic coefficient, and the hydrodynamic coefficient was evaluated from the previously developed relation for an ordered system. The long-range electrostatic interactions between the particles are determined by using a singularity method, which provides accurate solutions to the linearized electrostatic field. The predictions on the variation of concentration polarization layer have been presented, from which we found that both the permeate flux and the particle diffusion are related to determine the concentration distribution above the cake layer.     

Potential and flux decomposition for dynamical systems and non-equilibrium thermodynamics: curvature, gauge field, and generalized fluctuation-dissipation theorem

The driving force of the dynamical system can be decomposed into the gradient of a potential landscape and curl flux (current). The fluctuation-dissipation theorem (FDT) is often applied to near equilibrium systems with detailed balance. The response due to a small perturbation can be expressed by a spontaneous fluctuation. For non-equilibrium systems, we derived a generalized FDT that the response function is composed of two parts: (1) a spontaneous correlation representing the relaxation which is present in the near equilibrium systems with detailed balance and (2) a correlation related to the persistence of the curl flux in steady state, which is also in part linked to a internal curvature of a gauge field. The generalized FDT is also related to the fluctuation theorem. In the equal time limit, the generalized FDT naturally leads to non-equilibrium thermodynamics where the entropy production rate can be decomposed into spontaneous relaxation driven by gradient force and house keeping contribution driven by the non-zero flux that sustains the non-equilibrium environment and breaks the detailed balance. On any particular path, the medium heat dissipation due to the non-zero curl flux is analogous to the Wilson lines of an Abelian gauge theory.     

Diffusion in glassy polymers: A model using a homogenization method and the effective medium theory

Glassy polymers are considered as inhomogeneous with regions in which the gas sorption follows Henry's law and others where it follows Langmuir's law. It is assumed that the linear dimensions of these regions are small compared with the macroscopic length of interest but large compared with the mean free path of the penetrant gas molecules. Applying an homogenization method it is shown that the average flux is directly proportional to the concentration gradient in the polymer. This relationship can be expressed in terms of an effective diffusion coefficient D_eff, which depends on the details of the microstructure. D_eff is evaluated in the framework of the effective medium theory and compared with experimental data for diffusion of five vapors in ethylcellulose.     

Drift velocity in non-isothermal inhomogeneous systems

Drift velocity and driving force are not directly proportional in the case of inhomogeneous suspensions, where a space dependent mobility induces an additional contribution to the drift velocity. Similarly, particle flux and drift velocity are related not only by the gradient of density but also by an additional contribution given by the gradient of the self-diffusion coefficient. We provide quantitative support to this scenario in a non-equilibrium system by means of computer simulations with a temperature gradient. Moreover, our simulation results demonstrate that the temperature gradient-induced mass transport coefficient, namely thermal diffusion coefficient, is not directly proportional to the drift velocity so that the well-accepted relation of proportionality is just an approximation.