Macroscopic Concentration Gradient Research Articles

Formation of a nanoscale two-phase microstructure in Cu–Zn( Al) samples with macroscopic concentration gradient

In binary Cu Zn and ternary Cu–Zn–Al with 1.5 wt% Al, the formation of a nanoscale two-phase α + ordered β 2 microstructure is observed after quenching samples with a macroscopic solute concentration gradient from 875 °C in iced brine. The two-phase microstructure has formed from high temperature β phase and is located at higher Zn contents, after the concentration invariant massive or martensitic transformation, respectively. Its occurrence further substantiates the high competition between different types of solid-state reactions in brass. The nanoscopic α + β 2 microstructure was investigated by a combination of EBSD and TEM. Experimentally measured Kikuchi patterns were correlated with dynamically simulated ones by a pattern deconvolution which compensates the limitation of existing commercial EBSD software. The employed combination of methods revealed changing phase fractions of α and β 2 along the Zn gradient. Cellular reaction, coupled growth and spinodal decomposition were discussed as possible mechanisms for the formation of the nanoscale structure, with coupled growth identified as the most likely. • Combination of state-of-the-art EBSD and (HR)TEM analysis for phase identification • Determination of c-dependent phase fractions by a comparative study of measured and dynamically simulated Kikuchi patterns • Observation of nanoscopic α + β 2 microstructure in α/β brass in direct competition with massive/martensitic transformation • Classification of competitive solid state reactions depending on the level of undercooling classification of competitive solid state reactions depending on the level of undercoolings

Formation of a Nanoscale Two-Phase Microstructure in Cu– Zn–Al Samples with Macroscopic Concentration Gradient

Formation of a Nanoscale Two-Phase Microstructure in Cu– Zn–Al Samples with Macroscopic Concentration Gradient

Phase Identification in Multi-Phase Cu-Zn/Cu-Al Alloys with Macroscopic Concentration Gradients

Abstract In order to study concentration invariant phase transitions in Cu-Zn-Al alloys which occur in a very narrow concentration range samples with concentration gradients were produced. Cylinders made of brass and Cu-Al bronze were joined by diffusion bonding and subsequently annealed. The samples exhibit multi-phase regions (α, β, γ1, martensite) with changing compositions and very different chemical properties. Standard techniques for the metallographic preparation of Cu alloys usually target specific phases within limited concentration ranges. To evaluate the microstructure, a two-stage chemical etching method using sodium hydroxide solution and subsequent selective colour etching was successfully used to gain an overview of the phases present in different areas along the concentration gradient. For the characterization of martensite, its birefringent optical properties were utilized. In polarized light, details of the martensitic structure are revealed which would otherwise only be accessible by scanning electron methods.

Design of polygalacturonate hydrogels using iron(II) as cross-linkers: A promising route to protect bioavailable iron against oxidation

We designed stable and highly reproducible hydrogels by external unidirectional diffusion of Fe2+ ions into aqueous solutions of polygalacturonate (polyGal) chains. The Fe2+ ions act as cross-linkers between the Gal units in such a way that both the molar ratio R ([Fe2+]/[Gal units] = 0.25) and the mesh size of the polyGal network at the local scale (ξ = 75 ± 5 Å) have constant values within the whole gel, as respectively determined by titration and Small Angle Neutron Scattering. From macroscopic point of view, there is a progressive decrease of polyGal concentration from the part of the gel formed in the early stages of the gelation process, which is homogeneous, transparent and whose Young modulus has a high value of ∼105 Pa, up to the part of the gel formed in the late stages, which is heterogeneous, highly turbid and has a much lower Young modulus of ∼103 Pa. Since the local organization of the polyGal chains remains identical all along the hydrogels, this macroscopic concentration gradient originates from the formation of heterogeneities at a mesoscopic length scale during the gelation process. In addition, X-ray Absorption Spectroscopy measurements remarkably reveal that Fe2+ ions keep their +II oxidation state in the whole gels once they have cross-linked the Gal units. These polyGal hydrogels thus protect iron against oxidation and could be used for iron fortification.

Equilibrium and non-equilibrium concentration fluctuations in a critical binary mixture.

When a macroscopic concentration gradient is present across a binary mixture, long-ranged non-equilibrium concentration fluctuations (NCF) appear as a consequence of the coupling between the gradient and spontaneous equilibrium velocity fluctuations. Long-ranged equilibrium concentration fluctuations (ECF) may be also observed when the mixture is close to a critical point. Here we study the interplay between NCF and critical ECF in a near-critical mixture aniline/cyclohexane in the presence of a vertical concentration gradient. To this aim, we exploit a commercial optical microscope and a simple, custom-made, temperature-controlled cell to obtain simultaneous static and dynamic scattering information on the fluctuations. We first characterise the critical ECF at fixed temperature T above the upper critical solution temperature Tc, in the wide temperature range [Formula: see text] °C. In this range, we observe the expected critical scaling behaviour for both the scattering intensity and the mass diffusion coefficient and we determine the critical exponents [Formula: see text], [Formula: see text] and [Formula: see text], which are found in agreement with the 3D Ising values. We then study the system in the two-phase region (T < T c). In particular, we characterise the interplay between ECF and NCF when the mixture, initially at a temperature Ti, is rapidly brought to a temperature T f > T i. During the transient, a vertical diffusive mass flux is present that causes the onset of NCF, whose amplitude vanishes with time, as the flux goes to zero. We also study the time dependence of the equilibrium scattering intensity I eq, of the crossover wave vector q co and of the diffusion coefficient D during diffusion and find that all these quantities exhibit an exponential relaxation enslaved to the diffusive kinetics.

Diffusiophoresis in suspensions of charged porous particles.

An analysis of the diffusiophoretic motion in a suspension of charged porous spheres in an electrolytic solution with a macroscopic concentration gradient is presented. Each porous particle can be a solvent-permeable and ion-penetrable charged floc or polyelectrolyte molecule, in which the densities of the fixed charges and frictional segments are constant, surrounded by an arbitrary electric double layer. The multiparticle interaction effects are considered through the use of a unit cell model, which allows the overlap of adjacent double layers. The differential equations governing the electric potential, ionic concentration, and fluid velocity distributions inside and outside the porous particle in a unit cell are linearized by assuming that the system is only slightly deviated from equilibrium and then solved as power expansions in its dimensionless fixed-charge density. A closed-form expression for the diffusiophoretic velocity of the porous particle correct to the second order of the fixed charge density is obtained from a balance between the electrostatic and hydrodynamic forces acting on it. Detailed comparisons of the results for the multiparticle diffusiophoresis obtained from the cell model with various boundary conditions are made. The effect of particle interactions on the diffusiophoresis, which is a linear combination of electrophoresis and chemiphoresis, can be significant and complicated in typical situations. Although the electrophoretic mobility of the particles decreases with an increase in the particle volume fraction, their chemiphoretic mobility is not necessarily a monotonic function of it.

Dynamics of mixing and bimolecular reaction kinetics in aquifers

Descriptions of chemical transformation kinetics and hydrologic transport need to be coupled to understand the composition of flowing waters. The coupling of a bimolecular transformation reaction (reactant 1 + reactant 2 → product; rate \( r = \kappa c_{1} c_{2} \)) with spatially heterogeneous subsurface flows is addressed here. The flow microstructure—that controls the spreading rate of solutes and the mean reactant concentrations (C 1, C 2)—creates concentration microstructure whose intensity is characterized by the variances \( \sigma_{{c_{1} }}^{2} \), \( \sigma_{{c_{2} }}^{2} \), and the cross-covariance \( \overline{{c_{1} c_{2} }} \). In addition to the macroscopic overlap of the reactants, that is quantified by the product of the mean concentrations that are routinely modeled using effective dispersion coefficients \( D_{ij} \), the concentration microstructure plays an important role in determining reaction macro-kinetics as the mean reaction rate is \( \overline{r} = \kappa (C_{1} C_{2} + \overline{{c_{1} c_{2} }} ) \). For initially non-overlapping reactants \( \overline{{c_{1} c_{2} }} + C_{1} C_{2} = 0 \) and \( \overline{r} = 0 \) under pure advection. It is shown that due to the action of local dispersion, at large time, the \( \overline{{c_{1} c_{2} }} \) budget is characterized by a balance between its rate of production and dissipation, which results in \( \overline{{c_{1} c_{2} }} \approx 2\tau D_{ij} ({{\partial C_{1} } \mathord{\left/ {\vphantom {{\partial C_{1} } {\partial x_{i} )({{\partial C_{2} } \mathord{\left/ {\vphantom {{\partial C_{2} } {\partial x_{j} )}}} \right. \kern-0pt} {\partial x_{j} )}}}}} \right. \kern-0pt} {\partial x_{i} )({{\partial C_{2} } \mathord{\left/ {\vphantom {{\partial C_{2} } {\partial x_{j} )}}} \right. \kern-0pt} {\partial x_{j} )}}}} \), where τ is the dissipation time-scale characteristic to the destruction of concentration fluctuation by local dispersion. This results in \( \overline{r} = \kappa_{\text{eff}} C_{1} C_{2} \), where \( \kappa_{\text{eff}} \approx \kappa [1 + 2\tau D_{ij} ({{\partial { \ln }C_{1} } \mathord{\left/ {\vphantom {{\partial { \ln }C_{1} } {\partial x_{i} )({{\partial { \ln }C_{2} } \mathord{\left/ {\vphantom {{\partial { \ln }C_{2} } {\partial x_{j} )}}} \right. \kern-0pt} {\partial x_{j} )}}}}} \right. \kern-0pt} {\partial x_{i} )({{\partial { \ln }C_{2} } \mathord{\left/ {\vphantom {{\partial { \ln }C_{2} } {\partial x_{j} )}}} \right. \kern-0pt} {\partial x_{j} )}}}}] \), which accounts for the influence of concentration microstructure and small-scale mixing on the macroscopic bimolecular kinetics. The effective rate parameter κeff is greater than the intrinsic rate constant κ measured under well-mixed conditions if the macroscopic concentration gradients have the same sign (initially overlapping reactants). For the initially non-overlapping reactants which result in macroscopic gradients having opposite signs, κ eff < κ. The macroscopic reactant concentration gradients, effective dispersion coefficients, and the dissipation time-scale control the reaction macro-kinetics, in addition to the intrinsic rate constant κ and the mean reactant concentrations. The formulation for reaction macro-kinetics developed here helps explain previously reported disparities between laboratory and field-scale transformation rates and also provides a way to represent the influence of reactant concentration microstructure in large-scale descriptions of reactive transport.

Electro-diffusive transport in macroscopic porous media: Estimation of effective transport properties using numerical upscaling

In engineering approaches, the electro-diffusive transport behavior of ions through charged porous media is often described by phenomenological theories, formulated on the observation scale of the macroscopic material, without considering the underlying physics on the particle scale. We propose a new approach, by implementing a generalized multiscale framework, based on the classical Poisson–Nernst–Planck theory, which is valid for both charged and uncharged porous media. A numerical upscaling scheme is employed for evaluation of the transport properties in the governing macroscopic equations. Macroscale estimates of the effective diffusion coefficients and of the effective fixed charge concentration are found based on the electrolyte background concentration and on the surface charge applied on the microscale. To demonstrate this new methodology, we implement the upscaling scheme for three different pore geometries: a linear cylindrical pore, a linear slit, and a spherical inclusion. For the chosen geometries and boundary conditions, we find a significant dependence of the effective macroscale properties on the magnitude of prescribed concentrations, the particle surface charge, and the shape of the pore space. Parametric studies reveal effective diffusion coefficients in charged porous media exceeding the corresponding self-diffusion coefficients by up to 34%. For a constant macroscopic concentration gradient, decreasing background concentrations result in a moderate increase of effective diffusion coefficients. The magnitude of the surface charge turns out to strongly influence the effective diffusion coefficients, that is increasing the magnitude of surface charge leads to significantly increased effective diffusion coefficients. These results suggest that rigorously considering electrochemical couplings will lead to a deeper understanding as to the origins of macroscopic experimental observations, and with future inclusion of effects such as cation exchange, to improved computer simulation-based modeling of ion transport through charged porous materials such as clay soils.

Steady-State Local Diffusive Fluxes in Porous Geo-Materials Obtained by Pore-Scale Simulations

Computer simulations of non-sorbing tracers diffusing in fluid-saturated porous sediment/rock were performed using pore-scale X-ray microtomographic images to reveal the following. (i) The histogram of the magnitude of the local diffusive flux vector obeys a unimodal log-normal distribution having a long positive tail. Simulations using model images were also performed to show that the flux broadening in large pores and the flux mixing at the pore network junctions are responsible for the log-normal shape. (ii) The simulation enabled us to directly visualize pore voxels with large and small fluxes, confirming the existence of transport pores and stagnant pores. Because of the unimodal nature, however, it was difficult to distinguish transport pores from stagnant pores using an objective threshold in the histogram. (iii) Another histogram of the flux vector component along the direction of the macroscopic concentration gradient was analyzed. A negative tail was found in the histogram, indicating that local counter diffusion exists in the porous geo-materials. However, the population and intensity of the counter diffusion fluxes are too small and weak to contribute to the overall diffusive transport across the porous media system. A long positive tail representing a large-flux diffusive pathway was also observed in the histogram. However, again, the population of the large-flux transport pores is small. As a result, the main conveyer of the tracer is the stagnant pores (not the transport pores), which have small positive flux values but a large population.

Diffusion in Solutions of Micelles. What Does Dynamic Light Scattering Measure?

Dynamic light scattering (DLS) is routinely used to measure the diffusion of surfactant micelles. Theory and experiment suggest, however, that the microscopic concentration fluctuations monitored by DLS obey the same mutual diffusion equations that describe the decay of macroscopic concentration gradients. According to this interpretation, DLS provides mutual diffusion coefficients for the total surfactant components, including contributions from micelles, free surfactant monomers, and counterions. An attempt is made to decide the correct interpretation of DLS measurements by comparing DLS diffusion coefficients (DDLS) with mutual diffusion coefficients measured by macroscopic gradient techniques for binary aqueous solutions of ionic and zwitterionic surfactants. Possible contributions to DDLS from free surfactant monomers are investigated by extending DLS measurements into the critical micelle (cmc) region where substantial portions of the surfactants diffuse as free monomers. The widely held assumption ...

Gradient-driven fluctuations experiment: fluid fluctuations in microgravity

We describe an experimental breadboard developed for the investigation of nonequilibrium fluctuations induced by macroscopic temperature and concentration gradients under microgravity conditions. Under these conditions the amplitude of the fluctuations diverges strongly for long wavelengths. The setup was developed at the University of Milan and at the University of California at Santa Barbara within the gradient-driven fluctuations experiment (GRADFLEX) project of the European Space Agency, in collaboration with the National Aeronautics and Space Administration. The apparatus uses a quantitative shadowgraph technique for characterization of the static power spectrum of the fluctuations S(q) and the measurement of their dynamics. We present preliminary experimental results for S(q) obtained in the presence of gravity for gradient-driven fluctuations for two cases, those induced in a liquid mixture with a concentration gradient produced by the Soret effect and those induced in a single-component fluid by a temperature gradient.

Protein Diffusion Coefficients Determined by Macroscopic-Gradient Rayleigh Interferometry and Dynamic Light Scattering

Dynamic light scattering (DLS) is extensively used for measuring macromolecule diffusion coefficients. Contrary to classical techniques based on macroscopic concentration gradients, DLS probes microscopic fluctuations in concentration. DLS accuracy and its concordance with macroscopic-gradient techniques remains an outstanding important issue. We measured lysozyme diffusion coefficients in aqueous salt using both DLS and Rayleigh interferometry, a highly accurate macroscopic-gradient technique. The precision of our results is unprecedented. We find that our DLS values were systematically 2% higher than interferometry values. We believe that our interferometric measurements have produced the most accurate diffusion data ever reported for a protein, providing a new standard for quality control of DLS measurements. Furthermore, by interferometry, we have determined the whole diffusion coefficient matrix required for rigorously describing lysozyme-salt coupled diffusion. For the first time, we experimentally demonstrate that DLS does not provide the protein diffusion coefficient but one eigenvalue of the diffusion coefficient matrix.

Scale dependence of reaction rates in porous media

Elemental turnover in porous media depends on substrate concentrations at the pore-scale. In this study, the effect of small scale variability in concentration fields on reaction rate estimates and the validity of the continuum approximation in reactive transport models are investigated via a pore-scale numerical model. Artificial porous media are generated using an identical overlapping sphere algorithm. By comparison between explicit pore-scale simulations and macroscopic continuum approximations, it is shown that inhomogeneous solute distribution within the pores can affect estimates of elemental turnover rates. The error associated with large scale rate estimates depends on the type of reaction, pore geometry, reaction kinetics and macroscopic concentration gradient. A correction term that involves a phenomenological parameter which can be evaluated numerically and macroscopic concentration gradients is introduced to improve the accuracy of upscaled homogeneous reaction rates. Implications for macroscopic descriptions of surface processes and surface attached microbial populations are discussed and it is shown that pore-scale heterogeneity can substantially affect estimates of heterogeneous reactions, while for homogeneous reactions, the error amounts to only a couple of percents.

Comparison of ternary diffusion coefficients obtained from dynamic light scattering and Taylor dispersion

Over the last decade dynamic light scattering (DLS) has developed into a powerful technique for the determination of transport properties in liquids and liquid mixtures. Its advantague is based on the fact that microscopic fluctuations in concentration and temperature are used to characterize the transport behavior and no external macroscopic gradients are needed. All classical methods of Fick's diffusion coefficient determination, like Taylor dispersion (TD) or interferometric methods, however, use macroscopic concentration gradients and give a diffusion coefficient matrix.In the present paper we report results of diffusion coefficient measurements in ternary liquid mixtures of glycerol(0)–acetone(1)–water(2) at 298.15K. Taylor dispersion technique is used to determine the mutual diffusion coefficients Dij along a path of constant mole fraction of water x2=0.4200 from the binary edge towards the critical solution point in the ternary mixture. These results are compared with dynamic light scattering data in the vicinity of the critical solution point and along the very same path of constant x2. The aim of the comparison is to identify the physical character of the mass diffusion mode Dm obtained in our previous DLS measurements. We found that in this ternary system the diffusion mode Dm is not related to any of the diffusion coefficients Dij but we can clearly show that it coincides with the lowest eigenvalue of Fick's diffusion coefficient matrix. There are theoretical arguments which, at least in the vicinity of the critical solution point, lead to the conjection that this behavior might be general for ternary systems with molecules of similar size.

Nearly irreversible, fast heterogeneous reactions in premixed flow

When heterogeneous chemical reaction is sufficiently fast, transport of reactants becomes limiting. In a fixed bed reactor, macroscopic concentration gradients cannot be eliminated as a factor limiting the rate of reaction, possibility coupling to the mesoscopic mass transfer of reactants to the surface of the catalyst as limiting, if the reaction does not occur inside a porous support. A theory for strictly irreversible binary reaction is developed that shows the possibility of regimes of kinetic asymmetry in which a crossover point occurs internally in the reactor, demarking a region of high supersaturation in which the surface is effectively depleted of one reagent, followed by a region of rapidly decaying supersaturation, where the surface is effectively depleted of the other reagent. The parametric dependence of this crossover point is given in terms of a transcendental equation which depends on operating parameters (superficial velocity and inlet concentrations) and ratios of transport properties of the reagents. These solutions are corroborated by full nonlinear numerical computations of the boundary value problem, for the case when asymmetric mass transfer coefficients admit the possibility that the mode of operation switches from relative surface depletion of one reactant to depletion of the other in a binary reaction. It is shown that X indicates the region of greatest molecular efficiency in the reactor, and if operating parameters are so chosen such that X is large, high conversion is achieved in the precrossover region. The modified Thiele modulus analysis of the numerical solutions identifies the crossover point and the region of greatest product formation.

Fast binary reactions in a heterogeneous catalytic batch reactor

When heterogeneous chemical reaction is sufficiently fast, transport of reactants becomes limiting. In a well stirred, batch reactor, macroscopic concentration gradients can be eliminated as a factor limiting the rate of reaction, leaving only the mesoscopic mass transfer of reactants to the surface of the catalyst as limiting, if the reaction does not occur inside a porous support. Here, a transformation of the governing equations for the time-dependence of bulk and surface concentrations results in second order ODE in time and a single nonlinear constraint with boundary values at the initial and infinite times for two auxiliary variables termed modified Thiele moduli. This system of two equations—one differential, one algebraic—and two unknowns is an exact consequence of the governing equations (three ODEs and three algebraic constraints). The power of this formulation is demonstrated by analytic solutions for irreversible and nearly irreversible theories. These solutions are corroborated by full nonlinear numerical computations of the boundary value problem, for the case when asymmetric mass transfer coefficients admit the possibility that the mode of operation switches from relative surface depletion of one reactant to depletion of the other in a binary reaction. The modified Thiele modulus formulation reveals the time scale for the switch over, as well as giving a reliable prediction for the time scale for 99% conversion based on the switch time identified from the irreversible theory.

Steady-state concentration profiles of hydrogen in tubular metallic membranes

This article analyzes the stress-induced diffusion (SID) model for hydrogen transport in thin cylindrical membranes when the planar approximation, usually adopted in the literature, is relaxed. The SID model, explicitly formulated for a cylindrical membrane predicts a non-linear steady-state concentration profile, characterized by a concentration gradient at the inner surface which can be significantly lower than the macroscopic concentration gradient imposed, depending on the initial hydrogen content of the membrane. We provide analytical arguments to show that this effect is due to the fact that concentration-induced stresses do not relax at steady state, and this is a peculiar feature of thin hollow cylinders, not occurring in thin plates.

Gravity-induced sedimentation during melting and liquid phase sintering of Pb–Sn alloys

Gravity-induced phase separation during melting of Pb–Sn alloys results in a macroscopic concentration gradient that must be eliminated before phase equilibrium is established. When Pb–Sn alloys are heat treated in solid-plus-liquid regions (akin to liquid phase sintering), recession of the mushy (solid-plus-liquid) region accompanies elimination of the concentration gradient. Sedimentation apparently ceases as phase equilibrium is approached. Some solid-plus-liquid alloys do not sediment. These have a solid volume fraction sufficient to sterically hinder the gravity-induced phase separation. This critical volume fraction is about 0.50 for hypoeutectic (Pb-rich) alloys and about 0.75 for hypereutectic (Sn-rich) alloys.

On the Use of the Hypothesis of Statistically Homogeneous Suspensions in the Calculation of Their Conductivity

In this paper the condition of a “statistically homogeneous” suspension is carefully analyzed. This condition requires that in the presence of a macroscopic gradient of the electric potential there are no macroscopic ion concentration gradients in the system. It is shown that the use of this condition, which should always be required in the calculation of the electrical conductivity of colloidal suspensions, is far from trivial. Furthermore, its hasty use can lead to substantial errors in the calculation of the conductivity increment.

Giant fluctuations in diffusion processes

Concentration fluctuations in a homogeneous mixture are in general small and confined to the molecular lengthscale. It has been recently predicted that stressed fluids should exhibit anomalously large fluctuations. We will show here that anomalously large fluctuations also occur when macroscopic concentration gradients relax to the uniform state via diffusion. The measurements have been taken by low angle static light scattering and shadowgraphs. We find that at larger wavevectors the amplitude of the fluctuations diverges as q-4. It is also found that gravity stabilizes at a constant value the fluctuations below a critical wavevector. We will present data on a mixture close to a consolution critical point. Recent results will also be presented on ordinary liquid mixtures, a polymer and a protein solution. These new results have been obtained by means of a quantitative shadowgraph technique. They confirm that giant fluctuations are always associated with mass flow due to diffusion across a macroscopic gradient.