Linear Concentration Profiles Research Articles

A generalized mathematical framework for thermal oxidation kinetics

We present a generalized mathematical model for thermal oxidation and the growth kinetics of oxide films. The model expands long-standing classical models by taking into account the reaction occurring at the interface as well as transport processes in greater detail. The standard Deal-Grove model (the linear-parabolic rate law) relies on the assumption of quasi-static diffusion that results in a linear concentration profile of, for example, oxidant species in the oxide layer. By relaxing this assumption and resolving the entire problem, three regimes can be clearly identified corresponding to different stages of oxidation. Namely, the oxidation starts with the reaction-controlled regime (described by a linear rate law), is followed by a transitional regime (described by a logarithmic or power law depending on the stoichiometry coefficient m), and ends with the well-known diffusion-controlled regime (described by a parabolic rate law). The theory of Deal-Grove is shown to be the lower order approximation of the proposed model. Various oxidation rate laws are unified into a single model to describe the entire oxidation process.

Biotinylation, a Post-translational Modification Controlled by the Rate of Protein-Protein Association

Biotin protein ligases catalyze specific covalent linkage of the coenzyme biotin to biotin-dependent carboxylases. The reaction proceeds in two steps, including synthesis of an adenylated intermediate followed by biotin transfer to the carboxylase substrate. In this work specificity in the transfer reaction was investigated using single turnover stopped-flow and quench-flow assays. Cognate and noncognate reactions were measured using the enzymes and minimal biotin acceptor substrates from Escherichia coli, Pyrococcus horikoshii, and Homo sapiens. The kinetic analysis demonstrates that for all enzyme-substrate pairs the bimolecular rate of association of enzyme with substrate limits post-translational biotinylation. In addition, in noncognate reactions the three enzymes displayed a range of selectivities. These results highlight the importance of protein-protein binding kinetics for specific biotin addition to carboxylases and provide one mechanism for determining biotin distribution in metabolism.

Simple and effective method to measure the diffusion coefficient of organic vapors in porous media

A quick and reliable method to measure a vapor concentration within porous substrate was developed. The technique consists of two steps, where a modified head space single drop microextraction (HS-SDME) sampling is used to entrap the vapor phase. In the second step, the entrapped vapor concentration is measured by gas chromatography (GC-FID). The technique is used to measure an effective diffusion coefficient of n-pentane in dry medium grain sand, with the sand partially saturated with water as an inert liquid. The measurements are carried out in a cylindrical sand holder on which the HS-SDME sampling ports are mounted. A linear vapor concentration profile along the bed thickness is found. From known concentration gradient and measuring the mass flux gravimetrically, the effective diffusion coefficient is determined. It turns out that the diffusion coefficient decreases from 8.49 × 10−6 for dry sand to 7.13 × 10−6 m2 s−1 as a function of water saturation. Additional hindrance to the vapor transport is observed from both the simple volumetric effect due to the porous medium void space reduction caused by the presence of water and increase of the tortuosity.

Carbon diffusion in friction welded joints of refractory metals in a liquid

The effect of carbon on materials joined by friction welding in a liquid was studied. Titanium and tantalum specimens resistance-heated in a liquid and then specimens friction-welded in a liquid were tested. Same-metal (titanium, vanadium, tantalum) joints and dissimilar-metal joints were friction welded. The distribution of microhardness in the friction welded joints and their microstructure and linear concentration profiles were determined. The carbon from shielding liquid decomposition interacts with the metals being welded. This applies to the surface layers on the flash. In vanadium–vanadium and vanadium–other metal joints a marked increase in microhardness in a narrow zone at the surface of the joint and carbide segregations in this zone on the vanadium side occur. In titanium joints an increase in carbon content in the weld zone occurs.

Generation of arbitrary monotonic concentration profiles by a serial dilution microfluidic network composed of microchannels with a high fluidic-resistance ratio

This paper reports a serial dilution microfluidic network composed of microchannels with a high fluidic-resistance ratio for generating linear concentration profiles as well as logarithmic concentration profiles spanning 3 and 6 orders of magnitude. The microfluidic networks were composed of thin fluidic-resistance microchannels with 160 to 730 microm(2) cross-sectional areas and thick diffusion-mixing microchannels with 3,600 to 17,000 microm(2) cross-sectional areas, and were fabricated from polydimethylsiloxane by multilayer photolithography and replica molding. We proposed a design algorithm of the microfluidic network for an arbitrary monotonic concentration profile by means of a hydrodynamic calculation. Because of the high fluidic-resistance ratio of the fluidic-resistance microchannels to the diffusion-mixing microchannels, appropriate geometry and dimensions of the fluidic-resistance microchannels allowed us to obtain desired concentration profiles. The fabricated microfluidic network was compact, occupying a 8 x 18 to 21.0 x 13.5 mm(2) area on the microchip. Both the linear and the logarithmic concentration profiles were successfully generated with the error less than 15% for the linear concentration profile, 22% and 35% for the logarithmic concentration profiles of 3 and 6 orders of magnitude, respectively. The generated linear concentration profiles of the small molecule, calcein, were independent of the flow rate within the range of 0.009 to 0.23 microL/min. The concentration profiles of the large molecules, dextrans, depended on the flow rate and molecular weight. The required residence time of large molecules in the diffusion-mixing microchannel was correlated with dimensionless diffusion time, Fick number, and was discussed based on the scaling law. These compact, stable serial dilution microfluidic networks are expected to be applied to various integrated on-chip analyses.

Optimisation and analysis of microreactor designs for microfluidic gradient generation using a purpose built optical detection system for entire chip imaging

This paper presents and fully characterises a novel simplification approach for the development of microsystem based concentration gradient generators with significantly reduced microfluidic networks. Three microreactors are presented; a pair of two-inlet six-outlet (2-6) networks and a two-inlet eleven-outlet (2-11) network design. The mathematical approach has been validated experimentally using a purpose built optical detection system. The experimental results are shown to be in very good agreement with the theoretical predictions from the model. The developed networks are proven to deliver precise linear concentration gradients (R(2) = 0.9973 and 0.9991 for the (2-6) designs) and the simplified networks are shown to provide enhanced performance over conventional designs, overcoming some of the practical issues associated with traditional networks. The optical measurements were precise enough to validate the linearity in each level of the conventional (2-6) networks (R(2) ranged from 0.9999 to 0.9973) compared to R(2) = 1 for the theoretical model. CFD results show that there is an effective upper limit on the operating flow rate. The new simplified (2-11) design was able to maintain a linear outlet profile up to 0.8 microl/s per inlet (R(2) = 0.9992). The proposed approach is widely applicable for the production of linear and arbitrary concentration profiles, with the potential for high throughput applications that span a wide range of chemical and biological studies.

Refractive index profile optimisation for the design of optical fibres

Owing to advanced manufacturing techniques, it is possible to produce cylindrical single-mode fibres with nearly arbitrary refractive index profiles. For the design of optical fibres automated optimisation schemes have yet to be exploited. We have employed deterministic local, and stochastic optimisation schemes for the minimisation of a cost function based on dispersion, dispersion slope, macro-bending losses and mode-field diameter, on the space of continuous piecewise linear dopant concentration profiles. For the local schemes (modified and quasi Newton), it appears possible to select a few initial profiles, such that the optimisation results are close to the global optima (within 8%), found using schemes (simulated annealing and differential evolution), while reducing computation times significantly (minutes instead of days). For the local schemes, the cost function gradient is required. Frechet derivatives are more efficient than finite-difference approximations. A sensitivity analysis provides useful information for manufacturers regarding the required profile accuracy. A comparison of our optimised fibre designs with commercially available optical fibres demonstrates that existing fibres can be improved.

A platform for assessing chemotactic migration within a spatiotemporally defined 3D microenvironment

While the quantification of cell movement within defined biochemical gradients is now possible with microfluidic approaches, translating this capability to biologically relevant three-dimensional microenvironments remains a challenge. We introduce an accessible platform, requiring only standard tools (e.g. pipettes), that provides robust soluble factor control within a three-dimensional biological matrix. We demonstrate long-lasting linear and non-linear concentration profiles that were maintained for up to ten days using 34.5 muL solute volume. We also demonstrate the ability to superimpose local soluble factor pulses onto existing gradients via defined dosing windows. The combination of long-term and transient gradient characteristics within a three-dimensional environment opens the door for signaling studies that investigate the migratory behavior of cells within a biologically representative matrix. To this end, we apply temporally evolving and long-lasting gradients to study the chemotactic responses of human neutrophils and the invasion of metastatic rat mammary adenocarcinoma cells (MtLN3) within three-dimensional collagen matrices.

Paste boriding process: evaluation of boron mobility on borided steels

The present study evaluates the boron mobility at the material surface of borided steels AISI 1045 and M2 during the paste boriding process. This surface hardening technique produces two characteristic phases on the material, FeB, Fe2B and a transition zone, denominated diffusion zone, in the layer/substrate interface. The thermochemical treatment was performed at three different temperatures: 1193, 1223, 1273 K and two treatment times: 2 and 6 h for the 1045 steel, and 1223, 1253 and 1273 K with the same treatment times for M2 steel, modifying the boron potentials in equilibrium at the substrates surfaces. By using the mass balance equation, and assuming a linear concentration profile at the interfaces, the new model determined the diffusion coefficients of boron in the phases formed at the surface of the substrates, included the diffusion zone for both steels.

Evaluation of boron mobility on the phases FeB, Fe2B and diffusion zone in AISI 1045 and M2 steels

The present study evaluates the growth kinetics of boride layers at the material surface on AISI 1045 and M2 steels during the paste boriding process. This surface hardening technique produces on the material two characteristic phases FeB, Fe2B and a transition zone, denominated diffusion zone, in the layer/substrate interface. The thermochemical treatment was done at three different temperatures: 1193, 1223 and 1273K with two treatment times: 2 and 6h for the 1045 steel, and 1223, 1253 and 1273K with the same treatment times for M2 steel, modifying the boron potentials in equilibrium at the substrates surfaces. Using the mass balance equation, and assuming a linear concentration profile at the interfaces, the mobility of boron was determined on both types of steels. The influence of boron potential, treatment time and temperatures is clearly observed on the growth kinetics of boride layers.

The crossed interdiffusion of sodium nitrate and sulfate through an anion exchange membrane, as studied by Raman spectroscopy

Raman spectroscopy is applied to visualize concentration profiles in solution adjacent to the top of DSV anion exchange membranes in horizontal positions. The concentration profiles of sulfates, nitrates and their fluxes through the membrane are recorded as a function of time during 13–15 h experiments. A three layer mathematical model based on extended Nernst–Planck equations with a convective term is developed. The membrane’s kinetic and static parameters, such as ionic diffusion and ion exchange coefficients, are determined by separate experiments. Only one parameter, the diffusion boundary layer thickness, δ, is fitted. It is found that the model described fluxes well, especially when the variation of δ is taken into account. At the same time, the model gives nearly linear concentration profiles, whereas the measured profiles are much more complicated. This complicated shape is explained by natural convection and the emergence of the Benard vortexes in the system.

Effect of temperature on the gas permeation properties of NaY-type zeolite formed on the inner surface of a porous support tube

Since the micropore size of Y-type zeolite is in the range of 0.74– 1.2 nm , small molecules are able to pass one another inside the micropores. Thus ion-exchanged Y-type zeolite membranes are expected to show catalytic activity as well as being useful for gas separation with high permeation rates. In this study, NaY-type zeolite membranes were synthesized by a hydrothermal reaction on the inner surface of a porous α-alumina support tube. The Y-type zeolite membranes were stable to heat at temperatures up to 400°C, and the permeances to single-component hydrocarbons showed a maximum at temperatures that were dependent on the heat of adsorption. The permeances to C 3H 6, n-C 4H 10 and i-C 4H 10 decreased with time at elevated temperatures. This can be explained by the deposition of carbonaceous matter, which was removed by a heat treatment in an atmosphere of O 2 at 300–400°C. The permeation mechanism for the NaY-type zeolite membrane was investigated for single-component CO 2. The Maxwell–Stefan diffusivities were compared with the Fickian diffusivities obtained using a sorption–diffusion model, in which linear concentration profiles across the membrane were assumed. The differences between the Fickian diffusivities and the Maxwell–Stefan diffusivities were negligible at elevated temperatures.

Facilitated Transport of CO2 in Liquid Membrane of CsHCO3 Aqueous Solution.

The facilitated transport of CO2 gas through liquid membranes of aqueous 0-6.27mol/L CsHCO3 solution was studied. The permeability ratio Pco2/Po2 was found to increase at high CsHCO3 concentration in the liquid membrane. The pressure dependence of the permeability coefficients was also measured. The concentration profiles of CO2, HCO3-, CO32- and H+ were calculated by assuming that the concentration of CO2 and HCO3- can be represented as polynomial functions of the position coordinate x in the membrane, and the Fick’s second law holds. The experimental results of the permeability coefficients of CO2 were explained satisfactory by assuming the linear concentration profiles of CO2 and HCO3- in the membrane.

Onset of stationary and oscillatory convection in a tilted porous cavity saturated with a binary fluid: Linear stability analysis

In the present work, we study the onset of double-diffusive convective regimes in a tilted rectangular cavity, filled with a porous medium, saturated by a binary fluid. Two opposite walls are maintained at different but uniform temperatures and concentrations while the two other walls are impermeable and adiabatic. When the thermal and solutal buoyancy forces are comparable in intensity but have opposite signs, the motionless double-diffusive regime with linear temperature and concentration profiles is a solution of the problem. The first part of the study consists of a linear stability analysis of the motionless regime. We determine the critical thermal Rayleigh number for the onset of stationary and oscillatory convection. Indeed, we point out that there exist primary Hopf bifurcations for the studied problem in porous medium, while in the same configuration with a fluid medium only primary stationary bifurcations exist. When the first primary bifurcation creates a steady state branch of solutions, the bifurcation is either transcritical or pitchfork depending on the aspect ratio, A and the tilt, φ of the cavity. The onset of oscillatory convection (Hopf bifurcation) depends not only on A and φ but also on the Lewis number, Le and the normalized porosity, ε. Then, we determine the parts of the (Le, ε) parameter space for which the first primary bifurcation is stationary or oscillatory. In particular, it is found that in the case Le⩾1 and for ε Le2<1 the first primary bifurcation is always a Hopf bifurcation for any A and φ except for φ=+90°. For ε Le2>1 only stationary primary bifurcations exist. In the case Le<1, zones where stationary and oscillatory primary bifurcations exist are separated by a curve depending on A and φ. The last part of this work consists of a series of numerical simulations. The onset of stationary and oscillatory convection is obtained numerically at the critical Rayleigh number predicted by linear analysis. We also verified the frequency of oscillations for several sets of dimensionless parameters. The numerical simulations show multiple subcritical solutions.

Convection in a Continuously Stratified Fluid

This paper begins by commemorating Count Rumford's observations of convection in 1797. Rumford investigated the effects of stewed apples, eiderdown, and other fibrous materials on retarding vertical circulation in liquids. In contrast to Count Rumford's historic experiments, the present paper describes a simple, visually instructive, physical chemistry experiment for demonstrating how a mathematical model can be applied to free convection in a vertically stratified salt solution. In the first part of the experiment, a linear salt concentration profile is created in a beaker of water at room temperature. The salt concentration (and density) is greatest at the bottom of the liquid. When the beaker containing the salt solution is then heated on a hot plate, a well defined convecting bottom layer forms almost immediately. The measured rate of growth in thickness of this convecting layer is compared to that predicted from a model based on density differences. The temperature of the liquid above the bottom conv...

Thickness and concentration profile of the boundary layer in electrodialysis

Back electrical motive force (emf) measurements with spiral electrodialysis (SpED) modules showed that obtaining the profile of the back emf transient curves during depolarization is difficult from the Nernst model, and the assumption of a linear concentration profile in a stirred polarized boundary layer is oversimplified. A non-linear concentration distribution model derived from the error function is introduced.

Ordering of block copolymer melts in confined geometry

Microphase separation of symmetric diblock copolymers between two rigid walls is simulated by a coarse-grained model to study the geometry of the microdomains without presupposing their basic shape. The geometry is found to be dependent on the wall separation and the domains can be either aligned horizontally or vertically. The horizontal geometry is induced by wetting at the walls and lamellas parallel to them develop quickly after a quench from the one-phase region. The vertical geometry involves domains oriented perpendicular to the walls and develops much more slowly. For identical walls, some segregation of the wetting phase to the walls is observed for the vertical geometry. This segregation does not significantly change the structure. If the vertical phase develops between walls that attract opposite phases, however, a linear concentration profile develops. In some cases the structure of this vertical phase is found to be quite different from simple, perpendicularly oriented lamellas.

On the nonlinear stability of a fluid layer of a mixture heated and salted from below

The nonlinear stability of the motionless state of a binary fluid mixture heated and salted from below, in the Oberbeck-Boussinesq scheme, is studied, in the stress-free boundary case. A stabilizing effect of gradient of solute on thermal convection is shown and a globally nonlinear exponential stability theorem is proved. In particular, when the ratio r of the Schmidt and the Prandtl number is less than 1, a region of coincidence of linear and nonlinear critical parameters is found. The stability of plane parallel convective flows (plane Couette and Poiseuille flows with linear temperature and concentration profiles) is also studied. Stability conditions independent of Reynolds number are found.

Simulated concentration profiles of redox centers in thin polymer films: Oxidation-state dependence of diffusion and partitioning between film and solu

Charge and mass transport in polymer films containing electrostatically bound redox centers are complicated by several factors: (1) the partitioning of the redox centers out of the film and into the adjacent solution, (2) the diffusion of redox centers within the film and in solution, and (3) the state of oxidation of the film, which affects the diffusivities locally. In this paper, digital simulations are used to analyze the effects of these factors separately on concentration profiles established laterally in a film between parallel cathodic and anodic band electrodes. The simulations are based on explicit finite difference methods and include partitioning of redox molecules between film and solution. The partitioning is governed by capacity factors for oxidized, A, and reduced, B, species, analogous to those defined in separation science, where solutes partition between stationary and mobile phases. Linear concentration profiles are obtained for cases where diffusion is (1) limited to the film without leaching of redox centers into the solution, and (2) limited to the solution, but with a partitioning equilibrium between the film and solution. The ratio of the slopes of the concentration profiles for B and A depends on the ratio of diffusion coefficients D A / D B and on the ratio of capacity factors k′ B′/ k′ A. Non-linear concentration profiles are established when transport depends on the state of oxidation of the film.

Gas dispersion in volume-cycled tube flow. I. Theory.

A mathematical theory is derived for the dispersion of a contaminant bolus introduced into a fully developed volume-cycled oscillatory pipe flow. The convection-diffusion equation is solved for a tracer gas bolus by expressing the local concentration field as a series expansion of derivatives of the area-averaged concentration. The local, as well as the area-averaged, concentration is determined for a uniform initial slug or Gaussian bolus. The effect of various flow parameters such as Womersley parameter, Schmidt number, and tidal volume is investigated. The overall dispersion is characterized by a time-averaged effective diffusion coefficient, which for long duration coincides with previous dispersion theories based on a constant linear axial concentration profile. The effective diffusion coefficient can be determined from the local time history of concentration, independent of the spatial location or the initial tracer bolus. Furthermore the local peaks of the concentration-time curve follow a decaying curve dictated by the time-averaged effective diffusion coefficient. Thus the theory is directly applicable for dispersion measurements in oscillatory tube flows, a basis for the pulmonary airways application, as shown by Gaver et al. (J. Appl. Physiol. 72: 321-331, 1992).