Concentration-dependent Coefficient Research Articles

Effect of concentration-dependent diffusion coefficient on dendrite growth in directional solidification

Solute diffusion is an important process that determines the dendrite growth during solidification. The theoretical model generally simplifies the solute diffusion coefficient in liquid phase into a constant. Nevertheless, the composition of the boundary layer changes greatly in the solidification process, the diffusion coefficient will no longer be a constant and is dependent on concentration. In this paper, the quantitative phase field model is used to simulate the effect of concentration-dependent diffusion coefficient on dendrite growth in directional solidification. In the model, the concentration-dependent diffusion process is investigated by coupling the concentration-dependent diffusion coefficient in the liquid solute diffusion equation. A series of simulation results confirms that the concentration-dependent diffusion process has a significant effect on the dendrite growth. The results show that the increase of the coupling intensity of solute concentration will enhance the diffusion of solute in the mushy zone between primary dendrites to the dendrite tip, resulting in the increase of solute enrichment at the dendrite tip, thereby increasing the tip undercooling. The variation of diffusion coefficient in liquid phase has little effect on the tip radius of dendrite, and the simulation results are in good agreement with those from the theoretical model. Moreover, the amplitude of dendritic side branches decreases with the increase of solute diffusion coefficient. In the study of dendrite arrays, it is found that the concentration-dependent diffusion coefficient increases the primary spacing and reduce the tip position. The results of this study indicate that for a system with a concentration-dependent coefficient significantly, the effect of concentration-dependent diffusion on tip undercooling and side branches should be considered in the quantitative and experimental verification of the existing model.

Multi-Temperature in Situ Magnetic Resonance Imaging of Polarization and Salt Precipitation in Li-Ion Battery Electrolytes

The parameterization and validation of robust electrochemical models for lithium-ion batteries in automotive applications, which are intended to address both the state of charge (SOC) and state of health (SOH) estimation problems, is an important milestone in their widespread deployment. Much effort has been focused on the mass transport characteristics of the electrode domains, but a significant fraction of overall polarization stems from concentration gradients that form in the electrolyte domain and their attendant effect on the mass transport parameters, which have recently been shown by in situ MRI to be quite significant, even within modest operational regimes.1 The effect of temperature on these concentration gradients remains to be explored by in situ MRI. The mass transport in the electrolyte domain is typically modelled as being governed by material balance of a binary salt in a concentrated solution, which can be realized by modifying the Nernst-Planck equation to have non-constant, concentration-dependent coefficients (Equation 1).2 This system has a steady-state concentration gradient (Equation 2) when migration flux driven by the potential and diffusion flux against the concentration gradient are in equilibrium (see Figure 1A). Using in situ MRI, the evolution of this gradient toward the steady-state upon constant-current charging can be monitored (Figure 1B). A surprising outcome of these studies was that at 10°C, an equal parts (v/v) mixture of EC/DEC with 1.00 M LiPF6 exhibited anomalous concentration gradient formation (Figure 2) with a current density corresponding to 6 A·m-2 (or ~C/4), whereas the same mixture and cell construction could be polarized without difficulty at room temperature. On comparing raw MR images from before the polarization and after the relaxation of the disrupted concentration gradient, it is clear that the overall signal intensity is lower whilst the shape of the profile is retained, which is a direct indicator that salt precipitation has occurred. There are two routes by which this salt precipitation phenomenon can be avoided: lowering the salt concentration, so that the absolute concentration at the upper portion of the gradient is decreased proportionately for an equivalent current density, and the selection of an electrolyte with better low-temperature solvation characteristics. Successful polarization and visualization of the electrolyte domain in situ was achieved with a 0.85 M variant of the equal-parts EC/DEC mixture utilized above, at a series of temperatures. This constitutes a means of avoiding the critical concentration at the low temperature where the 1.00 M variant exhibited precipitation, but it would come with a trade-off of reduced ionic conductivity. More significantly, a ‘full-strength’ 1.00 M LiPF6 in EC/PC/DMC, 5:2:3 (v/v) mixture was also successfully polarized at low, room, and elevated temperatures, with the same 6 A·m-2 applied current density. Previous ex situ experiments using the pulsed-field gradient NMR technique, in conjunction with conductivity measurements, had revealed that ion pairing at low temperature was significantly reduced in electrolyte mixtures where the fraction of high-dielectric solvent was maximized.3 A quantitative comparison of the steady-state concentration gradients at a series of inter-electrode distances and temperatures was also conducted with this electrolyte formulation, to assess how the magnitudes of the gradients varied with temperature, and to test the hypothesis inherent in Equation 2 that the inter-electrode spacing does not play a role in determining the magnitude of the steady-state gradient. Within 3σ confidence intervals, the inter-electrode spacing was not found to affect the inter-electrode spacing, which directly confirms that the in situ MRI technique can generate steady-state mass transport parameters which are relevant to real battery modelling efforts. A significant effect of the temperature on the magnitude of the steady-state concentration gradient is also illustrated in these multi-temperature imaging experiments, demonstrating the importance of characterizing the spatial variation and temperature-dependence of electrolyte mass transport characteristics for electrochemical model parameterization and validation. References (1) Krachkovskiy, S. A.; Bazak, J. D.; Werhun, P.; Balcom, B. J.; Halalay, I. C.; Goward, G. R. Visualization of Steady-State Ionic Concentration Profiles Formed in Electrolytes during Li-Ion Battery Operation and Determination of Mass-Transport Properties by in Situ Magnetic Resonance Imaging. J. Am. Chem. Soc. 2016, 138 (25), 7992–7999. (2) Sethurajan, A. K.; Krachkovskiy, S. A.; Halalay, I. C.; Goward, G. R.; Protas, B. Accurate Characterization of Ion Transport Properties in Binary Symmetric Electrolytes Using In Situ NMR Imaging and Inverse Modeling. J. Phys. Chem. B 2015, 119 (37), 12238–12248. (3) Krachkovskiy, S. A.; Bazak, J. D.; Fraser, S.; Halalay, I. C.; Goward, G. R. Determination of Mass Transfer Parameters and Ionic Association of LiPF 6 : Organic Carbonates Solutions. J. Electrochem. Soc. 2017, 164 (4), A912–A916. Figure 1

Adsorption reaction and diffusion based-models with solid-phase concentration-dependent coefficient for adsorption of reactive dye

ABSTRACTThe adsorption behavior for tartrazine removal was studied in batch and fixed-bed column operations with the commercially available anion exchange resin. The breakthrough curves were simulated according to mathematical models that considered the mass transfer resistance, such as the film diffusion, pore diffusion, and a combination of both. Analyses of adsorption reaction models in a batch experiment could be an efficient way to select an adsorption model in a fixed-bed operation, because in this study, such models represented the film and pore diffusion phenomena in a separate initial concentration range. The simulation analysis showed that the adsorption process could be controlled by a combination of film and internal particle resistance, and the tartrazine adsorption process was strongly affected by the internal mass transfer resistance. The concentration-dependent rate expression was used to improve the prediction accuracy of both the film and pore diffusion models, and its application to conventional mathematical models could not define meaningful segments in the adsorption process, but could provide a most useful and accurate way of quantitatively estimating the mass transfer effect.

Simultaneous measurements of the density and viscosity of 1-hexene + 1-decene mixtures at high temperatures and high pressures

The density and viscosity of binary 1-hexene+1-decene mixtures have been simultaneously measured over the temperature range from 298 to 470K and at pressures up to 196MPa using hydrostatic weighing and falling-body techniques, respectively. The measurements were made for three concentrations of 0.25, 0.50, and 0.75 mass fraction of 1-decene. The combined expanded uncertainty of the density, pressure, temperature, and viscosity measurements at the 95% confidence level with a coverage factor of k=2 is estimated to be 0.15 to 0.30%, 0.05%, 0.02K, and 1.5 to 2.0%, respectively, depending on temperature and pressure ranges. The measured densities and viscosities were used to calculate excess molar volumes and viscosity differences. It is shown that the values of excess molar volume for 1-hexene+1-decene mixtures are positive, while the viscosity differences are negative at all measured temperatures and pressures over the whole concentration range. The measured densities and viscosities were also used to develop a Tait-type equation of state and viscosity correlation model for liquid 1-hexene+1-decene mixtures. Theoretically based Arrhenius–Andrade type equations with concentration dependent coefficients were used to represent the temperature and concentration dependences of the measured viscosities for liquid 1-hexene+1-decene binary mixtures at atmospheric pressure.

TILT ANGLE AND THE TEMPERATURE SHIFTS CALCULATED AS A FUNCTION OF CONCENTRATION FOR THE AC* PHASE TRANSITION IN A BINARY MIXTURE OF LIQUID CRYSTALS

We study here the tilt angle and the temperature shifts as a function of concentration for the AC* phase transition in a binary mixture, using our mean field model with the biquadratic P2θ2 coupling — and also with the bilinear Pθ and P2θ2 couplings. By expanding the free energy in terms of the tilt angle and polarization, the tilt angle and the temperature shift are evaluated by using the coefficients given in the free energy expansion. By employing a concentration-dependent coefficient, the tilt angle and the temperature shift are calculated as a function of concentration of 10.O.4 for the SmAC* transition in a binary mixture of C7 and 10.O.4. Our calculated values of the tilt angle and the temperature shifts decrease as the concentration of 10.O.4 increases, as confirmed experimentally for the AC* transition in this binary mixture. This indicates that our mean field models studied here are satisfactory to explain the observed behavior of the AC* transition of the binary mixture of C7 and 10.O.4.

Hydrate film growth at the interface between gaseous CO2 and sodium chloride solution

Greenhouse gas CO2 has become a serious problem for human beings. The hydrate technology has been considered as a possible approach to sequester CO2. In this work, the lateral growth rates of a CO2 hydrate film in aqueous NaCl solutions of different concentrations were measured by means of suspending a single gas bubble in liquid. The results show that the film growth rates depended on not only the driving force, but also the NaCl concentration, and the film growth rates decreased with the increasing NaCl concentration. The simple relationship vf∝ΔT 5/2 could be used to correlate the hydrate film growth rate of a CO2 + NaCl + water system by introducing a NaCl concentration-dependent coefficient. The film thickness was investigated experimentally and evaluated theoretically; the results show that it became thicker at a higher NaCl concentration when the temperature and pressure were specified. In addition, a series of interesting phenomena, such as the occurrence of double hydrate films, were displayed and discussed.

Modeling phase behavior for quantifying micro-pervaporation experiments

We present a theoretical model for the evolution of mixture concentrations in a micro-pervaporation device, similar to those recently presented experimentally. The described device makes use of the pervaporation of water through a thin PDMS membrane to build up a solute concentration profile inside a long microfluidic channel. We simplify the evolution of this profile in binary mixtures to a one-dimensional model which comprises two concentration-dependent coefficients. The model then provides a link between directly accessible experimental observations, such as the widths of dense phases or their growth velocity, and the underlying chemical potentials and phenomenological coefficients. It shall thus be useful for quantifying the thermodynamic and dynamic properties of dilute and dense binary mixtures.

Simultaneous sorption and desorption of Cd, Cr, Cu, Ni, Pb, and Zn in acid soils: I. Selectivity sequences

The sorption and desorption of six heavy metals by and from the surface or immediately subsurface horizons of eleven acid soils of Galicia (N.W. Spain) were characterized by means of batch experiments in which the initial sorption solution contained identical mass concentrations of each metal. Concentration-dependent coefficients K d were calculated for the distribution of the metals between the soil and solution phases, and the values obtained for initial sorption solution concentrations of 100 mg L −1 of each metal ( K d100) were used, for each soil, to order the metals as regards their sorption and retention. Pb and Cu were sorbed and retained to a greater extent than Cd, Ni or Zn, which had low K d100 values. Pb was sorbed more than any other metal. Cr was generally sorbed only slightly more than Cd, Ni or Zn, but was strongly retained, with K d100 (retention) values greater than those of Pb and Cu in soils with very low CEC (<3 cmol (+) kg −1). The sorption of Pb and Cu correlated with organic matter content, while the retention of these and the other metals considered appeared to depend on clay minerals, especially kaolinite, gibbsite, and vermiculite.

Heat and water mass transfer modeling in polyimide based advanced composites

In this work, analytical and numerical models were utilized to analyze the absorption of water in polyimide resins. Past studies showed that the presence of water or other fluids often had deleterious effects on the long term performance of composite materials. Therefore, the issue of concern addressed here was the hydrophilic nature of polyimide resins and its impact on polyimide utilization as matrix resins in fiber reinforced composites designed for aircraft parts. To fully understand and characterize the behavior of the polyimide resins, it was necessary to evaluate the water concentration and thermal profiles that might be encountered during service conditions. Consequently, several models of heat and mass transfer were developed to investigate the absorption of water in these materials. Initially, analytical models based on Fickian diffusion assumptions were developed, with deviations from this ideal behavior expressed by various changing boundary conditions. As analytical models could not provide a solution to complex nonFickian cases, finite differential methods were applied to develop numerical models that accounted for concentration dependent coefficients of diffusion, as well as periodic boundary conditions relevant to hygrothermal cycling. Such models were found to provide excellent agreement with the experimental data. Additionally, thermal profiles for the simulation of an airplane engine core cowl were also calculated. Overall, this work provided the means to understand the hygrothermal conditions within a polyimide composite during laboratory simulated service conditions.

Field-assisted cation exchange: an independent support of the kinetic interaction model

The field-assisted exchange of cations is discussed in terms of the kinetic interaction model. As a consequence of this model we obtain a generalized diffusion equation with concentration dependent coefficients. From this point of view we get as a stable solution a shape invariant concentration profile growing into the bulk with a constant velocity. The predicted results are compared with experimental data.

Mathematical modeling of laser-induced diffusion processes

The possibility of using lasers to form semiconductor materials (cadmium and mercury tellurides) with a predetermined width of the band gap was explored using mathematical modeling. A one-dimensional physicomathematical model based on diffusion and heat conduction equations and transport equations for laser radiation with temperature- and concentration-dependent coefficients was used. Self-similar solutions were used in order to obtain certain qualitative regularities of the processes. In the general case the corresponding system of partial differential equations was integrated numerically.

First and Second Concentration-Dependent Coefficients of Translational Diffusion and Sedimentation for Poly(α-methylstyrene) in a Good Solvent

The first and second concentration-dependent coefficients of translational diffusion, kD and kD2, and of sedimentation, ks and ks2, for poly(α-methylstyrene) fractions of high molecular weight in benzene at 30 °C were determined by dynamic light scattering and sedimentation velocity measurements. In the molecular weight, Mw, range of 3.76 × 105 ≤ Mw ≤ 6.85 × 106, the first coefficients are positive, as reported previously, but the second are negative, which has not been previously observed. These molecular weight dependences are described by power laws (k ∝ Mwa), but the exponents for kD (0.769) and ks (0.663) and for kD2 (1.54) and ks2 (1.48) are smaller than the predicted values in the good-solvent limit, 0.80 and 1.60, respectively. Expressed in the volume-fraction frame of reference, the coefficients are not constant, which contradicts the hard sphere approximation for swollen chains, and are represented well by universal functions of X, the ratio of the thermodynamic interaction radius derived from the second virial coefficient, A2, and the hydrodynamic chain radius derived from the diffusion coefficient; the functions are as predicted by Akcasu and Benmouna with a Mw-independent thermodynamic g factor (=A3/A22Mw) = 0.22−0.23. The hydrodynamic g factors defined by kD2/kD2 and ks2/ks2 are constant, −0.14 and −0.05, respectively, for higher Mw. The two- and three-body effective hydrodynamic interaction radii deduced from ks and ks2 are larger than those from kD and kD2, indicating the long-range nature of the purely hydrodynamic interactions. The swollen chain dynamics in good solvents cannot be described by the hard sphere approximation but require a more detailed treatment of hydrodynamic interactions.

A mathematical interpretation of diffusion processes with concentration-dependent diffusion coefficient

The integral balance method has been used to obtain an approximate analytical solution of a nonlinear boundary value problem which arises in the theory of diffusion with a concentration-dependent coefficient. It is the purpose of this paper to give an interpretation of the supposition of interface reactions which obey the law of kinetic mass action.

The effects of concentration on partitioning of flexible chains into pores

Equilibrium partitioning between bulk solution and pores was investigated by computer experiment using the Monte Carlo technique. In a multichain athermal system, self-avoiding walks of up to 60 steps were generated on a simple cubic lattice with variable pore size and chain concentration. The results demonstrate that the constraints imposed by the pores and/or neighbouring chains reduce the chain entropy. The entropy related parameter, the partition coefficient K (the equilibrium pore-to-bulk concentration ratio) increases linearly with bulk concentration ø. A maximum concentration dependence of K was found for large pores characterized by a coil-to-pore dimension ratio λ of about 0.1–0.3. In this range the increase in bulk concentration brings about a dramatic enhancement of partitioning into pores. The results of computer experiments are compared with predictions of hard-sphere solutes theory and with static and gel-chromatographic measurements of the partitioning equilibrium. The implications of the concentration-dependent coefficient K for hindered transport of flexible polymers inside pores are briefly discussed.

Thermodynamic properties of ethanol and water— III. Description of the different excess functions ( gE, hE, cEP) using an empirical model

Published values of the thermodynamic excess functions ( g E, h E, c E P ) are used to fit the temperature-dependent parameters of an empirical g E model. A modified Redlich-Kister approach with six concentration-dependent coefficients is chosen as a model equation for the excess Gibbs energy. The temperature dependence of the coefficients is defined in such a way that the result is a quadratic dependence of the molar excess heat capacity c E P as a function of the temperature. The fitting is carried out over the temperature range 25–150°C using the maximum likelihood method combined with the Gauss-Newton procedure (Anderson et al.). The h E and c E P data, in particular, are clearly reproduced much better than when a fit based only on VLE data is used.

Thermodynamic prediction of the ae3 temperature of steels with additions of Mn, Si, Ni, Cr, Mo, Cu

Published binary phase diagrams and activity data for iron and binary and ternary alloys have been used to evaluate the general linear series expansion of the activity coefficient and the standard free energy changes and these have been employed in turn for the accurate thermodynamic determination of the Ae3 temperature in steels with additions of Mn, Si, Ni, Cr, Mo, and Cu. A computer program accurate for total additions up to 7.0 wt pct and an analytic formula accurate for total additions up to 2.5 wt pct have been developed. The predicted Ae3 temperatures compare favorably with observations on over 200 steels from international compendia. It is demonstrated that existing linear regression formulae are incorrect and that for low alloy additions they should be replaced by linear expressions with carbon concentration dependent coefficients.

Interference nature of the anomalous hall and Nernst-Ettingshausen effects below the Debye temperature

is satisfied worse than in [3], where T -> | The slightly t empera tu re and concentrat ion-dependent coefficients Bi differ in magnitude and sign f rom those found in [3]. The concentrat ion-dependent pa ramete r s R(0), b, c for powers of T and the specific e lec t r ica l res is t iv i ty p do not disclose the corre la t ion predicted by theory. The shear coefficient B 5 is hence almost s imi lar to that in [3]. For the ~ phase Bs/| D = -2 .14 9 10 -16, while for the T phase of the alloys Bs/| = -1 .66 9 10 -16 ~2 9 cm/G. Let us note that the rms deviation of the exper i mental values f rom those found is not worse than 1%. An est imate of the e r r o r in determining the pa ramete r s R~ ~ b, and c by using Chebyshev polynomials yields ar 10%, :~ ~30%, %~100%.

Adsorption in finite bath and countercurrent flow with systems having a concentration dependant coefficient of diffusion

In many adsorbents surface diffusion is the governing mechanism of transport. The coefficient of surface diffusion may be concentration dependent. A model based on the assumption that the coefficient of diffusion is exponentially dependent on concentration in the adsorbent is used. Filmresistance is included. The model equations for adsorption in finite bath and in countercurrent flow have been solved for some cases for a range of parameter values which may apply for many adsorption systems where activated carbon is used for adsorbing organic compounds from aqueous solution. The results are presented in generalized diagrams. A comparison is made with other models with constant coefficient of diffusion. It is shown that by proper choice of variables most models give very similar results. The very simple model based on pore diffusion and a saturation isotherm approximately describes other cases such as pore diffusion and surface diffusion with and without a concentration dependent coefficient of diffusion. The saturation isotherm model has a simple analytical solution which makes it very simple to use.

Variational solutions for two nonlinear boundary-value problems for diffusion with concentration-dependent coefficients

The theory of complementary variational principles is used to obtain variational solutions for two nonlinear boundary-value problems which arise in the theory of diffusion with concentration-dependent coefficients.

Regularity Properties of Flows Through Porous Media: A Counterexample

Regularity Properties of Flows Through Porous Media: A Counterexample