A Primer of Diffusion Problems

Ghez, R.

doi:10.1002/3527602836

Cited by 90 publications

(36 citation statements)

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“…Here G(σ → σ ) is viewed as an element of the transition matrix for the transition from state σ to state σ [2]. Alternatively, one can write an equivalent difference-differential equation [53] dσ…”

Section: Microscopic State Microscopic Processes and The Master Equmentioning

confidence: 99%

An overview of spatial microscopic and accelerated kinetic Monte Carlo methods

Chatterjee

Vlachos

2007

J Computer-Aided Mater Des

423

400

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The microscopic spatial kinetic Monte Carlo (KMC) method has been employed extensively in materials modeling. In this review paper, we focus on different traditional and multiscale KMC algorithms, challenges associated with their implementation, and methods developed to overcome these challenges. In the first part of the paper, we compare the implementation and computational cost of the null-event and rejection-free microscopic KMC algorithms. A firmer and more general foundation of the null-event KMC algorithm is presented. Statistical equivalence between the null-event and rejection-free KMC algorithms is also demonstrated. Implementation and efficiency of various search and update algorithms, which are at the heart of all spatial KMC simulations, are outlined and compared via numerical examples. In the second half of the paper, we review various spatial and temporal multiscale KMC methods, namely, the coarse-grained Monte Carlo (CGMC), the stochastic singular perturbation approximation, and the τ -leap methods, introduced recently to overcome the disparity of length and time scales and the one-at-a time execution of events. The concepts of the CGMC and the τ -leap methods, stochastic closures, multigrid methods, error associated with coarse-graining, a posteriori error estimates for generating spatially adaptive coarse-grained lattices, and computational speed-up upon coarse-graining are illustrated through simple examples from crystal growth, defect dynamics, adsorption-desorption, surface diffusion, and phase transitions.

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Section: Microscopic State Microscopic Processes and The Master Equmentioning

confidence: 99%

An overview of spatial microscopic and accelerated kinetic Monte Carlo methods

Chatterjee

Vlachos

2007

J Computer-Aided Mater Des

423

400

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“…Fick's equation establishes a relationship between the component flow and the concentration gradients (GHEZ, 1988;RAOULT-WACK, 1994). The ideas and theories on diffusion are well-established (CRANK, 1975;GEANKOPLIS, 1972).…”

Section: Introductionmentioning

confidence: 99%

Saline distribution during multicomponent salting in pre-cooked quail eggs

Borsato

Moreira

et al. 2012

Food Sci. Technol

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The relationship of NaCl with problems of arterial hypertension has led to a reduction in the levels of this salt in food production. KCl has been used as a partial substitute for NaCl since it cannot be completely substituted without affecting the acceptability of the end product. In this study, the diffusion that occurs during quail egg salting in static and stirred brine was simulated. The mathematical model used was based on a generalization of the Fick's 2nd law, and the COMSOL Multiphysics software was used to simulate the diffusion in the NaCl-KCl-water system. The deviations in the simulated data and experimental data were 2.50% for NaCl and 6.98% for KCl in static brine, while in the stirred brine they were 3.48% for NaCl and 4.72% for KCl. The simulation results presented good agreement with the experimental values and validated the predictive capacity of the model.

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“…16 To predict the c s and c p dependence of S͑c s , c p ͒, we use its relation to the diffusion coefficient D. Numerical methods were used to extract D from the motion of interfaces in a swelling L ␣ lamellar system. 14 Here, we adapt an analytical solution to this Stefan ͑moving boundary͒ problem, previously used for the precipitation of a solid phase from a supersaturated liquid, 17 and the growth of a colloidal crystal. 18 This requires two assumptions, both phases are semi-infinite and thus the concentrations c s and c p are fixed far from the interface ͑only at late times do we observe subdiffusive growth caused by the limited extent of the two phases͒ and polymer diffusion is slow into the lamellar phase, i.e., into the gaps between bilayers, but fast within the contacting ͑bulk͒ polymer solution relative to the movement of the interface, thus c p is constant in both time and space.…”

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confidence: 99%

Swelling and shrinking kinetics of a lamellar gel phase

Fairhurst¹,

Baker²,

Shaw³

et al. 2008

Applied Physics Letters

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We investigate the swelling and shrinking of L ␤ lamellar gel phases composed of surfactant and fatty alcohol after contact with aqueous poly͑ethyleneglycol͒ solutions. The height change ⌬h͑t͒ is diffusionlike with a swelling coefficient S: ⌬h = S ͱ t. On increasing polymer concentration, we observe sequentially slower swelling, absence of swelling, and finally shrinking of the lamellar phase. This behavior is summarized in a nonequilibrium diagram and the composition dependence of S quantitatively described by a generic model. We find a diffusion coefficient, the only free parameter, consistent with previous measurements. © 2008 American Institute of Physics. ͓DOI: 10.1063/1.2913762͔In everyday life and many industrial processes, materials swell by absorption of solvent, e.g., washing powder, 1 foodstuff, 2 diapers, 3 eyeballs, 4 and clay. 5 Conversely, if the solvent flow is reversed materials shrink, as for a hypertonic cell with a lower solute concentration than its environment. Model systems are often preferred for study. Swelling rates of L ␣ surfactant lamellar phases are observed to change when the chemical potential difference between lamellar phase and contacting solution is varied through polymer addition. 6 Artificial liposomes can be swollen or shrunk using glycerol solutions 7 and hard sphere colloidal suspensions shrink when contacted with high concentration polymer solutions. 8 Here, we quantitatively investigate the swelling and shrinking behavior of a complex surfactant system, namely an L ␤ lamellar gel phase, as used in cosmetics and pharmaceuticals. The volume change is initiated by contact with aqueous polymer solution. Our observations are in quantitative agreement with a generic model, which we expect to be applicable to a large variety of situations.The lamellar phase is prepared following an industrial procedure. 9,10 It consists of a cationic quaternary surfactant ͓behenyl trimethyl ammonium chloride ͑BTAC͔͒ and a fatty alcohol ͓1-octadecanol͔ at a molar ratio of 1:3 in water with different total surfactant concentrations c s . Electron and light microscopy reveal a disordered system. Numerous stacks of bilayers form an open structure with small pockets of water and excess fatty alcohol. 10 While the sample is prepared at elevated temperature T, the experiments are performed at T = 25°C, which is below the chain melting temperature ͑about 78°C͒. Approximately 0.5 g of this L ␤ lamellar gel phase is pipetted into a 2 cm 3 cylindrical glass cell and centrifuged for 1 min at 2500 rpm to ensure that the entire highly viscous sample is at the bottom of the cell with a smooth upper surface. Within a range of lamellar masses ͑0.45-1.25 g͒ no systematic change in behavior was observed within the ϳ12% experimental uncertainty. The experiment is started by adding 1.5 g of water or polymer solution on top of the lamellar phase. The use of polymer ͓poly͑ethyleneglycol͒, PEG-10000 with molar mass from 8500 to 11500 g / mol and an average radius of gyration of about 3 nm͔ allows us to vary the diffe...

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A Primer of Diffusion Problems

Cited by 90 publications

References 0 publications

An overview of spatial microscopic and accelerated kinetic Monte Carlo methods

An overview of spatial microscopic and accelerated kinetic Monte Carlo methods

Saline distribution during multicomponent salting in pre-cooked quail eggs

Swelling and shrinking kinetics of a lamellar gel phase

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