Thermal Physics, 2nd ed.

Kittel, C.; Kroemer, H.; Scott, H. L.

doi:10.1119/1.19072

Cited by 345 publications

(537 citation statements)

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“…(30). This is nontrivial for complex biomolecules since the discretization of the jump conditions is defined on the interface, and in most cases the interface points are off grid.…”

Section: Matched Interface and Boundary Methodsmentioning

confidence: 99%

“…In this work, we utilize the matched interface and boundary method [25][26][27][36][37][38] for the discretization of Eq. (30). The main idea of the MIB method is that to maintain the designed order of accuracy, the finite difference schemes for regular points (away from the interface) and irregular points (near the interface) are different.…”

Section: Matched Interface and Boundary Methodsmentioning

confidence: 99%

“…Under the conditions that the particles do not interact and that their concentrations are sufficiently low, the total chemical potential (electrochemical potential) of the αth ion species, denoted as μ α , can be expressed as 30 μ α (r) = k B T ln n α (r) − k B T ln n Qα + q α (r), (1) where is the electrostatic potential of the system, k B and T are the Boltzmann constant and the absolute temperature, respectively, n α and q α are the concentration and the valence of the αth ion species, respectively. Here, n Qα is the intrinsic concentration of the αth species, 30 note that the intrinsic concentration is independent of the concentration n α , and for a specific ion species, it is a constant under a given temperature. For example, n Q is the density of state for an ideal gas n Q = ((m g k B T )/(2π¯2)) 3/2 , where m g is the molecular mass of the gas and¯is the reduced Planck constant.…”

Section: A Poisson-nernst-planck (Pnp) Modelmentioning

confidence: 99%

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Poisson–Boltzmann–Nernst–Planck model

Zheng

Wei

2011

The Journal of Chemical Physics

141

111

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The Poisson-Nernst-Planck (PNP) model is based on a mean-field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential. It provides qualitative explanation and increasingly quantitative predictions of experimental measurements for the ion transport problems in many areas such as semiconductor devices, nanofluidic systems, and biological systems, despite many limitations. While the PNP model gives a good prediction of the ion transport phenomenon for chemical, physical, and biological systems, the number of equations to be solved and the number of diffusion coefficient profiles to be determined for the calculation directly depend on the number of ion species in the system, since each ion species corresponds to one Nernst-Planck equation and one position-dependent diffusion coefficient profile. In a complex system with multiple ion species, the PNP can be computationally expensive and parameter demanding, as experimental measurements of diffusion coefficient profiles are generally quite limited for most confined regions such as ion channels, nanostructures and nanopores. We propose an alternative model to reduce number of Nernst-Planck equations to be solved in complex chemical and biological systems with multiple ion species by substituting Nernst-Planck equations with Boltzmann distributions of ion concentrations. As such, we solve the coupled Poisson-Boltzmann and Nernst-Planck (PBNP) equations, instead of the PNP equations. The proposed PBNP equations are derived from a total energy functional by using the variational principle. We design a number of computational techniques, including the Dirichlet to Neumann mapping, the matched interface and boundary, and relaxation based iterative procedure, to ensure efficient solution of the proposed PBNP equations. Two protein molecules, cytochrome c551 and Gramicidin A, are employed to validate the proposed model under a wide range of bulk ion concentrations and external voltages. Extensive numerical experiments show that there is an excellent consistency between the results predicted from the present PBNP model and those obtained from the PNP model in terms of the electrostatic potentials, ion concentration profiles, and current-voltage (I-V) curves. The present PBNP model is further validated by a comparison with experimental measurements of I-V curves under various ion bulk concentrations. Numerical experiments indicate that the proposed PBNP model is more efficient than the original PNP model in terms of simulation time.

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“…(30). This is nontrivial for complex biomolecules since the discretization of the jump conditions is defined on the interface, and in most cases the interface points are off grid.…”

Section: Matched Interface and Boundary Methodsmentioning

confidence: 99%

Section: Matched Interface and Boundary Methodsmentioning

confidence: 99%

Section: A Poisson-nernst-planck (Pnp) Modelmentioning

confidence: 99%

See 1 more Smart Citation

Poisson–Boltzmann–Nernst–Planck model

Zheng

Wei

2011

The Journal of Chemical Physics

141

111

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“…The driving force for boundary migration in this case, denoted p g , is given by (21). An expression for the corresponding mobility, M g , can be obtained by stipulating that, in this case, the migration velocity in the direction of the local unit normal, n, should be given by (see Porter and Easterling [30])…”

Section: Grain-boundary Mobilitymentioning

confidence: 99%

Advances in the numerical treatment of grain-boundary migration: Coupling with mass transport and mechanics

Mourad

Garikipati

2006

Computer Methods in Applied Mechanics and Engineering

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This work is based upon a coupled, lattice-based continuum formulation that was previously applied to problems involving strong coupling between mechanics and mass transport; e.g. diffusional creep and electromigration [1, 2]. Here we discuss an enhancement of this formulation to account for migrating grain boundaries. The level set method is used to model grain-boundary migration in an Eulerian framework where a grain boundary is represented as the zero level set of an evolving higherdimensional function. This approach can easily be generalized to model other problems involving migrating interfaces; e.g. void evolution and freesurface morphology evolution. The level-set equation is recast in a remarkably simple form which obviates the need for spatial stabilization techniques. This simplified level-set formulation makes use of velocity extension and field re-initialization techniques. In addition, a least-squares smoothing technique is used to compute the local curvature of a grain boundary directly from the level-set field without resorting to higherorder interpolation. A notable feature is that the coupling between mass transport, mechanics and grain-boundary migration is fully accounted for. The complexities associated with this coupling are highlighted and the operator-split algorithm used to solve the coupled equations is described.

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“…In this work, the growth and dissolution rates were derived with matrix concentrations based on the diffusion-limited growth law and detailed balance equilibrium theory [1], And, continuity equations for interstitial oxygen, silicon interstitials and vacancies were solved considering annealing ambients and surface conditions. Generally, before thermal process,in orcdr to protect the surface of silicon thin oxide films are coated.…”

Section: Introductionmentioning

confidence: 99%

Modeling and Simulation of Oxygen Precipitation in CZ Silicon

Yoon

Kwack

1998

Simulation of Semiconductor Processes and Devices 1998

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This paper describes modeling and simulation for the growth and dissolution of oxygen precipitates in Czochralski silicon(CZ) wafers during heat treatment. Growth and dissolution rates arc newly derived and inserted into a set of chemical rate equations(CILEs) and a Fokkor-Planck equation(FPE). Annealing ambients and surface conditions are taken into account for solving continuity equations in order to calculate oxygon depth profile and oxygen precipitates density more precisely.

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Thermal Physics, 2nd ed.

Cited by 345 publications

References 0 publications

Poisson–Boltzmann–Nernst–Planck model

Poisson–Boltzmann–Nernst–Planck model

Advances in the numerical treatment of grain-boundary migration: Coupling with mass transport and mechanics

Modeling and Simulation of Oxygen Precipitation in CZ Silicon

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