A model of self-limiting residual acid diffusion for pattern doubling

Fuhrmann, Jürgen; Fiebach, André; Uhle, M.; Erdmann, Andreas; Szmanda, Charles R.; Truong, Chi

doi:10.1016/j.mee.2008.10.023

Cited by 3 publications

(7 citation statements)

References 11 publications

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“…This reaction system is well known in biology and chemistry, see . For an example occurring in the simulation of the postexposure‐bake step in optical lithography we refer to . The MMH‐mechanism is the reaction system

X_{1} + X_{2} ⇌ X_{3} ⇌ X_{2} + X_{4} .

Here, an enzyme

X_{1}

reacts with the substrate

X_{2}

to form a complex

X_{3}

which then decays to the product

X_{4}

and releases the substrate

X_{2}

.…”

Section: Numerical Examplementioning

confidence: 99%

“…At thermodynamic equilibrium the chemical activities are constant over all control volumes

K \in V

, therefore the solution can be obtained by solving

0 = R_{1} (a^{*}) = a_{1}^{*} a_{2}^{*} - a_{3}^{*}, 0 = R_{4} (a^{*}) = a_{2}^{*} a_{4}^{*} - a_{3}^{*}

together with . In the model that we have in mind , the diffusion coefficient of

X_{2}

is given by

D_{2} (a_{1}, a_{4}) = D_{20} \exp true(- φ_{3} \frac{φ_{1} {\overset{u}{true}}_{1} a_{1} + (1 - φ_{1}) {\overset{u}{true}}_{4} a_{4}}{φ_{2} {\overset{u}{true}}_{1} a_{1} + (1 - φ_{2}) {\overset{u}{true}}_{4} a_{4}} true),

where

φ_{1}, φ_{2}, φ_{3}

are so‐called lumped constants, see . All other diffusion coefficients are assumed to be piecewise constant and independent of the concentration.…”

Section: Numerical Examplementioning

confidence: 99%

“…This reaction system is well known in biology and chemistry, see [28][29][30]. For an example occurring in the simulation of the postexposure-bake step in optical lithography we refer to [8,9,31]. The MMH-mechanism is the reaction system…”

Section: Numerical Examplementioning

confidence: 99%

“…together with (5.1)-(5.2). In the model that we have in mind [9,8], the diffusion coefficient of X 2 is given by…”

Section: Numerical Examplementioning

confidence: 99%

“…where ϕ 1 , ϕ 2 , ϕ 3 are so-called lumped constants, see [9]. All other diffusion coefficients are assumed to be piecewise constant and independent of the concentration.…”

Section: Numerical Examplementioning

confidence: 99%

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Convergence of an implicit Voronoi finite volume method for reaction–diffusion problems

Fiebach

Glitzky

Linke

2015

Numerical Methods Partial

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We investigate the convergence of an implicit Voronoi finite volume method for reaction–diffusion problems including nonlinear diffusion in two space dimensions. The model allows to handle heterogeneous materials and uses the chemical activities of the involved species as primary variables. The numerical scheme works with boundary conforming Delaunay meshes and preserves positivity and the dissipative property of the continuous system. Starting from a result on the global stability of the scheme (uniform, mesh‐independent global upper, and lower bounds), we prove strong convergence of the chemical activities and their gradients to a weak solution of the continuous problem. To illustrate the preservation of qualitative properties by the numerical scheme, we present a long‐term simulation of the Michaelis–Menten–Henri system. Especially, we investigate the decay properties of the relative free energy over several magnitudes of time, and obtain experimental orders of convergence for this quantity. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 141–174, 2016

show abstract

X_{1} + X_{2} ⇌ X_{3} ⇌ X_{2} + X_{4} .

Here, an enzyme

X_{1}

reacts with the substrate

X_{2}

to form a complex

X_{3}

which then decays to the product

X_{4}

and releases the substrate

X_{2}

.…”

Section: Numerical Examplementioning

confidence: 99%

“…At thermodynamic equilibrium the chemical activities are constant over all control volumes

K \in V

, therefore the solution can be obtained by solving

0 = R_{1} (a^{*}) = a_{1}^{*} a_{2}^{*} - a_{3}^{*}, 0 = R_{4} (a^{*}) = a_{2}^{*} a_{4}^{*} - a_{3}^{*}

together with . In the model that we have in mind , the diffusion coefficient of

X_{2}

is given by

D_{2} (a_{1}, a_{4}) = D_{20} \exp true(- φ_{3} \frac{φ_{1} {\overset{u}{true}}_{1} a_{1} + (1 - φ_{1}) {\overset{u}{true}}_{4} a_{4}}{φ_{2} {\overset{u}{true}}_{1} a_{1} + (1 - φ_{2}) {\overset{u}{true}}_{4} a_{4}} true),

where

φ_{1}, φ_{2}, φ_{3}

are so‐called lumped constants, see . All other diffusion coefficients are assumed to be piecewise constant and independent of the concentration.…”

Section: Numerical Examplementioning

confidence: 99%

Section: Numerical Examplementioning

confidence: 99%

“…together with (5.1)-(5.2). In the model that we have in mind [9,8], the diffusion coefficient of X 2 is given by…”

Section: Numerical Examplementioning

confidence: 99%

“…where ϕ 1 , ϕ 2 , ϕ 3 are so-called lumped constants, see [9]. All other diffusion coefficients are assumed to be piecewise constant and independent of the concentration.…”

Section: Numerical Examplementioning

confidence: 99%

See 3 more Smart Citations