Complex dynamics in initially separated reaction-diffusion systems

Havlin, Shlomo; Araujo, Mariela; Lereah, Y.; Larralde, Hernán; Shehter, Arkady; Stanley, H. Eugene; Trunfio, Paul; Vilensky, Baruch

doi:10.1016/0378-4371(95)00246-4

Cited by 24 publications

(18 citation statements)

References 45 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As some aspects of this problem were already considered [4], we present briefly only those results which are relevant to our paper. In Fig.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…However, when one of the components is immobile (or 'static'), the asymptotic kinetics of the reaction front can change dramatically. For example, in the mean-field approximation the width of the front converges to a stationary value, and the reaction rate at x f decreases as t −1/2 , which corresponds to α = 0 and β = 1 2 [4,22]. We have therefore two asymptotic universality classes: one characteristic for the 'dynamic' systems in which both components diffuse, and the 'quasistatic' one observed if one of the diffusion constants is zero.…”

Section: Introductionmentioning

confidence: 99%

“…This is due to the fact that not only the kinetics of such systems exhibits many surprising features [1][2][3][4][5], but they are also amenable to experimental studies [3,6,7].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic behaviour of initially separated A + B(static) → 0 reaction-diffusion systems

Koza

1997

Physica A: Statistical Mechanics and its Applications

View full text Add to dashboard Cite

We examine the long-time behaviour of A + B(static) → 0 reaction-diffusion systems with initially separated species A and B. All of our analysis is carried out for arbitrary (positive) values of the diffusion constant D A of particles A and initial concentrations a 0 and b 0 of A's and B's. We derive general formulae for the location of the reaction zone centre, the total reaction rate, and the concentration profile of species A outside the reaction zone. The general properties of the reaction zone are studied with a help of the scaling ansatz. Using the mean-field approximation we find the functional forms of 'tails' of the reaction rate R and the dependence of the width of the reaction zone on the external parameters of the system. We also study the change in the kinetics of the system with D B > 0 in the limit D B → 0. Our results are supported by numerical solutions of the mean-field reaction-diffusion equation.

show abstract

“…As some aspects of this problem were already considered [4], we present briefly only those results which are relevant to our paper. In Fig.…”

Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Asymptotic behaviour of initially separated A + B(static) → 0 reaction-diffusion systems

Koza

1997

Physica A: Statistical Mechanics and its Applications

View full text Add to dashboard Cite

show abstract

“…Mathematically, the extra condition is called the Stefan condition [10][11][12][13][14][15][16]. If diffusion occurs inside a circular domain and the reaction occurs at the periphery of this domain, the solution to the Stefan problem in two dimensions is given by a scaling variable determined from the conservation of the total number of adsorbates on the substrate including those incorporated in the solid domain [11][12][13][17][18][19][20][21]. This situation is converse to the present case for which the diffusing adsorbates on the substrate are present outside the circular solid domain.…”

Section: Introductionmentioning

confidence: 99%

Scaling theory for two-dimensional single domain growth driven by attachment of diffusing adsorbates

Seki

2019

New J. Phys.

View full text Add to dashboard Cite

Epitaxial growth methods are a key technology used in producing large-area thin films on substrates but as a result of various factors controlling growth processes the rational optimization of growth conditions is rather difficult. Mathematical modeling is one approach used in studying the effects of controlling factors on domain growth. The present study is motivated by a recently found scaling relation between the domain radius and time for chemical vapor deposition of graphene. Mathematically, we need to solve the Stefan problem; when the boundary moves, its position should be determined separately from the boundary conditions needed to obtain the spatial profile of diffusing adsorbates. We derive a closed equation for the growth rate constant defined as the domain area divided by the time duration. We obtain approximate analytical expressions for the growth rate; the growth rate constant is expressed as a function of the two-dimensional diffusion constant and the rate constant for the attachment of adsorbates to the solid domain. In experiments, the area is decreased by stopping the source gas flow. The rate of decrease of the area is obtained from theory. The theoretical results presented provide a foundation to study controlling factors for domain growth.

show abstract

“…The results that do exist for early time behavior are primarily focused on the motion and width of the front when reactants are separated at a discrete interface. [32][33][34][35][36][37][38][39] In the engineering literature on diffusion-reaction equations, anomalous kinetics are characterized by the effectiveness factor, which is defined as the ratio of the observed reaction rate to the classical (i.e., non-transport-limited) reaction rate. 11 Although this parameter provides no information about the exponents for scaling in time, it is nevertheless a useful concept to describe the effective rate of reaction in the regimes where no simple scaling law is expected to exist.…”

Section: Introductionmentioning

confidence: 99%

A non-scale-invariant form for coarse-grained diffusion-reaction equations

Ostvar

Wood

2016

The Journal of Chemical Physics

View full text Add to dashboard Cite

The process of mixing and reaction is a challenging problem to understand mathematically. Although there have been successes in describing the effective properties of mixing and reaction under a number of regimes, process descriptions for early times have been challenging for cases where the structure of the initial conditions is highly segregated. In this paper, we use the method of volume averaging to develop a rigorous theory for diffusive mixing with reactions from initial to asymptotic times under highly segregated initial conditions in a bounded domain. One key feature that arises in this development is that the functional form of the averaged differential mass balance equations is not, in general, scale invariant. Upon upscaling, an additional source term arises that helps to account for the initial configuration of the reacting chemical species. In this development, we derive the macroscopic parameters (a macroscale source term and an effectiveness factor modifying the reaction rate) defined in the macroscale diffusion-reaction equation and provide example applications for several initial configurations.

show abstract

Complex dynamics in initially separated reaction-diffusion systems

Cited by 24 publications

References 45 publications

Asymptotic behaviour of initially separated A + B(static) → 0 reaction-diffusion systems

Asymptotic behaviour of initially separated A + B(static) → 0 reaction-diffusion systems

Scaling theory for two-dimensional single domain growth driven by attachment of diffusing adsorbates

A non-scale-invariant form for coarse-grained diffusion-reaction equations

Contact Info

Product

Resources

About