## 1. Introduction

[2] Continuum models and particle-based models have been developed and applied extensively to quantify fractal-like kinetics for chemical reactions due to incomplete mixing [*Kopelman*, 1988]. Effective parameters were usually used in these models to describe chemical reaction kinetics. For example, the continuum models typically assume well mixing for reactant particles at the supporting scale of the advection-dispersion-reaction (ADR) equation. This assumption leads to an effective rate coefficient (*K _{f}*) decreasing in time [

*Kapoor et al*., 1997;

*Grima and Schnell*, 2006;

*Sanchez-Vila et al*., 2010], or an effective dispersion coefficient (

*D*

_{eff}) smaller than the macrodispersion coefficient in the ADR equation [

*Cirpka*, 2002]. Complex settings and transport properties were also incorporated into the ADR continuum model, such as the perturbation concentration field [

*Luo et al*., 2008], the spatial concentration correlation [

*Tartakovsky et al*., 2012], and non-Fickian diffusion in time [

*Donado et al*., 2009;

*Willmann et al*., 2010;

*Dentz et al*., 2011] or space [

*Bolster et al*., 2012], to account for nonuniform mixing. In most cases, effective parameters

*K*or

_{f}*D*

_{eff}are involved [

*Tartakovsky et al*., 2012].

[3] Similarly, some particle-based models also define effective parameters, such as the interaction radius *R* that controls the number of reactant pairs in a potential reaction. The particle-based models simulate the packets of unmixed reactant molecules using individual, random-walking particles. They have been widely used by the chemical physics and biology communities (as reviewed by *Erban and Chapman* [2009]), including the compartment-based model (where only molecules within the same compartment can react) [*Hattne et al*., 2005; *Isaacson and Peskin*, 2006] and the grid-free simulation for the motion of individual molecules [*Andrews and Bray*, 2004; *Gillespie*, 2009]. The latter extends the pioneering work of *Smoluchowski* [1917] (where two molecules react whenever they are close to each other) and *Gillespie* [1976, 1977] (for dilute gas particles). Recently, *Benson and Meerschaert* [2008] developed a probability-based scheme to account for the overlapping effective reaction volume of two molecules. *Edery et al*. [2009, 2010] proposed a reactive, continuous time random-walk framework for non-Fickian transport. Other novel particle-based approaches have also been developed to simulate pore-scale reactive transport [*Meakin and Tartakovsky*, 2009], including Lattice-Boltzmann [*Kang et al*., 2006], computational fluid dynamics, and smoothed particle hydrodynamics [*Tartakovsky et al*., 2007, 2009]. Although different reaction schemes may be used in various particle-based approaches, effective parameters such as *R* sometimes are needed to control reaction kinetics [*Benson and Meerschaert*, 2008; *Edery et al*., 2009, 2010]. The parameter *R* has a different physical meaning from *K _{f}*, and the relationship between the two critical parameters remains obscure. In addition, can

*R*remain constant, or will it actually change with time (the same as

*K*) and depend on physical/chemical conditions? How to define correctly the reaction probability in Lagrangian schemes using a predefined

_{f}*R*? These questions need to be explored to reliably link

*R*and

*K*.

_{f}[4] This study attempts to evaluate and link the effective parameters used in the continuum model and the particle-based model for the second-order bimolecular reaction *A* + *B* → *C*. This knowledge gap was identified by *Sanchez-Vila et al*. [2010], who proposed a power-law rate coefficient in the continuum model to interpret efficiently the irreversible bimolecular reaction and suggested further investigation for the relationship between particle-based model parameters and the effective rate coefficient. Such a relationship can help us understand further the reactive kinetics. It can also help field applications, if the effective reaction parameter in the continuum model (commonly used for large-scale processes) can be defined.

[5] Several studies have focused on the comparison of microscopic and macroscopic models, where the dispersion coefficient was assumed to be different from that derived from conservative tracer transport experiments. For example, *Tartakovsky et al*. [2008] and *Tartakovsky* [2010] conducted a detailed comparison of the pore-scale Langevin model (solved with smoothed particle hydrodynamics) and continuum simulations (based on stochastic advection-dispersion equations) for reactions. They found that (1) the *D*_{eff} in the continuum model should be only the molecular diffusion coefficient and (2) the Darcy transport model with a constant rate coefficient overestimates the reaction rate for large Peclet numbers, motivating us to explore the effective rate coefficient in the continuum model for various conditions. In addition, *Porta et al*. [2012a] demonstrated the influence of reaction on effective dispersion through microscale simulation in a plane channel, where nonlocal terms due to time-dependent dispersion coefficient are not significant in the plane channel geometry. *Porta et al*. [2012b] found that the time-nonlocal dispersion terms are critical for reactions after upscaling the pore-scale process to mesoscale using the volume average method. Their models (using a constant rate coefficient) can explain not only the observed tailing behavior of tracer concentrations, but also the time-dependent effective rate coefficient used by others. Motivated by these studies, we will derive the effective parameter in the particle-based model proposed by *Benson and Meerschaert* [2008], and compare it with *K _{f}* in the macroscopic continuum model. In contrast to

*Porta et al*. [2012a, 2012b], here we focus on the time-independent dispersion coefficient in order to quantify the influence of Darcy-scale Fickian diffusion in homogeneous media on reaction kinetics, a deceptively simple case that has not been understood sufficiently yet [

*Benson and Meerschaert*, 2008]. Superdiffusion, subdiffusion, their mixtures, and/or transient anomalous dispersion due to various multiscale medium heterogeneity can be added to our particle-based approach, using, for example, the Langevin models proposed by

*Zhang et al*. [2007, 2008] and extended by

*Zhang et al*. [2013] (see further discussion in section 'Discussion: Extension of the Models Under Other Settings'). Therefore, further investigation is still needed to explore the effective model parameters when a constant dispersion coefficient is used in both the particle-based and continuum models [

*Sanchez-Vila et al*., 2010].

[6] Two commonly used reaction systems are considered in this study. The first one is the bulk system where reactants are initially well mixed and the domain size remains stable. Bulk bimolecular reaction has been the focus for various studies (see, for example, *Kang and Redner* [1984, 1985], among many others), where the known chemical kinetics for both the diffusion-controlled and the thermodynamic rate-limited reactions can be used to check the methods developed by this study. The second system is the sharp-concentration-gradient (SCG) system representing an open system with nonstationary chemical fronts. In the SCG bimolecular reaction, there is no overlap between the opposite reactants initially, and the chemical front expands with time. In laboratory experiments of bimolecular reactions [*Raje and Kapoor*, 2000; *Gramling et al*., 2002; *Oates and Harvey*, 2006], the two reactants *A* and *B* typically have a sharp contact initially, forming the SCG system. Laboratory measurements therefore can be used to define the effective parameters in both the continuum and particle-based models. To the best of our knowledge, the interaction radius used as the effective parameter in the particle-based models has not been systematically evaluated in the above two systems. In addition, the exact form of the effective rate coefficient in the continuum models for the bulk reaction remains obscure (which is also one of the motivations to link the effective parameters in particle-based models and continuum models). The objective of this study is to fill these specific knowledge gaps.

[7] The rest of this paper is organized as follows. In section 'The Bulk System', both the particle-based model and the continuum model with effective parameters are developed to quantify reactive transport in the bulk system. The effective rate coefficient in the continuum model is evaluated based on solutions of the particle-based model. In section 'The Sharp-Concentration-Gradient System', numerical methods are extended to simulate bimolecular reactions in the SCG system. In section 'Applications: Modeling SCG Reactions Observed in Laboratories', the two numerical models are applied to capture chemical kinetics observed in laboratories, and the relationship between effective model parameters is explored. In section 'Discussion: Extension of the Models Under Other Settings', the effective rate coefficient and model extensions are discussed. Finally, conclusions are drawn in section 'Conclusions'. Mathematical descriptions of the numerical methods are shown in Appendices Derivation of the Interaction Radius in the Particle-Based Model for Bimolecular Reaction in the Bulk System, Derivation of Equation (6), Defining the Forward Reaction Probability in the Particle-Based Model, Kf as a Function of Global Reactant Concentration to supplement the evaluation and linkage of model parameters, including the derivation of interaction radius *R* (Appendix Derivation of the Interaction Radius in the Particle-Based Model for Bimolecular Reaction in the Bulk System), the reaction kinetics due to various *K _{f}* (Appendix Derivation of Equation (6)), and the definition of forward reaction probability (Appendix Defining the Forward Reaction Probability in the Particle-Based Model).