Particle-based models and continuum models have been developed to quantify mixing-limited bimolecular reactions for decades. Effective model parameters control reaction kinetics, but the relationship between the particle-based model parameter (such as the interaction radius R) and the continuum model parameter (i.e., the effective rate coefficient Kf) remains obscure. This study attempts to evaluate and link R and Kf for the second-order bimolecular reaction in both the bulk and the sharp-concentration-gradient (SCG) systems. First, in the bulk system, the agent-based method reveals that R remains constant for irreversible reactions and decreases nonlinearly in time for a reversible reaction, while mathematical analysis shows that Kf transitions from an exponential to a power-law function. Qualitative link between R and Kf can then be built for the irreversible reaction with equal initial reactant concentrations. Second, in the SCG system with a reaction interface, numerical experiments show that when R and Kf decline as t−1/2 (for example, to account for the reactant front expansion), the two models capture the transient power-law growth of product mass, and their effective parameters have the same functional form. Finally, revisiting of laboratory experiments further shows that the best fit factor in R and Kf is on the same order, and both models can efficiently describe chemical kinetics observed in the SCG system. Effective model parameters used to describe reaction kinetics therefore may be linked directly, where the exact linkage may depend on the chemical and physical properties of the system.
 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 (Kf) decreasing in time [Kapoor et al., 1997; Grima and Schnell, 2006; Sanchez-Vila et al., 2010], or an effective dispersion coefficient (Deff) 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 Kf or Deff are involved [Tartakovsky et al., 2012].
 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 ), 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  (where two molecules react whenever they are close to each other) and Gillespie [1976, 1977] (for dilute gas particles). Recently, Benson and Meerschaert  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 Kf, 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 Kf) and depend on physical/chemical conditions? How to define correctly the reaction probability in Lagrangian schemes using a predefined R? These questions need to be explored to reliably link R and Kf.
 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. , 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.
 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.  and Tartakovsky  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 Deff 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 , and compare it with Kf 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.  (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].
 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.
where and are the forward and reverse kinetic coefficients of reaction, respectively; L [L] is the total domain size (i.e., length in one dimension); represents the initial concentration of reactant A; [dimensionless] denotes the initial number of A particles; Δt [T] is the time step (defined later); , , and [dimensionless] are the number of A, B, and C particles at time t, respectively; and mC and mB [M] denote the mass carried by each C and B particles, respectively. In this study, the superscript “0” means that the parameter is a constant, to distinguish it from the time-dependent parameter. The interaction radius has dimensions of [L] (i.e., length). The sign of indicates the direction of the reaction. If , the forward reaction will dominate, since the forward rate of reaction is larger than the reverse rate. Conversely, when , the reverse reaction dominates.
 For a 1-D irreversible reaction A + B → C, (1) reduces to a constant radius
where mA [M] denotes the mass carried by each A particle. The interaction radius (2) is a positive constant, implying that only the forward reaction can occur. The above interaction radius R identifies the pairs of opposite reactant particles that can collide. If the distance of two A and B particles is less than R, they have the chance to react. Hence, a smaller R means a slower reaction rate, if all the other properties (especially the mobility of reactants) are the same. The definition (2) is a 1-D simplification of the 3-D radius proposed by Pogson et al. . It also explains the following empirical probability of forward reaction proposed by Benson and Meerschaert :
where Ω [L] denotes the domain size (i.e., L in this study), N0 [dimensionless] is the initial number of particles , and is associated with the density of two reactant particles separated by a distance s.
 Two factors in (2) including the time step Δt and the number of particles have no predictions, limiting the model predictability. The time step Δt can be selected as
where so that the forward reaction probability and so that the reverse reaction probability . Numerical experiments (not shown here) reveal that Δt following (3) does not significantly change the reaction kinetics, due to the counterbalance between the increased interaction radius and the increased particle mobility. A smaller Δt does lead to a smaller R, but it also simultaneously decreases the displacement (i.e., the mobility) of reactants during each jump. Hence, the global reaction rate does not change significantly. On the other hand, (or the particle number density ) significantly affects the reaction kinetics [Benson and Meerschaert, 2008; Porta et al., 2012a] and therefore constrains the model predictability. Pogson et al.  suggested defining each particle as a reactant molecule, so that
where S [mol−1] is Avogadro's constant. Such definition however may result in too many particles. In Benson and Meerschaert , each particle can represent a packet of reactant molecules, to increase the computational efficiency. The only fitting factor in R, therefore, is the initial particle number .
2.2. Continuum Approximation of the Global Mass Evolution
 The continuum approximation solves the kinetic equation (i.e., the rate equation) with effective rate coefficient [Grima and Schnell, 2006]. The rate equation is
where , , and (with dimensions of ) denote the concentrations of A, B, and C, respectively. The mass evolution in a reversible reaction with time-dependent forward and/or reverse rate coefficients has not been focused on before. For simplicity, we assume that the forward and reverse rates follow the same functional forms (so that the analytical solution can be derived), such as the power-law function
where the exponent , and and are constant. The resultant reactant concentration is (see Appendix Derivation of Equation (6) for derivation):
where , , , and can be either zero or positive. When reaching equilibrium , the reactant A concentration becomes constant: , which can also be obtained by assuming that t is large in (6b).
 We first check the applicability of the particle-based model in capturing reaction kinetics described by the macroscopic approximations (section 'Continuum Approximation of the Global Mass Evolution'). Figure 1 shows that with a relatively large particle number and a relatively strong particle mobility (D = 0.1), the Lagrangian approach developed in section 'The Particle-Based Model to Simulate Reaction A + B ↔ C' simulates the rate-limited, irreversible and reversible, reactions with equal or unequal initial concentrations between A and B. Here D denotes the diffusion coefficient, to capture the pure diffusion in the bulk system (without advection).
 We then explore the effective parameters in both the particle-based and the continuum models for incomplete mixing. Figures 2a–2c show the modeled for irreversible reactions. The symbols in Figures 2d–2f show the overall rate coefficient simulated by the particle-based model, which can be generally matched by the continuum model with an effective rate coefficient Kf transferring from an exponential to a power-law function. This conclusion contains uncertainty, owning to the noise in the particle-based model results (the noise decreases with an increase of the particle number). Numerical tests also show that for the case of , the power-law exponent of is −3/4, following the well-known 1/4-decay law of reactant concentration [Kang and Redner, 1985]. For the other cases tested in Figure 2, the power-law exponent of is slightly larger than −3/4.
 Figure 3 shows the results for reversible reactions. A similar transition for is also observed. At an early time, the continuum model solution (8a), (8b) with an exponential and matches the particle-based solution (Figure 3a). Figure 3e shows that the value calculated by the particle-based model declines as a power law at the late time, implying that the effective rate coefficients and in the continuum model should be power-law functions defined as (5). This conclusion is validated by the lines in Figure 3e, which are the analytical solutions (6).
2.4. Discussion: Effective Parameters in the Two Models
 The continuum approximation discussed above reveals the transient decrease of the effective rate coefficient due to the anomalous decline of reactant mass. During an early period, numerical experiments show that the best fit effective in the continuum model does not remain constant but rather declines exponentially (due to the gradual deviation from well mixing), resulting in a reactant mass decay slower than the classical thermodynamic law. At a late time when the separated clusters of opposite reactants add additional delay to the reaction [Benson and Meerschaert, 2008], the effective rate coefficient declines even slower than exponential, following the power law. The power-law approach to equilibrium or full depletion is found for all test cases (including both the irreversible and reversible reactions, and various initial concentrations for reactants), implying that the power-law effective rate coefficient in the continuum model might be universal for reactions in the bulk system at late times.
 The link between parameters in the particle-based model and the continuum model cannot be built quantitatively, but the qualitative relationship between model parameters (or even the quantitative relationship for a few parameters) can be explored for most cases. For example, for the irreversible reaction with equal initial concentrations, the continuum model with the following effective rate coefficient can capture the full-range mass evolution simulated by the particle-based model:
 Here τ [T] and s [dimensionless] are fitting parameters, although they might be approximated roughly. For example, the time τ (the critical time recording the transition of from exponential to power-law functions) can be approximated by
if . Here defines the diffusion time across the interaction radius. The negative relationship between the transition time and the diffusion time described by (10) is expected, because a smaller interaction radius or a stronger mobility (i.e., diffusivity) leads to a higher reaction rate and therefore, a larger τ. A similar characteristic transition time was derived analytically by Tartakovsky et al. [2012, equation (52)] for a d-D domain.
 Applications of (9) are shown in Figure 4. The continuum model with the effective rate coefficient (9) produces a similar mass evolution as the particle-based model. It is also noteworthy that after the effective rate coefficient is built, one can write as a function of the global concentration of reactants (see Appendix Kf as a Function of Global Reactant Concentration for examples).
3. The Sharp-Concentration-Gradient System
 The SCG reaction has been simulated systematically using macroscopic continuum models [Sanchez-Vila et al., 2010; Willmann et al., 2010; among others]. Analytical analysis of reaction kinetics in the SCG system can also be found in a few studies [Gálfi and Rácz, 1988; Cornell and Droz, 1993; Howard and Cardy, 1995; among others]. Recently, Porta et al. [2012a] developed microscale simulation and numerical upscaling of a SCG bimolecular reaction A + B → C in a 3-D plane channel with a parabolic distribution of fluid velocity in the vertical direction. Here we constrain our attention to the 1-D model, to keep consistent with the bulk system considered above.
 We explore the kinetics for irreversible bimolecular reactions with equal initial concentrations for the SCG system. The same condition was used in laboratory experiments discussed in section 4.
3.1. The SCG Bimolecular Reaction Simulated by the Particle-Based Approach
3.1.1. Early- and Late-Time Approximations of Reaction Rate
 There is no analytical solution for the interaction radius for the SCG reaction. To evaluate the reaction rate at early or late times, we follow the arguments in O'Shaughnessy and Vavylonis . O'Shaughnessy and Vavylonis  found that the product mass (denoted as m(t) [M]) is related to the number of A-B pairs in the SCG interface
where the superscript “s” denotes the density at the SCG interface.
 At the initial time when the reactant concentrations around the SCG interface are close to the far-field concentrations (since the time is too short to deplete a significant amount of reactants), (11) leads to
where the superscript “∞” denotes the value far away from the interface (which remains constant in this case). Hence, the product mass increases linearly with time t initially:
where tinterface is a limit in time defined later.
 At late times, O'Shaughnessy and Vavylonis  found that the SCG interface particle number for reactant A-B pairs decreases as
where Xt is proportional to the diffusive displacement of particles, which follows the t1/2 law in our case:
 We will test the above theoretical results using numerical simulations next.
3.1.2. Particle-Based Modeling Scheme
 The particle-tracking scheme should remain the same as the one developed for the bulk system, except for the interaction radius for the SCG system (denoted as ). We propose and compare two methods to approximate .
 First, remains stable and it is assumed to be the same as (2), if we regard the whole model domain as the reaction zone of reactants. The motivation of this method is based on the work by Edery et al. [2009, 2010], where a constant interaction radius was fitted by laboratory data. This simple method can be used for all cases where observations are available, but the constant value (2) does not account for the potential influence of the reaction interface on the interaction radius.
where the unit parameter γ = 1 has dimensions of Tη, and βParticle therefore has dimensions of , which is the same as the rate constant used in the other models discussed later. The subscript “Particle” denotes the particle-based model. The exponent η is between 0 and 1. When η = 0.5, the resultant product mass increases as the square root of time, which follows (16). This implies that η = 0.5 might be used to build the interaction radius. Following the procedure in Appendix Derivation of the Interaction Radius in the Particle-Based Model for Bimolecular Reaction in the Bulk System, we find the corresponding interaction radius
 When η = 0 (e.g., no interface), (18) reduces to (2). Hence, the functional form of (18) might be valid for both the bulk and SCG systems.
 The t−0.5 decline rate of might be related to the expansion of the injected reactant front. When the two opposite reactants share the same advective velocity, the plume front of the injected reactant particles expands as , where α = 2 for Fickian diffusion (and for superdiffusion) [Zhang et al., 2008]. According to (16), the product mass increases as , which can be described by the rate equation . The corresponding interaction radius is , which is (18) when α = 2. Therefore, a faster expansion rate for the reactant front will result in a relatively slower decreasing rate of the interaction radius, which leads to a relatively faster reaction rate. This hypothesis is consistent with the conclusion in Bolster et al. .
 In addition, the flow velocity can affect βParticle (see further discussion later). Finally, (18) shows that the interaction radius is a function of the numerical resolution (e.g., the time step Δt and the initial particle number ), but βParticle should not depend on Δt or .
 Numerical tests (Figure 5a) show that the (18) captures the scaling law of product mass analyzed in section 'Early- and Late-Time Approximations of Reaction Rate'. When , the early-time (i.e., when time t < tinterface) product mass increases linearly with time. This linear growth cannot be captured when a constant Rs is used. At late times, the product mass simulated by a constant Rs is slightly higher than that using a time-dependent (since the latter has a smaller interaction radius). In addition, if the exponent η in (18) is smaller, the late-time product mass tends to be relatively higher, as expected (see Figure 5a for the case of ).
 Numerical experiments (Figure 5b) also show that the opposite reactant particles can survive at the diffusive front and then react at a very late time. Figure 5b shows the “birth” position of each product C particle due to reaction (not the spreading of C particles after reaction), which is located in the middle of each pair of A and B particles. This implies that the diffusive front might not deplete completely in the modeling time, and the injected reactant front expands as t1/2. In the following, we will check further both the constant radius (2) and the time-dependent radius (18) against experimental data. Note that the simulated t0.5-law increase of product mass is consistent with the rate observed by Porta et al. [2012a] using microscale simulation and numerical upscaling for a reactive flow in a plane channel.
3.1.3. The Transition Time
 We check the above particle-based scheme extensively, with several examples shown in Figures 6a and 6b. The Lagrangian model with captures the early-time (see (13)) and late-time behaviors (see (16)) of product mass revealed earlier. Figures 6c and 6d show that the simulated global reaction rate declines as t−0.5 at late times.
 We found empirically the transition time:
 Approximation (19) can be derived by assuming that tinterface is the time where the advective length is first reached by the diffusive displacement
 Hence, the time (19) defines the period for the depletion zone to be fully developed.
 At the early time , numerical experiments imply that the reaction kinetics is affected mainly by the rate constant βParticle (Figure 6a). A and B particles meet around the sharp interface at the very beginning, whose distance is small enough for opposite particles to react. In this short time and small domain, the rate constant βParticle affects the reactive capability more than the dispersion coefficient D does. At the late time , D affects both the expansion of the reactant front and the mobility of particles around the reaction interface. Therefore, D plays a more important role in the resultant reaction rate at the late time than at the early time. Between the early and the late times, there is a transition zone, whose duration increases with the decrease of the rate constant βParticle.
3.2. The Continuum Approach
 We develop a Galerkin finite element method (GFEM) to solve the ADR equation:
where vi and Di (i = A,B,C) are the flow velocity and the dispersion coefficient, respectively; T is the maximum time period; and and (I = A,B,C) are the boundary and initial value functions, respectively. Based on the variational formulation of the ADR, we discretize the temporal derivatives using the backward Euler scheme and the spatial terms using the finite element scheme. Picard's method is used to linearize the nonlinear forward reaction terms. The solver defined in a piecewise linear element space produces a stable and accurate approximation with the optimal convergence rate of first order in the energy norm. The program is available to readers upon request. The GFEM solver is slightly different from the continuum approach proposed by Sanchez-Vila et al. : it also models the reverse reaction, any types of rate coefficients, and species-dependent transport parameters (such as the dispersion coefficient).
 We first test the power-law rate coefficient proposed by Sanchez-Vila et al. 
where the dimensions for βContinuum is which is the same as βParticle in (18), and the function of γ is the same as the γ in (18). Here the subscript “Continuum” denotes the continuum model. The unit parameter γ in (17) and (21) is introduced for dimensional purposes and can be omitted if (i.e., the case for the standard rate equation). Here, γ can be used for the other cases. Figures 7a and 7b show that at an early time (when the Gaussian-shape plume front has not completely formed yet), the product mass grows as ∼t1. At a later time the product mass grows as ∼t0.5, following the same scaling rate as the case of complete mixing (which is ) [see Gramling et al., 2002]. Such evolution is also similar to that using the particle-based model. The interception at x axis is more controlled by βContinuum (than by D), while the transition zone from the t1 law to the t0.5 law is affected by D.
 We then test the influence of the power-law exponent m on the reactive kinetics. Figures 7c and 7d show that when 0 < m < 1, the product mass grows as ∼t0.5 at late times. When m > 1, the rate coefficient decreases rapidly with time, decreasing the growth rate of product. Therefore, we constrain the range of m between 0 and 1. Figures 7c and 7d also imply that various combinations of βContinuum and m may generate a similar mass evolution rate at the late time. This result is analogous to the comment in Porta et al. [2012b], who showed that different forms of can generate similar and acceptable fitting results. This dilemma motivates us to fix m, so that it can be eliminated from the fitting parameters. We will test further this selection and the resultant uniqueness of the best fit βContinuum using laboratory data in the next section.
 Next we take the exponential function for rate coefficient:
Numerical tests (not shown here) confirm that when b is relatively large (e.g., b ≥ 0.2), is very small at late times, canceling the reaction. When b is small (e.g., b < 0.0001), remains relatively stable with t, and the product mass grows as ∼t0.5. The power-law (21), however, can be more convenient in applications, because (1) the exponential does not generate a mass evolution significantly different from that using a power-law , and (2) the exponential has two fitting parameters.
4. Applications: Modeling SCG Reactions Observed in Laboratories
 Two well-known laboratory measurements of the SCG reaction are used here to test the applicability of the above physical models and explore the relationship between model parameters, especially the empirical interaction radius (18) and the effective rate coefficient (defined by both (21) and (22)).
4.1. Case 1: Irreversible Reactive Transport by Gramling et al. 
 In Gramling et al.'s  experiment, a sodium Ethylenediaminetetraacetic acid (EDTA) solution was displaced by a copper sulfate solute filled in a 36 cm long thin glass chamber, producing copper EDTA (CuEDTA2−) at the interface. The evolution of product mass in the chamber was monitored. Snapshots of CuEDTA2− (shown by the circles in Figure 8) along the relatively homogeneous sand column were also observed at various times and flow rates.
4.1.1. Particle-Based Model
 In the particle-based model, parameters v, D, L, and [A0] can be measured in the laboratory [Gramling et al., 2002]. In particular, Gramling et al.  calibrated D by CuEDTA2− snapshots (where the CuEDTA2− solution was injected directly into the chamber as a nonreactive tracer) using the second-order advection-dispersion equation. Hence, there are only two fitting parameters, the parameter βParticle and the number of particles . The earlier numerical analysis shows that βParticle affects directly the reaction rate, especially the product mass. controls particularly the resolution of chemical concentrations, and a small tends to enhance the incomplete mixing of reactant particles and therefore decreases the reaction rate. The best fit results using the interaction radius (18) are shown in Figures 8 and 9. A reasonable number of particles (i.e., ) is needed to simulate the experiment.
 For each group of observations (categorized by the flow rate), the first group was used to fit βParticle and . The remaining snapshots were then used for validation. The goodness of fit was judged visually. Note again that the dispersion coefficient is not fitted but calibrated using the tracer transport experiment. As discussed earlier, the particle number and the rate constant βParticle have distinct impacts on product mass, and therefore, they can be fitted conveniently. We did not identify multiple combinations of and βParticle that can generate a similar best fit result.
 Three behaviors related to reaction kinetics are observed. First, a constant Rs can overestimate the reaction rate at late times, as shown by Figure 9 (the dotted line). The expansion of the reactant front enhances the contact of opposite reactant particles. If the interaction radius remains stable, more pairs of particles can react and increase the global reaction rate continuously. Second, the best fit parameter βParticle increases as a power-law function of the flow velocity v: (Figure 10). The hydrologic explanation for the linkage between βParticle and v is the diffusion. A large velocity corresponds to a large dispersion coefficient, and a large dispersion coefficient will enhance the mixing of reactants and hence produce a relatively large rate coefficient. This behavior is generally consistent with previous studies [Porta et al., 2012a; Tartakovsky, 2010], where the Peclet number was found to affect reaction kinetics. Third, the transition time tinterface defined by (19) is so small that none of the measurements can capture any early-time behavior, as also commented by Sanchez-Vila et al. . Indeed, for advection-dominated transport processes (such as most laboratory experiments), tinterface is typically small. This conclusion however may not be true in the field where dispersion can be dominant.
4.1.2. Continuum Model
 We first use the continuum model with a power-law (21). The model contains two fitting parameters: the effective dispersion coefficient DFit which is typically smaller than the laboratory calibrated dispersion coefficient [Sanchez-Vila et al., 2010; Porta et al., 2012a], and the parameter βContinuum in (21). Salles et al.  found that a single dispersion coefficient cannot capture non-Gaussian transport if the travel distance is not much larger than the product of the Peclet number and the pore-scale dimension (see their equation (102)). For comparison purposes (the particle-based model uses a constant dispersion coefficient), we simplify DFit as a fitting constant. A constant dispersion coefficient was also used for nonlocal transport models [Zhang et al., 2008].
 Results show that the best fit DFit is 80% of the hydrodynamic dispersion coefficient (Figure 11), no matter m = 0.93 or 0.50. The other fitting parameter βContinuum affects more the product peak concentration (denoted as Cmax) than dispersion does. The reason is that the reactant concentration at the interface X = vt remains half of the initial concentration, no matter the magnitude of dispersion [Gramling et al., 2002]. The reaction rate is proportional to the reactant concentration, and hence, Cmax is affected significantly by βContinuum. This is consistent with the numerical results in Sanchez-Vila et al. , who found that Cmax changes significantly with the rate coefficient . In particular, the exponent m affects the power-law slope of the Cmax versus time curve, while the factor βContinuum shifts the curve up and down. Since m can be chosen as 0.5, here βContinuum controls the growth of Cmax. In addition, the dispersion of product can spread the peak concentration, but the advection-dominated transport does not provide enough time for the plume peak to spread significantly through the relatively short sand column. The resultant product peak concentration therefore is most likely to increase at time t > tinterface with 0 < m < 1, as shown by our numerical results and the work in Sanchez-Vila et al. , and also the measured product peak by Gramling et al. .
 Most importantly, we find that the best fit βContinuum in the continuum model with a rate coefficient is on the same order as the best fit βParticle in the particle-based model with an interaction radius (Figure 10). When , however, the best fit βContinuum is one order of magnitude larger than βParticle (note that βContinuum has the same dimensions as βParticle).
 Several reasons motivate us to fix m = 0.5 in (21). First, the continuum model with m = 0.5 fits the observed peak concentration slightly better than that using m = 0.93 (see Figure 11a at time t = 1510 s). Second, the effective rate coefficient is consistent with the empirical rate coefficient used in the rate equation (17). Third, after m is fixed, we can fit DFit and βContinuum conveniently. Otherwise multiple combinations of m and βContinuum can generate similar reaction kinetics. Fourth, when m = 0.5 in (21), we may link the rate constant used in the particle-based model and the continuum model.
 We also fit the data using an exponential in the continuum model. The numerical solutions (not shown here) match generally well the measured snapshot and mass evolution. For example, for the flow rate Q = 16 mL/min, the best fit rate coefficient M−1 s−1 and the best fit effective diffusion coefficient DFit = 0.01 cm2/s (which is the same as the best fit effective diffusion coefficient for a power-law ). Hence, DFit in the continuum model may not be sensitive to the form of , implying that the best fit DFit might be unique. However, the exponential has one more fitting parameter, and therefore, the power-law is preferable in applications. In addition, the exponential approaches zero much faster than the power-law . When t is large, there is a significant discrepancy between reaction rates when different are used.
 The selection of dispersion coefficient D for the SCG reaction is based on previous research. For example, Sanchez-Vila et al.  found that the D used in the continuum model is slightly lower than the calibrated value, while Benson and Meerschaert  suggested the calibrated D for the particle-based model. This discrepancy might be related to the nature of the model. On one hand, the continuum model assumes perfect mixing, where the reaction rate does not decline with the decrease of the degree of reactant particle mixing (for example, near the edge of the interface). The model D therefore can be decreased, to counterbalance the space-independent reaction rate assumed by the continuum approach. On the other hand, the particle-based model accounts for the incomplete mixing of reactant particles, where the actual reaction rate is space-dependent and hence, no adjustment of D is needed. We will further test the above hypothesis in a future study.
4.2. Case 2: Irreversible Reactive Transport by Raje and Kapoor 
 The bimolecular second-order reaction between 1,2-naphthoquinone-4-sulfonic acid (NQS) and aniline (AN) was measured by Raje and Kapoor  in a 18 cm long glass column packed with uniform glass beads. Two runs were conducted with different seepage velocities and initial concentrations for reactants. Run 1 injected 0.5 mM NQS into the column filled originally with 0.5 mM AN solution, and Run 2 injected 0.25 mM AN into the column filled originally with 0.25 mM NQS solution. Raje and Kapoor  also calculated the longitudinal dispersivity (αL = 0.33 cm) and the rate constant . The breakthrough curve (BTC) for the product 1,2-naphthoquinone-4-aminobenzene (NQAB) was monitored.
 The particle-based model fits generally well the observed BTCs. In Run 1, the best fit βParticle = 0.587 mM−1 s−1 (with η = 0.5 in (18)) and generate a BTC similar to the measurement (Figure 12a), while the other model parameters (velocity v = 0.096 cm/s and dispersion coefficient D = 0.032 cm2/s) were either measured directly or calibrated in the laboratory by Raje and Kapoor . Since D can be predefined, βParticle is the remaining factor that controls the peak of the BTC. For Run 2 (the figure is not shown here), the best fit βParticle = 0.533 mM−1 s−1 is slightly smaller than that in Run 1, probably due to the smaller flow rate in Run 2 (which is 0.07 cm/s). These behaviors are the same as we observed in fitting Gramling et al.'s  data. In particular, the best fit βParticle is on the same order as the measured rate constant and they have the same units.
 The simulated BTC using the continuum model with a constant rate coefficient Kf = 0.438 mM−1 s−1 (calculated in the laboratory) and the other model parameters measured in the laboratory (shown by the black line in Figure 12b) is almost identical to the BTC simulated by Raje and Kapoor  (the dashed line in Figure 12b) using the same model and parameters. It confirms the well-known fact that the laboratory calculated rate constant overestimates the observed reaction rate.
 We then fitted the BTC by adjusting the time-dependent and the effective dispersion coefficient DFit. The best fit DFit =0.025 cm2/s, which is 78% of the laboratory calculated dispersion coefficient. The magnitude of the best fit βContinuum = 0.50 mM−1 s−1 is slightly larger than the measured rate constant, and it is close to the best fit βParticle in the particle-based model (see also Figure 10).
 The same behavior was also observed for Run 2 (not shown here). The only discrepancy is that the best fit βContinuum = 1.25 mM−1 s−1 increases with the decrease of flow rate (see also Figure 10), which is opposite to what we observed above. This discrepancy might be related to the BTC data we used (additional data such as the snapshot are needed to define reliably the model parameters), or the uncertainty in the dispersion coefficient. For example, here the dispersion coefficient for Run 2 was estimated from Run 1, which may not represent the changed flow condition and therefore can cause uncertainty in the fitted βContinuum. In addition, the snapshots at different times (such as those used in Case 1) may provide more information of reactant dynamics (i.e., both the spatial and temporal variation) than the BTC does. Future laboratory experiments are therefore needed to explore further the relationship between βContinuum and the flow rate.
5. Discussion: Extension of the Models Under Other Settings
 The above analysis shows that the effective model parameters depend on both the chemical properties (such as the initial reactant concentrations and the rate constant) and physical properties (e.g., system setup, domain size, flow rate, and time regime). This study provides the fundamental methods for further extensions.
 Here we briefly discuss several extensions. First, for a random initial fluctuation in reactant concentration in the bulk system (which was quantified analytically by Tartakovsky et al. ), the particle-based model proposed in section 'The Particle-Based Model to Simulate Reaction A + B ↔ C' may be applicable if the initial particle positions represent the spatial fluctuation of initial concentrations. In other words, the spatial distribution of particle number density should follow the statistics of initial random fluctuations in concentration. As discussed above, when the size of the bulk system remains stable, the interaction radius remains unchanged (since the rate equation (A1) defined on the global concentration does not change with the spatial fluctuation of initial concentrations). This is consistent with the conclusion in Pogson et al. , who found that the local particle-based interaction method for a bulk system can be used “for any molecular distributions since the interaction radius obtained should hold whatever the circumstance.” Second, to capture the influence of spatiotemporally nonlocal transport [Neuman and Tartakovsky, 2009] on reaction kinetics, one can generalize the Fickian diffusion using superdiffusive displacement and capture the subsequent stagnant stage for particles using the time-Langevin approach proposed by Zhang et al. . Third, for kinetically controlled reactions (such as mineral dissolution), a time component can be added to particle dynamics to account for the nonnegligible reaction time. Reactants in the solid phase can be represented by immobile particles, using the mobile/immobile decomposition method [Zhang et al., 2008]. Fourth, for multidimensional transport processes, the 1-D length (1) can be extended conveniently to its multidimensional counterpart (such as a sphere for 3-D).
 Additional revisions are needed for the extension of the continuum model, where a different effective rate coefficient may be needed to account for the change of settings. The governing equation for reactive transport with temporal nonlocality also remains to be shown [Sokolov et al., 2006; Langlands and Henry, 2010].
 For natural porous media where heterogeneity is pervasive [Barber et al., 1992; de Marsily et al., 2005], the intrinsic physical and chemical heterogeneity may affect the reaction kinetics. For example, Li et al.  found that the spatial distribution of Fe(III) content and hydraulic conductivity (such as the preferential flow path) significantly affects uranium bioreduction rates at the scale of tens of meters. Li et al.  found that the average concentration tends to miss the spatial variations of mineral concentrations due to pore-to-pore heterogeneity and therefore overestimates the dissolution rate. Salehikhoo et al.  conducted laboratory column experiments and found that flow velocity, column length scale, and mineral distribution cause spatial variations of magnesite dissolution rates. To capture the observed reaction kinetics using the particle-based model developed above, we may need to separate the influence of physical and chemical heterogeneity. The influence of physical heterogeneity on reaction can be accounted by the transport properties of species, using either the fine-resolution grid-based numerical model and/or nonlocal transport models [Berkowitz et al., 2006; Neuman and Tartakovsky, 2009; Zhang et al., 2007]. The chemical heterogeneity might be captured by defining space-dependent effective parameters. We leave this extension for a future study.
 This study quantifies chemical kinetics in diffusion-controlled, bimolecular second-order reactions using the particle-based model and the macroscopic continuum model, where the chemical reaction occurs in either the bulk system or the sharp-concentration-gradient system. Effective parameters used in the two different types of models, including the interaction radius R and the effective rate coefficient Kf, are approximated, evaluated, and compared systematically. Analytical analysis, numerical experiments, and comparison to laboratory data lead to the following four main conclusions regarding R, Kf, and their potential relationship.
 (1) The interaction radius R used in the particle-based model depends on both the chemical and physical properties. For the bulk system, the agent-based approach shows that R remains constant for an irreversible reaction due to the stable volume, and it decreases nonlinearly in time for a reversible reaction to account for the competition between forward and reverse reactions. In the SCG system, R may decrease with time (e.g., ), probably because of the nonlinear expansion of the reactant front. Numerical tests show that when η = 0.5, the particle-based model captures the analytical and the observed reaction kinetics in the SCG system.
 (2) The effective rare coefficient in the macroscopic continuum model declines as a power-law function at late times for all cases considered in this study. First, for the bulk reactions, decreases exponentially at an early time due to less diffusion limitation, and transitions to a power-law form later due to the strong influence of diffusion on reaction. Such transition is found for both irreversible and reversible reactions, with equal or unequal initial reactant concentrations. The transition time is an inverse function of the diffusion time for the case of irreversible reaction with equal initial concentrations. Qualitative link between and R can be built for most cases, especially the irreversible reaction with equal initial concentrations where a quantitative approximation of the relationship is provided in this study. Second, in the SCG system, either a power-law (e.g., proposed by Sanchez-Vila et al. ) or an exponential can capture the observed reaction kinetics, but the exponential can quench the reaction relatively faster.
 (3) Applications in fitting the laboratory bimolecular reactions suggest the possible link between effective model parameters in the SCG system. For example, the parameter βContinuum (in ) might be on the same order as the laboratory measured rate constant. It is also on the same order as βParticle, the factor in the interaction radius . Therefore, βContinuum may be linked directly with βParticle for the SCG system, a conclusion that needs further validation using future laboratory experiments. Effective model parameters used to describe chemical kinetics for bimolecular reactions therefore can be linked for both the SCG and bulk systems, but the exact and complex relationship may depend on the chemical and physical properties of the system. In addition, the fitting parameter βParticle is reduced to the rate constant for a bulk reaction. The corresponding βContinuum was also treated as a fitting parameter and assumed to be a function of the system chemistry by Sanchez-Vila et al. . This study shows that βContinuum and βParticle also depend on the physical properties of the reaction system, such as the flow rate. Further work is needed to fully explore the physical meaning of the two parameters. It is also noteworthy that while the above applications show that the best fit βContinuum is on the same order of magnitude as the rate constant measured by Raje and Kapoor , more experiments are needed to check the generality of this finding.
 (4) Anomalous evolution for the product mass is found for the SCG reaction, using both the particle-based and continuum models. The product mass increases as ∼t1 at the early time (where tinterface≈ D/v2) when the reaction is affected mainly by the rate constant. At the late time , the product mass increases as ∼t0.5 (which is consistent with the result in Porta et al. [2012a]), where the diffusion affects both the plume front expansion and the particle mobility. Such transient power-law evolution of product mass can be characterized efficiently by the particle-based and continuum models where both the interaction radius R and the effective rate coefficient decline continuously as t−1/2.
Appendix A: Derivation of the Interaction Radius in the Particle-Based Model for Bimolecular Reaction in the Bulk System
 The particle-based model is similar to the one proposed by Benson and Meerschaert . In the following, we explain only the derivation of the interaction radius. Benson and Meerschaert  proposed directly an empirical formula for forward reaction (see their equation (1)). The interaction radius R for reversible reaction A + B ↔ C actually can be derived analytically, by extending the agent-based method proposed by Pogson et al. . A similar procedure can be found in Scheibe et al. . Irreversible reactions can be treated as a specific case of reversible reactions with a zero reverse rate. The interaction radius also represents the maximum distance for two molecules to collide. Any two molecules with separation exceeding R cannot interact during one time step, and hence, R should be related to the time step.
 The rate equation (i.e., the classical thermodynamic rate law) is
 At a typically small time step Δt, the change in concentration A is
 The proportion of B particles that interact with A at this time step therefore is
where due to the reaction A + B ↔ C considered in this study. Assuming that (1) the total spatial volume is V, and (2) particles A and B are distributed uniformly in the closed system with volume V, we obtain the correspondence between the following two dimensionless ratios [see Pogson et al., 2006; Scheibe et al., 2006]
where Vi denotes the proportion of volume in V where A interacts with B.
 The concentration of A at time t can be calculated by
 An A particle may combine with a B particle if and only if this B particle is within an interaction volume V* surrounding A of size
 Whether the reaction will occur following the collision A and B particles also depends on other conditions, such as the energies of particles A and B relative to the activation energy of the reaction and molecule orientation. A given interaction volume V* provides the interaction radius.
 For the case of a 1-D reversible bimolecular reaction, (A6) and (A7) give the signed interaction radius (1) and the simplification (2).
 The interaction radius in (2) can also be derived using an alternative approach. When reactants A and B have the same initial concentrations, the rate equation
 The well-known analytical solution for reactant A (assuming perfect mixing) is
 Inserting (A10) into (A9), one obtains the interaction radius
which is the same as (2) since the denominator . However, the derivation of (2) is more general, since it does not rely on the assumption of equal initial concentration of reactants.
 The constant (2) can also be used to model reversible reactions, but the reverse reaction (with a probability ) must be modeled first. Otherwise the particles undergoing the reverse reaction are forced to experience the forward reaction in the same time step. Dynamic equilibrium will be achieved when the number of C particles created in the forward reaction approaches the number of C particles destroyed in the reverse reaction.
Appendix B: Derivation of Equation (6)
 The reactant B and product C concentrations can be written as
where we assume that there is no product C initially. Inserting (B1a), (B1b) into (4a), we obtain
whose solution can be easily derived, which is (6a).
 Following the above arguments, one can derive the analytical solution (8) when rate coefficients are exponential functions defined by (7). Here we give only the final solution for under the condition (B4):
Appendix C: Defining the Forward Reaction Probability in the Particle-Based Model
 How to accurately define the forward reaction probability using a Lagrangian scheme is critical to the particle-based model. Here we test three empirical Lagrangian algorithms. The probability of reaction for two adjacent reactants, which may relate to both the interaction radius Rb (built in Appendix A) and the distance between reactants, will be analyzed to explore the feasibility of each empirical algorithm.
 First, we propose an apparent forward reaction probability motivated by Edery et al.  (where the interaction radius is a fitting parameter) and Smoluchowski :
where and denote the position of the ith A particle and the jth B particle at time t. (C1) limits the reaction for the closest pair of reactants, where the reaction must occur if two reactants A and B meet close enough. The resultant actual reaction probability depends on the percentage of reactants that are located inside of the interaction radius. To avoid confusion (with the apparent probability ), the actual reaction probability is denoted as . For equal and uniform initial concentrations , the proportion of reactants within Rb is
Hence, the actual forward reaction probability is
The B particle can be located either at the left or right side of the A particle, doubling the reaction probability (hence, the factor 2 is used in (C3)).
 The above scheme may overestimate the reaction rate. As explained by Pogson et al. , the reaction actually need not necessarily occur even if two particles are within the interaction radius, because other properties, such as the orientation and bimolecular attraction, also affect the reaction. The correct forward reaction probability is
The probability from (C3) overestimates the correct reaction probability (C4).
 Second, we propose the following apparent reaction probability :
where w is a factor for normalization. A uniform [0 1] random variable U* will then be generated to control whether the reaction will proceed for colliding particles. If , the A and B particles will combine to produce a C particle located in the middle of the two reactants. Otherwise, no reaction occurs. The apparent probability is similar to the inverse distance law proposed by Pogson et al.  (to account for the noncovalent attractive forces), where the closest pair of reactants A and B has the highest probability (which needs not be 1) to react. Compared to (C1), here fewer particles can interact. For equal and uniform initial concentrations , we have
where the bar denotes the average. Leading (C6) into (C5), one obtains the average of the actual reaction probability
which links the forward reaction probability to the percentage of reacting particles.
 Third, for comparison purposes, we propose one more apparent probability
There are two substeps for this scheme. First, if , then no reaction will occur. This substep is to account for the number of reactant pairs located in Rb. Second, if , then from (C8) is compared to a newly generated uniform [0 1] random variable U*. Reaction occurs only for , and the probability increases with the increase of interaction radius and the decrease of reactant distance. does not rely on the factor w and hence can be computationally more convenient than . The average of the actual reaction probability is therefore
 Extensive numerical tests are conducted to check the applicability of the above three empirical definitions of the apparent forward reaction probability (i.e., from (C1), from (C5), and from (C8)). Results (not shown here) reveal that overestimates the actual reaction rate. When is used, the simulated reaction rate is sensitive to the factor w. The superiority of is apparent: it does not overestimate the reaction rate at the early time as or require the trial-and-error method to obtain the empirical parameter as . Examples using are shown in Figure 1. Therefore, this study only uses from (C8) to calculate the forward reaction probability.
Appendix D: Kf as a Function of Global Reactant Concentration
 For reaction A + B → C with in the bulk system, equation (9b) approximates the late-time . The corresponding analytical solution is given by equation (6a). Combining (9b), (6a), and the rate equation, we find that is a power-law function of :
which implies that declines much faster than the reactant concentration.
 For the other cases, however, is no longer a simple power-law function of . This is consistent with the conclusion in Grima and Schnell , who found that the reaction kinetics with a time-dependent rate coefficient differs from generalized mass action. For example, for reaction A + B → C with , is
where ln denotes the natural logarithm and p is the power-law exponent in .
 This work was supported by the National Science Foundation (NSF) under DMS-0913757 (P.S.) and DMS-1025417 (Y.Z.). This paper does not necessary reflect the view of NSF. Y.Z. also thanks Mark M. Meerschaert and Boris Baeumer for helpful comments. We thank the Associate Editor and A.M. Tartakovsky and two anonymous reviewers for insightful suggestions that significantly improved this work.