Modeling coupled nanoparticle aggregation and transport in porous media: A Lagrangian approach
Introduction
Stability of nanoparticles (NPs) in aqueous systems is a key factor controlling their transport and ultimate fate in aqueous environments (D. Lin et al., 2010). Changes in particle shape and size resulting from particle–particle interactions (i.e. aggregation) may significantly alter NP transport potential (Solovitch et al., 2010) as well as their reactivity (Klaine et al., 2008). Indeed, there have been numerous reports of the effect of particle aggregation on NP transport in porous media (e.g., Bian et al., 2011, Fang et al., 2009, Klaine et al., 2008, Raychoudhury et al., 2012, Solovitch et al., 2010, Wiesner and Bottero, 2007). In general, aggregation will be controlled by the properties of particles (e.g., size, chemical composition, surface charge, surface roughness and heterogeneity), and chemistry of the aqueous phase (e.g., ionic strength, pH, and NOM) (Bian et al., 2011).
Theoretically, the kinetics of particle aggregation depends on the magnitude of the electrostatic energy barrier. Particles that overcome the energy barrier will aggregate with another particle in a deep primary energy minimum (Elimelech et al., 1995). The height of the energy barrier for NP deposition or aggregation depends on the size of the interacting particles, with smaller particles (i.e. NPs versus micrometer-sized particles) exhibiting much lower energy barriers (Elimelech and O'Meila, 1990, Petosa et al., 2010). Thus, it is expected that NPs will aggregate more in the primary minimum (Petosa et al., 2010), implying a higher propensity for aggregation than that observed for larger colloids. It has also been suggested that NPs which aggregate in primary energy minima are less likely to dissociate following changes in solution chemistry, such as ionic strength reduction or changes in pH (Chen and Elimelech, 2006).
Depending upon particle surface and solution chemistry, aggregation can influence particle–collector interactions (i.e. transport and retention) by increasing the effective particle hydrodynamic size. According to colloidal filtration theory (Yao et al., 1971), hydrodynamic size strongly influences the behavior of particles approaching collector surfaces. Thus, consideration of particle–particle interactions is imperative when attempting to predict the transport and retention of NPs in porous media (Petosa et al., 2010). For example, for many metallic particles, such as nZVI, aggregation can be one of the main processes governing the environmental fate of nanoparticles, particularly at high concentrations (Phenrat et al., 2009). The classical clean bed filtration theory however, does not incorporate particle–particle interactions that account for spatial and temporal variations in particle sizes observed during aggregation within a porous medium.
Smoluchowski (1917) first provided a mathematical foundation for the description of aggregation kinetics. In the past few decades, the various processes governing particle transport in porous media have been subject to extensive investigation (Chen and Elimelech, 2006, Elimelech and O'Meila, 1990, Li et al., 2008, Yao et al., 1971). A coupling of the mathematical formulations representing aggregation kinetics and particle transport is complicated by the fact that conventional Eulerian models employ a single particle to represent an entire population of particles and do not address non-uniform particle characteristics. Thus, very few modeling investigations have explored the coupling of NP transport and aggregation. Chatterjee and Gupta (2009) attempted to overcome the limitation of an Eulerian formulation by solving a separate transport equation for each family of aggregates (i.e. “n” separate partial differential equations (PDEs) for colloidal monomers, dimers, trimers, and “n”-mers) with second order production/decay terms estimated from the Smoluchowski equation. However, in their numerical implementation, they neglected particle–particle interactions that may occur during transport through a porous medium. More recently, using a similar multiple transport equation formulation, Raychoudhury et al. (2012) developed and applied an Eulerian finite difference-based model to simulate the transport of carboxymethyl cellulose (CMC)-modified nanometer sized particles of zero-valent iron (nZVI) in laboratory-scale, water-saturated sand columns. Their approach, unlike that of Chatterjee and Gupta (2009), incorporates particle–particle interactions during transport through porous medium, however, it is subject to two limitations: (1) the assumption of binary collisions, and (2) a finite number of primary particles for the largest cluster.
In this work, a more general Lagrangian approach for coupling particle–collector and particle–particle interactions is presented that does not share the limitations of the Eulerian approaches noted above. This approach, based upon the random-walk particle tracking (RWPT) method, can be easily coupled with any flow model because of the simplicity of the explicit equations. In addition, through the implementation of particles as discrete mass parcels, global mass conservation is readily satisfied (Salamon et al., 2006). Model development and implementation are described below, followed by a comparison of model simulations with those from an existing Eulerian model (Raychoudhury et al., 2012) to verify the accuracy of the Lagrangian model. A model sensitivity study is then presented to demonstrate model capabilities and to explore the effects of changes in physicochemical properties on particle–particle and, subsequently, particle–collector interactions.
Section snippets
Mathematical model
A general one-dimensional mass balance equation describing the transport of NPs in porous media is given as (Elimelech and O'Meila, 1990):where CwNP [mg/L] and ωsNP [mg/kg dry soil] are the aqueous and solid phase concentrations of NPs, respectively; θw [–]and ρb [kg/m3] are the medium moisture content and bulk density, respectively; and vw [m/s] and [m2/s] are the pore-water velocity and hydrodynamic dispersion coefficient of the NPs,
Numerical implementation
Consistent with the above formulation, an aggregation module was incorporated in the one-dimensional hybrid Eulerian–Lagrangian particle (HELP-1D) transport simulator, which was originally developed to describe the transport of nZVI particles reacting with halogenated organic compounds (Taghavy et al., 2010). HELP-1D implements a RWPT method to model particle transport, where the random path of a model particle p, representing a sub-population of particles, xp(t), is determined in discrete time
Model verification and application
To verify the HELP-1D simulator, it was first applied to reproduce a simulation of the transport of aggregating CMC-coated nZVI particles in a series of column experiments, as reported in Raychoudhury et al. (2012). That study employed a coupled Eulerian transport and aggregation model (herein called R-model) to model the continuous injection of 8 pore volumes of CMC-nZVI at three input concentrations (70, 200, and 725 mg/L) into 9 cm long columns containing water-saturated 40–45 mesh silica sand
Conclusions
An increase in the hydrodynamic diameter of NPs due to aggregation can potentially reduce the dominance of Brownian diffusion on NP attachment the relative contributions of and gravity (settling) forces acting on particle clusters and collector surfaces. Despite its potential importance, aggregation has not widely been included in conventional models of nanoparticle transport. In this study, a Lagrangian framework was developed and implemented in a RWPT simulator to model nanoparticle
Acknowledgments
This work was supported by the Advanced Energy Consortium: http://www.beg.utexas.edu/aec/ under projects BEG08-11 and BEG13-01. Member companies include BP America Inc., BG Group, Petrobras, Repsol, Schlumberger, Statoil, Shell, and Total. Additional support for this research was provided a grant from the National Science Foundation, Award no. CBET-1236653. The work has not been subject to NSF review, and therefore, does not necessarily reflect the views of the organization and no official
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