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314   Environmental Applications of Nanomaterials

        filtration (Yao et al. 1971) and the movement of colloids in natural sys-
        tems (Elimelech et al. 1995; Ryan and Elimelech 1996). Typically this
        has focused on particles on the order of 1 micrometer, but particles
        that are a few hundred nanometers have also been studied. As pre-
        viously discussed, the fate and transport of nanoparticles in porous
        media can be considered a filtration problem (Figure 8.1). Filtration
        theory indicates that the magnitude and rate of aggregation or dep-
        osition will depend on physical properties, including the nanoparti-
        cle size, pore size distribution of the media, and the flow velocity, and
        by the chemical properties such as the pH, ionic strength, and ionic
        composition, which control the magnitude and polarity of the attrac-
        tive and repulsive forces between the nanoparticles and between the
        nanoparticles and mineral grains. Thus, the hydrogeochemistry of the
        system, along with the properties of the nanoparticles, will deter-
        mine the transportability of nanoparticles at a specific site. Increasing
                                                                   2
        ionic strength or the presence of divalent cations such as Ca  and
           2
        Mg    can destabilize the particles by decreasing electrostatic double
        layer (EDL) repulsions between particles and allow aggregation
        (labeled A in Figure 8.1). This will also increase nanoparticle deposi-
        tion onto media grains (labeled C in Figure 8.1). Greater attachment
        efficiency to media grains (or membrane surfaces in a membrane
        filter) will limit nanoparticle transport. Alternatively, aggregation of
        nanoparticles to larger-sized aggregates may potentially serve to
        increase particle mobility. The diffusion rate of the larger aggregates
        will be slower than for the larger particles, which may decrease the
        rate of particle-media interactions. Nanoparticle aggregation or the
        presence of high concentrations of particles could also lead to strain-
        ing (labeled B in Figure 8.1), which will limit or retard transport.
        The flow velocity also plays a significant role. At a high porewater
        velocity, the residence time of the nanoparticles at the collector sur-
        face may be too short to allow for attachment to occur. Low attachment
        efficiency will result in longer transport distances. For nanoparticles
        (d < 100 nm) in environmental media (e.g., soil, sediments), it is dif-
          p
        ficult to predict a priori how the various interactions among the par-
        ticles (aggregation) and media grains will affect their transport.
          There are limitations to applying standard deep-bed (or clean-bed)
        filtration models in natural systems, and especially for remediation
        applications where concentrated suspensions of particles will be
        injected. First, typical filtration models have to make many simplify-
        ing assumptions, such as homogeneous media, monodisperse particle
        size distribution of nanomaterials, constant porewater velocity, and
        uniform surface properties of nanoparticles and filter media. These are
        typically not applicable in real environmental systems, which are phys-
        ically and chemically heterogeneous. Another assumption of filtration
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