Modelling the Release, Transport and Fate of Engineered Nanoparticles in the Aquatic Environment – A Review

Abstract

Engineered nanoparticles, that is, particles of up to 100 nm in at least one dimension, are used in many consumer products. Their release into the environment as a consequence of their production and use has raised concern about the possible consequences. While they are made of ordinary substances, their size gives them properties that are not manifest in larger particles. It is precisely these properties that make them useful. For instance titanium dioxide nanoparticles are used in transparent sunscreens, because they are large enough to scatter ultraviolet light but too small to scatter visible light.

To investigate the occurrence of nanoparticles in the environment we require practical methods to detect their presence and to measure the concentrations as well as adequate modelling techniques. Modelling provides both a complement to the available detection and measurement methods and the means to understand and predict the release, transport and fate of nanoparticles. Many different modelling approaches have been developed, but it is not always clear for what questions regarding nanoparticles in the environment these approaches can be applied. No modelling technique can be used for every possible aspect of the release of nanoparticles into the environment. Hence it is important to understand which technique to apply in what situation. This article provides an overview of the techniques involved with their strengths and weaknesses. Two points need to be stressed here: the modelling of processes like dissolution and the surface activity of nanoparticles, possibly under influence of ultraviolet light, or chemical transformation has so far received relatively little attention. But also the uncertainties surrounding nanoparticles in general—the amount of nanoparticles used in consumer products, what constitutes the appropriate measure of concentration (mass or numbers) and what processes are relevant—should be explicitly considered as part of the modelling.

A Mathematical Details

1.1 A.1 Population Balance Theory

The “free” nanoparticles and the nanoparticles in clusters are divided into size classes and equations are developed to describe the evolution of the number of particles and clusters in each size class (Quik et al. 2014):

$$ \frac{dN_j}{dt}=\frac{1}{2}{\displaystyle \sum_{i=1}^{i= j-1}}{k}_{i, j- i}{N}_i{N}_j-{N}_j{\displaystyle \sum_{i=1}^{i=\infty }}{k}_{i, j}{N}_i $$

(2)

where:

k i ,  j	rate coefficient for the (successful) collision of particles in size classes i and j
N i	concentration of particles in the size class i

The first term in this equation represents the formation of larger clusters from individual particles or smaller clusters. The second term represents the reduction in number of the particles and clusters due to the formation of these larger clusters. No provision is made here for the disintegration of these clusters.

1.2 A.2 DLVO Theory

In its simplest form the DLVO theory predicts the potential energy between a colloidal particle and a (macroscopic) surface or between two colloidal particles as the sum of electrostatic and van der Waals forces. If furthermore the particles are assumed to be identical and therefore have the same surface potential and radius, then the interaction energy can be expressed as (Wikipedia 2015; Macpherson et al. 2012):

$$ W(h)={W}_{vdW}(h)+{W}_{dl}(h)=-\frac{AR^{*}}{6\pi h}+2{\upepsilon \upepsilon}_0 R{\psi}_0^2{e}^{-\kappa h} $$

(3)

$$ {R}^{*}=\frac{R_1{R}_2}{R_1+{R}_2}=\frac{1}{2} R $$

(4)

where:

A	the Hamaker constant
h	distance between the particles’ surfaces
R	radius of the particles
ε 0	the electric permittivity of vacuum
ε	the dielectric constant of water
κ	the inverse Debye-Hückel length
ψ 0	the surface potential of the particles

In this equation the first term is the contribution of the van der Waals forces and the second term is the contribution of the electrostatic forces, as modelled via the double-layer theory (Macpherson et al. 2012). The Debye-Hückel length and the Hamaker constant both depend on the ionic strength of the medium. The Hamaker constant also depends on the characteristics of the colloidal particles and the surfaces in question. The theory is used to examine if there is a minimum in the potential energy, which indicates whether the colloidal particles remain separated or instead aggregate in this minimum (see Fig. 3).

1.3 A.3 Transport and Adsorption in Groundwater

The equations that link the concentration of nanoparticles in the porewater (C) to the concentration of nanoparticles retained in the soil (S) are:

$$ \frac{\partial C}{\partial t}+\frac{\rho_b}{n}\frac{\partial S}{\partial t}-\frac{\partial }{\partial z} D\frac{\partial C}{\partial z}+ v\frac{\partial C}{\partial z}=0 $$

(5)

and:

$$ \frac{\rho_b}{n}\frac{\partial S}{\partial t}={k}_{att} C-{k}_{det} S $$

(6)

where:

ρ b	soil bulk density
n	porosity
v	velocity of the porewater
C	concentration of nanoparticles in the porewater
S	concentration of adsorbed nanoparticles
D	diffusion coefficient
k att	adsorption (attachment) rate coefficient
k det	desorption (detachment) rate coefficient

This model formulation allows an arbitrarily high concentration of adsorbed nanoparticles, whereas in reality the adsorption capacity is finite. To accommodate a limited adsorption capacity, a blocking function may be introduced which effectively reduces the rate of adsorption as a function of the concentration of adsorbed nanoparticles (Liang et al. 2013; Kasel et al. 2013). Experience with such experiments has shown that the adsorption often depends on the distance from the entrance, leading to expressions like:

$$ \psi =\left(1-\frac{S}{S_{max}}\right){\left(\frac{d_{50}+ z}{d_{50}}\right)}^{-\beta} $$

(7)

where:

S max	capacity (maximum concentration) for the adsorption (deposition)
d 50	size of the soil particles
z	distance to the entrance
β	shape parameter

and k_att in Eq. (6) is replaced by k_attψ.

Tufenkji and Elimelech developed the following semi-empirical formula for the collision rates of colloidal and nano-sized particles with the soil as a consequence of various transport mchanisms (Tufenkji and Elimelech 2004):

$$ {k}_{att}\sim {10}^{-3.25}{N_{LO}}^{0.51}{N_{E1}}^{-0.27}{N_{DL}}^{1.06} $$

(8)

where:

NLO	the London number, relating the Hamaker constant, the viscosity of the fluid, the flow velocity and the particle diameter
NE1	the first electrokinetic parameter, which depends on the surface charge of the particles
NDL	the ratio of the particle diameter and the Debye-Hückel length