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INOM EXAMENSARBETE KEMIVETENSKAP, AVANCERAD NIVÅ, 30 HP
, STOCKHOLM SVERIGE 2016
Simulation of Metal and Metal Oxide Nanoparticle Sedimentation in Solution Using a Computational Model
SARA ISAKSSON
KTHSKOLAN FÖR KEMIVETENSKAP
Abstract
Nanoparticles are used in many different applications because of their small size and unique
properties. The usage is increasing rapidly, which will increase the nanoparticle exposure to
the environment. Up till now, environmental behavior and ecotoxicology of nanoparticles
have only been studied to a certain extent and because of the increasing usage, research
should focus more on nanoparticle behavior and ecotoxicology. An effective way of studying
nanoparticles in aqueous environments is to use mathematical models. In this study, the In
vitro Sedimentation, Diffusion, and Dosimetry (ISDD) model was investigated and applied to
copper, manganese, and zinc oxide nanoparticles to determine their sedimentation velocity in
1 mM NaClO4(aq).
The results show that the simulated sedimentation of nanoparticles in solution, i.e. the output
from the ISDD model, can vary a lot depending on some of the input parameters in the model.
The fact that some of these parameters have to be estimated increases the uncertainty of the
ISDD model, although it is possible to yield results in great agreement with experimentally
determined sedimentation velocities for the studied systems. The simulation results could
always be explained by the theory behind it, which increases the reliability of the ISDD
model.
The possibility of measuring the effective density of nanoparticle agglomerates using the
volumetric centrifugation method was also investigated. This method makes it possible to
avoid estimating the fractal dimension, an input parameter with great uncertainty in the ISDD
model. The results look promising, although further investigation is needed.
The ISDD model seems to be a promising model for future simulation work. The model
should be investigated further in order to minimize the uncertainties due to estimations. The
possibility to predict nanoparticle sedimentation using a mathematical model will save a lot of
time and money, and it can be a helpful tool in the extensive work of identifying the behavior
of nanoparticles in aqueous environments.
Contents
1 Nomenclature .................................................................................................................................. 1
2 Background ..................................................................................................................................... 3
2.1 Simulating the behavior of nanoparticles in solution .............................................................. 5
2.1.1 The In vitro Sedimentation, Diffusion, and Dosimetry (ISDD) model ........................... 5
2.2 Theory ..................................................................................................................................... 7
2.2.1 The behavior of particles in solution ............................................................................... 7
2.2.2 DLVO theory ................................................................................................................... 8
2.2.3 Effects of dissolved organic matter (DOM) on particle agglomeration ........................ 11
2.2.4 Permeability of agglomerates ........................................................................................ 12
2.2.5 Volumetric centrifugation method (VCM) .................................................................... 14
2.3 Nanoparticles ......................................................................................................................... 15
2.3.1 Zinc oxide nanoparticles ................................................................................................ 15
2.3.2 Copper nanoparticles ..................................................................................................... 15
2.3.3 Manganese nanoparticles ............................................................................................... 16
2.4 Purpose .................................................................................................................................. 16
3 Experimental ................................................................................................................................. 17
3.1 Materials and characterization ............................................................................................... 17
3.1.1 Nanoparticles ................................................................................................................. 17
3.1.2 Solutions ........................................................................................................................ 18
3.2 Exposure experimental plan .................................................................................................. 18
3.3 Nanoparticle sedimentation measurements using atomic absorption spectroscopy (AAS) .. 19
3.4 Agglomerate size measurements using photon cross-correlation spectroscopy (PCCS) ...... 20
3.5 Volumetric centrifugation method (VCM) ............................................................................ 20
3.6 Simulations of nanoparticle sedimentation in solution with the ISDD model ...................... 21
3.6.1 Simulations of nanoparticle sedimentation using effective densities measured with
VCM .............................................................................................................................. 22
4 Results and discussion ................................................................................................................... 23
4.1 Experimentally measured nanoparticle sedimentation in solution using AAS ..................... 23
4.1.1 ZnO nanoparticles in 1 mM NaClO4(aq) ...................................................................... 23
4.1.2 Cu nanoparticles in 1 mM NaClO4(aq) ......................................................................... 24
4.1.3 General .......................................................................................................................... 26
4.2 Agglomerate sizes measured with PCCS .............................................................................. 27
4.2.1 ZnO nanoparticles in 1 mM NaClO4(aq) ...................................................................... 27
4.2.2 Cu nanoparticles in 1 mM NaClO4(aq) ......................................................................... 30
4.3 Simulations of nanoparticle sedimentation in solution with the ISDD model ...................... 32
4.3.1 Input parameters ............................................................................................................ 32
4.3.2 Finding intervals of simulated fractions of sedimentation with the ISDD model ......... 41
4.3.3 Limitations with the ISDD model ................................................................................. 46
4.4 Volumetric centrifugation method (VCM) ............................................................................ 47
4.4.1 Agglomerate sizes measured with PCCS ...................................................................... 49
4.4.2 Simulations with the ISDD model in combination with VCM...................................... 50
4.5 Packing effects of particles in agglomerates ......................................................................... 52
4.6 DLVO forces ......................................................................................................................... 52
4.7 Dose tests ............................................................................................................................... 52
5 Conclusions ................................................................................................................................... 53
6 Future work ................................................................................................................................... 54
7 Acknowledgements ....................................................................................................................... 55
8 References ..................................................................................................................................... 56
9 Appendix ....................................................................................................................................... 60
9.1 PCCS correlation functions ................................................................................................... 60
9.2 Simulations of nanoparticle sedimentation in solution with the ISDD model ...................... 61
9.2.1 Fractal dimension (DF).................................................................................................. 61
9.2.2 Primary particle size (d) ................................................................................................ 62
9.3 The ISDD model Matlab® code ............................................................................................ 63
9.3.1 Calculate particle properties .......................................................................................... 63
9.3.2 Core particle model ....................................................................................................... 66
9.3.3 Core particle model input .......................................................................................... 68
1
1 Nomenclature
A Hamaker constant [J]
AAS Atomic absorption spectroscopy
c Packing coefficient of a particle agglomerate
D Diffusion rate [m2/s]
DF Fractal dimension
d Primary particle diameter [m]
da Agglomerate diameter [m]
dc Principal cluster diameter [m]
DOM Dissolved organic matter
e Elementary charge [C]
EDL Electrical double layer
ENP Engineered nanoparticle
Fb Buoyancy [N]
Fd Drag force [N]
Fg Gravity [N]
g Gravitational acceleration [m/s2]
ISDD In vitro Sedimentation, Diffusion, and Dosimetry
kB Boltzmann constant [J/K]
MENP Mass of engineered nanoparticles [mg]
MENPsol Solubilized mass of engineered nanoparticles [mg]
NA Avogadro's number
n Grouping factor
PCCS Photon cross-correlation spectroscopy
PCV Packed cell volume
PF Packing factor
PZC Point of zero charge
PSD Particle size distribution
R Gas constant [J/(mol∙K)]
r Distance [m]
SF Stacking factor
T Temperature [K]
V Settling velocity [m/s]
VCM Volumetric centrifugation method
vdW Van der Waals
Vpellet Pellet volume [µL]
VS Settling velocity predicted by Stokes' law [m/s]
z Charge number of ion
εa Agglomerate porosity
𝜖0 Dielectric permittivity of vacuum [F/m]
𝜖𝑟 Relative dielectric constant [F/m]
2
κ-1 Debye length [m]
Γ Settling velocity ratio
µ Viscosity [Pa∙s]
ξ Permeability factor
ρEV Effective density of agglomerates [g/cm3]
ρENP Density of engineered nanoparticles [g/cm3]
ρf Fluid density [g/cm3]
ρmedia Media density [g/cm3]
ρp Particle density [g/cm3]
ψ Electrostatic potential [V]
3
2 Background
Nanoparticles are defined as particles with at least one dimension in the size range 1-100 nm.
Traditionally, nanoparticles in air were referred to as ultrafine particles and nanoparticles in
soil and water as colloids (with a slightly different size range). Nanoparticles have been
present on earth for millions of years, continuously produced in natural processes as volcanic
eruptions, sea spray aerosols, and continental mineral dust. Mankind has used nanoparticles
for thousands of years and our increasing capability of synthesizing and manipulating
nanoparticles has contributed to a rapidly growing use [1, 2, 3]. The fact that the specific
surface area increases with decreasing particle size leads to nanoparticles having different
properties than bulk materials of same composition [4]. However, the different properties of
nanoparticles do not origin only from a relatively larger specific surface area. Materials in the
lower nanoscale region have unique optical, electronic, magnetic, and mechanical size- and
shape-dependent properties. The size- and shape-dependent properties are due to the quantum
confinement effect, i.e. strong confinement of electrons and holes when the radius of a
particle is below the exciton Bohr radius of the material [5].
Utilizing the nanoparticles of a certain material is an efficient way of using that material,
which is beneficial considering e.g. cost and environmental aspects. Besides from the
properties mentioned above, nanoparticles have other unique properties to take advantage of.
They are fairly mobile in solution and nanoparticles can be incorporated into another material,
producing a composite with unique properties [6]. The potential of nanotechnology is huge
and synthesized nanoparticles, so called engineered nanoparticles (ENPs), are found in e.g.
electronic, biomedical, pharmaceutical, cosmetic, energy, environmental, food packaging,
coating, catalytic, and material applications [1, 7].
The increasing use in industrial as well as household applications will most likely increase the
human and environmental exposure to ENPs and they are now the subject of a worldwide
interest [1, 8]. During production, usage, and disposal, ENPs might end up in air, soils, and
aquatic environments. The risks of ENPs are still largely unknown and today there are no
specific regulations for usage [1, 7]. There is a possibility that specific characteristics of ENPs
due to their small size will lead to harmful interactions with biological systems and the
environment, with the potential to be toxic [4]. Their small size and relatively large specific
surface area make them important binding phases for organic and inorganic contaminants.
ENPs reaching land can possibly contaminate soil, migrate into surface- and groundwater, and
interact with the biota. ENPs in solid wastes, wastewater effluents and other emissions, and
accidental spillages can end up in aquatic systems by wind or rainwater runoff. Since there is
an increasing control of volatile emissions from manufacturing processes, the biggest risks for
environmental release come from spillages during the transportation of ENPs, intentional
releases for environmental applications, and wear and erosion from general use (diffuse
releases) [3]. It is necessary to establish principles and test procedures to ensure safe
manufacture and use of ENPs [4]. Colvin emphasizes the importance to find out whether the
unknown risks of ENPs overshadow the benefits of using them [9]. Up till now, research has
mostly been focusing on toxicology and health effects of ENPs and even though the
information is restricted, environmental behavior and ecotoxicology of ENPs have been even
less studied [1, 8].
ENPs ending up in aquatic environments might remain as primary particles due to high
colloidal stability, but when in higher concentrations they tend to agglomerate. Removal of
4
ENPs from the water can be effects of sedimentation, dissolution processes, chemical
reactions, attaching to an immobile material, or being taken up by aquatic organisms.
Sedimentation, the key process of removal, will be significantly counteracted in the case when
a water flow occurs. Water may also inhibit agglomeration due to hydrophilic repulsion, i.e.
water forming a steric layer on ENPs with a hydrophilic surface. Agglomeration will also be
affected by environmental parameters such as temperature and water chemistry (ionic
strength, presence of dissolved organic matter (DOM) etc.). A more thoroughly explanation
on how ionic strength and DOM affects particle agglomeration is given in sections 2.2.2.2 and
2.2.3, respectively. There is a possibility that ENPs and their agglomerates will interact with
the aquatic fauna, which will alter the degree of agglomeration. For example, ENP
agglomerates in the micron-size region might dis-agglomerate under the influence of bacteria.
ENPs can also be taken up by water living organisms and enter their cells by diffusing
through cell membranes, endocytosis, and adhesion [3, 10].
Environmental risks of ENPs are evaluated by characterizing exposure levels and biological
receptor effects. The understanding of exposure levels is limited since ENPs are rarely
quantified in environmental samples [11].
In vitro studies have shown that nanoparticles are more biologically active than corresponding
micron-sized particles of the same chemical composition. The toxicity of nanoparticles can,
besides due to being an effect of their relatively large specific surface area, be derived from
physicochemical characteristics such as shape, primary particle size, agglomeration state,
surface potential, and surface chemistry. In biological fluids, protein adsorption on
nanoparticle surfaces is also an important factor [12].
Allouni et al studied titanium dioxide (TiO2) nanoparticles in cell culture medium and noted
that once in solution, the nanoparticles agglomerated rapidly and their size did not stay in the
nano-sized region. The agglomeration rate was affected by the nanoparticle concentration, and
the sedimentation, due to agglomeration, increased with increasing concentration. A higher
nanoparticle concentration increases the rate of particle-to-particle interactions, thus
increasing agglomeration [12].
There are concerns about how relevant hazard assessments are, considering that laboratory
experiments are often based on administered doses of ENPs. Research shows that
administered doses might exceed ENP concentrations predicted to occur in the environment
[11]. In a recently published paper, Liu et al stresses the question whether in vitro toxicity
testing of ENPs should consider the delivered dose instead of the administered dose. The
administered dose is defined as the initial ENP mass concentration, while the delivered dose is
the settled ENP mass per suspension volume, hence taking sedimentation into account. Liu et
al studied how particle size distribution (PSD) and permeability of agglomerates affected
ENP sedimentation using a model based on Stokes’ law. The model was used to calculate the
delivered dose of different ENPs and then compare the observed toxicity ranking to a ranking
based on the administered dose. The study showed that toxicity ranking based on the
calculated delivered dose was similar to the ranking based on the administered dose [13]. This
might be interpreted as using the administered dose being as relevant as using the delivered
dose. However, the conclusion was based on comparing the toxicity of seven toxic metal
oxide ENPs and does not say anything about the actual dose of the ENPs.
5
2.1 Simulating the behavior of nanoparticles in solution A way of examining the environmental risks of ENPs is to use mathematical models.
Mathematical models improve our fundamental understanding of environmental behavior,
fate, and transport of ENPs and facilitate risk assessments and management activities. An
article by Dale et al stated that the earliest approaches to simulate environmental fate of ENPs
relied on material flow analysis (MFA). MFA is a methodology that tracks the stocks and
flows of substances into and between technological compartments and environmental
compartments, and it helps when conceptualizing a material’s life cycle. To date, simulating
the fate of ENPs has mainly focused on heteroagglomeration, dissolution, and sedimentation.
However, the fate models are developing rapidly and in the near future it is likely that the
models will take other processes, such as dis-agglomeration, resuspension, and reactions with
ligands into account. Environmental conditions, e.g. pH, temperature, and ionic strength, have
an important effect on ENPs behavior but at present, they are difficult to quantify. The impact
of surface coatings and DOM is very complex, which makes it difficult to construct a general
model [14].
There exist several models to study the environmental behavior and toxicity of ENPs. For
example, Liu et al developed a sedimentation model for in vitro dosimetry of metal oxides.
The model considers PSD and the permeability of nanoparticle agglomerates, and it is based
on the “particle in a box” simulation approach. The studied nanoparticle behavior in media
considers diffusion and gravitational settling, i.e. Stokes-Einstein equation (see equation 1)
and Stokes’ law (see equation 3), respectively. Stokes’ law is modified with a correction
factor that accounts for the permeability of agglomerates [13]. Another dosimetry model,
developed by Arvidsson et al, considers how the behavior of nanoparticles in aqueous
environments is affected by the electrostatic potential barriers surrounding ENPs by including
a collision efficiency factor in their calculations. They add an equation accounting for a
continuous inflow of particles and also, they study the effect of natural colloids on ENPs by
adding a term describing agglomeration to the previously mentioned equation [15]. Mukherjee
et al studied the evolution of silver nanoparticles in biological media with their
agglomeration-diffusion-sedimentation-reaction model (ADSRM). The ADSRM model
describes the processes involved in the interaction between ENPs and their environment and it
takes the entire spectrum of kinetic and dynamic transformation processes relevant for ENPs
into account [16].
2.1.1 The In vitro Sedimentation, Diffusion, and Dosimetry (ISDD) model
In 2010, Hinderliter et al published an article about a mathematical model for calculating
particle behavior in media. It is called the In vitro Sedimentation, Diffusion, and Dosimetry
(ISDD) model. The ISDD model is a computational model of particle sedimentation,
diffusion, and dosimetry for non-interacting spherical particles and their agglomerates in a
common cell culture system. The three major processes transporting particles in static uniform
solutions are diffusion, sedimentation, and advection. Since there is no flow in a static
solution, advective forces are minor and are assumed to not affect the system. Hence, the
ISDD model is derived from Stokes’ law and Stokes-Einstein equation (see section 2.2.1 for
the theory behind the model). The resulting ISDD model is a partial differential equation that
dynamically simulates the transport of micro- and nanosized particles in solution in the
vertical dimension. To solve the partial differential equation numerically, Hinderliter et al
uses the PDE solver in Matlab® [17].
6
The ISDD model has some limitations that can lead to under- or overestimations of the
simulated particle sedimentation. The main limitation is that the ISDD model assumes
impermeable agglomerates of a single average size, i.e. it does not consider the permeability
of agglomerates and the PSD in a given solution. It assumes furthermore that all particles are
spherical and it does not consider the fact that the agglomerate size changes over time [13,
17].
Hirsch et al used the ISDD model to study agglomerates of nanoparticles in vitro. They stated
that the initial and boundary conditions of the ISDD model are simplified and hence that the
model gives an idealistic picture compared to real in vitro experiments. Despite that, their
research found that experimental trends in cellular uptake of nanoparticles or agglomerates
could be described using the ISDD model [18].
A recently performed study by Cohen et al calculates the delivered dose of ENPs to a cell
culture as a function of exposure time using the ISDD model. Toxicity tests of ENPs using
animal testing would be of great cost and pose ethical concerns, and reliable in vitro methods
are therefore attractive options. Today, some in vitro tests produce results conflicting with
animal data. One explanation for this is that researchers that use the administered dose in in
vitro tests ignore important processes such as particle diffusion and sedimentation. Diffusion
and sedimentation are strongly influenced by particle and media characteristics, e.g. how the
particles agglomerate. Cohen et al concluded that agglomerate characteristics (hydrodynamic
diameter and effective density) affect the dose delivered to cells and that measuring these
characteristics is important for in vitro toxicology testing [19]. This is expected from a
physical point of view and the approach can be transferred to investigating environmental
behavior and ecotoxicology of ENPs, since particle diffusion and sedimentation occurs in
those situations as well.
A problem when calculating diffusion and sedimentation of nanoparticles is the fact that
nanoparticle agglomerates have lower density compared to the primary particles due to
entrapped media (e.g. water or cell medium) between the particles in the agglomerates. The
effective density can be calculated using the fractal dimension, DF (see section 2.2.1.2).
However, DF can be neither measured nor verified and hence the estimation will lack validity.
Even though a combination of dynamic light scattering (DLS) and analytical
ultracentrifugation (AUC) could possibly give accurate measurements of the effective density,
AUC requires relatively expensive equipment and the process would be time consuming.
Recently, DeLoid et al developed a simple and low-cost method for estimating the effective
density. The method is called the volumetric centrifugation method (VCM) and the effective
density of nanoparticle agglomerates is determined by volumetric centrifugation of a
nanoparticle suspension in a packed cell volume (PCV) tube. The centrifugation produces a
pellet of packed agglomerates with the media remaining between them. Knowing the density
of the media and the nanoparticles, the effective density can be estimated [20]. VCM is
described more thoroughly in section 2.2.5. In 2015, Liu et al predicted nanoparticle
sedimentation by using the ISDD model in combination with the effective density measured
by VCM [13].
7
2.2 Theory The following sections provide the reader with theory helpful for the understanding of the
behavior of nanoparticles in solution, and the theory behind the methods used in this study.
2.2.1 The behavior of particles in solution
2.2.1.1 Diffusion and sedimentation
The solution dynamics of nanoparticles can be explained in terms of diffusion, gravitational
settling, and agglomeration [21].
Diffusion is a spontaneous process where particles move from an area of high concentration
to an area of low concentration. The diffusive transport is hence driven by a concentration
gradient and the rate depends on particle size and media viscosity [21]. The relationship
between diffusion rate (D [m2/s]) and particle diameter (d [m]) is described by Stokes-
Einstein equation, see equation 1. Besides the particle diameter, the diffusion rate also
depends on media temperature (T [K]) and media viscosity (µ [Pa∙s]). R is the gas constant
[J/(mol∙K)] and NA is Avogadro’s number [17].
D = 𝑅𝑇
3𝑁𝐴𝜋𝜇𝑑 (1)
Gravitational settling, which leads to particle sedimentation, is the net result of opposing
forces acting on a particle in solution, i.e. gravity (Fg), drag (Fd), and buoyancy (Fb). Gravity
is the force that drives the particles downward to sediment. The relationship between the
forces is described in equation 2 [22] and illustrated in figure 1. The sedimentation rate (VS
[m/s]) depends on particle diameter (d [m]), particle density (ρp [g/cm3]), media density (ρf
[g/cm3]), and media viscosity (µ [Pa∙s]. It is described by Stokes’ law, see equation 3 [17, 21].
Figure 1 – The forces acting on a particle in solution.
𝐹𝑔 − 𝐹𝑏 = 𝐹𝑑 (2)
𝑉𝑆 = 𝑔(𝜌𝑝−𝜌𝑓)𝑑2
18𝜇 (3)
While diffusion is the dominant form of transport processes for small particles, gravitational
settling is dominant for large, dense particles [21].
8
Particles moving through a fluid can cause fluid motion and turbulence, which is described by
Reynolds number (a dimensionless ratio of inertial to viscous forces). When Reynolds number
is less than one, the flow is considered to be laminar, and equations 1 and 3 define the only
terms which are necessary to consider. For spheres smaller than 100 nm in diameter,
Reynolds number is less than one and hence, any turbulence occurring will not be considered
[17].
2.2.1.2 Agglomeration
The phenomenon when particles suspended in liquid cluster into larger masses is called
agglomeration [21]. The process shifts the PSD towards a larger mean, which can affect the
particle transport since larger particles sediment more rapidly due to gravitation but diffuse
more slowly, depending on the amount of media entrapped within the agglomerates.
Agglomeration also reduces the total number of free particles and the particle surface area
available for interactions. The agglomeration process can be described by the Smoluchowski
equation [23].
Agglomeration affects the shape, density, and size of particle agglomerates [17]. When
particles pack into agglomerates there will be space between individual particles, i.e.
agglomerates are not solid. Solution media will be trapped within the agglomerates during
formation. This leads to agglomerates having lower effective density compared to the primary
particles [21, 19]. During rapid agglomeration, fractal agglomerates are often formed [15].
The interparticle pore space in fractal agglomerates is due to packing effects and the fractal
nature of agglomerates. Packing effects can be described by a packing factor (PF), and are
determined by particle shape and how the particles are packed into agglomerates. PF have a
value between 0 and 1, and PF = 1 reflects the absence of pore space in an agglomerate. The
fractal nature of agglomerates is represented by the fractal dimension (DF), and is determined
by flocculation processes when the agglomerates are formed. DF takes a value between 1 and
3, where 3 represents a perfect sphere with zero porosity, meaning no entrapped liquid
between the particles. DF is generally less than 3 for agglomerates in natural systems. PF can
be estimated to have a value of 0.637, which represents randomly packed spherical
monomers. DF can be defined according to equation 4 and represents the porosity of an
agglomerate on a macro level, while PF defines the agglomerate porosity on a micro level.
When determining agglomerate density and porosity, DF is usually more important than PF
[17, 24].
𝜀𝑎 = 1 − (𝑑𝑎
𝑑)
𝐷𝐹−1
(4)
Nanoparticle agglomerates can diffuse and settle differently depending on their hydrodynamic
diameter and effective density, hence affecting delivered dose as a function of exposure time
[19]. It is difficult to measure DF and PF for agglomerates, and previous publications about
nanoparticle diffusion and sedimentation have made assumptions regarding these factors [18].
2.2.2 DLVO theory
A colloidal dispersion is thermodynamically unstable and the colloids will always tend to
agglomerate and separate. Sometimes the agglomeration process is slow (hours to days),
which makes the dispersion look practically stable. A colloidal dispersion is regarded stable
when no significant agglomeration takes place. Particles in a colloidal dispersion affect each
other by attractive and repulsive forces acting on different length scales (fractions of
9
nanometer to several nanometers). These interaction forces can be explained by the DLVO
theory (named after the scientists developing it; Deryaguin, Landau, Verwey, and Overbeek).
The theory is based on the attractive van der Waals (vdW) forces and the repulsive electrical
double layer (EDL) forces between particles, and the stability of a colloidal dispersion can be
explained by their combined effect [23, 25].
2.2.2.1 Van der Waals (vdW) forces
The vdW forces are always attractive and consist of several terms; the London dispersion
force (induced dipole – induced dipole interactions), the Keesom force (dipole – dipole
interactions), and the Debye force (dipole – induced dipole interactions). The vdW forces
between surfaces separated by a medium can be seen as material constants and they vary little
between different materials. The most common way to calculate the vdW force is to assume
that the interaction is pairwise additive, which is called the Hamaker approach. Equation 5 is
used to calculate the van der Waals force between two infinite planar walls, where A [J] is the
Hamaker constant and r [m] is the distance between the walls.
𝑃𝑣𝑑𝑊 = −𝐴
6𝜋𝑟3 (5)
The assumption made by the Hamaker approach is not completely correct and instead, the
more accurate Lifshitz theory can be used. The Lifshitz theory requires data on the frequency-
dependent dielectric permittivity for all frequencies. Fortunately, the more simple Hamaker
approach is often sufficient for these calculations [25, 26].
2.2.2.2 Electrical double layer (EDL)
The EDL is the charged layer of ions and molecules at the surface of a particle and the electric
field generated by the charged surface. Depending on the surface ligands of the particle, this
can have a net negative or net positive charge. In general, these forces are repulsive and if
they are strong enough, the colloidal dispersion is virtually stable. The repulsive forces are
electrostatic and act on fairly large length scales. It is difficult to measure the net surface
charge of a particle and instead, the zeta potential (ζ) is usually measured. The zeta potential
is the potential at the shear plane which divides the ions and molecules that are fixed to the
particle surface from those that can move freely with the liquid relative to the bulk aqueous
phase, i.e. it is a voltage reflecting the effects of surface charge and flow dynamics near the
surface. Figure 2 pictures the two layers the EDL consists of, i.e. the Stern layer and the
diffusive layer. The figure schematically shows where the zeta potential is measured [25].
10
Figure 2 – A schematic diagram showing the different layers of the electrical double layer (EDL) on the surface of a particle, and the potential for each layer as well as the Debye length (1/κ). X represents the distance from the particle surface [m] and ψ represents the electrostatic potential [V]. The figure is redrawn from Handy et al [25].
Now consider the effect of adding salt ions to the medium, i.e. increasing the ionic strength.
Since opposite charges attract, some of the added salt ions will accumulate in the EDL and
screen some of the surface charges of the particle. The screening will compress the thickness
of the EDL, hence reducing the length scale which the repulsive forces act on. Since the
stabilizing forces are reduced, two particles can now approach each other more closely and
start to be affected by the attractive forces acting on shorter length scales, e.g. the vdW forces.
This can lead to particles colliding, attaching to each other, and eventually agglomerating.
This means that the colloidal dispersion is not stable any more. At low ionic strength, the
EDL extends beyond the range of the vdW force and the dispersion is stable. At high ionic
strength, the EDL shrinks and the resulting attractive net force will lead to agglomeration. The
Debye length (κ-1), which is the length where the potential has fallen to the value of 1/e of the
potential at the Stern layer (see figure 2), is defined in equation 6. The definition clarifies
what is stated above, increasing the ionic strength will compress the EDL while decreasing
the ionic strength will extend it. In summary, ionic strength influence the agglomeration, and
hence the stability, of a colloidal dispersion [23, 25, 27].
𝜅−1 = √𝜖0𝜖𝑟𝑘𝐵𝑇
∑ 𝑒2𝑧𝑖2𝜌∞,𝑖𝑖
(6)
Another important parameter regarding the stability of a colloidal system is how the surface
charge varies with the pH of the surrounding media. Some particles have a net negative
charge over wide pH ranges, while others have a net positive charge. The point of zero charge
(PZC), i.e. the pH value where the net charge of the particle is neutral, differs for different
materials. The PZC is hence affected by pH. It also depends on other factors, e.g. DOM
sorbed to the particle surface, but not on the ionic strength [25]. The relationship between pH
and ionic strength, and how it affects the stability of the colloidal system is pictured in a
11
stability map (see figure 3). As can be seen in the figure, the stability of a dispersion changes
depending on pH and salt concentration of the solution (i.e. ionic strength). If the ionic
strength is high enough, the EDL will collapse and agglomeration occurs irrespective of pH
[23].
Figure 3 – A stability map showing the effect of pH and ionic strength on colloidal stability. The figure is redrawn from Allen et al [23].
2.2.3 Effects of dissolved organic matter (DOM) on particle agglomeration
Dissolved organic matter (DOM) is present in almost all aquatic ecosystems and depending
on conditions and climate, the concentration typically ranges from 0.1 to 10 mg/L. The most
important DOM in surface waters can be divided into three categories; humic substances,
polysaccharides, and proteins. By binding to colloid surfaces, DOM can modify the surface
properties and hence influences their stability and transport in soils. The different DOM
sorption mechanisms onto colloids are; hydrophobic interactions (solvent exclusion),
Coulomb and van der Waals forces, ligand exchange (condensation with a hydroxyl group at
the surface), surface ion chelation, cation bridging, and hydrogen bonding. Usually, a
combination of several interactions is needed to describe the complex behavior of DOM [28].
DOM can affect the colloidal stability in several ways. DOM is predominantly consisting of
negative molecules, hence they bring negative charges to the particle surfaces when adsorbing
onto them. If the particles are initially positively charged, this can lead to electrostatic
destabilization and the particles will agglomerate. Instead, if the surface charge is initially
negative or if the amount of adsorbed DOM is enough to reverse the surface charge,
electrostatic stabilization will hinder agglomeration. Electrosteric repulsion, which induces
colloidal stabilization, is the combination of electrostatic effects and steric hindrance due to
large DOM molecules. The thickness of the adsorbed layer of DOM depends on the amount
of adsorbed molecules and the conformation of the molecules, i.e. it is affected by media
composition and DOM-surface interactions. If the adsorbed layer of DOM is thin,
electrostatic effects will dominate and if the layer is thick, steric effects will become more
important. The stabilization due to the repulsion of two macromolecular layers is very
efficient if the thickness of the adsorbed layer is larger than the Debye length, since particles
cannot approach each other over the distance where vdW forces are dominant [28].
While humic like molecules stabilize the particles electrostatically by coating their surfaces,
long polysaccharides and peptides induce flocculation by uniting the particles, which can lead
to sedimentation. Cation bridging should always be considered in media containing
multivalent cations. DOM adsorbing onto colloids can induce dis-agglomeration by
modifying their surface charges and forming a steric layer that destabilizes the particle-
particle interactions. This will lead to dis-agglomeration, a well-known phenomenon in soil
12
science. The different DOM-particle interactions mentioned above are general and should be
relativized since the DOM effect on colloidal stability depends on several parameters [28].
The different effects of DOM sorption on colloidal stability are schematically illustrated in
figure 4.
Figure 4 – A schematic description of the effects of DOM sorption on colloidal stability. The figure is redrawn from Philippe et al [28].
DOM consists of dynamic molecules that change conformation, surface charge etc. depending
on environmental parameters (e.g. pH and ionic strength). Hence, its impact on nanoparticle
sedimentation will be complex. It is furthermore difficult to find a general model able to
predict for interactions between DOM and nanoparticles [6].
2.2.4 Permeability of agglomerates
When simulating particle agglomeration and sedimentation it is important to take the
agglomerate permeability into account. Agglomerates will sediment faster than predicted by
Stokes’ law since the law assumes that the agglomerates are impermeable spheres. In reality,
the agglomerates consist of particles that are not tightly packed, i.e. they are fractal. The
fractal agglomerates increase in porosity as their size increases. However, the agglomerate
13
permeability is not solely determined by the fractal dimension, it also depends on how the
primary particles are packed within the agglomerates. Agglomerates sediment faster than
impermeable spheres since the liquid flow through the agglomerates will reduce their drag
force, hence they sink faster (see figure 1 and equation 2) [29].
Agglomerate permeability can be simulated using two different approaches. The first one
assumes a uniform distribution of the small spheres within the agglomerates (single-particle-
fractal model). The other approach accounts for the fact that fractal agglomerates consist of
smaller agglomerates, denoted smaller fractal clusters (cluster-fractal model). These smaller
fractal clusters are denser and less permeable than the large agglomerates. The pores that are
formed between the largest clusters control the overall agglomerate permeability, hence they
dictate the sedimentation velocity and how efficient the agglomerate captures other particles
[29].
Li et al have developed a model for predicting fractal agglomerate permeability based on
three commonly used permeability correlations (Brinkman, Carman-Kozeny, and Happel
equations) [29]. These models were used to compare the single-particle-fractal model and the
cluster-fractal model. The study showed that models based on Brinkman and Happel cluster-
fractal models give the best results. In this study, the Brinkman cluster-fractal model will be
used for simulating agglomerate permeability. The dimensionless permeability factor derived
from the Brinkman correlation for the cluster-fractal model (ξ) depends on, besides DF, a
grouping factor (n) and a packing coefficient (c), see equation 7. The grouping factor is
defined in equation 8, where da [m] is the agglomerate diameter and dc [m] is diameter of the
smaller fractal clusters, and can be interpreted as the number of smaller fractal clusters within
the large fractal agglomerate. For simplification, it is assumed that the largest clusters that
form an agglomerate are of the same size and, in turn, that they are composed of equally sized
smaller clusters [29].
𝜉 = 4,2 (𝑑𝑎
𝑑𝑐) [3 +
4
𝑐(
𝑑𝑎
𝑑𝑐)
3−𝐷𝐹
− 3√8
𝑐(
𝑑𝑎
𝑑𝑐)
3−𝐷𝐹
− 3]
−1/2
= 4,2 (𝑛
𝑐)
1/𝐷𝐹
[3 +4
𝑐(
𝑛
𝑐)
(3−𝐷𝐹)/𝐷𝐹
−
3√8
𝑐(
𝑛
𝑐)
(3−𝐷𝐹)/𝐷𝐹
− 3]
−1/2
(7)
𝑛 = 𝑐 (𝑑𝑎
𝑑𝑐)
𝐷𝐹
(8)
The calculated dimensionless permeability factor (ξ) will be used in equation 9 to find the
settling velocity ratio (Γ), which is also a dimensionless number. The settling velocity ratio is
multiplied with the sedimentation velocity predicted by Stokes’ law (VS, see equation 3) to
find a more accurate sedimentation velocity for permeable agglomerates (V) [29].
Γ =𝑉
𝑉𝑆=
ξ
ξ−tan ξ+
3
2ξ2 (9)
Since the permeability factor depends on the grouping factor but not on the primary particle
size, the settling velocity ratio does not change with the size of the agglomerate as long as DF
and the packing coefficient remain constant [29].
14
2.2.5 Volumetric centrifugation method (VCM)
Particles in solution generally form agglomerates, which are not dense but have media trapped
between the particles. The entrapped media often have lower density than the primary
particles, resulting in the agglomerates having an effective density lower than the bulk density
of the material. The ISDD model, based on the Sterling equation [30] takes account for this by
adding the DF parameter [17]. DF is a theoretical value that can neither be measured nor
verified. The volumetric centrifugation method (VCM), a method developed by DeLoid et al
[20], allows the user to measure the effective density of ENP agglomerates in solution. A
sample of ENPs in solution is centrifuged in a special PCV tube. The PCV tube ends in a
narrow part where a pellet of particle agglomerates can form during centrifugation (see figure
5). The pellet contains packed agglomerates and the media remaining between them (inter-
agglomerate media). After centrifugation, the volume of the pellet can be measured and used
in calculations to find the effective density [20].
Figure 5 – A description of the volumetric centrifugation method, where a sample of ENPs in solution is centrifuged in a PCV tube to produce a pellet. The volume of the pellet can be measured and used for calculating the effective density of the ENP agglomerates [20].
The stacking factor (SF) refers to the fraction of the pellet volume occupied by agglomerates.
SF can be calculated from experimental values, but since small differences in SF result in
even smaller differences in effective density, SF can be approximated to the theoretical values
of 0.634 for random stacking of uniform spheres, or 0.74 for ordered stacking of uniform
spheres (theoretical maximum). Multiplying SF with the pellet volume yields the volume of
the agglomerates (equation 10) [20].
𝑉𝑎𝑔𝑔 = 𝑉𝑝𝑒𝑙𝑙𝑒𝑡 ∙ 𝑆𝐹 (10)
The effective density is calculated according to equation 11. The mass of the ENPs that have
dissolved in the solution (MENPsol) subtracted from the mass of the ENPs (MENP) is divided
with the volume of the agglomerates. If the dissolved mass of particles would not be taken
into account, this would lead to an overestimation of the effective density. The resulting
density is then multiplied with the fraction of agglomerates in the solution, which is then
added to the media density. The result is hence the effective density of the agglomerates [20].
𝜌𝐸𝑉 = 𝜌𝑚𝑒𝑑𝑖𝑎 + [(𝑀𝐸𝑁𝑃−𝑀𝐸𝑁𝑃𝑠𝑜𝑙
𝑉𝑝𝑒𝑙𝑙𝑒𝑡∙𝑆𝐹) (1 −
𝜌𝑚𝑒𝑑𝑖𝑎
𝜌𝐸𝑁𝑃)] (11)
The effective density can be used in sedimentation calculations instead of estimating a value
for DF.
15
2.3 Nanoparticles In this section, the three different metal and metal oxide nanoparticles studied within this
master thesis are presented together with a short background about each material.
2.3.1 Zinc oxide nanoparticles
Zinc oxide (ZnO) nanoparticles are particularly used in sunscreens. In the near future, they
may overtake the usage of TiO2 nanoparticles in sunscreens since ZnO nanoparticles can
block both UV-A and UV-B radiation, offering better protection than TiO2 nanoparticles that
only absorb UV-B. ZnO nanoparticles are also used in e.g. ceramics and rubber processing,
dye-sensitized solar cells, coatings, wastewater treatment, and as a fungicide [31, 32].
In 2010, Wong et al estimated that 250 tons of metal oxide nanomaterials (TiO2, ZnO, and
Fe2O3) can be potentially discharged into the marine environment due to skincare products
[31].
ZnO nanoparticles are one of few nanomaterials currently used in large volumes, with the
likelihood of being released into the environment, and is therefore one of the main focuses in
ecotoxicology studies of nanoparticles [33].
Bian et al studied the agglomeration and dissolution of small ZnO nanoparticles (4 nm in
diameter) in aqueous environments. They investigated the influence of pH, ionic strength,
size, and adsorption of humic substances. Previous studies had already shown that ZnO
nanoparticles released into water systems can potentially harm aquatic organisms, especially
if dissolved Zn2+ ions are released. Experimental studies performed by Bian et al found that
increasing the ionic strength, hence reducing the thickness of the EDL, increased
agglomeration and sedimentation of ZnO nanoparticles. This conforms to results from
somewhat larger sized ZnO nanoparticles (>10 nm in diameter). However, the presence of
humic substances can inhibit agglomeration when the humic substance concentration is >3
mg/L. At low concentrations of humic substances (1.7 mg/L) the sedimentation seems to
increase, probably due to charge neutralization. Agglomeration and sedimentation shows a pH
dependence. The sedimentation rate was much higher at a pH close to the PZC for ZnO (pHpzc
= 9.2). Regarding dissolution, ZnO nanoparticles tend to dissolve to a greater extent than
larger sized particles. The addition of humic substances increased the dissolution only at high
pH. The researchers highlight that these results can be used when deducing the solution phase
behavior of ZnO nanoparticles in the size regime <10 nm [32].
2.3.2 Copper nanoparticles
Copper (Cu) nanoparticles are used in several industrial and commercial applications, e.g. as
additive in lubricants, polymers and plastics, metallic coatings, and inks. Cu nanoparticles are
for instance deposited on graphite surfaces to improve the charge-discharge property and
copper-fluoropolymer nanocomposites are used as bioactive coatings to inhibit the growth of
certain microorganisms [34]. Cu nanoparticles are used in nanofluids, which work as heat
transfer fluids with significantly high thermal conductivity. The use of nanofluids as heat
transfer fluids are in addition energy resource efficient [35, 36].
Chen et al studied the acute toxicological effects of Cu nanoparticles in vivo and found that
Cu nanoparticles induce toxicological effect and heavy injuries on kidney, liver, and spleen.
The tests were performed on mice. The study also stated that Cu nanoparticles are classified
16
as moderately toxic and are in the same toxicity class as copper ions, while copper particles of
micron-size are practically non-toxic, i.e. the toxicity is a matter of size [34].
2.3.3 Manganese nanoparticles
Manganese (Mn) nanoparticles are used in catalysis and battery technology [37]. MRI
(Magnetic Resonance Imaging) is one of the most important and most frequently used
imaging tools for diagnosis in clinics. Contrast agents are used to improve the visibility in
MRI and today, gadolinium (Gd) based agents are the most common ones. Recently, Mn
based agents have shown better performances in certain disease detections, e.g. pancreatic
lesions. Therefore, Zhen et al developed Mn based nanoparticles with the purpose to use them
as contrast agents instead of using Gd agents. Mn based nanoparticles are a relatively new
class of materials and Zhen et al concluded that more studies should be performed about the
biosafety of Mn based nanoparticles as well as a more thoroughly comparison with other
contrast agents [38].
Studies have indicated that elevated levels of Mn exposure to humans may lead to
Parkinsonism, hence there might be significant pathological consequences and risks to the
central nervous system when manufacturing nanoscale Mn. Also, in vitro studies show that
Mn specifically targets the dopaminergic system. Industries that typically produce or work
with large amounts of Mn metal or powder applications are steel and non-steel alloy
production, battery manufacture, colorants, pigments, ferrites, welding fluxes, fuel additives,
catalysts, and metal coatings [37].
2.4 Purpose The purpose of this study was to develop and apply the ISDD model for studying
sedimentation of nanoparticles in an aqueous environment. The output from the ISDD model,
sedimentation velocity, was correlated with experimental data in order to find the optimal
parameters for simulating nanoparticle sedimentation. Sedimentation velocities were
experimentally determined with atomic absorption spectroscopy (AAS) and PSD
measurements of the agglomerates by using photon cross-correlation spectroscopy (PCCS).
Two types of nanoparticles were investigated; ZnO and Cu.
The ambition is to have a computational model where the user can enter measured
agglomerate sizes after different time periods in solution and generate a graph that shows the
fraction of nanoparticles in solution that will sediment over time.
Complementary, the effective density of Cu and Mn nanoparticle agglomerates was to be
calculated with VCM and then used as an input in the ISDD model instead of estimating the
DF factor.
The ISDD model and VCM have previously been used in studies of nanoparticles in cell
medium. This study aimed to find out whether it is possible to use them for simulations of
nanoparticles in aqueous environments as well.
This study is a part of Mistra Environmental Nanosafety, a Swedish national research project
striving to develop new, improved methods for risk assessments of nanoparticles [39].
17
3 Experimental
3.1 Materials and characterization
3.1.1 Nanoparticles
The ZnO nanoparticles were supplied by the Institute for Reference Materials and
Measurements at Joint Research Centre, European Commission, Belgium. Supplier
information is found in [40]. The primary particle size of ZnO nanoparticles was estimated
from the TEM image in figure 6. The diameters of the measurable particles in the image (15
pcs) were combined to a mean value that was used for simulating sedimentation. The
maximum and minimum particle diameters were in addition used within the simulations,
giving a total of three primary particle sizes, in order to investigate which primary particle
size that in the simulations correlated best with the experimental data. Some of the particles in
the TEM image were not spherical. In those cases, the longest side of the particle was used as
the diameter.
Figure 6 – TEM image of ZnO nanoparticles [40].
The Cu nanoparticles were obtained from associate professor A. Yu Godymchuk, Tomsk
Polytechnic University, Russia and the Mn nanoparticles from American Elements, Los
Angeles, California (Lot# 1441393479-680). The primary particle sizes were determined to be
around 100 nm for Cu and around 20 nm for Mn, see TEM images in figure 7. Details about
the Cu and Mn nanoparticles are given in [41].
18
Figure 7 - TEM images of Cu and Mn nanoparticles in 1 mM NaClO4(aq) [41].
The material bulk densities used in the calculations are 5.6 g/cm3 (ZnO), 8.96 g/cm3 (Cu), and
7.3 g/cm3 (Mn) [42].
3.1.2 Solutions
Ultrapure MilliQ water (18.2 MΩ cm; Millipore, Solna, Sweden) was used as solvent in the
experiments and for rinsing the equipment.
1 mM NaClO4(aq) was prepared in a 2 L flask by solving 0.2449 g NaClO4 powder (Lot#
MKBS8852V, Sigma-Aldrich) in MilliQ water.
Dulbecco’s Modified Eagle Medium (DMEM) was purchased from Life Technologies,
Sweden (Lot# 1644395). Proteins were added to DMEM and hence the solution is denoted
DMEM+. Details about the preparation of DMEM+ are given in [41]. The media density
(1.005715 g/cm3) was calculated as a mean value from weighing 1, 2, 3, 4, and 5 mL of the
DMEM+ solution.
3.2 Exposure experimental plan The ISDD model was used to simulate the sedimentation of ZnO and Cu nanoparticles in 1
mM NaClO4(aq). In order to investigate the validity of the ISDD model, both the size of the
nanoparticle agglomerates and the nanoparticle concentration in solution were measured after
certain time periods. The agglomerate sizes were used as input in the ISDD model when
simulating the nanoparticle sedimentation over time. The measured nanoparticle
concentrations were used as experimental data to which the ISDD data could be compared
and validated.
To avoid estimating DF, VCM was employed for Cu and Mn nanoparticles in DMEM+. The
measured effective density was used as input in the ISDD model, instead of an estimated DF,
when simulating the extent of sedimentation over time.
The containers used in the exposure experiments (glass vials, Nalgene® jars, plastic tubes, and
plastic bottles) were cleaned following an acid-cleaning procedure. After cleaning the
19
containers with water and detergents using a dish brush, they were immersed in 10 % HNO3
for at least 24 h. The containers were then taken out from the HNO3 bath and rinsed with
MilliQ water four times before left to air-dry.
3.3 Nanoparticle sedimentation measurements using atomic absorption
spectroscopy (AAS) The nanoparticle concentration in solution and the metal release after a certain time were
measured using atomic absorption spectroscopy (AAS).
For each exposure, 6 ± 0.2 mg nanoparticles (Cu or ZnO) were weighted (XP26 DeltaRange
Microbalance, Mettler Toledo) in a glass vial. 6 mL MilliQ water was added to the vial (stock
solution, 1 g/L) and the nanoparticles were dispersed using sonication (Branson Sonifier 250,
constant mode, output 2) for 882 s. 1.5 mL respectively 0.15 mL of the sonicated stock
solution was pipetted to a 60 mL PMP Nalgene® jar and diluted with 1 mM NaClO4(aq) to 0.1
g/L respectively 0.01 g/L nanoparticle concentration (total volume 15 mL). This was repeated
twice, generating three replicates. A blank sample with 15 mL 1 mM NaClO4(aq) was
prepared in parallel. The replicates and the blank sample were incubated in 25 oC (Platform-
rocker incubator SI80, Stuart) for a certain exposure time (1, 2, 4, 24, 72, or 168 hours). A
new stock solution was prepared for every time measurement.
For ZnO samples with particle concentration 0.1 g/L, trials were made both with the incubator
standing still and with the incubator rocking the samples at 25 rpm (revolutions per minute).
For ZnO samples with particle concentration 0.01 g/L and both Cu trials, the samples were
kept still during incubation.
After incubation, 5 mL of each replicate were pipetted to a plastic tube and 5 mL were
filtrated through an inorganic membrane filter (0.02 µm, 25 mm diameter, Anotop 25 syringe
filter, GE Healthcare Life Sciences) to a plastic tube. 5 mL of the blank sample was also
pipetted to a plastic tube, resulting in a total amount of seven samples. The samples were
acidified to pH < 2 with 65 % HNO3 and stored in 20 oC before measuring the total metal
concentration with AAS (Perking‐Elmer AAnalyst 800), with the flame for high metal
concentrations [mg/L].
The measured total metal concentrations were used to calculate the fraction of nanoparticles
in solution that had sedimented after a certain time period according to equation 12. The total
particle concentration (ctotal) was measured for non-filtrated samples and the concentration of
dissolved particles (cdissolved particles) was measured in filtrated samples.
𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠 𝑠𝑒𝑑𝑖𝑚𝑒𝑛𝑡𝑒𝑑 = 𝐶𝑡𝑜𝑡𝑎𝑙−𝐶𝑑𝑖𝑠𝑠𝑜𝑙𝑣𝑒𝑑 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠
𝐶𝑡𝑜𝑡𝑎𝑙 (12)
In addition to the measurements described above, 0.5 mL was taken out from each stock
solution after sonication and mixed with 10 mL MilliQ water in a plastic bottle. The solution
was acidified to pH < 2 with 65 % HNO3 and the nanoparticle concentration was measured
with AAS to determine the real concentration of the stock solution. This procedure is referred
to as the dose test.
20
3.4 Agglomerate size measurements using photon cross-correlation spectroscopy
(PCCS) The nanoparticle agglomerate sizes were measured using photon cross-correlation
spectroscopy (PCCS) in order to investigate the stability of the Cu and ZnO nanoparticles in 1
mM NaClO4(aq). Unfortunately, the PCCS instrument got problems with the hardware during
the time frame of the thesis work and only ZnO nanoparticles (particle concentrations 0.1 g/L
and 0.01 g/L) were analyzed. The PCCS data for Cu nanoparticles (particle concentration 0.1
g/L) used in the simulations originates from previous PCCS measurements. Unfortunately, no
PCCS data was available for Cu nanoparticles in 1 mM NaClO4(aq) with a particle
concentration of 0.01 g/L.
A stock solution was prepared in the same way as for the AAS measurements. The difference
was that only one stock solution was prepared, compared with the AAS method where a new
stock solution was prepared for each exposure time. After diluting the stock solution with 1
mM NaClO4(aq) to desired nanoparticle concentration (0.1 g/L or 0.01 g/L), 1 mL of the
solution was added to a PCCS cuvette (Eppendorf AG, Germany, UVette Routine pack, Lot#
C153896Q). This was repeated twice, generating three replicates. The replicates were
analyzed with PCCS (Nanophox, Sympatec GmbH) after certain times (0, 1, 2, 4, 24, 72, and
168 h), and the samples were incubated in 25 oC (Cultura mini-incubator 13311, Merck)
between the measurements. The PCCS program was set to measure each replicate three times
for a time period of 3 min per measurement. Before the first measurement started, the
instrument waited 2 min in order to let the temperature of the replicates set to 25 oC. A latex
standard was tested prior to analysis in order to ensure the accuracy of the instrument.
In addition to the measured agglomerate sizes, the mass distributions were calculated using
the refractive indexes of 1.989 for ZnO and 1.590 for Cu*. The algorithm used to obtain the
mass size distribution was the non-negative least squares (NNLS) analysis (auto setting by the
instrument).
A dose test was prepared in the same way as for the AAS measurements (see section 3.3) and
measured with AAS to determine the real concentration of the sonicated solutions.
The measured agglomerate sizes were used as input data in the ISDD model to simulate
nanoparticle sedimentation (see section 3.6).
3.5 Volumetric centrifugation method (VCM) 2 mg nanoparticles (Cu or Mn) were weighted in a glass vial and 2 mL DMEM+ was added
(stock solution, 1 g/L). The stock solution was sonicated in a sonication bath (Ultrasonic
cleaner, VWR® symphonyTM, VWR International) for 20 min, during which the glass vial
was shaken by hand every fifth min. 100 µL stock solution and 900 µL DMEM+ were
pipetted to a PCV tube, generating a sample of 1 mL with particle concentration 0.1 g/L. This
was repeated twice, generating three replicates. The samples were centrifuged (Centrifuge
5702, Eppendorf) at 3000 g for 1 h to obtain a pellet of the nanoparticle agglomerates. Also, a
dose test was prepared in the same way as described before (see section 3.3).
*The mass distribution for the Cu trials was calculated for a refractive index of 1.590 by
mistake. A value of 0.309 should have been used instead, but calculating for another refractive
index led to minimal changes in the results and hence this mistake was not corrected for.
21
The pellet volumes (Vpellet) were measured using a sliding rule-like measure device obtained
from the PCV tube manufacturer. A mean value of the measured pellet volumes was used to
calculate the effective density of the agglomerates (ρEV) according to equation 11. Input data
is listed in table 1. The stacking factor (SF) was assumed to be the theoretical value of 0.634
for random stacking of uniform spheres [20].
Table 1 - Input parameters for calculating the effective density using VCM.
ENP Media density [g/cm3]
ENP mass [mg]
Solubilized mass [mg/L]
ENP density [g/cm3]
Pellet volume [µL]
Stackning factor [-]
Cu 1.005715 0.057162 4.4 8.96 0.125 0.634
Mn 1.005715 0.043859 2.3 7.3 0.150 0.634
The calculated effective densities were used as input data in the ISDD model to simulate
nanoparticle sedimentation (see section 3.6.1).
3.6 Simulations of nanoparticle sedimentation in solution with the ISDD model The ISDD model was used to simulate nanoparticle sedimentation, both for nanoparticles (Cu
and ZnO) in 1 mM NaClO4(aq) and for nanoparticles (Cu and Mn) in DMEM+. The ISDD
model Matlab® code was kindly provided from the developers (Hinderliter et al [17]) and
revised to make it more suitable for the purpose of this study. The main changes were making
the program take account for the fact that the agglomerate diameters change over time and the
addition of a settling velocity ratio. The ISDD model input data for the revised version are
particle and media characteristics, DF, PF, settling velocity ratio, and agglomerate diameters
measured for different exposure times. The revised Matlab® code is attached in appendix 9.3.
The program was run for Cu and ZnO nanoparticles in 1 mM NaClO4(aq) while changing
different parameters to test how much each parameter affects the outcome. The simulation
work also aimed to find an interval for how much the simulated sedimentation can vary using
the ISDD model and how well it conforms to experimental data.
To begin with, three different DF values were tested (1.7, 2.3, and 2.8). DF was then varied
around the DF value giving the best outcome in order to see how close to experimental data it
is possible to get with the ISDD model and to find out what DF value to be the most suitable
to use for a specific nanoparticle material and concentration. In some simulations, a settling
velocity ratio was added to find out the contribution of agglomerate permeability on the
outcome of the model and if it resulted in improved results.
Since experimentally determining PF was not possible, three different values (0.25, 0.44, and
0.637) were tested in order to see how this variable affects the outcome. Li et al assumed the
packing coefficient to be 0.25 when calculating the settling velocity ratio [29]. The value
0.637 was reported as PF for random cluster packing of spherical monomers by Sterling et al
[30]. This value is also what the developers of the ISDD model, Hinderliter et al, used in their
simulations [17]. A PF value of 0.44 was also tested since it is the mean value of 0.25 and
0.637.
22
For ZnO and Cu nanoparticles in 1 mM NaClO4(aq), three different primary particle sizes
were tested; a maximum value, a mean value, and a minimum value. The primary particle
sizes for ZnO nanoparticles (33.3 nm, 88.9 nm, and 176.7 nm) were determined from a TEM
image (see section 3.1.1). For Cu, the primary particle size was determined to be around 100
nm. The maximum and minimum values were estimated to be 50 nm and 200 nm to get a
broad variation for the primary particle size and to test the impact of primary particle size in
the ISDD model.
The agglomerate size measurements with PCCS were done in triplicates and the mean values
were used in the simulations. To test how the agglomerate sizes affect the results, the standard
deviations were added to or subtracted from the mean value, and the resulting agglomerate
sizes were used to simulate sedimentation as well.
When using the program, one important input parameter is the agglomerate size measured
over time. In this study, the agglomerate sizes were determined with PCCS. To yield an
agglomerate size for a certain exposure time, a mean value was calculated according to
equation 13, where da(t=x h) is the agglomerate size measured after x h and da(t=y h) is the
agglomerate size measurement done before t = x h.
𝑑𝑎(𝑡 = 𝑥 ℎ) =𝑑𝑎(𝑡= 𝑥 ℎ)+𝑑𝑎(𝑡= 𝑦 ℎ)
2 (13)
The following parameters were the same for all simulations in 1 mM NaClO4(aq):
temperature, 298 K; media density, 1.0 g/cm3; media viscosity, 0.00089 Pa∙s; media height, 1
cm.
3.6.1 Simulations of nanoparticle sedimentation using effective densities measured with
VCM
For the Cu and Mn nanoparticles in DMEM+, the effective densities of the agglomerates
calculated with VCM were used as input in the program instead of estimating DF.
Agglomerate sizes after certain times used in the simulations are found in [41].
The following parameters were the same for all simulations in DMEM+: temperature, 310 K;
media density; 1.005715 g/cm3; media viscosity, 0.00069 Pa∙s; media height, 3.1 mm.
23
4 Results and discussion
4.1 Experimentally measured nanoparticle sedimentation in solution using AAS The fraction of nanoparticles that had sedimented after a certain time, i.e. the sedimentation
velocity, was measured with AAS for Cu and ZnO nanoparticles in 1 mM NaClO4(aq). The
results are shown in figures 8 and 10, where the error bars represent the standard deviations
between the three replicates for each measurement. For some measurements the error bars are
not visual in the figures due to very small standard deviations.
The fraction of nanoparticles that had sedimented was calculated according to equation 12,
which takes account for nanoparticles that have dissolved in solution and hence do not
sediment. The filtrated samples were run through a 20 nm membrane, i.e. nanoparticles <20
nm could pass the membrane and hence not considered in the fraction of nanoparticles that
has sedimented.
4.1.1 ZnO nanoparticles in 1 mM NaClO4(aq)
Figure 8 shows the fraction of ZnO nanoparticles in 1 mM NaClO4(aq) that had sedimented
over time. Three trials were made with ZnO nanoparticles; particle concentration 0.1 g/L
(with rocking), 0.1 g/L and 0.01 g/L. With rocking means that the samples were rocked during
incubation, compared to the other trials where no rocking occurred. The three trials are
referred to as ZnO 0.1 g/L (with rocking), ZnO 0.1 g/L, and ZnO 0.01 g/L.
Figure 8 – The fraction of ZnO nanoparticles in 1 mM NaClO4(aq) that had sedimented over time for the particle concentrations 0.1 g/L and 0.01 g/L. The labeling “with rocking” for ZnO 0.1 g/L indicates that the samples were constantly rocked at the velocity 25 rpm during incubation. The samples in the other two trials were kept still during incubation, i.e. no rocking.
The rocking caused the particles to sediment faster during the first two hours (compare ZnO
0.1 g/L (with rocking) with ZnO 0.1 g/L). This was expected since the rocking led to
advection in the samples, which probably caused the nanoparticles to agglomerate faster and
hence sediment faster. This is only valid when the rocking velocity is fairly slow. Rocking at
higher speed will probably cause turbulence in the samples, which instead will make it more
difficult for the nanoparticles to agglomerate and sediment.
0
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ZnO nanoparticles in 1 mM NaClO4(aq)
ZnO 0.1 g/L(with rocking)
ZnO 0.1 g/L
ZnO 0.01 g/L
24
In common for all trials is that after 24 h, nearly all particles had sedimented. It is most
interesting to study what happens up till 4 h. ZnO 0.1 g/L (with rocking) sediments fastest,
then ZnO 0.01 g/L and the slowest sedimentation is for ZnO 0.1 g/L. In theory, ZnO 0.1 g/L
should sediment faster than ZnO 0.01 g/L. A possible explanation for this is discussed in
section 4.1.3. However, observed differences in sedimentation between the trials are not
always significant. Comparing ZnO 0.1 g/L (with rocking) with ZnO 0.1 g/L, and ZnO 0.1
g/L (with rocking) with ZnO 0.01 g/L, the differences are not significant (p > 0.05, two-tailed
distribution t-test that performed two-sample unequal variance) for most measurement times.
This means that two measurements not statistically necessarily differ, although it might look
like it in the figure. On the other hand, for ZnO 0.1 g/L and ZnO 0.01 g/L, the differences are
significant (p < 0.05, two-tailed distribution t-test that performed two-sample unequal
variance) for most measurement times.
The fraction of dissolved particles for the ZnO nanoparticle trials are shown in figure 9.
Figure 9 – The fraction of ZnO nanoparticles in 1 mM NaClO4(aq) that had dissolved over time for the particle concentrations 0.1 g/L and 0.01 g/L. The labeling “with rocking” for ZnO 0.1 g/L indicates that the samples were constantly rocked at the velocity 25 rpm during incubation. The samples in the other two trials were kept still during incubation, i.e. no rocking.
In principal, the same fraction of the nanoparticles dissolved for ZnO 0.1 g/L (with rocking)
and ZnO 0.1 g/L, i.e. the rocking did not affect the dissolution. For ZnO 0.01 g/L, roughly the
same amount of nanoparticles dissolved in the solution compared to the other trials but since
the particle concentration was ten times lower in ZnO 0.01 g/L, the fraction of particles that
had dissolved became almost ten times higher compared to ZnO 0.1 g/L (with rocking) and
ZnO 0.1 g/L. It seems unexpected that almost ten times more of the particles in ZnO 0.01 g/L
dissolve and more trials should be done in order to find out whether these results were just a
coincidence or not.
4.1.2 Cu nanoparticles in 1 mM NaClO4(aq)
Figure 10 shows the fraction of Cu nanoparticles in 1 mM NaClO4(aq) that had sedimented
over time. Two trials were made with the Cu nanoparticles; particle concentration 0.1 g/L and
0.01 g/L. The samples were kept still, i.e. no rocking occurred, during both trials. The trials
are referred to as Cu 0.1 g/L and Cu 0.01 g/L.
0
0,1
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1 10 100 1000
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lg (Exposure time [h])
ZnO nanoparticles in 1 mM NaClO4(aq)
ZnO 0.1 g/L(with rocking)
ZnO 0.1 g/L
ZnO 0.01 g/L
25
Figure 10 – The fraction of Cu nanoparticles in 1 mM NaClO4(aq) that had sedimented over time for the particle concentrations 0.1 g/L and 0.01 g/L.
Both trials show the same behavior, except that it seems like Cu 0.01 g/L sediments somewhat
faster than Cu 0.1 g/L during the first 2 h. After 4 h, it seems like Cu 0.1 g/L sediments faster
and this behavior was expected for all exposure times. A possible explanation for the
unexpected results during the first 2 h is discussed in section 4.1.3. However, observed
differences between Cu 0.1 g/L and Cu 0.01 g/l for the first three measurements (1 h, 2 h, and
4 h) are not significant (p > 0.05, two-tailed distribution t-test that performed two-sample
unequal variance). This means that the two trials not necessary differ, although it might look
like that in the figure.
For the higher particle concentration, almost all nanoparticles have sedimented after 24 h,
while for the lower particle concentration it takes longer time.
The fraction of dissolved particles for the Cu nanoparticle trials are shown in figure 11.
Figure 11 – The fraction of Cu nanoparticles in 1 mM NaClO4(aq) that had dissolved over time for the particle concentrations 0.1 g/L and 0.01 g/L.
0
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Cu nanoparticles in 1 mM NaClO4(aq)
Cu 0.1 g/L
Cu 0.01 g/L
0
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Cu nanoparticles in 1 mM NaClO4(aq)
Cu 0.1 g/L
Cu 0.01 g/L
26
The dissolution behavior for Cu nanoparticles in 1 mM NaClO4(aq) shows the same trend as
for ZnO nanoparticles, i.e. the fraction of dissolved particles were roughly ten times higher for
the lower particle concentration. Once again, further investigations should be done to find out
whether these results were a coincidence or not.
4.1.3 General
The results show a clear trend that after 24 h, essentially all nanoparticles have sedimented. It
seems like most of the sedimentation happens between 4 h and 24 h. It is quite a large gap in
measurement times between 4 h and 24 h, and it would have been interesting to study the
sedimentation closer to find out more exactly what happens during that time period. Another
interesting aspect is to expose the nanoparticle samples for more than 168 h to see if
eventually 100 % of the nanoparticles will sediment.
In theory, a higher particle concentration should lead to faster sedimentation since more
particles in solution increases the likelihood of agglomeration and hence sedimentation. The
AAS results show the opposite, for both ZnO and Cu the sedimentation velocity is higher for
the lower nanoparticle concentration. This could be just a coincidence since the size of
magnitude between the measured particle concentrations (0.1 g/L and 0.01 g/L) is pretty
small, it only differs with a factor of ten, which is relatively close from a kinetic point of
view. The results might have looked different if comparing two particle concentrations that
differ more. To be able to draw conclusions about the difference in sedimentation between the
two particle concentrations, more experiments should be performed where the replicates
originate from different stock solutions.
For some measurements, it seems like a smaller fraction of particles have sedimented than for
the previous measurement, i.e. sedimented particles have resuspended. However, the
differences between those measurements are not significant and hence there might not be a
difference. This needs further investigations.
When comparing the ZnO and Cu sedimentation data, it seems like before 24 h (during when
most of the particles have sedimented) Cu sediments slightly faster than ZnO. This indicates
that ZnO is more stable in 1 mM NaClO4(aq) than Cu. However, this might not be valid in the
environment since other factors, e.g. proteins and DOM in fresh water, will affect the
behavior of the nanoparticles [28]. ZnO nanoparticles dissolve to a greater extent in 1 mM
NaClO4(aq) compared to Cu nanoparticles.
Cu 0.01 g/L is the only trial where in principle all particles have not sedimented after 24 h.
This is a bit strange since Cu seems to be more unstable in solution compared to ZnO, as
discussed above.
The solution in the experiments, NaClO4(aq), was chosen because it is a simple electrolyte
that will have minor influence on the nanoparticles. In the experimental setup the
nanoparticles were kept in a stationary solution. In reality, for example in an environmental
setting, the water will not probably be completely still and sedimented particles can resuspend
due to some turbulent flow. If no outer factors (such as pH, ionic strength, presence of DOM)
change, the particles will most probably sediment with the same velocity again. If the
conditions adjacent the particles change, for example if the particles are moved to another part
of the water compartment, they could probably sediment faster or slower. This means that
27
sedimentation in the environment calculated with the ISDD model for certain conditions
might not be valid if the nanoparticles are moved or if the outer factors change in any other
way (e.g. due to acidic rain, overfertilization, or factory emissions) [6].
Even if the nanoparticles have sedimented, they will still dissolve and generate free ions (that
will react with available ligands) in the solution. In the environment, this means that even if
the nanoparticles are not moving around in the water any more, they will still release metals
that can affect the surrounding.
4.2 Agglomerate sizes measured with PCCS The agglomerate sizes for ZnO and Cu nanoparticles in 1 mM NaClO4(aq) were measured
with PCCS for the different exposure times. The results, which are measured as the middle of
the integral curve for the size distribution (x50), are shown in figures 12, 13, and 17. The error
bars represent the standard deviations between the three replicates of each measurement.
4.2.1 ZnO nanoparticles in 1 mM NaClO4(aq)
Figure 12 shows the agglomerate sizes measured with PCCS for ZnO nanoparticles in 1 mM
NaClO4(aq) with particle concentration 0.1 g/L (ZnO 0.1 g/L).
Figure 12 - Agglomerate sizes for ZnO nanoparticles in 1 mM NaClO4(aq) with particle concentration 0.1 g/L measured with PCCS over time.
The PCCS data for the 168 h measurement was too poor to be reliable, seen from the bad
correlation function (see appendix 9.1) as well as from the low count rate of the measurement
(see figure 16). However, the other measurements were good although the standard deviations
between the replicates were very large, especially for the 1, 2, and 4 h measurements. Poor
PCCS data after a long exposure time indicates that most of the nanoparticles have
sedimented from solution, hence there are no particles left in solution to be measured.
Considering this, it is strange that the PCCS data for the 24 h and 72 h measurements were
good enough to use, i.e. the correlation functions were good and the count rates were high
enough (see figure 16), since according to the AAS measurements most of the particles should
have sedimented at that point. It is possible that the PCCS instrument detected some dust or
one of the few particles that were still left in solution after 24 h and 72 h.
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ZnO 0.1 g/L
28
Figure 13 shows the agglomerate sizes measured with PCCS for ZnO nanoparticles in 1 mM
NaClO4(aq) with particle concentration 0.01 g/L (ZnO 0.01 g/L).
Figure 13 - Agglomerate sizes for ZnO nanoparticles in 1 mM NaClO4(aq) with particle concentration 0.01 g/L measured with PCCS over time.
For the 24, 72, and 168 h measurement, the PCCS data was too poor to be reliable, which can
be determined from the correlation functions (see appendix 9.1) as well as from the low count
rate of the measurements (see figure 16). As mention above, poor PCCS data most likely
depends on the fact that almost all particles in the solution have sedimented. For ZnO 0.01
g/L, this correlates well with the AAS measurements. The standard deviation was pretty high
for the 4 h measurement but overall it looks better than for ZnO 0.1 g/L.
The PCCS measurements are needed when simulating sedimentation with the ISDD model
and poor PCCS data means that no simulation is possible for those exposure times. However,
since nearly all nanoparticles had sedimented for the times yielding poor PCCS data any
simulation of the sedimentation velocity is not necessary for those exposure times anyway.
As the error bars in figures 12 and 13 show, there is a big difference in measured agglomerate
size between the replicates for some samples. This can be due to dust in the PCCS cuvettes or
an uneven distribution of particles between the replicates. Also, the particles might
agglomerate and sediment faster in some replicates compared to the others, possibly because
of the uneven particle distribution. Another possible explanation is that the particles have
different sizes when provided from the manufacturer.
The results from the AAS measurements for ZnO nanoparticles in 1 mM NaClO4(aq)
discussed in section 4.1.1 showed that the nanoparticles sedimented slightly faster in the
solution with a lower particle concentration (0.01 g/L) compared with the solution with a
higher particle concentration (0.1 g/L). This behavior could possibly be explained from the
PCCS results. If the measured agglomerate sizes were larger for ZnO 0.01 g/L than for ZnO
0.1 g/L, the sedimentation should be faster for ZnO 0.01 g/L since larger agglomerates tend to
sediment faster. Comparing figures 12 and 13, it seems like the agglomerate sizes were in the
same size region independently of the particle concentration. However, the standard
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Exposure time [h]
ZnO 0.01 g/L
29
deviations are very large for some of the measurements for ZnO 0.1 g/L and it is therefore
difficult to say if this explanation is valid or not.
4.2.1.1 Particle size distribution (PSD)
The PSDs for the samples are shown in figure 14 (ZnO 0.1 g/L) and figure 15 (ZnO 0.01 g/L).
In most cases, the PSD is narrow. This means that the samples are not very polydisperse. This
is beneficial when using these results for simulating sedimentation with the ISDD model,
since the model does not take polydispersity into account [13].
Figure 14 – Particle size distributions for agglomerate sizes for different exposure times measured with PCCS for ZnO nanoparticles in 1 mM NaClO4(aq) with particle concentration 0.1 g/L. The picture to the left represents replicate A, the middle picture represents replicate B, and the picture to the right represents replicate C. The x axis represents the measured agglomerate sizes [nm] and the y axis represents the mass distribution [%].
Figure 15 – Particle size distributions for agglomerate sizes for different exposure times measured with PCCS for ZnO nanoparticles in 1 mM NaClO4(aq) with particle concentration 0.01 g/L. The picture to the left represents replicate A, the middle picture represents replicate B, and the picture to the right represents replicate C. The x axis represents the measured agglomerate sizes [nm] and the y axis represents the mass distribution [%].
4.2.1.2 Scattered light intensity
The PCCS instrument detects the intensity of the light that scatters when the laser beam hits a
particle or an agglomerate, i.e. it detects the scattered light intensity. This is expressed as the
count rate, where count rate = 0 means that there are no particles left in solution. Figure 16
shows the count rates for the PCCS measurements of ZnO nanoparticles in 1 mM
30
NaClO4(aq), both for particle concentration 0.1 g/L (figure a) and 0.01 g/L (figure b). The
error bars represent the standard deviation between the three measured replicates.
Figure 16 – The scattered light intensity for the PCCS measurements expressed as count rate for (a) ZnO 0.1 g/L and (b) ZnO 0.01 g/L. The x axis represents exposure time [h] and the y axis represents scattered light intensity [kcps].
The count rate is more than a tenfold lower for ZnO 0.01 g/L than for ZnO 0.1 g/L. This
depends on the difference in particle concentrations, a lower particle concentration yields a
lower count rate. For some measurements (ZnO 0.1 g/L after 168 h, ZnO 0.01 g/L after 24,
72, and 168 h), the count rates were very low (< 20 kcps), which means that the scattered light
intensity was below the detection limit for the PCCS. Hence, those measurements are not
good enough to use as results, which is also seen in the correlation functions (see appendix
9.1).
4.2.2 Cu nanoparticles in 1 mM NaClO4(aq)
Due to problems with the PCCS instrument, it was not possible to complete the measurements
for Cu nanoparticles in 1 mM NaClO4(aq). Instead, agglomerate sizes measured with PCCS
for Cu nanoparticles in 1 mM NaClO4(aq) with particle concentration 0.1 g/L from previous
measurement were used. In contrast to the AAS measurements and the PCCS measurements
for ZnO nanoparticles, this data set only contains measurements for exposure times 0, 4, and
24 h. There was no data for the agglomerate sizes of Cu nanoparticles in 1 mM NaClO4(aq)
with particle concentration 0.01 g/L available, hence this trial will not be considered any
more.
Figure 17 shows the agglomerate sizes measured with PCCS for Cu nanoparticles in 1 mM
NaClO4(aq) with particle concentration 0.1 g/L (Cu 0.1 g/L).
31
Figure 17 – Agglomerate sizes for Cu nanoparticles in 1 mM NaClO4(aq) with particle concentration 0.1 g/L measured with PCCS over time.
It might look like the agglomerate sizes decrease over time for Cu 0.1 g/L, but this can
possibly be a result from the fact that larger, denser agglomerates sediment faster and hence
are not seen in the PCCS measurements.
4.2.2.1 Particle size distribution (PSD)
The PSDs for Cu 0.1 g/L are shown in figure 18. They are pretty narrow, although they are
not as narrow as for the ZnO nanoparticle measurements (see figures 14 and 15). For
example, the 4 h measurement for Cu 0.1 g/L (replicate C) shows two peaks. The ISDD
model does not account for PSD in the calculations [13], which will probably lead to some
errors in the cases where the samples show polydispersity.
Figure 18 – Particle size distributions for agglomerate sizes for different exposure times measured with PCCS for Cu nanoparticles in 1 mM NaClO4(aq) with particle concentration 0.1 g/L. The picture to the left represents replicate A, the middle picture represents replicate B, and the picture to the right represents replicate C. The x axis represents the measured agglomerate sizes [nm] and the y axis represents the mass distribution [%].
4.2.2.2 Scattered light intensity
The scattered light intensity detected with the PCCS instrument for Cu 0.1 g/L is shown as
count rates in figure 19. The error bars represent the standard deviations between the three
measured replicates.
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Cu 0.1 g/L
32
Figure 19 – The scattered light intensity for the PCCS measurements expressed as count rate for Cu 0.1 g/L. The x axis represents exposure time [h] and the y axis represents scattered light intensity [kcps].
The count rates are high enough (count rate >20 kcps) for all three measurements, which
means that the PCCS data should be reliable.
4.3 Simulations of nanoparticle sedimentation in solution with the ISDD model The ISDD model was used to simulate the sedimentation velocity of nanoparticles in a
solution, which was measured as the fraction of nanoparticles in solution that had sedimented
after a certain time. The results were then compared with experimental data (results from the
AAS measurements) in order to evaluate the ISDD model.
Different input parameters in the ISDD model were varied to investigate how much they
affect the results and to find out the most important parameters.
The simulations were done for three different trials; ZnO nanoparticles in 1 mM NaClO4(aq)
with particle concentration 0.1 g/L, ZnO nanoparticles in 1 mM NaClO4(aq) with particle
concentration 0.01 g/L, and Cu nanoparticles in 1 mM NaClO4(aq) with particle concentration
0.1 g/L. In the following section, the three different trials will be referred to as ZnO 0.1 g/L,
ZnO 0.01 g/L, and Cu 0.1 g/L. No simulations were done for the AAS trial ZnO 0.1 g/L (with
rocking), since the ISDD model does not consider advection in the solutions [17]. Simulations
for Cu nanoparticles in 1 mM NaClO4(aq) with particle concentration 0.01 g/L were not
performed either since no data for agglomerate sizes were available.
4.3.1 Input parameters
The following sections investigate how sensitive the ISDD model is for different input
parameters. One input parameter was varied at time while keeping the other parameters
constant during the simulation, in order to find out how that particular input parameter
affected the sedimentation results from the ISDD model. The sedimentation data,
experimentally determined with AAS (referred to as experimental data), is added to the
graphs for comparison.
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Cu 0.1 g/L
33
4.3.1.1 Fractal dimension (DF)
DF represents the fractal nature of an agglomerate. It can be defined as the porosity of the
agglomerate on a macro level [17]. It is an important parameter when simulating the
sedimentation, but unfortunately it cannot be measured experimentally [20].
Three different DF values (1.7, 2.3, and 2.8) were tested in the simulations, while keeping the
other parameters constant. DF can take values in the range 1 < DF < 3, and the three values
tested here were chosen since they are reasonable values in the higher, lower, and middle part
of that range. In the articles studied, DF was never lower than 1.7 or higher than 2.5 [17, 29].
A DF value between 2 and 2.5 corresponds to inefficient or moderately efficient space filling,
and the agglomerates will have an effective density that is significantly lower than that of the
primary particles [17].
Figure 20 shows the result for ZnO 0.1 g/L. The results for the other trials are found in
appendix 9.2.1.
Figure 20 – Sedimentation for ZnO nanoparticles in 1 mM NaClO4(aq) (particle concentration 0.1 g/L) simulated with the ISDD model for DF = 1.7; 2.3; 2.8. The primary particle size is 88.9 nm and no settling velocity ratio was added to the calculations. The experimental data is added to the figure for comparison.
Varying DF affects the outcome from the ISDD model a lot. For example, the fraction of
particles that had sedimented after 24 h were calculated to roughly 0.1 respectively 1 for the
DF value 1.7 respectively 2.8. This is a huge difference and hence, DF is a very important
parameter when simulating sedimentation with the ISDD model. This will lead to great
uncertainties, since DF cannot be measured experimentally and has to be estimated.
DF describes the porosity of agglomerates, i.e. it reflects their effective densities.
Sedimentation is a result of the gravitational force on agglomerates, hence the density of
agglomerates has an impact on the sedimentation velocity. This is probably why DF has such
a large impact on the results from the ISDD model. A higher DF value means that the
agglomerates are less porous and have a higher effective density, therefore they should
sediment faster, which is also seen in the results.
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Exposure time [h]
ZnO 0.1 g/L
DF=1.7
DF=2.3
DF=2.8
AAS data
34
4.3.1.2 Packing factor (PF)
PF describes how the particles in an agglomerate pack relative to each other. PF = 1 describes
the absence of pore space in the agglomerate [17]. To test the effect of the packing factor on
the ISDD results, three different PF values (0.25, 0.44, and 0.637) were tested. PF = 0.25 is
the value used in studies by Li et al [29], PF = 0.637 is the theoretical value for randomly
packed spherical monomers according to Hinderliter et al [17], and PF = 0.44 was tested
since it is the mean value between 0.25 and 0.637.
It turned out that varying PF did not affect the outcome of the simulations at all. According to
Hinderliter et al, DF is generally more important than PF when determining the density and
porosity of an agglomerate [17]. This could explain why varying PF had no impact on the
simulated sedimentation.
4.3.1.3 Primary particle size (d)
Another input parameter in the ISDD model is the primary particle size of the nanoparticles
(d). The primary particle sizes for the materials used in this study (Cu and ZnO nanoparticles)
were determined from TEM images (sees figures 6 and 7). For ZnO, a maximum, minimum,
and mean value of the primary particle size were measured while for Cu, only a mean size
was estimated. Therefore, smaller and larger primary particle sizes were assumed in order to
test the effect of the primary particle size in the simulations for the Cu nanoparticles as well.
The results for ZnO 0.1 g/L are shown in figure 21. The results for the other trials are found in
appendix 9.2.2.
Figure 21 – Sedimentation for ZnO nanoparticles in 1 mM NaClO4(aq) (particle concentration 0.1 g/L) simulated with the ISDD model for d = 33.3 nm; 88.9 nm; 176.7 nm. The fractal dimension is 2.4 and no settling velocity ratio was added to the calculations. The experimental data is added to the figure for comparison.
The primary particle size does not affect the simulation results as much as DF, but it still has a
great impact in the results.
These results are based on ISDD simulations where all parameters except for the primary
particle size are kept constant. This means that changing the primary particle size will change
how many particles that can fit in an agglomerate, and hence it will affect the porosity and the
0
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d=33.3 nm
d=88.9 nm
d=176.7 nm
AAS data
35
effective density of the agglomerates. As mentioned before, these are important factors
regarding the sedimentation and this is probably the explanation for why the primary particle
size has a big impact on the simulated sedimentation.
It can be pretty difficult to measure nanoparticle sizes from TEM images, it is more of an
estimation than an actual measurement, and considering how important this parameter is in
the simulations, it can lead to large errors if the primary particle size is estimated poorly.
Also, the nanoparticle samples are often polydisperse. The ISDD model does not take account
for the polydispersity, which can affect the simulated results negatively. Besides this, the
ISDD model assumes spherical particles, which not always is the case in reality.
4.3.1.4 Permeability of agglomerates
The settling velocity ratio, which accounts for the permeability of agglomerates, was not a
part of the ISDD model from the beginning, but it was added to the calculations since it is an
important parameter that can have a large impact on the sedimentation calculation [29].
The permeability factor was calculated using the Brinkman correlation function, which was
then used to calculated the dimensionless settling velocity ratio (Γ). The calculations depend
on DF, a packing coefficient (c), and a grouping factor (n). Figure 22 shows how the
calculated settling velocity ratio varies with the different parameters. The different values of c
(0.25, 0.44, and 0.637) were chosen with the same reasoning as for the packing factor (see
section 3.6) and the DF values tested here were chosen because they are used in the
sedimentation simulations for ZnO and Cu nanoparticles in 1 mM NaClO4(aq). For every set
of c and DF, the settling velocity ratio was calculated for 5 ≤ n ≤ 100. This was done to
investigate how much the settling velocity ratio depends on n, since that parameter cannot be
determined experimentally [29].
36
Figure 22 – Settling velocity ratios of agglomerates with different fractal dimension (a) DF=1.7; (b) DF=1.8; (c) DF=1.9; (d) DF=2.0; (e) DF=2.1; (f)1 DF=2.2; (g) DF=2.3; (h) DF=2.4; (i) DF=2.5; (j) DF=2.6; (k) DF=2.7; (l) DF=2.8 calculated for varying packing coefficients and grouping factors. The x axis represents the grouping factor (n) and the y axis represents the settling velocity ratio (Γ).
A lower grouping factor mostly yields a higher settling velocity ratio. Increasing the grouping
factor will eventually lead to the settling velocity ratio reaching a steady state. However, in
most cases the change in settling velocity ratio is not very large when changing the grouping
factor and hence it does not affect the outcome significantly. For the lowest c value, c = 0.25,
the settling velocity ratio varies markedly while for c = 0.637 the curve is basically flat. It is
difficult to know which c to use but according to Li et al [29], c = 0.25 should be used. Hence,
settling velocity ratios calculated for that c value were used in the simulations when
accounting for the agglomerate permeability in the ISDD model.
The value of the settling velocity ratio decreases with an increasing DF, meaning that the less
porous the agglomerates are, the less effect has the agglomerate permeability on the
sedimentation. This seems reasonable since the porosity of an agglomerate should correspond
with its permeability, hence less porous agglomerates should sediment slower than the more
porous agglomerates due to an increase in drag force. The same reasoning can be applied on
the relationship between the settling velocity ratio and c, where an increase in c leads to a
decrease in settling velocity ratio. A lower c value means that the particles in an agglomerate
are packed sparser with more pore space in between them. This will increase the
sedimentation velocity due to the reduction in drag force.
37
The equations used to calculate the settling velocity ratio are only valid for 1.7 < DF < 2.5.
For DF = 1.7, Li et al calculated the settling velocity ratio to be >1.7. This means that the
agglomerates can in theory sediment 70 % faster than predicted by Stokes’ law for
impermeable spheres. For less fractal agglomerates, DF = 2.5, the settling velocity ratio is
only 1.2 and there is less of an effect on the sedimentation [29]. This can also be seen in
figure 22, the settling velocity ratio is higher for lower DF values and also, it varies more with
the grouping factor when DF is lower. Since the settling velocity ratio equations are only
valid up to DF < 2.5, the results for DF = 2.6, 2.7, and 2.8 should not be reliable. However,
they follow the same trends as the results for lower DF values and the calculated settling
velocity ratios are within reasonable values. As stated above, the settling velocity ratio does
not have that much of an effect on the sedimentation for higher DF.
Previous studies show that the settling velocity ratio does not change much when n >30 [29].
This is seen in the results in figure 22 as well. Figures 23-25 show how varying the grouping
factor changes the settling velocity ratio and hence the simulation results. The experimental
data is added to the graphs for comparison and the error bars represent the standard
deviations. Since the grouping factor cannot be determined experimentally, three different n
values were tested; n = 5, 15, and 30. A grouping factor of 5 (n = 5) was chosen since it yields
a large settling velocity ratio, n = 30 because the settling velocity ratio does not change that
much for higher values, and n = 15 because it is close to the mean value between 5 and 30.
Figure 23 shows data for ZnO 0.1 g/L when DF = 2.4, figure 24 for ZnO 0.01 g/L when DF =
2.7, and figure 25 for Cu 0.1 g/L when DF = 2.8. For every DF, the three values of c
mentioned above (0.25, 0.44, and 0.637) were tested. The mean primary particle sizes (88.9
nm for ZnO and 100 nm for Cu) were used since they had proven to give the best results in
previous simulations.
For ZnO 0.1 g/L and DF = 2.4, c = 0.25 and n = 5 gives the best fit to experimental data. This
is illustrated in figure 23a.
38
Figure 23 – Fraction of particles sedimented over time for ZnO NPs (0.1 g/L) in 1 mM NaClO4(aq) with a primary particle size of 88.9 nm and fractal dimension of 2.4 for varying settling velocity ratios depending on the grouping factor and the packing coefficient. The tested c values are (a) 0.25, (b) 0.44, and (c) 0.637, and n = 5, 15, 30 is tested for all c values. The case where Γ = 1 is also plotted, as well as the experimental data. The x axis represents the exposure time [h] and the y axis represents the fraction of particles sedimented.
For ZnO 0.01 g/L and DF = 2.7 (see figure 24), c = 0.25 also gives the best fit to experimental
data, but it seems to be independent of the grouping factor. The tested grouping factors give
approximately the same fit.
39
Figure 24 – Fraction of particles sedimented over time for ZnO NPs (0.01 g/L) in 1 mM NaClO4(aq) with a primary particle size of 88,9 nm and fractal dimension of 2.7 for varying settling velocity ratios depending on the grouping factor and the packing coefficient. The tested c values are (a) 0.25, (b) 0.44, and (c) 0.637, and n = 5, 15, 30 is tested for all c values. The case where Γ = 1 is also plotted, as well as the experimental data. The x axis represents the exposure time [h] and the y axis represents the fraction of particles sedimented.
For Cu 0.1 g/L and DF = 2.8 (see figure 25), the results differ from the ZnO trials since in this
case, c = 0.637 gives the best fit. The results seem to be independent of the grouping factor
since the tested grouping factors give an equally good fit to experimental data.
40
Figure 25 – Fraction of particles sedimented over time for Cu NPs (0.1 g/L) in 1 mM NaClO4(aq) with a primary particle size of 100 nm and fractal dimension of 2.8 for varying settling velocity ratios depending on the grouping factor and the packing coefficient. The tested c values are (a) 0.25, (b) 0.44, and (c) 0.637, and n = 5, 15, 30 is tested for all c values. The case where Γ = 1 is also plotted, as well as the experimental data. The x axis represents the exposure time [h] and the y axis represents the fraction of particles sedimented.
The fact that the results are independent of the grouping factor for both ZnO 0.01 g/L (DF =
2.7) and Cu 0.1 g/L (DF = 2.8) goes well with the theory since the settling velocity ratio has a
smaller effect on the sedimentation of less fractal agglomerates, i.e. when DF is higher. It is
actually questionable whether it is possible to calculate the settling velocity ratio in these two
cases since the settling velocity ratio calculations are only valid for 1.7 < DF < 2.5. Therefore,
no conclusions should be drawn from these results.
In the following simulations, n = 5 will be used when adding a settling velocity ratio to the
calculations, to see how much it affects the results. According to Liu et al, the number of
principal clusters are typically n ≥ 4 [13], and a grouping factor of 4 has been used in previous
studies. Unfortunately, the previous studies do not state why they use n = 4 [13, 29]. It is
possible that n = 4 should have been used in these calculations instead, even though there is
not much of a difference between 4 and 5.
Based on simulation trials and the results for ZnO 0.1 g/L when DF = 2.4 (see figure 23), it
looks like c = 0.25 and n = 5 give the best results. Considering the fact that c = 0.25 has been
used in previous studies [29], c = 0.25 should probably be used in future simulation work
when adding a settling velocity ratio. It is difficult to recommend which n to use, but since n =
5 gave the best results in this study and n = 4 has been used in previous studies [29], an n
value in that size region should probably be used. There is a great chance that other values for
c and n in different combinations give good simulation data as well. If it is not possible to
experimentally determine c and n, more studies should be done on which values for c and n to
use when calculating the settling velocity ratio.
41
4.3.2 Finding intervals of simulated fractions of sedimentation with the ISDD model
Besides investigating the impact of the different input parameters on the simulated
sedimentation, the purpose with the simulation work was to find intervals for how much the
results from the ISDD model can vary and to see if the experimentally measured
sedimentation data fits in those intervals. The simulation work also aimed to see how close to
experimental data the ISDD results could get, i.e. to find an optimal fit for the experimental
data with the model.
The results are shown in figures 26-31, where in figure a, the stacks represent the
experimentally determined sedimentation data and the error bars represent the intervals in
which the ISDD data can vary within (the lower value is the minimum ISDD output and the
higher value is the maximum ISDD output). In figure b, the maximum and minimum ISDD
results are illustrated as line graphs together with experimental data and the optimal fit, where
the inputs in the ISDD model are regulated to get the best possible fit to experimental data.
For the experimental data, the standard deviations are added as error bars in figure b.
The results show two different intervals for each nanoparticle material and particle
concentration. First, the range 1.7 < DF < 2.5 was tested since, according to Li et al the
equations for calculating the settling velocity ratio are only valid in that DF range [29].
Second, a more narrow DF range was tested for which the optimal fit was found within, i.e.
the DF ranges were different for the different trials.
The intervals represent the extreme values for the sedimentation simulated with the ISDD
model. This means that for the minimum ISDD output, the lower DF value was used in
combination with the minimum primary particle size, the standard deviations for the measured
agglomerate sizes were subtracted from the PCCS data and no settling velocity ratio was
added. For the maximum ISDD output, the higher DF value was used in combination with the
maximum primary particle size, the standard deviations for the measured agglomerate sizes
were added to the PCCS data and a settling velocity ratio was added to the calculations. When
trying to find an optimal fit to the experimental data using the ISDD model, an appropriate DF
was used for simulation together with the mean primary particle size and no standard
deviations were added or subtracted to the PCCS data. A settling velocity ratio was added if it
improved the results.
The added sedimentation velocity ratios are calculated for c = 0.25 and n = 5 (see section
4.3.1.4).
4.3.2.1 ZnO nanoparticles in 1 mM NaClO4(aq)
Figure 26 shows the simulation results for ZnO 0.1 g/L sedimentation in the DF range 1.7 to
2.5. The resulting interval is very large, which is not desirable since it means that if the user
does not know all the input parameters when simulating sedimentation with the ISDD model,
the results can end up somewhere in that interval.
In the beginning, the experimental data is in the middle of the interval, which should be
around DF = 2.1, while in the end of the exposure times it correlates better with a higher DF.
42
Figure 26 – Sedimentation of ZnO nanoparticles in 1 mM NaClO4(aq) (particle concentration 0.1 g/L) simulated with the ISDD model for 1.7 < DF < 2.5. Figure a shows the experimental data where the error bars represent the interval for which the simulated sedimentation data vary within. Figure b shows the maximum and minimum simulated sedimentation data as graphs together with the experimental data. The error bars for the experimental data represents the standard deviations between the three replicates. The x axis represents the exposure time [h] and the y axis represents the fraction of particles sedimented.
Figure 27 shows the simulation results for ZnO 0.1 g/L in a narrower DF range, which was
found after several simulations to fit the experimental data best. The interval is still very large,
even if DF only varies between 2.3 and 2.5. The optimal fit is very close to experimental data.
In this case, DF = 2.4, a settling velocity ratio was added, no standard deviations were added
to or subtracted from the PCCS data, and the mean primary particle size was used in the
simulation. The standard deviations for the experimental data are relatively small and the
optimal fit is still close to experimental data within those deviations.
Figure 27 – Sedimentation of ZnO nanoparticles in 1 mM NaClO4(aq) (particle concentration 0.1 g/L) simulated with the ISDD model for 2.3 < DF < 2.5. Figure a shows the experimental data where the error bars represent the interval for which the simulated sedimentation data vary within. Figure b shows the maximum and minimum simulated sedimentation data as graphs together with the experimental data. The error bars for the experimental data represents the standard deviations between the three replicates. The x axis represents the exposure time [h] and the y axis represents the fraction of particles sedimented.
Figure 28 shows the simulation results for ZnO 0.01 g/L in the DF range 1.7 to 2.5. DF = 1.7
generates very low values and is nowhere near the experimental data. In the beginning of the
experiment, the experimental data shows larger sedimentation than the results from the ISDD
model, even for the higher DF (DF = 2.5), hence this simulated interval does not correlate
well with experimental data. However, the difference between experimental data and the
simulated data for DF = 2.5 is small and it is possible that it is a consequence of artefacts.
43
Figure 28 – Sedimentation of ZnO nanoparticles in 1 mM NaClO4(aq) (particle concentration 0.01 g/L) simulated with the ISDD model for 1.7 < DF < 2.5. Figure a shows the experimental data where the error bars represent the interval for which the simulated sedimentation data vary within. Figure b shows the maximum and minimum simulated sedimentation data as graphs together with the experimental data. The error bars for the experimental data represents the standard deviations between the three replicates. The x axis represents the exposure time [h] and the y axis represents the fraction of particles sedimented.
Figure 29 shows the simulation results for ZnO 0.01 g/L sedimentation in a more appropriate
DF range, which is 2.6 < DF < 2.8 for this trial. The optimal fit for the experimental data is
found when DF = 2.7, a settling velocity ratio was added, no standard deviations were added
to or subtracted from the PCCS data, and for the mean value of the primary particle size. As
for ZnO 0.1 g/L, the standard deviations for the experimental data are small.
Figure 29 – Sedimentation of ZnO nanoparticles in 1 mM NaClO4(aq) (particle concentration 0.01 g/L) simulated with the ISDD model for 2.6 < DF < 2.8. Figure a shows the experimental data where the error bars represent the interval for which the simulated sedimentation data vary within. Figure b shows the maximum and minimum simulated sedimentation data as graphs together with the experimental data. The error bars for the experimental data represents the standard deviations between the three replicates. The x axis represents the exposure time [h] and the y axis represents the fraction of particles sedimented.
4.3.2.2 Cu nanoparticles in 1 mM NaClO4(aq)
For the simulations with Cu nanoparticles (Cu 0.1 g/L), there were only two agglomerate
sizes that could be used in the ISDD model. In order to make a good evaluation of the ISDD
model, it would have been better to have a larger data set than only two measurement points.
The standard deviations for the experimental data are added to the graph in the b figures
(figures 30 and 31) but they cannot be seen because they are too small.
44
Figure 30 shows the simulation results for Cu 0.1 g/L in the DF range 1.7 to 2.5. As for ZnO
0.01 g/L, DF = 1.7 generates very low values and the experimental data correlates better with
a higher DF value.
Figure 30 – Sedimentation of Cu nanoparticles in 1 mM NaClO4(aq) (particle concentration 0.1 g/L) simulated with the ISDD model for 1.7 < DF < 2.5. Figure a shows the experimental data where the error bars represent the interval for which the simulated sedimentation data vary within. Figure b shows the maximum and minimum simulated sedimentation data as graphs together with the experimental data. The error bars for the experimental data represents the standard deviations between the three replicates. The x axis represents the exposure time [h] and the y axis represents the fraction of particles sedimented.
Figure 31 shows the simulation results for Cu 0.1 g/L in a more appropriate DF range, which
is 2.7 < DF < 2.9 for this trial. These are very high DF values since the maximum value of DF
is 3. The optimal fit is found when DF = 2.8, no settling velocity ratio was added, no standard
deviations were added to or subtracted from the PCCS data, and the experimentally
determined primary particle size for Cu nanoparticles was used.
Figure 31 – Sedimentation of Cu nanoparticles in 1 mM NaClO4(aq) (particle concentration 0.1 g/L) simulated with the ISDD model for 2.7 < DF < 2.9. Figure a shows the experimental data where the error bars represent the interval for which the simulated sedimentation data vary within. Figure b shows the maximum and minimum simulated sedimentation data as graphs together with the experimental data. The error bars for the experimental data represents the standard deviations between the three replicates. The x axis represents the exposure time [h] and the y axis represents the fraction of particles sedimented.
Unlike the ZnO experiments where the minimum, maximum, and mean primary particle sizes
were measured and used in the simulations, the primary particle size for Cu was
experimentally determined to only a mean value (d = 100 nm). The maximum and minimum
values used in these simulations for Cu were estimated in order to see how much changing the
primary particle size affects the results and, if there are deviations in the primary particle size,
how well it will correlate with experimental data. With that said, it seems promising that the
45
optimal fit is found for d = 100 nm since that is the experimentally determined primary
particle size of Cu nanoparticles.
4.3.2.3 General
The intervals found with the ISDD model are unfortunately very large, but it is important to
remember that they show the extreme values in order to find out the size of the interval, i.e.
the margins of error, when using the ISDD model.
The results show that using the mean value of the primary particle size (89.9 nm for ZnO and
100 nm for Cu) in the simulations gives a better fit and is therefore recommended in future
simulation work. For the optimal fits, the agglomerate sizes used as input in the ISDD model
are without adding or subtracting the standard deviations.
One source of error using the ISDD model is the estimation of DF. Instead of estimating a DF,
a range of DF values were tested. DF can have a value between 1 and 3, where 1 represents a
rod and 3 represents a perfect sphere [17]. First, the range 1.7 < DF < 2.5 was tested since the
permeability factor equations are only valid in that DF range [29]. As the results show, DF
ranging from 1.7 to 2.5 yields very large intervals and hence the ISDD model does not seem
reliable. After several simulation trials, a more appropriate range of DF was found for each
nanoparticle material and particle concentration: 2.3 < DF < 2.5 for ZnO 0.1 g/L, 2.6 < DF <
2.8 for ZnO 0.01 g/L, and 2.7 < DF < 2.9 for Cu 0.1 g/L. For the more customized DF ranges,
the intervals are still large, which shows how much impact DF has on the results. Although, as
discussed above, it is still important to keep in mind that the intervals show the extreme
values generated when using the extreme input parameters.
The DF values for ZnO 0.01 g/L and Cu 0.1 g/L are very high. According to literature, it
seems as DF is never higher than 2.5 [17, 29]. For comparison, Hinderliter et al simulated the
transport rates of iron oxide (Fe2O3) agglomerates in cell medium for 2.0 < DF < 2.3. They
found that simulations for DF = 2.3 gave results that were in best agreement with
experimental data [17].
A problem for ZnO 0.01 g/L and Cu 0.1 g/L is that the more appropriate DF range is not
within the range where the equations for calculating the settling velocity ratio are valid and
the question is whether it is possible to add a settling velocity ratio in these cases. The settling
velocity ratio equations are valid for 1.7 < DF < 2.5 [29]. The fact that the simulated data for
ZnO 0.01 g/L and Cu 0.1 g/L conforms best to experimental data for high DF values indicates
that the agglomerates are packed very dense, i.e. they are not very porous.
It is not clear whether DF only depends on the nanoparticle material and solution, or if the
particle concentration will affect DF as well. The DF values that were found to give the best
simulated data differed a lot between ZnO 0.1 g/L and ZnO 0.01 g/L, where DF was 2.4
respectively 2.7. The only difference between the two trials is the particle concentration,
hence it is tempting to draw the conclusion that the particle concentration in the solution
affects DF. The particles will collide more in a higher particle concentration, which possibly
could affect DF. However, there are several uncertainties in the measurements on which the
results are built on and more information is needed before any conclusions can be drawn. It
would seem reasonable if DF was around the same value for the same nanoparticle material in
the same solution, since the particles most likely interact in similar ways independently of the
46
particle concentration. This theory contradicts the simulation results presented here, provided
that 2.4 and 2.7 is a big difference in DF.
The measured agglomerate sizes used as inputs in the ISDD model as well as the
experimentally measured sedimentation were done in triplicates. However, the triplicates
came from the same stock solution and to be able to evaluate the ISDD model better, more
trials should be done where the different replicates are taken from different stock solutions.
The idea of using the ISDD model is to have a simple model where the user enters certain
input parameters and is provided with a result without having to do experimental work to test
whether the results from the model can be trusted or not. However, this study was a first trial
using the ISDD model and the idea of finding the ideal DF range for the different ENPs was
to see how accurate the ISDD model can be. In the future, it will hopefully not be necessary to
estimate DF since calculating the effective density of the agglomerates using VCM can
probably be used instead (see sections 2.2.5 and 4.4).
4.3.3 Limitations with the ISDD model
The original ISDD model as received from the developers [17] had some limitations since it
assumes impermeable agglomerates of a single average size. This means that it does not
account for the PSD and the permeability of nanoparticle agglomerates. It does further not
consider the fact that the agglomerate sizes change over time and that the nanoparticles not
necessary have a spherical shape. These assumptions can lead to a significant underestimation
of the calculated fraction of sedimented particles. Not considering the PSD can lead to that the
larger particles in the end of the spectrum are not taken into account, hence underestimating
sedimentation since larger particles sediment faster than smaller ones. Permeable particles
sediment faster than impermeable particles due to a reduced drag force, hence assuming
impermeable agglomerates will also lead to an underestimation of the sedimentation [29].
These observations are in close agreement with what was presented in a scientific paper about
computational sedimentation models written by Liu et al [13]. Hinderliter et al observed in
their simulation experiments that sometimes the ISDD model overestimates the sedimentation
and sometimes it underestimates the sedimentation [17]. Despite these under- and
overestimations, according to Hinderliter et al, the error in the ISDD model is low relative to
the potential errors associated with common assumptions applied to most in vitro particle
toxicity studies [17].
For the simulations in this study, a revised version of the ISDD model was used (see section
3.6). An array with the different agglomerate sizes for different exposure times was added to
the revised ISDD model in order to account for changes in agglomerate size over time. A
solution with nanoparticles is a dynamic solution where the nanoparticles will diffuse,
agglomerate, dissolve, and sediment. The sedimentation velocity is affected by the
agglomerate sizes and porosity, and not accounting for the changes in agglomerate size during
the trials will lead to under- or overestimations with the ISDD model. Another improvement
was the addition of the settling velocity ratio to the calculations in the ISDD model, which can
affect the simulation results a lot depending on the magnitude of the settling velocity ratio.
The PSD for the agglomerates in the samples were mostly very narrow, which can be seen in
figures 14, 15, and 18. Therefore, the PSD was not considered in the revised ISDD model.
Today, there is probably not much to do about the fact that the ISDD model assumes spherical
particles. Nanoparticles can have a lot of different shapes and even if the model does not
47
reflect the reality perfectly, the assumption of spherical particles is most likely the best
estimation so far.
Despite the fact that the ISDD model neglects the PSD for the agglomerates, DeLoid et al
found that using average values for the agglomerate diameters and effective densities led to a
systematic error of ~ 6 %, which is reasonable to neglect when calculating the sedimentation.
Their results suggested that faster-settling and slower-settling agglomerates roughly balanced
each other out when using the average values in the calculations [20].
Another limitation is that the ISDD model does not account for advection and should not be
applied when significant advection or mechanical mixing occur in the samples during the
experiment. As stated by Hinderliter et al, the particle sedimentation must not generate
turbulence (low Reynolds numbers) [17].
The ISDD model does not consider the fact that some of the particles in a solution will
dissolve, hence not sediment. It was not a problem in this study, since the ISDD model was
used to calculate the fraction of particles that had sedimented. If the intention is to use to
ISDD model to quantify how much particles that have sedimented after a certain time, the
amount of dissolved particles should be calculated for.
The results presented above are simulated for ZnO and Cu nanoparticles in 1 mM
NaClO4(aq). In another setting the results might look completely different. Nanoparticles in
the environment will experience other conditions than during laboratory controlled
experiments. The aqueous environment will probably have another temperature, ionic
strength, and pH, to mention some of the parameters that will change. In the environment,
humic substances and other DOM will be present and interact with the nanoparticles, hence
affect the agglomeration and sedimentation processes.
4.4 Volumetric centrifugation method (VCM) Attempts to measure the effective densities of the nanoparticle agglomerates with VCM were
done as a complement to the ISDD model simulations.
At first, VCM trials were made with Cu and ZnO nanoparticles in 1 mM NaClO4(aq).
Unfortunately it did not work, even when centrifuging for more than 8 h. According to
previous studies, 1 h centrifugation should be enough [13, 20]. The particle concentrations
0.01 g/L, 0.1 g/L, and 1 g/L were tested and they all resulted in the same type of failure. The
nanoparticles were stuck on the walls of the PCV tubes or sedimented above the narrow part
of the tubes. Sometimes, a small pellet started to form but there was still a lot of nanoparticles
left in the PCV tube (see figure 32). A possible explanation for this is that the nanoparticles
formed agglomerates of too large size in 1 mM NaClO4(aq) to be able to make their way
down the narrow part of the PCV tube. A solution to this problem could be to use larger PCV
tubes.
However, VCM worked successfully when centrifuging Cu and Mn nanoparticles in DMEM+.
This probably depends on the fact that in cell medium, the proteins stabilized the
nanoparticles and stopped them from agglomerating too fast, which made it possible for them
to form a pellet. These nanoparticles did not stick onto the walls of the PCV tubes when
centrifuged in cell medium.
48
Cu and Mn nanoparticles in DMEM+ were chosen for the centrifugation trial in cell medium
because the PCCS data needed for the simulations with the ISDD model was already available
for Cu and Mn nanoparticles in DMEM+ from a previous study [41].
Figure 32 - Cu nanoparticles in 1 mM NaClO4(aq) after several hours of centrifugation at 3000g.
The pellet volumes measured after centrifugation are listed in table 2 together with the other
parameters needed to calculate the effective densities (see equation 11) for Cu and Mn
nanoparticles in DMEM+. The calculated effective densities are listed in table 2 as well.
Comparing the results with previous studies, they seem reasonable since they are in the same
order of magnitude as the results from the previous studies [13, 20, 43]. Also, the effective
densities are considerably lower than the material bulk densities, which is good since
agglomerates are supposed to have a lower density than the bulk material due to entrapped
media between the particles in the agglomerates.
Table 2 - Input parameters and effective densities for Cu and Mn nanoparticles in DMEM+ (particle concentration 0.1 g/L) calculated with VCM.
ENP ENP mass [mg]
Solubilized mass [mg/L]
Sample volume [mL]
Pellet volume [µL]
Stackning factor [-]
Material bulk density [g/cm3]
Effective density in DMEM+ [g/cm3]
Cu 0.057162 4.4 1 0.125 0.634 8.96 1.60
Mn 0.043859 2.3 1 0.150 0.634 7.3 1.38
The benefit of calculating the effective density with VCM is to not have to estimate DF in the
ISDD model simulations. This is of course a progress since fewer estimated parameters make
the results more reliable. VCM is an easy method to use and it does not require a lot of
expensive instruments. It is further not very time consuming. One limitation with this method
is that it is fairly difficult to read the pellet volume with the sliding rule-like device since the
volumes are so small. In some replicates, the agglomerates formed a pellet with an uneven
top, which made it even more difficult to read the volume. Those replicates were remade to
get better results. The difficulties in reading the pellet volume make the results more as
estimations rather than actual measurements. However, using VCM is probably still more
accurate than estimating DF.
The samples were centrifuged for 1 h since that is what has been done in previous studies [13,
20]. In the previous studies, the centrifugation speed was set to 1000 g, 2000 g, 3000 g, or
49
4000 g. In this study, 3000 g was used, basically because that was the maximum speed of the
centrifuge in use.
When calculating the effective density, SF was set to the theoretical value of random stacking
for uniform spheres (0.634) [20]. This is an approximation since in reality, the nanoparticles
are not perfect spheres.
4.4.1 Agglomerate sizes measured with PCCS
The agglomerate sizes for Cu and Mn nanoparticles in DMEM+, both with particle
concentration 0.1 g/L, were measured with PCCS in a previous study [41]. Measurements
were done for exposure times 0, 1, 2, and 3 h for Cu nanoparticles, and 0, 1, 5, and 10 h for
Mn nanoparticles. The results are shown in figures 33 and 34, and the measured sizes are the
agglomerate sizes at maximum scattered intensity.
Figure 33 - Agglomerate sizes for Cu nanoparticles (0.1 g/L) in DMEM+ measured with PCCS over time.
Figure 34 – Agglomerate sizes for Mn nanoparticles (0.1 g/L) in DMEM+ measured with PCCS over time.
0
50
100
150
200
250
300
0 0,5 1 1,5 2 2,5 3 3,5
Par
ticl
e si
ze [
nm
]
Exposure time [h]
Cu in DMEM+
0
100
200
300
400
500
600
700
800
900
0 2 4 6 8 10 12
Par
ticl
e si
ze [
nm
]
Exposure time [h]
Mn in DMEM+
50
The agglomerate size measurements were done in triplicates. No data for the standard
deviations between the triplicates was available.
4.4.2 Simulations with the ISDD model in combination with VCM
Simulations of the sedimentation for Cu and Mn nanoparticles in DMEM+ with particle
concentration 0.1 g/L were performed using a combination of the ISDD model and the
effective densities of the agglomerates measured with VCM. Instead of estimating DF, the
ISDD model calculated DF using the measured effective density. The results are shown in the
sections below.
No settling velocity ratios were added in these simulations. The simulations were only
performed for one primary particle size (100 nm for Cu and 20 nm for Mn), but PF was varied
in the same way as for the previous simulations (see section 3.6).
4.4.2.1 Cu nanoparticles in DMEM+
Simulating sedimentation for Cu nanoparticles in DMEM+ with the ISDD model and the
measured effective density did not work. According to the error message from the program,
DF becomes smaller than one or larger than three, which is not possible, and the program
could not perform the calculations. The inaccurately calculated DF probably origins from the
fact that the agglomerate sizes measured with PCCS were very small, they were almost the
same size as the primary particles. If the agglomerates and primary particles are around the
same size, i.e. an agglomerate consists of roughly one or two particles, the effective density
for the agglomerates should be around the same magnitude as the bulk density of the material.
For Cu, the effective density was calculated to 1.60 g/cm3 while the bulk density is 8.96
g/cm3, hence there is a big difference. It is possible that the pellet volume was estimated
poorly, leading to a miscalculated effective density. Another possible source of error is SF,
which was approximated to the theoretical value of 0.634 for random stacking of uniform
spheres [20]. The Cu nanoparticles are most likely not only spheres but they have various
different forms, which would yield a lower SF since they cannot pack as tightly as spheres
can. According to equation 11, a lower SF leads to a larger value of the effective density and
hence, the calculated effective density for Cu nanoparticle agglomerates was most likely
higher than 1.60 g/cm3. However, DeLoid et al found that the calculations for the effective
density are relatively insensitive to errors in SF, using a SF value 50 % larger than the
measured SF resulted in only an 11 % change in the calculated effective density. Therefore,
using the theoretical value of SF (0.634) should not lead to any significant errors [20].
A third possible explanation to the problem with simulating sedimentation for Cu
nanoparticles in DMEM+ is that the agglomerate sizes or the primary particle size were
measured incorrectly.
The simulations did not work for Cu nanoparticles in DMEM+ independently of PF. The same
values for PF were tested as in the previous simulations (PF = 0.25, 0.44, and 0.637) and they
all gave the same error message in the ISDD model.
4.4.2.2 Mn nanoparticles in DMEM+
The sedimentation velocity for Mn nanoparticles in DMEM+ (particle concentration 0.1 g/L)
was simulated with the ISDD model using the effective density calculated with VCM. The
result is shown in figure 35.
51
Figure 35 – Sedimentation for Mn nanoparticles in DMEM+ (particle concentration 0.1 g/L) simulated with the ISDD model in combination with the effective density calculated with VCM.
The graph shows that after 10 h, around 16 % of the nanoparticles had sedimented. Due to
lack of data for the agglomerate sizes after a longer time, it was not possible to simulate the
sedimentation for more than 10 h. Otherwise, it would have been interesting to see how the
sedimentation had looked after a longer time, e.g. 24 h or a week. The sedimentation would
probably take longer time compared to the trials with Cu an ZnO nanoparticles in 1 mM
NaClO4(aq) since the Mn nanoparticles were dispersed in a cell medium, which stabilizes the
particles better than NaClO4(aq), due to proteins and other additives in the cell medium.
The difference to the simulations for Cu and ZnO nanoparticles in 1 mM NaClO4(aq) is that
the computer model uses the effective density to calculate DF, instead of the user having to
estimate DF. The calculated DF values for the different exposure times are listed in table 3.
DF is around 2.2 for all simulated exposure times, a value which seems to be in good
agreement with DF values used in previous studies [17, 29].
Table 3 - DF values calculated with the ISDD model for different exposure times.
Exposure time [h] DF (output from ISDD simulations)
1 2.22
5 2.23
10 2.19
For the simulations, PF = 0.637 was used as input. Changing PF did not affect either the
simulated sedimentation velocity or the calculated DF.
To be able to investigate whether the ISDD model in combination with VCM gives reliable
results, the simulated data should be compared to experimental data. Unfortunately, there is
no experimental data for the sedimentation of Mn nanoparticles in DMEM+ available at this
point.
0
0,02
0,04
0,06
0,08
0,1
0,12
0,14
0,16
0,18
0 2 4 6 8 10 12
Frac
tio
n o
f p
arti
cles
sed
imen
ted
Exposure time [h]
Mn 0.1 g/L in DMEM+
52
4.5 Packing effects of particles in agglomerates Different methods and equations applied in this study use a parameter that represents how the
nanoparticles pack within an agglomerate. This parameter is named differently for different
methods; packing factor (PF) in the ISDD model [17], stacking factor (SF) in VCM [20], and
packing coefficient of a particle agglomerate (c) in the settling velocity ratio calculations [29].
Based on the description of these parameters, it seems most likely that they are the same
parameter with different names. Since this parameter is difficult to measure experimentally,
different approximations have been used depending on method. In the ISDD model; PF =
0.637 for randomly packed spherical monomers [17]; in VCM, SF = 0.634 for random
stacking of uniform spheres [20]; and in the permeability factor calculations, c = 0.25 in most
cases, although it is not clarified from where this value originates [29].
The similarity between PF and SF, they are practically the same value, enhances the
hypothesis that they are the same parameter. The assumed value of c differs from the other
two parameters. Considering the fact that all three parameters are described very similar, it is
still likely that they are the same. Since it is believed that these parameters are the same, it
would have made sense to use the same value in all calculations but because of the different
values stated in the literature for this parameter, different values for PF and c (0.25, 0.44, and
0.637) were tested in the simulations. SF was kept at 0.634 all the time.
A bit surprisingly, the simulation results were found to be independent of PF, which is
discussed in section 4.3.1.2. The dimensionless settling velocity ratio, calculated using c,
varies with the parameter and it seems like c = 0.25 gives the best results. Hence, it might be a
good idea to use the different approximated values stated for each method in future simulation
work until this has been investigated further.
4.6 DLVO forces Sedimentation largely depends on DLVO forces. Metals have very high Hamaker constants,
i.e. the vdW forces dominate and there is barely any repulsion between the particles, which
will lead to fast particle agglomeration and sedimentation. This have been shown by
previously done DLVO measurements for Cu and Mn nanoparticles in 1 mM NaClO4(aq),
and the results correlate with data measured with PCCS. In the presence of oxygen, oxide
layers will form on the nanoparticles, which results in a lower Hamaker constant. Hence, this
should be taken into account when calculating the DLVO forces [44].
4.7 Dose tests During the experimental part of the study, a dose test was prepared for every sonicated sample
(see section 3.3). This was done in order to determine the real concentration in the samples,
since it is usually not the same as the theoretical concentration. The theoretical concentration
is calculated from the mass of added nanoparticles and the volume of added solution, and it
does not reflect the truth since in reality, some nanoparticles in the sonicated samples will
sediment before the three replicates have been prepared. Other factors will also affect the
concentration, e.g. the fact that not all nanoparticles will disperse during sonication. There
will also most probably be an uneven distribution of nanoparticles in the sonicated sample,
which will lead to an uneven nanoparticle distribution between the replicates. Therefore, it is
good to prepare dose tests. The dose tests were analyzed with AAS and the real
concentrations were used in the simulations instead of the theoretical data.
53
5 Conclusions
The aim of this study was to simulate sedimentation of nanoparticles in a solution using the
ISDD model. The simulation work investigated the model’s sensitivity for different input
parameters and evaluated the simulated sedimentation with experimentally measured
simulation data. The study also aimed to use VCM in combination with the ISDD model to
avoid estimating DF when simulating the nanoparticle sedimentation.
The input parameters in the ISDD model that was found to be important were DF, settling
velocity ratio, and primary particle size of the nanoparticles. DF has the largest impact on the
simulation results, which leads to large uncertainties since DF cannot be measured and
therefore has to be estimated. The settling velocity ratio is calculated using equations that
build on several assumptions, which also increases the possibility of errors. Varying the
primary particle size affects the results, although not as much as varying DF. However, this
highlights the importance of measuring the primary particle size correctly. The simulation
results were the same independently of PF, hence varying PF does not affect the results.
Simulating nanoparticle sedimentation with the ISDD model can lead to results that differ a
lot from experimental data depending on the input parameters, although it is possible to yield
results in good agreement with experimental data. In order to minimize the uncertainties due
to estimations, further simulations with the ISDD model should be done.
Calculating the effective density of agglomerates using the VCM and combine it with the
ISDD model, instead of estimating a DF value of the agglomerates seem to probably yield
good results. However, this approach needs further investigations before anything definite can
be stated.
In summary, the ISDD model seems to be a promising model for future simulation work. The
simulated results did conform to the experimental data in some trials and all simulation results
could be explained by the theory, which proves that the model works to some extent. The
possibility to predict nanoparticle sedimentation using a computational model will save a lot
of time and money, and it can be a helpful tool in the extensive work of identifying the
behavior of nanoparticles in solution.
54
6 Future work
Future simulation work with the ISDD model should include nanoparticles of other materials
and simulations in other solutions than NaClO4(aq) in order to get broader data, which will
hopefully increase the understanding of the ISDD model and help evaluating the model
further. Additional experiments where the replicates come from different stock solutions
should be performed.
The ISDD model should be improved to become a better reflection of the reality, e.g.
modifying the calculations to account for the PSD within the particle samples. On a long term,
it is desirable to have a model that considers how different factors in an aqueous environment
will affect the nanoparticles, e.g. interactions with DOM, ionic strength, and pH.
VCM in combination with the ISDD model gave promising results, although it needs further
investigations. Since this method makes it possible to avoid the DF value in the ISDD model
to be estimated, which is a major limitation, it should be developed and incorporated in future
simulation work.
This study was a first trial to test the possibility of using the ISDD model to simulate
nanoparticle behavior in aqueous environments. Even though it is a long way to go before this
model completely fulfills its purpose, this study has hopefully helped in the important work of
characterizing the environmental risks with nanoparticles.
55
7 Acknowledgements
I wish to thank Professor Inger Odnevall Wallinder for the great opportunity to do my master
thesis at the division of Surface and Corrosion Science. Not only did I learn a lot, but I also
had a great time meanwhile.
I feel very grateful to my supervisor Jonas Hedberg for always helping and encouraging me,
and for his great patience with answering all my questions. I want to thank my supervisor
Susanna Wold for her support and encouragement, and for giving me feedback during the
course of my master thesis. I also want to thank Eva Blomberg for being a part of this study
and for explaining the theory behind particle agglomeration.
A special thanks to Sulena Pradhan for her kind help, and for lending me her data on PCCS
measurements with Cu nanoparticles.
It has been a pleasure working with all of you, and I am thankful to everyone at the division
of Surface and Corrosion Science for making me feel welcome and always being there when I
needed help.
I need to mention the developers behind the ISDD model, and especially Dr. Justin
Teeguarden at Pacific Northwest National Laboratory in Richland, Washington who took his
time to make sure I used the ISDD model correctly. Also, thanks to Mr. Glen DeLoid at
Harvard T.H. Chan School of Public Health in Boston, Massachusetts who helped me with the
volumetric centrifugation method.
Finally I would like to thank the Mistra Environmental Nanosafety Program for letting me be
a part of their important work for a safer environment.
56
8 References
[1] B. Nowack and T. D. Bucheli, "Occurence, behavior and effects of nanoparticles in the
environment," Environmental pollution, pp. 5-22, 2007.
[2] S. M. Louie, R. Ma and G. V. Lowry, "Transformations of nanomaterials in the environment,"
Frontiers of nanoscience, vol. 7, pp. 55-87, 2014.
[3] S. J. Klaine, P. J. Alvarez, G. E. Batley, T. F. Fernandes, R. D. Handy, D. Y. Lyon, S.
Mahendra, M. J. McLaughlin and J. R. Lead, "Nanomaterials in the environment: behavior, fate,
bioavailability, and effects," Environmental toxicology and chemistry, vol. 27, no. 9, pp. 1825-
1851, 2008.
[4] A. Nel, T. Xia, L. Mädler and N. Li, "Toxic potential of materials at the nanolevel," Science, vol.
311, pp. 622-627, 2006.
[5] V. Biju, T. Itoh, A. Anas, A. Sujith and M. Ishikawa, "Semiconductor quantum dots and metal
nanoparticles: syntheses, optical properties, and biological applications," Analytical and
bioanalytical chemisty, vol. 391, pp. 2469-2495, 2008.
[6] S. Wold, Personal communication, Stockholm, 2016.
[7] H. A. Khan and R. Shanker, "Toxicity of nanomaterials," BioMed research international, 2005.
[8] T. M. Scown, R. v. Aerle and C. R. Tyler, "Review: Do engineered nanoparticles pose a
significant threat to the aquatic environment?," Critical reviews in toxicology, vol. 40, no. 7, pp.
653-670, 2010.
[9] V. L. Colvin, "The potential environmental impact of engineered nanomaterials," Nature
biotechnology, vol. 21, pp. 1166-1170, 2003.
[10] A. Bruinink, J. Wang and P. Wick, "Effect of particle agglomeration in nanotoxicology,"
Archives of toxicology, vol. 89, pp. 659-675, 2015.
[11] P. A. Holden, F. Klaessig, R. F. Turco, J. H. Priester, C. M. Rico, H. Avila-Arias, M. Mortimer,
K. Pacpaco and J. L. Gardea-Torresdey, "Evaluation of exposure concentrations used in
assessing manufactured nanomaterial environmental hazards: are they relevant?," Environmental
science & technology, vol. 48, pp. 10541-10551, 2014.
[12] Z. E. Allouni, M. R. Cimpan, P. J. Hol, T. Skodvin and N. R. Gjerdet, "Agglomeration and
sedimentation of TiO2 nanoparticles in cell culture medium," Colloids and surfaces B:
Biointerfaces, vol. 68, pp. 83-87, 2009.
[13] R. Liu, H. H. Liu, Z. Ji, C. H. Chang, T. Xia, A. E. Nel and Y. Cohen, "Evaluation of toxicity
ranking for metal oxide nanoparticles via an in vitro dosimetry model," ACS Nano, 2015.
[14] A. L. Dale, E. A. Casman, G. V. Lowry, J. R. Lead, E. Viparelli and M. Baalousha, "Modeling
nanomaterial environmental fate in aquatic systems," Environmental science & technology, vol.
49, pp. 2587-2593, 2015.
57
[15] R. Arvidsson, S. Molander, B. A. Sandén and M. Hassellöv, "Challenges in exposure modeling
of nanoparticles in aquatic environments," Human and ecological risk assessment, vol. 17, pp.
245-262, 2011.
[16] D. Mukherjee, B. F. Leo, S. G. Royce, A. E. Porter, M. P. Ryan, S. Schwander, K. F. Chung, T.
D. Tetley, J. Zhang and P. G. Georgopoulos, "Modeling physicochemical interactions affecting
in vitro cellular dosimetry of engineered nanomaterials: application to nanosilver," Journal of
nanoparticle research, vol. 16, no. 2616, 2014.
[17] P. M. Hinderliter, K. R. Minard, G. Orr, W. B. Chrisler, B. D. Thrall, J. G. Pounds and J. G.
Teeguarden, "ISSD: A computational model of particle sedimentation, diffusion and target cell
dosimetry for in vitro toxicity studies," Particle and fiber toxicology, vol. 7, no. 36, 2010.
[18] V. Hirsch, C. Kinnear, L. Rodriguez-Lorenzo, C. Monnier, B. Rothen-Rutishauser, S. Balog and
A. Petri-Fink, "In vitro dosimetry of agglomerates," Nanoscale, vol. 6, pp. 7325-7331, 2014.
[19] J. M. Cohen, J. G. Teeguarden and P. Demokritou, "An integrated approach for the in vitro
dosimetry of engineered nanomaterials," Particle and fibre toxicology, vol. 11, no. 20, 2014.
[20] G. DeLoid, J. M. Cohen, T. Darrah, R. Derk, L. Rojanasakul, G. Pyrigotakis, W. Wohlleben and
P. Demokritou, "Estimating the effective density of engineered nanomaterials for in vitro
dosimetry," Nature communications, vol. 5, no. 3514, 2014.
[21] J. G. Teeguarden, G. O. Paul M. Hinderliter, B. D. Thrall and J. G. Pounds, "Particokinetics in
vitro: Dosimetry considerations for in vitro nanoparticle toxicity assessments," Toxicological
sciences, vol. 95, no. 2, pp. 300-312, 2007.
[22] C. P. Johnson, X. Li and B. E. Logan, "Settling velocities of fractal aggregates," Environmental
science & technology, vol. 30, no. 6, pp. 1911-1918, 1996.
[23] E. Allen, P. Smith and J. Henshaw, "A review of particle agglomeration," 2001.
[24] M. C. S. Jr., J. S. Bonner, A. N. Ernest, C. A. Page and R. L. Autenrieth, "Application of fractal
flocculation and vertical transport model to aquatic sol-sediment systems," Water research, vol.
39, pp. 1818-1830, 2005.
[25] R. D. Handy, F. v. d. Kammer, J. R. Lead, M. Hassellöv, R. Owen and M. Crane, "The
ecotoxicology and chemistry of manufactured nanoparticles," Ecotoxicology, vol. 17, pp. 287-
314, 2008.
[26] K. Holmberg, B. Jönsson, B. Kronberg and B. Lindman, "Colloidal forces," in Surfactants and
polymers in aqueous solution, West Sussex, John Wiley & Sons, 2002, pp. 175-191.
[27] B. Kumar and S. R. Crittenden, "Stern potential and Debye length measurements in dilute ionic
solutions with electrostatic force microscopy," Nanotechnology, vol. 24, 2013.
[28] A. Philippe and G. E. Schaumann, "Interactions of dissolved organic matter with natural and
engineered inorganic colloids: a review," Environmental science & technology, vol. 48, pp.
8946-8962, 2014.
[29] X.-Y. Li and B. E. Logan, "Permeability of fractal aggregates," Water research, vol. 35, no. 14,
pp. 3373-3380, 2001.
58
[30] M. C. S. Jr, J. S. Bonner, A. N. Ernest, C. A. Page and R. L. Autenrieth, "Application of fractal
flocculation and vertical transport model to aquatic sol-sediment systems," Water research, vol.
39, pp. 1818-1830, 2005.
[31] S. W. Y. Wong, P. T. Y. Leung, A. B. Djurisic and K. M. Y. Leung, "Toxicities of nano zinc
oxide to five marine organisms: influence of aggregate size and ion solubility," Analytical and
bioanalytical chemistry, vol. 396, no. 2, pp. 609-618, 2010.
[32] S.-W. Bian, I. A. Mudunkotuwa, T. Rupasinghe and V. H. Grassian, "Aggregation and
dissolution of 4 nm ZnO nanoparticles in aqueous environments: Influence of pH, ionic strength,
size, and adsorption of humic acid," Langmuir, vol. 27, pp. 6059-6068, 2011.
[33] M. A. Maurer-Jones, I. L. Gunsolus, C. J. Murphy and C. L. Haynes, "Toxicity of engineered
nanoparticles in the environment," Analytical chemistry, vol. 85, pp. 3036-3049, 2013.
[34] Z. Chen, H. Meng, G. Xing, C. Chen, Y. Zhao, G. Jia, T. Wang, H. Yuan, F. Z. Chang Ye, Z.
Chai, C. Zhu, X. Fang, B. Ma and L. Wan, "Acute toxicological effects of copper nanoparticles
in vivo," Toxicology letters, vol. 163, no. 2, pp. 109-120, 2006.
[35] X. Li, D. Zhu and X. Wang, "Evaluation on dispersion behavior of the aqueous copper nano-
suspensions," Journal of colloid and interface science, vol. 310, no. 2, pp. 456-463, 2007.
[36] H.-t. Zhu, Y.-s. Lin and Y.-s. Yin, "A novel one-stop chemical method for preparation of copper
nanofluids," Journal of colloid and interface science, vol. 277, no. 1, pp. 100-103, 2007.
[37] S. M. Hussain, A. K. Javorina, A. M. Schrand, H. M. Duhart, S. F. Ali and J. J. Schlager, "The
interaction of manganese nanoparticles with PC-12 cells induces dopamine depletion,"
Toxicological sciences, vol. 92, no. 2, pp. 456-463, 2006.
[38] Z. Zhen and J. Xie, "Development of manganese-based nanoparticles as contrast probes for
magnetic resonance imaging," Theranostics, vol. 2, no. 1, pp. 45-54, 2012.
[39] M. -. T. S. f. f. s. e. research, "Mistra Environmental Nanosafety," [Online]. Available:
www.mistraenvironmentalnanosafety.org.
[40] C. Singh, S. Friedrichs, M. Levin, R. Birkedal, K. Jensen, G. Pojana, W. Wohllenben, S.
Schulte, K. Wiench, T. Turney, O. Koulaeva, D. Marshall, K. Hund-Rinke, W. Kördel, E. V.
Doren, P.-J. D. Temmerman, M. A. D. Fransisco, J. Mast and N. G. e. al, NM-series of
representative manufactured nanomaterials - ZnO; NM-110, NM-111, NM-112, NM-113;
Characterisation and test item preparation, European commission: Institute for reference
materials and measurements, 2011.
[41] Y. Hedberg, S. Pradhan, F. Cappellini, M. Karlsson, E. Blomberg, H. Karlsson, I. O. Wallinder
and J. Hedberg, Unpublished results, Stockholm, 2016.
[42] W. M. Haynes, CRC handbook of chemistry and physics, CRC press, 2014.
[43] J. M. Cohen, J. G. Teeguarden and P. Demokritou, "An integrated approach for the in vitro
dosimetry of engineered nanomaterials," Particle and fibre toxicology, vol. 11, no. 20, 2014.
[44] E. Blomberg, Personal communication, Stockholm, 2015.
59
[45] H. L. Karlsson, P. Cronholm, Y. Hedberg, M. Tornberg, L. D. Battice, S. Svedhem and I. O.
Wallinder, "Cell membrane damage and protein interaction induced by copper containing
nanoparticles - Importance of the metal release process," Toxicology, vol. 313, pp. 59-69, 2013.
[46] K. Midander, P. Cronholm, H. L. Karlsson, K. Elihn, L. Möller, C. Leygraf and I. O. Wallinder,
"Surface characteristics. copper release, and toxicity of nano- and micrometer-sized copper and
copper(II)oxide particles: a cross-disciplinary study," Small, vol. 5, pp. 389-399, 2009.
60
9 Appendix
9.1 PCCS correlation functions
Figure 36 – Correlation functions for the PCCS measurements after 168 h for ZnO nanoparticles in 1 mM NaClO4(aq) with particle concentration 0.1 g/L; where a represents replicate A, b represents replicate B, and c represents replicate C. The x axis represents the lag time [ms] and the y axis represents the correlation [%].
Figure 37 – Correlation functions for the PCCS measurements after 24 h for ZnO nanoparticles in 1 mM NaClO4(aq) with particle concentration 0.01 g/L; where a represents replicate A, b represents replicate B, and c represents replicate C. The x axis represents the lag time [ms] and the y axis represents the correlation [%].
Figure 38 – Correlation functions for the PCCS measurements after 72 h for ZnO nanoparticles in 1 mM NaClO4(aq) with particle concentration 0.01 g/L; where a represents replicate A, b represents replicate B, and c represents replicate C. The x axis represents the lag time [ms] and the y axis represents the correlation [%].
61
Figure 39 – Correlation functions for the PCCS measurements after 168 h for ZnO nanoparticles in 1 mM NaClO4(aq) with particle concentration 0.01 g/L; where a represents replicate A, b represents replicate B, and c represents replicate C. The x axis represents the lag time [ms] and the y axis represents the correlation [%].
9.2 Simulations of nanoparticle sedimentation in solution with the ISDD model
9.2.1 Fractal dimension (DF)
Figure 40 – Sedimentation for ZnO nanoparticles in 1 mM NaClO4(aq) (particle concentration 0.01 g/L) with the ISDD model for DF = 1.7; 2.3; 2.8. The primary particle size is 88.9 nm and no settling velocity ratio was added to the calculations. The experimental data is added to the figure for comparison.
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
0 1 2 3 4 5
Frac
tio
n o
f p
arti
cles
sed
imen
ted
Exposure time [h]
ZnO 0.01 g/L (ppd=88.9 nm, PerF=1)
DF=1.7
DF=2.3
DF=2.8
AAS data
62
Figure 41 – Sedimentation for Cu nanoparticles in 1 mM NaClO4(aq) (particle concentration 0.1 g/L) with the ISDD model for DF = 1.7; 2.3; 2.8. The primary particle size is 100 nm and no settling velocity ratio was added to the calculations. The experimental data is added to the figure for comparison.
9.2.2 Primary particle size (d)
Figure 42 – Sedimentation for ZnO nanoparticles in 1 mM NaClO4(aq) (particle concentration 0.01 g/L) simulated with the ISDD model for d = 33.3 nm; 88.9 nm; 176.7 nm. The fractal dimension is 2.7 and no settling velocity ratio was added to the calculations. The experimental data is added to the figure for comparison.
0
0,2
0,4
0,6
0,8
1
1,2
0 5 10 15 20 25
Frac
tio
n o
f p
arti
cles
sed
imen
ted
Exposure time [h]
Cu 0.1 g/L (ppd=100 nm, PerF=1)
DF=1.7
DF=2.3
DF=2.8
AAS data
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
0 1 2 3 4 5
Frac
tio
n o
f p
arti
cles
sed
imen
ted
Exposure time [h]
ZnO 0.01 g/L (DF=2.7, PerF=1)
d=33.3 nm
d=88.9 nm
d=176.6 nm
AAS data
63
Figure 43 – Sedimentation for Cu nanoparticles in 1 mM NaClO4(aq) (particle concentration 0.1 g/L) simulated with the ISDD model for d = 50 nm; 100 nm; 200 nm. The fractal dimension is 2.8 and no settling velocity ratio was added to the calculations. The experimental data is added to the figure for comparison.
9.3 The ISDD model Matlab® code
9.3.1 Calculate particle properties
% created by Dennis Thomas (October 2, 2013)
% last modified by Dennis Thomas (October 2, 2013)
function [A,B,DF,alpha,aggdiam,aggpor,aggdens,aggnp] =
calcparticleproperties(diamp,dp,iagg,parameter)
% Calculates agglomerate properties from particle numbers per agglomerate
% or agglomerate diameter
% Input:
% - diamp: monomer particle diameter (nm)
% - dp: monomer particle density (g/cc)
% - iagg: index of an element in the parameter.agg_NP or
% parameter.agg_diameter array
% - parameter: structure data type containing all the parameters
% specified by the user
% Output:
% - A: diffusivity (m^2/s)
% - B: sedimentation velocity (m/s)
% - alpha: dimensionless parameter in the Mason-Weaver equation
% - aggdiam: agglomerate diameter (m)
% - aggpor: agglomerate porosity
% - aggdens: agglomerate density (g/cc)
% - aggnp: particle numbers per agglomerate
% constants (use SI units)
diampm = diamp * 1e-9; % particle diameter (m)
rp = diampm/2.; % particle radius (m)
R=8.314472; % Gas Constant (L kPa/K/mol)
N=6.022e23; % Avogadro's Number
g=9.8; % Gravity (m/s2)
PF = parameter.PF;
DF = parameter.DF;
0
0,2
0,4
0,6
0,8
1
1,2
0 5 10 15 20 25 30
Frac
tio
n o
f p
arti
cles
sed
imen
ted
Exposure time [h]
Cu 0.1 g/L (DF=2.8, PerF=1)
d=50 nm
d=100 nm
d=200 nm
AAS data
64
PerF = parameter.PerF; %Permeability factor
dw = parameter.media_density;
visc = parameter.media_viscosity;
T = parameter.media_temperature;
L = parameter.media_height;
% Mason and Weaver constants
switch parameter.use_agginput
case 'aggNP'
if parameter.agg_NP(iagg) > 1
% compute agglomerate diameter, porosity, and density using
Sterling
% equations
NP = parameter.agg_NP(iagg);
aggnp = NP;
aggdiam = (NP/PF)^(1/DF)*diampm; % agg diameter
(Sterling equation 5) (m)
aggrad = aggdiam/2.; % (m)
aggpor = 1.-(aggdiam/diampm)^(DF-3.); % agg porosity
aggdens = dp * (1.-aggpor)+dw*aggpor; % agg density
A=R*T/(N*6*pi*visc*aggrad); % m2/s
B=PerF*(g*(aggdens-dw)*(aggdiam)^(DF-1.)*(diampm)^(3.-
DF)/18./visc*1e3); %m/s
else
aggnp = 1;
aggdiam = diampm;
aggrad = aggdiam/2.;
aggpor = 0.;
aggdens = dp;
A=R*T/(N*6*pi*visc*rp); % diffusion (m2/s)
B=PerF*(2.*g*(dp-dw)*rp*rp/9./visc*1e3); % sedimentation
(m/s)
end
case 'aggDia'
if parameter.agg_diameter(iagg) == diamp
aggnp = 1;
aggdiam = diampm;
aggrad = aggdiam/2.;
aggpor = 0.;
aggdens = dp;
A=R*T/(N*6*pi*visc*rp); % diffusion (m2/s)
B=PerF*(2.*g*(dp-dw)*rp*rp/9./visc*1e3); % sedimentation
(m/s)
elseif parameter.agg_diameter(iagg) > diamp
aggnp = int32(PF*(parameter.agg_diameter(iagg)/diamp)^DF);
aggdiam = parameter.agg_diameter(iagg)* 1e-9; % agg
diameter in m
aggrad = aggdiam/2.; % agg radius(m)
aggpor = 1.-(aggdiam/diampm)^(DF-3.); % agg porosity
aggdens = dp * (1.-aggpor)+dw*aggpor; % agg density
A=R*T/(N*6*pi*visc*aggrad); % m2/s
B=PerF*(g*(aggdens-dw)*(aggdiam)^(DF-1.)*(diampm)^(3.-
DF)/18./visc*1e3); %m/s
else
65
ME = MException('Check input value of agglomerate diameter: ',
...
'value should not be less than the monomer particle
diameter.');
throw(ME);
end
case 'aggDens'
if parameter.agg_density(iagg) == dp
aggnp = 1;
aggdiam = diampm;
aggrad = aggdiam/2.;
aggpor = 0.;
aggdens = dp;
A=R*T/(N*6*pi*visc*rp); % diffusion (m2/s)
B=PerF*(2.*g*(dp-dw)*rp*rp/9./visc*1e3); % sedimentation
(m/s)
elseif (parameter.agg_density(iagg) < dp &&
parameter.agg_diameter(iagg) > diamp)
aggdens = parameter.agg_density(iagg); % agg density (g/cc)
aggdiam = parameter.agg_diameter(iagg)*1e-9; % agg diameter
in m
aggrad = aggdiam/2.; % agg radius(m)
aggpor = (aggdens-dp)/(dw-dp); % agg porosity (unitless)
DF = 3+log(1-aggpor)/log(aggdiam/diampm);
if (DF < 1 || DF > 3)
err = MException('CheckDFValue:OutOfBounds',...
'DF value is not between 1 and 3');
errCause = MException('ResultChk:BadInput',...
'agglomerate density and diameter values are not consistent
with a DF value between 1 and 3.');
err = addCause(err, errCause);
throw(err);
%
end
% user-specified value of the porosity factor (PF) is used to
% calculate the number of particles per agglomerate (aggnp)
aggnp = int32(PF*(aggdiam/diampm)^DF);
A=R*T/(N*6*pi*visc*aggrad); % m2/s
B=(g*(aggdens-dw)*(aggdiam)^(DF-1.)*(diampm)^(3.-
DF)/18./visc*1e3); %m/s
else
err = MException('CheckDiaDensValue:OutOfBounds',...
'Check input value of agglomerate diameter and
density.');
errCause = MException('ResultChk:BadDiaDens',...
'agglomerate density should not be greater than primary
particle density.');
err = addCause(err, errCause);
errCause = MException('ResultChk:BadDiaDens',...
'agglomerate diameter should not be less than primary particle
diameter.');
66
err = addCause(err, errCause);
throw(err);
end
end
alpha = A/B/L; % dimensionless
9.3.2 Core particle model
% Last modified by Dennis Thomas.
% Last modified date: October 2,2013
% Note: The boundary conditions (B.Cs.)were not implemented
% correctly in the original version of the ISDD code because of an error
% in the expression used for ql and qr in the function 'partbc'.
% This version has the correct implementation of the B.Cs.
function [sst]= coreparticlemodel(k,B,alpha,parameter)
L = parameter.media_height; % Dish Depth (m).
tmaxh = parameter.tmaxh; % total simulation time (h)
numt = parameter.numt; % size of t array for data extraction (plots etc.)
numx = parameter.numx; % size of x array
m = 0; % PDE parameter, do not change
% setup time and distance arrays
tmax = tmaxh*60.*60.; %Simulation time (sec)
dimt = linspace(0,tmax,numt); % time array (s), numt points between 0
and Tmax
t = dimt*B/L; % dimensionless time
dimx = linspace(0,L,numx); % distance array (m)
x = dimx/L; % dimensionless distance
% Call the PDE solver
sol = pdepe(m,@partpde,@partic,@partbc,x,t,[],alpha,k);
% Extract the first solution component as u
u = sol(:,:,1);
sst=[dimt'/60./60. zeros(numt,1)];
%Calculates frac dep at each time step
for izx = 1:numt % Calculate results at each time step
PartInMed=trapz(x,u(izx,:)); % Integrated unitless sum of particles
in each "slice" of the media.
FracPartDep = (1-PartInMed); % Fraction of particles deposited
sst(izx,2)=FracPartDep; % Array of fraction deposited over time
end
% Sub functions for PDE solver
% -------------------------------------------------------------------------
function [c,f,s] = partpde(x,t,u,DuDx,alpha,k)
c = 1;
f = alpha*DuDx;
s = -DuDx;
67
% -------------------------------------------------------------------------
function u0 = partic(x,alpha,k)
u0 = 1; %This is the uniform initial condition
% -------------------------------------------------------------------------
function [pl,ql,pr,qr] = partbc(xl,ul,xr,ur,t,alpha,k)
% Note [Dennis Thomas]: ql = 1 and qr = k because 'alpha' is included in
% the 'f' function (see 'partpde' function).
pl = -ul;
ql = 1; % added by Dennis Thomas on October 1, 2013;
%previously "ql = alpha"
pr = -ur;
qr = k; % added by Dennis Thomas on October 1, 2013;
%previously "qr = k*alpha"
% Solution of equation from Mason and Weaver 1923
% The PDE is
%
% D(u)/Dt' = A D^2(u)/Dy^2 - B Du/Dy
%
% However, the solution of this equation can be quite stiff, largely due
% to the large orders of magnitude differences possible between A and B.
%
% To overcome this, the equation can be nondimensionalized as follows:
% Define:
% x = y/L
% alpha = A/B/L
% t = t'*B/L
%
% This will result in a revised PDE of
%
% D(u)/Dt = alpha D^2(u)/Dx^2 - Du/Dx
%
% In the form expected by PDEPE, the equation is
% D_ D_
% |1| * Dt |u| = Dy | alpha*Du/Dx | + | - Du/Dx |
%
% --- -------------- --------------
% c f(x,t,u,Du/Dx) s(x,t,u,Du/Dx)
%
% with m = 0.
%
% The initial condition is u(x,0) = 1 for 0 <= x <= 1.
%
% The top (left) boundary condition is A*Du/Dy = Bu,
% which translates to alpha*Du/Dx = u.
% In the form expected by PDEPE, the left bc is
%
% |-u| + |alpha| * | Du/Dx | = |0|
%
% --- ----- ---------------- ---
% p(0,t,u) q(0,t) f(0,t,u,Du/Dx) 0
%
% The bottom (right) boundary condition is u=0. In other words,
% concentration (u) goes to zero at the 'sticky' cell boundary.
% This is different from the Mason and Weaver reflective boundary.
% Set k=0 for perfectly sticky, k=alpha for reflective
%
% |-u| + |k| * | Du/Dx | = |0|
%
% ---- --- -------------- ---
% p(1,t,u) q(1,t) f(1,t,u,Du/Dx) 0
68
9.3.3 Core particle model input
% Coreparticleinput.m Modified 2.7.12 with corrected #/ml calculation
% Modified by Dennis Thomas on October 2, 2013
% - changed parameter variables
% - added options for calculating agglomerate properties from particle
% numbers or agglomerate diameter
% Last Modified by Justin Teeguarden, February 6, 2014
% - annotations and ease of reading
% Last Modified by Dennis Thomas, February 11, 2014
% - added use of directly measured agglomerate density
% to accomadate linkage to Harvard VCM method
% Last Modified by Justin Teeguarden, February 17, 2014
% Agglomerate number & SA deposted added. Calulations
% moved below call to calcparticleproperties.
% Last Modified by Sara Isakssom, November, 2015
% - Added array for the change in agglomerate size over time
%-----------------------HOW TO RUN ISDD----------------------
%This file accepts inputs that describe the particle and the
%experimental conditions. Right clicking and running this file
%will initiate model simulations and create the output files.
% 1. The user enters values for each parameter, all of which
% can be indentified by the naming convention: parameter.name
% 2. The user decides how to calculate or enter the agglomerate
% diameter (see note under "agglomerate characteristics") by
% selecting and entering 'aggDia" or 'aggNP'
% 3. The user (recommended) saves this file with a unique
% name associated with the experiment, for example:
% 25nmGoldMacrophage24hour.m
% 4. Right clicking and running this m file will initiate the
% simulations.
% -----------------PARAMETER INPUT BEGINS HERE-------------------------
% -----EXPERIMENTAL CONDITION INPUTS (Use SI units)
% User directly inputs these experimental conditions in this file
parameter.media_height = 0.01; % Dish Depth (m) 20 ul in 96 well.
parameter.media_volume = 1; % ml (20 ul for UCLA)
parameter.media_temperature = 298.0; % Temperature (K)
parameter.media_viscosity = 0.00089; % Water viscosity (N s/m2) with 5%
increase for serum @10% (S: Change to viscosity for water at 25 oC =
0.00089 Ns/m2)
parameter.media_density = 1.0; %Density of media (g/mL)
% Those values in brackets [ ] are arrays. Entering more than one value
% will initiate simulations for all entered values (e.g. 5 different
% particle diameters, 4 different particle densities).
69
%------PRIMARY PARTICLE CHARACTERISTICS INPUTS
parameter.ptcl_diameter = [88.9]; % Particle Diameter (nm)
parameter.ptcl_density = [5.6]; % Particle Density (g/cc)
parameter.conc = 10; % Concentration of particles (ug/mL)
%------AGGLOMERATE CHARACTERISTICS INPUTS
%ISDD allows the calculation of agglomerate density using either:
% 1. The measured agglomerate diameter (DLS), the primary particle size,
% primary particle density, liquid density and the assumed packing
factor
% and assumed fractal dimension. In this case, enter
% The diameter of the agglomerate in: parameter.agg_diameter
% The number 1, for the # of particles per agglomerate in:
parameter.agg_NP
% The assumed fractal dimension in parameter.DF
% The assumed packing factor in parameter.PF
% 'aggDia' for parameter.use_agginput
% OR
% 2. The number of particles per agglomerate, the primary particle size,
% primary particle density, liquid density and the assumed packing
factor
% and assumed fractal dimension. This method requires calcualtion of the
% number of particles per agglomerate using the Sterling equation.
% This is done external to the model by the user. In this case, enter
% The diameter of the agglomerate in: parameter.agg_diameter
% The # of particles per agglomerate (calculated by user) in:
parameter.agg_NP
% The assumed fractal dimension in parameter.DF
% The assumed packing factor in parameter.PF
% 'aggNP' for parameter.use_agginput
% OR
% 3. Experimentally measured agglomerate diameter (DLS) and experimentally
% measured agglomerate density (e.g. Harvards Volumetric
Centrifugation).
% The porosity is calculated from the density of the agglomerate,
primary
% particle and the denisty of water. The fractal dimension is then
% calculated from the perosity, particle diameter and agglomerate
diameter
% finally, the number of particles per agglomerate is calculated from
% the PF and the number of particles per agglomerate and the DF.
% In this case, enter
70
% The diameter of the agglomerate in: parameter.agg_diameter
% The value 'aggDens' for parameter.use_agginput
% The PF and primary particle size
% The measured agglomerate density in parameter.agg_density
% The value 1 for the # of particles/agglomerate in: parameter.agg_NP
% Note that entered values of DF are ignored
%
% NOTES
% 'aggDia' lets the 'calcparticleproperties' function use
% the agglomerate diameter (parameter.agg_diameter) to calculate
% agglomerate porosity(aggpor),density (aggrho), and
% particles numbers per agglomerate (aggnp)
%
% 'aggNP' lets the 'calcparticleproperties' function use
% the particle numbers per agglomerate (parameter.agg_NP) to calculate
% agglomerate diameter (aggdiam), porosity(aggpor), and
% density(aggdens)
% 'aggDens' lets the 'calcparticleproperties' function use
% the agglomerate density (measured) and the agglomerate diameter
% (measured) to simulate the sedimentation and diffusion. The porosity
(aggpor)
% and number of particles per agglomerate and the fractal dimension
% are calculated by ISDD.
parameter.DF = 2.7; % Fractal Dimension (only relevant to
agglomerates)
parameter.PF = 0.637; % Packing Factor (only relevant to
agglomerates)
parameter.PerF = 1; % Permeability factor (only relevant to
agglomerates) PerF = 1 --> Stokes' sedimentation velocity (impermeable
spheres)
%parameter.agg_diameter = [200]; % agglomerate diameter (nm)
parameter.agg_NP = [10]; % Particle numbers per agglomerate
parameter.agg_density = [1.3825]; % density of agglomerate class of
particles (g/cc)
parameter.use_agginput = 'aggDia';% allowed values: 'aggDia','aggNP',
'aggDens'
%--------SIMULATION TIME and TIME and DISTANCE ARRAYS -----------------
%parameter.tmaxh = 24.0; %Simulation time (hours)
% setup time and distance arrays
parameter.numt = 48; % size of t array for data extraction (plots etc.)
parameter.numx = 1501; % size of x array (distance)
%--------------BOUNDARY CONDITIONS-----------------
% Type of Boundary Condition at the cell surface (bottom)
parameter.bctype = 'sticky'; % allowed values: 'sticky', 'reflective'
71
% -----------------------PARAMETER INPUT ENDS HERE-------------------------
%----------------------SIMULATION CODE STARTS HERE-------------------------
dmp0=parameter.ptcl_diameter; %Particle Diameter (nm)
dp0=parameter.ptcl_density; %Particle Density (g/cc)
conc = parameter.conc; %Concentration of particles (ug/mL)
mediavolume = parameter.media_volume; %ml (20 ul for UCLA)
% Boundary Condition
switch parameter.bctype
case 'sticky' % Concentration = 0 at the bottom wall
k = 0;
case 'reflective' % flux = 0 at the bottom wall
k = 1;
end
% Todays Date
datestr(now)
% Initialize Arrays
sst=[]; sds=[]; ndeptc=[]; sdeptc=[]; mdeptc=[];
fdeptc=[];ndeptaggc=[];sdeptaggc=[];sedhast=[]; FS=[];
% setup time and distance arrays
numt = parameter.numt; % size of t array for data extraction (plots
etc.)
numx = parameter.numx; % size of x array
aggdia = [553.8266667 741.1466667 914.855];
t = [1 1 2];
for i = 1:length(aggdia)
parameter.agg_diameter = aggdia(i);
parameter.tmaxh = t(i);
%for j = 1:length(ppd)
%parameter.ptcl_diameter = ppd(j)
num_ptcldia = length(parameter.ptcl_diameter);
num_ptcldens = length(parameter.ptcl_density);
switch parameter.use_agginput
case 'aggNP'
num_agg = length(parameter.agg_NP);
case 'aggDia'
num_agg = length(parameter.agg_diameter);
case 'aggDens'
num_agg = length(parameter.agg_density);
end
% Loops for particle simulations
72
for issd=1:num_ptcldia % number of particles in dmp0 you want to run
diamp=dmp0(issd)
for issd2=1:num_ptcldens % number of densities in dp0 you want to
run
dp=dp0(issd2)
for issn=1:num_agg % number of agglomerate types to run
% Call the calcparticleproperties function to
% calculate diffusivity (A), sedimentation velocity (B),
% agglomerate parameters, and alpha
[A,B,DF,alpha,aggdiam,aggpor,aggdens,aggnp] =
calcparticleproperties(diamp,dp,issn,parameter);
concn = (conc/1000000.)*6./(dp0*(pi*power(diamp*0.0000001,3)));
%#/mL
concs = concn*pi*diamp*diamp*power(10,-14); % surface area
concentration cm2/ml
concaggn = concn/double(aggnp) % agglomerate cncentration #/ml
concaggs =
double(concaggn)*pi*parameter.agg_diameter*parameter.agg_diameter*power(10,
-14); % surface area concentration of agglomerates cm2/ml
out1 = ['primary particle diameter (nm), diamp =
',num2str(diamp),'.'];
disp(out1);
out1 = ['primary particle density (g/cc), dp =
',num2str(dp),'.'];
disp(out1);
out1 = ['aggregate particle diameter (m), aggdiam =
',num2str(aggdiam),'.'];
disp(out1);
out1 = ['aggregate particle porosity (unitless), aggpor =
',num2str(aggpor),'.'];
disp(out1);
out1 = ['aggregate particle density (g/cc), aggdens =
',num2str(aggdens),'.'];
disp(out1);
out1 = ['number of particles per aggregate, aggnp =
',num2str(aggnp),'.'];
disp(out1);
out1 = ['fractal dimension, DF = ',num2str(DF),'.'];
disp(out1);
out1 = ['packing factor, PF = ',num2str(parameter.PF),'.'];
disp(out1);
out1 = ['permeability factor, PerF =
',num2str(parameter.PerF),'.'];
disp(out1);
out1 = ['diffusivity (m^2/s), A = ',num2str(A),'.'];
disp(out1);
out1 = ['sedimentation velocity (m/s), B = ',num2str(B),'.'];
disp(out1);
out1 = ['alpha (A/(B*L)) = ',num2str(alpha),'.'];
disp(out1);
out1 = ['Exposure time [h] = ',num2str(parameter.tmaxh),'.'];
disp(out1);
% Call the ISDD model function within the particle loop
73
[ssts]= coreparticlemodel(k,B,alpha,parameter);
%Calculate deposition at each time step for each particle loop
% Total Deposited per cm2
% Time, number, surface area, mass
%sst=[sst ssts(:,2) ssts(:,2)*concn*mediavolume
ssts(:,2)*concs*mediavolume ssts(:,2)*conc*mediavolume];
sst=[sst ssts(:,2) ssts(:,2)*concn*mediavolume
ssts(:,2)*concs*mediavolume ssts(:,2)*conc*mediavolume
ssts(:,2)*concaggn*mediavolume ssts(:,2)*concaggs*mediavolume];
ndeptc=[ndeptc ssts(:,2)*concn*mediavolume]; % isolated array
of number deposited
sdeptc=[sdeptc ssts(:,2)*concs*mediavolume]; % isolated array
of surface area deposited
mdeptc=[mdeptc ssts(:,2)*conc*mediavolume]; % isoloated array
of mass deposited
fdeptc=[fdeptc ssts(:,2)]; % isoloated array
of fraction deposited
ndeptaggc=[ndeptaggc ssts(:,2)*concaggn*mediavolume]; %
isolated array of number of agglomerates deposited
sdeptaggc=[sdeptaggc ssts(:,2)*concaggs*mediavolume]; %
isolated array of surface area of agglomerates deposited
sedhast=[sedhast B]; %Stores sedimentaiton velocities in an
array when several runs are performed (e.g. several particle densities as
inputs)
%Calculate Cell AUC
fracAUC=trapz(ssts(:,1),ssts(:,2)) % AUC of fraction deposited
sds=[sds; k alpha dp diamp ssts(numt,2)
ssts(numt,2)*concn*mediavolume ssts(numt,2)*concs*mediavolume
ssts(numt,2)*conc*mediavolume fracAUC fracAUC*concn*mediavolume
fracAUC*concs*mediavolume fracAUC*conc*mediavolume];
end % of particle diameter loop
end % of particle density loop
end % of particle number per agglomerate loop
end
%end
fracsedh=[t' fdeptc(end,:)']
%----------------------DATA ARRAYS FOR PLOTTING-------------------------
%adds the time data to the arrays for easy plotting
sst=[ssts(:,1) sst];
ndept=[sst(:,1), ndeptc];
sdept=[sst(:,1), sdeptc];
mdept=[sst(:,1), mdeptc];
fdept=[sst(:,1), fdeptc];
save sst.dat sst -ascii
save sds.dat sds -ascii
hold on
plot(fracsedh(:,1),fracsedh(:,2))
axis([0 180 0 1.2]);
xlabel('Exposure time [h]');
74
ylabel('Fraction of particles sedimented');
title('ZnO 0,1 g/L');