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.

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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.

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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.

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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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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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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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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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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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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

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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

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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

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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');